Resummation of double-differential cross sections

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Published for SISSA by Springer

Received: October 30, 2014

Accepted: January 17, 2015 Published: February 18, 2015

Massimiliano Procura,a Wouter J. Waalewijnb,c and Lisa Zeuneb

aAlbert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern,CH-3012 Bern, Switzerland

bITFA, University of Amsterdam,

Science Park 904, 1018 XE, Amsterdam, The Netherlands

cTheory Group, Nikhef,

Science Park 105, 1098 XG, Amsterdam, The Netherlands

E-mail: mailto:[email protected]

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

Web End [email protected]

Abstract: LHC measurements involve cuts on several observables, but resummed calculations are mostly restricted to single variables. We show how the resummation of a class of double-di erential measurements can be achieved through an extension of Soft-Collinear E ective Theory (SCET). A prototypical application is pp ! Z + 0 jets, where the jet

veto is imposed through the beam thrust event shape T , and the transverse momentum

pT of the Z boson is measured. A standard SCET analysis su ces for pT m1/2ZT 1/2 and

pT T , but additional collinear-soft modes are needed in the intermediate regime. We

show how to match the factorization theorems that describe these three di erent regions of phase space, and discuss the corresponding relations between fully-unintegrated parton distribution functions, soft functions and the newly dened collinear-soft functions. The missing ingredients needed at NNLL/NLO accuracy are calculated, providing a check of our formalism. We also revisit the calculation of the measurement of two angularities on a single jet in JHEP 1409 (2014) 046, nding a correction to their conjecture for the NLL cross section at O( 2s).

Keywords: Resummation, E ective eld theories, QCD

ArXiv ePrint: 1410.6483

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP02(2015)117

Web End =10.1007/JHEP02(2015)117

Resummation of double-di erential cross sections and fully-unintegrated parton distribution functions

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Contents

1 Introduction 1

2 Factorization 52.1 E ective theory for the region between SCETI and SCETII boundaries 52.2 Factorization formulae 8

3 Ingredients at NNLL 113.1 Hard function 113.2 Beam functions 123.3 Soft functions 133.4 Collinear-soft function 153.5 Renormalization and anomalous dimensions 173.6 NLL cross section 18

4 Matching the e ective theories 19

5 NLO cross section 215.1 Ingredients 225.2 Cancellation of IR divergences 245.3 Result 255.4 Comparison to resummed predictions 26

6 Measuring two angularities on one jet 29

7 Conclusions 32

A Plus distributions 33

B Renormalization group evolution 33

1 Introduction

Experimental LHC analyses typically involve several kinematic cuts. Many of them are fairly harmless from a theoretical point of view. However, when these restrictions on initial-and/or nal-state radiation lead to widely separated energy scales, large logarithms can be induced in the corresponding cross section, requiring resummation. One example is given by the jet veto used to suppress backgrounds in Higgs analyses, where the resummation of jet-veto logarithms [16] greatly reduces the dominant source of theoretical uncertainty. A closely related process is Drell-Yan (or vector boson) production in the case the lepton

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pair has a small pT compared to their invariant mass Q [717]. Another example is the jet mass mJ spectrum of a jet with transverse momentum pJT , which requires resummation around the peak of the distribution where mJ pJT [1821].

In this paper we focus on double di erential measurements, where both observables lead to large logarithms. Using e ective eld theory methods, we derive new resummed expressions for a class of double di erential cross sections. Our results smoothly connect to the phase space boundaries, which require di erent e ective eld theories. This formalism has applications to jet cross sections and jet substructure studies, and we will consider an example of both in this paper.

As the eld of jet substructure has matured [2224], multivariate analyses have become common. Furthermore, some of the measurements with the best discrimination power are ratios of infrared and collinear safe observables, such as ratios of N-subjettiness [2527], energy correlation functions [2830] or planar ow [31, 32]. These quantities are themselves not infrared and collinear safe, and their calculation involves marginalizing over the resummed two-dimensional distribution [33]. The pioneering study in ref. [34], investigating the measurement of two angularities on one jet, inspired the present paper.

Our formalism can also be applied to pp ! H + 1 jet production, where in addition to

the jet veto the transverse momentum of the jet becomes small. This important contribution to the cross section is not yet fully understood [6]. In this paper, to better illustrate the features of our framework, we will mainly focus on a simpler (but related) problem in Z + 0 jet production, carrying out the simultaneous resummation of the jet veto and the transverse momentum of the Z boson.

Resummation is often achieved using the parton shower formalism. The great advantage of parton shower Monte Carlo event generators, such as Pythia [35] and Herwig [36], is that they produce a fully exclusive nal state, giving the user full exibility. On the other hand, this approach is limited to leading logarithmic (LL) accuracy, and it is di cult to estimate the corresponding theory uncertainty. It is also not clear to what extent correlations between resummed observables are correctly predicted by Monte Carlo models, see e.g. ref. [37]. By contrast, we predict these correlations and our resummed predictions have a theory uncertainty attached to it, whose reliability can be veried by comparing di erent orders in resummed perturbation theory. Note that there has been signicant progress by matching higher-order matrix elements with parton showers (see e.g. refs. [3845]) and (partially) including higher-order resummation [42].

We will illustrate the features of our framework in the specic case of pp ! Z + 0 jets,

where the transverse momentum pT of the Z boson is measured and a global jet veto is imposed using the beam thrust event shape [1, 46]

T =Xi

piT e[notdef] i[notdef] = Ximin[notdef]p+i, pi[notdef] . (1.1)

The sum on i runs over all particles in the nal state, except for the leptonic decay products of the Z. Here, piT is the magnitude of the transverse momentum and i the pseudorapidity of particle i in the center-of-mass frame of the hadronic collision. Light-cone coordinates

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Figure 1. The di erent regions for the double measurement of pT and beam thrust T in Z-boson

production from pp collisions.

are dened as

p[notdef]i = p+i

n[notdef]

2 + pi

n[notdef]

2 + p[notdef]i? , pi = n[notdef]pi , p+i = n[notdef]pi , (1.2)

where n[notdef] = (1, 0, 0, 1) and n[notdef] = (1, 0, 0, 1) are along the beam axis. Beam thrust imposes

a global veto on all radiation in an event, which is impractical in the LHC environment. This can be remedied by only including the contributions from jet regions in eq. (1.1) [47]. We will nevertheless consider the global veto to keep our discussion as simple as possible. At the end of section 2.2 we will comment on a special class of non-global measurements whose logarithms can easily be resummed within our approach.

We will perform resummations using Soft-Collinear E ective Theory (SCET) [4851]. Which version of SCET is the appropriate one, namely what the relevant degrees of freedom are, depends on the region of phase space probed by the measurement, as shown in gure 1 and discussed below. We nd that in the intermediate region, between the SCETI and

SCETII boundaries, the e ective eld theory involves additional collinear-soft modes. This type of mode was introduced in a di erent context in ref. [52], and has led us to also refer to our e ective theory as SCET+. Since we are considering di erent observables than ref. [52], there are of course important di erences, which will be discussed in section 2.1. We now comment on the theoretical description relevant for each region of phase space in the (pT , T ) plane.

Fixed Order: pT , T Q

When pT and T are parametrically of the same size as the hard scale Q2 = p2Z m2Z,

resummation is not necessary and a xed-order calculation su ces.

SCETI: pT Q1/2T 1/2

This case was discussed in ref. [53]. The collinear and soft modes, shown in the left panel of gure 2, interact. The SCETI scale hierarchy implies that the soft radiation contributes only to T (its contribution to pT is power suppressed), whereas

the collinear radiation contributes both to the pT and the T measurement. This

collinear radiation is described by fully-unintegrated parton distribution functions (PDFs) [5356], which depend on all momentum components of the colliding parton. By contrast, the standard PDFs only depend on the momentum fraction x.

SCET+: pT Q1rT r with 1/2 < r < 1

As pT is lowered, the collinear modes can no longer interact with the soft mode. They split o collinear-soft modes that do interact with the soft modes, see gure 2. (To have a distinct mode contribution requires su cient distance from the SCETI and SCETII boundaries.) In this scenario, the collinear radiation only contributes to pT , the soft radiation only to T , and the collinear-soft radiation enters in both

measurements. The SCET+ power counting will be given below in table 1.

SCETII: pT T

As pT is reduced further, the soft mode absorbs the two collinear-soft modes. In the resulting theory there are no interactions between the collinear and the soft modes, as shown in the left panel of gure 2. The collinear radiation, which in the SCETII

case is described by transverse-momentum dependent (TMD) PDFs, only a ects pT ,

whereas now the soft mode contributes to both measurements.

Z+forward jet: pT Q1/2T 1/2

As pT exceeds this bound, the QCD radiation becomes (much) more energetic than the invariant mass Q of the Z boson. This cannot be described as initial-state radiation, but rather as Z production in association with an energetic forward jet.

Terra incognita: pT T

Unlike the previous regions, the cross section no longer receives a contribution from a single emission. There is a small NNLO contribution from the region of phase space where the two emissions are (almost) back-to-back in the transverse plane. In double parton scattering (DPS) the production of the Z and the two jets are (largerly) independent of each other, causing the jets to naturally be back-to-back.1 The contribution from DPS is therefore also important. As the proper method for combining single and double parton scattering is still under debate [5964], we leave this for future work.

In this paper, we also show how to combine the SCETI, SCET+ and SCETII regions to achieve NNLL resummation throughout. The corresponding next-to-leading order cross section is calculated, providing a check of our results.

In most earlier studies of multi-dimensional observables in SCET, such as refs. [65, 66], the measurements concerned di erent regions of phase space (hemispheres, jets, etc.). There, resummation is achieved by assuming a single parametric relation between the observables, to avoid so-called non-global logarithms [67, 68]. In ref. [34] the two boundary theories for the measurement of two angularities on a single jet were identied. There an

1This feature is exploited to extract DPS experimentally, see e.g. refs. [57, 58].

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SCETI SCETII

SCET+

p2 p2T T Q

p2T

p2 p2T T 2

p2 p2T

T T T

T T

p2 T 2 p2 T 2

p+ Q

p2T /T

p+ Q

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Figure 2. The modes in SCETI, SCET+ and SCETII: collinear (green), collinear-soft (blue) and soft (orange). Interactions between modes in the e ective theory are shown with wiggly lines. These are removed by the decoupling transformations in eq. (2.7).

interpolating function across the intermediate region was derived, by requiring it to be continuous and have a continuous derivative at the boundaries. We revisit their NLL results and nd a discrepancy at O( 2s) in the bulk. It is worth mentioning that in this case both

boundaries involve SCETI-type theories, to which our framework can be applied as well.

