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Published for SISSA by Springer Received: May 20, 2014

Accepted: June 25, 2014 Published: July 17, 2014

J. Alwall,a R. Frederix,b S. Frixione,b V. Hirschi,c F. Maltoni,d O. Mattelaer,dH.-S. Shao,e T. Stelzer,f P. Torriellig and M. Zaroh,i

aDepartment of Physics, National Taiwan University, Taipei 10617, Taiwan

bPH Department, TH Unit, CERN, CH-1211 Geneva 23, Switzerland

cSLAC National Accelerator Laboratory,

2575 Sand Hill Road, Menlo Park, CA 94025-7090 U.S.A.

dCP3, Universit Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

eDepartment of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University,Beijing 100871, China

f University of Illinois, Urbana, IL 61801-3080, U.S.A.

gPhysik-Institut, Universitat Zrich, Winterthurerstrasse 190, 8057 Zurich, Switzerland

hSorbonne Universits, UPMC Univ. Paris 06, UMR 7589, LPTHE, F-75005, Paris, France

iCNRS, UMR 7589, LPTHE, F-75005, Paris, France

E-mail: mailto:[email protected]

Web End [email protected] , [email protected], [email protected], [email protected], [email protected], [email protected], mailto:[email protected]

Web End [email protected] , [email protected], mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We discuss the theoretical bases that underpin the automation of the computations of tree-level and next-to-leading order cross sections, of their matching to parton shower simulations, and of the merging of matched samples that di er by light-parton multiplicities. We present a computer program, MadGraph5 aMC@NLO, capable of handling all these computations parton-level xed order, shower-matched, merged in a unied framework whose dening features are exibility, high level of parallelisation, and human intervention limited to input physics quantities. We demonstrate the potential of the program by presenting selected phenomenological applications relevant to the LHC and to a 1-TeV e+e collider. While next-to-leading order results are restricted to QCD corrections to SM processes in the rst public version, we show that from the user viewpoint no changes have to be expected in the case of corrections due to any given renormalisable Lagrangian, and that the implementation of these are well under way.

Keywords: Monte Carlo Simulations, NLO Computations

ArXiv ePrint: 1405.0301

Open Access, c

The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP07(2014)079

Web End =10.1007/JHEP07(2014)079

The automated computation of tree-level and next-to-leading order di erential cross sections, and their matching to parton shower simulations

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Contents

1 Introduction 1

2 Theoretical bases and recent progress 32.1 Methodology of computation 42.2 General features for SM and BSM physics 52.2.1 Complex mass scheme 62.2.2 BSM-specic capabilities 72.3 LO computations 82.3.1 Generation of tree-level matrix elements 92.3.2 Width calculator 112.3.3 Event reweighting 122.3.4 Tree-level merging 152.4 NLO computations 162.4.1 NLO cross sections and FKS subtraction: MadFKS 212.4.2 One-loop matrix elements: MadLoop 252.4.3 Integration of one-loop contributions 392.4.4 Matching to showers: MC@NLO 432.4.5 Merging samples at the NLO: FxFx 502.5 Spin correlations: MadSpin 55

3 How to perform a computation 593.1 LO-type generation and running 603.2 NLO-type generation and running 623.2.1 (N)LO+PS results 643.2.2 f(N)LO results 673.3 Possibilities for LO simulations 69

4 Illustrative results 704.1 Total cross sections 704.2 Di erential distributions 834.3 One-loop SM and BSM results: a look ahead 100

5 Conclusions and outlook 106

A Technical prerequisites, setup, and structure 109

B Advanced usage 112B.1 Models, basic commands, and extended options 113B.2 Setting the hard scales at the NLO 118B.3 Scale and PDF uncertainties: the NLO case 120B.4 Scale and PDF uncertainties: the LO case 124

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B.5 Other LO reweighting applications 125B.6 Output formats and standalone libraries 126

C Features of one-loop computations 127C.1 TIR and IREGI 127C.2 Quantitative prole of MadLoop performances 130C.3 Computation of the integrand polynomial coe cients 131

D Third-party codes included in MadGraph5 aMC@NLO 134

1 Introduction

Quantum Chromo Dynamics is more than forty years old [1, 2], and perturbative calculations of observables beyond the leading order are almost as old, as is clearly documented in several pioneering works (see e.g. refs. [39]), where the implications of asymptotic freedom had been quickly understood. The primary motivation for such early works was a theoretical one, stemming from the distinctive features of QCD (in particular, the involved infrared structure, and the fact that its asymptotic states are not physical), which imply the need of several concepts (such as infrared safety, hadron-parton duality, and the factorisation of universal long-distance e ects) that come to rescue, and supplement, perturbation theory. On the other hand, the phenomenological necessity of taking higher-order e ects into account was also acknowledged quite rapidly, in view of the structure of jet events in e+e

collisions and of the extraction of S from data.

Despite this very early start, the task of computing observables beyond the Born level in QCD has remained, until very recently, a highly non-trivial a air: the complexity of the problem, due to both the calculations of the (tree and loop) matrix elements and the need of cancelling the infrared singularities arising from them, has generated a very significant theoretical activity by a numerous community. More often than not, di erent cases (observables and/or processes) have been tackled in di erent manners, with the introduction of ad-hoc solutions. This situation has been satisfactory for a long while, given that beyond-Born results are necessary only when precision is key (and, to a lesser extent, when large K factors are relevant), and when many hard and well-separated jets are crucial for the denition of a signature; these conditions have characterized just a handful of cases in the past, especially in hadronic collisions (e.g., the production of single vector bosons, jet pairs, or heavy quark pairs).

The advent of the LHC has radically changed the picture since, in a still relatively short running time, it has essentially turned hadronic physics into a high-precision domain, and one where events turning up in large-pT tails are in fact not so rare, in spite of being characterised by small probabilities. Furthermore, the absence so far of clear signals of physics beyond the Standard Model implies an increased dependence of discovery strategies upon theoretical predictions for known phenomena. These two facts show

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that presently the phenomenological motivations are extremely strong for higher-order and multi-leg computations of all observables of relevance to LHC analyses.

While a general solution is not known for the problem of computing exactly the perturbative expansion for any observable up to an arbitrarily large order in S, if one restricts oneself to the case of the rst order beyond the Born one (next-to-leading order, NLO henceforth), then such a solution does actually exist; in other words, there is no need for ad-hoc strategies, regardless of the complexity of the process under study. This remarkable fact results from two equally important theoretical achievements. Namely, a universal formalism for the cancellation of infrared singularities [1014], and a technique for the algorithmic evaluation of renormalised one-loop amplitudes [1525], both of which must work in a process- and observable independent manner. At the NLO (as opposed to the NNLO and beyond) there is the further advantage that xed-order computations can be matched to parton-shower event generators (with either the MC@NLO [26] or the POWHEG [27] method see also refs. [2837] for early, less-developed, or newer related approaches), thus enlarging immensely the scope of the former, and increasing signicantly the predictive power of the latter.

It is important to stress that while so far we have explicitly considered the case of QCD corrections, the basic theoretical ideas at the core of the subtraction of infrared singularities, of the computation of one-loop matrix elements, and of the matching to parton showers will require no, or minimal, changes in the context of other renormalisable theories, QCD being essentially a worst-case scenario. This is evidently true for tree-level multi-leg computations, as is proven by the exibility and generality of tools such as MadGraph5 [38], that is able to deal with basically any user-dened Lagrangian.

In summary, there are both the phenomenological motivations and the theoretical understanding for setting up a general framework for the computation of (any number of) arbitrary observables in an arbitrary process at the tree level or at the NLO, with or without the matching to parton showers. We believe that the most e ective way of achieving this goal is that of automating the whole procedure, whose technological challenges can be tackled with high-level computer languages capable of dealing with abstract concepts, and which are readily available.

The aim of this paper is that of showing that the programme sketched above has been realised, in the form of a fully automated and public computer code, dubbed Mad-Graph5 aMC@NLO. As the name suggests, such a code merges in a unique framework all the features of MadGraph5 and of aMC@NLO, and thus supersedes both of them (and must be used in their place). It also includes several new capabilities that were not available in these codes, most notably those relevant to the merging of event samples with di erent light-parton multiplicities. We point out that MadGraph5 aMC@NLO contains all ingredients (the very few external dependencies that are needed are included in the package) that are necessary to perform an NLO, possibly plus shower (with the MC@NLO formalism), computation: it thus is the rst public (since Dec. 16th, 2013) code, and so far also the only one, with these characteristics. Particular attention has been devoted to the fact that calculations must be doable by someone who is not familiar with Quantum Field Theories, and specically with QCD. We also show, in general as well as with explicit

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examples, how the construction of our framework lends itself naturally to its extension to NLO corrections in theories other than QCD, in keeping with the fact that such a exibility is one of the standard features of the tree-level computations which were so far performed by MadGraph, and that has been inherited by MadGraph5 aMC@NLO.

It is perhaps superuous to point out that approaches to automated computations constitute a eld of research which has a long history, but which has had an exponential growth in the past few years, out of the necessities and possibilities outlined above. The number of codes which have been developed, either restricted to leading order (LO henceforth) predictions [3853], or including NLO capabilities [5482] is truly staggering. The level of automation and the physics scope of such codes, not to mention other, perhaps less crucial, characteristics, is extremely diverse, and we shall make no attempt to review this matter here.

We have organized this paper as follows. In section 2, we review the theoretical bases of our work, and discuss new features relevant to future developments. In section 3 we explain how computations are performed. Section 4 presents some illustrative results, relevant to a variety of situations: total cross sections at the LHC and future e+e colliders, di erential distributions in pp collisions, and benchmark one-loop pointwise predictions, in the Standard Model and beyond. We nally conclude in section 5. Some technicalities are reported in appendices A to D.

2 Theoretical bases and recent progress

At the core of MadGraph5 aMC@NLO lies the capability of computing tree-level and one-loop amplitudes for arbitrary processes. Such computations are then used to predict physical observables with di erent perturbative accuracies and nal-state descriptions. Since there are quite a few possibilities, we list them explicitly here, roughly in order of increasing complexity, and we give them short names that will render their identication unambiguous in what follows.

1. fLO: this is a tree- and parton-level computation, where the exponents of the coupling constants are the smallest for which a scattering amplitude is non zero. No shower is involved, and observables are reconstructed by means of the very particles that appear in the matrix elements.

2. fNLO: the same as fLO, except for the fact that the perturbative accuracy is the NLO one. As such, the computation will involve both tree-level and one-loop matrix elements.

3. LO+PS: uses the matrix elements of an fLO computation, but matches them to parton showers. Therefore, the observables will have to be reconstructed by using the particles that emerge from the Monte Carlo simulation.

4. NLO+PS: same as LO+PS, except for the fact that the underlying computation is an NLO rather than an LO one. In this paper, the matching of the NLO matrix elements with parton showers is done according to the MC@NLO formalism.

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5. MLM-merged: combines several LO+PS samples, which di er by nal-state multiplicities (at the matrix-element level). In our framework, two di erent approaches, called kT -jet and shower-kT schemes, may be employed.

6. FxFx-merged: combines several NLO+PS samples, which di er by nal-state multiplicities.

We would like to stress the fact that having all of these di erent simulation possibilities embedded in a single, process-independent framework allows one to investigate multiple scenarios while being guaranteed of their mutual consistency (including that of the physical parameters such as coupling and masses), and while keeping the technicalities to a minimum (since the majority of them are common to all types of simulations). For example, one may want to study the impact of perturbative corrections with (NLO+PS vs LO+PS) or without (fNLO vs fLO) the inclusion of a parton shower. Or to assess the e ects of the showers at the LO (LO+PS vs fLO) and at the NLO (NLO+PS vs fNLO). Or to check how the inclusion of di erent-multiplicity matrix elements can improve the predictions based on a xed-multiplicity underlying computation, at the LO (MLM-merged vs LO+PS) and at the NLO (FxFx-merged vs NLO+PS).

In the remainder of this section we shall review the theoretical ideas that constitute the bases of the computations listed above in items 16. Since such a background is immense, we shall sketch the main characteristics in the briefest possible manner, and rather discuss recent advancements that have not yet been published.

2.1 Methodology of computation

The central idea of MadGraph5 aMC@NLO is the same as that of the MadGraph family. Namely, that the structure of a cross section, regardless of the theory under consideration and of the perturbative order, is essentially independent of the process, and as such it can be written in a computer code once and for all. For example, phase spaces can be dened in full generality, leaving only the particle masses and their number as free parameters (see e.g. ref. [83]). Analogously, in order to write the infrared subtractions that render an NLO cross section nite, one just needs to cover a handful of cases, which can be done in a universal manner. Conversely, matrix elements are obviously theory- and process-dependent, but can be computed starting from a very limited number of formal instructions, such as Feynman rules or recursion relations. Thus, MadGraph5 aMC@NLO is constructed as a meta-code, that is a (Python) code that writes a (Python, C++, Fortran) code, the latter being the one specic to the desired process. In order to do so, it needs two ingredients:

a theory model;

a set of process-independent building blocks.

A theory model is equivalent to the Lagrangian of the theory plus its parameters, such as couplings and masses. Currently, the method of choice for constructing the model given a Lagrangian is that of deriving its Feynman rules, that MadGraph5 aMC@NLO will eventually use to assemble the matrix elements. At the LO, such a procedure is fully automated in

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FeynRules [8489]. NLO cross sections pose some extra di culties, because Feynman rules are not su cient for a complete calculation one needs at least UV counterterms, possibly plus other rules necessary to carry out the reduction of one-loop amplitudes (we shall generically denote the latter by R2, adopting the notation of the Ossola-Papadopoulos-Pittau method [18]). These NLO-specic terms are presently not computed by FeynRules1 and have to be supplied by hand,2 as was done for QCD corrections to SM processes. Therefore, while the details are unimportant here, one has to bear in mind that there are LO and NLO models to be employed in MadGraph5 aMC@NLO the former being those that could also be adopted by MadGraph5, and the latter being the only ones that permit the user to exploit the NLO capabilities of MadGraph5 aMC@NLO.

Given a process and a model, MadGraph5 aMC@NLO will build the process-specic code (which it will then proceed to integrate, unweight, and so forth) by performing two di erent operations. a) The writing of the matrix elements, by computing Feynman diagrams in order to dene the corresponding helicity amplitudes, using the rules specied by the model. b) Minimal editing of the process-independent building blocks. In the examples given before, this corresponds to writing denite values for particles masses and the number of particles, and to select the relevant subtraction terms, which is simply done by assigning appropriate values to particle identities. The building blocks modied in this manner will call the functions constructed in a). Needless to say, these operations are performed automatically, and the user will not play any role in them.

We conclude this section by emphasising a point which should already be clear from the previous discussion to the reader familiar with recent MadGraph developments. Namely that, in keeping with the strategy introduced in MadGraph5 [38], we do not include among the process-independent building blocks the routines associated with elementary quantities (such as vertices and currents), whose roles used to be played by the HELAS routines [91] in previous MadGraph versions [39, 51]. Indeed, the analogues of those routines are now automatically and dynamically created by the module ALOHA [92] (which is embedded in MadGraph5 aMC@NLO), which does so by gathering directly the relevant information from the model, when this is written in the Universal FeynRules Output (UFO [93]) format. See section 2.3.1 for more details on this matter.

2.2 General features for SM and BSM physics

Since the release of MadGraph5 a signicant e ort, whose results are now included in MadGraph5 aMC@NLO, went into extending the exibility of the code at both the input and the output levels. While the latter is mostly a technical development (see appendix B.6), which allows one to use di erent parts of the code as standalone libraries and to write them in various computer languages, the former extends the physics scope of MadGraph5 aMC@NLO w.r.t. that of MadGraph5 in several ways, and in particular for what concerns the capability of handling quantities (e.g., form factors, particles with spin larger than one, and so forth) that are relevant to BSM theories. Such an extension, to be

1We expect they will in the next public version [90], since development versions exist that are already capable of doing so see e.g. section 4.3.

2Note that these are a nite and typically small number of process-independent quantities.

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discussed in the remainder of this section and partly in section 2.3.1, goes in parallel with the analogous enhanced capabilities of FeynRules, and focuses on models and on their use. Thus, it is an overarching theme of relevance to both LO and NLO simulations, in view of the future complete automation of the latter in theories other than the SM. Some of the topics to which signicant work has been lately devoted in MadGraph5 aMC@NLO, and which deserve highlighting, are the following:

1. Complex mass scheme (section 2.2.1).

2. Support of various features, of special relevance to BSM physics (section 2.2.2).

3. Improvements to the FeynRules/UFO/ALOHA chain (section 2.3.1).

4. Output formats and standalone libraries (appendix B.6).

5. Feynman gauge in the SM (section 2.4.2).

6. Improvements to the front-end user interface (the MadGraph5 aMC@NLO shell appendix A).

7. Hierarchy of couplings: models that feature more than one coupling constant order them in terms of their strengths, so that for processes with several coupling combinations at the cross section level only the assumed numerically-leading contributions will be simulated (unless the user asks otherwise) section 2.4 and appendix B.1.

We would nally like to emphasise that in the case of LO computations, be they fLO, LO+PS, or merged, one can always obtain from the short-distance cross sections a set of physical unweighted events. The same is not true at the NLO: fNLO cross sections cannot be unweighted, and unweighted MC@NLO events are not physical if not showered. This di erence explains why at the LO we often adopt the strategy of performing computations with small, self-contained modules whose inputs are Les Houches event (LHE henceforth) les [94, 95], while at the NLO this is typically not worth the e ort compare e.g. appendices B.3 and B.4, where the computation of scale and PDF uncertainties at the NLO and LO, respectively, is considered. Further examples of modules relevant to LO simulations are given in section 2.3.3.

2.2.1 Complex mass scheme

In a signicant number of cases, the presence of unstable particles in perturbative calculations can be dealt with by using the Narrow Width Approximation (NWA).3 However, when one is interested in studying either those kinematical regions that correspond to such unstable particles being very o -shell, or the production of broad resonances, or very involved nal states, it is often necessary to go beyond the NWA. In such cases, one needs to perform a complete calculation, in order to take fully into account o -shell e ects, spin correlations, and interference with non-resonant backgrounds in the presence of possibly large gauge cancellations. Apart from technical di culties, the inclusion of all resonant

3See section 2.5 for a general discussion of the NWA and of other related approaches.

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and non-resonant diagrams does not provide one with a straightforward solution, since the (necessary) inclusion of the widths of the unstable particles which amounts to a resummation of a specic subset of terms that appear at all orders in perturbation theory leads, if done naively, to a violation of gauge invariance. While this problem can be evaded in several ways at the LO (see e.g. refs. [96101]), the inclusion of NLO e ects complicates things further. Currently, the most pragmatic and widely-used solution is the so-called complex mass scheme [102, 103], that basically amounts to an analytic continuation in the complex plane of the parameters that enter the SM Lagrangian, and which are related to the masses of the unstable particles. Such a scheme can be shown to maintain gauge invariance and unitarity at least at the NLO, which is precisely our goal here.

In MadGraph5 aMC@NLO it is possible to employ the complex mass scheme in the context of both LO and NLO simulations, by instructing the code to use a model that includes the analytical continuation mentioned above (see the Setup part of appendix B.1 for an explicit example). For example, at the LO this operation simply amounts to upgrading the model that describes SM physics in the following way [102]:

The masses mk of the unstable particles are replaced by

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qm2k imk k .

The EW scheme is chosen where (mZ), mZ, and mW (the former a real number, the

latter two complex numbers dened as in the previous item) are input parameters.

All other parameters (e.g., GF and W ) assume complex values. In particular, Yukawa

couplings are dened by using the complex masses introduced in the rst item.

At the NLO, the necessity of performing UV renormalisation introduces additional complications. At present, the prescriptions of ref. [103] have been explicitly included and validated. As was already mentioned in section 2.1, this operation will not be necessary in the future, when it will be replaced by an automatic procedure performed by FeynRules.

2.2.2 BSM-specic capabilities

One of the main motivations to have very precise SM predictions, and therefore to include higher order corrections, is that of achieving a better experimental sensitivity in the context of New Physics (NP) searches. At the same time, it is necessary to have as exible, versatile, and accurate simulations as is possible not only for the plethora of NP models proposed so far, but for those to be proposed in the future as well. These capabilities have been one of the most useful aspects of the MadGraph5 suite; they are still available in MadGraph5 aMC@NLO and, in fact, have been further extended.

As was already mentioned, the required exibility is a direct consequence of using the UFO models generated by dedicated packages such as FeynRules or SARAH [104], and of making MadGraph5 aMC@NLO compatible with them. Here, we limit ourselves to listing the several extensions recently made to the UFO format and the MadGraph5 aMC@NLO code, which have a direct bearing on BSM simulations.

The possibility for the user to dene the analytic expression for the propagator of a

given particle in the model [105].

The implementation of the analytical formulae for two-body decay widths [106], which

allows one to save computing time when the knowledge of the widths of all unstable particles in the model is needed (see section 2.3.2).

The possibility to dene form factors (i.e., coupling constants depending on the kine

matic of the vertex) directly in the UFO model. Note that, since form factors cannot be derived from a Lagrangian, they cannot be dealt with by programs like FeynRules, and have therefore to be coded by hand in the models. In order to be completely generic, the possibility is also given to go beyond the format of the current UFO syntax, and to code the form factors directly in Fortran.4

Models and processes are supported that feature massive and massless particles of

spin 3/2 [105]. This implies that all spins are supported in the set {0, 1/2, 1, 3/2, 2}.

The support of multi-fermion interactions, including the case of identical particles

in the nal state, and of UFO models that feature vertices with more than one fermion ow. Multi-fermion interactions with fermion-ow violation, such as in the presence of Majorana particles, are not supported. Such interactions, however, can be implemented by the user by splitting the interaction in multiple pieces connected via heavy scalar particles, a procedure that allows one to dene unambiguously the fermion ow associated with each vertex.

While not improved with respect to what was done in MadGraph5, we remind the reader that the module responsible for handling the colour algebra is capable of treating particles whose SUc(3) representation and interactions are non-trivial, such as the sextet and ijk-type vertices respectively.

2.3 LO computations

The general techniques and strategies used in MadGraph5 aMC@NLO to integrate a tree-level partonic cross section, and to obtain a set of unweighted events from it, have been inherited from MadGraph5; the most recent developments associated with them have been presented in ref. [38], and will not be repeated here. After the release of MadGraph5, a few optimisations have been introduced in MadGraph5 aMC@NLO, in order to make it more e cient and exible than its predecessor. Here, we limit ourselves to listing the two which have the largest overall impact.

1. The phase-space integration of decay-chain topologies has been rewritten, in order to speed up the computations and to deal with extremely long decay chains (which can now easily extend up to sixteen particles). In addition, the code has also been optimised to better take into account invariant-mass cuts, and to better handle the case where interference e ects are large.

4Which obviously implies that this option is not available should other type of outputs be chosen (see appendix B.6). Further details are given at: https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/FormFactors

Web End =https://cp3.irmp.ucl.ac.be https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/FormFactors

Web End =/projects/madgraph/wiki/FormFactors .

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2. It is now possible to integrate matrix elements which are not positive-denite.5 This is useful e.g. when one wants to study a process whose amplitude can be written as a sum of two terms, which one loosely identies with a signal and a background. Rather than integrating |S + B|2, one may consider |S|2 + 2(SB) and |B|2 separately,

which is a more exible approach (e.g. by giving one the possibility of scanning a parameter space, in a way that a ects S while leaving B invariant), that also helps reduce the computation time by a signicant amount. One example of this situation is the case when |B|2 is numerically dominant, and thus corresponds to a very large

sample of events which can however be generated only once, while smaller samples of events, that correspond to |S|2 + 2(SB) for di erent parameter choices, can be

generated as many times as necessary. Another example is that of an e ective theory where a process exists (S) that is also present in the SM (B). In such a case, it is typically |B|2 + 2(SB) which is kept at the lowest order in 1/ (with being

the cuto scale). Finally, this functionality is needed in order to study the large-Nc expansion in multi-parton amplitudes, where beyond the leading 1/Nc terms the positive deniteness of the integrand is not guaranteed.

In the following sections, we shall discuss various topics relevant to the calculation of LO-accurate physical quantities. Section 2.3.1 briey reviews the techniques employed in the generation of tree-level amplitudes, emphasising the role of recent UFO/ALOHA developments. Section 2.3.2 presents the module that computes the total widths of all unstable particles featured in a given model. Section 2.3.3 describes reweighting techniques. Finally, in section 2.3.4 we review the situation of MLM-merged computations.

2.3.1 Generation of tree-level matrix elements

The computation of amplitudes at the tree level in MadGraph5 aMC@NLO has a scope which is in fact broader than tree-level physics simulations, since all matrix elements used in both LO and NLO computations are e ectively constructed by using tree-level techniques. While this is obvious for all amplitudes which are not one-loop ones, in the case of the latter it is a consequence of the L-cutting procedure, which was presented in detail in ref. [68] and which, roughly speaking, stems from the observation that any one-loop diagram can be turned into a tree-level one by cutting one of the propagators that enter the loop. Furthermore, as was also explained in ref. [68] and will be discussed in section 2.4.2, all of the companion operations of one-loop matrix element computations (namely, UV renormalisation and R2 counterterms) can also be achieved through tree-level-like calculations, which are thus very central to the whole MadGraph5 aMC@NLO framework.

The construction of tree-level amplitudes in MadGraph5 aMC@NLO is based on three key elements: Feynman diagrams, helicity amplitudes, and colour decomposition. Helicity amplitudes [107113] provide a convenient and e ective way to evaluate matrix elements for any process in terms of complex numbers, which is quicker and less involved than one

5We stress that this statement is non-trivial just because it applies to LO computations. In the context of NLO simulations this is the standard situation, and aMC@NLO has obviously been always capable of handling it.

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based on the contraction of Lorentz indices. As the name implies, helicity amplitudes are computed with the polarizations of the external particles xed. Then, by employing colour decompositions [114116], they can be organised into gauge-invariant subsets (often called dual amplitudes), each corresponding to an element of a colour basis. In this way, the complexity of the calculation grows linearly with the number of diagrams instead of quadratically; furthermore, the colour matrix that appears in the squared amplitude can be easily computed automatically (to any order in 1/Nc) once and for all, and then stored in memory. If the number of QCD partons entering the scattering process is not too large (say, up to ve or six), this procedure is manageable notwidthstanding the fact that the number of Feynman diagrams might grow factorially. Otherwise, other techniques that go beyond the Feynman-diagram expansion have to be employed [45, 49, 52, 117]. The algorithm used in MadGraph5 aMC@NLO for the determination of the Feynman diagrams has been described in detail in ref. [38]. There, it has been shown that it is possible to e ciently factorise diagrams, such that if a particular substructure shows up in several of them, it only needs to be calculated once, thus signicantly increasing the speed of the calculation. In addition, a not-yet-public version of the algorithm can determine directly dual amplitudes by generating only the relevant Feynman diagrams, thereby reducing the possible factorial growth to less than an exponential one.

The diagram-generation algorithm of MadGraph5 aMC@NLO is completely general, though it needs as an input the Feynman rules corresponding to the Lagrangian of a given theory. The information on such Feynman rules is typically provided by FeynRules, in a dedicated format (UFO). We remind the reader that FeynRules is a Mathematica-based package that, given a theory in the form of a list of elds, parameters and a Lagrangian, returns the associated Feynman rules in a form suitable for matrix element generators. It now supports renormalisable as well as non-renormalisable theories, two-component fermions, spin-3/2 and spin-2 elds, superspace notation and calculations, automatic mass diagonalization and the UFO interface. In turn, a UFO model is a standalone Python module, that features self-contained denitions for all classes which represent particles, parameters, and so forth. With the information from the UFO, the dedicated routines that will actually perform the computation of the elementary blocks that enter helicity amplitudes are built by ALOHA. Amplitudes are then constructed by initializing a set of external wavefunctions, given their helicities and momenta. The wavefunctions are next combined, according to the interactions present in the Lagrangian, to form currents attached to the internal lines. Once all of the currents are determined, they are combined to calculate the complex number that corresponds to the amplitude for the diagram under consideration. Amplitudes associated with di erent diagrams are then added (as complex numbers), and squared by making use of the colour matrix calculated previously, so as to give the nal result. We point out that versions of MadGraph earlier than MadGraph5 used the HELAS [91, 118] library instead of ALOHA. By adopting the latter, a signicant number of limitations inherent to the former could be lifted. A few examples follow here. ALOHA is not forced to deal with pre-coded Lorentz structures; although its current implementation of the Lorentz algebra assumes four space-time dimensions, this could be trivially generalised to any even integer, as the algebra is symbolic and its actual representation enters only at the nal stage of the

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output writing; its exibility has allowed the implementation of the complex mass scheme (see section 2.2.1), and of generic UV and R2 computations (see section 2.4.2); it includes features needed to run on GPUs, and the analogues of the Heget [119, 120] libraries can be automatically generated for any BSM model; nally, it caters to matrix-element generators other than MadGraph, such as those now used in Herwig++ [121, 122] and Pythia8 [123]. Since its release in 2011 [92], several important improvements have been made in ALOHA. On top of the support to the complex mass scheme and to the Feynman gauge, and of specic features relevant to one-loop computations (which are discussed in sects. 2.2.1 and 2.4.2), we would like to mention here that there has been a conspicuous gain in speed, both at the routine-generation phase as well as in the actual evaluation, thanks to the extensive use of caching. In addition, the user is now allowed to dene the form of a propagator (which is not provided by FeynRules), while previously this was determined by the particle spin: non-trivial forms, such as those relevant to spin-2 particles in ADD models [124], or to unparticles, can now be used.

2.3.2 Width calculator

Among the basic ingredients for Monte Carlo simulations of new-physics models are the masses and widths of unstable particles; which particle is stable and which unstable may depend on the particular parameter benchmark chosen. Masses are typically obtained by going to the mass eigenbasis and, if necessary, by evolving boundary conditions from a large scale down to the EW one. Very general codes exist that perform these operations starting from a FeynRules model, such as AsperGe [89]. The determination of the corresponding widths, on the other hand, requires the explicit calculation of all possible decay channels into lighter (SM or BSM) states. The higher the number of the latter, the more daunting it is to accomplish this task by hand. Furthermore, depending on the mass hierarchy and interactions among the particles, the computation of two-body decay rates could be insu cient, as higher-multiplicity decays might be the dominant modes for some of the particles. The decay channels that are kinematically allowed are highly dependent on the mass spectrum of the model, so that the decay rates need to be re-evaluated for every choice of the input parameters. The program MadWidth [106] has been introduced in order to address the above issues. In particular, MadWidth is able to compute partial widths for N-body decays, with arbitrary values of N, at the tree-level and by working in the narrow-width approximation.6 The core of MadWidth is based on new routines for diagram generation that have been specically designed to remove certain classes of diagrams:

Diagrams with on-shell intermediate particles. If the kinematics of an A n-

particles decay allows an internal particle B, that appears in an s-channel, to be on shell, the corresponding diagram can be seen as a cascade of two decays, A

B+(nk)-particles followed by B k-particles. It is thus already taken into account

in the calculation of lower-multiplicity decay channels, and the diagram is discarded.

6Even if those two assumptions are quite generic, there are particles for which they do not give su ciently-accurate results, such as the Standard Model Higgs, which has signicant loop-induced decay modes.

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Radiative diagrams. Roughly speaking, if one or more zero-mass particles are

radiated by another particle, the diagram is considered to be a radiative correction to a lower-multiplicity decay the interested reader can nd the precise denition in ref. [106]. Such a diagram is therefore discarded, because it should be considered only in the context of a higher-order calculation. Furthermore, all diagrams with the same coupling-constant combination and the same external states are also discarded, so that gauge invariance is preserved.

MadWidth begins by generating all two-body decay diagrams, and then iteratively adds extra nal state particles with the condition that any diagram belonging to either of the classes above is forbidden. This iterative procedure stops when all N = 4 modes have been considered, or estimated to be numerically irrelevant.7 All diagrams thus generated are integrated numerically. MadWidth uses several methods to reduce signicantly the overall computation time. Firstly, it features two fast (and conservative) estimators, one for guessing the impact of adding one extra nal-state particle before the actual diagram-generation phase, and another one for evaluating the importance of a single integration channel. Both of these estimators are used to neglect parts of the computation which are numerically irrelevant. Secondly, if the model is compatible with the recent UFO extension of ref. [106], and thus includes the analytical formulae for two-body decays, then the code automatically uses those formulae and avoids the corresponding numerical integrations.

We conclude this section by remarking that, although essential for performing BSM cross-section computations, MadWidth should be seen as a complement for the existing tools that generate models. This is because the information it provides one with must be available before the integration of the matrix elements is carried out but, at the same time, really cannot be included in a model itself, since it depends on the chosen benchmark scenario. For more details on the use of this model-complementing feature in Mad-Graph5 aMC@NLO, and of its mass-matrix diagonalisation analogue to which we have alluded above, see the Setup part of appendix B.1.

2.3.3 Event reweighting

The generation of large samples of events for experimental analyses can be a very time-consuming operation, especially if it involves a full simulation of the detector response. It is therefore convenient, whenever possible, to apply corrections, or to study systematics of theoretical or modelling nature, by using reweighting techniques. The reweighting of either one-dimensional distributions or that performed on a event-by-event basis are equally-common practices in experimental physics. Although the basic idea (that a non-null function can be used to map any other function dened in the same domain) behind these two procedures is the same, one must bear in mind that they are not identical, and in particular that the former can never be proven to be formally correct, since correlations (with other, non-reweighted variables) may be lost or modied, while the latter is correct in general, at least in the limit of a large number of events. For this reason, we consider only event-by-event reweighting approaches in what follows.

7Both of these conditions can be controlled by the user.

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Thanks to its exibility and the possibility of accessing a large amount of information in a direct and straightforward manner (model and running parameters, matrix elements, PDFs, and so forth) MadGraph5 aMC@NLO provides one with an ideal framework for the implementation of such approaches. The strategy is rather simple: one starts with a set of hard events, such as those contained in an LHE le, and rescales their weights:

wnew = r wold (2.1)

without modifying their kinematics. The rescaling factor r is not a constant, and may change on an event-by-event basis. This implies that, even when the original sample of events is unweighted (i.e., the wolds are all equal), the reweighted one will be in general weighted (i.e., the wnews may be di erent), and therefore degraded from a statistical point of view. If, however, the spread of the new weights is not too large (i.e., the rs are close to each other, and feature a small number of outliers), the reweigthing is typically advantageous with respect to generating a new independent sample from scratch.

While eq. (2.1) is completely general, its practical implementation depends on the kind of problems one wants to solve. We shall consider three of them in this section, two of which constitute a direct application of the basic formula (2.1), and a third one which is more sophisticated. Since these procedures address di erent types of physics, they are conveniently associated with di erent modules in MadGraph5 aMC@NLO, but they are all fully embedded into our framework and easily accessible through it, as we shall briey explain in what follows and in appendices B.4 and B.5.

The simplest example is that of the evaluation of the uncertainties through to the variations of the renormalisation and factorisation scales, and of the PDFs. In such a case r is easily determined by using the identities i and j of the initial-state partons, and the power (b) of S in the Born matrix elements:8

r = fnewi(x1, newF)fnewj(x2, newF)bS(newR) foldi(x1, oldF)foldj(x2, oldF) bS(oldR)

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. (2.2)

It should be stressed that, although scale and PDF systematics can also be computed with reweighting techniques at the NLO (see ref. [125]), in general they cannot be written in the very simple form of eq. (2.1), which is that of an overall rescaling. For a direct comparison of the NLO and LO techniques employed in this context by MadGraph5 aMC@NLO, and for fuller details about them, see appendices B.3 and B.4 respectively.

Another rather simple example is that of the case where one is interested in studying the implications of changing the modelling of a process, with the sole constraint that its initial and nal states be the same. Such a situation can for example occur when the numerical values of the coupling constants are modied, or when the contributions of classes of diagrams are included or eliminated (e.g., Higgs exchange in EW vector boson scattering). The common feature of all examples of this kind is that they are associated

8There may be cases where such matrix elements do not factorise a single S factor see e.g. section 2.4. We limit ourselves to discussing the simplest, and more common, situation here.

with changes to matrix elements, all computed with the same kinematic conguration. Therefore, one can simply write:

r = |Anew|2/|Aold|2 , (2.3)

which for obvious reasons is dubbed matrix-element reweighting. Despite its simplicity, this method has very many di erent applications, from parameter scanning in new-physics models (where one starts with a single sample of events, that corresponds to a given benchmark point, and generates as many new ones as the number of parameter congurations of interest), to more advanced approaches, such as the inclusion of exact loop e ects (Anew)

in processes that can be also, and more simply, described by e ective theories (Aold) see refs. [126, 127] for recent results of the latter type that make use of MadGraph5 and

MadGraph5 aMC@NLO. Some further comments on matrix-element reweighting and its practical usage in MadGraph5 aMC@NLO are given in appendix B.5.

We nally turn to discussing the matrix-element method [128130], which can be seen as a reweighting one because the weights determined at the parton level through matrix-element computations are possibly modied by a convolution to take into account a variety of blurring e ects (such as those due to a detector). On the other hand, from the practical point of view it turns out to be more convenient, rather than talking about reweighting factors, to introduce a likelihood P (q|) for the observation of a given kinematic congu-

ration (q) given a set of theoretical assumptions (). By doing so, one can just re-express eq. (2.3) in a di erent language:

P (q|) = V

|A(q)|2 , (2.4) with V a suitable volume factor, and

the total rate associated with the given assumptions . The advantage of eq. (2.4) is that it is suitable to handle cases which are much more complicated than the purely-theoretical exercise of eq. (2.3), which has led to its introduction here. For example, in the rst approximation one may think of q as the actual kinematic conguration measured by an experiment, whose accuracy is such that it can be directly used as an argument of the matrix elements, as is done in eq. (2.4). Note that this is a rather strong assumption, that in practice identies hadron-level with parton-level quantities, and assumes that the knowledge of the nal state is complete (such as that which one can ideally obtain in Drell-Yan production, pp Z +). It is clear that there are

many ways in which this simple approximation (which is used, for example, in refs. [131 135]) can break down: the e ects of radiation, of the imperfect knowledge of the detector, of the impossibility of a strict identication of parton- with hadron-level quantities, the uncertainties that plague the latter, the fact that all the relevant four-momenta cannot in general be measured, are but a few examples. It is therefore necessary to generalise eq. (2.4), which one can do as follows:

P (q|) =

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dx1dx2d(p)f(1)(x1)f(2)(x2) |A(p)|2 W (q, p) . (2.5)

In eq. (2.5), we have denoted by p the parton kinematic conguration. All e ects that may turn p into the detector-level quantity q (whose dimensionality therefore need not

coincide with that of p) are parametrised by W (q, p), called transfer function. As for any hadron-collision measurable quantity, eq. (2.5) features the convolution with the PDFs. The likelihood introduced in this way can be used in the context e.g. of an hypothesis test in order to determine which among various choices of is the most probable.

Although this method is both conceptually simple and very attractive, the numerical evaluation of P (q|) is di cult because the transfer function W (q, p) behaves in a

way which cannot be probed e ciently by phase-space parametrisations that work well for just the matrix elements. In order to address this issue, a dedicated program, dubbed MadWeight [136], has been introduced that includes an optimised phase-space treatment specically designed for eq. (2.5). The new version of the code [137] embedded in Mad-Graph5 aMC@NLO features several improvements w.r.t. that of ref. [136]. It includes the method for the approximate description of higher-order e ects due to initial-state radiation, as proposed in ref. [138]. Furthermore, several optimizations have been achieved that render the computations much faster (sometimes by orders of magnitude); this allows one to use this approach also in the case of very involved nal states, such as those relevant to Higgs production in association with a tt pair [139]. Further details on MadWeight and its use within MadGraph5 aMC@NLO can be found in appendix B.5.

2.3.4 Tree-level merging

The goal of merging is that of combining samples associated with di erent parton multiplicities in a consistent manner, that avoids double counting after showering, thus allowing one to e ectively dene a single fully-inclusive sample. The tree-level merging algorithms implemented in MadGraph5 aMC@NLO are a hybrid version of those available in Alpgen [45] and SHERPA [140]; they work for both SM and BSM hard processes, but are fully automated only when the shower phase is performed with either Pythia6 [141] or Pythia8 [123] (however, there are no reasons in principle which prevents these schemes from working with HERWIG6 [142, 143] or Herwig++ [121, 122]). They are based on the use of a kT -measure [144] to dene hardness and to separate processes of di erent multiplicities, and do not perform any analytic-Sudakov reweighting of events; rather, this operation is e ectively achieved by rejecting showered events under certain conditions (see later), which implies a direct use of the well-tuned showering and hadronization mechanisms of the parton shower Monte Carlos.

There are two merging schemes that can be used in conjunction with Pythia6 and Pythia8; in the case of the latter, one is also given the possibility of considering CKKWL approaches [145147] (after having generated the samples relevant to various parton multiplicities with MadGraph5 aMC@NLO); in what follows, we shall limit ourselves to briey describe the former two methods. Firstly, one has the kT -jet MLM scheme [148], where nal-state partons at the matrix-element level are clustered according to a kT jet algorithm to nd the equivalent parton shower history of the event. In our implementation the Feynman diagram information from MadGraph5 aMC@NLO is used to retain only those clusterings that correspond to actual Feynman diagrams. In order to mimic the behaviour of the parton shower, the kT value for each clustering vertex associated with a QCD branching is used as the renormalisation scale for S in that vertex. All factorisation

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scales squared, and the renormalisation scale squared for the hard process (the process with the zero-extra-parton multiplicity), are constructed by clustering back to the irreducible 2 2 system and by using the transverse mass in the resulting frame: 2 = p2T + m2. The smallest kT value found in the jet-reconstruction procedure is restricted to be larger than some minimum cuto scale, which we denote by QMEcut; if this condition is not satised, the event is rejected. The hard events are then showered by Pythia: at the end of the perturbative-shower phase, nal-state partons are clustered into jets, using the very same kT jet algorithm as before, with the jets required to have a transverse momentum larger than a given scale Qmatch, with Qmatch > QMEcut. The resulting jets are compared to the partons at the hard subprocess level (i.e., those that result from the matrix-element computations): a jet j is said to be matched to a parton p if the distance between the two, dened according to ref. [144], is smaller than the minimal jet hardness: kT (j, p) < Qmatch. The event is then

rejected unless each jet is matched to a parton, except in the case of the largest-multiplicity sample, where extra jets are allowed if softer than the kT of the softest matrix-element parton in the event, QMEsoftest. Secondly, and with the aim to give one a non-parametric way to study merging systematics, one has the shower-kT scheme, which can be used only with

Pythias pT -ordered shower. In this case, events are generated by MadGraph5 aMC@NLO as described above and then showered, but information is also retained on the hardest (which is also the rst, in view of the pT -ordered nature of Pythia here) emission in the shower, QPShardest; furthermore, one sets Qmatch = QMEcut, which cannot be done in the context of the kT -jet MLM scheme. For all samples but the largest-multiplicity one, events are rejected if QPShardest > Qmatch, while in the case of the largest-multiplicity sample events are rejected when QPShardest > QMEsoftest. This merging scheme is simpler than the kT -jet MLM one, but it rather e ectively mimics it. Furthermore, it probes the Sudakov form factors used in the shower in a more direct manner. Finally, the treatment of the largest-multiplicity sample is fairly close to that used in the CKKW-inspired merging schemes. In both the kT -jet MLM and shower-kT methods, merging systematics are associated with variations of Qmatch; in the former case, changes to Qmatch must be done by keeping Qmatch QMEcut a

constant. For applications of the two schemes described here, see e.g. refs. [126, 148151].

