Full Text

Turn on search term navigation

http://crossmark.crossref.org/dialog/?doi=10.1140/epjc/s10052-015-3480-z&domain=pdf

Web End = http://crossmark.crossref.org/dialog/?doi=10.1140/epjc/s10052-015-3480-z&domain=pdf

Web End = Eur. Phys. J. C (2015) 75:304

DOI 10.1140/epjc/s10052-015-3480-z

Regular Article - Theoretical Physics

Open source QCD t project

S. Alekhin1,2, O. Behnke3, P. Belov3,4, S. Borroni3, M. Botje5, D. Britzger3, S. Camarda3, A. M. Cooper-Sarkar6,K. Daum7,8, C. Diaconu9, J. Feltesse10, A. Gizhko3, A. Glazov3, A. Guffanti11, M. Guzzi3, F. Hautmann12,13,14,A. Jung15, H. Jung3,16, V. Kolesnikov17, H. Kowalski3, O. Kuprash3, A. Kusina18, S. Levonian3, K. Lipka3,B. Lobodzinski19, K. Lohwasser1,3, A. Luszczak20, B. Malaescu21, R. McNulty22, V. Myronenko3,S. Naumann-Emme3, K. Nowak3,6, F. Olness18, E. Perez23, H. Pirumov3, R. Plaakyte3,a, K. Rabbertz24,V. Radescu3,a, R. Sadykov17, G. P. Salam25,26, A. Sapronov17, A. Schning27, T. Schrner-Sadenius3,S. Shushkevich3, W. Slominski28, H. Spiesberger29, P. Starovoitov3, M. Sutton30, J. Tomaszewska31,O. Turkot3, A. Vargas3, G. Watt32, K. Wichmann3

1 Deutsches Elektronen-Synchrotron (DESY), Platanenallee 6, 15738 Zeuthen, Germany

2 Institute for High Energy Physics, 142281 Protvino, Moscow Region, Russia

3 Deutsches Elektronen-Synchrotron (DESY), Hamburg, Germany

4 Present address: Department of Physics, St. Petersburg State University, Ulyanovskaya 1, 198504 St. Petersburg, Russia

5 Nikhef, Science Park, Amsterdam, The Netherlands

6 Department of Physics, University of Oxford, Oxford, UK

7 Fachbereich C, Universitt Wuppertal, Wuppertal, Germany

8 Rechenzentrum, Universitt Wuppertal, Wuppertal, Germany

9 Aix Marseille Universite, CNRS/IN2P3, CPPM UMR 7346, 13288 Marseille, France

10 CEA, DSM/Irfu, CE-Saclay, Gif-sur-Yvette, France

11 Niels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark

12 School of Physics and Astronomy, University of Southampton, Southampton, UK

13 Rutherford Appleton Laboratory, Chilton OX11 0QX, UK

14 Department of Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK

15 FERMILAB, Batavia, IL 60510, USA

16 Elementaire Deeltjes Fysica, Universiteit Antwerpen, 2020 Antwerp, Belgium

17 Joint Institute for Nuclear Research (JINR), Joliot-Curie 6, 141980 Dubna, Moscow Region, Russia

18 Southern Methodist University, Dallas, TX, USA

19 Max Planck Institut fr Physik, Werner Heisenberg Institut, Fhringer Ring 6, Mnchen, Germany

20 T. Kosciuszko University of Technology, Krakw, Poland

21 Laboratoire de Physique Nuclaire et de Hautes Energies, UPMC and Universit, Paris-Diderot and CNRS/IN2P3, Paris, France

22 University College Dublin, Dublin 4, Ireland

23 CERN, European Organization for Nuclear Research, Geneva, Switzerland

24 Institut fr Experimentelle Kernphysik, Karlsruhe, Germany

25 CERN, PH-TH, 1211 Geneva 23, Switzerland

26 Leave from LPTHE, CNRS UMR 7589, UPMC Univ. Paris 6, 75252 Paris, France

27 Physikalisches Institut, Universitt Heidelberg, Heidelberg, Germany

28 Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krakw, Poland

29 PRISMA Cluster of Excellence, Institut fr Physik (WA THEP), Johannes-Gutenberg-Universitt, 55099 Mainz, Germany

30 Department of Physics and Astronomy, University of Sussex, Sussex House, Brighton BN1 9RH, UK

31 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland

32 Institute for Particle Physics Phenomenology, Durham University, Durham DH1 3LE, UK Received: 14 November 2014 / Accepted: 19 May 2015 / Published online: 2 July 2015 The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract HERAFitter is an open-source package that provides a framework for the determination of the parton distribution functions (PDFs) of the proton and for many

a e-mail: mailto:[email protected]

Web End [email protected]

different kinds of analyses in Quantum Chromodynamics (QCD). It encodes results from a wide range of experimental measurements in leptonproton deep inelastic scattering and protonproton (protonantiproton) collisions at hadron colliders. These are complemented with a variety of theo-

http://crossmark.crossref.org/dialog/?doi=10.1140/epjc/s10052-015-3480-z&domain=pdf

Web End = http://crossmark.crossref.org/dialog/?doi=10.1140/epjc/s10052-015-3480-z&domain=pdf

Web End = HERAFitter

123

304 Page 2 of 17 Eur. Phys. J. C (2015) 75 :304

retical options for calculating PDF-dependent cross section predictions corresponding to the measurements. The framework covers a large number of the existing methods and schemes used for PDF determination. The data and theoretical predictions are brought together through numerous methodological options for carrying out PDF ts and plotting tools to help to visualise the results. While primarily based on the approach of collinear factorisation, HERA-Fitter also provides facilities for ts of dipole models and transverse-momentum dependent PDFs. The package can be used to study the impact of new precise measurements from hadron colliders. This paper describes the general structure of HERAFitter and its wide choice of options.

1 Introduction

The recent discovery of the Higgs boson [1,2] and the extensive searches for signals of new physics in LHC proton proton collisions require high-precision calculations to test the validity of the Standard Model (SM) and factorisation in Quantum Chromodynamics (QCD). Using collinear factorisation, inclusive cross sections in hadron collisions may be written as

(s(2R), 2R, 2F) =

a,b

dx1 dx2 fa(x1, 2F) fb(x2, 2F)

ab(x1, x2; s(2R), 2R, 2F)

2QCD

(1)

where the cross section is expressed as a convolution of Parton Distribution Functions (PDFs) fa and fb with the parton cross section

ab, involving a momentum transfer q such that Q2 = |q2| 2QCD, where QCD is the QCD

scale. At Leading Order (LO) in the perturbative expansion of the strong-coupling constant, the PDFs represent the probability of nding a specic parton a (b) in the rst (second) hadron carrying a fraction x1 (x2) of its momentum. The indices a and b in Eq. 1 indicate the various kinds of partons,i.e. gluons, quarks and antiquarks of different avours that are considered as the constituents of the proton. The PDFs depend on the factorisation scale, F, while the parton cross sections depend on the strong-coupling constant, s, and the factorisation and renormalisation scales, F and R. The parton cross sections

ab are calculable in perturbative QCD (pQCD) whereas PDFs are usually constrained by global ts to a variety of experimental data. The assumption that PDFs are universal, within a particular factorisation scheme [37], is crucial to this procedure. Recent review articles on PDFs can be found in Refs. [8,9].

A precise determination of PDFs as a function of x requires large amounts of experimental data that cover a wide kinematic region and that are sensitive to different kinds of

partons. Measurements of inclusive Neutral Current (NC) and Charge Current (CC) Deep Inelastic Scattering (DIS) at the leptonproton (ep) collider HERA provide crucial information for determining the PDFs. The low-energy xed-target data and different processes from protonproton (pp) collisions at the LHC and protonantiproton (p p) collisions

at the Tevatron provide complementary information to the HERA DIS measurements. The PDFs are determined from 2 ts of the theoretical predictions to the data. The rapid ow of new data from the LHC experiments and the corresponding theoretical developments, which are providing predictions for more complex processes at increasingly higher orders, has motivated the development of a tool to combine them together in a fast, efcient, open-source framework.

This paper describes the open-source QCD t framework HERAFitter [10], which includes a set of tools to facilitate global QCD analyses of pp, p p and ep scattering data.

It has been developed for the determination of PDFs and the extraction of fundamental parameters of QCD such as the heavy quark masses and the strong-coupling constant. It also provides a common framework for the comparison of different theoretical approaches. Furthermore, it can be used to test the impact of new experimental data on the PDFs and on the SM parameters.

This paper is organised as follows: The general structure of HERAFitter is presented in Sect. 2. In Sect. 3 the various processes available in HERAFitter and the corresponding theoretical calculations, performed within the framework of collinear factorisation and the DGLAP [1115] formalism, are discussed. In Sect. 4 tools for fast calculations of the theoretical predictions are presented. In Sect. 5 the methodology to determine PDFs through ts based on various 2 denitions is described. In particular, various treatments of correlated experimental uncertainties are presented. Alternative approaches to the DGLAP formalism are presented in Sect. 6. The organisation of the HERAFitter code is discussed in Sect. 7, specic applications of the package are presented in Sect. 8, which is followed by a summary in Sect. 9.

2 The HERAFitter structure

The diagram in Fig. 1 gives a schematic overview of the HERAFitter structure and functionality, which can be divided into four main blocks:

Data: Measurements from various processes are provided in the HERAFitter package including the information on their uncorrelated and correlated uncertainties. HERA inclusive scattering data are directly sensitive to quark PDFs and indirectly sensitive to the gluon PDF through scaling violations and the longitudinal structure function FL. These data are the basis of any proton PDF extraction and are used in all current PDF sets from MSTW [16], CT [17], NNPDF

123

Eur. Phys. J. C (2015) 75 :304 Page 3 of 17 304

Fig. 1 Schematic overview of the HERAFitter program

Table 1 The list of experimental data and theory calculations implemented in the HERAFitter package. The references for the individual calculations and schemes are given in the text

Experimental data

Fig. 2 Distributions of valence (xuv, xdv), sea (x S) and the gluon (xg) PDFs in HERAPDF1.0 [21]. The gluon and the sea distributions are scaled down by a factor of 20. The experimental, model and parametrisation uncertainties are shown as coloured bands

ranges in x. Measurements from the xed-target experiments, the Tevatron and the LHC provide additional constraints on the gluon and quark distributions at high-x, better understanding of heavy quark distributions and decomposition of the light-quark sea. For these purposes, measurements from xed-target experiments, the Tevatron and the LHC are included.

The processes that are currently available within the HERAFitter framework are listed in Table 1.

Theory: The PDFs are parametrised at a starting scale, Q20, using a functional form and a set of free parameters p. These

PDFs are evolved to the scale of the measurements Q2, Q2 >

Q20. By default, the evolution uses the DGLAP formalism [1115] as implemented in QCDNUM [22]. Alternatively, the

CCFM evolution [2326] as implemented in uPDFevolv [27] can be chosen. The prediction of the cross section for a particular process is obtained, assuming factorisation, by the convolution of the evolved PDFs with the corresponding parton scattering cross section. Available theory calculations for each process are listed in Table 1. Predictions using dipole models [2830] can also be obtained.