The paper is structured as follows. In section 2.1 we introduce SCET+, perform the matching of QCD onto SCET+ currents, and comment on the (dis)similarities with the theory introduced in ref. [52]. Section 2.2 contains the factorization formulae for the Drell-Yan cross section with a simultaneous measurement of pT and T in the SCETI, SCET+

and SCETII regions of phase space, as well as the eld-theoretic denitions of the matrix elements involved. We calculate/collect all the ingredients necessary to achieve NNLL accuracy in section 3 and discuss the (all-order) matching of SCETI, SCET+ and SCETII in section 4. The corresponding NLO cross section is calculated in section 5, providing a verication of our resummed predictions. In section 6 we calculate the double angularity measurement on a single jet and compare with ref. [34]. Conclusions and outlook are presented in section 7.

2 Factorization

2.1 E ective theory for the region between SCETI and SCETII boundaries

Soft-Collinear E ective Theory (SCET) [4851] describes the collinear and soft limits of QCD. For a pedagogical introduction see e.g. refs. [69, 70]. SCET captures QCD in the infrared regime up to corrections that are suppressed by powers of the SCET expansion parameter 1, in exchange for enabling the resummation of large logarithms of . As

discussed in section 1, both the process and measurement determine which modes give the leading contributions to the cross section in a specic kinematic regime. In gure 2 we summarize the scalings and interactions between di erent degrees of freedom leading to the physical picture in section 1. These modes need to be well-separated, in order for to be small. The decoupling of modes in the SCET Lagrangian (at leading power) allows one to

Figure 3. The [notdef]-evolution resums double logarithms from separations in virtuality (between hyperbolae), while the -evolution resums single logarithms related to separations in rapidity (along hyperbolae). The collinear, collinear-soft and soft modes are depicted in green, blue and orange, respectively.

Mode: Scaling (, +, ?)

n-collinear Q(1, 2r, r) (Q, p2T /Q, pT ) n-collinear Q( 2r, 1, r) (p2T /Q, Q, pT )

n-collinear-soft Q( 2r1, , r) (p2T /T , T , pT )

n-collinear-soft Q( , 2r1, r) (T , p2T /T , pT )soft Q( , , ) (T , T , T )

Table 1. Modes and power counting in SCET+ with T /Q (pT /Q)1/r.

factorize multi-scale cross sections into products (or convolutions) of single-scale functions for each mode. At its natural scale, each of these function contains no large logarithms. By applying the renormalization group (RG) evolution from these natural scales to a common scale [notdef], we achieve resummation of logarithms of in the cross section. For modes that are not separated in virtuality but only in rapidity, we will sum the corresponding single logarithms through the -evolution of the rapidity renormalization group [71, 72].2 Pictorially, the [notdef]-evolution sums logarithms related to the separation between the mass hyperbolae of the modes, whereas the -evolution sums the logarithms related to the separation along them, see gure 3.

We will now discuss SCET+ in some detail, focussing on modes, matching of QCD onto SCET+ and factorization. We refrain from performing a full formal construction of the e ective theory. Factorization means there are no interactions between the various modes, and each mode is described by a (boosted) copy of QCD. In particular, one can use the standard QCD Feynman rules (rather than e.g. the collinear e ective Lagrangian of ref. [49]) to carry out the computations for each sector.

The measurement of beam thrust T and transverse momentum pT , with pT Q1rT r

and 1/2 < r < 1,3 suggests that the relevant modes are those listed in table 1 and shown

2For alternative approaches to rapidity resummation in SCET, see e.g. refs. [14, 73].

3Note that our analysis is independent of the parameter r, as is clear from the second way of writing the modes in table 1. However, we prefer to use a single power counting parameter .

p2T

p2T /T

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in the center panel of gure 2, with power counting parameter

TQ [parenleftBig]

A collinear mode only a ects the pT -distribution, as the contribution to T from its small

light-cone component is power suppressed. Similarly, a soft mode only contributes to T ,

whereas the collinear-soft modes contributes to both measurements. These on-shell modes are uniquely specied by these features. Of course, additional (redundant) modes may be included, as long as the double counting is removed (for example by an appropriate zero-bin subtraction [74]). As usual, we will assume the cancellation of (o -shell) Glauber modes. These account for initial-state hadron-hadron interactions taking place before the collision, which would ruin factorization [75]. This cancellation has only been rigorously proven for inclusive Drell-Yan [76], and could be spoiled due to our pT and T measurements [77].

The QCD quark and gluon elds are decomposed into several SCET elds which scale di erently with respect to the expansion parameter . By matching quark currents from QCD onto SCET+ we obtain

= C(Q2, [notdef])

nWnSnXnVn V nXnSnW nn . (2.2)

The matching coe cient C(Q2, [notdef]) captures the e ect of hard virtual gluon exchanges not present in the e ective theory. In eq. (2.2), n and

n are the elds for collinear (anti-) quarks moving in the n (n) direction and denotes a generic Dirac structure. The Wilson line Wn arises from n-collinear gluons emitted by

(which itself is n-collinear) [50]

. (2.3)

The Wilson line Vn is its direct analog for n-collinear-soft gluons (obtained by replacing An ! Ancs). Soft gluons emitted by are summed into the Wilson line Sn [51]

Sn = P exp

[bracketleftbigg]

0 du n[notdef]As(u n)

and the analog for n-collinear-soft gluons is Xn.

To x the ordering of Wilson lines, we exploit gauge invariance of SCET+. In order to preserve the scaling of the elds, separate collinear, collinear-soft and soft gauge transformations have to be introduced, see e.g. refs. [51, 52]. Only the n-collinear elds transform under n-collinear gauge transformations. The other elds are taken far o -shell and are thus unable to resolve the local change induced by this gauge transformation. This causes W nn

and

nWn to be grouped together. Under a n-collinear-soft gauge transformation Uncs

W nn ! W nn , Sn ! Sn , Vn ! UncsVn , Xn ! UncsXn ,

nWn !

nWn , Sn ! Sn , Vn ! Vn , Xn ! Xn , (2.5)

which groups V nXn together. Similarly, XnVn must be grouped together by n-collinear-soft gauge invariance. The e ect of a soft gauge transformation Us is given by

W nn ! W nn , Sn ! UsSn , Vn ! UsVnUs , Xn ! UsXnUs ,

nWn !

nWn , Sn ! UsSn , Vn ! UsVnUs , Xn ! UsXnUs . (2.6)

1/r

. (2.1)

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Wn = P exp

0 du n[notdef]An(u n)

[bracketleftbigg]

[integraldisplay]

[bracketrightbigg]

[integraldisplay]

[bracketrightbigg]

, (2.4)

The soft gluon eld acts as smooth background for collinear-soft modes, implying that the e ect of a soft gauge transformation on collinear-soft modes is similar to a global color rotation. This almost xes the ordering in eq. (2.2). There are still a few other possibilities that satisfy the constraints from gauge invariance, such as

nWnSnV nXn XnVnSnW nn. However, these can be ruled out by considering the tree-level matching of QCD onto SCET+.

At this point the soft elds still interact with the collinear-soft elds, as indicated in the middle panel of gure 2. By performing the analog of the BPS eld redenition [51], we decouple the soft elds from the collinear-soft elds,

Vn ! SnVnSn , Xn ! SnXnSn ,

Vn ! SnVnSn , Xn ! SnXnSn . (2.7)

This leads to

= C(Q2, [notdef])

nWnXnVnSn SnV nXnW nn . (2.8)

The various modes in this matching equation no longer interact and the derivation of factorization formulae now follows the standard procedure in SCET. In particular, establishing factorization to all orders in s requires decoupling of the di erent modes in the

Lagrangian, for which we refer to ref. [52].

One expects that this matching receives power corrections of the size 2r1 p2T /(QT ) and 22r T 2/p2T , which measure the distance from the respective SCETI and SCETII

boundary regions of phase space. In our NLO calculation in section 5 we nd corrections of the rst type but not of the second. However, we expect that this will no longer be the case at higher orders.

Finally, we briey comment on the (dis)similarities of our theory with the SCET+ introduced in ref. [52]. In that paper the dijet invariant mass (mj1j2) distribution for nearby jets is calculated, with the hierarchy mj1, mj2 mj1j2 Q. Their collinear-soft modes

can resolve the two nearby jets, whereas the soft modes do not, and the collinear modes are restricted to the individual jets. Their factorization theorem involves convolutions through the small collinear light-cone component. Since we consider di erent type of observables, our convolutions of collinear-soft modes with either collinear or soft radiation have a di erent structure. The matching in ref. [52] was (also) performed in two steps, where in the rst step the two nearby jets are not resolved from each other. Nevertheless, the similarities between the modes and Wilson lines in our and their approach seemed su cient to us to adopt the same name for our e ective theory.

2.2 Factorization formulae

We now discuss SCET factorization formulae for Drell-Yan cross sections that are di erential both in T and pT , both at the SCETI and SCETII phase space boundaries and in

the SCET+ bulk. In Drell-Yan production, pp ! Z/ ! [lscript]+[lscript], the lepton pair has a

large invariant mass Q. A proof of factorization at leading power in QCD/Q has been

given by Collins, Soper and Sterman [9], for any value of the transverse momentum pT of the lepton pair, namely for both pT Q and pT Q. Here we impose in addition a

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veto on hard central jets through a cut on beam thrust T in the center-of-mass frame of

the pp collision [1], see eq. (1.1). We consider di erent kinematic regimes for pT and T ,

as discussed in the introduction. We will not perform the joint resummation of threshold logarithms that becomes important as Q approaches the total CM energy Ecm [78].

If QCD pT (T Q)1/2 Q (SCETI case), we have the following leading-power factorization formula [46, 53]

dQ2 dY dp2T dT

0q H(Q2, [notdef])

[integraldisplay]

dt1 dt2

[integraldisplay]

d2[vector]k1? d2[vector]k2? [integraldisplay]

dk+ S(k+, [notdef])

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[notdef] hBq(t1, x1,[vector]k1?, [notdef]) Bq(t2, x2,[vector]k2?, [notdef]) + (q $ q) [bracketrightBig] [notdef]

eY t1 + eY t2Q k+[parenrightbigg]

p2T [notdef][vector]k1? +[vector]k2?[notdef]

[parenrightbig]

2 , (2.9)

whose ingredients we will describe below. The sum extends over the various quark avors, Y is the total rapidity of the leptons, and the momentum fractions of the colliding partons are

x1 = Q

Ecm eY , x2 =

Ecm eY . (2.10)

The quantities eY t1/Q, eY t2/Q and k+ in eq. (2.9) are the contributions to T from the n

collinear, n-collinear and soft radiation. For n-collinear radiation, we always have p+i < pi, for n-collinear radiation, p+i > pi, whereas the soft radiation can go into both hemispheres (p+i < pi and p+i > pi).