2.4 NLO computations

When discussing the problem of perturbative corrections, one should bear in mind that one usually considers an expansion in terms of a single quantity (which is a coupling constant for xed-order computations). However, this is just a particular case of the more general scenario in which that expansion is carried out simultaneously in two or more couplings, all of which are thus treated as small parameters we shall refer to such a scenario as mixed-coupling expansion. Despite the fact that there is typically a clear numerical hierarchy among these couplings, a mixed-coupling situation is far from being academic; in fact, as we shall show in the following, there are cases when one is obliged to work with it. In order to study a generic mixed-coupling expansion without being too abstract, let us consider an observable which receives contributions from processes that stem from both QCD and QED interactions. The specic nature of the interactions is in fact not particularly relevant (for example, QED here may be a keyword that also understands the

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pure-EW contributions); what matters, for the sake of the present discussion, is that may depend on more than one coupling constant. We assume that the regular function

(S, ) (2.6)

admits a Taylor representation:

(S, ) =

Xn=0

Xm=0 nSm n!m!

n+m

nSm

(S,)=(0,0), (2.7)

which is by denition the perturbative expansion of . The rst few terms of the sums in eq. (2.7) will be equal to zero, with the number of such vanishing terms increasing with the complexity of the process under consideration this is because n and m are directly related to the number of vertices that enter a given diagram. In general, it is clear that for a given pair (n, m) which gives a non-vanishing contribution to eq. (2.7), there may exist another pair (n, m), with n 6= n, m 6= m, and n + m = n + m whose contribution to eq. (2.7) is

also non zero. It appears therefore convenient to rewrite eq. (2.7) with a change of variables:

k = n + m , l = n m , (2.8)

whence:

(S, ) =

Xk=0

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Xl=k

Pk,l (k+l)/2S(kl)/2 ((k + l)/2)!((k l)/2)!

(k+l)/2S(kl)/2

(S,)=(0,0), (2.9)

where

mod(k, 2), mod(l, 2) , (2.10)

which enforces the sum over l to run only over those integers whose parity is the same as that of k (therefore, there are k + 1 terms in each sum over l for a given k). Equation (2.9) implies that we need to call Born the sum (over l) of all the contributions with the smallest k k0 which are non-vanishing. Hence, the NLO corrections will correspond to the sum

over l of all terms with k = k0 + 1. This notation is compatible with the usual one used in the context of the perturbation theory of a single coupling: the QCD- or QED-only cases are recovered by considering l = k or l = k respectively. In a completely general case, for any

given k there will exist two integers lm(k) and lM(k) which satisfy the following conditions:

k lm(k) lM(k) k , (2.11)

and such that all contributions to eq. (2.9) with

l < lm(k) or l > lM(k) (2.12)

vanish, and l = lm(k), l = lM(k) are both non-vanishing. This implies that in the range

lm(k) l lM(k) , (2.13)

Pk,l =

there will be at least one (two if lm(k) 6= lM(k)) non-null contribution(s) to eq. (2.9) (the

typical situation being actually that where all the terms in eq. (2.13) are non-vanishing). Given eq. (2.13), one can re-write eq. (2.9) in the following way:

(S, ) =

where

2 (k + lm(k)) , (2.15)

c(k) = 1

2 (k lM(k)) , (2.16)

(k) = 1

2 (lM(k) lm(k)) . (2.17)

The coe cients k,l of eq. (2.14) can be expressed in terms of the quantities that appear in eq. (2.9), but this is unimportant here. A typical situation is where:

lM(k + 1) = lM(k) + 1 , (2.18) lm(k + 1) = lm(k) 1 , (2.19)

so that:

cs(k + 1) = cs(k) , (2.20) c(k + 1) = c(k) , (2.21) (k + 1) = (k) + 1 , (2.22)

whence:

(S, ) = cs(k0)Sc(k0)

Xk=k0 cs(k)Sc(k)

(k)

Xq=0 k,q (k)qSq , (2.14)

cs(k) = 1

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(k0)+p

Xq=0 k0+p,q (k0)+pqSq , (2.23)

where the Born and NLO contributions correspond to p = 0 and p = 1 respectively. Note that eq. (2.23) is the most general form of the observable (S, ) if one allows the possibility of having k0+p,0 = 0 or k0+p, (k0)+p = 0 (or both) for p > k0, since this renders eqs. (2.18) and (2.19) always true. Equation (2.23) has the advantage of a straightforward interpretation of the role of NLO corrections.

An example may help make the points above more explicit. Consider the contribution to dijet production due to the partonic process uu uu; the corresponding lowest-order

t- and u-channel Feynman diagrams feature the exchange of either a gluon or a photon (or a Z, but we stick to the pure-U(1) theory here). The Born matrix elements will therefore be the sum of terms that factorise the following coupling combinations:

2S , S , 2 , (2.24)

which implies k0 = 2, (2) = 2, and cs(2) = c(2) = 0. Therefore, according to eq. (2.23), the NLO contribution p = 1 will feature the following coupling combinations:

3S , 2S , S2 , 3 . (2.25)

Xp=0

as2 asa a2

as a3

as3 as2a a2

Figure 1. QCD (blue, right-to-left arrows) corrections and QED (red, left-to-right arrows) corrections to dijet production. See the text for details.

From the procedural point of view, it is convenient to identify QCD and QED corrections according to the relationship between one coupling combination in eq. (2.24) and one in eq. (2.25), as follows:

nSm

QED

nSm+1 , (2.27)

which has an immediate graphic interpretation, depicted in gure 1. Such an interpretation has a Feynman-diagram counterpart in the case of real-emission contributions, which is made explicit once one considers cut-diagrams, like those presented in gure 2. Loosely speaking, one can indeed identify the diagram on the left of that gure as representing QED (since the photon is cut) real-emission corrections to the 2S Born contribution. On the other hand, the diagram on the right represents QCD (since the gluon is cut) real-emission corrections to the S Born contribution. This immediately shows that, in spite of being useful in a technical sense, QCD and QED corrections are not physically meaningful if taken separately: in general, one must consider them both in order to arrive at a sensible, NLO-corrected result. This corresponds to the fact that a given coupling combination in the bottom row of gure 1 can be reached by means of two di erent arrows when starting from the top row (i.e., the Born level). Therefore, gure 1 also immediately shows that when one considers only the Born term associated with the highest power of S (), then

QCD-only (QED-only) corrections are sensible (because only a right-to-left or left-to-right arrow is relevant, respectively): they coincide with the NLO corrections as dened above (see the paragraph after eq. (2.10)). It also should be clear that the above arguments have a general validity, whatever the values of cs(k0), c(k0), and (k0) in eq. (2.23) the former two quantities never play a role in the analogues of gure 1, while by increasing (k0)

one simply inserts more blobs (i.e., coupling combinations) in both of the rows of gure 1. Finally, note that reading eqs. (2.26) and (2.27) in terms of diagrams, as has been done for those of gure 2, becomes much harder when one considers virtual contributions. For example, the one whose O(2S) cut-diagram is shown in gure 3 (and its analogues) can

indeed be equally well interpreted as a QED loop correction to a QCDQCD O(2S) Born cut-diagram, or as a QCD loop correction to a QCDQED O(S) Born cut-diagram.

MadGraph5 aMC@NLO has been constructed by having eq. (2.23) in mind; although the majority of the relevant features are not yet available in the public version of the code,

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QCD

n+1Sm , (2.26) nSm

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Figure 2. Real-emission contributions to dijet production at the NLO and O(2S). The saw-shaped lines represent Cutkosky cuts.

Figure 3. Virtual contribution to dijet production at the NLO and O(2S). The saw-shaped line

represents a Cutkosky cut.

all of them have already been thoroughly tested in the module responsible for computing one-loop matrix elements (see sects. 2.4.2 and 4.3), which is by far the hardest from this point of view, and the checks on the real-emission part are also at quite an advanced stage. The basic idea is that of giving the user the choice of which coupling combinations to retain either at the Born or at the NLO level; this corresponds to choosing a set of blobs in the upper or lower row of gure 1, respectively. MadGraph5 aMC@NLO will then automatically also consider the blobs in the row not involved in the selection by the user, in order to construct a physically-meaningful cross section, compatible with both the users choices, and the constraints due to a mixed-coupling expansion (the arrows in gure 1). It should be stressed that, although the results for the coe cients k0+p,q can be handled separately by MadGraph5 aMC@NLO, such coe cients are not (all) independent from each other from a computational viewpoint, because a single Feynman diagram (an amplitude-level quantity) may contribute to several k0+p,qs (the latter being amplitude-squared quantities). For this reason, as far as the CPU load is concerned the choice of which

coupling combinations to consider can be equivalently made at the amplitude level. Indeed, this is the only option presently available in the public version of MadGraph5 aMC@NLO; more detailed explanations are given in appendix B.1.

2.4.1 NLO cross sections and FKS subtraction: MadFKS

In this section, we briey review the FKS subtraction [10, 12] procedure, and emphasise the novelties of its implementation in MadGraph5 aMC@NLO w.r.t. its previous automation in MadFKS [61], the dedicated module included in aMC@NLO.

We shall denote by n the number of nal-state particles relevant to the Born contributions to a given cross section. The set of all the partonic 2 n subprocesses that

correspond to these contributions will be denoted by Rn; each of these subprocesses can

be represented by the ordered list of the identities of its 2 + n partons, thus:

r = (I1, . . . In+2) Rn . (2.28)

The rst operation performed by MadGraph5 aMC@NLO is that of constructing Rn, given

the process and the theory model. For example, if one is interested in the hadroproduction of a W +Z pair in association with a light jet

pp W +Zj (2.29) as described by the SM, MadGraph5 aMC@NLO will obtain:

R3 =

. (2.30)

Since the processes in Rn are tree-level, MadGraph5 aMC@NLO will construct them very

e ciently using the dedicated algorithms (see section 2.3.1). Beyond the Born level, an NLO cross section receives contributions from the one-loop and real-emission matrix elements. As is well known, the set of the former subprocesses coincides9 with that of the Born, Rn. Real-emission processes are by nature tree-level, and can therefore be obtained

by using the very same algorithms as those employed to generate the Born contributions. This is achieved by making the code generate all tree-level processes that have the same nal-state as the Borns, plus one light jet using the example of eq. (2.29), these would correspond to:

Such was the strategy adopted in the original MadFKS implementation [61]. There is however an alternative procedure, which we have implemented in MadGraph5 aMC@NLO because it is more e cient than the previous one in a variety of ways. Namely, for any given r0 Rn, one considers all possible a bc branchings for each non-identical a r0

with a strongly-interacting (i.e., g gg, g qq, and q qg, but also Q Qg, with

Q a quark with non-zero mass). For each of these branchings, a new list is obtained

9This is because we are considering here only those cases where one-loop matrix elements are obtained by multiplying the one-loop amplitudes times the Born ones. Loop-induced processes, in which the LO contribution is a one-loop amplitude squared, are not to be treated as part of an NLO computation.

JHEP07(2014)079

d, W +, Z, g

, . . . u, g, W +, Z, d

, . . .

pp W +Zjj . (2.31)

by removing a from r0, and by inserting the pair (b, c) in its place. By looping over r0 one thus constructs the set10 of real-emission processes Rn+1. As a by-product, one also

naturally obtains, for each r Rn+1, the pairs of particles which are associated with a

soft and/or a collinear singularity of the corresponding matrix element (which we denote by M(n+1,0)(r)); by denition [61], these pairs form the set of FKS pairs, PFKS(r), which

is central in the FKS subtraction procedure. We point out, nally, that regardless of the type of strategy adopted to construct Rn+1 and PFKS, it is immediate to apply it in

MadGraph5 aMC@NLO to theories other than QCD, such as QED.After having obtained PFKS(r), MadGraph5 aMC@NLO constructs the S functions

that are used by FKS in order to achieve what is e ectively a dynamic partition of the phase space: in each sector of such a partition, the structure of the singularities of the matrix elements is basically the simplest possible, amounting (at most) to one soft and one collinear divergence. The properties of the S functions are:

Sij(r) 1 i, j collinear , (2.32)

Sij(r) 1 i soft , (2.33)

Sij(r) 0 all other IR limits , (2.34)

Sij(r) = 1 . (2.35)

One exploits eq. (2.35) by rewriting the real matrix elements as follows:

M(n+1,0)(r) =

X(i,j)PFKS(r) Sij(r)M(n+1,0)(r)

X(i,j)PFKS(r) M(n+1,0)ij. (2.36)

Thanks to eqs. (2.32)(2.34), M(n+1,0)ij has the very simple singularity structure mentioned

above. Furthermore, the terms in the sum on the r.h.s. of eq. (2.36) are independent of each other, and MadGraph5 aMC@NLO is thus able to handle them in parallel.

The FKS method exploits the fact that phase-space sectors associated with di erent Sij functions are independent of each other by choosing di erent phase-space parametrisations in each of them. There is ample freedom in such a choice, bar for two integration variables: the rescaled energy of parton i (denoted by i), and the cosine of the angle between partons i and j (denoted by yij), both dened in the incoming-parton c.m. frame. The idea is that these quantities are in one-to-one correspondence with the soft (i 0) and collinear (yij

1) singularities respectively, which renders it particularly simple to write the subtracted cross section. The (n + 1)-body phase space is then written as follows:

dn+1 = (n+1)ij

Kn+1((ij)n+1) d(ij)n+1 , (2.37)

where (ij)n+1 collectively denote the 3n 1 independent integration variables, with

{i, yij} (ij)n+1 , (2.38)

10The exceedingly rare cases of non-singular real-emission contributions can be obtained by crossing; one example is qq Hg, which is the crossed process of qg Hq.

JHEP07(2014)079

(i,j)PFKS(r)

X(i,j)PFKS(r)

and where

Kn+1 = {k3, k4, kn+3} (2.39) is the set of nal-state momenta. Given eq. (2.38), MadGraph5 aMC@NLO chooses the other 3n 3 integration variables and thus determines (n+1)ij and the functional depen

dence Kn+1((ij)n+1) according to the form of the integrand, gathered from the underlying

Feynman diagrams. In general, this implies splitting the computation in several integration channels, which are independent of each other and can be dealt with in parallel. Such multi-channel technique is irrelevant here, and will be understood in the following. More details can be found in refs. [152, 153] and [61]. Implicit in eq. (2.37) are the maps that allow one to construct soft and collinear kinematic congurations starting from a non-special conguration (i.e., one where no parton is soft, and no two partons are collinear). This we shall denote as follows. Given:

K(E)n+1 = {k3, k4, kn+3} non special (2.40)

MadGraph5 aMC@NLO constructs its soft, collinear, and soft-collinear limits with:

K(S)n+1 = K(E)n+1(i = 0) , (2.41)

K(C)n+1 = K(E)n+1(yij = 1) , (2.42)

K(SC)n+1 = K(E)n+1(i = 0, yij = 1) . (2.43)

Furthermore, all the phase-space parametrisations employed by MadGraph5 aMC@NLO are such that:11

K(S)n+1 = K(C)n+1 = K(SC)n+1 , (2.44) which is benecial from the point of view of the numerical stability of observables computed at the NLO, and is necessary in view of matching with the parton shower according to the MC@NLO formalism. As is usual in the context of NLO computations, we call the non-special and the IR-limit congurations (and, by extension, the corresponding cross-section contributions) the event and the soft, collinear, and soft-collinear counterevents respectively.

Given a real-emission process rR Rn+1 and an Sij contribution, the FKS-subtracted

cross section consists of four terms:

d(NLO)ij(rR) n

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d(NLO,)ij(rR)

o=E,S,C,SC (2.45)

d(NLO,)ij(rR) = L()

rR; (ij)Bj

W ()ij(rR) d(ij)Bj d(ij)n+1 , (2.46)

where d(ij)Bj is the integration measure over Bjorken xs, L() is the corresponding parton

luminosity factor,12 and the short-distance weights W ()ij are reported in refs. [61, 125].

11Equation (2.44) holds for all particles except the FKS-pair partons; for the latter, it is the sum of their four-momenta that is the same in the three congurations. This is su cient owing to the underlying infrared-safety conditions.

12Whose dependence on is a consequence of eq. (2.44) when j = 1, 2 see ref. [26] for more details.

In ref. [61], in particular, an extended proof is given that all contributions to an NLO cross section which are not naturally (n + 1)-body ones (such as the Born, virtual, and initial-state collinear remainders) can be cast in a form formally identical to that of the soft or collinear counterterms, and can thus be dealt with simultaneously with the latter. The fully di erential cross section that emerges from eqs. (2.45) and (2.46) is:

d(NLO)ij(rR)

whose elements give non-identical contributions to the sum over FKS pairs. Therefore:

XrRRn+1

with (n+1)ij a suitable symmetry factor. The sum on the r.h.s. of eq. (2.49) is obviously more convenient to perform than that on the l.h.s.; it is customary to include the symmetry factor in the short-distance weights (see e.g. ref. [125]). The number of elements in PFKS(rR) can indeed be much smaller than that in PFKS(rR). For example, when rR is a purely gluonic

process, we have #(PFKS(rR)) = 3 (i.e., independent of n), while #(PFKS(rR)) = (n+1)(n+

2). While the former gure typically increases when quarks are considered, it remains true that, for asymptotically large ns, #(PFKS) is a constant while #(PFKS) scales as n2.

The sum on the r.h.s. of eq. (2.49) is what has been originally implemented in MadFKS. It emphasises the role of the real-emission processes, which implies that for quantities which are naturally Born-like (such as the Born matrix elements themselves) one needs to devise a way to map unambiguously Rn onto Rn+1. The interested reader can nd the denition

of such a map in section 6.2 of ref. [61], which we summarise here using the Born cross

K K(E)n+1 d(NLO,E)ij(rR) d(ij)Bjd(ij)n+1 d(ij)Bjd(ij)n+1

+ K K(S)n+1 X=S,C,SCd(NLO,)ij(rR)d(ij)Bjd(ij)n+1d(ij)Bjd(ij)n+1 , (2.47)

where we have understood the complete integration over the measures on the r.h.s.. Mad-Graph5 aMC@NLO scans the phase space by generating randomly {(ij)Bj, (ij)n+1}. For each of these points, an event kinematic conguration K(E)n+1 and its weight, and a counterevent

kinematic conguration K(S)n+1 and its weight are given in output; with these, any number

of (IR-safe) observables can be constructed. As can be seen from eq. (2.47), the weight associated with the single counterevent kinematics is the sum of the soft, collinear, soft-collinear, Born, virtual, and initial-state collinear remainders contributions, which reduces the probability of mis-binning and thus increases the numerical stability of the result.

In order for the results of eq. (2.47) to be physical, they must still be summed over all processes rR Rn+1 and all FKS pairs (i, j) PFKS(rR). As far as the latter sum is

concerned, it is easy to exploit the symmetries due to identical nal-state particles, and thus to arrive at the set [61]:

PFKS PFKS , (2.48)

JHEP07(2014)079

X(i,j)PFKS(rR)d(NLO)ij(rR) dK=

XrRRn+1

X(i,j)PFKS(rR)(n+1)ijd(NLO)ij(rR)dK, (2.49)

section d(LO) as an example:

d(LO)(rR) =

X(i,j)PFKS(rR)gIiSij(i = 0)d(LO)(ri\R) , (2.50)

where

rR = (I1, . . . Ii, . . . Ij, . . . In+3) = ri\R = (I1, . . . I\i, . . . Ij, . . . In+3) . (2.51)

In other words, through the r.h.s. of eq. (2.50) one is able to dene a Born-level quantity as a function of a real-emission process. In MadGraph5 aMC@NLO we have followed the opposite strategy, namely that of dening real-emission level quantities as functions of a Born process. The proof that this can indeed be done is given in appendix E of ref. [61]; here, we limit ourselves to summarising it as follows, using the event contribution d(NLO,E)ij

to the NLO cross section as an example. One has the identity:

XrRRn+1

JHEP07(2014)079

X(i,j)PFKS(rR)(n+1)ij d(NLO,E)ij(rR) =

XrBRnd(NLO,E)(rB) . (2.52)

Here we have dened:

X(i,j)PFKS(rR)

(n+1)ij(rR) d(NLO,E)ij(rR) , (2.53)

where the generalised Kronecker symbol (. . .) is equal to one if its arguments are equal, and to zero otherwise, and

rji,i\R = (I1, . . . I\i, . . . Iji, . . . In+3) . (2.54)

Although the rst sum in eq. (2.53) might seem to involve a very large number of terms,

MadGraph5 aMC@NLO knows immediately which terms will give a non-zero contribution, thanks to the procedure used to construct Rn+1 which was outlined at the beginning of

this section. On top of organising the sums over processes and FKS pairs in a di erent way w.r.t. the rst version of MadFKS, MadGraph5 aMC@NLO also performs some of these (and, specically, those in eq. (2.53)) using MC techniques (whereas all sums are performed explicitly in ref. [61]). In particular, for any given rB Rn, one real-emission process and

one FKS pair are chosen randomly among those which contribute to eq. (2.53); these choices are subject to importance sampling, and are thus adaptive. In summary, while the procedure adopted originally in MadFKS takes a viewpoint from the real-emission level, that adopted in MadGraph5 aMC@NLO emphasises the role of Born processes. The two are fully equivalent, but the latter is more e cient in the cases of complicated processes, and it o ers some further advantages in the context of matching with parton showers.

2.4.2 One-loop matrix elements: MadLoop

Both MadGraph5 aMC@NLO and its predecessor aMC@NLO are capable of computing the virtual contribution to an NLO cross section in a completely independent manner (while

d(NLO,E)(rB) =

XrRRn+1

rB, rji,i\R

still allowing one to interface to a third-party one-loop provider if so desired), through a module dubbed MadLoop [68]. However, there are very signicant di erences between the MadLoop embedded in aMC@NLO (i.e., the one documented in ref. [68]), and the MadLoop currently available in MadGraph5 aMC@NLO; hence, in order to avoid confusion between the two, we shall call the former MadLoop4, and the latter MadLoop5. The aim of this section is that of reviewing the techniques for automated one-loop numerical computations, and of giving the rst public documentation of MadLoop5.

As the above naming scheme suggests, core functionalities relevant to the handling of tree-level amplitudes were inherited from MadGraph4 in MadLoop4, while MadLoop5 uses MadGraph5. This was a necessary improvement in view of the possibility of computing virtual corrections in arbitrary renormalisable models (i.e., other than the SM). More in general, one can identify the following three items as strategic capabilities, that were lacking in MadLoop4, and that are now available in MadLoop5:13

A. The adoption of the procedures introduced with MadGraph5, and in particular the UFO/ALOHA chain for constructing amplitudes starting from a given model.

B. The possibility of switching between two reduction methods for one-loop integrals, namely the Ossola-Papadopoulos-Pittau (OPP [18]) and the Tensor Integral Reduction (TIR [154, 155]) procedures.

C. The organization of the calculation in a way consistent with the mixed-coupling expansion described in section 2.4, and in particular with eq. (2.23).

It should be clear that these capabilities induce an extremely signicant broadening of the scope of MadLoop (in particular, extending it beyond the SM). As a mere by-product, they have also completely lifted the limitations a ecting MadLoop4, which were described in section 4 of ref. [68]. It is instructive to see explicitly how this has happened. Item A. is responsible for lifting MadLoop4 limitation #1 (MadLoop4 cannot generate a process whose Born contains a four-gluon vertex, because the corresponding R2 routines necessary in the OPP reduction were never validated, owing to the technically-awkward procedure for handling them in MadGraph4; this step is now completely bypassed thanks to the UFO/ALOHA chain). Limitation #2 (MadLoop4 cannot compute some loops that feature massive vector bosons, which is actually a limitation of CutTools [156], in turn due to the use of the unitary gauge) is now simply absent because of the possibility of adopting the Feynman gauge, thanks again to item A. Limitation #4 (MadLoop4 cannot handle nite-width e ects in loops) is removed thanks to the implementation of the complex mass scheme, a consequence of item A. Finally, item C. lifts MadLoop4 limitation #3 (MadLoop4 cannot generate a process if di erent contributions to the Born amplitudes do not factorise the same powers of all the relevant coupling constants).

The advances presented in items A.C. above are underpinned by many technical di erences and improvements w.r.t. MadLoop4. Here, we limit ourselves to listing the most signicant ones:

13Some of them are not yet public, but are fully tested.

JHEP07(2014)079

MadLoop5 is written in Python, the same language which was adopted for Mad-

Graph5 (MadLoop4 was written in C++).

The UV-renormalisation procedure has been improved and rendered fully general

(that of MadLoop4 had several hard-coded simplifying solutions, motivated by QCD).

An extensive use is made of the optimisations proposed in ref. [24] (OpenLoops).

The self-diagnostic numerical-stability tests, and the procedures for xing

numerically-unstable loop-integral reductions, have been redesigned.

More details on these points will be given in what follows. Before turning to that, we shall discuss the basic principles used by MadLoop5 for the automation of the computation of one-loop integrals.

Generalities. Given a 2 n partonic process r (see eq. (2.28)), MadLoop computes

the quantity:

V (r) =

Xcolour spin

nA(n,1)(r)A(n,0)(r)o, (2.55)

with A(n,0) and A(n,1) being the relevant tree-level and UV-renormalised one-loop ampli

tudes respectively; the averages over initial-state colour and spin degrees of freedom are understood. The result for V (r) is given as a set of three numbers, corresponding to the residues of the double and single IR poles, and the nite part, all in the t Hooft-Veltman scheme [157]. In the case of a mixed-coupling expansion, each of these three numbers is replaced by a set of coe cients, in the form given by eq. (2.23). There may be processes for which A(n,0) is identically equal to zero, and A(n,1) is nite; for these processes (called

loop-induced), MadLoop computes the quantity:

VLI(r) =

Xcolour spin

JHEP07(2014)079

A(n,1)(r)

2 . (2.56)

Only eq. (2.55) is relevant to NLO computations proper, and we shall mostly deal with it in what follows. In the current version of MadGraph5 aMC@NLO, the loop-induced VLI(r)

cannot be automatically integrated (for want of an automated procedure for multi-channel integration), and hence in this case the code output by MadLoop5 must be interfaced in an ad-hoc way to any MC integrator (including MadGraph5 aMC@NLO see e.g. ref. [127] for a recent application).

The basic quantity which MadLoop needs to compute and eventually renormalise in order to obtain A(n,1) that enters eq. (2.55) is the one-loop UV-unrenormalised amplitude:

A(n,1)U =

XdiagramsC , (2.57)

where C denotes the contribution of a single Feynman diagram after loop integration, whose

possible colour, helicity, and Lorentz indices need not be specied here, and are understood;

they will be re-instated later. A standard technique for the evaluation of C is a so-called

reduction procedure, pioneered by Passarino and Veltman [154], which can be written as follows:

C = Red [C] =Xici(C)J (Red)i + R(C) . (2.58)

The quantities J (Red)i are one-loop integrals, independent of C. The essence of any reduction

procedure is that it is an algebraic operation that determines the coe cients ci and R (which are functions of external momenta and of masses, and some of which may be equal to zero); the intricacies of loop integration are dealt once and for all in the computations of the J (Red)is (which are much simpler than any C). As the leftmost equality in eq. (2.58)

indicates, from the operator point of view Red[ ] is the identity; its meaning is that of replacing C with the linear combination in the rightmost member of eq. (2.58). As the

notation {J (Red)i} suggests, di erent reduction procedures can possibly make use of di erent

sets of one-loop integrals.Equation (2.58) is basically what one would do if one were to compute C in a non-

automated manner. In automated approaches, however, additional problems arise, for example due to the necessity of relying on numerical methods, which have obvious di -culties in dealing with the non-integer dimensions needed in the context of dimensional regularisation, and with the analytical information on the integrand of C, which is ex

tensively used in non-automated reductions. In order to discuss how these issues can be solved, let us write C in the following form:

C =

) , C(

) =

JHEP07(2014)079

)

Qm1 i=0 Di

, (2.59)

where d = 4 2, and we have assumed the diagram to have m propagators in the loop

and have dened:

Di = (

+ pi)2 m2i , 0 i m 1 , (2.60)

with mi the mass of the particle relevant to the ith loop propagator, and pi some linear combination of external momenta. For any four-dimensional quantity x, its (4 2)-

dimensional counterpart is denoted by x, and its (2)-dimensional one by ~x. The fact

that pi, rather than pi, enters eq. (2.60) is a consequence of the use of the t Hooft-Veltman scheme. The loop momentum is decomposed as follows:

= + ~

with

= 0 , (2.61)

with similar decompositions holding for the Dirac matrices

and metric tensor g. One can thus dene [158] the purely four-dimensional part of the numerator that appears in eq. (2.59):

N() = lim

0 N(

= ;

= , g = g) , (2.62)

from whence one can obtain its (2)-dimensional counterpart:

eN(, ~) = N() N() . (2.63)

The quantity dened in eq. (2.62), not involving non-integer dimensions, can be treated by a computer with ordinary techniques. By using eq. (2.63) in eq. (2.59) one obtains:

C = CnonR2 + R2 , (2.64)

, (2.65)

. (2.66)

Both integrals in eq. (2.65) and eq. (2.66) still depend on (42)-dimensional quantities, but

they do so in a way that allows one to further manipulate and cast them in a form suitable for a fully numerical treatment. In particular, one can show [158] that the computation of R2 is equivalent to that of a tree-level amplitude, constructed with a universal set of theory-dependent rules (see ref. [159], refs. [160162], and refs. [163165] for the QCD, QED+EW, and some BSM cases respectively), analogous to the Feynman ones and that can be derived once and for all (for each model) by just considering the one-particle-irreducible amplitudes with up to four external legs [166]. On the other hand, eq. (2.65) is still potentially divergent in four dimensions. The details of how this is dealt with may vary, but the common characteristic is that all of them are entirely dened by a reduction procedure. In other words, we shall use the following identity:

C = Red [CnonR2] + R2 , (2.67) which is a consequence of eqs. (2.58) and (2.64). The general idea is that all things that are inherently (4 2)-dimensional in Red[CnonR2] can be parametrized in terms of the

one-loop integrals J (Red)i, so that any piece of computation that would require an analytical knowledge of the integrand and an analytical treatment of the (2)-dimensional terms is

indeed treated analytically, but in a universal manner through J (Red)i.

We emphasise that, although the decomposition of eq. (2.64) is inspired by the OPP reduction method, it is universal, in the sense that the operator Red[ ] in eq. (2.67) does not need to be OPP-inspired, and that the denition of R2 has nothing to do with the OPP procedure as such, but rather with the interplay of (42)-dimensional quantities and their

four-dimensional counterparts. There are of course several alternative approaches, but the majority of them do not lend themselves to the numerical computation of the rational part R(CnonR2) + R2. The two methods which have been used for complicated numerical simulations are bootstrap [17] and D-dimensional unitarity [19, 20, 23]. However, they also involve rather non-trivial issues, such as the presence of spurious singularities (for bootstrap), or the necessity of performing additional computations in 6 and 8 dimensions (for D-dimensional unitarity). The latter problem can be bypassed by means of a mass shift [167], which however might imply additional complications in the case of axial couplings in massive theories. In summary, while it is true that there are advantages and disadvantages in each of these approaches, we point out that R2 must not really be seen as an extra issue in the context of a complete calculation, simply because one has to carry out UV renormalisation anyhow, which is similar to R2 but more involved.

where

CnonR2 =

N()

Qm1 i=0 Di

eN(, ~ )

Qm1 i=0 Di

R2 =

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Integral reduction in MadLoop. We now turn to discussing the way in which the previous formulae are handled by MadLoop. In order to do so, we shall re-instate in the notation the dependence of the amplitudes on the relevant quantities; in particular, we work with scalar sub-amplitudes that have denite helicities (i.e., all Lorentz indices are contracted away, and all Dirac matrices are sandwiched between spinors), and that factorise a single colour factor. The latter condition implies that, in general, our sub-amplitudes are not in one-to-one correspondence with Feynman diagrams (typically when these feature at least a four-gluon vertex), but that they can be written as follows:

A(n,0)h =Xb(0)bBh,b , (2.68)

for the Born amplitude. Here, h denotes a given helicity conguration, and b runs over all possible single-colour factors. The quantity (0)b is one such colour factor, that collects all the colour indices, which are understood. Hence, Bh,b is a scalar quantity which does not

contain any colour index. In the case of one-loop diagrams, we shall use a similar notation, thus replacing the quantities that appear in eqs. (2.65) and (2.66) with:

N() (1)lNh,l() , (2.69)Di

Di,l , (2.70)

R2 R2,h,l . (2.71) The quantity (1)l in eq. (2.69) has the same meaning as (0)b in eq. (2.68), but is relevant to one-loop amplitudes rather than tree-level ones, hence the di erent notation. Since the index l unambiguously identies a single-colour-structure subamplitude, and the latter has a non-trivial kinematic dependence, di erent ls may correspond to di erent one-loop Feynman diagrams, and thus the necessity of inserting a dependence on l on the r.h.s. of eq. (2.70). Finally, we did not factor out the colour structure in eq. (2.71), since this will not be relevant in the following. We remark that decompositions such as those on the r.h.s.s of eqs. (2.68) and (2.69) are easily handled by MadGraph5 aMC@NLO, which has inherited the treatment of the colour algebra from MadGraph5 (see section 2.3 of ref. [38]). It should be stressed that such a treatment is symbolic and that, although introduced to deal with tree-level quantities, is perfectly capable of computing one-loop ones such as (1)l. Furthermore, the colour algebra is internally decomposed in terms of colour ows (for which several di erent representations are available); although this information is presently not used in the integral-reduction procedure, it will be trivial to exploit it in the future, should this need arise. By using eqs. (2.68)(2.71) one obtains:

A(n,1)UA(n,0) =Xcolour Xh

JHEP07(2014)079

Xl(1)l

Nh,l()

Qml1i=0 Di,l

R2,h,l

! Xb

(0)bBh,b!

(2.72)

As was already mentioned, the R2 term gives rise to what is e ectively a tree-level computation; therefore, we shall drop it in what follows, and just deal with:

A(n,1)UA(n,0) nonR

2 =

Nh,l()

Qml1i=0 Di,l

lb Bh,b , (2.73)

Xcolour(1)l(0)b . (2.74)

The integral-reduction procedure of MadLoop4 [68] can be read directly from eq. (2.73), and is as follows:

A(n,1)UA(n,0) nonR

2 =

lb =

lb Bh,b , (2.75)

where Red OPP, since in MadLoop4 only OPP reduction has been considered. Equa

tion (2.75) is not particularly satisfactory from an e ciency point of view, for two reasons, both of which have to do with the fact that the integral-reduction operation is quite time-consuming. Firstly, the Red[ ] operator is called #h #l number of times (we recall that,

by construction, #l is equal to or larger than the number of loop diagrams). Secondly, each of these calls involves the recomputation of Nh,l() a large number of times (determined by the OPP procedure), but which involve only changing the numerical value of , without a ecting any other quantity entering it. The rst issue is solved by reducing the number of integral-reduction operations, while the second by rendering more e cient the computation of Nh,l(). The strategies adopted in MadLoop5 are the following. One begins by

observing that the operator Red[ ] acts on the space of one-loop integrals. Therefore, Born amplitudes must be seen as c-numbers as far as this operator is concerned. One can thus exploit the fact that Red[ ] is linear. Furthermore, what really drives integral reduction is the structure of the denominators (numerators are just numbers computed with suitable values of , specic to the given Red[ ] operator). Hence, one can organize loop integrals in sets of topologies, the latter being dened as subsets of integrals with the same denominator combinations:

ml1

Yi=0Di,l =

mp1

Nh,l()

Qml1i=0 Di,l

XbRed

JHEP07(2014)079

Yi=0Di,p , l, p t , t topologies . (2.76)

By exploiting eq. (2.76), one can rewrite eq. (2.75) as follows:

A(n,1)UA(n,0) nonR

2 =

XtRed

Nh,l() lb Bh,b

Qmlt1i=0 Di,lt

Plt

, (2.77)

for any lt t. Eq. (2.77) is optimal from the viewpoint of reducing the number of calls to

Red[ ], and thus addresses the rst of the issues mentioned before. As far as the second of those issues is concerned, MadLoop5 makes a systematic use of the fact that any numerator

Nh,l() admits the following representation:

Nh,l() =

rmax

Xr=0C(r)1...r;h,l 1 . . . r , (2.78)

where the coe cients C(r) are independent of the loop momentum; when r = 0, we understand that no Lorentz indices and no loop momenta appear on the r.h.s. of eq. (2.78). The quantity rmax is the largest rank in Nh,l, and in the Feynman gauge in renormalisable

theories is always lower than or equal to the number of loop propagators. However, since di erent Nh,l functions appear in the inner sums in eq. (2.77), it is actually convenient

(just for the sake of using the present simplied notation) to regard it as the largest rank in the whole one-loop amplitude; for a given Nh,l, this simply implies that some of the

C(r) coe cients will be equal to zero. Equation (2.78) can be exploited in two ways. One starts by determining the C(r)s once and for all. Then, in the context of the OPP reduction, the numerical computation of Nh,l() becomes much faster, as suggested originally

in OpenLoops (see ref. [24]), because for each new value of generated within the OPP reduction one simply needs to perform the sums and multiplications explicit in eq. (2.78), without recomputing the C(r)s. Furthermore, when eq. (2.78) is symbolically (as opposed to numerically) replaced in eq. (2.77), the structures of tensor integrals naturally emerge. Thus, eq. (2.78) paves the way to performing a Tensor Integral Reduction (TIR) as well. Therefore, regardless of whether an OPP or TIR procedure will be applied, the inputs to the Red[ ] operator in MadLoop5 are the sets:

JHEP07(2014)079

XbC(r)1...r;h,l lb Bh,b)rmaxr=0. (2.79)

Then, when using OPP the Lorentz indices of the two members of these sets are contracted in order to give OPP the scalar functions it needs. On the other hand, when using TIR the rst members in eq. (2.79) are all that is needed for this type of reduction to work. It is clear that the practical success of the decomposition in eq. (2.78) relies on the capability of a fast and e cient computation of the coe cients C(r). MadLoop5 has a fully independent implementation of the recursion-construction procedure presented in ref. [24] (thanks to a dedicated treatment by ALOHA), and its own internal system of caching and retrieving the C(r)s. At variance with what is done in ref. [24], MadLoop5 does not assume rmax to be less than or equal to the number of loop propagators, i.e. it admits the possibility that the contribution of any given vertex to the rank of Nh,l() be larger than one. This is useful, for example, in the context of the computation of QCD corrections to processes stemming from a Higgs EFT Lagrangian. See appendix C.3 for more details.

We conclude this part with a few diverse observations. Firstly, given the advantages of an e cient caching and recycling of the coe cients C(r), in the case of a mixed-coupling expansion MadLoop5 starts from determining all of the coe cients that will contribute to k0+1,0 (we recall that we always associate the largest power of the dominant coupling constant, as dened by the hierarchy of the model, with the terms q = 0: see section 2.4).

Some of the C(r)s thus computed will also contribute to k0+1,1, for which MadLoop5 will only calculate those C(r)s not yet found, and just recycle the others. This procedure is then iterated till necessary, in the sense that it can be stopped when reaching the largest qs among those selected by the user; it is maximally e cient when the smallest user-selected q is equal to zero, which is justied from a physics viewpoint given the coupling hierarchy of the model. Secondly, in general the kinematic congurations which are potentially unstable in TIR are rather di erent than with OPP; therefore, the possibility of using TIR as an alternative to OPP before turning to quadruple-precision calculations (see below) is very benecial for reducing the overall computing time when numerically-unstable situations

1 . . . r

Qmlt1i=0 Di,lt

Xlt

are encountered. Finally, since TIR essentially performs integral reduction at the level of amplitudes, rather than at that of amplitude squared as OPP, it allows one to use e ciently the decomposition of eq. (2.78) and its caching-and-recycling system also in the case of loop-induced matrix elements, eq. (2.56).

UV renormalisation and R2 contribution. In order to obtain V (r) as dened in eq. (2.55), MadLoop must sum to the result of the integral-reduction procedure, eq. (2.77), the R2 (see eq. (2.72)) and the UV-renormalisation contributions. Both of these can be cast in the form of a tree-level-like amplitude A(n,X) times the Born amplitude, whose

contribution to eq. (2.55) will therefore be:

Xcolour spin

nA(n,X)(r)A(n,0)(r)o, X = R2 , UV . (2.80)

In an automated approach, the computation of A(n,X) may be performed in the same man

ner as that of A(n,0), provided that the usual Feynman rules are supplemented by new UV

and R2 rules, and by imposing that A(n,X) contain one and only one UV- or R2-type ver

tex.14 This was indeed the procedure adopted by MadLoop4 for the R2 computation. As far as UV renormalisation was concerned, the fact that MadLoop4 was limited to considering QCD corrections to SM processes allowed signicant simplications, and eq. (2.80) was e ectively computed in a simpler way, by taking the Born amplitude squared multiplied by suitable UV factors. Mass insertions cannot be accounted for in this way; however, their structure being identical to that of the R2 two-point vertex, the two could always be treated together (see ref. [68] for more details).

The above solution is not tenable when considering an arbitrary renormalisable theory, and therefore MadLoop5 must explicitly compute eq. (2.80) for both the UV and R2 contributions. This is not the only signicant di erence between MadLoop4 and MadLoop5. According to the general philosophy of the current MadGraph5 aMC@NLO approach, the UV and R2 rules are part of the NLO theory model15 chosen by the user, and that Mad-

Loop adopts when performing a computation: they are not (as opposed to what happened with MadLoop4) hard-coded UV or R2 computer routines corresponding to n-point counterterms, but a set of instructions in UFO, that ALOHA will dynamically translate into the latter routines.

After including the UV and R2 rules into a UFO model (an operation that, we remind the reader, has to be performed once and for all per theory and per type of corrections, and which FeynRules will soon be able to perform automatically), it should be clear that the computation by MadGraph5 aMC@NLO of A(n,X) becomes identical to that of a regular

tree-level amplitude for which only Feynman rules are relevant; thus, we shall call this a tree-matching construction. In order to increase the exibility of MadLoop5, in particular for what concerns the exclusion of the contributions of certain loop integrals, and to allow developers more freedom when debugging, an alternative but fully equivalent procedure

14Note that in this context a vertex can have two external legs.

15In fact, it is their very presence that tells NLO and LO models apart. See also section 2.1.