QCD analysis: The PDFs are determined in a least squares t: a 2 function, which compares the input data and theory predictions, is minimised with the MINUIT [31] program. In HERAFitter various choices are available for the treatment of experimental uncertainties in the 2 denition. Correlated experimental uncertainties can be accounted for using a nuisance parameter method or a covariance matrix

Process Reaction Theory schemes calculations

HERA, xed target

DIS NC ep eX

p X

TR , ACOT,ZM (QCDNUM),FFN (OPENQCDRAD, QCDNUM),TMD (uPDFevolv)

HERA DIS CC ep e X ACOT, ZM (QCDNUM),

FFN (OPENQCDRAD)

DIS jets ep e jetsX NLOJet++ (fastNLO)

DIS heavy quarks

ep ec cX,

ep eb bX

TR , ACOT,ZM (QCDNUM),FFN (OPENQCDRAD, QCDNUM)

Tevatron, LHC

DrellYan pp( p) l l X,

pp( p) l X

MCFM (APPLGRID)

Top pair pp( p) t t X MCFM (APPLGRID),

HATHOR, DiffTop Single top pp( p) tl X, MCFM (APPLGRID)

pp( p) t X,

pp( p) tW X

Jets pp( p) jetsX NLOJet++ (APPLGRID),

NLOJet++ (fastNLO)

LHC DY heavy quarks

pp V hX MCFM (APPLGRID)

[18], ABM [19], JR [20] and HERAPDF [21] groups. Measurements of charm and beauty quark production at HERA are sensitive to heavy quark PDFs and jet measurements have direct sensitivity to the gluon PDF. However, the kinematic range of HERA data mostly covers low and medium

123

304 Page 4 of 17 Eur. Phys. J. C (2015) 75 :304

method as described in Sect. 5.2. Different statistical assumptions for the distributions of the systematic uncertainties, e.g. Gaussian or LogNormal [32], can also be studied (see Sect. 5.3).

Results: The resulting PDFs are provided in a format ready to be used by the LHAPDF library [33,34] or by TMDlib [35].HERAFitter drawing tools can be used to display the PDFs with their uncertainties at a chosen scale. As an example, the rst set of PDFs extracted using HERAFitter from HERA I data, HERAPDF1.0 [21], is shown in Fig. 2 (taken from Ref. [21]). Note that following the conventions, the PDFs are displayed as parton momentum distributions x f (x, 2F).

3 Theoretical formalism using DGLAP evolution

In this section the theoretical formalism based on DGLAP [1115] equations is described.

A direct consequence of factorisation (Eq. 1) is that the scale dependence or evolution of the PDFs can be predicted by the renormalisation group equations. By requiring physical observables to be independent of F, a representation of the parton evolution in terms of the DGLAP equations is obtained:

d fa(x, 2F)d log 2F =

b=q, q,g

, (4)

where Y = 1 (1 y)2 and is the electromagnetic-

coupling constant. The generalised structure functions F2,3 can be written as linear combinations of the proton structure functions F2 , F Z2,3 and F Z2,3, which are associated with pure photon exchange terms, photonZ interference terms and pure Z exchange terms, respectively. The structure function F2 is the dominant contribution to the cross sec

tion, x F3 becomes important at high Q2 and FL is siz

able only at high y. In the framework of pQCD, the structure functions are directly related to the PDFs: at LO F2 is the weighted momentum sum of quark and antiquark distributions, F2 x e2q(q + q) (where eq is the

quark electric charge), x F3 is related to their difference, x F3 x 2eqaq(q q) (aq is the axial-vector quark

coupling), and FL vanishes. At higher orders, terms related to the gluon distribution appear, in particular FL is strongly related to the low-x gluon.

The inclusive CC ep cross section, analogous to the NC ep case, can be expressed in terms of another set of structure functions, W:

d2e p

dxdQ2 =

1 P

= F2

Y+

x F3

y2Y+ FL

fb(z, 2F), (2)

where the functions Pab are the evolution kernels or splitting functions, which represent the probability of nding parton a in parton b. They can be calculated as a perturbative expansion in s. Once PDFs are determined at the initial scale 2F = Q20, their evolution to any other scale Q2 > Q20

is entirely determined by the DGLAP equations. The PDFs are then used to calculate cross sections for various different processes. Alternative approaches to the DGLAP evolution equations, valid in different kinematic regimes, are also implemented in HERAFitter and will be discussed in Sect. 6.

3.1 Deep inelastic scattering and proton structure

The formalism that relates the DIS measurements to pQCD and the PDFs has been described in detail in many extensive reviews (see, e.g., Ref. [36]) and it is only briey summarised here. DIS is the process where a lepton scatters off the partons in the proton by the virtual exchange of a neutral ( /Z) or charged (W) vector boson and, as a result, a scattered lepton and a hadronic nal state are produced. The common DIS kinematic variables are the scale of the process Q2, which is the absolute squared four-momentum of the exchanged boson, Bjorken x, which can be related in the parton model to the momentum fraction that is carried by the struck quark,

and the inelasticity y. These are related by y = Q2/sx, where

s is the squared centre-of-mass energy.

The NC cross section can be expressed in terms of generalised structure functions:

d2e p

dxdQ2 =

22Y+ x Q4 e

pr,NC , (3)

e p

r,NC

dzz Pab

xz ; 2F

G2F 2x

m2W

m2W + Q2

e p

r,CC (5)

e p

r,CC

= Y+ W2 Yx W3 y2 WL, (6)

where P represents the lepton beam polarisation. At LO in s, the CC e+ p and e p cross sections are sensitive to different combinations of the quark avour densities:

e+ p

r,CC

x[u + c] + (1 y)2x[d + s], (7)

e p

r,CC

x[u + c] + (1 y)2x[d + s]. (8)

Beyond LO, the QCD predictions for the DIS structure functions are obtained by convoluting the PDFs with appropriate hard-process scattering matrix elements, which are referred to as coefcient functions.

The DIS measurements span a large range of Q2 from a few GeV2 to about 105 GeV2, crossing heavy quark mass thresholds, thus the treatment of heavy quark (charm and beauty) production and the chosen values of their masses become important. There are different schemes for the treatment of heavy quark production. Several variants of these

123

Eur. Phys. J. C (2015) 75 :304 Page 5 of 17 304

schemes are implemented in HERAFitter and they are briey discussed below.

Zero-mass-variable avour number (ZM-VFN): In this scheme [37], the heavy quarks appear as partons in the proton at Q2 values above m2h (heavy quark mass) and they

are then treated as massless in both the initial and the nal states of the hard-scattering process. The lowest-order process is the scattering of the lepton off the heavy quark via electroweak boson exchange. This scheme is expected to be reliable only in the region where Q2 m2h, and it is inac

curate for lower Q2 values since it misses corrections of order m2h/Q2, while the other schemes mentioned below are accurate up to order 2QCD/Q2 albeit with different perturbative orderings. In HERAFitter this scheme is available for the DIS structure function calculation via the interface to the QCDNUM [22] package, thus it benets from the fast QCDNUM convolution engine.

Fixed avour number (FFN): In this rigorous quantum eld theory scheme [3840], only the gluon and the light quarks are considered as partons within the proton and massive quarks are produced perturbatively in the nal state.The lowest-order process is the heavy quark-antiquark pair production via boson-gluon fusion. In HERAFitter this scheme can be accessed via the QCDNUM implementation or through the interface to the open-source code OPENQCDRAD [41] as implemented by the ABM group. This scheme is reliable only for Q m2h, since it does not resum log

arithms of the form ln(Q2/m2h) which become important for Q2 m2h. In QCDNUM, the calculation of the heavy

quark contributions to DIS structure functions are available at Next-to-Leading Order (NLO) and only electromagnetic exchange contributions are taken into account. In the OPENQCDRAD implementation the heavy quark contributions to CC structure functions are also available and, for the NC case, the QCD corrections to the coefcient functions in Next-to-Next-to Leading Order (NNLO) are provided in the best currently known approximation [42,43].The OPENQCDRAD implementation uses in addition the running heavy quark mass in the MS scheme [44]. It is sometimes argued that this MS scheme reduces the sensitivity of the DIS cross sections to higher-order corrections. It is also known to have smaller non-perturbative corrections than the pole mass scheme [45].

General-mass variable avour number (GM-VFN): In this scheme (see [46] for a comprehensive review), heavy quark production is treated for Q2 m2h in the FFN scheme and

for Q2 m2h in the massless scheme with a suitable inter

polation in between. The details of this interpolation differ between implementations. The groups that use GM-VFN schemes in PDFs are MSTW, CT (CTEQ), NNPDF, and HERAPDF. HERAFitter implements different variants of the GM-VFN scheme.

GM-VFN ThorneRoberts scheme: The ThorneRoberts (TR) scheme [47] was designed to provide a smooth transition from the massive FFN scheme at low scales Q2 m2h to the massless ZM-VFNS scheme at high

scales Q2 m2h. Because the original version was

technically difcult to implement beyond NLO, it was updated to the TR scheme [48]. There are two variants of the TR schemes: TR standard (as used in MSTW PDF sets [16,48]) and TR optimal [49], with a smoother transition across the heavy quark threshold region. Both TR

variants are accessible within the HERAFitter package at LO, NLO and NNLO. At NNLO, an approximation is needed for the massive O(3s) NC coefcient functions relevant for Q2 m2h, as for the FFN scheme.

GM-VFN ACOT scheme: The AivazisCollinsOlness Tung (ACOT) scheme belongs to the group of VFN factorisation schemes that use the renormalisation method of CollinsWilczekZee (CWZ) [50]. This scheme unies the low scale Q2 m2h and high scale Q2 > m2h

regions in a coherent framework across the full energy range. Within the ACOT package, the following variants of the ACOT MS scheme are available at LO and NLO: ACOT-Full [51], S-ACOT- [52,53] and ACOTZM [51]. For the longitudinal structure function higher-order calculations are also available. A comparison of PDFs extracted from QCD ts to the HERA data with the TR and ACOT-Full schemes is illustrated in Fig. 3 (taken from [21]).

3.2 Electroweak corrections to DIS

Calculations of higher-order electroweak corrections to DIS at HERA are available in HERAFitter in the on-shell scheme. In this scheme, the masses of the gauge bosons mW and mZ are treated as basic parameters together with the top, Higgs and fermion masses. These electroweak corrections are based on the EPRC package [54]. The code calculates the running of the electromagnetic coupling using the most recent parametrisation of the hadronic contribution [55] as well as an older version from Burkhard [56].