At leading order in the electroweak interactions,

0q = 4 2em

9Q2E2cm

Q2q +(v2q + a2q)(v2[lscript] + a2[lscript]) 2Qqvqv[lscript](1 m2Z/Q2)(1 m2Z/Q2)2 + m2Z 2Z/Q4 [bracketrightbigg]

, (2.11)

where Qq is the quark charge in units of [notdef]e[notdef], v[lscript],q and a[lscript],q are the standard vector and

axial couplings of the leptons and quarks, and mZ and Z are the mass and width of the Z boson.

The hard function H(Q2) is the square of the Wilson coe cient C(Q2) for the matching of QCD onto SCET vector and axial quark currents4

H(Q2, [notdef]) = [notdef]C(Q2, [notdef])[notdef]2 . (2.12)

It does not depend on pT , since we only consider pT Q.5 Since lepton masses are

neglected, there is no contribution from gluon operators in the matching of the (axial) currents [46]. The gluon PDF only appears through its contribution to the quark beam function, see eq. (3.2).

Due to the SCETI hierarchy of scales, the e ect of soft radiation on the pT -distribution is power suppressed, so only the fully-unintegrated (FU) PDFs account for the recoil of the energetic initial-state radiation against the nal-state leptons. Because we consider

4As compared to eq. (2.2), in SCETI only collinear and (ultra-)soft Wilson lines enter the matching.

5The leptonic tensor in the Drell-Yan process does not depend on pT at leading order.

perturbative pT , T QCD, we will refer to these as FU beam functions in the following.

At the bare level, these are dened as the following proton matrix element of collinear elds [53]

Bq(t, x,[vector]k?) =

Dpn(p)

[vextendsingle][vextendsingle][vextendsingle]

[vector]n(0) n/

(kp+P) (tkP+) 2([vector]k?

[vector]P?)[vector]n(0)

[bracketrightbig] [vextendsingle][vextendsingle][vextendsingle] pn(p)

(2.13)

The light-like vector n[notdef] is along the direction of the incoming proton (i.e. p[notdef] = Ecmn[notdef]/2) and the operator P returns the momentum of the intermediate state.6 By boost invariance along the n-direction, these functions only depend on the momentum fraction x = k/p, the transverse virtuality t = kk+ of the colliding parton, and the transverse momentum

[vector]k? [53, 79].

The (ultra-)soft radiation is described by the beam thrust soft function S(k) [46]. This is given in terms of a soft Wilson-line correlator as

S(k+) = 1Nc [angbracketleft]0[notdef]Tr

[notdef]0[angbracketright] , (2.14)

where (T) T denotes (anti)time ordering and the operator P1 (P2) gives the momentum of the soft radiation going into the hemisphere dened by p+i < pi (p+i > pi).

In the region of phase space described by SCET+ ( QCD T pT (T Q)1/2 Q),

dQ2 dY dp2T dT

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T(Sn(0)Sn(0)) (k+ P+1 P2) T(Sn(0)Sn(0))

0q H(Q2, [notdef])

[integraldisplay]

d2[vector]k1? d2[vector]k2? d2[vector]k

cs1? d2[vector]k

cs 2?

[integraldisplay]

dk+1 dk+2 dk+ S(k+, [notdef])

[notdef] Bq(x1,[vector]k1?, [notdef], ) Bq(x2,[vector]k2?, [notdef], ) S k+1,[vector]k

cs1?, [notdef],

[parenrightbig]

S k+2,[vector]k

cs2?, [notdef],

[parenrightbig]

+(q $ q) .(2.15)

The contribution from collinear radiation is now encoded in TMD beam functions,

Bq(x,[vector]k?) =

Dpn(p)

[notdef] T k+1 k+2 k+

p2T [notdef][vector]k1? +[vector]k2? + [vector]k

cs1? +[vector]k

cs 2?[notdef]2

[vextendsingle][vextendsingle][vextendsingle]

[vector]n(0) n/

(k p + P) 2([vector]k?

[vector]P?) [vector]n(0)

[bracketrightbig] [vextendsingle][vextendsingle][vextendsingle]

(2.16)

Their naive denition using dimensional regularization is known to su er from light-cone singularities (rapidity divergences), which we regulate following refs. [71, 72]. There are separate but identical collinear-soft functions for the n and n direction,

S (k+,[vector]k?) =

1Nc [angbracketleft]0[notdef]Tr

T(Xn(0)Vn(0)) (k+P+) 2([vector]k?

[vector]P?)T(V n(0)Xn(0))

[notdef]0[angbracketright] ,

pn(p)

[angbracketrightBig]

= 1

Nc [angbracketleft]0[notdef]Tr

T(V n(0)Xn(0)) (k+P) 2([vector]k?

[vector]P?) T(Xn(0)Vn(0))

[notdef]0[angbracketright] , (2.17)

which are also a ected by rapidity divergences.

For the hierarchy QCD pT T Q, soft modes have the same virtuality and transverse momentum as the collinear ones, and contribute both to T and pT measure

ments. The corresponding SCETII factorization theorem has the form

6We can avoid using the label-momentum formalism employed in e.g. refs. [53, 79] since after factorization the collinear sector is simply a boosted copy of QCD.

dQ2 dY dp 2T dT

0q H(Q2, [notdef])

[integraldisplay]

d2[vector]k1? d2[vector]k2? d2[vector]k? [integraldisplay]

dk+ p 2T [notdef][vector]k1?+[vector]k2?+[vector]k?[notdef]

2 T k+ [parenrightbig]

[notdef] hBq(x1,[vector]k1?, [notdef], ) Bq(x2,[vector]k2?, [notdef], ) + (q $ q) [bracketrightBig]

S(k+,[vector]k?, [notdef], ) . (2.18)

The new ingredient is given by the FU soft function, which is dened as

S(k+,[vector]k?) =

1Nc [angbracketleft]0[notdef]Tr

T(Sn(0)Sn(0)) (k+ P+1 P2) 2([vector]k?

[vector]P?) T(Sn(0)Sn(0))

[notdef]0[angbracketright] .(2.19)

It is natural to ask to what extent our approach can be used to calculate non-global logarithms, which arise when di erent restrictions are applied to distinct regions of phase space [67, 68]. If instead of the transverse momentum of the Z boson one measures the pT,ISR of the initial-state radiation that recoils against it, we could e.g. restrict ourselves to the ISR in one hemisphere. In this case the factorization theorem in the region of phase space described by SCET+ is simply modied to

dQ2 dY dp2T,ISR dT

0q H(Q2, [notdef])

[integraldisplay]

dt2

[integraldisplay]

d2[vector]k1? d2[vector]k

cs 1?

[integraldisplay]

dk+1 dk+ S(k+, [notdef])

JHEP02(2015)117

[notdef] Bq(x1,[vector]k1?, [notdef], ) Bq(t2, x2, [notdef]) S k+1,[vector]k

cs1?, [notdef],

[parenrightbig]

(2.20)

+(q $ q) .

However, this does not address the problem arising when the soft function contains multiple scales (see for example [8082]), which occurs when e.g. the beam thrust measurement is restricted to one hemisphere.

3 Ingredients at NNLL

In this section we collect the expressions for the ingredients entering the factorization formulae in section 2.2, to the accuracy needed for NNLL resummations: the hard function at one loop is discussed in section 3.1, the FU and TMD beam function in section 3.2, the FU and beam thrust soft function in section 3.3 and the collinear-soft function in section 3.4. The FU soft function and collinear-soft function are calculated for the rst time. RG equations and anomalous dimensions for NNLL resummation are given in section 3.5 and appendix B. The anomalous dimensions of the collinear-soft function and FU soft function satisfy the consistency requirement imposed by the [notdef] and independence of the factorized cross sections in eqs. (2.15) and (2.18). In section 3.6 we combine these ingredients to obtain a compact expression for the NLL cross section.

3.1 Hard function

The one-loop Wilson coe cient C(Q2, [notdef]) from matching the quark current in QCD onto SCET was computed in refs. [83, 84]. Here Q2 is the square of the partonic center of mass

[notdef] T k+1

eY t2

Q k+

p2T,ISR [notdef][vector]k1? + [vector]k

cs 1?[notdef]2

energy. The matching is the same for SCETI, SCET+ and SCETII, because all e ective eld theory diagrams are scaleless and vanish in dimensional regularization. At one loop,

H(Q2, [notdef]) =

[vextendsingle][vextendsingle]C(Q2,

[notdef])

[vextendsingle][vextendsingle]2

= 1 + sCF 2

ln2 [parenleftbigg]

Q2[notdef]2[parenrightbigg]+ 3 ln

Q2[notdef]2[parenrightbigg] 8 + 726[bracketrightbigg]

. (3.1)

3.2 Beam functions

The FU beam function was dened in eq. (2.13), and its arguments t and [vector]k 2

? are restricted

to be of the same parametric size. As we assume that these scales are perturbative, the FU beam function can be matched onto PDFs [16, 46, 53, 79]

Bq(t, x,[vector]k?, [notdef])=Xj=u,,d,g...

[integraldisplay]

JHEP02(2015)117

dx[prime]

x[prime] Iqj[parenleftbigg]

t, x

x[prime] ,[vector]k?, [notdef][parenrightbigg]

fj(x[prime], [notdef])

1 + O

2QCDt , 2QCD [vector]k 2

[parenrightbigg][bracketrightbigg]

. (3.2)

Because of the kinematic bound [vector]k 2

? (1 x)t/x (see eq. (1.1) of ref. [53]), the renormal

ization is the same as the standard beam function and

[integraldisplay]

d2[vector]k? Bq(t, x,[vector]k?, [notdef]) = Bq(t, x, [notdef]) . (3.3)

Up to NLO, the matching coe cients in eq. (3.2) are [53]

I(0)qq(t, x,[vector]k?, [notdef]) = (t) (1 x) 2([vector]k?) ,

I(0)qg(t, x,[vector]k?, [notdef]) = 0 ,

I(1)qq(t, x,[vector]k?, [notdef]) =

s([notdef])CF

2[notdef]2 L1

[parenleftbigg]

t [notdef]2

(1x) ([vector]k2 ?)