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has been implemented, which we shall call loop-matching construction. The tree-matching and loop-matching constructions (an explanation of which will be given below) can be understood by considering the physics contents of the generic UV and R2 counterterm G, which we shall denote as follows:

G = nI(e)1

, . . . I(e)mo

. (2.81)

The quantity X determines the kind of counterterm one is working with UV wave-function, or internal n-point UV and R2 functions; it also allows the builder of the model to specify whether G will be used in the context of a tree-matching or loop-matching construction. In other words, and in order to stress this point again: MadLoop5 can handle both types of construction, which are simply seen as attributes of the model used for the computation. This exibility is important for example because models constructed by hand are set up in a di erent way w.r.t. that adopted by FeynRules. The quantity c is symbolically (i.e., not numerically) set equal to a coupling constant, with that implying that G is a counterterm relevant to NLO corrections in the theory whose perturbative expansion is governed by that coupling. Note that in a model there may be several subsets of counterterms, each associated with a di erent type of correction; in the example of the mixed QCD-QED case discussed at the beginning of section 2.4, when c = S one has counterterms for QCD corrections (eq. (2.26)), and when c = one has counterterms for QED corrections (eq. (2.27)). The quantities L and in eq. (2.81) denote the Lorentz and colour structures of G respectively. The set {I(e)1, . . . I(e)m} is the list of external particle identities associated with G. For example, if G is the contribution to S renormalisation due to the gu vertex, then this set is equal to {g, u,}; if G corresponds to the mass

insertion for the top quark, then one has16 {t, t}. The counterterm G in general receives

contributions from kloop 1 di erent types of loop diagrams, and {I(lk)1, . . . I(lk)nk} is called

the kth loop topology, i.e. the set of the identities of the particles circulating in the kth type of loop.17 When counting these loops, one needs to take into account the physical meaning of G. By using again the example of the gu vertex, one may be tempted to conclude that kloop = 1, which corresponds to the triangle corrections to such a vertex.

This would be incorrect: in fact, since G is a contribution to S renormalisation, it must include wave-function renormalisation factors; therefore, in the kloop types of loops one

must include the bubble diagrams relevant to the g, u, and external legs. Thus, in this example one will need to consider both {g, u} (for triangles and the u self-energy) and {g}, {q}, {b}, {t} (for the gluon self-energy; q is a massless quark, and b and t are heavy

quarks). It should be clear that this complication (there are more loop topologies that contribute to G than the list of its external particles would suggest) is basically due to wave-function renormalisation, which physically corresponds to the fact that renormalised

16A mass insertion is treated in a similar manner as vertices; therefore, all external particles must be outgoing, whence the t.

17The term set implies that two or more identical particles contribute one particle identity to this set, as opposed to the case of external particles, which are all explicitly present in the list {I(e)1, . . . I(e)m}. Note that, when circulating in a loop, a quark and its antiquark can be identied. Furthermore, it should be clear that the present topology has nothing to do with that introduced in eq. (2.76).

; nI(lk)1, . . . I(lk)nkokloopk=1 ; W, L, , c, X

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couplings are dened in terms of renormalised Green functions. When this is not the case, and notably for mass insertions, and for R2 contributions (because the latter are directly dened in terms of specic loop integrals see eq. (2.66)), loop topologies are indeed in one-to-one correspondence with those naively deduced from the list of external particles. Finally, in eq. (2.81) W represents the actual value of the counterterm G. By following

the usual textbook procedure which makes use of renormalised coupling constants, and by ignoring the Lorentz and colour structures, one may e.g. have:

W 1 Z1coupling = 1 Z1vertexYi

Zwf,i , W = 0 , (2.82)

for coupling and internal 2-point (bubble) renormalisation respectively, and

W m , W WR2,vertex , (2.83)

for UV mass insertions and an R2 correction, respectively. From the discussion presented before, it follows that the Z and m terms in eqs. (2.82) and (2.83) will be related to the sets

{I(lk)1, . . . I(lk)nk}. On the other hand, MadLoop5 leaves the model builder the possibility of

implementing coupling-constant renormalisation by directly working at the level of vertex renormalisation. This can be simply done for example by adopting:

W 1 Z1vertex , W 1 Zwf,i , (2.84)

instead of the settings of eq. (2.82). One may (slightly improperly, since the physics contents are exactly the same) refer to the procedures induced by eq. (2.82) and eq. (2.84) as coupling-constant and vertex renormalisation respectively. The latter is the current method of choice for the extension of FeynRules to NLO. Note that, when working with vertex renormalisation, wave-function factors for external legs need not be included (as opposed to the usual case of coupling-constant renormalisation), which can be easily specied in the model denition. Again, no assumption is made in MadLoop5 on the treatment of external legs, and full exibility is maintained by reading the relevant information from the model.

Now we suppose that an NLO model is given, and thus eq. (2.81) is fully specied for all counterterms relevant to all types of corrections dealt with by the model. We shall now discuss how MadLoop5 exploits such information. First of all, one may want to exclude some loops from the calculation (this can be motivated by physics requirements, for example when leaving out heavy-avour contributions, or done for debugging purposes); we call this operation loop-content ltering (not to be confused with diagram ltering, which discards over-counting L-cut diagrams at generation time see section 3.2.1 of ref. [68]). This ltering is basically trivial in the generation of diagrams: one simply does not include the undesired particles in the list of L-cut particles. The presence of {I(lk)1, . . . I(lk)nk} in eq. (2.81) allows one to do the same when computing the UV and R2 contributions: if this set contains the particle(s) to be discarded, the corresponding contributions are not included in eqs. (2.82)(2.84). Apart from loop-content ltering, eq. (2.81) can be exploited in the context of the tree-matching construction in the same way as all other elementary

JHEP07(2014)079

building blocks derived from ordinary Feynman rules, by using the information on external particles, and Lorentz and colour structures. As far as the loop-matching construction is concerned, one starts from a given loop integral C, and determines what we call its

associated tree topology:

C (C) =

, nI(lC)1, . . . I(lC)no . (2.85)

The notation understands that there are T (C) trees T attached to the loop18 (see gure 2

of ref. [68] for a graphical example of such trees); {I(lC)1, . . . I(lC)n} is the set of particles that ow in the loop. Since di erent loops can have the same associated tree topology,

MadLoop5 rst collects all of the di erent tree topologies relevant to the computation being performed. Next, for each of these, all counterterm vertices G (that have survived loop-content ltering) are found that fulll the following equation:

nI(e)1, . . . I(e)mo=

T1, . . . TT(C)

JHEP07(2014)079

TT(C) , (2.86)

where by R(T) we have denoted the root of the th tree T (the root being obviously the

single particle that stems from the loop, and branches into the tree); note that eq. (2.86) implies m = T (C). Among the counterterms thus found, one further considers the UV

mass insertion and R2 ones, and discards those that do not fulll the equation:

nI(lk)1, . . . I(lk)nko=

nI(lC)1, . . . I(lC)no. (2.87)

Equation (2.87) guarantees a rather strict correlation between a generated diagram and its UV and R2 counterterms, which is quite useful for example when establishing the correctness of a model. It should be pointed out, however, that such a correlation can never be turned into a one-to-one map, because of coupling-constant or vertex renormalisation, in which case eq. (2.87) cannot be imposed (since vertex corrections and wave-function renormalisation are always strictly related, and this is true also when carrying out a vertex-renormalisation procedure). Finally, for each counterterm G selected as explained above, MadLoop5 builds the corresponding tree amplitude by attaching to G the o -shell currents relevant to the tree structures {T1, . . . TT(C)}. When performing coupling-constant renor

malisation, MadLoop5 also constructs additional tree-amplitude counterterms by multiplying each Born amplitude by the suitable combination of external wave-function factors.

We conclude this part by re-iterating its main message: MadLoop5 has a fully-exible structure that allows it to handle on equal footing several di erent strategies, such as tree-matching vs loop-matching construction, or coupling vs vertex renormalisation, thanks to its capability of obeying the relevant instructions encoded in the model. This includes the prescription for the renormalisation scheme, which now must be simply seen as one of the model characteristics.

18A four-gluon vertex, with two gluons belonging to the loop, gives rise to two trees, both of which are attached to the loop at the same point.

R (T1) , . . . R

Checks, stability, and recovery of numerically-unstable integral reductions. MadLoop4 featured many self-consistency checks, that served to establish the correctness of any one-loop matrix element generated by the code; they are described extensively in section 3.3 of ref. [68], and all of them are inherited by MadLoop5. On top of those, in MadLoop5 we have included two new checks: rstly, we verify that the matrix elements are Lorentz scalars, by recomputing them using a kinematic conguration obtained by boosting and rotating the original one; and secondly, in the case of QCD corrections we test whether the matrix elements computed in the unitary gauge are identical to those computed in the Feynman gauge.19 The experience with MadLoop4 and MadLoop5 has shown that there is a vast amount of redundancy in all of these checks; therefore, in MadLoop5 we have decided to perform only one of them before proceeding to integrate the matrix elements. We have chosen the most complete one, namely that on the residues of the infrared poles (where the numerical values of such residues as returned by MadLoop are compared to those known analytically from the subtraction of real-emission singularities), which has the virtue of being an indirect test on the UV-renormalisation procedure as well. The other checks can still be performed if need be, by simply executing a single command from the MadGraph5 aMC@NLO interactive shell.20

Integral-reduction procedures are fairly involved, and some kinematic congurations may give rise to numerically-unstable results. Any automated approach must therefore have solid self-diagnostic and recovery strategies: those of MadLoop4 have been presented in section 3.4 of ref. [68]. In view of the extended scope of MadLoop5 w.r.t. that of MadLoop4, both of these strategies have been completely redesigned, for the technical reasons which we shall now discuss. Firstly, MadLoop4 based the instability diagnostics on the results of tests performed within CutTools, on a loop-by-loop basis. Therefore, the capability of MadLoop5 to exploit both the OPP and TIR methods has forced us to set up a CutTools-independent diagnostic tool. Furthermore, a loop-by-loop method is in any case not ideal, because it is never trivial to determine the threshold that decides when a computation is agged as unstable (e.g., a single loop integral may be unstable, but give a totally negligible contribution to the full amplitude). This problem is exacerbated when increasing the nal-state multiplicity, and indeed suggests to use an inclusive (i.e., at the level of the amplitude, rather than of the individual loop integral) type of test in MadLoop5, which can handle more complicated processes than MadLoop4. Secondly, the recovery strategy used by MadLoop4 (which involved a small deformation of the kinematics) is on the one hand related to the loop-by-loop diagnostic (because when one considers two or more integrals simultaneously, the deformation of the kinematics that renders stable an unstable integral can turn a stable integral into an unstable one), and on the other hand not particularly satisfactory when increasing the process multiplicity (because it becomes more di cult to obtain a stable result out of the deformed kinematic conguration).

The diagnostic procedure of MadLoop5 works as follows. Let us write the result of eq. (2.55), obtained by MadLoop after integral reduction and UV renormalisation, as

19Incidentally, these tests can also be performed in the context of tree-level matrix-element computations.

20Apart from that on the dependence on the mass of an heavy quark, which is still available but too process-specic to be worth automating.

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follows:

V (r) = (4)

(1 )

2F Q2

c22 +c1 + c0

. (2.88)

For any given kinematic conguration, the coe cients cj of eq. (2.88) are evaluated 1 + ntest times (in a way specied below), which we denote as follows:

cj c(i)j , i = 0, . . . ntest , for j = 2, 1, 0 . (2.89) These coe cients are used to dene the following quantities:

cj = 12

max

c(i)j

ontesti=0 + min

c(i)j

ontest i=0

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, (2.90)

ontesti=0 min

cj = max

c(i)j

ontesti=0 , (2.91)

which in turn enter the denition of the relative accuracy of the MadLoop evaluation:

P0j=2 cj

. (2.92)

A computation is deemed unstable, and the corresponding kinematic conguration called an Unstable Phase-Space point (UPS), when:

> , (2.93)

with a quantity which can be dened by the user, but whose default value is 103.

The c(i=0)j results in eqs. (2.89)(2.91) are those obtained by applying the OPP reduction with the given kinematic conguration. The c(i>0)j are obtained in two di erent ways, by performing again the integral reduction either: a) by using a kinematic conguration obtained by rotating the original one (hence, by following the same procedure as is used in one of the self-consistency checks previously discussed); or b) by using a di erent ordering of the loop propagators Di as input to OPP (this changes the inner workings of the reduction procedure, and is thus numerically di erent from, although physically completely equivalent to, what one does with the original ordering). These two re-computation procedures are called Lorentz test and Direction test respectively. By default, MadLoop5 sets ntest =

2, and performs one Lorentz test and one Direction test. Both ntest and the type of

tests performed can be controlled by the user. Note that any Direction test re-uses the coe cients C(r) of eq. (2.78) computed in the context of the rst evaluation (c(i=0)j), and is thus less time-consuming than Lorentz tests, despite the fact that both require the integral reduction to be performed from scratch. In the case of a mixed-coupling expansion, each of the c(i)j is expanded as is done in eq. (2.23), so that it will correspond to a set {c(i)j,q}. By

xing q (which is associated with a given combination of coupling constants, see section 2.4), one denes q as in eqs. (2.90)(2.92); a kinematic conguration is a UPS if, following eq. (2.93), q > for any q.

When a UPS is found, MadLoop5 has two main methods for recovery, which are attempted in turn. It starts by changing the integral-reduction procedure, from OPP to

TIR. The results of TIR depend on the specic TIR library MadLoop5 is linked to. In principle, any library might be used; in practice, so far we have considered IREGI [168] and PJFry++ [169, 170]. A given TIR library has a maximal number of propagators it can handle (presently, up to hexagons for IREGI and up to pentagons for PJFry++); in the case one particular loop integral exceeds that number, OPP is used again for it and only for it.21 More than one TIR library can be linked to MadLoop5 at the same time. After having found a UPS with OPP, MadLoop5 switches to the rst of such TIR libraries, and repeats the diagnostic tests mentioned above. If the result is again classied as a UPS, the next TIR library is used, and so forth. If none of the available TIR libraries is able to give a numerically-stable reduction, MadLoop5 resorts to the second method of recovery, namely the OPP integral reduction with all relevant quantities (Nh,l and the internal CutTools algebra) computed in quadruple precision. This is usually extremely e ective, but has the disadvantage of being extremely slow. In the case when the recovery in quadruple precision fails as well, MadLoop5 gives up, sets c0 = 0, and proceeds to the next kinematic conguration; the user is warned when this happens. We emphasise that the order in which the various integral-reduction procedures are used in the context of UPS recovery (OPP, TIR library #1 to TIR library #n) can be controlled through an input card. So in the present public version of MadGraph5 aMC@NLO, where TIR reduction is not yet included, only OPP and quadruple-precision calculations are employed.

2.4.3 Integration of one-loop contributions

The way in which the virtual contributions are integrated by MadGraph5 aMC@NLO in either an fNLO or an NLO+PS computation is quite di erent w.r.t. what was done in aMC@NLO. In the latter, one-loop matrix elements were integrated separately from the other contributions, and eventually combined with them at the level of either distributions (in the case of fNLO), or unweighted events (in the case of NLO+PS); on the other hand, in MadGraph5 aMC@NLO all contributions are integrated simultaneously. The original strategy of aMC@NLO had been adopted because it allowed one to control, in a very direct manner, the number of phase-space points for which the virtual corrections were computed, and by doing so to reduce such a number, without this implying a degradation of the overall accuracy of the physical results.22 The fact that the accuracy of the nal result does not change signicantly despite the reduction mentioned above stems from the following two observations. a) n-body phase-space integrals are signicantly simpler than (n + 1)-body ones, and therefore require to be sampled a smaller number of times than the latter. b) Virtual corrections are usually smaller than the Born, which implies that a smaller number of phase-space points has to be used to integrate the former than the latter, in order to obtain the same absolute precision for the two resulting integrals. The

21This being a single integral, it should be clear that this procedure will not necessarily result again in being classied as a UPS the original UPS was due to all integrals being reduced with OPP.

22The reduction of the number of evaluations of the one-loop matrix elements was (and still is) highly desirable because virtual contributions are typically the numerical bottleneck in our NLO computations (owing to the e ciency of the FKS subtraction, which leads to a relatively fast convergence of the real-emission contributions).

JHEP07(2014)079

possibility of exploiting observation a) in a exible manner was the main reason why in aMC@NLO the virtual contributions were integrated separately. In fact, without a separate integration, n-body matrix elements were previously evaluated the same number of times as (n + 1)-body ones (see eq. (2.50) for an explicit example, relevant to the Born). This could of course be bypassed in several ways, none of which however is simpler than a separate treatment, and better suited to an adaptive multi-channel integration. As explained in section 2.4.1, MadGraph5 aMC@NLO combines n- and (n + 1)-body contributions in the opposite way w.r.t. that of aMC@NLO, taking an n-body viewpoint. This is what allows MadGraph5 aMC@NLO to naturally use observation a) while integrating one-loop matrix elements together with all other contributions.

The simultaneous versus separate integration is only one of the di erences between the current treatment in MadGraph5 aMC@NLO and what was done previously. While the former has several advantages over the latter,23 if applied straightforwardly it implies that the same number of evaluations are performed for the one-loop as for the Born matrix elements, which is not ideal in view of observation b). In order to amend this situation, and thus to increase the speed of MadGraph5 aMC@NLO without a loss of accuracy, several solutions have been devised. They are based on the properties of the following quantity:24

A(n,0)h

2 , (2.94)

where A(n,0)h has been introduced in eq. (2.68), and Vh can be obtained e.g. from eq. (2.72)

by not performing the sum over h there (and similarly for the corresponding UV counterterms). The ratio in eq. (2.94) is a slowly-varying function over the phase space (behaving essentially as logarithms or dilogarithms), and is to a good extent independent of the helicity conguration h (note that the same helicity conguration h is used in the numerator and in the denominator): in other words, one-loop and Born matrix elements have very similar dependencies on helicity congurations and, in particular, there are no helicity congurations for which A(n,0)h is null while Vh is not. This also implies that the ratio of

eq. (2.94) is numerically of the same order as its helicity-summed counterpart:

A(n,0) 2

2 . (2.95)

In MadGraph5 aMC@NLO we exploit the behaviour w.r.t. to h of eq. (2.94) by performing the sum over helicities implicit in V by means of MC methods: for each phase-space point, a single helicity conguration is chosen, according to the relative weights of

A(n,0)h

2 . What

23On top of item a) discussed above: all other things being equal, two or more contributions integrated together lead to a better accuracy than when integrated separately, in the case of cancellation among them, as it often happens with NLO cross sections; also, a smaller number of integration channels implies a reduction of negative-weighted NLO+PS events.

24With some abuse of notation, V here denotes only the nite part of the one-loop contribution, i.e. the coe cient c0 of eq. (2.88) up to overall factors, which are irrelevant for the present discussion.

A(n,1)hA(n,0)h

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A(n,0)h

Ph Vh

A(n,0)h

has been said above guarantees the e ciency and the fast convergence of this procedure, as well as a reduction of the time spent in computing the one-loop contribution (see eq. (2.79) such a reduction is due to the fact that the numerator is simpler and therefore less time-consuming: the time spent carrying out the integral reduction is not a ected).

Let us nally see how the fact that the quantity in eq. (2.95) is a slowly-varying function of the kinematics helps reduce further the CPU load necessary to compute the integral of V . In order to shorten the notation introduced in section 2.4.1, we symbolically write the integral of the NLO cross section as follows:

dn (EV + V ) , (2.96)

where EV denotes all contributions other than V (the integration over the extra degrees of freedom relevant to the real matrix elements plays no role here, and is understood). Integrals such as that of eq. (2.96) are performed by adaptive methods, which entail successive estimates (called iterations) of quantities relevant to the integrals. For the generic integral:

dnF (2.97)

we shall denote by

Ik(F ) , k(F ) , (2.98)

the results of the kth iteration for the mean (i.e., the integral itself) and the standard deviation. One expects that:

lim

k Ik(F ) =

JHEP07(2014)079

dnF , lim

k k(F ) = 0 . (2.99)

It will be convenient for what follows to have an explicit expression for the mean:

Ik(F ) = 1 pk

Xi=1 n

(k,i)n

. (2.100)

Here, we have denoted by (k,i)n the ith phase-space point, generated at random during the course of the kth iteration; a total of pk points are considered. The quantity n collects all normalisation and jacobian factors. We point out that, when applying eqs. (2.97)(2.100) to the case of interest, eq. (2.96), one is able to obtain not only the integral of the sum EV + V , but also those of EV and V individually, by keeping track of Ik(EV ) and Ik(V )

respectively (despite the fact that the two terms are still integrated simultaneously). This is useful in view of the following manipulation: we introduce an approximant of V , that we denote by

eVk, which we use in the identity:

dnV =

Vk +

eVk

. (2.101)

As the notation suggests, the approximant

eVk is a function of the adaptive-integration iteration, where it is used according to the following formula:

Ik(V ) = 1 pk

Xi=1

(k,i)n e Vk

(k,i)n + 1pkfkpkfk

Xi=1

(k,i)n h V

(k,i)n e Vk

(k,i)n i.

(2.102)

A number 0 < fk 1 has been introduced in eq. (2.102), which implies that the di erence

eVk is computed only in a fraction fk of the total number of point thrown.25 For an explicit evaluation of eq. (2.102), we need to dene what enters it. We have:

eVk = ck

A(n,0)

2 , (2.103)

with ck a quantity to be determined iteration-by-iteration, similarly to what happens for fk. The initial conditions are:f1 = 1 , c1 = 0 , (2.104)

and for k > 1 we dene:

ck = grid {Ik1(V )}

grid

nIk1

(min

approximant of V . Secondly, if indeed

eVk1) that appears in eq. (2.106) to decrease faster than the estimated error on the integral of EV + V , thus inducing the values of fk to decrease. On the other hand, eq. (2.106) prevents the series of fks to be uctuating: w.r.t. the preceding value fk1, fk can be at most a factor of 2 larger, or a factor 1/4 smaller these values are simply sensible, but can of course be changed, as the absolute minimum for fk given in eq. (2.107).

The rationale behind eqs. (2.101)(2.107) should now be clear, and it has to do with the fact that one can compute

eVk much faster than V . One starts in the rst iteration by always computing V ; while doing so, MadGraph5 aMC@NLO gathers the information that will allow it to construct the approximant

eV2 to be used in the next iteration. While this procedure is iterated, the relative26 number of times V (

eVk) is computed is decreased (increased). The procedure is exact, being based on the local identity (2.101). Furthermore,

25Although eq. (2.102) literally implies that these are the rst pkfk points, in the actual computation they are chosen randomly in the whole set of the pk points, so that biases are avoided.

26MadGraph5 aMC@NLO, following MadGraph, starts with a relatively small number of points p1 80Ndim, and doubles it at each iteration.

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, (2.105)

eVk1) k1(EV + V ) , 2), 1 4

A(n,0)

2 o

fk = fk1 max

(2k1(V

)

. (2.106)

The value of fk obtained from eq. (2.106) is further constrained to be in the range

0.005 fk 1 . (2.107) A few explanations are in order. Firstly, ck and fk are dynamically constructed, using the information that the numerical integrator (we use a modied version of MINT [171])

has gathered during the previous iteration. One such piece of information is a grid, which among other things stores the averages of the function that is being integrated in non-overlapping phase-space regions which cover the whole phase space. Therefore, ck as dened in eq. (2.105) is a piecewise-constant function. Because of the properties of eq. (2.95), we expect it to be close to an overall constant, and

eVk dened in eq. (2.103) to be a good

eVk is an increasingly (with k) good approximant of V , we expect the quantity k1(V

the code is protected against any pathological behaviour: if, for example,

eVk does not turn out to be a good approximant of V , one will have fk 1 for all ks, so that

eVk will not play any role (see eq. (2.102)). In practice, this situation has not been encountered so far.

2.4.4 Matching to showers: MC@NLO

In this section, we review the MC@NLO formalism [26] and its implementation in Mad-Graph5 aMC@NLO, making extensive use of the results given in section 2.4.1. We start by considering the formulation where the short-distance cross sections are dened for a given real-emission process rR Rn+1, which also allows one to symplify the notation, since the

dependence on rR can thus be easily understood. We shall eventually arrive at expressions which lend themselves to the same manipulations as those carried out at the end of section 2.4.1, which MadGraph5 aMC@NLO exploits in order to deal with MC@NLO cross sections dened at given Born processes, precisely as for their NLO counterparts.

In essence, MC@NLO denes two short-distance cross sections, associated with real-emission-type kinematics (i.e., (n + 1)-body) and Born-type kinematics (i.e., n-body), dubbed H- and S-event contributions respectively. Their forms are written as follows:

d(H) =

X(i,j)PFKSd(NLO,E)ij d(MC) , (2.108)

d(S) = d(MC) +

X(i,j)PFKS

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X=S,C,SCd(NLO,)ij , (2.109)

where d(NLO,E)ij and d(NLO,)ij are exactly the same quantities (eqs. (2.45) and (2.46)) that appear in the NLO cross section (eq. (2.47)). The only new (w.r.t. the NLO) ingredient in MC@NLO is thus d(MC), which is the cross section one obtains from the parton shower Monte Carlo (PSMC) one interfaces to by truncating the perturbative expansion at O(b+1S)

(the Born matrix elements being of O(bS)), in the case of resolved emission (eq. (2.108))

and of no resolved emission (eq. (2.109)) indeed, as is implicit in the notation these two cases result in the same cross section, up to a sign. The crucial point is that, since the leading IR behaviour of any PSMC must be the same as that resulting from an exact matrix-element computation in QCD, eqs. (2.108) and (2.109) are locally nite.27 This is the reason why the d(MC) terms are called the MC counterterms, and the MC@NLO cross section, contrary to the NLO one, can be unweighted.

The denition of the MC counterterms immediately implies that their actual expressions depend on the specic PSMC one interfaces to. These expressions have therefore to be worked out case-by-case, which has been done for the following PSMCs, whose matching with NLO calculations has been fully validated in MadGraph5 aMC@NLO: Pythia8 [123], Herwig++ [121, 122], HERWIG6 [142, 143], and Pythia6 [141] (in the case of pT -ordered Pythia6, only processes with no strongly-interacting particles in the nal state are supported). The details of the construction of the MC counterterms for some of these PSMCs are given in refs. [26, 172175]. On the other hand, the general structure of the MC counterterms is actually PSMC-independent, and it is easy to convince oneself that they can

27In fact, this locality property may be spoiled by certain approximations inherent in the PSMC. It is not di cult to restore it [26], as we shall briey discuss later.

always be written in the following way:

d(MC) =

X(i,j)PFKSd(MC)ij , (2.110)

since FKS pairs are in one-to-one correspondence with IR singularities, which in turn are at the core of the shower mechanism. It is important to bear in mind that this implies that eq. (2.110) is therefore valid not only for those PSMCs based on a 1 2 branching

picture (such as those just mentioned), but more generally for any PSMC consistent with QCD (in particular, those that adopt a dipole picture [34, 176182]). The functional form of the terms d(MC)ij is the same for all of the PSMCs considered here,28 and we shall briey describe its construction in what follows. One starts with the PSMC cross section that results from a single branching:29

d(MC,0)ij =

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Xlcd(MC,0)ij,cl , (2.111)

d(MC,0)

ij,cl = L(MC)

2 dn . (2.112)

As the notation suggests, although d(MC,0)ij does not necessarily coincide with d(MC)ij (the possible di erences between the two will be explained below), it does fully include its physics contents, which we now turn to describing.

The sums in eq. (2.111) run over all possible planar colour congurations (c), and the individual colour lines belonging to them (l). In MadGraph5 aMC@NLO we represent a colour conguration as a list, c = {l1, . . . lm}, where the individual colour line is represented

as an ordered pair, lk = (s(k), e(k)), whose meaning is that of a connection between particle

Is(k) (the starting point of the line) and particle Ie(k) (the end point of the line). This

implies that, for any given c, a quark or an antiquark will belong to a single colour line (through its colour or anticolour respectively), while a gluon will belong to two colour lines (one for colour and one for anticolour). This is the reason for the factor Nij in eq. (2.112), which is equal to 1(2) if ij is a quark or an antiquark (a gluon). MadGraph5 aMC@NLO

constructs the colour congurations during an initialisation phase, by gathering the relevant information from the underlying matrix elements. (l)ij and z(l)ij are the PSMC shower variables; as the notation indicates, in general their forms depend on the branching particle i j (in particular, on whether it is in the nal or initial state), and on the colour line

(which determines the colour partner of i j). The actual shower variables are very

PSMC-dependent, and they are coded in MadGraph5 aMC@NLO for all the PSMCs one may match with. The colour connections in general also determine the choice of Bjorken xs made by the PSMC (see e.g. ref. [26]), which is the reason for the dependence on l in the

28Di erent classes of PSMCs may be conceived, for example dipole-shower-based or by going beyond the leading-Nc approximation (see e.g. refs. [183186]), which could induce a di erent form. However, the idea of MC counterterms in general, and of eq. (2.110) in particular, would still be valid.

29In order to simplify the notation, we understand the universal, azimuthal-dependent part of the branching (see e.g. appendix B of ref. [10]).

x(l)1,2 ijlNijS2

PIjIij(z(l)ij)(l)ij M(n,0)c

(MC)d(l)ijdz(l)ij

argument of the luminosity factor L(MC) in eq. (2.112). Pba(z) is the Altarelli-Parisi one-loop kernel [187], for parton b emerging from the branching of parton a with momentum fraction z. M(n,0)c is the Born matrix element multiplied by a factor determined by the

colour conguration c, according to the prescription of ref. [188]. Finally, (MC) symbolically denotes all kind of kinematics constraints, such as generation-level cuts (an n-body Born must have n well-separated partons), possible dead-zone conditions, and so forth.

Implicit in eq. (2.112) is the choice of a shower scale, which roughly speaking sets an upper bound for the hardness of each branching. Since PSMCs are based on a small-scale approximation, it is clear that the larger the shower scale, the worse the description of physics by any PSMC. While in the context of standalone-PSMC simulations it may be necessary to consider shower scales that stretch that approximation (simply to ll phase-space regions otherwise inaccessible), such an attitude is not justied when PSMC are matched with NLO computations, since the latter provide a much better description of hard-emission regions. Note that in MC@NLO these undesirable large-shower-scale e ects are indeed removed completely at O(b+1S) by the MC counterterms (see eqs. (2.108) and (2.109)).

However, at O(b+2S) and beyond the PSMC may still radiate in the hard regions, poten

tially giving e ects which are simply not sensible from the physics viewpoint. Furthermore, even at O(b+1S) it does not make much sense to allow the PSMC to produce radiation

only to eventually remove it. Fortunately, it is possible to give the PSMC an external mass scale in input; during the course of the shower, the PSMC will generate branchings after choosing the smallest between this external scale and its internally-generated shower scale. In MadGraph5 aMC@NLO, we exploit this possibility in the following way.30 Firstly, we introduce a function of a mass scale :

D() =

(2.113)

with 1 2 two given mass scales. While we typically regard D as a smooth function, it

is perfectly ne to consider its sharp version:

D() = (Q ) , Q = 1 = 2 , (2.114)

which is a particular case of eq. (2.113). Secondly, on an event-by-event basis we determine a mass scale by using:

r = D1(r) , (2.115)

with r a at random number (note that with eq. (2.114) one obtains r Q). Thirdly, we

give r in input to the PSMC, where it acts as an upper bound to the internally-generated shower scales as explained before. The physical meaning of r (be it a relative transverse momentum, a virtuality, or whatever else) depends on the specic PSMC chosen, but is irrelevant here and need not be specied. The crucial thing is the following: by means of this procedure, we are e ectively changing the shower w.r.t. what the PSMC would do if

30This technique has been used sparingly in MC@NLO v3.3 and higher.

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1 1 , monotonic 1 < 2 , 0 > 2 ,

left alone. This change must therefore correspond to a change in the MC counterterms, because of the very denition of the latter. This amounts to:

d(MC,D)ij = D ((Kn+1)) d(MC,0)ij . (2.116)

Note that the argument of D in eq. (2.116) is computed by using the underlying kinematic conguration (after having taken into account its PSMC-specic form: pT ,

pQ2, and so forth), and must not be generated randomly. Equation (2.116) can always be used in place of eq. (2.112), the latter being a particular case of the former, which one can formally obtain by setting 1 = 2 = .

As was discussed before, the function D controls perturbative e ects higher than NLO; hence, its variations can be used to assess the NLO+PS matching systematics (which is, by denition, the size of terms beyond the formal accuracy of the computation) of the MC@NLO method. Although this is expected to be small,31 its actual size is observable-, process- and (especially) PSMC-dependent,32 and it is therefore convenient to be able to study it in a straightforward way. This is the case in MadGraph5 aMC@NLO, where one can control the values of the scales i of eq. (2.113) through the external parameters fi, with i = fis0, and0 the Born-level partonic c.m. energy squared.

Equation (2.116) would give the desired MC counterterms if the corresponding PSMC behaved as expected in the IR regions, namely if it gave exactly the same result as a QCD matrix-element computation in both the collinear and the soft limits, whence the local cancellations in eqs. (2.108) and (2.109). Unfortunately, this is not the case in the soft limit, at least for Herwig and Pythia, where this deciency is basically a consequence of the necessity of having a Markovian shower. What is true, however, is that the amount of soft radiation predicted by the PSMCs is correct; in other words, only its angular pattern is not consistent with the one required by QCD. Fortunately, such an undesirable feature of certain PSMCs will not have dramatic consequences on physical observables, because of the infrared-safety of the latter (the interested reader can nd a fuller discussion of this issue in section A.5 of ref. [26]). The technical problem of the local niteness of the MC@NLO short-distance cross sections can be solved by the following denition of the MC counterterms:

d(MC)ij = (1 G) d(MC,D)ij + G d(NLO,S)ij|real . (2.117)

Here, G is a smooth function dened so that G 1 in the soft limit, and G = 0 outside

of the soft region; d(NLO,S)ij|real is the soft part of the NLO cross section, eq. (2.46), where only the real-emission matrix element contribution is kept.33

A few comments concerning eq. (2.117) are in order. Given that what the PSMC is supposed to do is d(MC,D)ij, while what MC@NLO assumes the PSMC does is d(MC) of

31The main reason being that MC@NLO short-distance cross sections have no contributions of O(b+2S) or higher; terms of these orders in the physical cross sections can only be generated through MC radiation.

32On top of being, obviously, matching-method dependent. We stress that the results of D variations in MadGraph5 aMC@NLO can not be used, even as a mere indication, of the matching systematics that a ects other matching methods, such as POWHEG.

33An analogous solution is adopted when the azimuthal part of the PSMC branching kernel does not agree with that predicted by the matrix elements.

JHEP07(2014)079

eq. (2.117), there is a mismatch of O(b+1S) between the two. This mismatch, however, is

utterly irrelevant for several reasons. Firstly, because of the properties of G, it is conned

to the soft regions, where e ects of all orders in S are equally important. Secondly, in practice in the soft region the PSMC does not even correspond to d(MC,D)ij if not in a

fully inclusive sense, since the PSMC is unable to handle emissions below the IR cuto s, which are of the order of the typical hadron mass (and this for a very fundamental reason: QCD does not have innite resolution power). Thirdly, because of the previous point all NLO+PS matching schemes are liable to have O(b+1S) e ects in small-scale regions which

are not in formal agreement with xed-order results at the NLO, even if the second term on the r.h.s. of eq. (2.117) were not present (see appendix B.3 of ref. [26] for a discussion specic to the MC@NLO formalism). Ultimately, then, the di erences driven by the G

function are power suppressed (see e.g. ref. [189]); eq. (2.117) is nothing but a formal trick to render the O(b+1S) MC@NLO cross sections non-divergent (which is important in view

of their numerical integration) in a region where not even the perturbative predictions of PSMCs are sensible, let alone those of xed-order computations.

Equation (2.117) also gives us the opportunity of commenting briey on the alternative implementation of the MC@NLO method presented in ref. [36]. There, the function G

does not appear, for the simple reason that the two short-distance cross sections on the r.h.s. of eq. (2.117) coincide in the soft limit and, as explained above, this is a su cient condition for not having to introduce G. In turn, this situation occurs because the shower

used in ref. [36], and in subsequent papers, in the context of NLO-matched simulations is constructed to have the same soft behaviour of the matrix elements (see, in particular, eq. (2.5) of ref. [190]). Once this is done, the form of the MC counterterms is uniquely determined, lest one has a mismatch of O(b+1S) (everywwhere in the phase space). One

may opt (as is done in ref. [36]) to see this as a choice made at the level of short-distance cross sections (the soft behaviour of the MC counterterms), that forces one to modify the shower in order to preserve the perturbative accuracy. While the point of view of ref. [26] and of this paper is the opposite one (namely that it is the chosen PSMC which determines the MC counterterms), the fundamental idea is just the same, and therefore the MC@NLO subtractions of ref. [36] do not di er in any signicant way from those that had been adopted in MC@NLO and are now used in MadGraph5 aMC@NLO. What has been changed in the approach of ref. [36] (w.r.t. the previous default) is the shower adopted in conjunction with the NLO matching performed there (which happens not to be the same as that used in simulations which are not matched to NLO results). We point out that should a version of the Pythia or Herwig showers become available with a matrix-element-type soft behaviour, the relevant MC counterterms would be constructed without a G function.

Furthermore, we stress that the NLO accuracy of the MC@NLO method (including, in particular, the whole 1/Nc expansion) is maintained, and that the O(b+1S) results are thus

in agreement with those of the corresponding matrix elements (up to power-suppressed e ects, as explained above), regardless of the behaviour of the PSMC and thus of the presence of a G function. On the other hand, if some aspect of the PSMC is decient (for

example the treatment of subleading-colour contributions), the MC@NLO method itself cannot provide an improvement in the MC-dominated kinematic regions any issue of this kind must be addressed at the level of the PSMC itself.

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Equation (2.117) is the nal form to be used in eq. (2.110); when the latter is in turn replaced in eqs. (2.108) and (2.109), one immediately realises that it is convenient to dene the H- and S-event contributions at xed FKS pair:

d(H)ij = d(NLO,E)ij d(MC)ij , (2.118)

d(S)ij = d(MC)ij +

X=S,C,SCd(NLO,)ij . (2.119)

It is easy to see that these quantities are locally nite (by construction of the shower variables (l)ij and z(l)ij), precisely as their summed counterparts of eqs. (2.108) and (2.109).

Eqs. (2.118) and (2.119) can now be used to dene the MC@NLO generating functional. In order to do that we also introduce, for consistency with the standard notation and in view of the discussion to be given in section 2.4.5, the H- and S-event kinematic congurations:

K(H)n K(E)n+1 , K(S)n K(S)n+1 , (2.120) so that the generating functional nally reads as follows:

(2.121)

where FMC is the generating functional of the PSMC one interfaces to, and its argument

indicates the parton conguration to be adopted as the starting condition for the shower. In eq. (2.121) we have reinstated the formal dependence of the short-distance cross sections on the real-emission process rR. In this way, the complete similarity between the MC@NLO and NLO cross sections in terms of sums over partonic processes and FKS pairs (i.e., between the r.h.s. of eq. (2.121) and the l.h.s. of eq. (2.49)) is evident. Thus, as was anticipated at the beginning of this section, all the manipulations performed at the end of section 2.4.1 apply to the MC@NLO case as well. In particular, the formulation where the MC@NLO short-distance cross sections are dened by xing the Born-level process is the one adopted in MadGraph5 aMC@NLO. Among other things, this renders it particularly easy to obtain S events by integrating over i and yij before unweighting (which is expected to reduce the number of negative-weight events, as advocated in ref. [26] in the context of the MC@NLO formalism; the same idea, dubbed folding, has been independently proposed and implemented in POWHEG).

Before concluding this section, we present two variants of what was discussed so far, that we shall want to consider for future MadGraph5 aMC@NLO developments. The rst one concerns the denition of the MC counterterms. The advantage of using eqs. (2.118) and (2.119) is a one-to-one correspondence between the shower variables (z(l)ij, (l)ij) and the

FKS integration variables (i, yij). Apart from a transparent way of identifying the IR singular structure (which in turn is related to d(H)ij and d(S)ij being locally nite at xed (i, j)), this implies that, when the integration measure over the MC variables is expressed in terms of d(ij)n+1, as it must according to eq. (2.121), one can factorise the jacobian

z(l)ij, (l)ij

(i, yij) ; (2.122)

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FMC@NLO =

XrRRn+1

FMC

K(H)n d(H)ij(rR) d(ij)Bjd(ij)n+1+ FMC

K(S)n d(S)ij(rR) d(ij)Bjd(ij)n+1

X(i,j)PFKS

in the current version of MadGraph5 aMC@NLO, this jacobian is computed analytically. The denitions of eqs. (2.118) and (2.119) might however have a numerical drawback, due to the presence of the factor Sij only in the terms d(NLO,)ij. Its absence in the MC countert

erms could induce di erences in the damping of singularities not due to the FKS pair (i, j), that in turn could result in an unnecessary large fraction of events with negative weights. This situation can be amended as follows: by using eq. (2.35) one obtains the identity:

d(MC) =

(MC)ij in place of d(MC)ij in eqs. (2.118) and (2.119), one can factor out a term

Sij. The disadvantage of course is that the pair (z(l)kp, (l)kp) is not in one-to-one correspondence with (i, yij) any longer, which implies that the relevant jacobians will be much more involved than that in eq. (2.122). However, this is clearly only a technical problem, which can be overcome by giving up the requirement that jacobians be computed analytically. With modern routines for the numerical evaluation of derivatives this appears denitely feasible, and it would also pave the way for a leaner interface to new PSMCs.

We now turn to the second variant, which concerns the MC@NLO formulation proper. A good feature of the MC@NLO cross section is that it gives a clear separation of matrix element and Monte Carlo e ects. A drawback is that one is forced to ignore the fact that in the MC-dominated region (i.e., at small scales) real-emission matrix elements do not give a good description of the underlying physics, which implies that the contribution of H events there is important only in terms of total rate, but not in terms of shapes, which indeed are dominated by showered S events. From the viewpoint of nal (physical) results this is irrelevant, but it entails a loss of e ciency, since typically H events have negative weights in the MC-dominated region. A possible way to address this problem is the following. Consider a function with the properties:

= 1 + O(S) , (2.126)

0 IR limits . (2.127) It is immediate to see that the following denitions:

d(H)ij =

d(NLO,E)ij d(MC)ij , (2.128)

d(S)ij = d(MC)ij +

X=S,C,SCd(NLO,)ij + d(NLO,E)ij (1 ) . (2.129)

result in a generating MC@NLO functional with the same formal accuracy as that of eq. (2.121). It is clear that, while this conclusion holds regardless of the form of , provided

X(i,j)PFKSd(MC)ij =

d(MC) =

X(i,j)PFKS

X(k,p)PFKS Skpd(MC)ij (2.123)

which implies

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X(i,j)PFKSd(MC)ij , (2.124)

(MC)ij = Sij

X(k,p)PFKSd(MC)kp . (2.125)

By using d

that the conditions in eqs. (2.126) and (2.127) are satised, from the physical viewpoint one would identify with a suitable combination of Sudakovs, whose explicit forms would ideally be extracted from the same PSMC one interfaces to. Although at present there is no straightforward way to obtain (numerically) the Sudakovs from a PSMC, there is no reason of principle which prevents the implementation of such a possibility in future versions of modern PSMCs. In the meanwhile, MadGraph5 aMC@NLO will have the means to test eqs. (2.128) and (2.129), by using the analytical expression for the Sudakovs which are currently used in the context of the FxFx NLO merging method (see section 2.4.5).