3.3 Diffractive PDFs

About 10 % of deep inelastic interactions at HERA are diffractive, such that the interacting proton stays intact (ep

eXp). The outgoing proton is separated from the rest of the nal hadronic system, X, by a large rapidity gap. Such events are a subset of DIS where the hadronic state X comes from the interaction of the virtual photon with a colour-neutral cluster stripped off the proton [57]. The process can be described analogously to the inclusive DIS, by means of the diffrac-

123

304 Page 6 of 17 Eur. Phys. J. C (2015) 75 :304

Fig. 3 Distributions of valence (xuv, xdv), sea (x S) and the gluon (xg) PDFs in HERAPDF1.0 [21] with their total uncertainties at the scale of Q2 = 10 GeV2 obtained using the TR scheme and compared to the

PDFs obtained with the ACOT-Full scheme using the k-factor technique (red). The gluon and the sea distributions are scaled down by a factor of 20

tive parton distributions (DPDFs) [58]. The parametrization of the colour-neutral exchange in terms of factorisable hard Pomeron and a secondary Reggeon [59], both having a hadron-like partonic structure, has proved remarkably successful in the description of most of the diffractive data. It has also provided a practical method to determine DPDFs from ts to the diffractive cross sections.

In addition to the usual DIS variables x, Q2, extra kinematic variables are needed to describe the diffractive process. These are the squared four-momentum transfer of the exchanged Pomeron or Reggeon, t, and the mass mX of the diffractively produced nal state. In practice, the variable mX is often replaced by the dimensionless quantity =

Q2 m2X +Q

(12)

where s is the squared centre-of-mass beam energy, the parton momentum fractions are given by x1,2 =

s exp(y), fq(x1, m2) are the PDFs at the scale of the invariant mass, and

q is the partonparton hard-scattering cross section. The corresponding triple differential CC cross section has the form

dmdyd cos =

2 48s sin4 W

2 t . In models based on a factorisable Pomeron, may be viewed at LO as the fraction of the Pomeron longitudinal momentum, xIP, which is carried by the struck parton, x = xIP, where P denotes the momentum of the proton.

For the inclusive case, the diffractive cross section reads

d4d dQ2dxIP dt = 22Q4 1 + (1 y)2 D(4)(, Q2, xIP, t)

(9)

with the reduced cross section:

D(4) = F D(4)2

y2 1+(1y)

m3(1 cos )2

(m2 m2W ) + 2W m2W

q1,q2

V 2q1q2 fq1(x1, m2) fq2(x2, m2), (13)

where Vq1q2 is the CabibboKobayashiMaskawa (CKM) quark mixing matrix and mW and W are the W boson mass and decay width, respectively.

The simple LO form of these expressions allows for the analytic calculations of integrated cross sections. In both NC and CC expressions the PDFs depend only on the boson rapidity y and invariant mass m, while the integral in cos can be evaluated analytically even for the case of realistic kinematic cuts.

2 F D(4)L. (10)

The diffractive structure functions can be expressed as convolutions of calculable coefcient functions with the

diffractive quark and gluon distribution functions, which in general depend on xIP, Q2, and t.

The DPDFs [60,61] in HERAFitter are implemented as a sum of two factorised contributions:

IP(xIP, t) f IPa(, Q2) + IR(xIP, t) f IRa(, Q2), (11) where (xIP, t) are the Reggeon and Pomeron uxes. The

Reggeon PDFs, f IRa are xed as those of the pion, while the Pomeron PDFs, f IPa, can be obtained from a t to the data.

3.4 DrellYan processes in pp or p p collisions

The DrellYan (DY) process provides valuable information about PDFs. In pp and p p scattering, the Z/ and W

production probe bi-linear combinations of quarks. Complementary information on the different quark densities can be obtained from the W asymmetry (d, u and their ratio), the ratio of the W and Z cross sections (sensitive to the avour composition of the quark sea, in particular to the s-quark distribution), and associated W and Z production with heavy quarks (sensitive to s, c- and b-quark densities). Measurements at large boson transverse momentum pT [greaterorsimilar] mW,Z are potentially sensitive to the gluon distribution [62].

At LO the DY NC cross section triple differential in invariant mass m, boson rapidity y and lepton scattering angle cos in the parton centre-of-mass frame can be written as [63,64]:

dmdyd cos =

2 3ms

q(cos , m)

fq(x1, m2) f q (x2, m2) + (q q) ,

123

Eur. Phys. J. C (2015) 75 :304 Page 7 of 17 304

Beyond LO, the calculations are often time-consuming and Monte Carlo generators are employed. Currently, the predictions for W and Z/ production are available up to

NNLO and the predictions for W and Z production in association with heavy-avour quarks are available to NLO.

There are several possibilities to obtain the theoretical predictions for DY production in HERAFitter. The NLO and NNLO calculations can be implemented using k-factor or fast grid techniques (see Sect. 4 for details), which are interfaced to programs such as MCFM [6567], available for NLO calculations, or FEWZ [68] and DYNNLO [69] for NLO and NNLO, with electroweak corrections estimated using MCSANC [70,71].

3.5 Jet production in ep and pp or p p collisions

The cross section for production of high pT hadronic jets is sensitive to the high-x gluon PDF (see, e.g., Ref. [16]).Therefore this process can be used to improve the determination of the gluon PDF, which is particularly important for Higgs production and searches for new physics. Jet production cross sections are currently known only to NLO. Calculations for higher-order contributions to jet production in pp collisions are in progress [7274]. Within HERAFitter, the NLOJet++ program [75,76] may be used for calculations of jet production. Similarly to the DY case, the calculation is very demanding in terms of computing power. Therefore fast grid techniques are used to facilitate the QCD analyses including jet cross section measurements in ep, pp and p p

collisions. For details see Sect. 4.

3.6 Top-quark production in pp or p p collisions

At the LHC, top-quark pairs (t t) are produced dominantly via

gg fusion. Thus, LHC measurements of the t t cross section

provide additional constraints on the gluon distribution at medium to high values of x, on s and on the top-quark mass, mt [77]. Precise predictions for the total inclusive t t

cross section are available up to NNLO [78] and they can be computed within HERAFitter via an interface to the program HATHOR [79].

Fixed-order QCD predictions for the differential t t cross

section at NLO can be obtained by using the program MCFM [67,8083] interfaced to HERAFitter with fast grid techniques.

Single top quarks are produced by exchanging electroweak bosons and the measurement of their production cross section can be used, for example, to probe the ratio of the u and d distributions in the proton as well as the b-quark PDF. Predictions for single-top production are available at the NLO accuracy by using MCFM.

Approximate predictions up to NNLO in QCD for the differential t t cross section in one-particle inclusive kinematics

are available in HERAFitter through an interface to the program DiffTop [84,85]. It uses methods of QCD threshold resummation beyond the leading logarithmic approximation. This allows the users to estimate the impact of the recent t t differential cross section measurements on the uncertainty

of the gluon density within a QCD PDF t at NNLO. A fast evaluation of the DiffTop differential cross sections is possible via an interface to fast grid computations [86].

4 Computational techniques

Precise measurements require accurate theoretical predictions in order to maximise their impact in PDF ts. Perturbative calculations become more complex and time-consuming at higher orders due to the increasing number of relevant Feynman diagrams. The direct inclusion of computationally demanding higher-order calculations into iterative ts is thus not possible currently. However, a full repetition of the perturbative calculation for small changes in input parameters is not necessary at each step of the iteration. Two methods have been developed which take advantage of this to solve the problem: the k-factor technique and the fast grid technique. Both are available in HERAFitter.

4.1 k-factor technique

The k-factors are dened as the ratio of the prediction of a higher-order (slow) pQCD calculation to a lower-order (fast) calculation using the same PDF. Because the k-factors depend on the phase space probed by the measurement, they have to be stored including their dependence on the relevant kinematic variables. Before the start of a tting procedure, a table of k-factors is computed once for a xed PDF with the time-consuming higher-order code. In subsequent iteration steps the theory prediction is derived from the fast lower-order calculation by multiplying by the pre-tabulated k-factors.

This procedure, however, neglects the fact that the k-factors are PDF dependent, and as a consequence, they have to be re-evaluated for the newly determined PDF at the end of the t for a consistency check. The t must be repeated until input and output k-factors have converged. In summary, this technique avoids iteration of the higher-order calculation at each step, but still requires typically a few re-evaluations.

In HERAFitter, the k-factor technique can also be used for the fast computation of the time-consuming GM-VFN schemes for heavy quarks in DIS. FAST heavy-avour schemes are implemented with k-factors dened as the ratio of calculations at the same perturbative order but for massive vs. massless quarks, e.g. NLO (massive)/NLO (massless). These k-factors are calculated only for the starting PDF and hence, the FAST heavy-avour schemes should only be

123

304 Page 8 of 17 Eur. Phys. J. C (2015) 75 :304

used for quick checks. Full heavy-avour schemes should be used by default. However, for the ACOT scheme, due to exceptionally long computation times, the k-factors are used in the default setup of HERAFitter.

4.2 Fast grid techniques

Fast grid techniques exploit the fact that iterative PDF tting procedures do not impose completely arbitrary changes to the types and shapes of the parameterised functions that represent each PDF. Instead, it can be assumed that a generic PDF can be approximated by a set of interpolating functions with a sufcient number of judiciously chosen support points.The accuracy of this approximation is checked and optimised such that the approximation bias is negligibly small compared to the experimental and theoretical accuracy. This method can be used to perform the time-consuming higher-order calculations (Eq. 1) only once for the set of interpolating functions. Further iterations of the calculation for a particular PDF set are fast, involving only sums over the set of interpolators multiplied by factors depending on the PDF. This approach can be used to calculate the cross sections of processes involving one or two hadrons in the initial state and to assess their renormalisation and factorisation scale variation.

This technique serves to facilitate the inclusion of time-consuming NLO jet cross section predictions into PDF ts and has been implemented in the two projects, fastNLO [87,88] and APPLGRID [89,90]. The packages differ in their interpolation and optimisation strategies, but both of them construct tables with grids for each bin of an observable in two steps: in the rst step, the accessible phase space in the parton momentum fractions x and the renormalisation and factorisation scales R and F is explored in order to optimise the table size. In the second step the grid is lled for the requested observables. Higher-order cross sections can then be obtained very efciently from the pre-produced grids while varying externally provided PDF sets, R and F, or s(R). This approach can in principle be extended to arbitrary processes. This requires an interface between the higher-order theory programs and the fast interpolation frameworks. For the HERAFitter implementations of the two packages, the evaluation of s is done consistently with the PDF evolution code. A brief description of each package is given below:

The fastNLO project [88] has been interfaced to the NLOJet++ program [75] for the calculation of jet production in DIS [91] as well as 2- and 3-jet production in hadronhadron collisions at NLO [76,92]. Threshold corrections at 2-loop order, which approximate NNLO for the inclusive jet cross section for pp and p p, have

also been included into the framework [93] following Ref. [94].