+ 1

[notdef]2 L0

[parenleftbigg]

t [notdef]2

(1+x2) L0(1x) [parenleftbigg]

[vector]k 2? (1x)t

[parenrightbigg]

+ (t) ([vector]k 2

(1+x2)L1(1x)

26 (1 x) 1+x21xln x + 1 x

[bracketrightbigg]

I(1)qg(t, x,[vector]k?, [notdef]) =

s([notdef])TF

[bracketleftbigg]

1[notdef]2 L0

t [notdef]2

[vector]k 2? (1x)t

[parenrightbigg]+ (t) ([vector]k 2?) ln1 x x

[bracketrightbigg]

[notdef]

x2+(1x)2[bracketrightbig]+ 2 (t) ([vector]k 2?) x(1 x)

, (3.4)

where some additional factors of 1/ are due to

2([vector]k?) =

1 ([vector]k

?) . (3.5)

The matching coe cients at NNLO have recently been calculated in ref. [85].

The TMD beam function satises a similar equation [9, 14, 72, 86]

Bq(x,[vector]k?, [notdef], ) =

[integraldisplay]

dx[prime]

x[prime] Iqj[parenleftbigg]

x[prime] ,[vector]k?, [notdef], [parenrightbigg]

fj(x[prime], [notdef])

1 + O

2QCD [vector]k 2

[parenrightBig][bracketrightbigg]

, (3.6)

with coe cients [87]

I(0)qq(x,[vector]k?, [notdef], ) = (1 x) 2([vector]k?) ,

I(0)qg(x,[vector]k?, [notdef], ) = 0 ,

I(1)qq(x,[vector]k?, [notdef], ) =

sCF

1[notdef]2 L0

[vector]k 2?

[notdef]2[parenrightbigg][bracketleftbigg](1+x2)L0(1x) + 2 (1x) lnp

+ ([vector]k 2 ?)(1x)

I(1)qg(x,[vector]k?, [notdef], ) =

sTF

1[notdef]2 L0

[vector]k 2?

[notdef]2[parenrightbigg][bracketleftbig]

x2 + (1 x)2

[bracketrightbig]

. (3.7)

Most approaches (such as in refs. [14, 17, 88]) do not (need to) separate the TMD beam and TMD soft function. In the SCET+ regime, instead, we need the TMD beam function but have a di erent soft function.

3.3 Soft functions

The (beam) thrust soft function was determined at NLO in refs. [46, 89, 90]

S(k+, [notdef]) = (k+) + s CF

8 [notdef]L1

k+[notdef]

[parenrightbigg]+ 26 (k+)[bracketrightbigg]+ O( 2s) . (3.8)

The NNLO contribution is known as well [80, 81].

We now calculate the FU soft function, which is di erential in both k+ and [vector]k?, with

k+ [notdef][vector]k?[notdef].

+ 2 ([vector]k 2

?)x(1 x)

JHEP02(2015)117

7 Starting from the denition in eq. (2.19), the tree-level result is

S(0)(k+,[vector]k?) = (k+) 2([vector]k?) . (3.9)

7This di ers from the FU soft function in ref. [56], because their k+ measurement is independent of the hemisphere the gluon goes into.

Using the rapidity regulator of refs. [71, 72], at one-loop order we nd

S(1)(k+,[vector]k?) =

4g2w2CF

(2)32[epsilon1]

e E[notdef]2 4

[epsilon1]

[integraldisplay]

dd[lscript] ([lscript]0) ([lscript]2) [notdef]2[lscript]3[notdef] [lscript][lscript]+

[notdef] 2(

[vector][lscript]? [vector]k?) [lscript]+ ([lscript] [lscript]+) + [lscript] ([lscript]+ [lscript]) k+

= sw2CF 21 e[epsilon1] E [notdef]2[epsilon1] 2[epsilon1]

[integraldisplay]

[parenrightbig]

d2[epsilon1][lscript][epsilon1] 1 [vector]k 2

? +

[vector][lscript]2

[epsilon1]

[integraldisplay]

[notdef] 0 d[lscript]3

([lscript]3)

q[vector]k 2? +[vector][lscript]2[epsilon1] + ([lscript]3)2

[parenleftBig][radicalBig]

[vector]k 2

? +

[vector][lscript]2

[epsilon1] + ([lscript]3)2 [lscript]3 k+[parenrightBig]

JHEP02(2015)117

= sw2CF 2 e[epsilon1] E [notdef]2[epsilon1] 2[epsilon1]

1 (k+)1

[integraldisplay]

d2[epsilon1][lscript][epsilon1] [vector]k

? (k+)2 +

[vector][lscript]2

[epsilon1]

( [vector]k 2

? +

[vector][lscript]2

[epsilon1] )[[vector]k 2

? (k+)2 +

[vector][lscript]2

[epsilon1] ]

= sw2CF 2 e[epsilon1] E [notdef]2[epsilon1] 2 ([epsilon1])

?)1+[epsilon1]+ [integraldisplay]

(x + 1 (k+)2/[vector]k

(k+)1 ([vector]k 2

1 x1+[epsilon1](x+1)[x+1(k+)2/[vector]k

= sw2CF 2 e[epsilon1] E [notdef]2[epsilon1]+ 2 ([epsilon1])

1 (k+)1 ([vector]k 2

?)1+[epsilon1]+ [integraldisplay]

0 dx

1 x1+[epsilon1](x + 1)1+

+ 1

[notdef]L0

k+[notdef]

[parenrightbigg]1 ([vector]k 2

?)1+[epsilon1] [bracketleftbigg]

([vector]k 2

? (k+)2) [integraldisplay]

0 dx

1 x1+[epsilon1](x + 1)

+ ((k+)2 [vector]k

[integraldisplay]

(k+)2/[vector]k 2

dx 1

x(x + 1)

[bracketrightbigg]

+ O( , [epsilon1])

= sw2CF

[bracketleftbigg] 1[epsilon1] ([vector]k2?) +1[notdef]2 L0

[vector]k 2?

[notdef]2[parenrightbigg][bracketrightbigg]

(k+) + 2[epsilon1]2 ([vector]k2 ?) (k+)

+ 2

[epsilon1] ln

[vector]k 2

?>0

([vector]k

[notdef] ?) (k+) + 2 ([vector]k

? (k+)2)

1 [notdef]L0

k+[notdef]

[parenrightbigg]1[notdef]2 L0

[vector]k 2? [notdef]2[parenrightbigg]

+ (k+)

[bracketleftbigg]

2[notdef]2 L1

[vector]k 2?

[notdef]2[parenrightbigg]+ 2[notdef]2 L0

[vector]k 2 ? [notdef]2

[notdef]

26 ([vector]k2?)

[bracketrightbigg]+ O( , [epsilon1])

[vector]k 2

? 0

sw2CF

[bracketleftbigg] 1[epsilon1] ([vector]k2?) +1[notdef]2 L0

[vector]k 2?

[notdef]2[parenrightbigg][bracketrightbigg]

(k+) + 1[epsilon1]2 ([vector]k2 ?) (k+)

+ 2

[epsilon1] ln

([vector]k

[notdef] ?) (k+) +

2[notdef]3 L

k+[notdef] ,

[vector]k 2?

[notdef]2[parenrightbigg]+ (k+)

[bracketleftbigg] 2[notdef]2 L1

[vector]k 2? [notdef]2[parenrightbigg]

+ 2

[notdef]2 L0

[vector]k 2 ? [notdef]2

[notdef]

212 ([vector]k2?)

[bracketrightbigg]+ O( , [epsilon1])

. (3.10)

Here longitudinal momenta get regulated by , which can be thought of as the analog for rapidity divergences of the UV regulator [epsilon1], with the dimensionful parameter acting like a renormalization scale. Both 1/ and 1/[epsilon1] divergences get absorbed in renormalization constants and give rise to [notdef]- and -RG equations. The bookkeeping parameter w is used to derive the anomalous dimensions (see eq. (3.12)) and will be eventually set equal to 1.

In eq. (3.10) we introduce x = [vector][lscript]2

[epsilon1] /[vector]k 2

? in intermediate steps, to simplify notation. In

the second to last step, we rst assume [vector]k 2

? > 0 to simplify the expansion in [epsilon1]. We then

extend the distributions to include [vector]k 2

? = 0 and x the coe cient of the (k+) ([vector]k

?) by

integrating the unexpanded result. The nite terms contain the following two-dimensional plus distribution

L (x1, x2) = lim !0

d dx1

d dx2

(x2 x21) (x1 ) ln x1 (ln x2 ln x1)

+ 14 (x21 x2) (x2 2) ln2 x2[bracketrightbigg]

. (3.11)

The 1/[epsilon1] and 1/ poles are renormalized. We obtain the one-loop anomalous dimension in eq. (3.25) by using [71, 72]

d sd ln [notdef] = 2[epsilon1] s + O( 2s) , dwd ln =

2 w + O(w2) , (3.12)

and setting w = 1 afterwards. These are the same as for the TMD soft function. The renormalized FU soft function is given by the remaining nite terms,

S(1)(k+,[vector]k?, [notdef], ) =

sCF

2[notdef]3 L

k+[notdef] ,

[vector]k 2? [notdef]2[parenrightbigg]

+ (k+)

JHEP02(2015)117

[notdef]

212 ([vector]k2?)

[bracketrightbigg]

. (3.13)

Its integral over k+ reproduces the TMD soft function in refs. [72, 87]

[integraldisplay]

dk+ S(1)(k+,[vector]k?, [notdef], ) =

[bracketleftbigg]

2[notdef]2 L1

[vector]k 2?

[notdef]2[parenrightbigg]+ 2[notdef]2 L0

[vector]k 2 ? [notdef]2

ln 2[notdef]2

212 ([vector]k2 ?) [bracketrightbigg]

= S(1)([vector]k?, [notdef], ) , (3.14)

which parallels eq. (3.3) for the FU beam function. Here we used that for x21 > x2,

[integraldisplay]

0 dx[prime]1 L (x[prime]1, x2) = lim

sCF

[bracketleftbigg]

1[notdef]2 L1

[vector]k 2?