2.4.5 Merging samples at the NLO: FxFx

In this section, we present a brief review of the FxFx procedure [191], whose aim is that of improving, by systematically including PSMC matching at the NLO, the multi-leg tree-level merging techniques established in the course of the past decade, such as CKKW, CKKW-L, and MLM (see refs. [145148, 192197]). The key word here is merging, which identies the following problem. If one obtains hard events from the processes:

I1 + I2 S + i partons , (2.130) where S is a set of p particles which does not contain any QCD massless partons, how does one match them to PSMCs when di erent i values (i.e., di erent nal-state hard-process multiplicities) are simultaneously considered? The core of the problem is the avoidance of double counting; note that this is on top of, and more complicated than, the double-counting problem that one faces when xing i in eq. (2.130), and which is a matching (not a merging) issue. While the formulation of the problem of merging is independent of the perturbative order (i.e., of the accuracy to which the cross sections of the processes in eq. (2.130) are computed), its solution is not, being strictly connected with the adopted matching strategy. NLO (and beyond) mergings are thus inherently more complicated than LO ones, because LO-matching is basically trivial; they have attracted a signicant amount of attention lately [184, 191, 198207]. The quickest way to realise this is that of considering the tree-level matrix element associated with the process in eq. (2.130); this will give the Born contribution to that process, but at the same time it will also be needed at the NLO as real-emission contribution to a process whose Born features a S + (i 1) partons nal

state. Such a double role of a given matrix element is specic to an NLO merging, and is absent at the LO.

This example of the tree-level matrix-element double role suggests a way to tackle the NLO-merging problem. In particular, in view of the fact that, in the context of a given calculation, the perturbative accuracy of the prediction for an observable is larger the more inclusive the observable, one wants to use as much as is possible an i-parton tree-level matrix element as a Born, rather than as a real-emission, contribution. The role of hard emissions will thus be mainly played by the Borns associated with processes with larger multiplicities, while for any given multiplicity the real-emission contributions will mostly provide the correct (NLO) normalisation. Because the MC@NLO formalism is designed to perturb in a minimal way both the underlying matrix-element description and the PSMC one uses for showering events, the above scheme essentially corresponds to

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limiting the hardness of H-event emissions. Technically, this can be achieved by simply exploiting the D function introduced in eq. (2.113), and by applying analogous conditions at the matrix element level. In order to do so, it is convenient (and particularly sensible physics-wise) to re-interpret the parameters i of eq. (2.113) in the following way:

1 soft (MC dominated),

1 < 2 intermediate,

> 2 hard (ME dominated).

Note that in the case of a sharp D function, eq. (2.114), the intermediate region collapses to a zero-measure set. Once one has dened hard and soft mass scales, one needs to dene a way to measure the hardness; because of the fact that NLO corrections will be computed, it is mandatory that such a measure be IR-safe. The easiest way to achieve this is that of employing quantities that arise from a jet-reconstruction algorithm. We denote by dj the scale (with canonical dimension equal to one, i.e. mass) at which a given S+partons conguration passes from being reconstructed as a j-jet one to being reconstructed as a (j 1)-jet one, according to a kT jet-nding algorithm [144] (in other words, there are j

jets of hardness dj , and (j 1) jets of hardness dj + , with arbitrarily small). In

general, for n nal-state partons one will have

dn dn1 . . . d2 d1 . (2.131)

It will also turn out to be convenient to dene

dj = s, j 0 , (2.132)

with s the parton c.m. energy, i.e. the largest energy scale available event-by-event. Since the function D determines rather directly the way in which the various partonic processes of eq. (2.130) are combined, the results of variations of the parameters that enter into it can be associated with the systematics of the merging procedure (and not of the matching one, as is the case for the unmerged cross sections discussed in section 2.4.4). This is particularly straightfoward, and totally analogous to what is typically done at the LO, when one chooses a sharp D function, in which case Q has to be seen as the merging scale.

We now limit ourselves to reporting the nal forms of the short-distance cross sections necessary to implement the FxFx merging scheme; the interested reader can nd more details in ref. [191]. We denote by N the largest light-parton multiplicity at the Born level that we consider (therefore, N is the largest value that i can possibly assume in eq. (2.130)). The cross sections given below are the analogues of those in eqs. (2.118) and (2.119), i.e., at xed real-emission process and FKS pair; in the present section, we understand these quantities, lest we clutter the notation with unnecessary details. On the other hand, each

H- or S-event contribution or short-distance cross section will carry an index, equal to the number of nal-state particles in the corresponding hard process (at the Born level), this

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information being crucial in all merging procedures. We have:

(S)p+i =

h X

=S,C,SC

d(NLO,)p+i + d(MC,0)p+iD(di+1(K(H)p+i))i

(2.133)

1 D(di(K(S)p+i))

di1(K(S)p+i) 2 ,

(H)p+i =

hd(NLO,E)p+i

1 D(di(K(H)p+i))

di1(K(H)p+i) 2

(2.134)

d(MC,0)p+i

1 D(di(K(S)p+i))

di1(K(S)p+i) 2 i D(di+1(K(H)p+i)) ,

(S)p+N =

h X

d(NLO,)p+N + d(MC,0)p+N

(2.135)

=S,C,SC

1 D(dN(K(S)p+N))

dN1(K(S)p+N) 2 ,

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(H)p+N = d(NLO,E)p+N

1 D(dN(K(H)p+N))

dN1(K(H)p+N) 2

(2.136)

dN1(K(S)p+N) 2 ,

where in eqs. (2.133) and (2.134) one has 0 i < N, and in writing the MC countert

erms we have understood the G dependence as given in eq. (2.117), which is irrelevant for

the sake of the present discussion. The MC@NLO cross section dened by eqs. (2.133) and (2.134) (eqs. (2.135) and (2.136)), and its analogue that we shall introduce later, is called the i-parton (N-parton) sample. With the cross sections above, one denes the FxFx generating functional:

FFxFx =

p+N

d(MC,0)p+N

1 D(dN(K(S)p+N))

Xn=pF(n)FxFx , (2.137)

F(n)FxFx = FMC K(H)n

dBjdn+1 + FMC K(S)n

(H)n

(S)n

dBjdn+1 . (2.138)

Note that eq. (2.138) and eq. (2.121) are formally identical (as was said above, the sums on the r.h.s. of eq. (2.121) are understood here), the di erence being in the form of the short-distance cross sections. Equation (2.137) implies that the FxFx generating functional is the incoherent sum of MC@NLO generating functionals, each of which incorporates FxFx-specic type of cuts but that are otherwise fully analogous to their non-merged counterparts. This renders it straightforward to implement the FxFx merging prescription into MadGraph5 aMC@NLO.

As was discussed in ref. [191], the formulation of FxFx according to eq. (2.138) can be supplemented by a Sudakov suppression, in keeping with what is done at the LO in the CKKW(-L) and MLM methods. The modications of the short-distance cross sections are in fact rather minimal, and amount to what follows:

(S)p+i =

hd(S)p+i + d( )p+ii i

(S)i,min, (S)i,max , (2.139)

(H)p+i = d

(H)p+i i

(H)i,min, (H)i,max , (2.140)

(S)p+i and d

(H)p+i dened in eq. (2.133) (for i < N) or eq. (2.135) (for i = N), and in eq. (2.134) (for i < N) or eq. (2.136) (for i = N) respectively. i is a suitable combination

with d

of Sudakov form factors, the construction of which follows closely the CKKW prescription. Further details, including the denition of the hard scales that enter these formulae can be found in ref. [191]. Here, we limit ourselves to stressing the following fact: while for the processes studied in ref. [191] the avour structure of i had been worked out by hand, it has now been fully automated in MadGraph5 aMC@NLO. The term d( )p+i is necessary in order not to spoil the formal NLO accuracy of the formalism:

d( )p+i = d(NLO,S)p+i|Born

1 D(di(K(S)p+i))

(S)i,min, (S)i,max ,(2.141)

where d(NLO,S)p+i|Born is the soft part of the NLO cross section, eqs. (2.45) and (2.46), where only the Born matrix element contribution is kept. By (1)i we have denoted the

O(S) term in the perturbative expansion of i. It should be clear that i satises eqs. (2.126) and (2.127); therefore, a by-product of the Sudakov-improved FxFx merging procedure (which we regard as our best FxFx scheme, and is thus the default in Mad-Graph5 aMC@NLO) is the possibility of testing the alternative (non-merged) MC@NLO formulation presented in eqs. (2.128) and (2.129).

We conclude this section by discussing two arguments of general relevance to NLO-merging techniques, and which may have signicant bearings on their phenomenological predictions: unitarity, and merging systematics. In the context of merging, imposing a unitarity condition means that the fully-inclusive merged cross section (i.e., the sum over all is of the i-parton-sample predictions) is equal to the total rate of the unmerged 0-parton sample.34 In FxFx unitarity is not imposed, for the reasons we shall now explain. One of the advantages of unitarity is the fact that the merging scale Q can be chosen35 in an arbitrarily-large range, whereas in non-unitary approaches this is not possible, and a sensible choice (which takes into account the hardness of the process, and the fact that Q must itself be hard) is always arbitrary to a certain extent. Such a benet of unitarity has however a less-pleasant side. Namely, in the general context of resummed computations matched with xed orders, constraints on total rates contribute to signicant modications of matrix-elements predictions, in shape and absolute value, also in regions where one would expect large-logarithms e ects to be suppressed (the Higgs pT spectrum is a spectacular example of this phenomenon see e.g. gures 1 and 2 of ref. [208]). This is of course acceptable, and actually constitutes a dening prediction of a matched formalism, if all contributions to the latter are perturbatively consistent: an example is that of an MC@NLO unmerged cross section, where the total rate is of O(b+1S), which is the same

perturbative order to which both S and H events contribute. When unitarised-merging is considered, the total rate is still imposed to be that of O(b+1S); however,

S- and H-event iparton samples (and their analogues within any merging formalism) are of O(b+i+1S); this mismatch of perturbative orders present for any i 1 might result in a bias (uniquely due

to the total-rate constraint) at the level of shapes, the stronger the larger i. Note that one could impose the unitarity constraint using an NNLO total-rate prediction (O(b+2S)), were

34We neglect here possible complications due to heavy-avour thresholds, and to contributions to i-parton samples that cannot be shower-generated starting from a lower multiplicity.

35In principle; in practice, there may be e ciency issues, which are however not of concern here.

di1(K(S)p+i) 2 (1)i

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that available. This would marginally alleviate the problem above, but not solve it: rstly, it merely shifts it to i 2, and secondly one still uses NLO-matched (and not NNLO-

matched) i-parton samples which thus require, at the very least, to be re-normalised. The counter argument is that, although a bias might indeed be present, it appears at perturbative orders which are in any case beyond accuracy. This is correct, but does not have any implications on the presence of large logarithmic terms, that could enhance such contributions. More importantly, it is also an argument that can be used in the context of a non-unitary approach, where it would apply chiey to total rates (and, by construction, in non-unitary approaches there is no bias due to total-rate constraints). In fact, the dependence on Q of inclusive results can be used e ectively in non-unitary mergings as a way to arrive at a sensible Q range, to be employed to assess the merging systematics of di erential distributions. Examples of this, and of the fact that FxFx exhibits a rather small Q dependence, will be given in section 4.2. In conclusion, the arguments above can be argued in di erent ways; it should be clear, however, that regardless of whether a merging approach imposes or not unitarity, in those phase-space regions where matrix elements and PSMCs will be vastly di erent the Q dependence is bound to be large. In order to be able to study such e ects as locally as is possible, we prefer not to use unitarity arguments in FxFx.

For what concerns the study of merging-scale systematics, we emphasise that the discussion presented above by no means justies the practice of not being conservative with the choice of Q in non-unitary approaches. There is a particularly common misconception, relevant when the physics one wants to study features a threshold for jet hardness (that we shall denote by p(cut)T henceforth): such a misconception entails choosing Q < p(cut)T. The (implicit) argument for this choice is that a tagged jet is by denition a hard quantity, and by setting Q > p(cut)T one might spoil the underlying NLO accuracy36 for the corresponding jet cross section. Several observations are in order here. Firstly, the use of p(cut)T as a

criterion to determine Q is a contradiction in terms: by denition, a merging prescription is what allows one to use samples of hard events without knowing a priori for which observables they will be employed, in particular which minimal jet hardness will be imposed. The fact that the merging is not perfect (i.e., it does not have zero systematics) just implies that both Q p(cut)T and Q p(cut)T are not particularly sensible, and nothing else. Secondly,

a criterion based solely on p(cut)T misunderstands the meaning of hardness, which is not absolute, but relative. For example, a jet with p(cut)T = 40 GeV can rightly be dened to be hard

in inclusive W production; the same jet is less hard in Higgs production, basically soft in tt production, and certainly soft in the production of a 1-TeV Z resonance. Indeed, it should be obvious that it is always a ratio of mass scales (one of which is of the order of the intrinsic hardness of the production process, essentially dened by the masses of the nal-state particles, and the other of the order of p(cut)T), and never the absolute value of one such scale,

that matters: the argument of a logarithm is a dimensionless quantity. Thirdly, and related to the previous item. When choosing Q > p(cut)T one might indeed spoil some underlying

NLO description, and for a very good reason: such matrix-element-driven prediction may

36Or the tree-level matrix element accuracy in the case of LO mergings, where this argument is also applied.

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simply be irrelevant in the case of a strong scale hierarchy where p(cut)T is much smaller

than the intrinsic hardness of the process, because there one believes the correct type of prediction for jet observables to be rather a PSMC-dominated one. Even if the presence of Sudakov suppression factors in the merged matrix elements may allow one to employ a merging scale which is smaller than what naive expectations would suggest, still by choosing Q < p(cut)T one runs the risk of spoiling the underlying PSMC description of the lowest-multiplicity sample. In conclusion, given that a merging-scale choice necessarily represents a non-perfect compromise between a matrix-element- and a PSMC-dominated prediction, for processes where a scale hierarchy is not overwhelmingly clear (i.e., when p(cut)T is a

non-negligible fraction of the hardness of the production process), one must not decide beforehand whether it is either a matrix-element or a PSMC description which is suited best: they are in principle both valid alternatives, and a su ciently large range of Q in the surrounding of p(cut)T must be probed, lest one underestimates the merging-scale systematics.

A nal technical remark: given that the work of ref. [191] has used HERWIG6 as PSMC, and that the FxFx formalism naturally matches a pT -ordered shower, its use with Pythia8 and Herwig++ does not pose any conceptual problems. In fact, a fully automatic FxFx interface with Pythia8 has now been achieved, but is not part of the current public MadGraph5 aMC@NLO release (the related, specic routine inside the Pythia8 code has become available starting from v8.185). For this reason, the sample FxFx-merged results which will be presented in section 4.2 will make use of HERWIG6. The FxFx-Pythia8 interface also paves the way for a fully analogous procedure, that will be carried out with Herwig++. We also remind the reader that the FxFx method has so far been formulated only for processes that do not feature light jets at the Born level of the lowest-multiplicity sample, and that the merging of b-quark production processes has not been explicitly studied. Although we believe that these cases can be treated with only minor modications (if any) w.r.t. the present implementation, we postpone their discussion to a future work.

2.5 Spin correlations: MadSpin

In both SM physics and BSM searches the role of unstable particles, that are not directly observable but (some of) whose decay products may be seen in detectors, is very prominent. Let us consider the production of p unstable particles uk (k = 1, . . . p; for example, u1 = Z, u2 = t, u3 = t, and so forth), each of which decays into nk particles d1,k, . . . dnk,k, in association with l stable particles s1, . . . sl:

x + y u1( d1,1 + . . . dn1,1 + X1) + . . . up( d1,p + . . . dnp,p + Xp) +s1 + . . . sl + X0 . (2.142)

It is convenient to regard eq. (2.142) as a parton-level quantity, so that x, y, and all the particles in the sets Xk are gluons or light quarks; the contents of Xk depend on the perturbative order considered,37 and need not be specied here. Equation (2.142) does not properly dene a process, but has the following intuitive meaning: it corresponds to the

37And, in general, on the type of corrections. In order to be denite, we consider here the case of QCD.

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contributions to the process

x + y d1,1 + . . . dn1,1 + . . . d1,p + . . . dnp,p + s1 + . . . sl + X , (2.143)

X =

[k=0

Xk , (2.144)

whose Feynman diagrams feature an s-channel propagator for each of the p unstable particles uk, with one end of the propagator attached to a subdiagram that contains at least the decay products d1,k, . . . dnk,k (such a subdiagram is a tree at the leading order). These diagrams may be called p-resonant diagrams and, by extension, any diagram that features n such propagator-plus-subdiagram structures will be called n-resonant (so that, by including the case n = 0, all diagrams contributing to eq. (2.143) can be classied in this way). This implies that it would be natural to associate with eq. (2.142) the matrix elements obtained by considering only the p-resonant diagrams in their computations. Unfortunately, this is not straightforward, since in general it violates gauge invariance. So the only possibility is that of an operative meaning: thus, eq. (2.142) stands for the matrix elements relevant to the process of eq. (2.143), subject to selection cuts whose purpose is that of forcing the p-resonant diagrams to be numerically dominant. While this approach is the cleanest possible from a theoretical viewpoint, it has an obvious problem of e ciency: the non-p-resonant contributions to eq. (2.143) might swamp the p-resonant ones; furthermore, the matrix elements of eq. (2.143) are usually very involved. The di culties mentioned above can be overcome by observing that a computation that uses only p-resonant diagrams is formally correct in the limit where all the widths of unstable particles vanish, uk 0, k (narrow-width approximation, which can be systematically improved in the context of a pole expansion [101, 209, 210]). An immediate consequence of the narrow-width approximation is that the amplitudes associated with parton emissions (i.e., beyond leading order) from the decay products dik,k do not interfere with those associated with emissions either from dik ,k

, for any k 6= k, or from any particle which is not a decay product. Hence,

higher-order corrections factorise, and can thus be sensibly considered separately for production and decay. This results in a signicant simplication of the calculation, since in the narrow-width approximation one can therefore write:

x + y u1 + . . . up + s1 + . . . sl + X0 , (2.145) uk d1,k + . . . dnk,k + Xk k = 1, . . . p . (2.146)

As this notation suggests, the particles uk in eqs. (2.145) and (2.146) are regarded as naland initial-state objects respectively, rather than as intermediate ones as in eq. (2.143). The calculation of amplitudes in the narrow-width approximation can be done by employing well-established spin-density-matrix techniques, which allow one to account for all spin correlations (we remind the reader that the process of eq. (2.143) is said to have decay spin correlations if its matrix elements depend non-trivially on the invariants di,kdj,k for some k;

production spin correlations are present in the case of non-trivial dependences on dik,kdi

k ,k

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for any ik and ik with k 6= k, or on dik,ksq, dik,kx, dik,ky and dik,kX0 for some ik and k).

When spin correlations e ects are small or can be neglected, a further simplication can be made, where one replaces the squared amplitudes associated with the p-resonant diagrams relevant to eq. (2.143) with those relevant to the production (eq. (2.145)) and decays (eq. (2.146)) separately.38 Despite being a priori rather crude, such a simplication is very widely used in the context of PSMCs, since the narrow-width approximation is more di cult to automate already at LO, and more so beyond LO. Besides, unweighted-event generation is signicantly more e cient for the process of eq. (2.145) than it is for that of eq. (2.143). The procedure is therefore that of generating (unweighted) events that correspond to eq. (2.145), and then let the PSMC decay the unstable particles according to eq. (2.146) (with Xk = , i.e., at the leading order) during the shower phase. Note that

the PSMC must know how to handle these decays, which sometimes involve non-trivial matrix elements (e.g., in top decays, or for H0 22). If this is not the case, decay spin

correlations are also incorrectly predicted.

In order to retain the advantages of the separation of production from decays at the level of squared amplitudes, such as e ciency and ease of automation, without losing the capability of predicting spin correlations, MadGraph5 aMC@NLO uses the method introduced in ref. [211],39 and studied there for top and W decays in the SM, which has been fully automated and extended to generic models in ref. [213]; the corresponding module in the code has been dubbed MadSpin.40 The method is based on the following identity:

lim

{ uk }0 M x

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+ y d1,1 + . . . dnp,p + s1 + . . . sl + X

Yk=1 1uk U {uk}pk=1

, (2.147)

where U is a universal factor, and the matrix elements at the numerator and denominator

on the l.h.s. are the tree-level ones relevant to the processes of eqs. (2.143) (with p-resonant diagrams only) and (2.145) (with X0 = X) respectively, and:

1uk = (p2uk m2uk)2 + (muk uk)2 . (2.148)

Note that the factors 1uk in eq. (2.147) cancel exactly the denominators of the unstable-particle propagators, so that the limit is indeed a nite quantity. The fact that the set X of radiated partons is the same at the numerator and denominator in eq. (2.147) implies that the two matrix elements are computed at the same relative order w.r.t. the leading one; this allows one to correctly take into account the e ects of hard radiation at the production level. Finally, the factor U depends only on the identities of the unstable particles

(and possibly on the decay kinematics), but is independent of the production process. It is computed by considering the decays of eq. (2.146) at the LO (i.e., with Xk = ), in a fully numerical manner by MadSpin.

38In other words, the narrow-width approximation allows one to get rid of the non-p-resonant diagrams, whereas the simplication mentioned here gives a prescription for the actual computation of the p-resonant diagrams where one does not use spin-density matrices.

39For an alternative method, based on the knowledge of the polarization states of the unstable particles and on spin-density matrices, and which is adopted in Herwig++, see ref. [212].

40When working at the LO, production spin correlations can be recovered not only by using MadSpin, but also by adopting the so-called decay-chain syntax see appendix B.1 for more details.

M(x + y u1 + . . . up + s1 + . . . sl + X)

Equation (2.147) is used within a standard hit-and-miss procedure, that determines the kinematics of the decay products dik,k given that of the unstable particles uk. In practice, unweighted events are rst obtained for the process in eq. (2.145); then, for each of these the phase-space of the decay products is sampled, and through hit-and-miss unweighted events for the process in eq. (2.143) are obtained. It should be clear that the latter events thus correctly incorporate the information on both production and decay spin correlations. Furthermore, there is evidence that, at the NLO, eq. (2.147) gives a general better description of the exact result for eq. (2.143) than that of an NLO narrow-width prediction, in spite of the LO-only treatment of the decays in the former. This has to do with the fact that, in the narrow-width approximation, congurations where the virtuality of an unstable particle (as reconstructed from nal-state objects) is larger than the particle mass are suppressed both by S (being due to hard NLO corrections) and by a kinematical factor for a recent discussion relevant to top physics, see section 3 of ref. [214]. Furthermore, the NLO origin of these congurations implies that, in the case of the production of more than one unstable particle, such o -shell e ects can be possibly included only for one particle at a time in the NWA. On the other hand, eq. (2.147) trivially allows one to include o -shell e ects without any kinematics suppression, already at the leading order, and for all unstable particles simultaneously. In particular, one starts by generating the virtualities of uk according to a Breit-Wigner distribution (rather than using the pole masses), and employs those in the generation of the momenta of the decay particles that enter the hit-and-miss procedure mentioned above. In so doing, one may introduce back into the problem gauge-violating terms formally of O( uk/muk), i.e., within the accuracy of the method. So the

only potential issue might result from the numerical coe cients in front of such terms not being of O(1). However, one can use eq. (2.147) to check that this is not the case; indeed,

the bound of that equation is a fairly good one, even for congurations where the virtuality of some unstable particle is more than ten widths away from the pole mass.

We have so far tacitly assumed to deal with the most common and numerically-relevant case in the SM, namely that of QCD corrections to the production of weakly-decaying unstable particles. When one starts considering EW NLO e ects, one may face a self-consistency problem, due e.g. to the fact that the loop propagators relevant to EW vector bosons would have to feature non-zero widths (through the complex mass scheme prescription), while the same particles would have to be treated as on-shell (owing to the uk 0 limits) when appearing in the nal state. Hence, in the context of a mixed-coupling expansion, or in general when width e ects are induced by the same type of interactions which are responsible for higher-order corrections, the method discussed in this section has to be employed by carefully considering the characteristics of the production process.

The MadSpin module included in MadGraph5 aMC@NLO features a number of upgrades w.r.t. that presented in ref. [213]. The most important of these is that the phase-space generation is now handled by the Fortran code (rather than by a Python routine). The determination of the maximum weight has been fully restructured as well; on top of that, it has been noticed that using more events, but less phase-space points per event, is more e cient (the defaults have changed from 20 and 104 to 75 and 400, respectively). These two major improvements have resulted in a dramatic decrease of the running time

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(by a factor of about 10). Furthermore, the former one has paved the way for lifting the present limitation that restricts the code to dealing only with (a succession of) 1 2

decays. From the phenomenology viewpoint, two important novelties are the following. Firstly, MadSpin now always writes the information on the polarization of nal-state particles, gathered from the matrix elements, onto the LHE les. This implies, in particular, that PSMCs, if equipped with a suitable module, can handle decays including exactly all decay spin correlations, and also the production ones due to the diagonal terms of the spin-density matrix. Secondly, one can now use a UFO model in the determination of the decays which can be di erent from the one adopted in the computation of the short-distance undecayed matrix elements.

3 How to perform a computation

The theoretical background discussed in section 2 can be ignored if one is only interested in obtaining phenomenological predictions. This is the attitude that will be taken in this section, whose aim is that of giving the briefest documentation for the basic running of the code, and thus to show its extreme simplicity and level of automation.

The MadGraph5 aMC@NLO package is self-contained (third-party codes are included in the tarball see appendix D for their complete list). The local directory where its tarball is unpacked is called main directory; there is a miniminal (and optional) setup to be done, as described in appendix A. In essence, the computation of a cross section by MadGraph5 aMC@NLO consists of three steps: generation, output, and running. All three steps are performed by executing on-line commands in the MadGraph5 aMC@NLO shell,41 which can be accessed by executing from a terminal shell in the main directory the following command:./bin/mg5 aMCThe prompt now reads:MG5 aMC>which signals the fact that one is inside the MadGraph5 aMC@NLO shell. The three steps mentioned above correspond to the following commands:MG5 aMC> generate processMG5 aMC> outputMG5 aMC> launchrespectively; here, process denotes the specic process one is interested into generating (see appendix B.1 for full details on the syntax).

When generating a process, one must decide whether he/she is interested in including or not including NLO e ects. In fact, although by denition an NLO cross section does include an LO part (the Born contribution), if NLO e ects are not an issue it does not make much sense to include their contributions only to discard them at run time, especially in view of their being much more involved than their Born counterparts. Furthermore, the majority of physics models (see section 2.1) have not yet been extended to include NLO corrections, and in these cases the whole procedure could not even be conceived.

41As in MadGraph5, scripting commands is of course possible.

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For these reasons, we talk about LO-type and NLO-type generations, and we note that these two types of generation give access to di erent running options. We shall discuss their di erent merits in the following sects. 3.1 and 3.2 respectively. Before going into that, we point out that an LO-type generation produces the same short-distance cross section code (and hence the same type of physics) as that one would have obtained by running MadGraph5. All possibilities that were available in MadGraph5 are still available in MadGraph5 aMC@NLO; at the same time, it should be clear that, even in the context of an LO-type generation, MadGraph5 aMC@NLO has a much wider scope than MadGraph5.

3.1 LO-type generation and running

[diamondsolid] Generation

In the generation phase, MadGraph5 aMC@NLO constructs the cross section relevant to the process given as input by the user, and thus performs the operations sketched in section 2.1. For example, if one is interested in ttW + production in pp collisions at the LO, one will need to execute the following command:MG5 aMC> generate p p > t t~ w+When generating a process, MadGraph5 aMC@NLO assumes the model to be the SM (with massive b quarks); a di erent model can be adopted, by importing it before the generation (see appendix B.1). For example, the generation of a pair of top quarks in association with a pair of neutralinos (the latter denoted by n1) can be achieved as follows: MG5 aMC> import model mssmMG5 aMC> generate p p > t t~ n1 n1

[diamondsolid] Output

When the process generation is complete, the relevant information is still in the computer memory, and needs to be written on disk in order to proceed. This is done by executing the command:MG5 aMC> output MYPROCwhere MYPROC is a name chosen by the user42 that MadGraph5 aMC@NLO will in turn assign to the directory under whose tree the cross section of the process just generated is written. We call such a directory the current-process directory, which is where all subsequent operations will be performed. For more details on this and for an overview of the structure of the current-process directory, see appendix A.

[diamondsolid] Running

The running stage allows one to accomplish a variety of tasks, the most important of which are the production of unweighted events, and the plotting of user-dened physical observables. Regardless of the nal product(s) of the run, MadGraph5 aMC@NLO will

42Such a name may be omitted; in this case, MadGraph5 aMC@NLO will choose one. On the other hand, there are a few names which are reserved, since they are interpreted as options of the output command. See appendix B.6 for more details.

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start by integrating the cross section generated and written in the two previous steps. In order to run MadGraph5 aMC@NLO, one executes the command:MG5 aMC> launchWhat is prompted afterwards opens an interactive talk-to (which, again, can be scripted)

that allows the user to choose among various options. This looks as follows:

The following switches determine which programs are run:1 Run the pythia shower/hadronization: pythia=OFF2 Run PGS as detector simulator: pgs=OFF3 Run Delphes as detector simulator: delphes=OFF4 Decay particles with the MadSpin module: madspin=OFF5 Add weight to events based on coupling parameters: reweight=OFF Either type the switch number (1 to 5) to change its default setting,or set any switch explicitly (e.g. type madspin=ON at the prompt)

Type 0, auto, done or just press enter when you are done.[0, 4, 5, auto, done, madspin=ON, madspin=OFF, madspin, reweight=ON,...]

We would like to emphasise that the structure of the above prompt will evolve in the near future, and be made more similar to its NLO counterpart (see section 3.2); however, the general idea that underpins its use will remain the same, so that what follows has to be regarded as an exemplication of the general features of the MadGraph5 aMC@NLO talk-to phase.

By entering 1, 2, 3, 4, or 5 at the prompt one toggles between the two values ON and OFF of the corresponding feature. For example, by entering 1 one is prompted again with the display above, except for the fact that pythia=OFF has now become pythia=ON. By entering 1 again, one gets back to pythia=OFF. By entering 0, or done, or by simply hitting return, the talk-to phase ends, and MadGraph5 aMC@NLO starts the actual run. The defaults shown above imply that MadGraph5 aMC@NLO will simply limit itself to integrating the cross section, and to producing the required number of unweighted events. On the other hand, by turning the various switches above to ON, one enables the following features. pythia=ON: with such a setting MadGraph5 aMC@NLO will steer the showering of the hard events previously generated, by employing Pythia6. However, it is crucial to bear in mind that the same hard events can be showered with PSMCs other than Pythia6, but in this case MadGraph5 aMC@NLO is not capable of steering the shower43 (the steering of Pythia8 will soon become available). The user may have an independent installation of Pythia6, but also install the code using the MadGraph5 aMC@NLO shell, by executing the command install pythia-pgs.pgs=ON: in this case, MadGraph5 aMC@NLO will also steer the run of the Pretty

Good Simulator (PGS) [215] after that of Pythia6 (i.e., rst all events are showered and hadronised, and next they are passed through the basic detector simulation as implemented by PGS). For this reason, when pgs=ON MadGraph5 aMC@NLO automatically sets pythia=ON. Note, also, that when the MadGraph5 aMC@NLO shell is used to install Pythia with the install pythia-pgs command, PGS is installed too.

43This is not the case for NLO simulations see later.

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delphes=ON: this allows one to steer the run of Delphes 3 [216] for a fast detector simulation. Delphes can also be installed through the MadGraph5 aMC@NLO shell with the command install Delphes.44 As for switch number 2, when delphes=ON Mad-Graph5 aMC@NLO sets automatically pythia=ON.madspin=ON: by doing so, one includes production spin correlations by means of Mad-

Spin (see section 2.5). Note that the decay-chain syntax (see appendix B.1) is actually faster and features a better approximation of the exact cross section than MadSpin, which is thus more conveniently used in the context of NLO simulations.reweight=ON: instructs MadGraph5 aMC@NLO to store in the LHE le information to be used later within the matrix-element reweighting procedure, for example relevant to assessing the impact of di erent theoretical assumptions (see section 2.3.3 and appendix B.5).

When switches 15 are set as desired by the user and 0, or done, or <return> are entered, MadGraph5 aMC@NLO proceeds with the run, whose rst stage is that of giving the user the possibility of modifying the various inputs relevant to the options selected above. Such inputs, and the * card.dat les where they are stored, have a self-explanatory meaning, and we will not discuss them in detail here.

3.2 NLO-type generation and running

[diamondsolid] Generation

The generation phase of an NLO-type generation has the same conceptual meaning of that relevant to the LO case, described at the beginning of section 3.1. The syntax is also very similar: adopting again the example of ttW + production in pp collisions, one will need to execute the following command in order to include QCD NLO e ects:MG5 aMC> generate p p > t t~ w+ [QCD]

As one can see, the only di erence w.r.t. the case of the LO-generation is in the presence of the keyword [QCD] here. Fuller details on the syntax for NLO-type generation are given in appendix B.1.

[diamondsolid] Output

There is no conceptual or technical di erence w.r.t. the case of an LO-type generation in the output phase. Specically, the command one will need to execute is the same as that described in section 3.1:MG5 aMC> output MYPROCThe same comments concerning the choice of MYPROC as made for LO simulations apply here (see footnote 42).

[diamondsolid] Running

Also in the case of an NLO-type generation, the running phase begins by integrating the cross section generated and written in the two previous steps. There is an important

44Or install Delphes2, if one wanted to use the older Delphes 2 [217] version.

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di erence w.r.t. the LO case that must be stressed here. Namely, at the LO the short-distance cross sections relevant to LO+PS and to fLO computations are identical. This is not the case at the NLO: the MC@NLO cross sections used in NLO+PS computations are di erent from the xed-order ones used in fNLO computations (the former contain the MC counterterms, the latter do not; the former can be unweighted, the latter cannot see section 2.4.4 for more details). However, when MadGraph5 aMC@NLO generates and outputs a process, it writes both cross sections, and in so doing allows the user to choose at runtime which type of computation to perform. In other words, after having generated a process and used it to obtain (say) fNLO results, there is no need to re-generate it to obtain NLO+PS results: it is su cient to run MadGraph5 aMC@NLO again.

In order to run MadGraph5 aMC@NLO, one executes the same command as that relevant to the LO-type generation:MG5 aMC> launchHowever, what is prompted afterwards opens an interactive talk-to which is di erent from the one of the LO-type generation. In particular, one now obtains what follows:

The following switches determine which operations are executed:1 Perturbative order of the calculation: order=NLO2 Fixed order (no event generation and no MC@[N]LO matching):fixed order=OFF3 Shower the generated events: shower=ON4 Decay particles with the MadSpin module: madspin=OFF Either type the switch number (1 to 4) to change its default setting,or set any switch explicitly (e.g. type order=LO at the prompt)

Type 0, auto, done or just press enter when you are done.[0, 1, 2, 3, 4, auto, done, order=LO, order=NLO,...]

By entering 1, 2, 3, or 4 at the prompt one toggles between the two values of the corresponding feature (which are NLO and LO for 1, and ON or OFF for 24). For example, by entering 2 one is prompted again what is displayed above, except for the fact that fixed order=OFF has now become fixed order=ON. By entering 2 again, one gets back to fixed order=ON. By entering 0, or done, or by simply hitting return, the talk-to phase ends, and MadGraph5 aMC@NLO starts the actual runs.

It is the combinations of the values of the switches 1 and 2 that control which kind of computation the program will perform. More explicitly, we have:

(order=NLO,fixed order=OFF) NLO+PS and FxFx-merged45 (order=NLO,fixed order=ON) fNLO

(order=LO,fixed order=OFF) LO+PS

(order=LO,fixed order=ON) fLO

One need not be suprised by the fact that LO+PS and fLO results can be obtained following an NLO-type generation, since all the LO information is obviously there, being part of the

45We remind the reader that i-parton samples, the contributions to an FxFx cross section of a given multiplicity, are nothing but unmerged NLO+PS samples with some extra damping factors, which are included by MadGraph5 aMC@NLO through a parameter (ickkw) in an input card.

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NLO cross section. Rather, one may wonder why these options are not disabled, since they lead to the same physics as a run that follows an LO-type generation. A discussion on this point will be given in section 3.3; here, we limit ourselves to giving the shortest answer, which is the following: the access to both LO and NLO results within the same generation procedure (i.e., in the same current-directory process) guarantees an extremely easy and swift comparison between them.

Switch 3 controls whether one employs MadGraph5 aMC@NLO to steer the showering of hard-subprocess events previously generated (shower=ON) or not (shower=OFF). We shall give a more extended discussion about this point in section 3.2.1. Here, we would like to stress one crucial point: if the hard events are relevant to an NLO+PS run, they do not have any physical meaning unless they are showered. Hence, one is free to use or not use MadGraph5 aMC@NLO for showering them, but shower them he/she must. Note that this is not the case for LO+PS hard events, although of course shower or not shower them leads to di erent types of physics (in particular, observables constructed with unshowered LO+PS hard events are the same as those resulting from an fLO computation).

Finally, switch 4 allows one to decide whether to include production spin correlations by means of MadSpin (see section 2.5). Since the method works starting from un-weighted events, MadSpin is disabled when a xed-order run is selected (in other words, the inputs fixed order=ON and madspin=ON are incompatible, and MadGraph5 aMC@NLO will automatically prevent the user from making such a choice; this is not necessary when order=LO, but it is done anyway in order to simplify things).

As in the case of an LO-type generation, after setting switches 14 to the desired values and entering 0, or done, or <return>, MadGraph5 aMC@NLO proceeds with the run by giving the user the possibility of modifying the various input cards relevant to the options selected.

3.2.1 (N)LO+PS results

The aim of this section is that of giving some details on the (N)LO+PS runs that follow an NLO-type generation. We recall that (N)LO+PS results are obtained by setting fixed order=OFF, and the perturbative order is assigned according to the value of the switch order (see section 3.2).

There are two types of objects46 that may be obtained with an (N)LO+PS run:

1. One or two les of unweighted events at the hard-subprocess level.

2. One le that collects results at the end of the shower, be them in the form of histograms, or n-tuples, or events; we call it MC.output.

The rst of the les in 1 is the result of the integration of the short-distance partonic cross section performed by MadGraph5 aMC@NLO. The second le is present only in the case when MadSpin is used (madspin=ON), and results from feeding the former le to MadSpin. These two les will be found under the current-process directory tree:

46There are actually several other auxiliary les produced by MadGraph5 aMC@NLO, whose role is however not important here. See appendix A for more details.

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MYPROC/Events/run nn/events.lhe.gzMYPROC/Events/run nn decayed mm/events.lhe.gzAlthough of course MadSpin may not be used (or it may simply be not relevant, if the generated process does not feature particles that decay), the above structure is the most general, and hence we shall discuss it so as to encompass all cases. We shall call the two les above the undecayed and decayed (hard-subprocess) event les, respectively; both les are fully compliant with the recent Les Houches Accord v3.0 [218].

The undecayed event le will contain events whose particle content is the same as that given as input during the generation step (up to one nal-state parton at the NLO), i.e., using the example of section 3.2:

x + y t + t+ W +(+z) , (3.1) where x, y, and z are quarks or gluons.47 The integration of the cross section that results in the actual events is performed once the values of the relevant parameters (such as particles masses, collider energy, PDFs) are given in input by the user. The two primary input cards are:MYPROC/Cards/run card.datMYPROC/Cards/param card.datOne can obtain an undecayed event le with some parameter settings, then change these settings, integrate the cross section again, and obtain a second event le; the procedure can be iterated as many times as one likes. Each run is identied by an integer number, chosen by MadGraph5 aMC@NLO, which unambiguously names the directory where the event les will be stored. So, in the example given above, we shall have nn=01 for the rst run, nn=02 for the second run, and so forth.

Each undecayed event le can be fed to MadSpin in order to obtain a decayed event le. In order to do so, the user is expected to give in input to MadSpin the actual decay(s) he/she is interested into, which can be done by means of the input card: MYPROC/Cards/madspin card.datUsing again the generation example given before, and supposing that one wants to study the decays:

e , W + + , (3.2)

then the decayed event le will contain events of the following kind:

x + y b + e+ + e +

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t b e+e , t

b + e +

e + + + (+z) . (3.3)

It is possible to run MadSpin multiple times, by feeding it with the same undecayed event le and by changing the type of decays considered (e.g., semileptonic top decays after the di-leptonic ones of eq. (3.2)). For each MadSpin run, MadGraph5 aMC@NLO will store the decayed event le in a di erent directory this is the reason for the integer number mm in the example above, which is automatically assigned to each new run: so mm=1 will identify the rst MadSpin run, mm=2 the second, and so forth.

47The parton z is present in the case of H events, and absent in S events [26].

In summary, directories that contain undecayed event les are identied by an integer nn, which roughly speaking corresponds to a given choice of settings in run card.dat and param card.dat. On the other hand, directories that contain decayed event les are identied by a pair of integers (nn,mm), which correspond to a given choice of settings in run card.dat, param card.dat, and madspin card.dat.

Both undecayed and decayed hard-subprocess event les are non physical (at the NLO), and must be showered in order to obtain physical results. Such a showering can be done without using MadGraph5 aMC@NLO, since it is nothing but the nowadays typical PSMC run starting from an external (to the PSMC) LH-compliant event le. On the other hand, MadGraph5 aMC@NLO can steer PSMC runs: this is convenient for a varienty of reasons, among which the most important are probably those of ensuring a full consistency among the parameters used when integrating the cross section and those adopted by the PSMC, and the correct setting of a few control switches in the PSMC itself.48 As was

explained in section 3.2, MadGraph5 aMC@NLO will steer the shower when shower=ON: in this case, several of the features of the PSMC run can be controlled through a Mad-Graph5 aMC@NLO input card:MYPROC/Cards/shower card.datThe parameters not explicitly included in this card must be controlled directly in the relevant PSMC, in the same way as one would follow in a PSMC standalone run.

The steering of the PSMC by MadGraph5 aMC@NLO also guarantees that the PSMC adopted is consistent with that assumed during the integration of the short-distance cross section. Indeed, it is important to keep in mind that NLO+PS events obtained with the MC@NLO formalism are PSMC-dependent, and for this reason one of the input parameters in run card.dat is the name of the PSMC which will be eventually used to shower the hard-subprocess events. This kind of consistency is the users responsibility in the case of a PSMC not steered by MadGraph5 aMC@NLO. Finally, we point out that the most recent versions of the source codes of the Fortran77 PSMCs (HERWIG6 and Pythia6) are included in the MadGraph5 aMC@NLO tarball.49 On the other hand, the modern C++ PSMC (Pythia8 and Herwig++) must be installed locally prior to running Mad-Graph5 aMC@NLO. The paths to their executables/libraries have to be included in the setup le mg5 configuration.txt (see appendix A).

When steering of the PSMC by MadGraph5 aMC@NLO one will obtain, at the end of the PSMC run, the le MC.output mentioned in item 2 at the beginning of this section. By default (i.e., if the user does not write his/her own analysis) this le will contain the full event record of each showered event (in StdHEP format for Fortran MCs, and in HepMC format for C++ PSMCs), and it will be named as follows:MYPROC/Events/run */events MCTYPE ll.hep.gz

48See http://amcatnlo.cern.ch

Web End =http://amcatnlo.cern.ch Help and FAQs Special settings for the parton shower.