The latest version of the fastNLO convolution program [95] allows for the creation of tables in which renormalisation and factorisation scales can be varied as a function of two predened observables, e.g. jet transverse momentum p and Q for DIS. Recently, the differen

tial calculation of top-pair production in hadron collisions at approximate NNLO [84] has been interfaced to fastNLO [86]. The fastNLO code is available online [96]. Jet cross section grids computed for the kinematics of various experiments can be downloaded from this site. The fastNLO libraries and tables with theory predictions for comparison to particular cross section measurements are included in the HERAFitter package. The interface to the fastNLO tables from within HERA-Fitter was used in a recent CMS analysis, where the impact on extraction of the PDFs from the inclusive jet cross section is investigated [97]. In the APPLGRID package [90,98], in addition to jet cross sections for pp(p p) and DIS processes, calcu

lations of DY production and other processes are also implemented using an interface to the standard MCFM parton level generator [6567]. Variation of the renormalisation and factorisation scales is possible a posteriori, when calculating theory predictions with the APPLGRID tables, and independent variation of S is also allowed.

For predictions beyond NLO, the k-factors technique can also be applied within the APPLGRID framework.

As an example, the HERAFitter interface to APPL-GRID was used by the ATLAS [99] and CMS [100] Collaborations to extract the strange quark distribution of the proton. The ATLAS strange PDF extracted employing these techniques is displayed in Fig. 4 together with

Fig. 4 The strange antiquark distribution versus x for the ATLAS epWZ free s NNLO t [99] (magenta band) compared to predic

tions from NNPDF2.1 (blue hatched) and CT10 (green hatched) at Q2 = 1.9 GeV2. The ATLAS t was performed using a k-factor

approach for NNLO corrections

123

Eur. Phys. J. C (2015) 75 :304 Page 9 of 17 304

a comparison to the global PDF sets CT10 [17] and NNPDF2.1 [18] (taken from [99]).

5 Fit methodology

When performing a QCD analysis to determine PDFs there are various assumptions and choices to be made concerning, for example, the functional form of the input parametrisation, the treatment of heavy quarks and their mass values, alternative theoretical calculations, alternative representations of the t 2 and for different ways of treating correlated systematic uncertainties. It is useful to discriminate or quantify the effect of a chosen ansatz within a common framework and HERAFitter is optimally designed for such tests. The methodology employed by HERAFitter relies on a exible and modular framework that allows independent integration of state-of-the-art techniques, either related to the inclusion of a new theoretical calculation, or of new approaches to the treatment of the data and their uncertainties.

In this section we describe the available options for the t methodology in HERAFitter. In addition, as an alternative approach to a complete QCD t, the Bayesian reweighting method, which is also available in HERAFitter, is described.

5.1 Functional forms for PDF parametrisation

Careful consideration must be taken when assigning the PDF freedom via functional forms. The PDFs can be parametrised using several predened functional forms and avour decompositions, as described briey below. The choice of functional form can lead to a different shape for the PDF distributions, and consequently the size of the PDF uncertainties can depend on the exibility of the parametric choice.

Standard polynomials: The standard-polynomial form is the most commonly used. A polynomial functional form is used to parametrise the x-dependence of the PDFs, where the index j denotes each parametrised PDF avour:

x f j (x) = A j x Bj (1 x)Cj Pj(x). (14)

The parametrised PDFs are the valence distributions xuv and xdv, the gluon distribution xg, and the light sea quark distributions, x, x d, x s, at the starting scale, which is chosen

below the charm mass threshold. The form of polynomials Pj(x) can be varied. The form (1 + j x + Dj x + E j x2)

is used for the HERAPDF [21] with additional constraints relating to the avour decomposition of the light sea. This parametrisation is termed HERAPDF-style. The polynomial can also be parametrised in the CTEQ-style, where Pj(x)

takes the form ea3x(1 + ea4x + ea5x2) and, in contrast to

the HERAPDF-style, this is positive by construction. QCD number and momentum sum rules are used to determine the normalisations A for the valence and gluon distributions, and the sum-rule integrals are solved analytically.

Bi-Log-normal distributions: This parametrisation is motivated by multi-particle statistics and has the following functional form:

x f j (x) = aj x pjbj log(x)(1 x)qjdj log(1x). (15)

This function can be regarded as a generalisation of the standard polynomial form described above, however, numerical integration of Eq. 15 is required in order to impose the QCD sum rules.

Chebyshev polynomials: A exible parametrisation based on the Chebyshev polynomials can be employed for the gluon and sea distributions. Polynomials with argument log(x) are considered for better modelling the low-x asymptotic behaviour of those PDFs. The polynomials are multiplied by a factor of (1 x) to ensure that they vanish as x 1. The

resulting parametric form reads

xg(x) = Ag (1 x)

Ng1

i=0

Agi Ti

2 log x log xmin log xmin

(16)

, (17)

where Ti are rst-type Chebyshev polynomials of order i. The normalisation factor Ag is derived from the momentum sum rule analytically. Values of Ng,S to 15 are allowed; however, the t quality is already similar to that of the standard-polynomial parametrisation from Ng,S 5 and has a similar

number of free parameters [101].

External PDFs: HERAFitter also provides the possibility to access external PDF sets, which can be used to compute theoretical predictions for the cross sections for all the processes available in HERAFitter. This is possible via an interface to LHAPDF [33,34] providing access to the global PDF sets. HERAFitter also allows one to evolve PDFs from LHAPDF using QCDNUM. Figure 5 illustrates a comparison of various gluon PDFs accessed from LHAPDF as produced with the drawing tools available in HERAFitter.

5.2 Representation of 2

The PDF parameters are determined in HERAFitter by minimisation of a 2 function taking into account correlated and uncorrelated measurement uncertainties. There are various forms of 2, e.g. using a covariance matrix or providing nuisance parameters to encode the dependence of each cor-

x S(x) = (1 x)

NS1

i=0

ASi Ti

2 log x log xmin log xmin

123

304 Page 10 of 17 Eur. Phys. J. C (2015) 75 :304

)

xg(x,Q

2 = 4.0 GeV

Q CT10_NNLO

MSTW2008_NNLO ABM12_4N_NNLO HERAPDF1.5_NNLO NNPDF2.3_NNLO

-3

10 -2

-1

Fig. 5 The gluon PDF as extracted by various groups at the scale of Q2 = 4 GeV2, plotted using the drawing tools from HERAFitter

related systematic uncertainty for each measured data point. The options available in HERAFitter are the following:

Covariance matrix representation: For a data point i with a corresponding theory prediction mi, the 2 function can be expressed in the following form:

2(m) =

i,k

(mi i)C1ik(mk k), (18)

where the experimental uncertainties are given as a covariance matrix Cik for measurements in bins i and k.

The covariance matrix Cik is given by a sum of statistical, uncorrelated and correlated systematic contributions:

Cik = Cstatik + Cuncorik + Csysik. (19)

Using this representation one cannot distinguish the effect of each source of systematic uncertainty. Nuisance parameter representation: In this case, 2 is

expressed as

2 (m, b) =

i mi 1 j ijbj 2

2i,uncm2i + 2i,stat imi 1 j ijbj

b2j, (20)

where i,stat and i,unc are relative statistical and uncor-related systematic uncertainties of the measurement i. Further, ij quanties the sensitivity of the measurement to the correlated systematic source j. The function 2 depends on the set of systematic nuisance parameters bj .

This denition of the 2 function assumes that systematic

uncertainties are proportional to the central prediction

values (multiplicative uncertainties, mi(1 j ijbj )),

whereas the statistical uncertainties scale with the square root of the expected number of events. However, additive treatment of uncertainties is also possible in HERA-Fitter.

During the 2 minimisation, the nuisance parameters bj

and the PDFs are determined, such that the effect of different sources of systematic uncertainties can be distinguished.

Mixed form representation: In some cases, the statistical and systematic uncertainties of experimental data are provided in different forms. For example, the correlated experimental systematic uncertainties are available as nuisance parameters, but the bin-to-bin statistical correlations are given in the form of a covariance matrix. HERAFitter offers the possibility to include such mixed forms of information.

Any source of measured systematic uncertainty can be treated as additive or multiplicative, as described above. The statistical uncertainties can be included as additive or following the Poisson statistics. Minimisation with respect to nuisance parameters is performed analytically, however, for more detailed studies of correlations individual nuisance parameters can be included into the MINUIT minimisation.

5.3 Treatment of the experimental uncertainties

Three distinct methods for propagating experimental uncertainties to PDFs are implemented in HERAFitter and reviewed here: the Hessian, Offset and Monte Carlo method.

Hessian (Eigenvector) method: The PDF uncertainties reecting the data experimental uncertainties are estimated by examining the shape of the 2 function in the neighbourhood of the minimum [102]. Following the approach of Ref. [102], the Hessian matrix is dened by the second derivatives of 2 on the tted PDF parameters. The matrix is diagonalised and the Hessian eigenvectors are computed. Due to orthogonality these vectors correspond to independent sources of uncertainty in the obtained PDFs.

Offset method: The Offset method [103] uses the 2 function for the central t, but only uncorrelated uncertainties are taken into account. The goodness of the t can no longer be judged from the 2 since correlated uncertainties are ignored. The correlated uncertainties are propagated into the PDF uncertainties by performing variants of the t with the experimental data varied by 1 from

the central value for each systematic source. The resulting deviations of the PDF parameters from the ones obtained in the central t are statistically independent, and they

123

Eur. Phys. J. C (2015) 75 :304 Page 11 of 17 304

can be combined in quadrature to derive a total PDF systematic uncertainty.

The uncertainties estimated by the offset method are generally larger than those from the Hessian method.Monte Carlo method: The Monte Carlo (MC) technique [104,105] can also be used to determine PDF uncertainties. The uncertainties are estimated using pseudo-data replicas (typically >100) randomly generated from the measurement central values and their systematic and statistical uncertainties taking into account all point-to-point correlations. The QCD t is performed for each replica and the PDF central values and their experimental uncertainties are estimated from the distribution of the PDF parameters obtained in these ts, by taking the mean values and standard deviations over the replicas.

The MC method has been checked against the standard error estimation of the PDF uncertainties obtained by the Hessian method. Good agreement was found between the methods provided that Gaussian distributions of statistical and systematic uncertainties are assumed in the MC approach [32]. A comparison is illustrated in Fig. 6. Similar ndings were reported by the MSTW global analysis [106].

Since the MC method requires large number of replicas, the eigenvector representation is a more convenient way to store the PDF uncertainties. It is possible to transform MC to eigenvector representation as shown by [107].Tools to perform this transformation are provided with

Fit vs H1PDF2000, Q

HERAFitter and were recently employed for the representation of correlated sets of PDFs at different perturbative orders [108].

The nuisance parameter representation of 2 in Eq. 20 is derived assuming symmetric experimental errors, however, the published systematic uncertainties are often asymmetric. HERAFitter provides the possibility to use asymmetric systematic uncertainties. The implementation relies on the assumption that asymmetric uncertainties can be described by a parabolic function. The nuisance parameter in Eq. 20 is modied as follows:

ij ijbj + ij, (21)

where the coefcients ij, ij are dened from the maximum and minimum shifts of the cross sections due to a variation of the systematic uncertainty j, Si j,

ij =

2 S+i j + Si j , ij =

2 S+i j Si j . (22)

5.4 Treatment of the theoretical input

The results of a QCD t depend not only on the input data but also on the input parameters used in the theoretical calculations. Nowadays, PDF groups address the impact of the choices of theoretical parameters by providing alternative PDFs with different choices of the mass of the charm quarks, mc, mass of the bottom quarks, mb, and the value of s(mZ ).