[notdef]2[parenrightbigg]+ 1[notdef]2 L0

[vector]k 2 ? [notdef]2

d dx2

14 (x2 2) ln2 x2[bracketrightbigg]= 12L1(x2) . (3.15)

3.4 Collinear-soft function

The calculation of the collinear-soft function, dened in eq. (2.17), is actually quite similar to that of the FU soft function. The main di erence is that collinear-soft radiation only goes into one hemisphere, leading to the change

[lscript]+ ([lscript] [lscript]+) + [lscript] ([lscript]+ [lscript]) k+

[parenrightbig]

! ([lscript]+ k+) . (3.16)

We conveniently separate out a contribution 12S(1)(k+,[vector]k?) from the hemisphere where the

measurement in the FU soft function and collinear-soft function are the same. The remain-

der does not contain rapidity divergences, allowing us to set = 0 from the beginning,

S (1)(k+,[vector]k?) =

4g2w2CF

(2)32[epsilon1]

e E[notdef]2 4

[epsilon1]

[integraldisplay]

dd[lscript] ([lscript]0) ([lscript]2) [notdef]2[lscript]3[notdef]

[lscript][lscript]+ 2(

[vector][lscript]? [vector]k?) [lscript]+ k+

[parenrightbig]

= 12 S(1) + sw2CF

e[epsilon1] E [notdef]2[epsilon1]

2[epsilon1]

[integraldisplay]

d2[epsilon1][lscript][epsilon1] 1 [vector]k 2

? +

[vector][lscript]2

[epsilon1]

[integraldisplay]

[notdef] 0 d[lscript]3

q[vector]k 2? +[vector][lscript]2[epsilon1] + ([lscript]3)2 + [lscript]3 k+

[parenrightbig]

q[vector]k 2? +[vector][lscript]2[epsilon1] + ([lscript]3)2

= 12 S(1) + sw2CF

JHEP02(2015)117

e[epsilon1] E [notdef]2[epsilon1]

2[epsilon1]

1 k+

[integraldisplay]

d2[epsilon1][lscript][epsilon1] (k+)2 [vector]k

[vector][lscript]2

[epsilon1]

[vector]k 2

? +

[vector][lscript]2

[epsilon1]

= 12 S(1) + sw2CF

e[epsilon1] E [notdef]2[epsilon1]

2 ([epsilon1])

(k+ [notdef][vector]k?[notdef])

[integraldisplay]

(k+)2[vector]k

0 d

[vector][lscript]2

? [epsilon1] 1 ([vector][lscript]2

[epsilon1] )1+[epsilon1]([vector]k 2

? +

[vector][lscript]2

[epsilon1] )

= 12 S(1) + sw2CF

e[epsilon1] E [notdef]2[epsilon1]

2 (1 [epsilon1])

(k+ [notdef][vector]k?[notdef])

[(k+)2/[vector]k 2? 1][epsilon1]k+ ([vector]k

?)1+[epsilon1]

[notdef] 2F1

1, [epsilon1], 1 [epsilon1], 1 (k+)2 [vector]k 2

[parenrightbigg]

(3.17)

= 12S(1) +

sw2CF

2[epsilon1]2 (k+) ([vector]k

[epsilon1]

1 [notdef]L0

k+ [notdef]

([vector]k 2?) +1[notdef]3 Lr

k+[notdef] ,

[vector]k 2? [notdef]2[parenrightbigg]

+ (k+ [notdef][vector]k?[notdef])[bracketleftbigg]

2 [notdef]L1

k+[notdef]

[parenrightbigg] 121[notdef]2 L1

[vector]k 2?

[notdef]2[parenrightbigg][bracketrightbigg]

212 (k+) ([vector]k2?) + O([epsilon1])

= sw2CF

[bracketleftbigg] 1[epsilon1] ([vector]k2?) +1[notdef]2 L0

[vector]k 2?

[notdef]2[parenrightbigg][bracketrightbigg]

(k+) + 1[epsilon1]2 (k+) ([vector]k2 ?)

[epsilon1]

1 [notdef]L0

k+ [notdef]

([vector]k 2?) +1[epsilon1] ln[notdef]

([vector]k2?) (k+) +1 [notdef]L0

k+[notdef]

[parenrightbigg]1[notdef]2 L0

[vector]k 2? [notdef]2[parenrightbigg]

+ (k+)

[bracketleftbigg]

1[notdef]2 L1

[vector]k 2?

[notdef]2[parenrightbigg]+ 1[notdef]2 L0

[vector]k 2 ? [notdef]2

[notdef]

212 ([vector]k2?)

[bracketrightbigg]+ O( , [epsilon1])

The expansion in [epsilon1] is again subtle at (k+,[vector]k 2

?) = (0, 0). Similar to section 3.3, we rst

expand assuming k+ > 0 and then extend the plus distributions to k+ = 0, xing the coe cient of (k+) ([vector]k 2

?) by integration. In an intermediate expression, the following two-

dimensional plus distribution arises

Lr(x1, x2) = lim

d dx1

d dx2

(x21 x2) (x2 2)[parenleftbigg]ln x1 14 ln x2

ln x2

+ (x2 x21) (x1 ) ln2 x1[bracketrightbigg]

. (3.18)

In the nal expression this combines with L in eq. (3.10) to give

L (x1, x2) + Lr(x1, x2) = L0(x1)L0(x2) . (3.19)

The divergences in eq. (3.17) lead to the one-loop anomalous dimensions in eq. (3.26). This satises the relation among anomalous dimensions required by consistency of the factorization theorem in eq. (2.15) at this order. The nite terms give

S (1)(k+,[vector]k?, [notdef], ) =

sCF

1 [notdef]L0

k+[notdef]

[parenrightbigg]1[notdef]2 L0

[vector]k 2? [notdef]2[parenrightbigg]

+ (k+)

[bracketleftbigg]

1[notdef]2 L1

[vector]k 2?

[notdef]2[parenrightbigg]+ 1[notdef]2 L0

[vector]k 2 ? [notdef]2

[notdef]

212 ([vector]k2?)

[bracketrightbigg]

. (3.20)

3.5 Renormalization and anomalous dimensions

In this section we write down the RG equations (RGEs) for all these ingredients, which are well-known except for the FU soft function and collinear-soft function. Their anomalous dimensions are constrained by consistency of the factorization theorems in section 2.2 and agree with the one-loop calculations in sections 3.3 and 3.4. For completeness we give the expressions for both the quark and gluon case, as indicated by an additional index i = q, g in this section. The anomalous dimensions involve the cusp anomalous dimension icusp and non-cusp anomalous dimensions iH, iJ, i , which are tabulated in appendix B.

The anomalous dimension of the Wilson coe cient C is

[notdef] d

d[notdef] C(Q2, [notdef]) = H(Q2, [notdef]) C(Q2, [notdef]) ,

H(Q2, [notdef]) = qcusp( s) ln Q2 i0

[notdef]2 + qH( s) , (3.21)

from which the evolution of the hard function H(Q2, [notdef]) = [notdef]C(Q2, [notdef])[notdef]2 directly follows.The FU beam function satises the following RGE8

[notdef] d

d[notdef] Bi(t, x,[vector]k?, [notdef]) = [integraldisplay]

JHEP02(2015)117

0 dt[prime] iB(t t[prime], [notdef]) Bi(t[prime], x,[vector]k?, [notdef]) ,

iB(t, [notdef]) = 2 icusp( s)

1[notdef]2 L0

[parenleftbigg]

t [notdef]2

[parenrightbigg]

+ iJ( s) (t) . (3.22)

The TMD beam function also involves a evolution (rapidity resummation)9

[notdef] d

d[notdef] Bi(x,[vector]k?, [notdef], ) = iB(p, [notdef], ) Bi(x,[vector]k?, [notdef], ) ,

d Bi(x,[vector]k?, [notdef], ) = [integraldisplay]

d2[vector]k[prime]

? i ([vector]k? [vector]k

[prime]

?, [notdef]) Bi(x,[vector]k

[prime]

?, [notdef], ) ,

iB(p, [notdef], ) = 2 icusp( s) ln

p + iJ( s) ,

i ([vector]k?, [notdef]) = icusp( s)

[vector]k 2?

[notdef]2[parenrightbigg]+ i ( s) 2([vector]k?) . (3.23)

8The additional spin structure [56] for the gluon beam function does not mix under renormalization and satises the same RGE.

9Its non-cusp -anomalous dimension has not yet been calculated at two loops and is not xed by consistency. However, the remaining degeneracy is irrelevant, since the TMD beam function has the same scale as the collinear-soft function (in SCET+) or FU soft function (in SCETII).

1[notdef]2 L0

The RGE of the (beam) thrust soft function is given by

[notdef] d

d[notdef] Si(k+, [notdef]) = [integraldisplay]

0 dk[prime]+ iS(k+ k[prime]+, [notdef]) Si(k[prime]+, [notdef]) ,

iS(k+, [notdef]) = 4 icusp( s) 1

[notdef]L0

k+[notdef]

[parenrightbigg] 2

iH( s) + iJ( s)

[bracketrightbig]

(k+) , (3.24)

and for the FU soft function it is given by,

[notdef] d

d[notdef] Si(k+,[vector]k?, [notdef], ) = iS([notdef], ) Si(k+,[vector]k?, [notdef], ) ,

d Si(k+,[vector]k?, [notdef], ) = 2 [integraldisplay]

d2[vector]k[prime]

? i ([vector]k? [vector]k

[prime]

?, [notdef]) Si(k+,[vector]k

[prime]

?, [notdef], ) ,

JHEP02(2015)117

iS([notdef], ) = 4 icusp( s) ln [notdef]

iH( s) + iJ( s)

[bracketrightbig]

, (3.25)

with i given in eq. (3.23).

The anomalous dimensions of the n-collinear-soft function and n-collinear-soft function are identical. Using the [notdef] and independence of the cross section in eq. (2.15), they are constrained by consistency to be

[notdef] d

d[notdef] Si(k+,[vector]k?, [notdef], ) = [integraldisplay]

0 dk[prime]+ iS (k+ k[prime]+, [notdef], ) Si(k[prime]+,[vector]k?, [notdef], ) ,

d Si(k+,[vector]k?, [notdef], ) = [integraldisplay]

d2[vector]k[prime]

? i ([vector]k? [vector]k

[prime]

?, [notdef]) Si(k+,[vector]k

[prime]

?, [notdef], )

iS (k+, [notdef], ) = 2 icusp( s)[bracketleftbigg]

1 [notdef]L0

k+[notdef]

[parenrightbigg]+ ln

[notdef] (k+)[bracketrightbigg]

. (3.26)

3.6 NLL cross section

At NLL, the cross section is generated by evolving the tree-level functions from their natural scale10

[notdef]H = iQ ,

[notdef]B = pT , B = Q ,

[notdef]S = pT , S = p2T /T ,

[notdef]S = T . (3.27) to a common scale using the RG equations in section 3.5. Evolving all functions to the collinear-soft scale ([notdef]S , S ), using results from refs. [72, 90, 9599], we obtain

[integraldisplay]

0 dp[prime]T [integraldisplay]

0 dT [prime]

ddQ2 dY dp[prime]T dT [prime]

Xq (0)q

fq(x1, [notdef]B)fq(x2, [notdef]B) + fq(x1, [notdef]B)fq(x2, [notdef]B)

[bracketrightbig]

[notdef]

S , (3.28)

10The inclusion of the factor of i in the hard scale H follows from eq. (3.21) and allows us to resum a

series of 2-terms [9194], thereby improving convergence.