49These codes are essentially frozen, so we expect no or very minor changes in the future. Should the need arise to use versions di erent w.r.t. those included here, copy them to MYPROC/MCatNLO/srcHerwig and MYPROC/MCatNLO/srcPythia, and write their names in MYPROC/MCatNLO/MCatNLO MadFKS.inputs. In the case of Pythia6, the routines UPINIT, UPEVNT, PDFSET, STRUCTP, and STRUCTM need also be commented out of the source code.

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Here, the run * directory is one of those introduced before; MC.output is moved to the same directory where one nds the hard-subprocess event le fed to the PSMC during the course of the run. MCTYPE is a string equal to the name of the PSMC used, and ll is an integer, chosen by MadGraph5 aMC@NLO, that allows one to distinguish di erent les obtained by showering the same hard-event le multiple times (for example by changing the shower parameters). Any other output created by the PSMC (such as its standard output, which would be printed on the screen during an interactive run) will be found in the directory: MYPROC/MCatNLO/RUN MCTYPE ppwith pp an integer that enumerates the various runs.50

While StdHEP full event records may be analysed o -line, they have the disadvantage of using a very signicant amount of disk space. It is very often more convenient to analyse showered events on-the-y, using the event kinematics and weight to construct observables and ll the corresponding histograms. This can be done in a very exible manner in the PSMC runs steered by MadGraph5 aMC@NLO. The users analysis can be stored in one of the directories51

MYPROC/MCatNLO/XYZAnalyzerwhere the string XYZ depends on the PSMC adopted. The name of such an analysis, and those of all its dependencies (be they in the form of either source codes or libraries) can be given in input to MadGraph5 aMC@NLO at runtime, as parameters in shower card.dat. It is clear that it is the users analysis that determines the format of MC.output. Since Mad-Graph5 aMC@NLO cannot know it beforehand, it will treat MC.output as any other standard le produced by the PSMC, which will thus be found in the directory RUN MCTYPE pp. An exception is that of the topdrawer format (which is human readable); in this case, MC.output will be named as follows:MYPROC/Events/run */plot MCTYPE kk *.topwith again kk an integer that allows one to distinguish the outputs relevant to di erent showering of the same hard-event le.

We conclude this section by mentioning the fact that when the launch command is executed with madspin=ON and shower=ON (i.e., both undecayed and decayed events are produced, the PSMC runs immediately follows the integration of the cross section, and it is steered by MadGraph5 aMC@NLO) only the decayed event le will be showered. In order to shower the undecayed event le, or to perform other shower runs, one needs to use the MadGraph5 aMC@NLO shell command shower. Please see appendix B.1 for more details.

3.2.2 f(N)LO results

As was discussed in section 3.2, xed-order results can be obtained with Mad-Graph5 aMC@NLO by setting fixed order=ON, with the perturbative order assigned according to the value of the switch order.

We remind the reader that an NLO computation not matched to a parton shower cannot produce unweighted events, since the matrix elements are not bounded from

50Note that in general pp 6= ll: for example, a PSMC run may not create an StdHEP event record, in which case pp would be increased, while ll would not.

51Which contain several ready-to-run templates.

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above in the phase space; unweighting events would require the introduction of unphysical cuto s that would bias predictions, and must therefore be avoided. The only thing one can do is that of considering weighted events, namely parton-level kinematic congurations supplemented by a weight (which is basically equal to the matrix element times a phase-space factor). Such events, which are readily obtained when sampling the phase space in the course of the integration of the cross section, can be either stored in a le or used on-the-y to predict physical observables. While we choose the latter option as our default, we remark that the former can be easily implemented by the interested user, as it will become clear in what follows.

The input parameters relevant to f(N)LO runs can be found in the same cards that control NLO+PS runs, namely run card.dat and param card.dat. Since observables will be plotted on the y, the user is expected to write his/her own analysis (which is a trivial task, in view of the rather small nal-state multiplicities of parton-level computations). This analysis must be written in Fortran (or at least it must include a Fortran front-end interface to the users core analysis, written in a language other than Fortran) and stored in the directory:

MYPROC/FixedOrderAnalysiswhich contains several templates, meant to be used as examples. As the templates show,

MadGraph5 aMC@NLO at present supports two output formats for users analyses Root and topdrawer (see appendix. A for more details about this). The analysis name and the format of its output must be given in input to MadGraph5 aMC@NLO, which is done by including them in the input card:

MYPROC/Cards/FO analyse card.datwhere the user will also be able to specify any other source le or library needed by the analysis. Upon running, MadGraph5 aMC@NLO will produce the nal-result le:

MYPROC/Events/run nn/MADatNLO.root if the Root format has been chosen, and:

MYPROC/Events/run nn/MADatNLO.topfor the topdrawer format; these will contain plots for the observables dened by the user. As in the case of NLO+PS runs, the integer nn will be chosen by MadGraph5 aMC@NLO in a non-ambiguous way (it should be obvious that both (N)LO+PS and f(N)LO simulations can be performed any number of times starting from a given generation, and nn will allow one to distinguish their results). By default, if the user does not write an analysis, MadGraph5 aMC@NLO will produce a le which will contain the predictions for the total rate (possibly within the cuts, as specied in run card.dat and cuts.f).

In essence, the users analysis will have to construct the desired observables, given in input the pair composed of a kinematic conguration and its corresponding weight,52

which are provided by MadGraph5 aMC@NLO. These pairs can be read from eq. (2.47): for each choice of the random variables {(ij)Bj, (ij)n+1} that scan the phase space and Bjorken

52Such a weight becomes an array of weights when the user asks MadGraph5 aMC@NLO to compute scale and PDF uncertainties. See section B.3 for more details.

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xs, MadGraph5 aMC@NLO will call the users analysis three times, with input pairs:

K(E)n+1 ,

, (3.5)

, (3.6)

d(NLO,)ij|nonBorn = d(NLO,)ij = C, SC , (3.7) d(NLO,S)ij|Born + d(NLO,S)ij|nonBorn = d(NLO,S)ij , (3.8)

and d(NLO,S)ij|Born is the contribution of the Born matrix elements to the soft-counterevent

weight. The users analysis must treat eqs. (3.4)(3.6) precisely in the same way: this is an essential condition which guarantees that infrared safety is not spoiled. In particular, to any given histogram all of these weights must contribute (obviously, to the bins and subject to the cuts determined by the respective kinematics congurations, K(E)n+1 and K(S)n+1), if the

NLO accuracy is to be maintained. Note, also, that the weights (3.4)(3.6) are correlated, and thus cannot be individually used in a statistical analysis as they would if they had been the result e.g. of an unweighted-event procedure.

Thanks to the fact that the Born weight, eq. (3.6), is kept separate from the other contributions to the NLO cross section, MadGraph5 aMC@NLO allows the user to plot, during the course of the same run, a given observable both at the NLO accuracy (by using eqs. (3.4)(3.6)) and at the LO accuracy (by using eq. (3.6) only). In order to allow the users analysis to tell eq. (3.6) apart from eqs. (3.4) and (3.5), MadGraph5 aMC@NLO will tag these weights with an integer, equal to 3, 1, and 2 respectively. For explicit examples, see one of the template analyses in MYPROC/FixedOrderAnalysis.

We conclude this section by pointing out that the writing of weighted events can be seen as a special type of analysis. The exibility inherent in the structure sketched above should easily allow a user to write and exploit such an analysis. In this case, we also note that, rather than associating a single weight with each event, one can use all of the scale- and PDF-independent ones dened in ref. [125] (which are available as variables in a common block) in order to be able to compute scale and PDF uncertainties through reweighting. These uncertainties can obviously be included when plotting observables on the y see appendix B.3.

3.3 Possibilities for LO simulations

As was mentioned in section 3, MadGraph5 aMC@NLO o ers two ways to obtain LO results. One is through an LO-type generation, where when executing the command generate one does not include the keyword [QCD]; this is completely equivalent to

d(NLO,E)ij

d(ij)Bjd(ij)n+1

, (3.4)

K(S)n+1 ,

X=S,C,SC d(NLO,)ij|nonBorn d(ij)Bjd(ij)n+1

K(S)n+1 ,

d(NLO,S)ij|Born

d(ij)Bjd(ij)n+1

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where

what one would do if using MadGraph5. The other is accessed through an NLO-type generation, by using order=LO.

We stress once again that all of the capabilities of MadGraph5 are inherited by MadGraph5 aMC@NLO. Therefore, if one is interested only in LO physics, there is no reason to simulate it within an NLO-type generation (which in that case would simply be a waste of time). This is also in view of the fact that many of the options available in the context of an LO-type generation are disabled when an LO process is computed after an NLO-type generation. The idea of LO runs in the context of NLO-type generation is that of rendering the comparison between LO and NLO a completely trivial a air. Note, for example, that the input cards relevant to LO- and NLO-type generations are di erent: many of the cuts that are present in the former would simply not make sense in the latter (owing to the necessity of being compliant with infrared safety). Also, one should bear in mind that the functional form of S used by MadGraph5 aMC@NLO (i.e., at one or two loops) is determined by the set of PDFs employed in the runs. Therefore, when working with an NLO-type generation, and assuming that fLO (or LO+PS) results are obtained during the same run as their fNLO (or NLO+PS) counterparts, the former will be based on NLO PDFs and two-loop S (which is of course fully consistent with perturbation theory, and actually the best option if one is interested in assessing the perturbative behaviour of the partonic short-distance matrix elements).

In summary, both options for LO runs lead to exactly the same physics. However, by running the code at the LO with its default inputs one generally does not obtain exactly the same numbers, since the relevant input cards are tailored either to LO-type or to NLO-type generations, which have di erent necessities and emphases. Hence, we recommend to consider fLO and/or LO+PS results obtained through NLO-type generation mainly in the context of studies that feature fNLO and/or NLO+PS simulations as well.

4 Illustrative results

4.1 Total cross sections

In this section we present benchmark results for total rates (possibly within the cuts which will be specied later), at both the LO and the NLO. These are fLO and fNLO results respectively, with the former computed in the context of an NLO-type generation (see section 3.3). On the one hand, the aim of this section is that of showing the extreme exibility and reach of MadGraph5 aMC@NLO: in keeping with our general philosophy, no part of the code has been customised to the computation of any of the processes below. The code has been run as is extracted from the tarball (apart from the occasional necessity of dening some nal-state cuts in cuts.f, in some cases relevant to (b-)jet production, which are explicitly mentioned in the captions of tables 111). We stress that several of these cross sections have never been computed before to NLO accuracy.

We summarise here the main physics parameters used in our runs; the complete list of inputs, in the form of the input cards of MadGraph5 aMC@NLO, can be found by visiting http://amcatnlo.cern.ch/cards_paper.htm

Web End =http://amcatnlo.cern.ch/cards paper.htm , which should render it particularly easy to reproduce the results that follow.

JHEP07(2014)079

mH = 125 GeV, mt = 173.2 GeV.

MSTWnlo2008 [219] PDFs with errors at 68%CL, nf = 4, 5. Note that these PDFs

are used to obtain the fLO results as well, and that they set the value of S(mZ).

Central scale choice: 0 = HT /2, with HT the scalar sum of the transverse masses

pp2T + m2 of all nal state particles.

Scale variations: independent, 1/20 < R, F < 20.

Diagonal CKM matrix.

Final-state objects are dened as follows:

Jets: anti-kT algorithm [220] with R = 0.5, pT (j) > 30 GeV, |(j)| < 4.

Photons: Frixione isolation [221] with R = 0.7, pT () > 20 GeV, |()| < 2.

While MadGraph5 aMC@NLO can handle intermediate resonances by using the complex mass scheme, in this section we consider W s, Zs, and top quarks as stable, and thus set their widths equal to zero. Furthermore, although matrix elements for loop-induced processes (e.g. such as gg ZZ), can be obtained with MadLoop, they have not been con

sidered in this section. Apart from reporting the absolute values of the total cross sections, the following tables also show (as percentages) scale and PDF uncertainties. These are computed exactly but without re-running, by exploiting the reweighting method presented in ref. [125]. Following ref. [68], Yukawas are renormalised with an on-shell scheme.

We tag the processes for which we are not aware of any fNLO result being available in the literature with an asterisk. We stress that, in those cases where an fNLO prediction had been previously obtained, the corresponding matching to PSMCs might not necessarily have been achieved; this is the typical situation for several of the high-multiplicity reactions. On the other hand, for all processes presented in the tables below the corresponding NLO+PS event samples can be obtained with MadGraph5 aMC@NLO. Some non-exhaustive examples will be given in section 4.2.

The following results are organised into broad classes of processes that share some dening characteristic. In hadroproduction (with a c.m. energy of 13 TeV), we have considered vector bosons plus up to three light jets (table 1), vector boson pairs plus up to two light jets (table 2), three vector bosons plus up to one light jet (table 3), four vector bosons (table 4), light jets, b-jets, and top quarks, possibly in association with each other (table 5), vector bosons in association with top or bottom quarks and light jets (table 6), single top quarks in association with b quarks (four-avour scheme53) and vector bosons (table 7), Higgs and double-Higgs in association with (possibly multiple) light jets, vector bosons and heavy quarks (tables 8 and 9). In e+e collisions (with a c.m. energy of 1 TeV), we have considered light jets, possibly in association with heavy quarks (table 10), and top quark pairs in association with (possibly multiple) vector or Higgs bosons (table 11).

53Throughout this paper, we adopt standard denitions for the four- and ve-avour schemes. See e.g. section 1 of ref. [222] for a recent short review.

JHEP07(2014)079

Process Syntax Cross section (pb)

Vector boson +jets LO 13 TeV NLO 13 TeV

a.1 pp ! W p p > wpm 1.375 0.002 105

+15.4% 16.6%

+2.0%1.6% 1.773 0.007 105

+5.2% 9.4%

+1.9% 1.6%

a.2 pp ! W j p p > wpm j 2.045 0.001 104

+19.7% 17.2%

+1.4%1.1% 2.843 0.010 104

+5.9% 8.0%

+1.3% 1.1%

a.3 pp ! W jj p p > wpm j j 6.805 0.015 103

+24.5% 18.6%

+0.8%0.7% 7.786 0.030 103

+2.4% 6.0%

+0.9% 0.8%

a.4 pp ! W jjj p p > wpm j j j 1.821 0.002 103

+41.0% 27.1%

+0.5%0.5% 2.005 0.008 103

+0.9% 6.7%

+0.6% 0.5%

a.5 pp ! Z p p > z 4.248 0.005 104

+14.6% 15.8%

+2.0%1.6% 5.410 0.022 104

+4.6% 8.6%

+1.9% 1.5%

a.6 pp ! Zj p p > z j 7.209 0.005 103

+19.3% 17.0%

+1.2%1.0% 9.742 0.035 103

+5.8% 7.8%

+1.2% 1.0%

a.7 pp ! Zjj p p > z j j 2.348 0.006 103

+24.3% 18.5%

+0.6%0.6% 2.665 0.010 103

+2.5% 6.0%

+0.7% 0.7%

a.8 pp ! Zjjj p p > z j j j 6.314 0.008 102

+40.8% 27.0%

+0.5%0.5% 6.996 0.028 102

+1.1% 6.8%

+0.5% 0.5%

a.9 pp ! j p p > a j 1.964 0.001 104

+31.2% 26.0%

+1.7%1.8% 5.218 0.025 104

+24.5% 21.4%

+1.4% 1.6%

a.10 pp ! jj p p > a j j 7.815 0.008 103

+32.8% 24.2%

+0.9%1.2% 1.004 0.004 104

+5.9% 10.9%

+0.8% 1.2%

Table 1. Sample of LO and NLO rates for vector-boson production, possibly within cuts and in association with jets, at the 13-TeV LHC; we also report the integration errors, and the fractional scale (left) and PDF (right) uncertainties. Where relevant, the notation understands the sum of the W + and W cross sections, and wpm is a label that includes both W + and W , dened from the shell with define wpm = w+ w-. All cross sections are calculated in the ve-avour scheme. Results at the NLO accuracy for W/Z plus jets are also available in MCFM for up to two jets [223225], including heavy-avour identication [226230], and in POWHEG [231233]. NLO cross sections for W plus three jets have appeared in refs. [234, 235]. The BlackHat+SHERPA collaboration has provided samples and results for up to Z plus four jets and W plus ve jets at the NLO [236240]. NLO+PS merged samples for W plus up to three jets are also available in SHERPA [241]. plus up to three jets calculations have been presented in refs. [242, 243]. We do not show cross sections for EW-induced V plus two jets processes with V = , Z, W ,

which are available in VBFNLO [244] and have been studied in ref. [245].

JHEP07(2014)079

Process Syntax Cross section (pb)

Vector-boson pair +jets LO 13 TeV NLO 13 TeVb.1 pp ! W +W (4f) p p > w+ w- 7.355 0.005 101

+5.0% 6.1%

+2.0%1.5% 1.028 0.003 102

+4.0% 4.5%

+1.9% 1.4%

b.2 pp ! ZZ p p > z z 1.097 0.002 101

+4.5% 5.6%

+1.9%1.5% 1.415 0.005 101

+3.1% 3.7%

+1.8% 1.4%

b.3 pp ! ZW p p > z wpm 2.777 0.003 101

+3.6% 4.7%

+2.0%1.5% 4.487 0.013 101

+4.4% 4.4%

+1.7% 1.3%

b.4 pp ! p p > a a 2.510 0.002 101

+22.1% 22.4%

+2.4%2.1% 6.593 0.021 101

+17.6% 18.8%

+2.0% 1.9%

b.5 pp ! Z p p > a z 2.523 0.004 101

+9.9% 11.2%

+2.0%1.6% 3.695 0.013 101

+5.4% 7.1%

+1.8% 1.4%

b.6 pp ! W p p > a wpm 2.954 0.005 101

+9.5% 11.0%

+2.0%1.7% 7.124 0.026 101

+9.7% 9.9%

+1.5% 1.3%

b.7 pp ! W +W j (4f) p p > w+ w- j 2.865 0.003 101

+11.6% 10.0%

+1.0%0.8% 3.730 0.013 101

+4.9% 4.9%

+1.1% 0.8%

b.8 pp ! ZZj p p > z z j 3.662 0.003 100

+10.9% 9.3%

+1.0%0.8% 4.830 0.016 100

+5.0% 4.8%

+1.1% 0.9%

b.9 pp ! ZW j p p > z wpm j 1.605 0.005 101

+11.6% 10.0%

+0.9%0.7% 2.086 0.007 101

+4.9% 4.8%

+0.9% 0.7%

b.10 pp ! j p p > a a j 1.022 0.001 101

+20.3% 17.7%

+1.2%1.5% 2.292 0.010 101

+17.2% 15.1%

+1.0% 1.4%

b.11 pp ! Zj p p > a z j 8.310 0.017 100

+14.5% 12.8%

+1.0%1.0% 1.220 0.005 101

+7.3% 7.4%

+0.9% 0.9%

b.12 pp ! W j p p > a wpm j 2.546 0.010 101

+13.7% 12.1%

+0.9%1.0% 3.713 0.015 101

+7.2% 7.1%

+0.9% 1.0%

b.13 pp ! W +W +jj p p > w+ w+ j j 1.484 0.006 101

+25.4% 18.9%

+2.1%1.5% 2.251 0.011 101

+10.5% 10.6%

+2.2% 1.6%

b.14 pp ! W W jj p p > w- w- j j 6.752 0.007 102

+25.4% 18.9%

+2.4%1.7% 1.003 0.003 101

+10.1% 10.4%

+2.5% 1.8%

b.15 pp ! W +W jj (4f) p p > w+ w- j j 1.144 0.002 101

+27.2% 19.9%

+0.7%0.5% 1.396 0.005 101

+5.0% 6.8%

+0.7% 0.6%

b.16 pp ! ZZjj p p > z z j j 1.344 0.002 100

+26.6% 19.6%

+0.7%0.6% 1.706 0.011 100

+5.8% 7.2%

+0.8% 0.6%

b.17 pp ! ZW jj p p > z wpm j j 8.038 0.009 100

+26.7% 19.7%

+0.7%0.5% 9.139 0.031 100

+3.1% 5.1%

+0.7% 0.5%

b.18 pp ! jj p p > a a j j 5.377 0.029 100

+26.2% 19.8%

+0.6%1.0% 7.501 0.032 100

+8.8% 10.1%

+0.6% 1.0%

b.19 pp ! Zjj p p > a z j j 3.260 0.009 100

+24.3% 18.4%

+0.6%0.6% 4.242 0.016 100

+6.5% 7.3%

+0.6% 0.6%

b.20 pp ! W jj p p > a wpm j j 1.233 0.002 101

+24.7% 18.6%

+0.6%0.6% 1.448 0.005 101

+3.6% 5.4%

+0.6% 0.7%

Table 2. Sample of LO and NLO rates for vector-boson pair production, possibly within cuts and in association with jets, at the 13-TeV LHC; we also report the integration errors, and the fractional scale (left) and PDF (right) uncertainties. See table 1 for the meaning of wpm. All cross sections are calculated in the ve-avour scheme, except for processes b.1, b.7, and b.15, which are obtained in the four-avour scheme to avoid resonant-top contributions. NLO results for V V production have been known for some time [246254, 256], are publicly available in MCFM and in VBFNLO [244], and are matched to parton showers in MC@NLO [26] and POWHEG [257]. NLO results for V V with up to an extra jet have been made available in POWHEG [258, 259]. NLO corrections to plus up to three jets are also known [260264]. Other available results are: W +W jj [255, 359], W W jj [265], W W jj (EW+QCD) [266], Zj [267], W jj [268], W Zjj [269], W j [270, 271], W Zj [272]. We do not show results for NLO corrections to EW-induced production of V V plus two jets, such as W W jj [273], W Zjj [274], and ZZjj [275], which can also be obtained with POWHEG and VBFNLO.

JHEP07(2014)079

Process Syntax Cross section (pb)

Three vector bosons +jet LO 13 TeV NLO 13 TeVc.1 pp ! W +W W (4f) p p > w+ w- wpm 1.307 0.003 101

+0.0% 0.3%

+2.0%1.5% 2.109 0.006 101

+5.1% 4.1%

+1.6% 1.2%

c.2 pp ! ZW +W (4f) p p > z w+ w- 9.658 0.065 102

+0.8% 1.1%

+2.1%1.6% 1.679 0.005 101

+6.3% 5.1%

+1.6% 1.2%

c.3 pp ! ZZW p p > z z wpm 2.996 0.016 102

+1.0% 1.4%

+2.0%1.6% 5.550 0.020 102

+6.8% 5.5%

+1.5% 1.1%

c.4 pp ! ZZZ p p > z z z 1.085 0.002 102

+0.0% 0.5%

+1.9%1.5% 1.417 0.005 102

+2.7% 2.1%

+1.9% 1.5%

c.5 pp ! W +W (4f) p p > a w+ w- 1.427 0.011 101

+1.9% 2.6%

+2.0%1.5% 2.581 0.008 101

+5.4% 4.3%

+1.4% 1.1%

c.6 pp ! W p p > a a wpm 2.681 0.007 102

+4.4% 5.6%

+1.9%1.6% 8.251 0.032 102

+7.6% 7.0%

+1.0% 1.0%

c.7 pp ! ZW p p > a z wpm 4.994 0.011 102

+0.8% 1.4%

+1.9%1.6% 1.117 0.004 101

+7.2% 5.9%

+1.2% 0.9%

c.8 pp ! ZZ p p > a z z 2.320 0.005 102

+2.0% 2.9%

+1.9%1.5% 3.118 0.012 102

+2.8% 2.7%

+1.8% 1.4%

c.9 pp ! Z p p > a a z 3.078 0.007 102

+5.6% 6.8%

+1.9%1.6% 4.634 0.020 102

+4.5% 5.0%

+1.7% 1.3%

c.10 pp ! p p > a a a 1.269 0.003 102

+9.8% 11.0%

+2.0%1.8% 3.441 0.012 102

+11.8% 11.6%

+1.4% 1.5%

c.11 pp ! W +W W j (4f) p p > w+ w- wpm j 9.167 0.010 102

+15.0% 12.2%

+1.0%0.7% 1.197 0.004 101

+5.2% 5.6%

+1.0% 0.8%

c.12 pp ! ZW +W j (4f) p p > z w+ w- j 8.340 0.010 102

+15.6% 12.6%

+1.0%0.7% 1.066 0.003 101

+4.5% 5.3%

+1.0% 0.7%

c.13 pp ! ZZW j p p > z z wpm j 2.810 0.004 102

+16.1% 13.0%

+1.0%0.7% 3.660 0.013 102

+4.8% 5.6%

+1.0% 0.7%

c.14 pp ! ZZZj p p > z z z j 4.823 0.011 103

+14.3% 11.8%

+1.4%1.0% 6.341 0.025 103

+4.9% 5.4%

+1.4% 1.0%

c.15 pp ! W +W j (4f) p p > a w+ w- j 1.182 0.004 101

+13.4% 11.2%

+0.8%0.7% 1.233 0.004 103

+18.9% 19.9%

+1.0% 1.5%

c.16 pp ! W j p p > a a wpm j 4.107 0.015 102

+11.8% 10.2%

+0.6%0.8% 5.807 0.023 102

+5.8% 5.5%

+0.7% 0.7%

c.17 pp ! ZW j p p > a z wpm j 5.833 0.023 102

+14.4% 12.0%

+0.7%0.6% 7.764 0.025 102

+5.1% 5.5%

+0.8% 0.6%

c.18 pp ! ZZj p p > a z z j 9.995 0.013 103

+12.5% 10.6%

+1.2%0.9% 1.371 0.005 102

+5.6% 5.5%

+1.2% 0.9%

c.19 pp ! Zj p p > a a z j 1.372 0.003 102

+10.9% 9.4%

+1.0%0.9% 2.051 0.011 102

+7.0% 6.3%

+1.0% 0.9%

c.20 pp ! j p p > a a a j 1.031 0.006 102

+14.3% 12.6%

+0.9%1.2% 2.020 0.008 102

+12.8% 11.0%

+0.8% 1.2%

Table 3. Sample of LO and NLO rates for triple-vector-boson production, possibly within cuts and in association with one jet, at the 13-TeV LHC; we also report the integration errors, and the fractional scale (left) and PDF (right) uncertainties. See table 1 for the meaning of wpm. All cross sections are calculated in the ve-avour scheme, except for processes with at least two W bosons, where the four-avour scheme is adopted to avoid resonant-top contributions. Triple-vector-boson cross sections at the NLO have been computed in: Z [267, 276], W [277], ZW [278], W W and ZZ [279], ZW W [358], ZZW and W W W [280, 358], [281, 282], ZZZ [283, 358]. The complete set of triple-vector-boson cross sections at the NLO is also available in VBFNLO [244]. Except for W j and W +W W j that have appeared in ref. [284] and ref. [285] respectively, V V V j cross sections at the NLO have been computed here for the rst time.

JHEP07(2014)079

Process Syntax Cross section (pb)

Four vector bosons LO 13 TeV NLO 13 TeV

c.21 pp ! W +W W +W (4f) p p > w+ w- w+ w- 5.721 0.014 104

+3.7% 3.5%

+2.3%1.7% 9.959 0.035 104

+7.4% 6.0%

+1.7% 1.2%

c.22 pp ! W +W W Z (4f) p p > w+ w- wpm z 6.391 0.076 104

+4.4% 4.1%

+2.4%1.8% 1.188 0.004 103

+8.4% 6.8%

+1.7% 1.2%

c.23 pp ! W +W W (4f) p p > w+ w- wpm a 8.115 0.064 104

+2.5% 2.5%

+2.2%1.7% 1.546 0.005 103

+7.9% 6.3%

+1.5% 1.1%

c.24 pp ! W +W ZZ (4f) p p > w+ w- z z 4.320 0.013 104

+4.4% 4.1%

+2.4%1.7% 7.107 0.020 104

+7.0% 5.7%

+1.8% 1.3%

c.25 pp ! W +W Z (4f) p p > w+ w- z a 8.403 0.016 104

+3.0% 2.9%

+2.3%1.7% 1.483 0.004 103

+7.2% 5.8%

+1.6% 1.2%

c.26 pp ! W +W (4f) p p > w+ w- a a 5.198 0.012 104

+0.6% 0.9%

+2.1%1.6% 9.381 0.032 104

+6.7% 5.3%

+1.4% 1.1%

c.27 pp ! W ZZZ p p > wpm z z z 5.862 0.010 105

+5.1% 4.7%

+2.4%1.8% 1.240 0.004 104

+9.9% 8.0%

+1.7% 1.2%

c.28 pp ! W ZZ p p > wpm z z a 1.148 0.003 104

+3.6% 3.5%

+2.2%1.7% 2.945 0.008 104

+10.8% 8.7%

+1.3% 1.0%

c.29 pp ! W Z p p > wpm z a a 1.054 0.004 104

+1.7% 1.9%

+2.1%1.7% 3.033 0.010 104

+10.6% 8.6%

+1.1% 0.8%

c.30 pp ! W p p > wpm a a a 3.600 0.013 105

+0.4% 1.0%

+2.0%1.6% 1.246 0.005 104

+9.8% 8.1%

+0.9% 0.8%

c.31 pp ! ZZZZ p p > z z z z 1.989 0.002 105

+3.8% 3.6%

+2.2%1.7% 2.629 0.008 105

+3.5% 3.0%

+2.2% 1.7%

c.32 pp ! ZZZ p p > z z z a 3.945 0.007 105

+1.9% 2.1%

+2.1%1.6% 5.224 0.016 105

+3.3% 2.7%

+2.1% 1.6%

c.33 pp ! ZZ p p > z z a a 5.513 0.017 105

+0.0% 0.3%

+2.1%1.6% 7.518 0.032 105

+3.4% 2.6%

+2.0% 1.5%

c.34 pp ! Z p p > z a a a 4.790 0.012 105

+2.3% 3.1%

+2.0%1.6% 7.103 0.026 105

+3.4% 3.2%

+1.6% 1.5%

c.35 pp ! p p > a a a a 1.594 0.004 105

+4.7% 5.7%

+1.9%1.7% 3.389 0.012 105

+7.0% 6.7%

+1.3% 1.3%

Table 4. Sample of LO and NLO rates for quadruple-vector-boson production, possibly within cuts, at the 13-TeV LHC; we also report the integration errors, and the fractional scale (left) and PDF (right) uncertainties. See table 1 for the meaning of wpm. All cross sections are calculated in the ve-avour scheme, except the processes with at least two W bosons, where the four-avour scheme is adopted to avoid resonant-top contributions. For all processes in this table NLO QCD corrections have never been computed before.

JHEP07(2014)079

Process Syntax Cross section (pb)

Heavy quarks and jets LO 13 TeV NLO 13 TeVd.1 pp ! jj p p > j j 1.162 0.001 106

+24.9% 18.8%

+0.8%0.9% 1.580 0.007 106

+8.4% 9.0%

+0.7% 0.9%

d.2 pp ! jjj p p > j j j 8.940 0.021 104

+43.8% 28.4%

+1.2%1.4% 7.791 0.037 104

+2.1% 23.2%

+1.1% 1.3%

d.3 pp ! b

b (4f) p p > b b 3.743 0.004 103

+25.2% 18.9%

+1.5%1.8% 6.438 0.028 103

+15.9% 13.3%

+1.5% 1.7%

d.4 pp ! b

bj (4f) p p > b b j 1.050 0.002 103

+44.1% 28.5%

+1.6%1.8% 1.327 0.007 103

+6.8% 11.6%

+1.5% 1.8%

d.5 pp ! b

bjj (4f) p p > b b j j 1.852 0.006 102

+61.8% 35.6%

+2.1%2.4% 2.471 0.012 102

+8.2% 16.4%

+2.0% 2.3%

d.6 pp ! b

bbb (4f) p p > b b b b 5.050 0.007 101

+61.7% 35.6%

+2.9%3.4% 8.736 0.034 101

+20.9% 22.0%

+2.9% 3.4%

d.7 pp ! tt p p > t t 4.584 0.003 102

+29.0% 21.1%

+1.8%2.0% 6.741 0.023 102

+9.8% 10.9%

+1.8% 2.1%

d.8 pp ! ttj p p > t t j 3.135 0.002 102

+45.1% 29.0%

+2.2%2.5% 4.106 0.015 102

+8.1% 12.2%

+2.1% 2.5%

d.9 pp ! ttjj p p > t t j j 1.361 0.001 102

+61.4% 35.6%

+2.6%3.0% 1.795 0.006 102

+9.3% 16.1%

+2.4% 2.9%

d.10 pp ! tttt p p > t t t t 4.505 0.005 103

+63.8% 36.5%

+5.4%5.7% 9.201 0.028 103

+30.8% 25.6%

+5.5% 5.9%

d.11 pp ! ttb

b (4f) p p > t t b b 6.119 0.004 100

+62.1% 35.7%

+2.9%3.5% 1.452 0.005 101

+37.6% 27.5%

+2.9% 3.5%

Table 5. Sample of LO and NLO total rates for the production of heavy quarks and/or jets, possibly within cuts, at the 13-TeV LHC; we also report the integration errors, and the fractional scale (left) and PDF (right) uncertainties. Processes d.1 and d.2, as well as processes involving at least a top pair, are computed in the ve-avour scheme. Processes that explicitly involve b-quarks in the nal state are calculated in the four-avour scheme. For processes d.3d.6 we require 2 (or 4) b-jets in the nal state with || < 2.5. For processes d.1d.6, we require the (b)-jets to

have pT > 80 GeV, with at least one of them with pT > 100 GeV. Calculations of cross sections at the NLO for this class of processes are available in the literature as well as in public codes: from the seminal results for the hadroproduction of a heavy quark pair [286290], to their NLO+PS implementation in MC@NLO [172] and POWHEG [291], to ttj [292] (also including top decays [258, 293] and parton shower e ects [294, 365]), to the computation of ttjj [295]. Merged NLO+PS results for tt plus jets are also available [191, 296, 297]. NLO results for three jets [298], four jets [74], and up to ve jets [299, 300] have been published. Two- and three-jet event generation is available in POWHEG [301, 302]. Calculations for bbbb [303, 304], ttbb [305307, 362], and tttt [308] production have appeared in the literature.

JHEP07(2014)079

Process Syntax Cross section (pb)

Heavy quarks+vector bosons LO 13 TeV NLO 13 TeVe.1 pp ! W b

b (4f) p p > wpm b b 3.074 0.002 102

+42.3% 29.2%

+2.0%1.6% 8.162 0.034 102

+29.8% 23.6%

+1.5% 1.2%

e.2 pp ! Z b

b (4f) p p > z b b 6.993 0.003 102

+33.5% 24.4%

+1.0%1.4% 1.235 0.004 103

+19.9% 17.4%

+1.0% 1.4%

e.3 pp ! b

b (4f) p p > a b b 1.731 0.001 103

+51.9% 34.8%

+1.6%2.1% 4.171 0.015 103

+33.7% 27.1%

+1.4% 1.9%

e.4 pp ! W b

bj (4f) p p > wpm b b j 1.861 0.003 102

+42.5% 27.7%

+0.7%0.7% 3.957 0.013 102

+27.0% 21.0%

+0.7% 0.6%

e.5 pp ! Z b

bj (4f) p p > z b b j 1.604 0.001 102

+42.4% 27.6%

+0.9%1.1% 2.805 0.009 102

+21.0% 17.6%

+0.8% 1.0%

e.6 pp ! b

bj (4f) p p > a b b j 7.812 0.017 102

+51.2% 32.0%

+1.0%1.5% 1.233 0.004 103

+18.9% 19.9%

+1.0% 1.5%

e.7 pp ! ttW p p > t t wpm 3.777 0.003 101

+23.9% 18.0%

+2.1%1.6% 5.662 0.021 101

+11.2% 10.6%

+1.7% 1.3%

e.8 pp ! ttZ p p > t t z 5.273 0.004 101

+30.5% 21.8%

+1.8%2.1% 7.598 0.026 101

+9.7% 11.1%

+1.9% 2.2%

e.9 pp ! tt p p > t t a 1.204 0.001 100

+29.6% 21.3%

+1.6%1.8% 1.744 0.005 100

+9.8% 11.0%

+1.7% 2.0%

e.10 pp ! ttW j p p > t t wpm j 2.352 0.002 101

+40.9% 27.1%

+1.3%1.0% 3.404 0.011 101

+11.2% 14.0%

+1.2% 0.9%

e.11 pp ! ttZj p p > t t z j 3.953 0.004 101

+46.2% 29.5%

+2.7%3.0% 5.074 0.016 101

+7.0% 12.3%

+2.5% 2.9%

e.12 pp ! ttj p p > t t a j 8.726 0.010 101

+45.4% 29.1%

+2.3%2.6% 1.135 0.004 100

+7.5% 12.2%

+2.2% 2.5%

e.13 pp ! ttW W + (4f) p p > t t w+ w- 6.675 0.006 103

+30.9% 21.9%

+2.1%2.0% 9.904 0.026 103

+10.9% 11.8%

+2.1% 2.1%

e.14 pp ! ttW Z p p > t t wpm z 2.404 0.002 103

+26.6% 19.6%

+2.5%1.8% 3.525 0.010 103

+10.6% 10.8%

+2.3% 1.6%

e.15 pp ! ttW p p > t t wpm a 2.718 0.003 103

+25.4% 18.9%

+2.3%1.8% 3.927 0.013 103

+10.3% 10.4%

+2.0% 1.5%

e.16 pp ! ttZZ p p > t t z z 1.349 0.014 103

+29.3% 21.1%

+1.7%1.5% 1.840 0.007 103

+7.9% 9.9%

+1.7% 1.5%

e.17 pp ! ttZ p p > t t z a 2.548 0.003 103

+30.1% 21.5%

+1.7%1.6% 3.656 0.012 103

+9.7% 11.0%

+1.8% 1.9%

e.18 pp ! tt p p > t t a a 3.272 0.006 103

+28.4% 20.6%

+1.3%1.1% 4.402 0.015 103

+7.8% 9.7%

+1.4% 1.4%

Table 6. Sample of LO and NLO total rates for the production of heavy quarks in association with vector bosons, possibly within cuts and in association with jets, at the 13-TeV LHC; we also report the integration errors, and the fractional scale (left) and PDF (right) uncertainties. Processes that explicitly involve b-quarks in the nal state, and process e.13, are calculated in the four-avour scheme, while all of the others are in the ve-avour scheme. Results are available in the literature for W bb [68, 309312], Zbb [68, 311, 313], tt [314], ttZ [68, 315, 316, 363, 364], ttW [68, 316, 317] production. For the majority of the processes in this table, NLO corrections are calculated in this work for the rst time.

JHEP07(2014)079

Process Syntax Cross section (pb)

Single-top LO 13 TeV NLO 13 TeV

f.1 pp ! tj (t-channel) p p > tt j $$ w+ w- 1.520 0.001 102

+9.4% 11.9%

+0.4%0.6% 1.563 0.005 102

+1.4% 1.8%

+0.4% 0.6%

f.2 pp ! tj (t-channel) p p > tt a j $$ w+ w- 9.956 0.014 101

+6.4% 8.8%

+0.9%1.0% 1.017 0.003 100

+1.3% 1.2%

+0.8% 0.9%

f.3 pp ! tZj (t-channel) p p > tt z j $$ w+ w- 6.967 0.007 101

+3.5% 5.5%

+0.9%1.0% 6.993 0.021 101

+1.6% 1.1%

+0.9% 1.0%

f.4 pp ! tbj (t-channel, 4f) p p > tt bb j $$ w+ w- 1.003 0.000 102

+13.8% 11.5%

+0.4%0.5% 1.319 0.003 102

+5.8% 5.2%

+0.4% 0.5%

f.5 pp ! tbj (t-channel, 4f) p p > tt bb j a $$ w+ w- 6.293 0.006 101

+16.8% 13.5%

+0.8%0.9% 8.612 0.025 101

+6.2% 6.6%

+0.8% 0.9%

f.6 pp ! tbjZ (t-channel, 4f) p p > tt bb j z $$ w+ w- 3.934 0.002 101

+18.7% 14.7%

+1.0%0.9% 5.657 0.014 101

+7.7% 7.9%

+0.9% 0.9%

f.7 pp ! tb (s-channel, 4f)

p p > w+ > t b, p p > w- > t b 7.489 0.007 100

+3.5% 4.4%

+1.9%1.4% 1.001 0.004 101

+3.7% 3.9%

+1.9% 1.5%

f.8 pp ! tb (s-channel, 4f)

p p > w+ > t b a, p p > w- > t b a 1.490 0.001 102

+1.2% 1.8%

+1.9%1.5% 1.952 0.007 102

+2.6% 2.3%

+1.7% 1.4%

f.9 pp ! tbZ (s-channel, 4f)

p p > w+ > t b z, p p > w- > t b z 1.072 0.001 102

+1.3% 1.5%

+2.0%1.6% 1.539 0.005 102

+3.9% 3.2%

+1.9% 1.5%

Table 7. Sample of LO and NLO total rates for the production of a single top, possibly in association and within cuts, at the 13-TeV LHC; we also report the integration errors, and the fractional scale (left) and PDF (right) uncertainties. The notation understands the sum of the t and t cross sections for all processes, and tt is a label that includes both t and t, dened from the shell with define tt = t t~ (and analogously for the label bb). Processes that explicitly involve b-quarks in the nal state are calculated in the four-avour scheme, while all of the others are in the ve-avour scheme. Being an EW-induced process, single-top production requires special care for the MadGraph5 aMC@NLO generation syntax: $$ means excluding particles in the s-channel, while the > w+ > (> w- > ) forces a W + (W ) to be present in the s-channel (see appendix B.1). Total NLO cross sections for t- and s-channel single-top production have been known for some time [318, 319]. All single-top channels are also available in MCFM [320323], MC@NLO [173, 324], and POWHEG [325, 326]. An NLO calculation for tZj production has appeared in ref. [327].