Other important aspects are the choice of the functional form for the PDFs at the starting scale and the value of the starting scale itself. HERAFitter provides the possibility of different user choices of all this input.

5.5 Bayesian reweighting techniques

As an alternative to performing a full QCD t, HERAFitter allows the user to assess the impact of including new data in an existing t using the Bayesian Reweighting technique. The method provides a fast estimate of the impact of new data on PDFs. Bayesian Reweighting was rst proposed for PDF sets delivered in the form of MC replicas by [104] and further developed by the NNPDF Collaboration [109,110]. More recently, a method to perform Bayesian Reweighting studies starting from PDF ts for which uncertainties are provided in the eigenvector representation has also been developed [106]. The latter is based on generating replica sets by introducing Gaussian uctuations on the central PDF set with a variance determined by the PDF uncertainty given by the eigenvectors. Both reweighting methods are implemented in HERAFitter. Note that the precise form of the weights used by both methods has recently been questioned [111,112].

2 = 4. GeV2

xG(x)

0 -4 10

-3

-2 10

-1 1

Fig. 6 Comparison between the standard error calculations as employed by the Hessian approach (black lines) and the MC approach (with more than 100 replicas) assuming Gaussian distribution for uncertainty distributions, shown here for each replica (green lines) together with the evaluated standard deviation (red lines) [32]. The black and red lines in the gure are superimposed because agreement of the methods is so good that it is hard to distinguish them

123

304 Page 12 of 17 Eur. Phys. J. C (2015) 75 :304

The Bayesian Reweighting technique relies on the fact that MC replicas of a PDF set give a representation of the probability distribution in the space of PDFs. In particular, the PDFs are represented as ensembles of Nrep equiprobable(i.e. having weights equal to unity) replicas, { f }. The central

value for a given observable, O({ f }), is computed as the

average of the predictions obtained from the ensemble as

O({ f }) =

1 Nrep

Nrep

k=1

O( f k), (23)

and the uncertainty as the standard deviation of the sample.

Upon inclusion of new data the prior probability distribution, given by the original PDF set, is modied according to the Bayes Theorem such that the weight of each replica, wk, is updated according to

wk =

(2k)

2 (Ndata1)e

2 2k

1 Nrep

Nrepk=1(2k)12 (Ndata1)e12 2k , (24)

where Ndata is the number of new data points, k denotes the specic replica for which the weight is calculated and 2k is the 2 of the new data obtained using the kth PDF replica. Given a PDF set and a corresponding set of weights, which describes the impact of the inclusion of new data, the prediction for a given observable after inclusion of the new data can be computed as the weighted average,

O({ f }) =

1 Nrep

Nrep

k=1

wkO( f k). (25)

To simplify the use of a reweighted set, an unweighted set (i.e. a set of equiprobable replicas which incorporates the information contained in the weights) is generated according to the unweighting procedure described in [109]. The number of effective replicas of a reweighted set is measured by its Shannon Entropy [110],

Neff exp

r2 4R20(x)

1 Nrep

Nrep

k=1

wk ln(Nrep/wk)

, (26)

which corresponds to the size of a retted equiprobable replica set containing the same amount of information. This number of effective replicas, Neff, gives an indicative measure of the optimal size of an unweighted replica set produced with the reweighting/unweighting procedure. No extra information is gained by producing a nal unweighted set that has a number of replicas (signicantly) larger than Neff. If Neff is much smaller than the original number of replicas the new data have great impact, however, it is unreliable to use the new reweighted set. In this case, instead, a full ret should be performed.

6 Alternatives to DGLAP formalism

QCD calculations based on the DGLAP [1115] evolution equations are very successful in describing all relevant hard-scattering data in the perturbative region Q2 [greaterorsimilar] few GeV2.

At small-x (x < 0.01) and small-Q2 DGLAP dynamics may be modied by saturation and other (non-perturbative) higher-twist effects. Various approaches alternative to the DGLAP formalism can be used to analyse DIS data in HERA-Fitter. These include several dipole models and the use of transverse-momentum dependent, or unintegrated PDFs (uPDFs).

6.1 Dipole models

The dipole picture provides an alternative approach to protonvirtual photon scattering at low x which can be applied to both inclusive and diffractive processes. In this approach, the virtual photon uctuates into a q q (or q qg)

dipole which interacts with the proton [113,114]. The dipoles can be considered as quasi-stable quantum mechanical states, which have very long life time 1/m px and a size which is

not changed by scattering with the proton. The dynamics of the interaction are embedded in a dipole scattering amplitude.

Several dipole models, which show different behaviours of the dipoleproton cross section, are implemented in HERA-Fitter: the Golec-BiernatWsthoff (GBW) dipole saturation model [28], a modied GBW model which takes into account the effects of DGLAP evolution, termed the Bartels GolecKowalski (BGK) dipole model [30] and the colour glass condensate approach to the high parton density regime, named the IancuItakuraMunier (IIM) dipole model [29]. GBW model: In the GBW model the dipoleproton cross section dip is given by

dip(x, r2) = 0

1 exp

, (27)

where r corresponds to the transverse separation between the quark and the antiquark, and R20 is an x-dependent scale parameter which represents the spacing of the gluons in the proton. R20 takes the form, R20(x) = (x/x0)1/ GeV2, and is

called the saturation radius. The cross-section normalisation 0, x0, and are parameters of the model tted to the DIS data. This model gives exact Bjorken scaling when the dipole size r is small.

BGK model: The BGK model is a modication of the GBW model assuming that the spacing R0 is inverse to the gluon distribution and taking into account the DGLAP evolution of the latter. The gluon distribution, parametrised at some starting scale by Eq. 14, is evolved to larger scales using DGLAP evolution.

123

Eur. Phys. J. C (2015) 75 :304 Page 13 of 17 304

BGK model with valence quarks: The dipole models are valid in the low-x region only, where the valence quark contribution to the total proton momentum is 5 to 15 % for x from0.0001 to 0.01 [115]. The inclusive HERA measurements have a precision which is better than 2 %. Therefore, HERA-Fitter provides the option of taking into account the contribution of the valence quarksIIM model: The IIM model assumes an expression for the dipole cross section which is based on the BalitskyKovchegov equation [116]. The explicit formula for dip can be found in [29]. The alternative scale parameter R, x0 and

are tted parameters of the model.

6.2 Transverse momentum dependent PDFs

QCD calculations of multiple-scale processes and complex nal-states can necessitate the use of transverse-momentum dependent (TMD) [7], or unintegrated parton distribution and parton decay functions [117125]. TMD factorisation has been proven recently [7] for inclusive DIS. TMD factorisation has also been proven in the high-energy (small-x) limit [126128] for particular hadronhadron scattering processes, like heavy-avour, vector boson and Higgs production.

In the framework of high-energy factorisation [126,129, 130] the DIS cross section can be written as a convolution in both longitudinal and transverse momenta of the TMD parton distribution function A x, kt, 2F with the off-shell

parton scattering matrix elements as follows:

j (x, Q2) =

The cross section j ( j = 2, L) is calculated in a FFN

scheme, using the boson-gluon fusion process ( g q q).

The masses of the quarks are explicitly included as parameters of the model. In addition to g q q, the contribu

tion from valence quarks is included via q q by using

a CCFM evolution of valence quarks [137139].

CCFM grid techniques: The CCFM evolution cannot be written easily in an analytic closed form. For this reason, a MC method is employed, which is, however, time-consuming and thus cannot be used directly in a t program.

Following the convolution method introduced in [139, 140], the kernel

A x , kt, p is determined from the MC

solution of the CCFM evolution equation, and then folded with a non-perturbative starting distribution A0(x)

xA (x, kt, p)

= x

dx dx A0(x )

A x , kt, p (x x x)

dx A0(x ) xx

A x

, kt, p , (29)

where kt denotes the transverse momentum of the propagator gluon and p is the evolution variable.

The kernel

A incorporates all of the dynamics of the evolution. It is dened on a grid of 50 50 50 bins in x, kt, p.

The binning in the grid is logarithmic, except for the longitudinal variable x for which 40 bins in logarithmic spacing below 0.1, and 10 bins in linear spacing above 0.1 are used.

Calculation of the cross section according to Eq. 28 involves a time-consuming multidimensional MC integration, which suffers from numerical uctuations. This cannot be employed directly in a t procedure. Instead the following equation is applied:

(x, Q2) =

j (x, Q2, z, kt)A z, kt, 2F ,

(28)

where the DIS cross sections j ( j = 2, L) are related to the

structure functions F2 and FL by j = 42Fj/Q2, and the

hard-scattering kernels

1 d2kt

x dxgA (xg, kt, p)

(x, xg, Q2)

= x dx A0(x )

(x/x , Q2), (30)

j of Eq. 28 are kt-dependent.

The factorisation formula in Eq. 28 allows for resummation of logarithmically enhanced small-x contributions to all orders in perturbation theory, both in the hard-scattering coefcients and in the parton evolution, fully taking into account the dependence on the factorisation scale F and on the factorisation scheme [131,132].

Phenomenological applications of this approach require matching of small-x contributions with nite-x contributions. To this end, the evolution of the transverse-momentum dependent gluon density A is obtained by combining the resummation of small-x logarithmic corrections [133135] with medium-x and large-x contributions to parton splitting [11,14,15] according to the CCFM evolution equation [2326]. Sea quark contributions [136] are not yet included at transverse-momentum dependent level.

(x , Q2) is calculated numerically with a MC integration on a grid in x for the values of Q2 used in the t.

Then the last step in Eq. 30 is performed with a fast numerical Gauss integration, which can be used directly in the t. Functional forms for TMD parametrisation: For the starting distribution A0, at the starting scale Q20, the following form is used:

xA0(x, kt) = N xB(1 x)C 1 Dx + Ex

exp[k2t/2], (31)

where 2 = Q20/2 and N, B, C, D, E are free parameters.

Valence quarks are treated using themethod of Ref. [137] as

where rst

123

304 Page 14 of 17 Eur. Phys. J. C (2015) 75 :304

/dx

1.05

1.8

1.6

Theory/Data

1.4

1.2

0.95

1.04

Data

Theory+shifts

1.02

0.8

0.6

0.98

0.4

0.96

0.2

pulls

-0.2

-2

-0.4

0.01 0.1

Fig. 7 An illustration of the consistency of HERA measurements [21] and the theory predictions, obtained in HERAFitter with the default drawing tool

described in Ref. [139] with a starting distribution taken from any collinear PDF and imposition of the avour sum rule at every scale p.

The TMD parton densities can be plotted either with HERAFitter tools or with TMDplotter [35].