(1 B) eRe(KH)+KB+KS2 E B E S (1 + B) (1 + S)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

Q2 i0 [notdef]2H

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

pT [notdef]B

2 B[parenleftBig]

T [notdef]S

where x1,2 = (Q/Ecm)e[notdef]Y and Re(. . . ) denotes the real part of a complex number. The evolution kernels are given by

KH([notdef]H, [notdef]S ) = 4Kq ([notdef]H, [notdef]S ) + 2K

H ([notdef]H, [notdef]S ) , H([notdef]B, [notdef]S ) = 2 q ([notdef]B, [notdef]S ) ,

S ([notdef]S, [notdef]S ) , S([notdef]S, [notdef]S ) = 4 q ([notdef]S, [notdef]S ) , (3.29)

in terms of functions given in appendix B. Since [notdef]B = [notdef]S there is no [notdef]-evolution for the beam functions. Because the scale of s in the -evolution is [notdef], the evolution of the non-cusp -anomalous dimension takes the simpler form shown in KB.

It is worth emphasizing that eq. (3.28) continuously merges with the SCETI and SCETII boundaries. This is no longer automatically achieved at NNLL, but can still be arranged, as discussed in the next section. We also stress that eq. (3.27) represents a naive choice of scales as these do no smoothly turn o at the boundaries leading to a discontinuity in the derivative of the cross section (see also the discussion around eq. (6.14)). This will be remedied by using prole functions [97, 100] in ref. [101], where a full analysis at NNLL will be presented.

4 Matching the e ective theories

We now show that the continuous description of the cross section across the SCETI, SCET+ and SCETII regions discussed in section 3.6 can naturally be extended to all orders. Specifically, in the SCET+ region of phase space,

Iij(t, x,[vector]k?, [notdef]) = [integraldisplay]d2[vector]k[prime]

? Iij(x,[vector]k

[prime]

?, [notdef], ) S (t/p,[vector]k? [vector]k

[prime]

?, [notdef], )

[notdef] S(k+ k[prime]+ k[prime][prime]+, [notdef]) , (4.1)

1 ([vector]k

S B

[parenrightBig]

, B( B, S ) = 2 q ( B, S ) ,

KS([notdef]S, [notdef]S ) = 4Kq ([notdef]S, [notdef]S ) + K

KB( B, S ) = 2 q ( s) ln

JHEP02(2015)117

[prime]

?, [notdef], ) ,

S(k+,[vector]k?, [notdef], ) = [integraldisplay]

d2[vector]k[prime]

[integraldisplay]

dk[prime]+ dk[prime][prime]+ S (k[prime]+,[vector]k[prime]

?, [notdef], )S (k[prime][prime]+,[vector]k? [vector]k

up to power corrections of O([vector]k

?), respectively. This follows directly

from the consistency of the factorization theorems in section 2.2: when the resummation is turned o , i.e. a common renormalization scale is chosen for all functions in the factorization theorem, the SCETI and SCETII factorization theorems simply produce the full xed-order cross section up to power corrections. As the SCET+ regime involves an additional expansion, its xed-order cross section can be obtained from either. Due to the many common ingredients between the SCET+, SCETI and SCETII factorization theorems, this then implies eq. (4.1).

We now restrict our attention to NNLL, for which eq. (4.1) reduces to

I(1)qq(t, x,[vector]k?, [notdef]) = (t) I(1)qq(x,[vector]k?, [notdef], ) + (1 x) S (1)(t/p,[vector]k?, [notdef], )

I(1)qg(t, x,[vector]k?, [notdef]) = (t) I(1)qg(x,[vector]k?, [notdef]) ,

S(1)(k+,[vector]k?, [notdef], ) =

?/t) and O((k+)2/[vector]k

?)S(1)(k+, [notdef]) + 2S (1)(k+,[vector]k?, [notdef], ) . (4.2)

?/t), whereas the last one holds exactly

for k+ < [notdef][vector]k?[notdef]. This naturally suggests the following procedure for patching together the

cross section at NNLL,11

dQ2dY dp2T dT

The rst equations are valid up to corrections of O([vector]k

0q H(Q2, [notdef])

[integraldisplay]

dt1 dt2

[integraldisplay]

d2[vector]k1? d2[vector]k2? d2[vector]k

cs1? d2[vector]k

cs2? d2[vector]k?[integraldisplay]

dk+1 dk+2 dk+

hBq(t1, x1,[vector]k1?, [notdef]) S (1) t1eY/Q,[vector]k1?, [notdef],

[parenrightbig][bracketrightBig]

S k+1,[vector]k

cs1?, [notdef],

[parenrightbig]

[notdef]

[notdef] hBq(t2, x2,[vector]k2?, [notdef]) S (1) t2eY/Q,[vector]k2?, [notdef],

[parenrightbig][bracketrightBig]

S k+2,[vector]k

cs2?, [notdef],

[parenrightbig]

JHEP02(2015)117

[notdef] hS(k+,[vector]k?, [notdef], )2S (1)(k+,[vector]k?, [notdef], )[bracketrightBig]

eYt1+eYt2Q k+1k+2k+[parenrightbigg]

p2T [notdef][vector]k1? + [vector]k2? + [vector]k

cs1? + [vector]k

2 + (q $ q) . (4.3)

Here the S (1)-term subtracted from the beam functions (soft function) are evaluated at the beam (soft) scale. From eq. (4.2) it follows that this reproduces the SCETI, SCET+ and SCETII factorization theorems in eqs. (2.9), (2.15) and (2.18), up to power corrections.

We now derive eq. (4.2), using cumulants to avoid subtleties related to distributions. Starting with the boundary between SCETI and SCET+,

[integraldisplay]

cs2? + [vector]k?[notdef]

[parenrightbig]

0 dt[prime] [integraldisplay]

[vector]k 2

0 d[vector]k[prime]

? ? I(1)qg(t[prime], x,[vector]k

[prime]

?, [notdef]) =

sTF

[bracketleftbigg]

ln min

(1 x)t x[notdef]2 ,

[vector]k 2 ? [notdef]2

Pqg(x) + 2x(1 x)

[bracketrightbigg]

Pqg(x) + 2x(1 x)

[bracketrightbigg]

. (4.4)

We thus obtain the second line in eq. (4.2) for 0 < x < 1 , where

= [vector]k

[integraldisplay]

[vector]k 2

0 d[vector]k[prime]

? ? I(1)qg(x,[vector]k

[prime]

?, [notdef], ) =

sTF

[bracketleftbigg]

[vector]k 2 ? [notdef]2

2 . (4.5)

In the SCET+ region of phase space, the size of the remaining interval 1 x 1 is

parametrically small, implying that the contribution from this region to the cross section is power suppressed.

Similarly, we nd that for 0 < x < 1 the rst line of eq. (4.2) is satised,

[integraldisplay]

0 dt[prime] [integraldisplay]

[vector]k 2

0 d[vector]k[prime]

? ? I(1)qq(t[prime], x,[vector]k

[prime]

?, [notdef]) =

sCF

[bracketleftbigg]

[vector]k 2 ? [notdef]2

Pqq(x) + 1 x [bracketrightbigg]

[integraldisplay]

[vector]k 2

= 0 d[vector]k[prime]

[prime]

?, [notdef], ) . (4.6)

11This has a natural generalization beyond NNLL in Fourier/Laplace space, where one can take the full inverse of S rather than the expanded version employed here.

? ? I(1)qq(x,[vector]k

Although 1 x 1 is again parametrically small, the integral over this region is not,

due to the presence of delta functions and plus distributions at x = 1,

[integraldisplay]

0 dt[prime] [integraldisplay]

[vector]k 2

0 d[vector]k[prime]

[integraldisplay]

? 2 1 dx I(1)qq(t[prime], x,[vector]k

[prime]

?, [notdef]) =

sCF

[bracketleftbigg]

ln2

[notdef]2[parenrightbigg]

26 + O( )

[bracketrightbigg]

[integraldisplay]

[vector]k 2

0 d[vector]k[prime]

[integraldisplay]

? 2 1 dx I(1)qq(x,[vector]k

[prime]

?, [notdef]) =

sCF

2 ln

[vector]k 2?

[notdef]2[parenrightbigg]ln

[parenrightbigg]+ O( )

[bracketrightbigg]

. (4.7)

The mismatch is captured by the collinear-soft function

[integraldisplay]

t/p

0 dk+[integraldisplay]

[vector]k 2

0 d[vector]k[prime]

? ? S (1)(k+,[vector]k

[prime]

?, [notdef], )

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= sCF 22

ln2 [parenleftbigg][vector]k2 ? [notdef]2

+2 ln

[vector]k 2?

[notdef]2[parenrightbigg][parenleftbigg]ln[parenleftbigg]

t [notdef]p

+ln [notdef]

26[bracketrightbigg]

, (4.8)

up to a power suppressed contribution

[integraldisplay]

[vector]k 2

0 d[vector]k[prime]

[integraldisplay]

[bracketleftbigg] [integraldisplay]

0 dt[prime] I(1)qq(t[prime], x,[vector]k

[prime]

?, [notdef]) I(1)qq(x,[vector]k

[prime]

?, [notdef])

[bracketrightbigg]

[integraldisplay]

t/p

0 dk+[integraldisplay]

[vector]k 2

0 d[vector]k[prime]

sCF

? ? S (1)(k+,[vector]k

[prime]

?, [notdef], ) =

22 ln2

t [vector]k 2

+ O( ) = O( ) . (4.9)

Note that in the last line it important that is not arbitrary but given by eq. (4.5). Combined with eq. (4.6), this establishes the rst line of eq. (4.2).