JHEP07(2014)079

Process Syntax Cross section (pb)

Single Higgs production LO 13 TeV NLO 13 TeVg.1 pp ! H (HEFT) p p > h 1.593 0.003 101

+34.8% 26.0%

+1.2%1.7% 3.261 0.010 101

+20.2% 17.9%

+1.1% 1.6%

g.2 pp ! Hj (HEFT) p p > h j 8.367 0.003 100

+39.4% 26.4%

+1.2%1.4% 1.422 0.006 101

+18.5% 16.6%

+1.1% 1.4%

g.3 pp ! Hjj (HEFT) p p > h j j 3.020 0.002 100

+59.1% 34.7%

+1.4%1.7% 5.124 0.020 100

+20.7% 21.0%

+1.3% 1.5%

g.4 pp ! Hjj (VBF) p p > h j j $$ w+ w- z 1.987 0.002 100

+1.7% 2.0%

+1.9%1.4% 1.900 0.006 100

+0.8% 0.9%

+2.0% 1.5%

g.5 pp ! Hjjj (VBF) p p > h j j j $$ w+ w- z 2.824 0.005 101

+15.7% 12.7%

+1.5%1.0% 3.085 0.010 101

+2.0% 3.0%

+1.5% 1.1%

g.6 pp ! HW p p > h wpm 1.195 0.002 100

+3.5% 4.5%

+1.9%1.5% 1.419 0.005 100

+2.1% 2.6%

+1.9% 1.4%

g.7 pp ! HW j p p > h wpm j 4.018 0.003 101

+10.7% 9.3%

+1.2%0.9% 4.842 0.017 101

+3.6% 3.7%

+1.2% 1.0%

g.8 pp ! HW jj p p > h wpm j j 1.198 0.016 101

+26.1% 19.4%

+0.8%0.6% 1.574 0.014 101

+5.0% 6.5%

+0.9% 0.6%

g.9 pp ! HZ p p > h z 6.468 0.008 101

+3.5% 4.5%

+1.9%1.4% 7.674 0.027 101

+2.0% 2.5%

+1.9% 1.4%

g.10 pp ! HZ j p p > h z j 2.225 0.001 101

+10.6% 9.2%

+1.1%0.8% 2.667 0.010 101

+3.5% 3.6%

+1.1% 0.9%

g.11 pp ! HZ jj p p > h z j j 7.262 0.012 102

+26.2% 19.4%

+0.7%0.6% 8.753 0.037 102

+4.8% 6.3%

+0.7% 0.6%

g.12 pp ! HW +W (4f) p p > h w+ w- 8.325 0.139 103

+0.0% 0.3%

+2.0%1.6% 1.065 0.003 102

+2.5% 1.9%

+2.0% 1.5%

g.13 pp ! HW p p > h wpm a 2.518 0.006 103

+0.7% 1.4%

+1.9%1.5% 3.309 0.011 103

+2.7% 2.0%

+1.7% 1.4%

g.14 pp ! HZW p p > h z wpm 3.763 0.007 103

+1.1% 1.5%

+2.0%1.6% 5.292 0.015 103

+3.9% 3.1%

+1.8% 1.4%

g.15 pp ! HZZ p p > h z z 2.093 0.003 103

+0.1% 0.6%

+1.9%1.5% 2.538 0.007 103

+1.9% 1.4%

+2.0% 1.5%

g.16 pp ! Htt p p > h t t 3.579 0.003 101

+30.0% 21.5%

+1.7%2.0% 4.608 0.016 101

+5.7% 9.0%

+2.0% 2.3%

g.17 pp ! Htj p p > h tt j 4.994 0.005 102

+2.4% 4.2%

+1.2%1.3% 6.328 0.022 102

+2.9% 1.8%

+1.5% 1.6%

g.18 pp ! Hb

b (4f) p p > h b b 4.983 0.002 101

+28.1% 21.0%

+1.5%1.8% 6.085 0.026 101

+7.3% 9.6%

+1.6% 2.0%

g.19 pp ! Httj p p > h t t j 2.674 0.041 101

+45.6% 29.2%

+2.6%2.9% 3.244 0.025 101

+3.5% 8.7%

+2.5% 2.9%

g.20 pp ! Hb

bj (4f) p p > h b b j 7.367 0.002 102

+45.6% 29.1%

+1.8%2.1% 9.034 0.032 102

+7.9% 11.0%

+1.8% 2.2%

Table 8. Sample of LO and NLO total rates for the production of a single SM Higgs, possibly in association and within cuts, at the 13-TeV LHC; we also report the integration errors, and the fractional scale (left) and PDF (right) uncertainties. See table 1 for the meaning of wpm, and table 7 for the meaning of tt, bb, and the generation syntax. Processes that explicitly involve b-quarks in the nal state are calculated in the four-avour scheme, while all of the others are in the ve-avour scheme, except for g.12. A complete set of references relevant to NLO rates for Higgs production can be found in refs. [328330, 360, 361]. The W -boson width is set equal to 2.0476 GeV for process g.17. Cross sections at the NLO for HV jj and HV V production appear in this work for the rst time.

JHEP07(2014)079

Process Syntax Cross section (pb)

Higgs pair production LO 13 TeV NLO 13 TeV

h.1 pp ! HH (Loop improved) p p > h h 1.772 0.006 102

+29.5% 21.4%

+2.1%2.6% 2.763 0.008 102

+11.4% 11.8%

+2.1% 2.6%

h.2 pp ! HHjj (VBF) p p > h h j j $$ w+ w- z 6.503 0.019 104

+7.2% 6.4%

+2.3%1.6% 6.820 0.026 104

+0.8% 1.0%

+2.4% 1.7%

h.3 pp ! HHW p p > h h wpm 4.303 0.005 104

+0.9% 1.3%

+2.0%1.5% 5.002 0.014 104

+1.5% 1.2%

+2.0% 1.6%

h.4 pp ! HHW j p p > h h wpm j 1.922 0.002 104

+14.2% 11.7%

+1.5%1.1% 2.218 0.009 104

+2.7% 3.3%

+1.6% 1.1%

h.5 pp ! HHW p p > h h wpm a 1.952 0.004 106

+3.0% 3.0%

+2.2%1.6% 2.347 0.007 106

+2.4% 2.0%

+2.1% 1.6%

h.6 pp ! HHZ p p > h h z 2.701 0.007 104

+0.9% 1.3%

+2.0%1.5% 3.130 0.008 104

+1.6% 1.2%

+2.0% 1.5%

h.7 pp ! HHZj p p > h h z j 1.211 0.001 104

+14.1% 11.7%

+1.4%1.1% 1.394 0.006 104

+2.7% 3.2%

+1.5% 1.1%

h.8 pp ! HHZ p p > h h z a 1.397 0.003 106

+2.4% 2.5%

+2.2%1.7% 1.604 0.005 106

+1.7% 1.4%

+2.3% 1.7%

h.9 pp ! HHZZ p p > h h z z 2.309 0.005 106

+3.9% 3.8%

+2.2%1.7% 2.754 0.009 106

+2.3% 2.0%

+2.3% 1.7%

h.10 pp ! HHZW p p > h h z wpm 3.708 0.013 106

+4.8% 4.5%

+2.3%1.7% 4.904 0.029 106

+3.7% 3.2%

+2.2% 1.6%

h.11 pp ! HHW +W (4f) p p > h h w+ w- 7.524 0.070 106

+3.5% 3.4%

+2.3%1.7% 9.268 0.030 106

+2.3% 2.1%

+2.3% 1.7%

h.12 pp ! HHtt p p > h h t t 6.756 0.007 104

+30.2% 21.6%

+1.8%1.8% 7.301 0.024 104

+1.4% 5.7%

+2.2% 2.3%

h.13 pp ! HHtj p p > h h tt j 1.844 0.008 105

+0.0% 0.6%

+1.8%1.8% 2.444 0.009 105

+4.5% 3.1%

+2.8% 3.0%

h.14 pp ! HHb

b p p > h h b b 7.849 0.022 108

+34.3% 23.9%

+3.1%3.7% 1.084 0.012 107

+7.4% 10.8%

+3.1% 3.7%

Table 9. Sample of LO and NLO total rates for Higgs-pair production, possibly in association and within cuts, at the 13-TeV LHC; we also report the integration errors, and the fractional scale (left) and PDF (right) uncertainties. See table 1 for the meaning of wpm, and table 7 for the meaning of tt. All cross sections are calculated in the ve-avour scheme, except for process h.11 which is obtained in the four-avour scheme to avoid resonant-top contributions. Processes h.1, h.2, h.3, h.6, h.12, and h.13 have appeared in ref. [127] as NLO+PS results; some of these were already known at the NLO [331]. The W -boson width is set equal to 2.0476 GeV for process h.13. Previous to the release of MadGraph5 aMC@NLO, the only available public code for Higgs pair production was HPAIR [332, 333], relevant to process h.1 (see ref. [127] for more details on the di erent approach adopted by MadGraph5 aMC@NLO). Process h.2 has been recently added to VBFNLO.

JHEP07(2014)079

Process Syntax Cross section (pb)

Heavy quarks and jets LO 1 TeV NLO 1 TeVi.1 e+e ! jj e+ e- > j j 6.223 0.005 101

+0.0%0.0% 6.389 0.013 101

+0.2% 0.2%

i.2 e+e ! jjj e+ e- > j j j 3.401 0.002 101

+9.6%8.0% 3.166 0.019 101

+0.2% 2.1%

i.3 e+e ! jjjj e+ e- > j j j j 1.047 0.001 101

+20.0%15.3% 1.090 0.006 101

+0.0% 2.8%

i.4 e+e ! jjjjj e+ e- > j j j j j 2.211 0.006 102

+31.4%22.0% 2.771 0.021 102

+4.4% 8.6%

i.5 e+e ! tt e+ e- > t t 1.662 0.002 101

+0.0%0.0% 1.745 0.006 101

+0.4% 0.4%

i.6 e+e ! ttj e+ e- > t t j 4.813 0.005 102

+9.3%7.8% 5.276 0.022 102

+1.3% 2.1%

i.7 e+e ! ttjj e+ e- > t t j j 8.614 0.009 103

+19.4%15.0% 1.094 0.005 102

+5.0% 6.3%

i.8 e+e ! ttjjj e+ e- > t t j j j 1.044 0.002 103

+30.5%21.6% 1.546 0.010 103

+10.6% 11.6%

i.9 e+e ! tttt e+ e- > t t t t 6.456 0.016 107

+19.1%14.8% 1.221 0.005 106

+13.2% 11.2%

i.10 e+e ! ttttj e+ e- > t t t t j 2.719 0.005 108

+29.9%21.3% 5.338 0.027 108

+18.3% 15.4%

i.11 e+e ! b

b (4f) e+ e- > b b 9.198 0.004 102

+0.0%0.0% 9.282 0.031 102

+0.0% 0.0%

i.12 e+e ! b

bj (4f) e+ e- > b b j 5.029 0.003 102

+9.5%8.0% 4.826 0.026 102

+0.5% 2.5%

i.13 e+e ! b

bjj (4f) e+ e- > b b j j 1.621 0.001 102

+20.0%15.3% 1.817 0.009 102

+0.0% 3.1%

i.14 e+e ! b

bjjj (4f) e+ e- > b b j j j 3.641 0.009 103

+31.4%22.1% 4.936 0.038 103

+4.8% 8.9%

i.15 e+e ! b

bbb (4f) e+ e- > b b b b 1.644 0.003 104

+19.9%15.3% 3.601 0.017 104

+15.2% 12.5%

i.16 e+e ! b

bbbj (4f) e+ e- > b b b b j 7.660 0.022 105

+31.3%22.0% 1.537 0.011 104

+17.9% 15.3%

i.17 e+e ! ttb

b (4f) e+ e- > t t b b 1.819 0.003 104

+19.5%15.0% 2.923 0.011 104

+9.2% 8.9%

i.18 e+e ! ttb

bj (4f) e+ e- > t t b b j 4.045 0.011 105

+30.5%21.6% 7.049 0.052 105

+13.7% 13.1%

Table 10. Sample of LO and NLO rates for the production of light jets in association with heavy quarks, possibly within cuts, at a 1-TeV e+e

collider; we also report the integration errors, and the fractional scale uncertainties. Cross sections for processes i.1i.10 are calculated in the veavour scheme. For processes i.11i.18 we use the four-avour scheme, and require the presence of at least two (four in i.15i.16) b-jets in the nal state. b-jets are clustered with the same parameters as light jets. Results at NLO accuracy for up to seven light jets can be found in refs. [9, 70, 334 338], and for a heavy-quark-pair plus up to one jet in refs. [339345]. All other processes are computed here for the rst time at the NLO.

JHEP07(2014)079

Process Syntax Cross section (pb)

Top quarks +bosons LO 1 TeV NLO 1 TeVj.1 e+e ! ttH e+ e- > t t h 2.018 0.003 103

+0.0%0.0% 1.911 0.006 103

+0.4% 0.5%

j.2 e+e ! ttHj e+ e- > t t h j 2.533 0.003 104

+9.2%7.8% 2.658 0.009 104

+0.5% 1.5%

j.3 e+e ! ttHjj e+ e- > t t h j j 2.663 0.004 105

+19.3%14.9% 3.278 0.017 105

+4.0% 5.7%

j.4 e+e ! tt e+ e- > t t a 1.270 0.002 102

+0.0%0.0% 1.335 0.004 102

+0.5% 0.4%

j.5 e+e ! ttj e+ e- > t t a j 2.355 0.002 103

+9.3%7.9% 2.617 0.010 103

+1.6% 2.4%

j.6 e+e ! ttjj e+ e- > t t a j j 3.103 0.005 104

+19.5%15.0% 4.002 0.021 104

+5.4% 6.6%

j.7 e+e ! ttZ e+ e- > t t z 4.642 0.006 103

+0.0%0.0% 4.949 0.014 103

+0.6% 0.5%

j.8 e+e ! ttZj e+ e- > t t z j 6.059 0.006 104

+9.3%7.8% 6.940 0.028 104

+2.0% 2.6%

j.9 e+e ! ttZjj e+ e- > t t z j j 6.351 0.028 105

+19.4%15.0% 8.439 0.051 105

+5.8% 6.8%

j.10 e+e ! ttW jj e+ e- > t t wpm j j 2.400 0.004 107

+19.3%14.9% 3.723 0.012 107

+9.6% 9.1%

j.11 e+e ! ttHZ e+ e- > t t h z 3.600 0.006 105

+0.0%0.0% 3.579 0.013 105

+0.1% 0.0%

j.12 e+e ! ttZ e+ e- > t t a z 2.212 0.003 104

+0.0%0.0% 2.364 0.006 104

+0.6% 0.5%

j.13 e+e ! ttH e+ e- > t t a h 9.756 0.016 105

+0.0%0.0% 9.423 0.032 105

+0.3% 0.4%

j.14 e+e ! tt e+ e- > t t a a 3.650 0.008 104

+0.0%0.0% 3.833 0.013 104

+0.4% 0.4%

j.15 e+e ! ttZZ e+ e- > t t z z 3.788 0.004 105

+0.0%0.0% 4.007 0.013 105

+0.5% 0.5%

j.16 e+e ! ttHH e+ e- > t t h h 1.358 0.001 105

+0.0%0.0% 1.206 0.003 105

+0.9% 1.1%

j.17 e+e ! ttW +W e+ e- > t t w+ w- 1.372 0.003 104

+0.0%0.0% 1.540 0.006 104

+1.0% 0.9%

Table 11. Sample of LO and NLO rates for the production of top quarks in association with bosons, possibly within cuts and in association with jets, at a 1-TeV e+e collider, and the fractional scale uncertainties. Cross sections are calculated in the ve-avour scheme; see table 1 for the meaning of wpm. Results at NLO accuracy for ttH production can be found in ref. [346]. All of the other processes are computed here for the rst time at the NLO.

JHEP07(2014)079

4.2 Di erential distributions

In this section we present sample results for di erential observables relevant to several processes, which we have simulated at the 8, 13, or 14 TeV LHC. While some of these have never been computed before at the NLO+PS accuracy (or even at fNLO; see section 4.1), and appear here for the rst time, we do not aim to present a series of phenomenological analyses, which would be out of the scope of this work, but rather at showing yet again the exibility of MadGraph5 aMC@NLO, and the type of results that one can obtain with it. For this reason, the various PSMCs which we shall use have been run with their default parameters, and no underlying events have been generated. Having said that, some of the predictions given here are motivated by recent measurements by ATLAS and CMS. Furthermore, the present section constitutes a complement in particular to section 2.4.5, since we shall discuss, using explicit examples, several features of the FxFx merging procedure which have been outlined before in a general fashion. We shall be mainly concerned with (N)LO+PS results, but we shall also consider f(N)LO ones where necessary. As was the case for the total rates presented in section 4.1, the computation of scale and PDF uncertainties has been carried out by using the reweighting procedure introduced in ref. [125] (see also appendix B.3). NLO+PS results that have never appeared in the literature are: six-lepton, ttW +W , and SM Higgs in VBF+1j production; furthermore, double-Higgs production in association with either a tt pair or a Z boson has been solely computed with MadGraph5 aMC@NLO, in ref. [127]. Finally, FxFx-merged results for ZZ and He+e production are also presented here for the rst time.

[diamondsolid] Six-lepton production. We start by studying the (N)LO+PS production of six lep-tons:

, (4.1)

which we have computed by using the complex mass scheme; the lepton is set stable, and its mass is kept at the physical value, while the electron and the muon are treated as massless. On top of the computation carried out with the exact six-lepton matrix elements of eq. (4.1), we have also considered the production of the ZW +W triplet, with the subsequent decays of the vector bosons performed with either MadSpin or by the PSMC (in this case, HERWIG6):

pp Z( e+e) W +( +) W (

) . (4.2)

While MadSpin multiplies the undecayed matrix elements by the branching ratios of the relevant decays, so that the rates resulting from eq. (4.2) are in absolute value directly comparable to those of eq. (4.1), the PSMC does not; in that case, we have therefore manually included such an overall factor. To all samples, we have applied the following cut:

M(+()) > 30 GeV , (4.3)

on all opposite-charged lepton pairs; given the lepton avours considered here, not surprisingly the vastly dominant e ect of such a cut is that due to the e+e pair. We

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0 250 500 750 1000 1250 1500 1750 2000 m(e+e-+nt-nt) [GeV]

Figure 4. Transverse momentum (left panel) and invariant mass (right panel) of the six-lepton system, for the processes of eqs. (4.1) and (4.2). See the text for details.

shall call eq. (4.3) the generation cut.54 On top of eq. (4.3), we have also imposed (also at the analysis level):

(e+e) mZ

< 20 GeV , (4.4)

(+) mW

< 20 GeV , (4.5)

(

) mW

< 20 GeV , (4.6)

which we shall call V -reco cuts. Since eq. (4.2) features only 3-resonant contributions (see section 2.5 about the notation used here for resonant and non-resonant diagrams), the results of the MadSpin- and PSMC-decayed samples are basically the same if one considers only the generation cut, or the generation and V -reco cuts together; for this reason, we shall discuss only the latter scenario. On the other hand, one of the reasons for comparing eqs. (4.1) and (4.2) is precisely that of assessing the importance of non-3-resonant contributions to six-lepton matrix elements; hence, in this case we shall present both the generation-cut-only and the generation-plus-V -reco cuts results.

In gure 4 we show observables relevant to the six-lepton system, i.e. obtained by summing the four-momenta of the leptons: the transverse momentum (left panel) and the invariant mass (right panel). Both the NLO+PS (solid histograms) and LO+PS (dashed histograms, rescaled as indicated in order to t into the layout) are displayed. The green histograms are the results of eq. (4.1) with only the generation cut (denoted by (N)LO ME); the results for the generation-plus-V -reco cuts are shown as yellow (eq. (4.1), denoted by (N)LO ME V -reco), red (eq. (4.2) with MadSpin, denoted by (N)LO MS V -reco), and blue (eq. (4.2) with PSMC decays, denoted by (N)LO PSMC V -reco) histograms respectively. In the middle insets, the ratios of all the NLO results over the NLO ME V -reco ones

54Despite the fact that it has been imposed at the analysis level, and the true generation cut is marginally lower so as to avoid biases.

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cosc t

Figure 5. Cosine of correlation angles for the + pair, in six-lepton production, eqs. (4.1) and (4.2). See the text for details.

are presented. Finally, in the lower insets each LO prediction is divided by its NLO counterpart (so these are essentially the inverse of the K factors). The plots show clearly the large impact of non-3-resonant contributions, which induce dramatic shape modications for M(6) < 500 GeV and 10 < pT (6) < 250 GeV (the very small pT region being dominated by PSMC radiation e ects). On the other hand, by imposing V -reco cuts the three predictions agree rather well with each other, which is the signal that spin correlations are unimportant for these observables (and, more importantly in view of the aim of this paper, that all is ne from a technical viewpoint, in the context of a very involved production process). We have performed similar comparisons for a large number of observables; here, we limit ourselves to reporting the results for the cosine of the angle dened by the directions of ight of the + and leptons, which we denote by + (left panel of gure 5) when it is computed in the laboratory frame, and by + (right panel of gure 5) when it is computed by rst boosting the four-momentum of the + and leptons to the rest frame of the + and

systems respectively (i.e., to the virtual-W + and W rest frames in the case of resonant contributions); the latter observable is known to be particularly suitable for the study of spin-correlation e ects. The same conclusions as for the observables of gure 4 apply here, bar for the + NLO PSMC V -reco one that is fairly di erent from both the NLO ME V -reco and NLO MS V -reco predictions, which in turn agree with each other quite well. As it was expected, this is a manifestation of the importance of spin correlations for such an observable, and a direct validation of the MadSpin procedure.

The overall messages that one can obtain from the present study are the following. Firstly, we did verify that the conclusions reached above are not qualitatively modied if one replaces the (N)LO ME results with those obtained by imposing the generation cut and eq. (4.4) only in other words, it is the simultaneous action of the three cuts of eqs. (4.4)(4.6) that brings the predictions for eq. (4.1) in agreement with those for eq. (4.2) and MadSpin; this is obviously because it is important that all three vector

bosons be near their respective mass shell. Secondly, the e ects of NLO corrections are non negligible, in both rate and shape; however, the patterns of comparison among the various calculations are to a large extent independent of the perturbative accuracy of the latter. Thirdly, production spin correlations are present, that can be properly described only by the full six-lepton computation, and by MadSpin as well if one limits oneself to the 3-resonant region. It is clear that the cuts of eqs. (4.5) and (4.6), and the denition of the observables considered here, bar + , cannot be achieved experimentally, owing to the presence of the neutrino four-momenta. However, they have helped us reach conclusions which have a general validity, and in particular in the case of a fully realistic analysis: namely, that non-resonant e ects in six-lepton production may be quite large and that, for all those cuts that render the 3-resonant contributions dominant, the undecayed-plus-MadSpin simulation provides one with a very good approximation of the exact calculation.

[diamondsolid] ttW W + production. We now turn to considering the process:

pp t( e+eb) t( e

eb) W (

) W +( +) , (4.7)

which we have simulated at the (N)LO+PS accuracy, by only considering the undecayed matrix elements with ttW +W nal states, and by using Pythia8 as PSMC and either MadSpin or the internal Pythia8 routine (which correctly accounts for decay spin correlations) for the decays of the top quarks and W bosons. In gure 6 we present the transverse momentum of the ttW +W system, which is the typical observable whose small-pT behaviour is dominated by MC e ects (whose systematics will not be studied here), and which is thus unreliable if computed at fNLO accuracy. Both the NLO+PS (solid histograms) and LO+PS (dashed histograms) results are displayed, with the respective scale-uncertainty bands (in dark and light shades respectively). The very signicant reduction of such theoretical systematics when higher-order corrections are included is evident in the whole range considered (see also entry e.13 in table 6 for its total-rate counterpart). While for asymptotically-large transverse momenta one expects the NLO+PS scale dependence to be of LO type (because in that region the computation is dominated by tree-level contributions), for such a massive system these pT s are confortably in the TeV-range. In gure 7 we show the transverse momenta of the four-hardest charged leptons in the events; the leptons are required to have pT () > 20 GeV and |()| < 2.5. At variance with those of gure 6,

these plots include the branching ratios of the decays reported in eq. (4.7). The LO+PS results in the main frames are rescaled so as to t into the layout. Each plot displays four histograms, that correspond to NLO+PS (solid, with MadSpin decays; dashed, with Pythia8 decays) and to LO+PS (dot-dashed, with MadSpin decays; dotted, with Pythia8 decays) results. The ratios of these predictions over the NLO+PS, MadSpin-decayed ones are shown in the insets. The radiative corrections are large, but relatively at in the pT ranges considered here; as in the case of the pT of the system, their inclusion reduces the scale uncertainty in a dramatic manner. Production spin correlations are sizable, so much so that MadSpin- and Pythia8-decayed results have shapes which are barely within, or slightly outside of, the theoretical systematics bands. This is true at the NLO; at the LO, the two predictions are compatible within uncertainties, but this is solely due to the fact that the

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+W-) [GeV]

Figure 6. Transverse momentum of the system of the four primary nal-state particles in ttW +W

production.

LO scale dependence is rather large: in fact, the pattern of the inclusion of production spin correlations is basically independent of the perturbative order at which one is working. This is nothing but another manifestation of the benets inherent to the increased predictive power of simulations that include both NLO and production spin correlation e ects.

[diamondsolid] Double-Higgs production. We now consider double-Higgs production in the SM at the 14 TeV LHC. This process has been investigated recently with MadGraph5 aMC@NLO in ref. [127], where all the six dominant channels at the LHC have been computed up to NLO+PS accuracy, some of them for the rst time. For all channels, the results of ref. [127] have improved what was available in the literature in at least one respect. In ref. [127] we have only presented (N)LO+PS distributions. Here, we amend this by showing also f(N)LO spectra; we use the transverse momentum of the Higgs pair, and the ttHH and ZHH channels, as a denite example; as PSMCs, we adopt Pythia8 and HERWIG6. The results are shown in gure 8, as solid (for NLO-accurate) and dashed (for LO-accurate) histograms. The main frames display the NLO predictions in absolute value, while the LO ones are rescaled in order to t into the layout; the K factors can be read from the insets, where we present the ratios of all results over those at the NLO+PS obtained with Pythia8. The common feature of the two plots is that NLO results are mutually closer than the corresponding LO ones; the two NLO+PS predictions are extremely similar for both processes, while the fNLO spectrum in ZHH production is only marginally softer (an overall e ect smaller than 20%). It is interesting to see that this stabilisation due to the in-

tt-W+W- production at 13 TeV LHC

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0.8

0 50 100 150 200 250

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0 50 100 150 200 250

Figure 7. Transverse momentum of the hardest four charged leptons in ttW +W production see eq. (4.7). The LO+PS results that appear in the insets have not been rescaled.

clusion of higher-order corrections follows di erent patterns for the two channels considered here. ZHH is predominantly a qq-initiated process: therefore, the di erence between the two standalone PSMCs (i.e., LO+PS) is expected to be smaller than in the case of ttHH production, which mainly proceeds through gg fusion. This is precisely what we see in gure 8 (compare the purple and red histograms in the insets). On the other hand, at fLO the Higgs pair recoils against a Z boson and a tt-system in ZHH and ttHH production respectively; the kinematics of the tt pair being non-trivial (at variance with that of a single Z) implies that the fLO prediction for pT (HH) is farther away from the corresponding LO+PS ones in the ZHH channel than in the ttHH channel (see the green dashed histograms in the insets, and compare the position of the peaks of the fLO and fNLO results).

[diamondsolid] Single-Higgs production. We now turn to discussing the production of a single SM Higgs at the 13 TeV LHC. The aims of gure 9, where we show the Higgs trans-verse momentum, are twofold. Firstly, we compare LO+PS with NLO+PS predictions; secondly, we present results for all PSMCs which are matched to NLO calculations in

10-4

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JHEP07(2014)079

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0 50 100 150 200 250 300 350 400 pT(HH) [GeV]

Figure 8. Transverse momentum of the two-Higgs system in ttHH (left panel) and ZHH (right panel) production. (N)LO+PS and f(N)LO results are shown. We have used Pythia8 and HERWIG6.

MadGraph5 [email protected] The comparison between the two panels of the gure shows the two expected behaviours: at large pT s, all NLO+PS predictions coincide (and are just on top of the corresponding fNLO result, not displayed here), while the LO+PS are vastly di erent; on the other hand, at small pT s, the relative behaviour of the various PSMCs is the same, regardless of the perturbative order of the underlying matrix-element computations. We point out that all PSMCs are treated on equal footing, i.e. they are given the same numerical values as shower-scales parameters (such scales are equal to mH at the

LO, and controlled by the D function at the NLO); so while di erent scales for di erent PSMCs could bring them in better agreement at the LO, this is actually a negative implication of the loss of predictivity at this perturbative order, and it is unnecessary when NLO corrections are included in the simulations; a further example of this pattern will be given below, in the study of Higgs production through VBF. We conclude this part by showing, in gure 10, the Higgs pT that results from the ve dominant production channels at the 13 TeV LHC (whose total rates are reported in lines g.1, g.4, g.6, g.9, and g.16 of table 8, where one can also nd the MadGraph5 aMC@NLO shell commands relevant to their generation); the thickness of the bands represent the combined scale and PDF uncertainties; all the predictions are obtained at the NLO+PS, with the use of Pythia8. This plot is another demonstration of the exibility of the MadGraph5 aMC@NLO framework.

[diamondsolid] Higgs production in VBF. As a further example of the stabilisation of the predictions that result from including higher-order matrix elements (which in turn serves as a validation exercise for the whole NLO+PS machinery in MadGraph5 aMC@NLO), we consider Higgs production in VBF, which we compute in two ways: by considering the process whose Born is of O(3) (which we denote by VBF+0j [347], and whose nal state at

55We remind the reader that Pythia6(pT ) is available for ISR-only processes.

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0 0 50 100 150 200 pT(H) [GeV]

0 0 50 100 150 200 250 300 pT(H) [GeV]

Figure 9. Higgs pT spectrum in gg fusion (HEFT), for various PSMCs, at the LO+PS (left panel) and NLO+PS (right panel) accuracy. Note the larger pT range in the right panel.

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ggH VBF

ZH ttH

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Figure 10. Higgs pT spectrum for the ve dominant production channels at the LHC, at the NLO+PS accuracy with Pythia8.

the Born level thus features a Higgs-plus-two-parton system), and the process whose Born is of O(3S) (which we denote by VBF+1j, and whose nal state at the Born level thus

features a Higgs-plus-three-parton system). These are the processes reported in lines g.4 and g.5 of table 8, where the interested reader can also nd the MadGraph5 aMC@NLO shell commands that one must use in order to generate them. The only analysis cuts we impose here are on the transverse momenta of the (anti-kT , R = 0.5) jets, by requiring that pT (j) > 20 GeV. The only observables for which a direct comparison between VBF+0j and VBF+1j is sensible, and allow one to assess the impact of perturbative corrections, are those related to the third jet, where one expects to have an e ective LO and NLO description respectively. In gure 11 we present predictions for the transverse momen-

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Figure 11. Third-jet observables in Higgs VBF+0j and VBF+1j production, both at the NLO+PS, with Pythia8, HERWIG6, and Pythia6(Q2).

tum spectrum of the third-hardest jet, and for the rate as a function of the transverse momentum of the veto jet:

veto (pT (jveto)) =

pT (jveto) dpT

ddpT . (4.8)

The veto jet is the hardest jet which is not one of the two tagging jet (which are dened to be the two hardest ones overall, and which we denote by j1 and j2 respectively), and whose rapidity obeys the condition:

min (yj1, yj2) y(jveto) max (yj1, yj2) . (4.9)

The quantity dened in eq. (4.8) is related to Pveto, dened e.g. in eq. (41) of ref. [330], by a simple normalisation factor, veto = NLO Pveto. For both VBF+0j (dashed histograms)

and VBF+1j (solid histograms) the results for three PSMCs (Pythia8 (red), Pythia6(Q2) (green), and HERWIG6 (black)) are displayed, with the VBF+0j ones rescaled by a factor 1/5 in order for them to t into the layout. The ratios of the Pythia predictions over the HERWIG6 ones are presented in the insets. In the inset relevant to VBF+1j, we also report the ratio of the HERWIG6 VBF+0j result over the VBF+1j one (black dashed histogram), which is related to the inverse of the K factor for both observables, the latter is of the order of 1.051.15. In the insets we also display the scale uncertainties as gray bands: it is evident that the inclusion of the contributions of relative O(S) in

VBF+1j signicantly reduces the theoretical systematics. Apart from this, the most striking consequence of such an inclusion is the fact that the three PSMCs in VBF+1j are fairly close to each other; this is not the case in VBF+0j, where Pythia8 has a much softer shape than either Pythia6 or HERWIG6. We point out that this is a feature of quantities related to the third jet: other observables which have an NLO nature in

VBF+0j are much better behaved, with the three PSMCs in good agreement already at

O(3(1 + S)). Therefore, although one could possibly nd settings for the PSMCs that would bring the predictions for the pT (j3) and veto VBF+0j spectra in better agreement than in gure 11, this would simply be the signal of an unsatisfactory predictive capability, which is restored by considering these observables in VBF+1j production.

[diamondsolid] Top-pair production. While di erential distributions relevant to tt production measured at the LHC by the ATLAS (at 7 TeV in lepton+jets events [348]) and the CMS (at 7 and 8 TeV, in lepton+jets and dilepton events [349351]) collaborations are generally in very good agreement with theoretical predictions, the CMS data for the reconstructed transverse momentum of the top quark (pT (t)) are visibly softer than NLO+PS predictions, and in disagreement with those of ATLAS for pT (t) < 200 GeV (ATLAS data are harder). Given this inconsistency between measurements it is premature to speculate on the origin of a possible discrepancy between data and theory; it is however of some interest to discuss the theoretical systematics that a ect the NLO+PS spectrum. Among these, those due to scale, PDFs, and choice of top-quark mass have been studied by the experimental collaborations, and shown to be smaller than the disagreement between data and theory [349]. Here, we therefore concentrate on other sources of systematics. One of these is due to missing higher orders, since the NLO+PS predictions used by the experiments include only up to O(3S) terms, namely tt+0j samples at the NLO. While the impact of missing higher or

ders in perturbation theory is estimated by scale variations, an important and independent check of this assessment may be obtained by considering NLO-merged prections. In the left panel of gure 12 we thus compare the unmerged tt+ 0j prediction with the FxFx one, where the tt+ 0j and tt+ 1j samples are combined with Q = 100 GeV. Both merged and unmerged results have been obtained with HERWIG6, by setting the collider energy equal to 8 TeV; the latter curve has been rescaled56 in order for its visible integral to coincide with that of the former (since in this case we are specically interested in a shape comparison: in absolute values, the two cross sections di er however by only 2.5%). As one can gather from the plot, the two predictions are close to each other; the FxFx prediction is sightly softer than the unmerged one, but this does not appear to be su cient to bring it in agreement with the CMS measurement. The variation of the merging scale in a large range (30155 GeV) does not induce any signicant change. It therefore appears that the systematics due to higher-order corrections are fully under control, since scale variations and NLO-merging give consistent results for this observable, and we thus conrm the previous ndings that it cannot explain the discrepancy between theory and CMS data. Another source of theoretical systematics in NLO+PS predictions is that due to the choice of the PSMC. This is presented in the right panel of gure 12, where we compare the (tt+ 0j unmerged) predictions obtained with Pythia8 (cyan), HERWIG6 (red), and Pythia6(Q2) (grey); the lower inset presents the bin-by-bin ratios of the latter two predictions over that of Pythia8. It is clear from the plot that the three PSMCs are amply consistent with each other; this is expected, since the pure-NLO result for pT (t) must not be dramatically

56If visually that may not seem to be the case, it is because the bin widths are not equal: note that the cross section is di erential.

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Figure 12. Transverse momentum of the top quark in tt production. Left panel: comparison between FxFx-merged (blue) and unmerged (red) predictions; the binning is the same as that of ref. [350]. Right panel: NLO+PS predictions obtained with di erent PSMCs, compared to the fNLO result.

modied by shower e ects; such an expectation is conrmed by the fNLO prediction, also reported in the right panel of gure 12 as the dashed histogram overlaid with full circles, which is in fair agreement with all the other curves. We therefore conclude that it does not seem possible to get NLO+PS predictions to agree with CMS data by changing the PSMC used in the simulations. We also point out that this statement by no means implies that acceptance corrections, which we do not compute here, are PSMC-independent; a careful investigation of these may be necessary should the discrepancy discussed here persist, given that pT (t) is not a quantity that can be measured directly.

Inclusive quantities that stem from either or both tops in tt events are an ideal testing ground for NLO+PS predictions, which should give a good description of the data for absolute normalisation as well as for shapes. On the other hand, tt events are characterised by large c.m. energies which imply a large amount of QCD radiation (only a tiny fraction of which originate from the top quarks, owing to their being very massive). The study of such radiation and of its topological properties is an interesting subject, particularly in view of the large statistical accuracy that can be obtained at the LHC. At variance with the case of inclusive observables, there is no reason to expect that all radiation-related observables will be well described by NLO+PS tt+ 0j predictions. In particular, those for which large jet multiplicities are important should be sensibly compared only to merged results, or at least to unmerged samples whose underlying matrix elements feature a su ciently large number of hard partons the most obvious example is that of Njet. On the other hand, for radiation-related observables which are also inclusive enough, NLO+PS simulations should do relatively well. One such case is that of the so-called gap fractions, which have been measured in the dilepton channel by both ATLAS (at 7 TeV [352]) and CMS (at 7 [353] and 8 TeV [354]), and compared to various theoretical predictions MC@NLO with HERWIG6, POWHEG with both HERWIG6 and Pythia, MadGraph with Pythia, Alpgen

with HERWIG6, and SHERPA. There are di erent levels of agreement among generators and with the data, whose discussion is outside the scope of this work; here, we concentrate on MC@NLO, since the relevant formalism is the same as that used in NLO+PS simulations in MadGraph5 aMC@NLO. Before going into the details, let us dene the main quantity that we shall study, namely the gap fraction for the pT of the hardest jet. In order to be denite, we shall use the same setup as in ref. [354] (which is CMSs at the 8 TeV LHC): jets are reconstructed with the anti-kT algorithm with R = 0.5, and the following cuts are imposed:

pT () 20 GeV , |()| 2.4 , pT (jb) 30 GeV , |(jb)| 2.4 , (4.10) on both charged leptons, and on the two hardest b-jets. Other type of cuts (e.g. lepton isolation) are seen to be unimportant, and are not imposed here. We then dene:

GFpT (j1)(Q) = 1

d (Q pT (j1))

dd , (4.11)

where is the cross section within the cuts of eq. (4.10), and for the notation of the argument of the gap fraction we adopt one similar to that of ref. [352], which is not liable to generate confusion. We have also introduced:

pT (j1) =

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(pT (j1) pT (j1) 30 GeV and min |(j1)| max ,0 otherwise , (4.12)

with j1 the hardest jet which is neither of the two b-jets on which the cuts of eq. (4.10) are applied. Note that if there is no jet harder than 30 GeV in the pseudorapidity interval (min, max), eq. (4.12) implies that the function in eq. (4.11) is identically equal to one.

Therefore, GFpT (j1)(Q) is a constant for Q < 30 GeV, equal to the fraction of events which do not have any jet harder than 30 GeV in the relevant pseudorapidity interval: for this reason, gap fractions are not displayed in this range. The quantity pT (j1) in eq. (4.11)

can be replaced by a function, dened analogously to what is done in eq. (4.12) in terms of any observable O with mass dimension equal to one; in this way, one constructs a

di erent type of gap fraction, GFO(Q). The transverse momentum of the second-hardest jet [354], and HT [352, 354] have been considered in the literature; they are rather strongly correlated with pT (j1), and will not be investigated any further here.

We now aim to compare the NLO+PS predictions of the MC@NLO program (v4.09) with those obtained with MadGraph5 aMC@NLO (both NLO+PS and FxFx-merged), which is interesting in two respects. Firstly, the unmerged NLO+PS results of the two codes would be identical, were it not for the fact that in the latter we have included the e ect of the function D (see eq. (2.113)); therefore, any di erence between the two is the signal of matching systematics.57 Secondly, the comparison between unmerged tt + 0j and FxFx simulations helps assess the impact of the inclusion of matrix elements of higher order in the latter. We have adopted HERWIG6 as PSMC, and the FxFx-merged

57There may be other very tiny di erences between MC@NLO and MadGraph5 aMC@NLO NLO+PS predictions, owing to possibly non-identical choices of parameters, which are negligible for all purposes.

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Figure 13. Gap fraction for the pT of the hardest jet, in four di erent pseudorapidity intervals, as predicted by FxFx-merged, unmerged (labelled aMC@NLO), and MC@NLO simulations. The setup follows closely that of ref. [354]. See the text for details.

simulations have been performed for two extreme choices of the merging scale, Q = 30

and 155 GeV. The results for the hardest-jet gap fraction are presented in gure 13, for the same four pseudorapidity intervals (min, max)=(0, 2.4), (0, 0.8), (0.8, 1.5), and (1.5, 2.1) as in ref. [354]; the insets display the ratios of the MC@NLO v4.09 and FxFx-merged results over the unmerged MadGraph5 aMC@NLO ones. We rst observe that all predictions are quite close to each other, the largest deviation being about 3%. Interestingly, some of the largest relative di erences are between the two unmerged predictions, which implies that the matching systematics is not negligible. In general, MadGraph5 aMC@NLO predicts more jet activity (i.e., a lower curve) at NLO+PS than MC@NLO v4.09 in the central and widest pseudorapidity regions, this di erence decreasing with increasing min. This

appears to be fairly consistent with what is seen in the 8-TeV CMS data (compare the upper left corner of gure 13 with gure 7 in ref. [354], and the other panels of gure 13 with the upper row of gure 8 in ref. [354]), whose analysis we have followed here. A similar trend as NLO+PS of MadGraph5 aMC@NLO is seen in the FxFx-merged result with Q = 155 GeV, while that obtained with Q = 30 GeV follows a slightly di erent pattern. Although these ndings are encouraging, it is certaintly too early to draw any rm conclusions; preliminarly, we can observe that the inclusion of more matrix-elements results into matched predictions seems to be benecial (be either through an NLO-merging

Q =20 GeV Q =30 GeV Q =50 GeV Q =70 GeV

14.77(+1.5%)9.08(+1.8%)

1.60(2.4%)

1.01(9.5%)

no cuts

FxFx

Alpgen

FxFx

Alpgen

14.47(0.6%)

8.84(0.9%)

1.65(+0.8%)1.27(+13.2%)

14.78(+1.5%)9.07(+1.7%)

14.568.92

1.55(5.4%)

0.92(18.4%)

Table 12. Total rates (in pb) for single-Higgs production in gluon-gluon fusion in HEFT, resulting from FxFx-merged (with up to two extra partons at the NLO) and Alpgen (with up to three extra partons at the LO) samples. Fractional di erences w.r.t. the corresponding results obtained with Q = 30 GeV are also reported.

procedure, or because the D function limits the impact of the HERWIG6 shower to smaller scales than it happens in MC@NLO). Furthermore, even by choosing an extremely large range for the merging scale, the FxFx systematics is smaller than that of the MLM-type merging in MadGraph5 aMC@NLO (see the results labelled MadGraph in ref. [354]). We shall in fact see below another clear example of the pattern of the reduction of the merging systematics when going from the LO to the NLO.