7 HERAFitter code organisation

HERAFitter is an open-source code under the GNU general public licence. It can be downloaded from a dedicated webpage [10] together with its supporting documentation and fast grid theory les (described in Sect. 4) associated with data les. The source code contains all the relevant information to perform QCD ts with HERA DIS data as a default set.1 The execution time depends on the tting options and varies from 10 min (using FAST techniques as described in Sect. 4) to several hours when full uncertainties are estimated. The HERAFitter code is a combination of C++ and Fortran 77 libraries with minimal dependencies, i.e. for the default tting options no external dependencies are required except the QCDNUM evolution program [22]. The ROOT libraries are only required for the drawing tools and when invoking APPLGRID. Drawing tools built into HERAFitter provide a qualitative and quantitative assessment of the results. Figure 7 shows an illustration of a comparison between the inclusive NC data from HERA I with the predictions based on HERAPDF1.0 PDFs. The consistency of the measurements and the theory can be expressed by pulls, dened as the difference between data and theory divided by the uncorrelated error of the data. In each kinematic bin of the measurement, pulls are provided in units of standard deviations. The pulls are also illustrated in Fig. 7.

In HERAFitter there are also available cache options for fast retrieval, fast evolution kernels, and the OpenMP (Open

1 Default settings in HERAFitter are tuned to reproduce the central HERAPDF1.0 set.

Multi-Processing) interface which allows parallel applications of the GM-VFNS theory predictions in DIS.

8 Applications of HERAFitter

The HERAFitter program has been used in a number of experimental and theoretical analyses. This list includes several LHC analyses of SM processes, namely inclusive DrellYan and Wand Z production [99,100,141143], inclusive jet production [97,144], and inclusive photon production [145]. The results of QCD analyses using HERA-Fitter were also published by HERA experiments for inclusive [21,146] and heavy-avour production measurements [147,148]. The following phenomenological studies have been performed with HERAFitter: a determination of the transverse-momentum dependent gluon distribution using precision HERA data [139], an analysis of HERA data within a dipole model [149], the study of the low-x uncertainties in PDFs determined from the HERA data using different parametrisations [101]. It is also planned to use HERA-Fitter for studying the impact of QED radiative corrections on PDFs [150]. A recent study based on a set of PDFs determined with HERAFitter and addressing the correlated uncertainties between different orders has been published in [108]. An application of the TMDs obtained with HERAFitter to W production at the LHC can be found in [151].

The HERAFitter framework has been used to produce PDF grids from QCD analyses performed at HERA [21,152] and at the LHC [153], using measurements from ATLAS [99,144]. These PDFs can be used to study predictions for SM or beyond SM processes. Furthermore, HERAFitter provides the possibility to perform various benchmarking exercises [154] and impact studies for possible future colliders as demonstrated by QCD studies at the LHeC [155].

9 Summary

HERAFitter is the rst open-source code designed for studies of the structure of the proton. It provides a unique and exible framework with a wide variety of QCD tools to facilitate analyses of the experimental data and theoretical calculations.

The HERAFitter code, in version 1.1.0, has sufcient options to reproduce the majority of the different theoretical choices made in MSTW, CTEQ and ABM ts. This will potentially make it a valuable tool for benchmarking and understanding differences between PDF ts. Such a study would, however, need to consider a range of further questions, such as the choices of data sets, treatments of uncertainties, input parameter values, 2 denitions, nuclear corrections, etc.

123

Eur. Phys. J. C (2015) 75 :304 Page 15 of 17 304

The further progress of HERAFitter will be driven by the latest QCD advances in theoretical calculations and in the precision of experimental data.

Acknowledgments HERAFitter developers team acknowledges the kind hospitality of DESY and funding by the Helmholtz Alliance Physics at the Terascale of the Helmholtz Association. We are grateful to the DESY IT department for their support of the HERAFitter developers. We thank the H1 and ZEUS Collaborations for the support in the initial stage of the project. Additional support was received from the BMBF-JINR cooperation program, the HeisenbergLandau program, the RFBR Grant 12-02-91526-CERN a, the Polish NSC project DEC-2011/03/B/ST2/00220 and a dedicated funding of the Initiative and Networking Fond of Helmholtz Association SO-072. We also acknowledge Nathan Hartland with Luigi Del Debbio for contributing to the implementation of the Bayesian Reweighting technique and would like to thank R. Thorne for fruitful discussions.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/

Web End =http://creativecomm http://creativecommons.org/licenses/by/4.0/

Web End =ons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP3.

References

1. G. Aad et al. (ATLAS Collaboration), Phys. Lett. B 716, 1 (2012). http://arxiv.org/abs/1207.7214

Web End =arXiv:1207.7214

2. S. Chatrchyan et al. (CMS Collaboration), Phys. Lett. B 716, 30 (2012). http://arxiv.org/abs/1207.7235

Web End =arXiv:1207.7235

3. J.C. Collins, D.E. Soper, Nucl. Phys. B 194, 445 (1982)4. J.C. Collins, D.E. Soper, G.F. Sterman, Phys. Lett. B 134, 263 (1984)

5. J.C. Collins, D.E. Soper, G.F. Sterman, Nucl. Phys. B 261, 104 (1985)

6. J.C. Collins, D.E. Soper, G.F. Sterman, Adv. Ser. Dir. High Energy Phys. 5, 1 (1988). http://arxiv.org/abs/hep-ph/0409313

Web End =hep-ph/0409313

7. J. Collins, Foundations of Perturbative QCD, vol. 32. Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology (2011)

8. E. Perez, E. Rizvi, Rep. Prog. Phys. 76, 046201 (2013). http://arxiv.org/abs/1208.1178

Web End =arXiv:1208.1178

9. S. Forte, G. Watt, Ann. Rev. Nucl. Part. Sci. 63, 291 (2013). http://arxiv.org/abs/1301.6754

Web End =arXiv:1301.6754

10. HERAFitter. https://www.herafitter.org

Web End =https://www.heratter.org 11. V.N. Gribov, L.N. Lipatov, Sov. J. Nucl. Phys. 15, 438 (1972a)12. V.N. Gribov, L.N. Lipatov, Sov. J. Nucl. Phys. 15, 675 (1972b)13. L.N. Lipatov, Sov. J. Nucl. Phys. 20, 94 (1975)14. Y.L. Dokshitzer, Sov. Phys. JETP 46, 641 (1977)15. G. Altarelli, G. Parisi, Nucl. Phys. B 126, 298 (1977)16. A. Martin, W. Stirling, R. Thorne, G. Watt, Eur. Phys. J. C 63, 189 (2009). http://arxiv.org/abs/0901.0002

Web End =arXiv:0901.0002 . http://mstwpdf.hepforge.org/

Web End =http://mstwpdf.hepforge.org/

17. J. Gao, M. Guzzi, J. Huston, H.-L. Lai, Z. Li et al., Phys. Rev. D 89, 033009 (2014). http://arxiv.org/abs/1302.6246

Web End =arXiv:1302.6246 . http://hep.pa.msu.edu/cteq/public/

Web End =http://hep.pa.msu.edu/cteq/ http://hep.pa.msu.edu/cteq/public/

Web End =public/

18. R.D. Ball et al., Nucl. Phys. B 867, 244 (2013). http://arxiv.org/abs/1207.1303

Web End =arXiv:1207.1303 . https://nnpdf.hepforge.org/

Web End =https://nnpdf.hepforge.org/

19. S. Alekhin, J. Bluemlein, S. Moch, Phys. Rev. D 89, 054028 (2014). http://arxiv.org/abs/1310.3059

Web End =arXiv:1310.3059

20. P. Jimenez-Delgado, E. Reya, Phys. Rev. D 89, 074049 (2014). http://arxiv.org/abs/1403.1852

Web End =arXiv:1403.1852

21. F. Aaron et al. (H1 and ZEUS Collaborations), JHEP 1001, 109 (2010). http://arxiv.org/abs/0911.0884

Web End =arXiv:0911.0884

22. M. Botje, Comput. Phys. Commun. 182, 490 (2011). http://arxiv.org/abs/1005.1481

Web End =arXiv:1005.1481 . http://www.nikhef.nl/user/h24/qcdnum/index.html

Web End =http://www.nikhef.nl/user/h24/qcdnum/index. http://www.nikhef.nl/user/h24/qcdnum/index.html

Web End =html

23. M. Ciafaloni, Nucl. Phys. B 296, 49 (1988)24. S. Catani, F. Fiorani, G. Marchesini, Phys. Lett. B 234, 339 (1990a)

25. S. Catani, F. Fiorani, G. Marchesini, Nucl. Phys. B 336, 18 (1990b)26. G. Marchesini, Nucl. Phys. B 445, 49 (1995)27. F. Hautmann, H. Jung, S.T. Monfared, Eur. Phys. J. C 74, 3082 (2014). http://arxiv.org/abs/1407.5935

Web End =arXiv:1407.5935

28. K. Golec-Biernat, M. Wsthoff, Phys. Rev. D 59, 014017 (1999). http://arxiv.org/abs/hep-ph/9807513

Web End =hep-ph/9807513

29. E. Iancu, K. Itakura, S. Munier, Phys. Lett. B 590, 199 (2004). http://arxiv.org/abs/hep-ph/0310338

Web End =hep-ph/0310338

30. J. Bartels, K. Golec-Biernat, H. Kowalski, Phys. Rev. D 66, 014001 (2002). http://arxiv.org/abs/hep-ph/0203258

Web End =hep-ph/0203258

31. F. James, M. Roos, Comput. Phys. Commun. 10, 343 (1975)32. M. Dittmar, S. Forte, A. Glazov, S. Moch, G. Altarelli et al. (2009). http://arxiv.org/abs/0901.2504

Web End =arXiv:0901.2504

33. M. Whalley, D. Bourilkov, R. Group (2005). http://arxiv.org/abs/hep-ph/0508110

Web End =hep-ph/0508110 34. LHAPDF. http://lhapdf.hepforge.org

Web End =http://lhapdf.hepforge.org 35. F. Hautmann, H. Jung, M. Kramer, P. Mulders, E. Nocera et al. (2014). http://arxiv.org/abs/1408.3015

Web End =arXiv:1408.3015

36. R. Devenish, A. Cooper-Sarkar, Deep Inelastic Scattering (2011). Oxford University Press (United Kingdom). ISBN 0199602255,9780199602254

37. J.C. Collins, W.-K. Tung, Nucl. Phys. B 278, 934 (1986)38. E. Laenen et al., Phys. Lett. B 291, 325 (1992)39. E. Laenen et al., Nucl. Phys. B 392(162), 229 (1993)40. S. Riemersma, J. Smith, W.L. van Neerven, Phys. Lett. B 347, 143 (1995). http://arxiv.org/abs/hep-ph/9411431