Lastly, we consider the boundary between SCET+ and SCETII, which involves the following ingredients

[integraldisplay]

0 dk[prime]+[integraldisplay]

[vector]k 2

0 d[vector]k[prime]

? ? S(1)(k[prime]+,[vector]k

[prime]

?, [notdef], ) =

sCF

2 ln2[parenleftbigg][vector]k2 ? [notdef]2

+4 ln

[vector]k 2 ? [notdef]2

k+ [notdef]

4 ln2[parenleftbigg]

k+[notdef]

[parenrightbigg]

+ 4 ln

[vector]k 2 ? [notdef]2

[notdef]

26 + ((k+)2[vector]k2?) ln2[parenleftbigg]

[vector]k 2? (k+)2[parenrightbigg][bracketrightbigg] ,

[integraldisplay]

sCF

[bracketleftbigg]

0 dk[prime]+ S(1)(k[prime]+, [notdef]) =

4 ln2

k+[notdef]

[parenrightbigg]+ 26[bracketrightbigg]

[integraldisplay]

0 dk[prime]+[integraldisplay]

[vector]k 2

0 d[vector]k[prime]

? ? S (1)(k[prime]+,[vector]k

[prime]

?, [notdef], ) =

sCF

ln2[parenleftbigg][vector]k2 ? [notdef]2

+2 ln

[vector]k 2?

[notdef]2[parenrightbigg][parenleftbigg] ln

k+ [notdef]

+ln [notdef]

2 6

(4.10)

It is straightforward to verify that for k+ < k? this satises the last line in eq. (4.2).

5 NLO cross section

In this section we determine the NLO cross section for Z + 0 jet production, di erential in the invariant mass Q2, the rapidity Y and pT of the Z and beam thrust T . We start by

collecting the relevant ingredients in section 5.1, check the cancellation of IR divergences

in section 5.2 and present the nal result in section 5.3. In section 5.4 we verify that this agrees with SCETI, SCET+ and SCETII, up to power corrections. This provides an important cross check of our formalism. We will match our resummed prediction onto these xed-order corrections in ref. [101].

5.1 Ingredients

The partonic cross section for the one-loop real and virtual corrections in MS are given by

(1)q,R = Q2

(0)q 8 sCF

e E[notdef]2 4

[epsilon1] 1 sqqg

sqg

sqg +sqg

sqg + 2sqqsqqg

sqgsqg [epsilon1][parenleftbigg]2 + sqg

sqg +sqg

sqg

[parenrightbigg][bracketrightbigg]

[epsilon1]

[bracketleftbigg] 1[epsilon1]2 32[epsilon1] 4 +7212 + O([epsilon1])

[bracketrightbigg]

. (5.1)

The Lorentz invariants that enter here are dened as

sij = (pi + pj)2 = 2pi[notdef]pj , sijk = (pi + pj + pk)2 = sij + sik + sjk , (5.2)

using an incoming momentum convention for pi. Due to the avor dependence of the tree-level partonic cross section

(0)q, we will for simplicity restrict ourselves to a single quark avor. The full cross section can be obtained by summing over quark avors.

We now discuss kinematics and phase space. The incoming partons have momenta

p1 = (x1Ecm, 0, 0) , p2 = (0, x2Ecm, 0) , (5.3)

in (, +, ?) light-cone coordinates (see eq. (1.2)), with x1,2 the momentum fractions of the

partons with respect to the colliding hadrons. At LO the nal state consists of a Z boson with momentum q[notdef], and the phase space integral yields

[integraldisplay]

d (0)ij =

JHEP02(2015)117

(1)q,V = Q2

(0)q sCF

[parenleftbigg]

[notdef]2 sqqg

[integraldisplay]

ddq

(2)d (2)d (q p1 p2)

dx1 x1

dx2x2 fi(x1, [notdef])fj(x2, [notdef]) [integraldisplay]

[notdef] (Q2 q2) [bracketleftbigg]

Y 12 ln [parenleftbigg]

q q+

[parenrightbigg][bracketrightbigg]

(p2T [vector]q2?) (T )

= 1

Q2 (p2T ) (T ) fi[parenleftbigg]

Ecm eY , [notdef][parenrightbigg]

[parenleftbigg]

Ecm eY , [notdef][parenrightbigg]

. (5.4)

At this order, the momentum fractions x1,2 and the momentum of the Z are thus

x1 = Q

Ecm eY , x2 =

Ecm eY , q = (QeY , QeY , 0) . (5.5)

At NLO, there is an additional massless parton that the Z-boson can recoil against. To be consistent with eq. (5.1), we use an incoming convention for the momentum p3 of this parton. Assuming for simplicity that this parton goes into the right hemisphere,

p+3 < p3, the phase space is given by

[integraldisplay]

d (1)ij,R =

[integraldisplay]

dx1 x1

dx2x2 fi(x1, [notdef])fj(x2, [notdef])[integraldisplay]

ddq

(2)d

[integraldisplay]

ddp3

(2)d (p03)2 (p23)(2)d (qp1p2p3)

[notdef] (Q2 q2) [bracketleftbigg]

Y 12 ln [parenleftbigg]

q q+

[parenrightbigg][bracketrightbigg]

(p2T [vector]q2?) (p+3 p3) (T + p+3)

= 1

(4)2[epsilon1] (1 [epsilon1])

(pT T )

p2[epsilon1]T

[parenleftBig][radicalBig]

Q2 + p2T eY T + p2T [parenrightBig][parenleftBig][radicalBig]

Q2 + p2T eY + T [parenrightBig]

[parenleftbigg][radicalBig]

Q2 + p2T eY

Ecm + T

Ecm , [notdef][parenrightbigg]

. (5.6)

The contribution for the other hemisphere d (1)ij,L can be obtained in a similar manner. From this we can read o

x1 =

qQ2 + p2T eY

Ecm +

JHEP02(2015)117

[notdef] fi

[parenleftbigg][radicalBig]

Q2 + p2T eY

Ecm +

p2T EcmT

, [notdef]

p2T EcmT

, x2 =

qQ2 + p2T eY

Ecm + T

Ecm ,

q =

[parenleftBig][radicalBig]

Q2 + p2T eY ,

qQ2 + p2T eY , pT

[parenrightBig]

, p3 =

p2T

, T , pT

. (5.7)

The (irrelevant) azimuthal angle in the transverse plane is not xed by the measurement. It is straightforward to evaluate the invariants in eq. (5.1) in terms of eq. (5.7). For qq ! Zg,

sqq = x1x2E2cm , sqg = x1EcmT , sqg = x2Ecm

p2T

. (5.8)

The other cases can be obtained by permutations. For gq ! Zq we have

sqq = x2Ecm

p2T

, sqg = x1x2E2cm , sqg = x1EcmT , (5.9)

and for qg ! Zq we have

sqq = x1EcmT , sqg = x1x2E2cm , sqg = x2Ecm

p2T

. (5.10)

Lastly, there is the NLO contribution from the PDFs, which consists of pure IR poles in dimensional regularization. This can be e ectively described as

f(1)q(x, [notdef]) = s

[epsilon1]

[integraldisplay]

dx[prime]

x[prime] CjPqj[parenleftbigg]

x x[prime]

fj(x[prime], [notdef]) , (5.11)

where the color factor Cj is CF (TF ) for j = q (j = g) and the splitting functions are

Pqq(z) = (1 + z2)L0(1 z) +

32 (1 z) , Pqg(z) = (1 z)2 + z2 . (5.12)

5.2 Cancellation of IR divergences

In this section we combine these ingredients and verify the cancellation of IR divergences. We assume pT , T Q to simplify the calculation, though we do not restrict to any

particular relative hierarchy between pT and T . This leads to

qq!Zg = Q2

(0)q8 sCF

e E[notdef]2 4

[epsilon1]

[bracketleftbigg]2p2T +(1[epsilon1])

p2T x1Ecm(x1EcmT p2T )T [bracketrightbigg][bracketleftbigg] 1+O

TQ,

p2T

Q2 [parenrightbigg][bracketrightbigg]

gq!Zq = Q2

(0)q 8 sTF

e E[notdef]2 4

[epsilon1]

[bracketleftbigg]1x1EcmT p2T 21 [epsilon1]

p2T

x21E2cmT 2 [bracketrightbigg][bracketleftbigg]1 + O

TQ,

p2T

Q2 [parenrightBig][bracketrightbigg]

TQ,

p2T

Q2[parenrightBig][bracketrightbigg]

. (5.13)

For qg ! Zq and gq ! Zq there is a fermion minus sign from crossing eq. (5.1) and we have

taken into account that we need to average over incoming gluon polarizations and colors instead of quark spins and colors, resulting in the overall factor 2Nc/[(2 2[epsilon1])(N2c 1)].

The phase space in eq. (5.6) simplies as well

[integraldisplay]

d (1)ij,R =

qg!Zq = Q2

(0)q 8 sTF

e E[notdef]2 4

[epsilon1] x1T 2x2p2T (x1EcmT p2T )[bracketleftbigg]1 + O

1 (4)2[epsilon1] (1 [epsilon1])

(pT T )

Q(QT + p2T eY )p2[epsilon1]T

Q eY

Ecm +p2T EcmT

, [notdef]

Q eY

Ecm , [notdef][parenrightbigg]

(5.14)

To avoid subtleties related to distributions, we calculate the cumulative cross section in pT and T ,

[integraldisplay]

p2T

JHEP02(2015)117

0 dp[prime]2T [integraldisplay]

0 dT [prime]

d4(1)q

dQ2 dY dp[prime]2T dT [prime]

[integraldisplay]

= 0 dp[prime]2T [integraldisplay]

0 dT [prime] [integraldisplay]

d (0)qq

(1)q,V +

(0)q f(1)q(x1, [notdef]) fq(x1, [notdef])

[bracketrightbigg]+

[integraldisplay]

p2T

d (1)qq,R

qq!Zg

[integraldisplay]

+ d (1)qg,R

qg!Zq +

[integraldisplay]

d (1)gq,R

gq!Zq + (x1 $ x2) + (q $ q)

(0)q fq(x1, [notdef])fq(x2, [notdef]) s

[parenleftbigg]

[notdef]2 Q2

[epsilon1]

[bracketleftbigg] 1[epsilon1]2 32[epsilon1] 4 +7212 + O([epsilon1]) [bracketrightbigg]

+ 1

[epsilon1]

[integraldisplay]

dx[prime]1 x[prime]1

fj(x[prime]1, [notdef])

fq(x1, [notdef]) CjPqj[parenleftbigg]

x1 x[prime]1

[parenrightbigg]

[epsilon1]

[integraldisplay]

dx[prime]1 x[prime]1

fq(x[prime]1, [notdef])

fq(x1, [notdef]) CF [bracketleftbigg]2x1(x[prime]1 x1)1+2[epsilon1]+ (1 [epsilon1])(x[prime]1 x1)12[epsilon1] x[prime]1[bracketrightbigg]

+ fg(x[prime]1, [notdef]) fq(x1, [notdef]) TF [bracketleftbigg]

1(x[prime]1 x1)2[epsilon1]

21 [epsilon1]

x1(x[prime]1 x1)12[epsilon1] x[prime]21

[bracketrightbigg]

e[epsilon1] E

(1 [epsilon1])

[parenleftbigg]

[notdef]2 E2cm

[epsilon1]

[notdef]

[bracketleftbigg]

min

1, T(x[prime]1 x1)Ecm, p2T(x[prime]1 x1)2E2cm [bracketrightbigg][epsilon1]

[integraldisplay]

dx[prime]1 x[prime]1

fq(x[prime]1, [notdef])fg(x2, [notdef]) fq(x1, [notdef])fq(x2, [notdef]) TF

x[prime]1 x2(x[prime]1 x1)2[epsilon1]

e[epsilon1] E

(2 [epsilon1])

[parenleftbigg]

[notdef]2 E2cm

[epsilon1]

[notdef]

[bracketleftbigg]

min

1, T(x[prime]1 x1)Ecm, p2T(x[prime]1 x1)2E2cm [bracketrightbigg]1[epsilon1][parenrightbigg]+ (x1 $ x2) + (q $ q) , (5.15)

where in the nal expression we used the shorthand notation

x1 = QEcm eY , x2 =

Ecm eY , (5.16)

which should not to be confused with eq. (5.7). The contribution from d L is included through (x1 $ x2).