[diamondsolid] Multi-parton merged predictions. We now turn to illustrate some results of the FxFx merging which are directly relevant to the unitarity and the merging-scale-choice arguments which have been discussed in general at the end of section 2.4.5. We shall do so by using an example which one expects to be critical from these viewpoints, namely Higgs production in gluon-gluon fusion (in HEFT) at the 8 TeV LHC, since such a process is characterised by very large higher-order corrections and by a very signicant amount of radiation in PSMCs. As was done in ref. [191], we shall also compare to the predictions obtained with Alpgen, which we shall take as a benchmark for the typical behaviour of LO merging procedures; we have used HERWIG6 as PSMC. We start by presenting in table 12 the results for the fully inclusive rates, both in the absence of cuts (upper two rows), and by imposing the presence of at least two jets (lower two rows): the latter are dened by means of the anti-kT algorithm with R = 0.4, and have to obey the following conditions:

pT (j) 25 GeV , |(j)| 5 . (4.13)

The rates have been obtained by considering four di erent values for the merging scale, which cover the very large range Q (20, 70) GeV. In order to be denite, we shall

take Q = 30 GeV as our central value; in table 12, we report in parenthesis the fractional di erence of all results obtained with Q 6= 30 GeV w.r.t. those obtained with Q = 30 GeV

that appear in the same row. The rates in absence of jet cuts are seen to be extremely stable against merging-scale variations. LO results have an only marginally-larger Q dependence, which is in any case much smaller than the scale uncertainty (not shown here); the same applies to NLO predictions. The NLO fully-inclusive rate for the unmerged H + 0j sample is 13.40 pb: it is therefore from 8% to 10% lower than the FxFx-merged results. Thus, despite the fact that FxFx does not impose any unitarity condition, the merged predictions

two jets

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Merged Higgs production at the 8 TeV LHC Alpgen, MLM merging (3 partons, LO)

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Figure 14. Higgs transverse momentum in single-Higgs production (gluon-gluon fusion in HEFT), as predicted by FxFx (left panel) and Alpgen (right panel), for various choices of the merging scale Q.

are naturally quite close to the unmerged one. They are not equal, nor they should be at the end of the day, we are including here contributions up to O(5S). The increase in

the cross section when passing from unmerged to merged results is rather consistent with expectations based on perturbative scaling, and the known large NNLO/NLO K factor that characterises the Higgs-production mechanism we study here. We stress that this feature is not an accident of this process, since we have observed it in all cases studied so far (see later for further examples). When considering rates obtained within the cuts of eq. (4.13), we see that the NLO results are still quite stable, while the merging-scale dependence of the LO ones becomes sizable. In order to further investigate this matter, we present in gure 14 the Higgs transverse momentum spectra, obtained with FxFx and Alpgen for the same four merging scales as before; the insets display the ratios of the various curves over the Q = 30 GeV corresponding ones. Obviously, the Q-dependence pattern reects that of table 12. However, it is interesting to see that when no jet cuts are applied there is a signicant compensation in terms of rates in the Alpgen curves (this can be clearly seen in the ratio plot, with the presence of a crossing point at a pT (H) of

about 70 GeV) in other words, the very small Q dependence of the LO total rates is partly an artifact, since locally in the phase space the various predictions di er by a larger amount. To some extent, the same is true for the FxFx-merged results, but the e ects are much more modest there. Note that, when jet cuts are applied, this phenomenon does not occur any longer, and local and global merging-scale dependences are quite similar: in particular, one sees how all the NLO curves are close to each other also within these cuts. The signicance of this (in)dependence is tightly related to the range chosen for the merging scale variation. The function log(Q/mH), which one may take as an indicator of

()Q (c)Q ()Q unmerged

pp e+e 7059(0.9%) 7121 7160(+0.5%) 7067(0.8%)

pp ZZ 7.383(0.01%) 7.384 7.387(+0.04%) 7.355(0.4%)

pp He+e 0.05180 (+0.9%) 0.05131 0.05117(0.2%) 0.05066(1.3%) Table 13. Total rates (in pb) for three processes, computed with FxFx by using three di erent merging scales (whose values are process-dependent, see eqs. (4.15)(4.17)), and with the unmerged lowest-multiplicity samples. Relative di erences w.r.t the FxFx results obtained with the central merging scales are also reported.

the typical quantity relevant when the merging scale is varied, changes by a factor of 3.16 in the range (20, 70) GeV considered here; we believe that this is a su ciently-large range to give a sensible indication of merging-scale systematics. As the results presented here clearly show, the supposed spoiling of some underlying NLO accuracy that occurs when large values of Q are adopted is simply not an issue if NLO and MC predictions are properly merged, and are reasonably consistent with each other. The latter is in fact a key point: we have observed that, by imposing VBF-type cuts, e.g.:

Mj1j2 400 GeV , | yj1j2| 2.8 , (4.14) the mild dependences shown in table 12 become huge (of the order of 80% and 70% at the

LO and NLO respectively). It is clear that the invariant-mass cut of eq. (4.14) introduces a third scale in the game which renders its treatment a complicated matter. Given that such a large merging-scale dependence is basically driven by the largest Qs, the problem is likely due to intrinsic di erences between the PSMC and matrix-element descriptions of the VBF region. However, we are able to immediately notice this only because we have considered a relatively large range of Q, which does not give any issues for su ciently inclusive quantities, but it does when VBF cuts are applied. We conclude by pointing out that the uncovering of this issue by means of merging-scale systematics does not imply its most naive solution, which would be that of restricting, to small values, the range of Q in this kinematic region, thus relying on a matrix-element-dominated description: in fact, such a description is not necessarily better than a PSMC one in the context of the multi-scale dynamics induced by eq. (4.13) and (4.14).

In order to further the previous arguments, and as a way to validate the FxFx merging procedure with a special attention to cases where the construction of the Sudakovs that enter eqs. (2.139)(2.141) is involved owing to the avour structure of the hard process, we consider here three di erent nal states, namely e+e, ZZ, and He+e, which we simulate at the 8 TeV LHC. In FxFx we include the 0-, 1-, and 2-parton samples for the former process, and the 0- and 1-parton samples for the latter two. In all cases, the unmerged 0-parton results are also presented. All merged and unmerged samples are showered with HERWIG6. We start with the fully-inclusive rates, reported in table 13; the values of the three merging scales are as follows:

(()Q, (c)Q, ()Q) = (15, 25, 45) GeV pp e+e , (4.15)

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Figure 15. One-jet rate in e+e production. The main frame presents the three FxFx-merged predictions as well as the unmerged one. The lower inset displays the ratios of these curves over the central FxFx-merged one. The insets to the right show the separate contributions of the unphysical 0- (long-dashed), 1- (dashed), and 2-parton (dotted) samples, for the three merging scales.

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Figure 16. As in gure 15, for the two-jet rate in e+e production.

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Figure 17. As in gure 15, for the one-jet rate in ZZ production. Only the 0- and 1-parton samples have been considered here.

= (45, 65, 105) GeV pp ZZ , (4.16)

= (50, 75, 100) GeV pp He+e , (4.17) which cover quite wide ranges. The message emerging from the table is analogous to that relevant to Higgs production, which we have discussed previously: the merging-scale dependence is very small. Furthermore, the unmerged results are extremely close to the merged one, in fact much closer than in the case of gg H; this is a natural consequence of

the relatively small (compared to Higgs) K factors for the present processes. Again, this fact emerges naturally, without the need to impose any unitarity condition in the merging. We conclude by showing some selected di erential distributions, and in particular the j-jet rates of the kT algorithm (denoted by dj, see eq. (2.131)); these quantities are known to be critical in the context of merging procedures, since they are very sensitive to artefacts of the latter, which show up as discontinuities in the spectra. Our results are presented in gures 1518, where the main frames display the predictions in absolute values. The lower insets display the ratios of the FxFx-merged and unmerged predictions over the FxFx one obtained with the central merging-scale values. The insets at the right of the gures show the way in which the i-parton samples combine in order to give the physical curves. All results for all processes behave as expected.

4.3 One-loop SM and BSM results: a look ahead

All the results presented in sects. 4.1 and 4.2 exploit the one-loop computations performed by the public version of MadLoop5, which amply demonstrates the reach and exibility of this code. The aim of this section is on the other hand that of showing that the current, still-private MadLoop5 program has a much larger scope, being able to handle

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Figure 18. As in gure 15, for the one-jet rate in He+e production. Only the 0- and 1-parton samples have been considered here.

computations of very high complexity also in the context of a mixed-coupling expansion (see section 2.4), and in theories other than the SM. As was already discussed, while the corresponding capabilities on the real-emission side have not yet been fully validated in MadGraph5 aMC@NLO, they do not pose any problem of principle, and only minor ones from the technical point of view; therefore, the results presented below constitute the proof that the major obstacles have been cleared which prevent MadGraph5 aMC@NLO from performing NLO computations in arbitrary renormalisable theories.

We shall give here benchmark results for given phase-space congurations. They will be presented in the form of the coe cients cj, j = 2, 1, 0, introduced in eq. (2.88) for

V , the colour- and helicity-summed virtual amplitude contracted with the corresponding Born one (see eq. (2.55)). We shall also denote by a0 the Born amplitude squared:

a0 =

Xcolour spin

A(n,0)

1 10 100

10-5

2 , (4.18)

where, similarly to V , the averages over initial-state colour and spin degrees of freedom are understood. Throughout this section, we have set Q = F = R = s, and all particles widths equal to zero for simplicity; the leptons that circulate in the loops are taken to be massless. In order to maximize the numerical accuracy, the computations reported here have been performed by using quadruple-precision arithmetics; the stability tests described in section 2.4.2 have shown that these results are numerically stable beyond the seventeen digits quoted below. The coe cients cj are computed in the t

Hooft-Veltman scheme and, in the case of the pole residues c2 and c1, compared to their analytically-known forms (see e.g. eq. (B.2) of ref. [61], whose generalisation to cases

101

other than QCD is straighforward). Units for quantities of canonical dimension equal to one are understood to be GeV. The CKM matrix is diagonal. The integral-reduction OPP procedure has been adopted in all cases; processes C) and D) have also been computed with TIR (using IREGI), and perfect agreement with the OPP results has been found. We point out that processes A) and B) feature 8- and 7-point rank-5 loop integrals respectively, and are therefore beyond the present capabilities of any TIR library. Further technical details relevant to the calculations presented in this section are given in appendix C.2. We emphasise that the one-loop results shown here have never been presented in the literature, and can serve as benchmarks for comparisons with future independent computations.

A) High-multiplicity multi-scale QCD process: gg ! d dbbtt. This process involves up to 8-point loop diagrams with three external scales: the top- and bottom-quark masses, and the partonic c.m. energy. A total of 54614 loop diagrams contribute, and all pure-QCD UV renormalisation counterterms are necessary, which makes it an excellent test case for MadLoop5. The following parameters have been used:

Parameter value Parameter value S 0.118 nlf 4mt 173.0 mb 4.7

The kinematic conguration considered is (we use an (E, px, py, pz) notation):

pg = ( 500 , 0 , 0 , 500 )

pg = ( 500 , 0 , 0 , -500 )

pd = ( 159.884957663500 , -100.187853644511 , 83.9823400815702 , 92.0465111972672 )

p d = ( 203.546206153656 , -154.329441032052 , -0.512510195103158 , 132.714803257139 ) pb = ( 81.9036633616240 , 4.56741073895954 , -80.4386221767117 , 13.9601895942747 )

pb = ( 41.5312244194448 , 6.99982274816896 , 9.96034329509376 , 39.4277395334349 ) pt = ( 239.961310957973 , 84.0110736983121 , 18.3862699981019 , -142.325385396572 )

pt = ( 273.172637443802 , 158.938987491122 , -31.3778210029510 , -135.823858185543 )

For the O(6S) Born matrix element and O(7S) V coe cients we have obtained: gg d

dbbtta0 1.7614866952133752e-14 c0 7.1888721656398052e-14 c1 -3.8948541926529643e-15 c2 -2.8670389920110557e-15

The relevant MadGraph5 aMC@NLO-shell generation command is:MG5 aMC> generate g g > d d~ b b~ t t~ [virt=QCD]

More details on this syntax can be found in appendix B.1. After the generation phase, the usual output and launch commands are executed. We point out, however, than in the present context (i.e., when one computes one-loop matrix elements pointwise), launch allows the user to specify the kinematic conguration for which the said matrix elements are to be computed.

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B) Mixed-coupling expansion: u d ! d dW +ZH. The Born matrix elements for this process receive contributions at O(2S3), O(S4), and O(5), resulting from

O(g2Se3) and O(e5) amplitudes. In the notation of section 2.4, this corresponds to k0 = 5,

cs(k0) = 0, c(k0) = 3, and (k0) = 2. We have considered the full set of NLO corrections, thus obtaining four terms of O(nSm), with 0 n 3 and m = 6 n (see eqs. (2.22)

and (2.23)). This process features essentially all complications one faces in the case of a mixed QCD-EW expansion, and in particular it tests fully the UV- and R2-related machinery, described in section 2.4.2, beyond the pure-QCD cases considered so far. From a technical viewpoint, the major challenges are represented by the fact that genuine EW corrections (as opposed to QCD corrections to processes that may feature EW external particles) signicantly complicate the avour structure of the diagrams (whose number therefore grows and which always pose a multi-scale problem), and by the necessity of keeping separate track of the k0+1,q coe cients of eq. (2.23). We have performed our computation in the Feynman gauge, with an ((mZ), mZ, mW ) input scheme for the EW parameters, and adopted the (mZ) renormalisation scheme [355] (we point out that the G scheme [355, 356] is also available in MadLoop5). The full set of inputs is thus:

Parameter value Parameter valueS 0.118 nlf 5mt 173.0 yt 173.0 mW 80.419 mZ 91.188 mH 125.0 1 132.507

The kinematical conguration is:

pu = ( 500 , 0 , 0 , 500 )

p d = ( 500 , 0 , 0 , -500 ) pd = ( 77.3867935143263 , -13.6335837243927 , 33.7255664483738 , -68.3039338032245 )

p d = ( 251.029839835656 , -74.4940380485791 , -235.871950829717 , 42.7906718212678 ) pW+ = ( 139.739680522225 , -81.0565319364851 , -74.5408139008771 , 30.5527158347332 )

pZ = ( 382.164100735946 , 208.038848497860 , 298.200182616267 , -74.3682536477996 )

pH = ( 149.679585391847 , -38.8546947884028 , -21.5129843340470 , 69.3287997950232 )

The corresponding Born results are:

u d d

dW +ZH a0

O(2S3) 2.8791434190645365e-16

O(S4) -4.2378807039987007e-17

O(5) 5.8013051661550053e-18

where the O(S4) interference term happens to be negative, while at one loop we obtain:

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u d d

dW +ZH O(3S3) O(2S4)c0 -4.9670212643498834e-17 3.5197577360529166e-18 c1 -1.0437771535958436e-16 1.5619709675879874e-17 c2 -2.8837935481452971e-17 3.9757576347989499e-18

O(S5) O(6)c0 2.3220780285374270e-18 -1.4592469761033279e-18 c1 -1.8146075843176133e-18 -5.0799804067050324e-21 c2 -5.4147748433007504e-19 -5.4195415714279579e-21

The relevant MadGraph5 aMC@NLO-shell generation command is:MG5 aMC> generate u d~ > d d~ w+ z h QCD=99 QED=99 [virt=QCD QED]where the QCD=99 QED=99 bit instructs MadLoop5 to consider the corrections to all Born-level contributions, and not only to the leading (QCD) O(2S3) one, while the [virt=QCD

QED] syntax implies that both QCD and QED/EW corrections need to be included. More details on this extended syntax can be found in appendix B.1.

C) Mixed-coupling expansion: u ! d dtt. Although the one-loop corrections to this process feature a smaller number of diagrams than those relevant to u d d

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dW +ZH

studied above, owing to the fact that the corresponding Born amplitudes have three quark lines the mixed-coupling expansion ladder depicted in gure 1 is wider: we have in fact (k0) = 4, with k0 = 4, cs(k0) = 0, c(k0) = 0. The input-parameter and scheme choices are the same as those adopted in the case of u d d

dW +ZH production, while

the kinematic conguration is:

pu = ( 500 , 0 , 0 , 500 )

pu = ( 500 , 0 , 0 , -500 )

pd = ( 77.6887158960956 , -19.3895923374881 , 35.1636848900680 , -66.5063572263756 )

p d = ( 288.053156184158 , -91.1103505191485 , -264.895455921162 , 67.1112676698377 ) pt = ( 218.623451637725 , -92.8925122931906 , -85.7235692614867 , 43.4702707482150 )

pt = ( 415.634676282022 , 203.392455149827 , 315.455340292580 , -44.0751811916771 )

We thus obtain at the Born level:

u d

dtt a0

O(4s) 8.0443110796911884e-10

O(3s) -4.1964024114099949e-11

O(2s2) 3.2368049995513863e-11

O(s3) -7.9030872133243511e-13

O(4) 1.8667390802029741e-13

while the one-loop coe cients turn out to be:

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u d

dtt O(5s) O(4s)c0 2.7744575300036875e-10 -6.1309409133299879e-11 c1 -2.4891722473717473e-10 5.1973614496390480e-12 c2 -8.0573035150936874e-11 3.1296167547367972e-12

O(3s2) O(2s3)c0 1.2122291790182845e-11 -4.0611498141889722e-12 c1 -8.6161115635612362e-12 4.3209683736654367e-15 c2 -3.1860291930204890e-12 3.5961341456741816e-14

O(s4) O(5)c0 -3.8642357648130340e-14 -1.1866388556893426e-14 c1 -3.6050223887148020e-14 -4.7983631557836333e-16 c2 -1.7642824564621470e-14 -2.4912793041300221e-16

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The relevant MadGraph5 aMC@NLO-shell generation command is:

MG5 aMC> generate u u~ > d d~ t t~ QCD=99 QED=99 [virt=QCD QED]

D) A BSM case study: QCD corrections to gg ! ~t1~t1g. While MadLoop has been used to compute one-loop corrections to a very large number of processes in the SM, its applications to other theories have been pretty limited so far. Here, we present the rst MadLoop5 results in a fully-edged BSM model; namely, we compute QCD corrections to ~t1~t1g production in the MSSM. From the technical point of view, the MSSM UFO model at the NLO is immensely complicated, and its writing by hand (which has been the procedure adopted for the SM) is inconceivable: we have therefore obtained it by using a development version of FeynRules. All massive modes are subtracted at zero momentum, following the same strategy as e.g. in ref. [357]. It should be pointed out that some of the elementary expressions and structures of such a model (related e.g. to the presence of Majorana fermions) are not featured in any other UFO model employed so far. For this reason, we have double checked our MadLoop5 results against a completely independent calculation performed with Mathematica. This guarantees that the elementary building blocks are correct, thus rendering analogous tests more and more irrelevant in the future. We have chosen the input parameters as follows:

105

Parameter value Parameter valueS 0.118 nlf 4mb 4.75 mt 175mW 79.82901 mZ 91.1876 m~g 607.7137 tan 9.748624 m~u1 561.119 m~u2 549.2593 m~c1 561.119 m~c2 549.2593 m~t1 399.6685 m~t2 585.7858 m ~d1 568.4411 m ~d2 545.2285 m~s1 568.4411 m~s2 545.2285 m~b1 513.0652 m~b2 543.7267

with a diagonal squark-mixing matrix. By using the following kinematic conguration:

pg = ( 500 , 0 , 0 , 500 )

pg = ( 500 , 0 , 0 , -500 )

p~t1 = ( 465.457552338590 , 88.1561012782457 , 197.510478842819 , -100.667451003198 ) p~t

1 = ( 442.275748385439 , -9.53590501776566 , -180.889189039748 , 55.3271680251616 ) pg = ( 92.2666992759711 , -78.6201962604800 , -16.6212898030706 , 45.3402829780365 )

we obtain:

gg ~t1~t1ga0 2.839872059757065e-4 c0 -2.081163174420354e-5 c1 -1.550338075591894e-4 c2 -4.800024159745521e-5

The relevant MadGraph5 aMC@NLO-shell commands are:MG5 aMC> import model loop MSSMMG5 aMC> generate g g > t1 t1~ g [virt=QCD]

We point out that, similarly to the version of MadLoop5 used to derive the results presented in this section, the loop MSSM model used here is not yet public.

5 Conclusions and outlook

The motivation for pursuing this project stems from the fact that all aspects of the computations of tree-level and NLO cross sections, including their matching to parton shower simulations, are well understood, in a manner which is fully independent of the process under consideration. Therefore, the best way to make use of this understanding is that of the full automation of such computations. Automation has indeed already proven to be a very successful strategy for obtaining tree-level results, as documented by the theoretical and experimental activities based on, and spurred by, MadGraph. In

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this paper we have presented the successor of MadGraph5, a code that we have named MadGraph5 aMC@NLO, which extends the capabilities of the former by giving the user the possibility of computing NLO QCD corrections, if desired in association with parton-shower matching. MadGraph5 aMC@NLO is indeed the successor, and not just a plugin, of MadGraph5, its main virtue being that of treating tree-level and NLO QCD computations on the very same footing as far as the user is concerned, the di erence between them is a switch in input. In particular, irrespective of the perturbative order of the computation, MadGraph5 aMC@NLO features the following characteristics: very lean dependencies, simplicity of use, and exibility. The information that the user has to provide is only related to physics, such as values of masses, couplings, and scales, and the denitions of observables, as well as the hard process one wants to generate.

In the current public version of MadGraph5 aMC@NLO the inclusion of higher-order e ects is restricted to QCD corrections to SM processes. Such a limitation is mostly due to the fact that, in the context of one-loop computations (performed by MadLoop or by any other one-loop provider), one needs to take care of the UV renormalisation procedure and (typically, but not always necessarily) of that related to the so-called R2 counterterms, which are in any case just a simpler version of the former. Both procedures are expressed as a set of rules that can be worked out directly from the Lagrangian, an operation that has to be done only once for a given theory, and that so far has been performed by means of analytical computations. However, all obstacles preventing the automatic computation of the UV and R2 rules have now essentially been cleared, as hinted by the results we have presented in section 4.3. Given that all remaining obstacles are minor and of technical nature, this will allow MadGraph5 aMC@NLO to evaluate, in the near future, any type of NLO corrections, starting from the same user-dened Lagrangians that are used in tree-level calculations.

This work shows clearly that automated techniques at the NLO are well past the developmental phase, and are indeed fully established, as was already the case for their tree-level counterparts; we believe that this is amply demonstrated by the results of section 4. For the non-trivial cases now of relevance to collider phenomenology, automated computations are more robust, faster by orders of magnitude, and less error-prone than analytical, process-by-process traditional approaches. Furthermore, an increase in complexity generally only requires more CPU power but no conceptually new solutions, one example of such a situation being that of the computation of EW or QCD corrections to supersymmetric processes. As a counterexample, one may mention the calculation of cross sections that feature nal states with a very large number of QCD partons, for which dedicated optimisations (such as recursive relations and colour reorganisation, which are being investigated by us) will be needed in order to go beyond what is currently feasible by MadGraph5 aMC@NLO.

The ready availability in MadGraph5 aMC@NLO of perturbatively-accurate and realistic predictions for an extremely large range of processes of signicant complexity should be seen as both solving a few problems, and opening up new and exciting possibilities. Among the former, we would like to mention explicitly the fact that automated tools help free experts in perturbative calculations from spending their resources in the increasingly involved computations necessary to the experiments, thus allowing them to concentrate on obtaining other, cutting-edge results (such as, to limit oneself to perturbative QCD, cal-

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culations of NNLO accuracy with universal subtraction methods, improvements to parton showers, and so forth). As far as future possibilities are concerned, one important characteristic of MadGraph5 aMC@NLO to bear in mind is its modularity: we shall be happy to support and help those interested in improving parts of the code and the underpinning physics strategies (such as recursive relations, alternative matching and merging schemes, integral-reduction techniques and libraries). From a phenomenological viewpoint, many di erent applications of MadGraph5 aMC@NLO can be foreseen. The capability of the code to assess systematically and in an easy manner the theoretical uncertainties due to scales, PDFs, and matching and merging methods should be routinely exploited by both theorists and experimentalists. The fact of having a basically unlimited set of processes predicted at the NLO accuracy has two immediate consequences: it gives one the chance of extracting the PDFs by using a much wider set of observables than employed at present, at the same time possibly including EW and parton-shower e ects; and that of nally achieving PSMC tunings that properly include NLO results. Exploratory studies at future colliders, such as circular or linear e+e ones, or very-high-energy hadron machines, can also be performed without the need of a dedicated e ort. Finally, it is hard to predict the kind of applications that will be relevant to BSM physics. In any case, extending the current exibility of MadGraph5 aMC@NLO for SM processes to new-physics models, be they renormalisable or e ective, and thus being able to readily investigate the implications of any theory, will certainly be crucial in current and future analyses.

Acknowledgments

We are grateful to Roberto Pittau for his support of CutTools, to Stefan Prestel for his dedicated help with the NLO matching to Pythia8 and for discussions on related topics, to Matteo Cacciari for his help and support with FastJet (core), to Andreas Papaefstathiou for his dedicated help with Herwig++, to Simon Badger, Daniel Maitre, and Markus Schulze for clarications concerning bootstrap and D-dimensional unitarity methods, to Markus Cristinziani and Roberto Chierici for information on top-physics measurements by ATLAS and CMS, to Adam Alloul, Neil Christensen, Celine Degrande, Claude Duhr and Benjamin Fuks for their collaboration to BSM developments and support of FeynRules, to Frank Krauss for explanations on the use of the MC@NLO method in SHERPA, to Simon de Visscher for his help with LO merging, to Pierre Artoisenet and Robbert Rietkerk for their work on MadSpin, to Valery Yundin and Andreas van Hameren for their support of PJFry++ and OneLoop, to Diogo Buarque Franzosi for his collaboration on the implementation of the complex mass scheme, to Alexis Kalogeropoulos for his contributions in developing and testing large-scale event production, to Kentarou Mawatari for many fruitful collaborations, to Stefano Pozzorini, Peter Skands and Torbjrn Sjstrand for useful clarications, to Josh Bendavid, Vitaliano Ciulli, and Sanjay Padhi for the very useful feedback on the use of the code, and to Michelangelo Mangano for his support and insight. We are thankful to our cluster management team, Jerme de Favereau, Pavel Demin, and Larry Nelson, for their help and collaboration over the years. FM and TS are particularly thankful to Kaoru Hagiwara for the constant support and many

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insightful suggestions since the very beginning of the MadGraph project. SF wishes to thank Bryan Webber for his collaboration, over the course of many years, on several of the topics discussed in this paper. The authors not based at CERN are grateful to the PH/TH Unit for the support and hospitality in many di erent occasions. This work has been supported in part by and performed in the framework of the ERC grant 291377 LHCtheory: theoretical predictions and analyses of LHC physics: advancing the precision frontier, supported in part by the Swiss National Science Foundation (SNF) under contract 200020-149517, by the European Commission through the LHCPhenoNet Initial Training Network PITN-GA-2010-264564, by the Research Executive Agency (REA) of the European Union under Grant Agreement number PITN-GA-2012-315877 (MCNet). The work of FM and OM is supported by the IISN MadGraph convention4.4511.10, by the IISN Fundamental interactions convention 4.4517.08, and in part by the Belgian Federal Science Policy O ce through the Interuniversity Attraction Pole P7/37. OM is Chercheur scientique logistique postdoctoral F.R.S.-FNRS, and wishes to thank the IPPP Durham for the kind hospitality extended to him during his stay there. The work of FM has been supported by the Francqui Foundation via a Francqui Research Professorship. The work of MZ has been partly supported by the ILP LABEX (ANR-10-LABX-63), in turn supported by French state funds managed by the ANR within the Investissements dAvenir programme under reference ANR-11-IDEX-0004-02. VH is supported by the SNF with grant PBELP2 146525. This material is based upon work supported in part by the US National Science Foundation under Grant No. PHY-1068326.

A Technical prerequisites, setup, and structure

MadGraph5 aMC@NLO is being developed and is routinely run on a variety of Linux platforms and on Mac OS-X systems. The basic requirements for running the code are the following:

A bash shell;

perl 5.8 or higher;

Python 2.6 or higher, but lower than 3.0;

gfortran/gcc 4.6 or higher; any other modern Fortran/C++ compiler should work,

provided it supports computations in quadruple precision.

After downloading the tarball, and upacking it in what will be called the main directory (which will contain several sub-directories, such as aloha, apidoc, bin, and so forth), the code is ready to run. No installation of external packages is mandatory, thanks to the fact that the tarball includes copies of the third-party codes listed in appendix D.

Setup. From a terminal shell in the main directory, type:./bin/mg5 aMC

At this point, one has entered the MadGraph5 aMC@NLO shell, which is made evident by the fact that the prompt now reads as follows:

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MG5 aMC>

A minimal setup phase may take place here, before generating and running the rst process. This phase, that must be done at most once (i.e., it does not have to be repeated before the generation of any process after the rst), consists essentially in dening conguration variables. For example, in the case where a local installation of FastJet were available,58 the path to it must be known by MadGraph5 aMC@NLO: this is achieved by executing the following command:MG5 aMC> set fastjet /<PATH TO FASTJET>/bin/fastjet-configLikewise, for the local installation of LHAPDF [366] to be found, one should execute the command:MG5 aMC> set lhapdf /<PATH TO LHAPDF>/bin/lhapdf-configEach of these commands associates the given value with a variable in the le: input/mg5 configuration.txtThe user can nd a list of all possible conguration variables by visiting that le (or by auto-completion with the <TAB> key after typing set in the MadGraph5 aMC@NLO shell). We point out that each of these variables can be directly edited in the le, as an alternative to executing the set command as shown above. More advanced setup options are described in appendix B.1.

Structure. The various subdirectories of the main directory will be of no interest to the regular user. The only possible exceptions areTemplate/NLOTemplate/LOa copy (with minor di erences) of which is created in the subdirectoryMYPROCof the main directory upon executing the command:MG5 aMC> output MYPROCafter having executed one of the two following commands:MG5 aMC> generate a b > c 1...c n [QCD]

MG5 aMC> generate a b > c 1...c nfor the NLO and LO case respectively (see section 3). The minor di erences in the copy alluded to before are due to the fact that, after the generate command has been issued, the program knows e.g. the number of nal-state particles (equal to n in the examples given here), which is thus explicitly written in some include les in the directory tree of MYPROC. In any case, these include les and their analogues must not be modied by the user.

The directory MYPROC will be called the current-process directory, and all the operations relevant to the process whose generation gave rise to it are performed somewhere in its directory tree. Such operations can roughly be arranged in two classes: input-type, to be performed by the user before the launch command, and output-type, performed by MadGraph5 aMC@NLO after the launch command.

The user may consider input-type operations in the following subdirectory: MYPROC/Cards

58We point out that FastJet (core) is part of the MadGraph5 aMC@NLO tarball.

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which contains the input cards (these are in plain text format, and liberally commented) that steer the MadGraph5 aMC@NLO run (e.g., run card.dat) or control the physical parameters of the theory (e.g., param card.dat). Contrary to other input-type operations, which will be mentioned below, the values of the entries in the input cards can not only be modied by the user by visiting the appropriate cards before the launch command, but can also be accessed after the launch command through an interactive talk-to within the MadGraph5 aMC@NLO shell. Both accessing modes can be used in the same run; note that the values of the inputs used in the actual run will be those stored in the input cards at the end of the talk-to phase.

Further input-type operations are specic to NLO-type generations, require a minimal knowledge of Fortran, and must be completed before the launch command is issued. They will involve editing les in the following subdirectories: MYPROC/FixedOrderAnalysisMYPROC/MCatNLOMYPROC/SubProcessesThe FixedOrderAnalysis subdirectory will need to contain the users xed-order analysis le(s) relevant to fLO or fNLO runs. The MCatNLO subdirectory is used only in the case when the user chooses to steer the shower phase within the MadGraph5 aMC@NLO framework; when the LHE les produced by MadGraph5 aMC@NLO are showered externally, such a subdirectory is ignored. If also the showering is steered by MadGraph5 aMC@NLO, the user will be able to access the drivers of the various event generators, and to write his/her own analysis inside the MCatNLO directory tree (whose structure is analogous to that of the MC@NLO package, for those familiar with it). Finally, the subdirectory SubProcesses contains the codes necessary for the computation of the cross section proper, and specic to the process that has been constructed in the generation step. Typically, the user will not need to modify any of these les; exceptions are those of setscales.f, where one denes the functional forms used for dynamic-scale computations (see section B.2), and of cuts.f, where one sets any desired parton-level cuts (on top of those accessible through run card.dat); both of these les are amply commented.

As far as output-type operations are concerned, these are by denition dealt with by MadGraph5 aMC@NLO; we shall mainly describe in what follows those relevant to an NLO-type generation. The relevant directories areMYPROC/Events/run *There will be as many run * subdirectories as number of runs;59 the string * will feature a run-identication number. These subdirectories will contain the nal outputs of the corresponding MadGraph5 aMC@NLO runs, provided that such outputs are in one of the formats recognised by the code. In particular, one will have:

1. For all types of runs: various plain-text les that summarise the inputs used and the results of the integration of the short-distance cross sections.

59Runs may e.g. di er by choices of input parameters, or type of physics simulated, such as f(N)LO vs(N)LO+PS, or the inclusion of spin correlations as predicted by MadSpin vs stable-particle production.

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2. In (N)LO+PS runs: the Les Houches event le(s) that contain hard-subprocess un-weighted events which are to be showered.

3. In (N)LO+PS runs when MadGraph5 aMC@NLO is used to steer the shower phase: the nal results after shower, provided that these are either: a) an StdHEP/HepMC le that contains the event records; or b) a topdrawer le that contains histograms dened by the user in his/her analysis.

4. In f(N)LO runs: the histograms dened by the user in his/her analysis, provided that their format be either Root or topdrawer.

There is no problem if the format of the output of the users analysis relevant to the shower phase (when MadGraph5 aMC@NLO is used to steer the shower) is not compliant with either 3.a or 3.b. Simply, such an output will not be moved into MYPROC/Events/run *, but will be kept in the directory where the PSMC run has actually been performed. This directory will be named:MYPROC/MCatNLO/RUN MCTYPE nnwith MCTYPE=PYTHIA8 and so forth (depending on the PSMC used), and nn an integer number increased by one unity for each new PSMC run.

On the other hand, the use of formats other than Root or topdrawer in f(N)LO runs is deprecated, since it implies some manual operations and the writing of code by the user. The reason is the following: cross sections are integrated by MadGraph5 aMC@NLO through multi-channel techniques this ensures optimal convergence and high degree of parallelisation, but each channel is non-physical (only their sum is). Analysis routines (regardless of the output format) are used by individual channels; hence, their outputs are to be summed.60 Summing the results of the individual channels is performed automatically by MadGraph5 aMC@NLO for Root and topdrawer outputs, via dedicated auxiliary codes; any new format would thus require the user to write a new such code, and a script that nds all single-channel outputs and feed them to his/her summing code.

We conclude this section by stressing again that the source codes which are compiled after executing the command launch, and the input cards used throughout the run, are those in the current-process directory tree, and not in the Template directory tree. Therefore, any modications to les in the Template directory tree will have no e ect on the current run. However, they will a ect all subsequent process generations, since as claried at the beginning of this section it is the les in Template that form the core of the contents of each current-process directory. Hence, this procedure is reserved to the very experienced users, and we strongly deprecate it.

B Advanced usage

This section reports on some of the features of MadGraph5 aMC@NLO whose understanding allows the user to exploit the full physics potential of the code. It is not meant to be

60This assumes the output is a set of histograms. In case of n-tuples, these will need to be combined, possibly after having rescaled their weights.

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a usage manual, but only to briey expand on some of the subjects which have been only touched upon in the main text.

B.1 Models, basic commands, and extended options

We start with the following general comment: within the MadGraph5 aMC@NLO shell, the <TAB> key plays the same role as in a normal terminal shell: when hitting it, a list of possible completions (e.g., commands relevant to the current context, or completion of the command syntax) is printed on the screen. Note that the shell commands help and tutorial can be used as well, and will provide the user with some minimal guidance.

Models. As was explained in section 2.1, MadGraph5 aMC@NLO needs a model in order to generate a process. When one enters the MadGraph5 aMC@NLO shell, the default is that of assuming the SM: however one can choose to work in another theory by loading a new model, by simply executing the command:MG5 aMC> import model ModelNamewhere ModelName is the name of the desired model. The list of available models (to which as usual the user can add his/her own) can be obtained by hitting the <TAB> key after import model. Each of these models is associated with a directory (under the main directory):models/ModelClassIn the directory ModelClass, one collects the denition of all those models which are tightly connected with each other, for example by having the same Lagrangian and di ering by the choice of some fundamental parameter. To give an explicit example: the default model for the SM in MadGraph5 aMC@NLO assumes the charm quark to be massless, but there is a model where the charm quark is massive. For both of these, we have: models/ModelClass models/sm

The massless-charm or massive-charm SM is explicitly loaded by typing:MG5 aMC> import model smMG5 aMC> import model sm-c massrespectively. Technically, these two commands are in one-to-one correspondence with the two les:models/sm/restrict default.datmodels/sm/restrict c mass.datThe user interested in some non-extensive modication of the SM can thus simply create his/her own le models/sm/restrict XXX.dat, which may be eventually loaded by executing import model sm-XXX. For more details on these matters, please visit: https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/Models/USERMOD

Web End =https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/Models/USERMOD .

We nally stress again that not all models support the computation of NLO corrections. In the rst public version of MadGraph5 aMC@NLO such corrections are restricted to QCD to SM processes. The relevant models are all found in the directory: models/loop smNote that, by default, when performing an NLO-type generation (i.e., by using the [QCD] keyword) the code switches automatically from the LO-type model to the corresponding

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NLO-type one (i.e., in the SM it switches from sm to loop sm). If the latter is not available, a warning is issued and the code proceeds no further.

Setup. A short description of the setup procedure has been already given at the beginning of appendix A. Here, we wish to point out that, on top of the environment variables found in input/mg5 configuration.txt, the user can also control other options, which for example a ect the physics schemes used by MadGraph5 aMC@NLO during the various computations. All such options can be listed with the following command:MG5 aMC> display optionswhich will display all options and their current values. In order to change the latter, one executes the set command, whose general syntax is:MG5 aMC> set Option ValueFor example, MadGraph5 aMC@NLO by default does not use the complex mass scheme in its computations. In order to change this, it is su cient to execute:MG5 aMC> set complex mass scheme TrueOther examples relevant to environment variables have already been given in appendix A.

The operation of computing the widths of the unstable particles present in the imported model can be seen as part of the setup procedure, being a complement to the model itself, independent of the generation procedure, and mandatorily performed before the running phase (see section 2.3.2). Such an operation is carried out by issuing the shell command: MG5 aMC> compute widths [{Options}]

which in turn executes the MadWidth module; the possible options of the above command can be as usual explored by hitting the <TAB> key. We remind the reader that MadWidth works at tree level and in the narrow-width approximation (in other words, the manual setting of widths in the context of an NLO simulation may be necessary).

Finally, another setup-type operation is the diagonalisation of the mass matrix. This is only available in a restricted class of UFO models which include the AsperGe [89] module, whose inputs are accessible to the user during the interactive talk-to phase.

Generation. The most general form of the generate command is the following: MG5 aMC> generate Process {AmpOrders} [{{Mode} Couplings}]

the only mandatory option being Process, i.e. the actual process one needs to generate. We shall now comment on the four options above in turn.

The option Process is simply the list of initial- and nal-state particles, separated by the conventional > sign:

Process a b > c 1...c n

One can further rene the syntax above in order to include in the computation only some of the contributions that one would normally obtain. Such renements are reported in table 14, and have the following meaning:

s.1 A production process is generated that features x in the nal state, with x subsequently decaying into the list of particles that follow the x > string; more in general, there may be p primary particles that play the same role as x. Only p-resonant diagrams (see section 2.5) are included in the computation. In the example of table 14,

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syntax example meaning x, x> p p > z j, z > b b~ s.1$ x p p > e+ e- $ z s.2/ x p p > e+ e- / z s.3> x > p p > z > e+ e- s.4$$ x p p > e+ e- $$ z s.5

Table 14. Process-generation syntax renements, also exemplied in the case of various processes that involve a Z boson. See the text for the explanation of the keywords s.1 s.5. Only syntaxess.3 s.5 are supported for NLO-type generations.

one has the associated production of a Z and a jet, with the Z further decayed into a bb pair. Spin correlations and x o -shell e ects are taken into account exactly, but the virtuality mx of x is forced to be in the following range:

|mx mx| bwcutoff x , (B.1) where mx is the pole mass of x, x its width, and bwcutoff is a parameter controlled by the user (through run card.dat). Syntax s.1 thus loosely imposes an on-shell condition; it is called decay-chain syntax, and can be iterated: any decay product can be decayed itself by using this syntax (e.g. x > y z, y > w s).

s.2 If x appears as an intermediate particle in the generated process, its virtuality is forced to be in the range:

|mx mx| > bwcutoff x , (B.2) which is the region complementary to that of eq. (B.1), and thus loosely imposes an o -shell condition. All diagrams are kept. In the example of table 14, one has Drell-Yan production with the invariant mass of the e+e pair larger than or smaller than the Z mass by at least bwcutoff Z. A consequence of the complementarity mentioned above is that, while cross sections generated with either s.1 or s.2 are bwcutoff-dependent, their sum is not (up to interference terms, which are neglected by the process of discarding non-resonant diagrams in s.1 ), and corresponds to the process generated with the simplest syntax. For example:

ddO (p p > z)

for any observable O.

s.3 All diagrams that feature (anywhere) the particle x are discarded.

s.4 The process is generated by demanding that at least one particle of type x be in an s-channel.

s.5 All diagrams that feature the particle x in an s-channel are discarded.

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ddO (p p > z, z > e + e) +

ddO (p p > e + e $ z) , (B.3)

We stress that all syntaxes but s.2 produce in general results which are non physical, because gauge invariance might be violated (although there are exceptions: see e.g. ref. [214]), and have therefore to be used with extreme caution. The situation becomes more involved at the NLO, so even more care is required. Syntaxes s.3, s.4, and s.5 are supported; more rened selections of individual loop diagrams in MadLoop5 can be imposed by editing the function user filter in the source code loop diagram generation.py.

The option AmpOrders allows the user to specify the upper bounds on the powers of the coupling constants that enter the scattering amplitudes (i.e., not the amplitudes squared); therefore, the coe cients k0+p,q are selected only in an indirect manner

see section 2.4. Furthermore, in the case of an NLO-type generation, such amplitudes are the Born ones: the couplings of the one-loop and real-emission amplitudes are then automatically determined according to the type of corrections to be included. The syntax for this option is the following:

AmpOrders coupling1 = p1 coupling2 = p2 . . . couplingn = pn

where couplingi is the name of the ith coupling in the model currently used, and p1 is an integer which represents the upper bound mentioned above. It should be obvious that couplingi is an arbitrary name, chosen by the author of the model in use. In order to see the list of all coupling names, one has just to type (after having imported the model):

MG5 aMC> display coupling orderFor example, by executing this command in the context of the SM, one obtains what follows:QCD : weight = 1QED : weight = 2which implies that the internal names of the SM couplings gS and gW is QCD and QED respectively, with the latter being hierarchically suppressed w.r.t. the former. As an explicit example of the use of the AmpOrders option in the SM, let us consider the case of:

p + p W + + Jb + Jlight , (B.4)

where Jb and Jlight are a b- and a light-jet respectively, and the ve-avour scheme is adopted. Numerically, the dominant contributions to such a process are due to diagrams whose corresponding amplitudes factorise the couplings g2SgW . However, amplitudes of order g3W are more interesting, since they feature diagrams with one top-propagator exchange, and are thus identied with single-top production (although of course at the same order one has non-resonant diagrams as well). By executing the command (after including the b-quark in the proton)

MG5 aMC> generate p p > w+ b jone would obtain only the amplitudes of O(g2SgW ), the choice being made by the code auto

matically according to the hierarchy shown above. In order to study single-top production, one can execute what follows instead [214]:MG5 aMC> generate p p > w+ b j QED=3 QCD=0which will force the code to consider only O(g3W ) amplitudes. Note that, by entering

QED=3 QCD=2, one will generate both O(g2SgW ) and O(g3W ) amplitudes. In future version of

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MadGraph5 aMC@NLO, the syntax of the option AmpOrders will be extended, so as to give the user the possibility of selecting directly cross-section-level quantities ( k0+p,qs), at both the leading and the next-to-leading order.