Web End =hep-ph/9411431

41. S. Alekhin, J. Blmlein, S. Moch, OPENQCDRAD. http://www-zeuthen.desy.de/~alekhin/OPENQCDRAD

Web End =http:// http://www-zeuthen.desy.de/~alekhin/OPENQCDRAD

Web End =www-zeuthen.desy.de/~alekhin/OPENQCDRAD

42. H. Kawamura, N. Lo Presti, S. Moch, A. Vogt, Nucl. Phys. B 864, 399 (2012)

43. I. Bierenbaum, J. Blumlein, S. Klein, Nucl. Phys. B 820, 417 (2009). http://arxiv.org/abs/0904.3563

Web End =arXiv:0904.3563

44. S. Alekhin, S. Moch, Phys. Lett. B 699, 345 (2011). http://arxiv.org/abs/1011.5790

Web End =arXiv:1011.5790

45. M. Beneke, Phys. Rep. 317, 1 (1999). http://arxiv.org/abs/hep-ph/9807443

Web End =hep-ph/9807443 46. R. Thorne, W. Tung (2008). http://arxiv.org/abs/0809.0714

Web End =arXiv:0809.0714 47. R.S. Thorne, R.G. Roberts, Phys. Rev. D 57, 6871 (1998). http://arxiv.org/abs/hep-ph/9709442

Web End =hep-ph/9709442

48. R.S. Thorne, Phys. Rev. D 73, 054019 (2006). http://arxiv.org/abs/hep-ph/0601245

Web End =hep-ph/0601245 49. R.S. Thorne, Phys. Rev. D 86, 074017 (2012). http://arxiv.org/abs/1201.6180

Web End =arXiv:1201.6180 50. J.C. Collins, Phys. Rev. D 58, 094002 (1998a). http://arxiv.org/abs/hep-ph/9806259

Web End =hep-ph/9806259 51. M. Aivazis, J.C. Collins, F.I. Olness, W.-K. Tung, Phys. Rev. D 50, 3102 (1994). http://arxiv.org/abs/hep-ph/9312319

Web End =hep-ph/9312319

52. M. Kramer, F.I. Olness, D.E. Soper, Phys. Rev. D 62, 096007 (2000). http://arxiv.org/abs/hep-ph/0003035

Web End =hep-ph/0003035

53. S. Kretzer, H. Lai, F. Olness, W. Tung, Phys. Rev. D 69, 114005 (2004). http://arxiv.org/abs/hep-ph/0307022

Web End =hep-ph/0307022

54. H. Spiesberger, Private communication55. F. Jegerlehner, in Proceedings, LC10 Workshop DESY, pp. 11117 (2011). DESY-PROC-2010-04

56. H. Burkhard, F. Jegerlehner, G. Penso, C. Verzegnassi, in CERN Yellow Report on Polarization at LEP (1988). CERN 88-06

57. A. Hebecker, Acta Phys. Polon. B 30, 3777 (1999). http://arxiv.org/abs/hep-ph/9909504

Web End =hep-ph/9909504

58. J.C. Collins, Phys. Rev. D 57, 3051 (1998). http://arxiv.org/abs/hep-ph/9709499

Web End =hep-ph/9709499 59. G. Ingelman, P.E. Schlein, Phys. Lett. B 152, 256 (1985)60. A. Aktas et al. (H1 Collaboration), Eur. Phys. J. C 48, 715 (2006). http://arxiv.org/abs/hep-ex/0606004

Web End =hep-ex/0606004

123

304 Page 16 of 17 Eur. Phys. J. C (2015) 75 :304

61. S. Chekanov et al. (ZEUS Collaboration), Nucl. Phys. B 831, 1 (2010). http://arxiv.org/abs/hep-ex/09114119

Web End =hep-ex/09114119

62. S.A. Malik, G. Watt, JHEP 1402, 025 (2014). http://arxiv.org/abs/1304.2424

Web End =arXiv:1304.2424 63. S.D. Drell, T.-M. Yan, Phys. Rev. Lett. 25, 316 (1970)64. M. Yamada, M. Hayashi, Nuovo Cim. A 70, 273 (1982)65. J.M. Campbell, R.K. Ellis, Phys. Rev. D 60, 113006 (1999). http://arxiv.org/abs/hep-ph/9905386

Web End =hep-ph/9905386

66. J.M. Campbell, R.K. Ellis, Phys. Rev. D 62, 114012 (2000). http://arxiv.org/abs/hep-ph/0006304

Web End =hep-ph/0006304

67. J.M. Campbell, R.K. Ellis, Nucl. Phys. Proc. Suppl. 205206, 10 (2010). http://arxiv.org/abs/1007.3492

Web End =arXiv:1007.3492

68. Y. Li, F. Petriello, Phys. Rev. D 86, 094034 (2012). http://arxiv.org/abs/1208.5967

Web End =arXiv:1208.5967

69. G. Bozzi, J. Rojo, A. Vicini, Phys. Rev. D 83, 113008 (2011). http://arxiv.org/abs/1104.2056

Web End =arXiv:1104.2056

70. D. Bardin, S. Bondarenko, P. Christova, L. Kalinovskaya, L.

Rumyantsev et al., JETP Lett. 96, 285 (2012). http://arxiv.org/abs/1207.4400

Web End =arXiv:1207.4400

71. S.G. Bondarenko, A.A. Sapronov, Comput. Phys. Commun. 184, 2343 (2013). http://arxiv.org/abs/1301.3687

Web End =arXiv:1301.3687

72. A. Gehrmann-De Ridder, T. Gehrmann, E. Glover, J. Pires, Phys.

Rev. Lett. 110, 162003 (2013). http://arxiv.org/abs/1301.7310

Web End =arXiv:1301.7310

73. E. Glover, J. Pires, JHEP 1006, 096 (2010). http://arxiv.org/abs/1003.2824

Web End =arXiv:1003.2824 74. J. Currie, A. Gehrmann-De Ridder, E. Glover, J. Pires, JHEP 1401, 110 (2014). http://arxiv.org/abs/1310.3993

Web End =arXiv:1310.3993

75. Z. Nagy, Z. Trocsanyi, Phys. Rev. D 59, 014020 (1999). http://arxiv.org/abs/hep-ph/9806317

Web End =hep-ph/9806317

76. Z. Nagy, Phys. Rev. Lett. 88, 122003 (2002). http://arxiv.org/abs/hep-ph/0110315

Web End =hep-ph/0110315 77. S. Chatrchyan et al. (CMS Collaboration), Phys. Lett. B 728, 496 (2014). http://arxiv.org/abs/1307.1907

Web End =arXiv:1307.1907

78. M. Czakon, P. Fiedler, A. Mitov, Phys. Rev. Lett. 110, 252004 (2013). http://arxiv.org/abs/1303.6254

Web End =arXiv:1303.6254

79. M. Aliev, H. Lacker, U. Langenfeld, S. Moch, P. Uwer et al., Comput. Phys. Commun. 182, 1034 (2011). http://arxiv.org/abs/1007.1327

Web End =arXiv:1007.1327

80. J.M. Campbell, R. Frederix, F. Maltoni, F. Tramontano, Phys. Rev.

Lett. 102, 182003 (2009). http://arxiv.org/abs/0903.0005

Web End =arXiv:0903.0005

81. J.M. Campbell, F. Tramontano, Nucl. Phys. B 726, 109 (2005). http://arxiv.org/abs/hep-ph/0506289

Web End =hep-ph/0506289

82. J.M. Campbell, R.K. Ellis, F. Tramontano, Phys. Rev. D 70, 094012 (2004). http://arxiv.org/abs/hep-ph/0408158

Web End =hep-ph/0408158

83. J.M. Campbell, R.K. Ellis, Report FERMILAB-PUB-12-078-T (2012). http://arxiv.org/abs/1204.1513

Web End =arXiv:1204.1513

84. M. Guzzi, K. Lipka, S.-O. Moch (2014). http://arxiv.org/abs/1406.0386

Web End =arXiv:1406.0386 85. M. Guzzi, K. Lipka, S. Moch (2014). https://difftop.hepforge.org/

Web End =https://difftop.hepforge.org/ 86. D. Britzger, M. Guzzi, K. Rabbertz, G. Sieber, F. Stober,M. Wobisch, in DIS 2014 (2014). http://indico.cern.ch/event/258017/session/1/contribution/202

Web End =http://indico.cern.ch/event/ http://indico.cern.ch/event/258017/session/1/contribution/202

Web End =258017/session/1/contribution/202 87. C. Adloff et al. (H1 Collaboration), Eur. Phys. J. C 19, 289 (2001). http://arxiv.org/abs/hep-ex/0010054

Web End =hep-ex/0010054

88. T. Kluge, K. Rabbertz, M. Wobisch (2006). http://arxiv.org/abs/hep-ph/0609285

Web End =hep-ph/0609285 89. T. Carli, G.P. Salam, F. Siegert (2005). http://arxiv.org/abs/hep-ph/0510324

Web End =hep-ph/0510324 90. T. Carli et al., Eur. Phys. J. C 66, 503 (2010). http://arxiv.org/abs/0911.2985

Web End =arXiv:0911.2985 91. Z. Nagy, Z. Trocsanyi, Phys. Rev. Lett. 87, 082001 (2001). http://arxiv.org/abs/hep-ph/0104315

Web End =hep-ph/0104315

92. Z. Nagy, Phys. Rev. D 68, 094002 (2003). http://arxiv.org/abs/hep-ph/0307268

Web End =hep-ph/0307268 93. M. Wobisch, D. Britzger, T. Kluge, K. Rabbertz, F. Stober (2011). http://arxiv.org/abs/1109.1310

Web End =arXiv:1109.1310

94. N. Kidonakis, J. Owens, Phys. Rev. D 63, 054019 (2001). http://arxiv.org/abs/hep-ph/0007268

Web End =hep-ph/0007268

95. D. Britzger, K. Rabbertz, F. Stober, M. Wobisch (2012). http://arxiv.org/abs/1208.3641

Web End =arXiv:1208.3641

96. FastNLO. http://fastnlo.hepforge.org

Web End =http://fastnlo.hepforge.org 97. V. Khachatryan et al. (CMS Collaboration) (2014). http://arxiv.org/abs/1410.6765

Web End =arXiv:1410.6765

98. APPLGRID. http://applgrid.hepforge.org

Web End =http://applgrid.hepforge.org 99. G. Aad et al. (ATLAS Collaboration), Phys. Rev. Lett. 109, 012001 (2012). http://arxiv.org/abs/1203.4051

Web End =arXiv:1203.4051

100. S. Chatrchyan et al. (CMS Collaboration), Phys. Rev. D 90,

032004 (2014). http://arxiv.org/abs/1312.6283

Web End =arXiv:1312.6283 101. A. Glazov, S. Moch, V. Radescu, Phys. Lett. B 695, 238 (2011). http://arxiv.org/abs/1009.6170

Web End =arXiv:1009.6170 102. J. Pumplin, D. Stump, R. Brock, D. Casey, J. Huston et al., Phys.

Rev. D 65, 014013 (2001). http://arxiv.org/abs/hep-ph/0101032

Web End =hep-ph/0101032 103. M. Botje, J. Phys. G 28, 779 (2002). http://arxiv.org/abs/hep-ph/0110123

Web End =hep-ph/0110123 104. W.T. Giele, S. Keller, Phys. Rev. D 58, 094023 (1998). http://arxiv.org/abs/hep-ph/9803393

Web End =hep-ph/9803393 105. W.T. Giele, S. Keller, D. Kosower (2001). http://arxiv.org/abs/hep-ph/0104052

Web End =hep-ph/0104052 106. G. Watt, R. Thorne, JHEP 1208, 052 (2012). http://arxiv.org/abs/1205.4024

Web End =arXiv:1205.4024 107. J. Gao, P. Nadolsky, JHEP 1407, 035 (2014). http://arxiv.org/abs/1401.0013

Web End =arXiv:1401.0013 108. HERAFitter Developers Team, M. Lisovyi (2014). http://arxiv.org/abs/1404.4234

Web End =arXiv:1404.4234 109. R.D. Ball, V. Bertone, F. Cerutti, L. Del Debbio, S. Forte et al.,

Nucl. Phys. B 855, 608 (2012). http://arxiv.org/abs/1108.1758

Web End =arXiv:1108.1758 110. R.D. Ball et al. (NNPDF Collaboration), Nucl. Phys. B 849, 112

(2011). http://arxiv.org/abs/1012.0836

Web End =arXiv:1012.0836 111. N. Sato, J. Owens, H. Prosper, Phys. Rev. D 89, 114020 (2014). http://arxiv.org/abs/1310.1089

Web End =arXiv:1310.1089 112. H. Paukkunen, P. Zurita (2014). http://arxiv.org/abs/1402.6623

Web End =arXiv:1402.6623 113. N.N. Nikolaev, B. Zakharov, Z. Phys. C 49, 607 (1991)114. A.H. Mueller, Nucl. Phys. B 415, 373 (1994)115. F. Aaron et al. (H1 Collaboration), Eur. Phys. J. C 71, 1579 (2011). http://arxiv.org/abs/1012.4355

Web End =arXiv:1012.4355 116. I. Balitsky, Nucl. Phys. B 463, 99 (1996). http://arxiv.org/abs/hep-ph/9509348

Web End =hep-ph/9509348 117. S.M. Aybat, T.C. Rogers, Phys. Rev. D 83, 114042 (2011). http://arxiv.org/abs/1101.5057

Web End =arXiv:1101.5057 118. M. Bufng, P. Mulders, A. Mukherjee, Int. J. Mod. Phys. Conf.

Ser. 25, 1460003 (2014). http://arxiv.org/abs/1309.2472

Web End =arXiv:1309.2472 119. M. Bufng, A. Mukherjee, P. Mulders, Phys. Rev. D 88, 054027

(2013). http://arxiv.org/abs/1306.5897

Web End =arXiv:1306.5897 120. M. Bufng, A. Mukherjee, P. Mulders, Phys. Rev. D 86, 074030

(2012). http://arxiv.org/abs/1207.3221

Web End =arXiv:1207.3221 121. P. Mulders, Pramana 72, 83 (2009). http://arxiv.org/abs/0806.1134

Web End =arXiv:0806.1134 122. S. Jadach, M. Skrzypek, Acta Phys. Polon. B 40, 2071 (2009). http://arxiv.org/abs/0905.1399

Web End =arXiv:0905.1399 123. F. Hautmann, Acta Phys. Polon. B 40, 2139 (2009)124. F. Hautmann, M. Hentschinski, H. Jung (2012). http://arxiv.org/abs/1205.6358

Web End =arXiv:1205.6358 125. F. Hautmann, H. Jung, Nucl. Phys. Proc. Suppl. 184, 64 (2008). http://arxiv.org/abs/0712.0568

Web End =arXiv:0712.0568 126. S. Catani, M. Ciafaloni, F. Hautmann, Phys. Lett. B 242, 97

(1990c)127. J.C. Collins, R.K. Ellis, Nucl. Phys. B 360, 3 (1991)128. F. Hautmann, Phys. Lett. B 535, 159 (2002). http://arxiv.org/abs/hep-ph/0203140

Web End =hep-ph/0203140 129. S. Catani, M. Ciafaloni, F. Hautmann, Nucl. Phys. B 366, 135

(1991)130. S. Catani, M. Ciafaloni, F. Hautmann, Phys. Lett. B 307, 147

(1993)131. S. Catani, F. Hautmann, Nucl. Phys. B 427, 475 (1994). http://arxiv.org/abs/hep-ph/9405388

Web End =hep-ph/9405388 132. S. Catani, F. Hautmann, Phys. Lett. B 315, 157 (1993)133. L. Lipatov, Phys. Rep. 286, 131 (1997). http://arxiv.org/abs/hep-ph/9610276

Web End =hep-ph/9610276 134. V.S. Fadin, E. Kuraev, L. Lipatov, Phys. Lett. B 60, 50 (1975) 135. I.I. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978) 136. F. Hautmann, M. Hentschinski, H. Jung, Nucl. Phys. B 865, 54

(2012). http://arxiv.org/abs/1205.1759

Web End =arXiv:1205.1759 137. M. Deak, F. Hautmann, H. Jung, K. Kutak, Forward-

Central Jet Correlations at the Large Hadron Collider (2010). http://arxiv.org/abs/1012.6037

Web End =arXiv:1012.6037 138. M. Deak, F. Hautmann, H. Jung, K. Kutak, Eur. Phys. J. C 72,

1982 (2012). http://arxiv.org/abs/1112.6354

Web End =arXiv:1112.6354 139. F. Hautmann, H. Jung, Nucl. Phys. B 883, 1 (2014). http://arxiv.org/abs/1312.7875

Web End =arXiv:1312.7875 140. H. Jung, F. Hautmann (2012). http://arxiv.org/abs/1206.1796

Web End =arXiv:1206.1796

123

Eur. Phys. J. C (2015) 75 :304 Page 17 of 17 304

141. G. Aad et al. (ATLAS Collaboration), Phys. Lett. B 725, 223

(2013). http://arxiv.org/abs/1305.4192

Web End =arXiv:1305.4192 142. G. Aad et al. (ATLAS Collaboration), JHEP 1406, 112 (2014). http://arxiv.org/abs/1404.1212

Web End =arXiv:1404.1212 143. G. Aad et al. (ATLAS Collaboration), JHEP 1405, 068 (2014). http://arxiv.org/abs/1402.6263

Web End =arXiv:1402.6263 144. G. Aad et al. (ATLAS Collaboration), Eur. Phys. J. 73, 2509

(2013). http://arxiv.org/abs/1304.4739

Web End =arXiv:1304.4739 145. G. Aad et al. (ATLAS Collaboration), Technical Report ATL

PHYS-PUB-2013-018, CERN, Geneva (2013)146. F. Aaron et al. (H1 Collaboration), JHEP 1209, 061 (2012). http://arxiv.org/abs/1206.7007

Web End =arXiv:1206.7007 147. H. Abramowicz et al. (H1 and ZEUS Collaborations), Eur. Phys.J. C 73, 2311 (2013). http://arxiv.org/abs/1211.1182

Web End =arXiv:1211.1182 148. H. Abramowicz et al. (ZEUS Collaboration) (2014), http://arxiv.org/abs/1405.6915

Web End =arXiv:1405.6915

149. A. Luszczak, H. Kowalski, Phys. Rev. D 89, 074051 (2013). http://arxiv.org/abs/1312.4060

Web End =arXiv:1312.4060

150. R. Sadykov (2014). http://arxiv.org/abs/1401.1133

Web End =arXiv:1401.1133 151. S. Dooling, F. Hautmann, H. Jung, Phys. Lett. B 736, 293 (2014). http://arxiv.org/abs/1406.2994

Web End =arXiv:1406.2994 152. HERAPDF1.5LO, NLO and NNLO (H1prelim-13-141 and

ZEUS-prel-13-003, H1prelim-10-142 and ZEUS-prel-10-018, H1prelim-11-042 and ZEUS-prel-11-002). Available via: http://lhapdf.hepforge.org/pdfsets

Web End =http:// http://lhapdf.hepforge.org/pdfsets

Web End =lhapdf.hepforge.org/pdfsets 153. Atlas, NNLO epWZ12, Available via: http://lhapdf.hepforge.org/pdfsets

Web End =http://lhapdf.hepforge.org/ http://lhapdf.hepforge.org/pdfsets

Web End =pdfsets 154. J. Butterworth, G. Dissertori, S. Dittmaier, D. de Florian, N.

Glover et al. (2014). http://arxiv.org/abs/1405.1067

Web End =arXiv:1405.1067 155. J. L. Abelleira Fernandez et al. (LHeC Study Group), J. Phys. G,

075001 (2012). http://arxiv.org/abs/1206.2913

Web End =arXiv:1206.2913

123

Word count: 12844

Show less

SIF and Springer-Verlag Berlin Heidelberg 2015

Abstract

Translate

HERAFitter is an open-source package that provides a framework for the determination of the parton distribution functions (PDFs) of the proton and for many different kinds of analyses in Quantum Chromodynamics (QCD). It encodes results from a wide range of experimental measurements in lepton-proton deep inelastic scattering and proton-proton (proton-antiproton) collisions at hadron colliders. These are complemented with a variety of theoretical options for calculating PDF-dependent cross section predictions corresponding to the measurements. The framework covers a large number of the existing methods and schemes used for PDF determination. The data and theoretical predictions are brought together through numerous methodological options for carrying out PDF fits and plotting tools to help to visualise the results. While primarily based on the approach of collinear factorisation, HERAFitter also provides facilities for fits of dipole models and transverse-momentum dependent PDFs. The package can be used to study the impact of new precise measurements from hadron colliders. This paper describes the general structure of HERAFitter and its wide choice of options.

Details

Title

HERAFitter

Author

Alekhin, S; Behnke, O; Belov, P; Borroni, S; Botje, M; Britzger, D; Camarda, S; Cooper-sarkar, A M; Daum, K; Diaconu, C; Feltesse, J; Gizhko, A; Glazov, A; Guffanti, A; Guzzi, M; Hautmann, F; Jung, A; Jung, H; Kolesnikov, V; Kowalski, H; Kuprash, O; Kusina, A; Levonian, S; Lipka, K; Lobodzinski, B; Lohwasser, K; Luszczak, A; Malaescu, B; Mcnulty, R; Myronenko, V; Naumann-emme, S; Nowak, K; Olness, F; Perez, E; Pirumov, H; Placakyt, R; Rabbertz, K; Radescu, V; Sadykov, R; Salam, G P; Sapronov, A; Schöning, A; Schörner-sadenius, T; Shushkevich, S; Slominski, W; Spiesberger, H; Starovoitov, P; Sutton, M; Tomaszewska, J; Turkot, O; Vargas, A; Watt, G; Wichmann, K

Pages

1-17

Publication year

2015

Publication date

Jul 2015

Publisher

Springer Nature B.V.

ISSN

14346044

e-ISSN

14346052

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1140/epjc/s10052-015-3480-z

ProQuest document ID

1707494875

SIF and Springer-Verlag Berlin Heidelberg 2015

HERAFitter: Open source QCD fit project

Jump to:

Full Text

Abstract

Details

Suggested sources