The option Mode61 allows the user to select which contributions to an NLO cross section (on top of that due to the Born, which is always present) will be included in the computation. The possible settings are the following:

Mode all= = both one-loop and FKS-subtracted real-emission

Mode real= = only FKS-subtracted real-emission

Mode virt= = only one-loop

The setting all= is equivalent to omitting the option Mode altogether, and should be the only one considered by the non-expert user, being the only one that leads to physical results. The setting real= instructs MadGraph5 aMC@NLO to not generate the part of the code relevant to virtual matrix elements with MadLoop. In this way, the cross section can still be dealt with as explained in section 3, but the results will be non-physical, unless an external one-loop provider is linked to MadGraph5 aMC@NLO (such an external OLP will thus e ectively play the same role as MadLoop). Finally, the setting virt= corresponds to the MadLoop standalone mode. In such a mode, the commands output and launch will behave di erently w.r.t. what is described in section 3, the idea being that of using the code so generated in order to obtain the pole residues and nite part of the virtual corrections for user-dened kinematic congurations see section 4.3 for explicit examples.

The option Couplings allows the user to specify which kind of NLO corrections Mad-Graph5 aMC@NLO will compute. The general syntax for this option is the following:

Couplings coupling1 coupling2 . . . couplingn

However, in the current version only QCD corrections can be computed, and therefore the only valid option read as follows:

Couplings QCD

as already mentioned several times in this paper. For examples of the more general syntax that can be used in a still-private MadGraph5 aMC@NLO version, see section 4.3.

Output. The most general form of the output command is the following:MG5 aMC> output [OutputForm] [MYPROC]

As was already mentioned in section 3 (see in particular footnote 42) the target-directory name MYPROC may be omitted, in which case MadGraph5 aMC@NLO will choose automatically a name (and print it out on the screen for the user to know). As the full syntax above implies, however, there are a few names that are reserved because, if used, the code will interpret them as one of the (optional) OutputForm keywords. These essentially serve to create executables (or standalone libraries) which are not those typically used for the integration of the cross sections and the unweighting of the events. They are described in appendix B.6.

Running. The most general form of the launch command is the following: MG5 aMC> launch {ProcDir} {RunMode} {Options}

61The use of which requires the use of the option Couplings as well. The opposite is not true.

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We cannot possibly describe in this paper all possibilities implied by the syntax above, which we plan to do in a forthcoming user manual; we urge the interested reader to explore them by either using the on-line tutorial, or executing the help launch command, or exploiting the <TAB> key.

Here, we limit ourselves to point out that ProcDir, if present, must coincide with one of the current-process directories previously generated. The implication is that, after the generation and output phase, a user may not immediately run the process, but rather generate a second one, or also quit the MadGraph5 aMC@NLO shell. Being saved on disk, the full information of a generated and outputed process can be retrieved at any later time. In order to do this, one needs simply to execute:MG5 aMC> launch MYPROC -iwhere MYPROC is the name of the current-process directory used throughout this paper. It should be clear that, upon executing the command above, one is again dealing with a specic current process. For this reason, many early-stage commands (such as generate) are disabled. In order to help the user remind this fact, the prompt itself is actually changed, and reads:MYPROC>

In short, we shall call the environment accessed by executing the launch -i command the running mode. One can re-enter the running mode of a given process an unlimited amount of times. Again, we shall not attempt to give here a full description of the various implications of this fact. However, it is interesting to discuss a specic usage in connection to what has been described in section 3.2.1, and namely how to shower hard-subprocess event les previously generated. For example, we have discussed in section 3.2.1 how, in the case of a process which features particles whose decay products are dealt with MadSpin, one obtains (at least) one undecayed and one decayed hard-event les, for example: MYPROC/Events/run 01/events.lhe.gzMYPROC/Events/run 01 decayed 1/events.lhe.gzonly the latter of which is showered by default when MadGraph5 aMC@NLO steers the shower. In order to shower the former, one simply has to execute:MG5 aMC> launch MYPROC -iMYPROC> shower run 01Namely, to access the running mode of the relevant process directory, and then execute the command shower followed by the name of the subdirectory of the Events directory where the events to be showered are stored. We point out that the shower command can be executed any number of times with the same argument, e.g. if one desires to change the seeds or the parameters in the PSMC.

B.2 Setting the hard scales at the NLO

The short distance NLO(+PS) cross sections in MadGraph5 aMC@NLO feature three hard scales: the renormalisation (R), factorisation (F ), and Ellis-Sexton (QES) scales. The latter appears only at the NLO and, at variance with the former two, its variations do not induce any changes in the cross sections. For this reason, QES is only useful in the context of validation studies, performed by developers; regular users are recommended to

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always set it equal to the factorisation scale. In view of the standard way of performing scale variations, it is convenient to write the hard scales as follows:

R = fRR0 , F = fF F0 , QES = fESQES0, (B.5)

where R0, F0, and QES0 are called reference scales.

Hard scales must be set by the user prior to compiling and running the code. They can be organised into two categories, typically called xed and dynamical; the scales belonging to the former have constant values, independent of the event kinematics, while those belonging to the latter depend on the four-momenta of the nal-state particles, and hence have values that change event-by-event during the course of the run. Whether a scale is xed or dynamical is determined by an input parameter. For example, in the case of R, the following setting in run card.dat:

T = fixed ren scale ! if .true. use fixed ren scalewill instruct the code to use a xed renormalisation scale. Such a xed value is also given by the user in input by setting the pre-factor fR and the reference scale R0 that appear in eq. (B.5). For example, the following entries in run card.dat:91.188 = muR ref fixed ! fixed ren reference scale2 = muR over ref ! ratio of current muR over reference muR set R0 = 91.188 GeV and fR = 2 respectively, and hence R = 182.376 GeV.

We now address the case of dynamical scales, again using the renormalisation scale to give denite examples. A dynamical R is used when:

F = fixed ren scale ! if .true. use fixed ren scaleWith this setting, one still has fR =muR over ref. On the other hand, the input muR ref fixed is ignored, and the reference scale is dened as follows:

R0 = muR ref dynamic . (B.6)

The quantity muR ref dynamic is a function of the four-momenta of the nal-state particles; its body is found in the le MYPROC/SubProcesses/setscales.f, where the user may include his/her denition of the dynamical scale best suited to the process of interest. Although a few examples of typical dynamical scales are given in the version of setscales.f which is included in the tarball of the package, we urge the user to study the structure of that le (which is amply commented note, in particular, the role of the variable temp scale id in that le, which helps keep track of the functional forms used), and to change it if need be. It should be clear that setscales.f follows the same rules as all the other core les of the package (see section A). Hence, in order for any modications to it to be taken into account in the current run, one must edit the le before the launch command is issued.

What said above for the renormalisation scale applies without changes to the factorisation and Ellis-Sexton scales; the relevant parameters and functions in run card.dat and setscales.f have self-explanatory names. Note that, in the case of the factorisation scale, one can in principle assign two di erent values to the two incoming hadrons hence, in the input cards one can nd the variables * ref fixed and * over ref, with *=muF1 and

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muF2, which are respectively relevant to the hadron coming from the left and the right. At the LO, this is unambiguous the scales only enter the relevant PDFs. At the NLO, however, there are several ways to write the logarithmic terms whose arguments are ratios of scales. In order to avoid complications, at the NLO the two factorisation scales must always be chosen to be equal; when this is not the case, the code stops.

We point out that the structure outlined above will allow the user to choose one scale to be xed and another one to be dynamical; however, in the vast majority of cases one will want the scales to be either all xed (not necessarily to the same value), or all dynamical. In the latter case, the functions in setscales.f are such that di erent functional forms can be adopted for di erent scales. Again, this is a somehow infrequent situation. Having this in mind, the default version of setscales.f sets all * ref dynamic functions (with *=muR, muF, and QES) equal to the same function scale global reference, and one can limit oneself to modifying the latter for a standard usage.

We conclude this section by summarising schematically what has been described above.

1. Decide whether to use xed (fixed * scale=T) or dynamical (fixed * scale=F) scales. This is done at runtime, either by editing run card.dat before executing the launch command, or directly at the prompt after having executed it.

2a. If xed scales are chosen: the relevant input parameters are * over ref (these are the f factors in eq. (B.5)) and * ref fixed (these are the reference scales 0 in eq. (B.5)). Both can be set at runtime in the same way as fixed * scale.

2b. If dynamical scales are chosen: the relevant quantities are the input parameters * over ref as in case 2a., and the functions responsible for dening the reference scales, to be found in setscales.f. Modications to the latter le must be carried out before executing the launch command (i.e., no modications are possible at runtime); by default, the reference dynamical scales are set equal to HT /2.

Note that the situation of FxFx-merged simulations is somewhat di erent, owing to the specic prescriptions for the settings of the scales which are inherent to the method; for more details, the user is encouraged to check http://amcatnlo.cern.ch/FxFx_merging.htm

Web End =http://amcatnlo.cern.ch/FxFx merging.htm .

B.3 Scale and PDF uncertainties: the NLO case

Among all the dependencies of a cross section, those due to hard scales and PDFs are special, since it is always possible to write

Xiwibi , (B.7)

where the coe cients wi (typically called weights) are independent of both scales and PDFs, while the basis members bi contain all the information on scales and PDFs in simple forms such as:bi = f(i)H1f(i)H2kiS

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n1, log R

QES , logFQES

. (B.8)

The key point is that, while the weights might be very expensive to compute CPU-wise, the basis members are straightforward. A convenient strategy is therefore that of rst evaluating the wis, and then of using them in eq. (B.7) for all the desired choices of scales and PDFs; each of these will therefore result in a basically instantaneous evaluation of the corresponding . In practice, what is done in MadGraph5 aMC@NLO is to compute for a given choice of scales and PDFs (the central or default choice), while at the same time storing the values of the wis (if this is required by the user in input see later), in order to re-use them at a later stage, typically for the assessment of the theoretical uncertainties.

We stress that eq. (B.7) is exact,62 and therefore so is the computation of a cross section starting from the weights for any given scale and PDF choice. A complete discussion, which includes all the relevant denitions of wi and bi for both xed-order and MC@NLO cross sections, is given in ref. [125] and will not be repeated here. The aim of this appendix is rather that of giving some details on the way in which eq. (B.7) is exploited when computing scale and PDF uncertainties in the context of (N)LO+PS and f(N)LO simulations.

The basic idea relevant to (N)LO+PS is the following. In the LHE le and event-by-event, the values of will be stored that correspond to all the combinations of scales and PDFs selected by the user in input (we denote the numbers of these combinations by N

and NPDF respectively). These s will have to be treated in the same way as the cross section associated with the central scales and PDFs (which is part of the standard LHE information, and corresponds to the variable XWGTUP); namely, each of them will constitute an entry in a histogram associated with that particular combination of scales and PDFs. In other words, for each observable of interest, one will have to ll 1 + N + NPDF histograms (the 1 being for the central choices). At the end of the run, and for each observable, the envelope of the 1 + N histograms will give the scale uncertainty a ecting that observable, and the envelope of the 1 + NPDF histograms will be the PDF uncertainty. The precise denition of these two envelopes is the users responsibility. In the case of the scales, one will typically want to consider the largest and smallest cross sections bin-by-bin, possibly excluding from the computation of such extremes some of the (R, F ) combinations (see e.g. [367] for a discussion on this point). In the case of the PDFs, the envelope must be dened following the prescription of the PDF authors.

We now show how the user can choose in input the scales and PDFs that will be used in the calculation of the uncertainties; we start from the former. In the present version of MadGraph5 aMC@NLO we have xed N = 8, which corresponds to the following combinations:

(fR, fF) , (fcR, fF) , (fR, fF) , (B.9)

(fR, fcF) , (fR, fcF) , (B.10)

(fR, fF) , (fcR, fF) , (fR, fF) , (B.11)

these being the pre-factors introduced in eq. (B.5) to dene the renormalisation and fac-

62The dependence on PDFs of the (parton-shower) Sudakovs cannot be accounted for by eq. (B.7). However, this is expected to be rather small, and particularly so when computing PDF uncertainties. See ref. [125] for more details.

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torisation scales given the reference scales:

R, F =

, {, } {, c, } . (B.12)

The combination (fcR, fcF) missing in eqs. (B.9)(B.11) is obviously that corresponding to the central scales, which is always computed. Equation (B.12) implies that scale variations are dened by choosing the reference renormalisation and factorisation scales (which is done as explained in section B.2), and by varying the pre-factors in front of them. Such prefactors are dened by means of some input parameters found in run card.dat. More specically, we have:

fR = muR over ref rw Rscale down , (B.13)

fcR = muR over ref , (B.14) fR = muR over ref rw Rscale up , (B.15)

fF = muF over ref rw Fscale down , (B.16)

fcF = muF over ref , (B.17) fF = muF over ref rw Fscale up . (B.18)

Given these inputs, MadGraph5 aMC@NLO will consider all the combinations given in eqs. (B.9)(B.11), compute the corresponding bis of eq. (B.8) using the central PDFs, and combine them with the weights according to eq. (B.7). The resulting (which could be denoted by (,) for consistency with eq. (B.12)) will be stored in the LHE le, provided the user sets:.true. = reweight scale ! reweight to get scale dependencein input.

As far as PDF uncertainties are concerned, we have assumed that one will use LHAPDF, where all members of an error set are identied by adjacent integer numbers. Using NNPDF 2.0 [368] as an example to be denite, one will have the following inputs: lhapdf = pdlabel ! PDF set90800 = lhaid ! if pdlabel=lhapdf, this is the lhapdf number90801 = PDF set min ! First of the error PDF sets90900 = PDF set max ! Last of the error PDF setsThe rst two lines instruct the code to use LHAPDF and to choose the central NNPDF2.0 set as the default. The last two lines will set NPDF = 100; for each of these hundred

NNPDF 2.0 error sets, MadGraph5 aMC@NLO will compute the bis of eq. (B.8) using the central scales, and combine them with the weights according to eq. (B.7). The resulting will be stored in the LHE le, provided the user sets:

.true. = reweight PDF ! reweight to get PDF uncertainty in input.

For what concerns the storage in the LHE le of the s computed as described above, we use a format which is fully compatible with the LHA v2.0 [95], and which is now o cially adopted as v3.0 [218]. In the le header, there will be a part (which we call the reweight

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section) where a description is given of the meaning of the weights that will appear in each event. Its structure may read as follows:<initrwgt><weight id=1> This is the central weight </weight><weightgroup type=scale variation combine=envelope><weight id=2> muR=0.5 muF=0.5 </weight><weight id=3> muR=0.5 muF=1.0 </weight>...<weight id=9> muR=2.0 muF=2.0 </weight></weightgroup><weightgroup type=NNPDF20 combine=gaussian><weight id=10> set001 </weight>...<weight id=109> set100 </weight></weightgroup></initrwgt>

Thus, each choice of scales and of PDFs is uniquely identied by an ID number. For example, id=1 will correspond to the central scale and PDF choices, (fR, fF ) = (fcR, fcF)

and PDF number 90800 in the examples given above (this is the same as XWGTUP, and hence is redundant, but it is convenient to include it in the reweight section as well). As far as id=2 to id=9 are concerned, these will correspond to the N combinations given in eqs. (B.9)(B.11), the numerical values reported in the header next to muR and muF being those of fR and fF respectively. Finally, id=10 to id=109 correspond to the NPDF error sets 9080190900.

In keeping with the reweight section in the LHE le header, there will be a reweight section event-by-event, which contains the actual numbers to be used to ll the histograms as described previously. Its structure will read as follows:<event>...<rwgt><weight id=1> 3.905e+01</wgt><weight id=2> 4.142e+01</wgt>...<weight id=109> 3.876e+01</wgt></rwgt></event>

The presence of the ID numbers in the reweight section of each event facilitates debugging, does not require the weights to be always in the same order, and is especially convenient for on-the-y manipulations (such as discarding some contributions, sorting them, and so forth).

We conclude this section with some general comments. The condition N = 8 can be trivially relaxed (which requires the inclusion of N itself in the list of the inputs); we have refrained from doing so in the rst public version of the code since three choices for each

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of the hard scales typically give a good estimate of the uncertainties. A more extended set of input parameters would also give one the possibility of performing scale variations by changing the functional forms of the reference scales w.r.t. those adopted for the central results. Finally, the complete generality could be achieved by storing in the LHE le the weights wis. This would require, however, that additional information be stored as well (such as the values of the Bjorken xs more details can be found in ref. [125]). While this can be trivially done, it would render the computation of the basis members bis of eq. (B.8), and their subsequent combination with weights according to eq. (B.7), a much more involved operation than the present one if performed by an external user.63 Such a possibility is left open for future developments.

In the case of f(N)LO runs, the information on scale and PDF variations is also available on an event-by-event basis. In keeping with the fact that here one cannot have unweighted events, MadGraph5 aMC@NLO will associate to each kinematic conguration an array of weights (rather than a single weight), each of which gives eq. (B.7) with the basis elements of eq. (B.8) recomputed for all desired scale and PDF choices. This implies that the second elements of the pairs in eqs. (3.4)(3.6) will be turned into arrays, with dimensionality:

1 + (1 + N) + NPDF reweight scale = .true. reweight PDF = .true. (B.19) 1 + (1 + N) reweight scale = .true. reweight PDF = .false. (B.20)

1 + NPDF reweight scale = .false. reweight PDF = .true. (B.21)

The rst entry thus always gives the value of the weight associated with central parameters. In the case of scale variations, this weight is present twice (the 1 in (1 + N)) essentially for reasons of backward compatibility which should not concern the user. The information of which weight is which is returned as the array of strings weights info and made available to the users initialisation routine analysis begin this is equivalent to the id number discussed before in the context of LHE les.

B.4 Scale and PDF uncertainties: the LO case

At the LO, the structure of the cross section, eq. (B.7), is trivial the sum contains one term i = 1 (see however footnote 8), which corresponds to the single PDF-and-coupling combination (i.e., to a single basis member, eq. (B.8)) relevant to this perturbative order: this is the reason for the simplicity of eq. (2.2). Thus here, at variance with the NLO case, from the users viewpoint it is therefore as simple to handle w1 as is to handle . This suggests the following strategy: when unweighted events are produced, they are stored in an intermediate LHE le with a non-standard format, that features the weights w1. After the end of the run, this LHE le is read by a standalone module, dubbed SysCalc, that converts the weights w1 into the corresponding cross sections , and stores them in a new

LHE le, which has this time the standard format already described in appendix B.3. This procedure is advantageous because it allows one to run SysCalc as many times as desired using the same intermediate LHE le obtained at generation time; this implies that the

63This not being the case at the LO, in LO-type generations we can adopt a di erent strategy see appendix B.4.

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type of scale or PDF variations need not be chosen before the event-generation phase, as is the case at the NLO. The SysCalc module can be installed64 with:MG5 aMC> install syscalcfrom the MadGraph5 aMC@NLO shell. At runtime, MadGraph5 aMC@NLO is instructed to save the weights w1 in the intermediate LHE le by setting

T = use syst ! Enable systematics studiesin run card.dat. When this is done, SysCalc is automatically called at the end of the run (although, as was said before, it can also be run independently afterwards). The type of scale and PDF variations considered are determined by the following entries in run card.dat:

0.5 1 2 = sys_scalefact # Central scale factors-1 = sys_scalecorrelation # for renormalization/scale variate

# -1: make all combination# -2: only correlated variation0.5 1 2 = sys_alpsfact # \alpha_s emission scale factors30 50 = sys_matchscale # variation of merging scale# PDF sets and number of members (0 or none for all members). CT10nlo.LHgrid = sys_pdf #

The list of values to the left of sys scalefact collects the multiplicative factors in front of the reference (xed or dynamic65) scales, and are thus analogous to the quantities fR and fF that appear in eq. (B.12); at variance with the current NLO implementation, such a list can contain more than three numbers. We stress that, when use syst=T, the value of scalefact in run card.dat is ignored. The ag sys scalecorrelation allows one to choose which of the renormalisation/factorisation scale combinations are considered. When -1 is entered, then all R and F values determined by the list sys scalefact are taken into account, while with -2 one restricts to code to dealing with R = F . More sophisticated options are also available (allowing one to select only some of the possible (R, F )

combinations), which will be documented elsewhere. The entry sys pdf is associated with the study of PDF systematics; if set equal to a PDF error set, all PDF members in that set will be considered. Alternatively, one can specify the individual PDF members to be taken into account. Finally, through SysCalc one can also investigate the systematics relevant to tree-level merging (see section 2.3.4), with sys alpsfact and sys matchscale corresponding to S-argument and Qmatch variations respectively. More details on SysCalc can be found at: https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/SysCalc

Web End =https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/SysCalc .

B.5 Other LO reweighting applications

In this appendix we comment in the briefest of manners on some technicalities relevant to the matrix-element reweighting (eq. (2.3)) and the matrix-element method (eq. (2.5)) discussed in section 2.3.3.

64SysCalc requires that LHAPDF be installed as well.

65Note that the setscales.f code relevant to LO calculations is di erent w.r.t that used in NLO ones.

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As was already mentioned in section 3.1, the former procedure is straightforwardly accessed through the MadGraph5 aMC@NLO shell, by simply setting reweight=ON in the interactive talk-to phase. By doing so, MadGraph5 aMC@NLO will use the le reweight card.dat to modify the parameters (found in param card.dat) used for the benchmark computation (that essentially corresponds to the denominator of eq. (2.3)). One may enter any number of such modications, each of which may contain any number of parameter changes, to be done through the set command (see the comments inside the le reweight card.dat). There are currently two main limitations to this procedure. Firstly, the changes must occur within one given model (i.e., one cannot reweight to matrix elements computed in a model di erent w.r.t. that adopted in the benchmark calculation). Secondly, the accessible kinematical region in the new hypothesis must be equal to or smaller than the original one. The interested reader can nd more information by visiting: https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/Reweight

Web End =https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/Reweight .

The matrix-element method is handled by MadWeight. As was discussed in section 3.1, the current version embedded in MadGraph5 aMC@NLO has vastly increased the speed of the previous version [136]. Such an increase is mainly due to a better combination of subprocesses, to a Monte-Carlo-type sum over jet-parton assigments (as opposed to an exact sum), and to the possibility of performing the simultaneous computation of P (q|) in the case of multiple choices of the transfer function. The MadWeight executable

specic to the process generated by the user simply corresponds to employing one of the reserved output keywords (see appendix B.6), namely to executing, after the generation phase, the command:MG5 aMC> output madweight MYPROCAfter that, one may continue with using the MadGraph5 aMC@NLO shell interface by executing:MG5 aMC> launch MYPROCFurther details can be found at: https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/MadWeight

Web End =https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/

https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/MadWeight

Web End =MadWeight .

B.6 Output formats and standalone libraries

The main purpose of MadGraph5 aMC@NLO is that of providing a self-contained framework where to compute cross sections and generate events at the desired level of perturbative accuracy, in both the SM and new-physics theories. On the other hand, the code may also be used to provide one with only a given ingredient of a calculation (e.g., a matrix element) which is to be performed elsewhere. This has been one of the dening characteristic of MadGraph, and has been wholly inherited by MadGraph5 aMC@NLO. Another example is the computation of a quantity which is not a cross section; a case in point is the likelihood dealt with by MadWeight (see section 2.3.3 and appendix B.5).

All possible executables, libraries, or more elementary objects that can be produced by MadGraph5 aMC@NLO can simply be seen as outputs of the (meta-)code; they are in fact in one-to-one correspondence with the optional rst keyword of the shell command: MG5 aMC> output [OutputForm] [MYPROC]

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The possible choices of OutputForm can be readily obtained by using the autocompletion <TAB> key in the shell after having typed output (or with help output). By doing so, one will notice that they all apply, bar one case (standalone, see below), to LO-type generations. Here we shall not list them all, but limit ourselves to commenting on those which are the most useful from the users point of view. These are:

standalone: self-contained Fortran77 library for the computation of either tree-level

matrix elements (after an LO-type generation), or one-loop matrix elements (after an NLO-type generation with the keyword virt=coupling1 ... see appendix B.1).

The directory structure thus created contains a simple program (check sa.f) which allows one to evaluate the matrix elements pointwise for test purposes.

standalone cpp: the same as above, but in C++ rather than in Fortran77. Works

only for tree-level matrix elements.

pythia8: self-contained library for the computation of tree-level matrix elements, in

a format which can be directly used in the Pythia8 PSMC. It includes a simple driver that one can employ to steer Pythia8 sample runs, and a Processes * directory that contains the said matrix elements. More details can be found in section 3.2 of ref. [38].

madweight: this output keyword allows one to set up a computation with Mad-

Weight (see appendix B.5).

We conclude this appendix by stressing that the above list (supplemented by that of the other output keywords not explicitly given here) is by no means all-inclusive. The reader who is interested in a particular output format specically suited to his/her need is encouraged to contact us in order for that to be developed and included in future versions of Mad-Graph5 aMC@NLO. For instance, dedicated outputs for matrix elements to be used in the MatchBox [181] framework and by EventDeconstruction [369, 370] are being developed.

C Features of one-loop computations

C.1 TIR and IREGI

The aim of this section is that of presenting the basic procedures used in TIR, and in particular by the program IREGI;66 more details on the latter will be given elsewhere [168]. IREGI computes the integral that appears as rst element in the set of eq. (2.79); owing to the fact that we apply TIR to a given loop topology (see eq. (2.77)), we can exploit eq. (2.76) to simplify the notation here and drop the dependence on lt. On the other hand, at variance with what was done in section 2.4.2, in order to make things more explicit it is convenient to insert in the notation the dependence on the external (four-dimensional, owing to the t Hooft-Veltman scheme) four-momenta pi and on the masses mi that circulate in the loop (see eq. (2.60)), so that the integral we are interested in reads as follows:

I1...r({pi}, {mi}) = Z

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. (C.1)

66The acronym stands for Integral REduction with General positive propagator Indices.

1 . . . r

Qm1 i=0 Di

Lorentz-covariance then guarantees that eq. (C.1) can be re-written as follows:

I1...r({pi}, {mi}) = (C.2)

X2 j+i0+i1++im1=r

{[g]j[p0]i0 [pm1]im1}1...rIji0...im1({pi}, {mi}) ,

for certain scalar integrals Iji0...im1, and where the symmetric tensor form

{[g]j[p0]i0 . . . [pm1]im1}1...r is dened in such a way that all non-equivalent permutations

of the Lorentz indices 1, . . . r on j metric tensors g and is external momenta ps contribute with weight one. For example:

{[g]2[p0]0[p1]0}1...4 = g12g34 + g13g24 + g14g23,

{[g]1[p0]0[p1]2}1...4 = g12p31p41 + g13p21p41 + g14p21p31,

{[g]0[p0]0[p1]4}1...4 = p11p21p31p41,

{[g]0[p0]1[p1]2}123 = p10p21p31 + p20p11p31 + p30p11p21. (C.3) Given eq. (C.2), the computation of the original tensor integral of eq. (C.1) is reduced to that of the scalar integrals Iji0...im1. One starts by observing that the latter are independent of the number of dimensions used in the numerator of eq. (C.1), because the decomposition of eq. (C.2) would hold, with the formal replacement g g, if one

had replaced

in eq. (C.1). Therefore, since working with the same number of dimensions in all parts of a computation is algebraically convenient, one determines the Iji0...im1 directly in d dimensions. This is done by recursively expressing such scalar integrals in terms of lower-point ones, till only integrals that cannot be further reduced are obtained these are just a few and well known: IREGI makes use of those tabulated in OneLoop [371] and QCDloop [372].

One way of performing such recursive reduction stems from the pioneering work of Passarino and Veltman [154]: by contracting both sides of eq. (C.1) (in d-dimensions) with metric tensors and external four-momenta, one relates the Iji0...im1 integrals to lower-rank tensor or lower-point scalar integrals. This lowering is due to the fact that, when contracting, one obtains the scalar products

2 and

pi, which are then re-expressed as follows:

2 = D0 + m20 , (C.4)

pi = (

D0 + m2i m20)/2 , (C.5) thus either cancelling some of the denominators or simplifying the dependence on

the numerator. The procedure is algebraic, and one ends up with a system of equations where the unknowns are the scalar integrals, and the coe cients known functions of the kinematic variables. The solution of such a system is non-trivial, owing for example to the presence of special kinematic congurations; IREGI implements to a large extent the strategies proposed in ref. [373].

An alternative and independent way for performing the recursive reduction follows the work of Davydychev [155]. One introduces the generalised loop-tensor and basic-scalar integrals:

1...r(d, {i}, {pi}, {mi}) =

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(2)2d/2

(2)d

1 . . .

Qm1i=0 Dii

, (C.6)

I0(d, {i}, {pi}, {mi}) =

(2)2d/2

(2)d

Qm1i=0 Dii

, (C.7)

where the scale-dependent prefactor is conventional, and the indices 0, 1, , m1 are positive integers. By using a Feynman-parameter representation and the d-dimensional analogue of eq. (C.2):

1...r(d, {i}, {pi}, {mi}) =

i(4)(42)2d/2 (C.8)

X2 j+i0+...+im1=r

{[g]j[p0]i0 . . . [pm1]im1}1...r

JHEP07(2014)079

i=0 i+rj (

Pm1i=0 i d/2 j) 2j

Qm1i (i)

0 Yi=0 dyiyi+ii1i

(1)

Pm1

j+d/2[summationtext]m1

i=0 i

Xi=0 yi1

Xi=0 yim2i

Xi<j yiyj(pipj)2+

(

Pm1i=0 i d/2)

Qm1i=0 (i)

I0(d, {i}, {pi}, {mi}) =

i (4)(42)2d/2

(1)

Pm1

i=0 i

0 Yi=0 dyiyi1i(

Xi=0yi 1)

d/2[summationtext]m1

i=0 i

, (C.9)

Xi<jyiyj(pi pj)2 +

Xi=0 yim2i

one arrives at:

1...r(d, {i}, {pi}, {mi}) =

X2 j+i0++im1=r

{[g]j[p0]i0 [pm1]im1}1r

(42)rj

(2)j

Yi=0 (i + ii) (i)

I0(d + 2(r j), {i + ii}, {pi}, {mi}) . (C.10)

By using eqs. (C.2) (in d dimensions) and (C.10), one nally obtains the relation:

Iji0...im1({pi}, {mi}) =

"(2)2d/2 (2)d

#1 (42)rj (2)j

Yi=0 (1 + ii) (1)

I0(d + 2(r j), {1 + ii}, {pi}, {mi})|d=42 . (C.11)

With eq. (C.9) one is also able to derive relationships among scalar integrals in di erent dimensions. For instance, eq. (6) of ref. [155] can be easily obtained:

I0(d 2, {i}, {pi}, {mi}) = 42

Xs=0s I0(d, {i + is}, {pi}, {mi}) , (C.12)

129

gg d

dbbtt ud

dtt u dd

dW +ZH gg ~t1~t1g

# Feynman diagrams 54614 10947 187138 3952 # topologies 8190 811 8098 437 (k0) 0 4 2 0

# non-zero hel. congs. 128 16 27 8 Generation time 15h 28min 25h 1min 38s Running time 18.6s (19%) 895ms (72%) 26.4s (32%) 83.6ms (68%) Output code size 600 Mb 20 Mb 700 Mb 6 Mb Runtime RAM usage 3.6 Gb 152 Mb 8.3 Gb 81 Mb Stability 2 108 1 107 4 107 1 107

Table 15. Performances of MadLoop5 in the context of the computations presented in section 4.3. See the text for details.

where by is we have denoted the Kronecker symbol. Furthermore, other recursion relations for the scalar integrals I0(d, {i}, {pi}, {mi}) can be obtained with the help of

the integration-by-parts method [374, 375], which exploits the fact that integrals are translation-invariant in dimensional regularization. The practical implementation in IREGI of the method discussed here follows ref. [375].

IREGI can use either of the two methods presented above for the recursive reduction, the actual choice being made by the calling code (in our case, MadLoop5). The current default in MadGraph5 aMC@NLO is the use of Passarino and Veltman; this is straightforward to change, since it is simply controlled by a parameter in an input card. IREGI has a minimal internal stability control: should e.g. Passarino and Veltman procedure be agged unstable, the code will turn to using Davydychevs. We stress that this by no means replaces the stability control performed by MadLoop5, described in section 2.4.2.

C.2 Quantitative prole of MadLoop performances

We have already stressed that the results presented in section 4.3 can be used as benchmarks for the validation of other codes. However, they reveal only one aspect of the performances of MadLoop5, which we complement in this appendix by reviewing some quantitative characteristics of the handling of the scattering processes considered before. A summary of such characteristics is reported in table 15.

The number of topologies is the upper bound of the sum over the index t in eq. (2.77), and it corresponds to the number of independent loop reductions (i.e., of evaluations of the Red[ ] operator introduced in eq. (2.58)) for one kinematic conguration. As one can see from the table, such a number is much smaller than the number of Feynman diagrams, which emphasises the importance of the optimization induced by eq. (2.76). The quantity (k0) is equal to the number of coupling-constant combinations, minus one, at the Born level, which is larger than zero in the case of a mixed-coupling expansion. See the beginning of section 2.4, and eqs. (2.17) and (2.22) in particular, for more details. The generation

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time includes the compilation of the source code. The running time corresponds to the time taken by the code output by MadLoop5 to compute the one-loop squared matrix element summed over colours but for a single helicity conguration.67 The percentage in parenthesis species the fraction of the running time spent in the loop reduction which, we remind the reader, is independent of the number of non-zero helicity congurations considered. The complementary fraction of the time is spent in the computation of the coe cients C(r)1...r;h,l of eq. (2.79), and scales linearly with the number of helicity combinations. The size of the output code includes external data les (that essentially contain the colour coe cients lb) loaded by the library which is in itself much lighter. The RAM measure reported is the runtime peak of residential memory allocated. The gures given in the last row are the relative accuracies estimated by the MadLoop5 internal stability tests in the context of double-precision computations that use the kinematic congurations considered in section 4.3 (however, we stress again that the matrix-element results have been obtained in quadruple precision); obviously, these pointwise accuracies have only an indicative value, since the real gure of merit necessitates averaging over a large statistical sample of independent kinematic congurations.

We conclude this appendix by stressing that the data reported in table 15 can be obtained by the user by using the command check profile in the MadGraph5 aMC@NLO shell. For example, one would haveMG5 aMC> check profile g g > t t~ z [virt=QCD]in the case of the virtual corrections to ttZ production (i.e., process e.8 of table 6).

C.3 Computation of the integrand polynomial coe cients

We have discussed in section section 2.4.2 how the use of the loop-integrand representation of eq. (2.78) increases the speed of OPP-based integral reductions, as well as giving one the possibility of using TIR methods. This appendix elaborates on the techniques adopted in MadLoop5 for the computation of the coe cients C(r)1...r;h,l. The key fact is that such coe cients are fully symmetric tensors of rank r with only 3+r

independent entries. In renormalisable theories and in the Feynman gauge the number of loop propagators sets the maximal rank in of that loop-integrand numerator, so that the total number of coe cients necessary to express the numerator of any loop of a, say, 2 6 process, is at

most Ncoeff(rmax = 8)

3+r r

= 495 which is well within the reach of modern computers. We start by rewriting the analogue of eq. (2.78) for any polynomial P (rmax)()

of maximal rank rmax with the following shorthand and symbolic notation:

P (rmax)() = C(rmax)kk , (C.13)

where k takes values between 1 and Ncoeff(rmax), and e ectively denes a map between the sets of Lorentz indices i and integer numbers. The choice of such a map is arbitrary, and we use what follows:

C(r)1,,r , 1 r C(rmax)k(1, ,r) , (C.14)

67The timing indicated is for a single core of a 2.7 GHz i7 CPU, with the gfortran compiler v4.8.1 without optimization ags (which have been shown to have a negligible impact).

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Prmax=8 r=0

W (1)i,0, k1,j

W (2)i,1, k2,k

V (1)j,1, s1,k

V (2)k,0, s2, V (3),1, s3,

V (4),1, s4,m

W (5)i,3, k5,m

w(1)

W (4)i,2, k4,

w(4)p

JHEP07(2014)079

W (3)i,1, k3,

w(2)n w(3)

Figure 19. An example of the MadLoop5 construction of the coe cients C(rmax)k. W denotes the

loop currents and V the vertex polynomials (see the text). The gure depicts an L-cut diagram; the original box loop is obtained by sewing the two fermion lines at the top.

k(1, , r) = Ncoeff(r 1) +

Xi=1(1 0i)(i + i 1)!i!(i 1)!, (C.15)

where we dene Ncoeff(1) = 0 (relevant to r = 0). To make things more explicit with

one example, eq. (C.13) reads, with rmax = 2:

C(2)kk C(2)0 + C(2)10 + C(2)21 + C(2)32 + C(4)43

+C(2)500 + C(2)601 + C(2)711 + C(2)802 + C(2)912

+C(2)1022 + C(2)1103 + C(2)1213 + C(2)1323 + C(2)1433 . (C.16)

The computation of the coe cients C(rmax)k with MadLoop5 is entirely numerical and follows the MadGraph procedure for evaluating Feynman diagrams. At tree level, the internal currents of a given Feynman diagram are denoted by w(n)j, with the integer n labeling them and the index j spanning the representation of the particle associated with the current. In

MadLoop5 the loop currents are promoted to more general objects embedding polynomials in and they are denoted by W (n)i,r,k,j, with the additional index i specifying the choice of the external current for the rst L-cut particle (which is non-physical and whose only constraint is to reproduce the Lorentz trace when summed over). The index k labels the coe cients of the polynomial of maximal rank r according to the convention of eq. (C.15). Figure 19 shows a complete example of the di erent numerical objects manipulated by MadLoop5 in order to construct the polynomial coe cients for the integrand of a box loop diagram . Note that since the currents w(#) attached to the loop are independent of the loop momentum , it is irrelevant to know whether they are external currents or originate from larger trees. The starting loop current W (1)i,0, k1,j is a polynomial of rank 0 since it does not have any loop momentum dependence; its index k1 can therefore only take the value0. For all loop diagrams, the starting current is always W (1)i,0, k1,j = ij k10.

132

The objects V (#)a,r,s,b are the representations (polynomials of rank r in the loop momentum) of the vertex plus propagator structures, with a and b the incoming and outgoing loop current indices respectively. For example, the explicit expression of V (1)j,1, s1,k for a massless quark in gure 19 reads:

V (1)j,1, s1,k s1 = gsjiw(1)

| {z }

, (C.17)

where the denominator of the propagator is removed since it is already accounted for in the integral-reduction procedure. To be more denite, we show here the expression of each coe cient of the vertex polynomial of eq. (C.17):

V (1)j,1,0,k = gs(/

vertex

w(1)( s11))jk for s1 = 1, . . . , 4 . (C.18)

In renormalisable theories, the rank of the vertex polynomials is maximally equal to one as only one power of the loop momentum can arise from either the propagator or the vertex itself (but not from both). This constraint in not enforced by MadLoop5, since it would make it impossible to compute one-loop diagrams in e ectives theories (such as HEFT). The numerical routines for the evaluation of vertex polynomials are generated automatically by ALOHA for each process, using the UFO model specications; this is what renders the optimisation discussed in this appendix applicable to any model.

Each subsequent loop current W (n+1) is obtained from the previous one W (n) and the vertex polynomial V (n) placed in between via the dening implicit relation:

W (n+1)i,r

1+r2, k1,j k1 = (W (n)i,r1, k2,m k2)(V (n)m,r2,s,js). (C.19)

The r.h.s. of eq. (C.19) is a multiplication of two polynomials and each coe cient of W (n+1)

is obtained by summing the corresponding terms in the expanded product. This implies that a symmetrisation of the coe cients is performed after each loop vertex and this step is crucial in order to limit their proliferation and the resulting computing time. To illustrate this, eq. (C.19) is rewritten here for the case r1 = r2 = 1 with k1 = 9, corresponding to the term multiplying 12 which can come either from k2 = 2, s = 3 or k2 = 3, s = 2:

W (n+1)i,2,9,j = W (n)i,1,2,mV (n)m,1,3,j + W (n)i,1,3,mV (n)m,1,2,j . (C.20)

Note the implicit summation on the index m which spans the representation of the particle circulating in the loop just before the vertex V (n). Eq. (C.19) is then iteratively used to compute all loop currents, until the second L-cut leg, which includes the last vertex and the L-cut propagator, is reached. The coe cients Ck of an n-point loop diagram are then simply obtained by closing the Lorentz trace:

Ck = W (n+1)i,r,k,i. (C.21)

In the example of gure 19, this translates into:

Ck = W (5)i,3,k,mmi =

133

ik( + p1)

| {z }

propagator

JHEP07(2014)079

w(1) /

p1)jk , V (1)j,1, s1,k = gs(/

Xi=1

W (5)i,3,k,i , (C.22)

with the sum written explicitly for clarity. It is clear that many loop diagrams can share some of their constituting loop currents; for example the L-cut box diagram of gure 19 can be prolonged to form loops with more propagators. MadLoop5 takes advantage of this and computes each loop current only once, for the rst loop diagram in which it appears. An algorithm analogous to the one described in ref. [24] is used for choosing the L-cut location in order to maximise the number of loop currents recycled.

We conclude this appendix by stressing that all of the optimisations inherited from the integrand representation of eq. (2.78) can be turned o in MadGraph5 aMC@NLO via the option loop optimized output of the interactive interface and that, when this is done, the structure of the code output by MadLoop5 is completely di erent. For this reason and despite being signicantly slower, the non optimized output mode provides a powerful self-consistency check and is useful for debugging purposes.

D Third-party codes included in MadGraph5 aMC@NLO

The tarball of MadGraph5 aMC@NLO is self-contained, and ready-to-run. This is also thanks to the fact that several third-party codes are included into it. We list all of them here: ALOHA [92], CutTools [156], FastJet (core) [376], HELAS [91], HERWIG6 [142], MINT [171], OneLoop [371], Pythia6 [141], QCDloop [372], RAMBO [377], StdHEP [378], Vegas [379, 380].

In the case of FastJet, what is included in MadGraph5 aMC@NLO is a stripped version of that code (visit www.fastjet.fr for more details). For more extended jet-reconstruction capabilities, the user might want to install FastJet proper (thus including all the relevant plugins).

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Abstract

We discuss the theoretical bases that underpin the automation of the computations of tree-level and next-to-leading order cross sections, of their matching to parton shower simulations, and of the merging of matched samples that differ by light-parton multiplicities. We present a computer program, MadGraph5 aMC@NLO, capable of handling all these computations -- parton-level fixed order, shower-matched, merged -- in a unified framework whose defining features are flexibility, high level of parallelisation, and human intervention limited to input physics quantities. We demonstrate the potential of the program by presenting selected phenomenological applications relevant to the LHC and to a 1-TeV e ^sup +^ e ^sup -^ collider. While next-to-leading order results are restricted to QCD corrections to SM processes in the first public version, we show that from the user viewpoint no changes have to be expected in the case of corrections due to any given renormalisable Lagrangian, and that the implementation of these are well under way.

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Title

The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations

Author

Alwall, J; Frederix, R; Frixione, S; Hirschi, V; Maltoni, F; Mattelaer, O; Shao, H-s; Stelzer, T; Torrielli, P; Zaro, M

Pages

1-157

Publication year

2014

Publication date

Jul 2014

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP07(2014)079

ProQuest document ID

1708014008

SISSA, Trieste, Italy 2014

The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations

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