We obtained eq. (5.15) by rst rewriting the p[prime]2T integral in terms of x[prime]

p[prime]2T = (x[prime]1 x1)Ecm T [prime] . (5.17)

such that

[integraldisplay]

p2T

JHEP02(2015)117

0 dp[prime]2T = [integraldisplay]

x1 dx[prime]1 Ecm T [prime] p2T (x[prime]1 x1)EcmT [prime]

[parenrightbig]

. (5.18)

For the subsequent T [prime] integral we nd

[integraldisplay]

0 dT [prime]

p2T (x[prime]1 x1)EcmT [prime]

(x[prime]1 x1)Ecm T [prime]) (T [prime])1+[epsilon1]

1, T(x[prime]1 x1)Ecm, p2T(x[prime]1 x1)2E2cm [bracketrightbigg][epsilon1]

, (5.19)

and similarly for the term that goes like (T [prime])[epsilon1]. The cancellation of IR divergences becomes

clear once we use the following expressions to expand in [epsilon1],

2x1 (x[prime]1x1)1+2[epsilon1]

+ (1[epsilon1])

1 (x[prime]1 x1) (x[prime]1 x1)[epsilon1]E[epsilon1]cm [bracketleftbigg]

min

(x[prime]1x1)12[epsilon1] x[prime]1

1[epsilon1] 32 +2 ln x1

x1x[prime]1[parenrightbigg]+ Pqq

x1x[prime]1[parenrightbigg]+ O([epsilon1]) ,

x1x[prime]1[parenrightbigg]+ O([epsilon1]) , (5.20)

which follow from eq. (A.3). Note that the ln x1 term on the rst line and the corresponding term from (x1 $ x2) combine with ln(E2/[notdef]2) to give ln(Q2/[notdef]2) = ln x1+ln x2+ln(E2/[notdef]2).

5.3 Result

We now present the cross section for pp ! Z + 0 jets, di erential in the invariant mass

and rapidity of the Z, and with cuts on the transverse momentum of the Z and on beam thrust. This is given by the nite O([epsilon1]0) terms in eq. (5.15), which we rearrange into the

1(x[prime]1 x1)2[epsilon1]

21 [epsilon1]

x1(x[prime]1 x1)12[epsilon1] x[prime]21

= Pqg

following form

[integraldisplay]

p2T

0 dp[prime]2T [integraldisplay]

0 dT [prime]

d4(1)q

dQ2 dY dp[prime]2T dT [prime]

(0)q fq(x1, [notdef])fq(x2, [notdef]) s

2[bracketleftbigg] ln2

x1Ecm[notdef]

[parenrightbigg] ln2

[notdef] [parenrightbigg][bracketrightbigg]

[integraldisplay]

1 dz1 z1

fj(x1/z1, [notdef])

fq(x1, [notdef]) CjPqj(z1) ln min

x21E2cm z21[notdef]2

, T x1Ecm z1(1z1)[notdef]2

, p2T

(1z1)2[notdef]2

[integraldisplay]

1 dz1 z1

fq(x1/z1, [notdef])

fq(x1, [notdef]) CF [bracketleftbigg]2(1+z21)L1(1z1) + [parenleftbigg]4 + 2 2

(1 z1) + 1 z1 [bracketrightbigg]

JHEP02(2015)117

+ fg(x1/z1, [notdef])

fq(x1, [notdef]) TF

2Pqg(z1) ln(1 z1) + 2z1(1 z1)

[bracketrightBig]

[integraldisplay]

, z21p2T

x21(1 z1)2E2cm [parenrightbigg]

+ (x1 $ x2) + (q $ q) . (5.21)

Here we changed variables to z1 = x1/x[prime]1.

5.4 Comparison to resummed predictions

We will now expand eq. (5.21) in the SCETI, SCET+ and SCETII regions of phase space, and verify that this agrees with the predictions from factorization theorems, up to power corrections. The second-to-last line of eq. (5.21) could never be produced by the factorization theorems, but is power-suppressed and does not need to be considered. Since the cross section in eq. (5.21) is a cumulative distribution, we benet from the cumulative expressions for the ingredients of the factorization formulae in section 4.

The minimum in eq. (5.21) cuts the z1 interval into three regions

min

[parenleftBig]

p1 z1[notdef]

1 dz1 z1

fq(x1/z1, [notdef])fg(x2, [notdef]) fq(x1, [notdef])fq(x2, [notdef]) TF

x1z1x2 min

1, z1T

x1(1 z1)Ecm

2, T p1z1(1 z1)[notdef]2, p2T(1 z1)2[notdef]2

[p1/(z1[notdef])]2 1 z1 za T p1/[z1(1 z1)[notdef]2] za z1 zb

p2T /[(1 z1)2[notdef]2] zb z1 0

(5.22)

with p1 = x1Ecm = QeY and boundaries at

za = 1

1 + T /p1

, zb = 1

1 + p2T /(T p1)

. (5.23)

Because the size of the interval 1 z1 za is parametrically small, O(T /Q), we only

need to keep the logarithmically enhanced contributions. From the z1 ! 1 behavior of

the splitting functions Pqj(z1) in eq. (5.12), it is clear that only the contribution from the

diagonal j = q term is not suppressed:

[integraldisplay]

1 dz1 z1

fj(x1/z1, [notdef])

fq(x1, [notdef]) CjPqj(z1) ln min [parenleftbigg]

p1 z1[notdef]

2, T p1z1(1 z1)[notdef]2, p2T(1 z1)2[notdef]2

= 2CF ln

p1[notdef]

[parenrightbigg] [integraldisplay]1za dz1 Pqq(z1)[bracketleftbigg]1 + O

[parenrightBig][bracketrightbigg]

= 2CF ln

p1[notdef]

[parenrightbigg][bracketleftbigg] ln

(p1)2

T 2

[parenrightbigg]

+ 32[bracketrightbigg][bracketleftbigg]

1 + O

[parenrightBig][bracketrightbigg]

[parenrightBig][bracketrightbigg](5.24)

In the SCETI region of phase space, the interval za z1 zb is not parametrically

small. We therefore do not give the boundary zb any special treatment. It is convenient

to rewrite the remaining integral over 1 z1 x1 and subtract the contribution from

1 z1 za. This requires us to extend Pqq(z) ln(1 z) to z ! 1, which we do as follows: Pqq(z) ln(1 z) ! (1 + z2)L1(1 z) . (5.25)

We thus obtain

= CF

[bracketleftbigg]

4 ln2

p1[notdef]

[parenrightbigg]+ 3 ln

p1[notdef]

[parenrightbigg]+ 4 ln

p1 [notdef]

T[notdef] [parenrightbigg][bracketrightbigg][bracketleftbigg]1 + O

[integraldisplay]

dz1 z1

fj(x1/z1, [notdef])

fq(x1, [notdef]) CjPqj(z1) ln min [parenleftbigg]

p1 z1[notdef]

2, T p1z1(1 z1)[notdef]2, p2T(1 z1)2[notdef]2(5.26)

JHEP02(2015)117

[integraldisplay]

1 dz1 z1

fj(x1/z1, [notdef])

fq(x1, [notdef]) CjPqj(z1) [bracketleftbigg]

ln min

T p1z1[notdef]2 ,

p2T(1 z1)[notdef]2 ln(1 z1) [bracketrightbigg]

[integraldisplay]

za dz1 CF Pqq(z1) [bracketleftbigg]

T p1[notdef]2[parenrightbigg] ln(1 z1)

[bracketrightbigg][bracketleftbigg]1 + O

[parenrightBig][bracketrightbigg]

[integraldisplay]

1 dz1 z1

fj(x1/z1, [notdef])

fq(x1, [notdef]) CjPqj(z1) [bracketleftbigg]

ln min

T p1z1[notdef]2 ,

p2T(1 z)[notdef]2 ln(1 z1) [bracketrightbigg]

+ CF

[parenrightBig][bracketrightbigg]

Combining eqs. (5.21), (5.24) and (5.26), it is straightforward to verify that this agrees with the SCETI factorization formula in eq. (2.9), using the results in section 4.

In the SCET+ and SCETII region of phase space, the interval za z1 zb is also

parametrically small, O(p2T /(T Q)). In fact, for SCETII both zb < za and zb > za are

allowed. We start by assuming zb < za,

3 ln2

p1 [notdef]

32 ln

p1 [notdef]

2 ln

p1 [notdef]

T[notdef] [parenrightbigg] ln2

T[notdef] [parenrightbigg]32 ln

T[notdef] [parenrightbigg][bracketrightbigg][bracketleftbigg] 1+O

[integraldisplay]

dz1 z1

fj(x1/z1, [notdef])

fq(x1, [notdef]) CjPqj(z1) ln min [parenleftbigg]

p1 z1[notdef]

2, T p1z1(1 z1)[notdef]2, p2T(1 z1)2[notdef]2

= CF

[integraldisplay]

dz1 Pqq(z1)

[bracketleftbigg]

T p1[notdef]2[parenrightbigg] ln(1 z)

[bracketrightbigg][bracketleftbigg]1 + O

[parenleftBig]

p2T

T Q

[parenrightBig][bracketrightbigg]

= CF

8 ln

p1 [notdef]

T[notdef] [parenrightbigg]4 ln2

T[notdef] [parenrightbigg]+4 ln

p2T

[notdef]2[parenrightbigg][bracketleftbigg] ln

p1 [notdef]

+ln

T[notdef] [parenrightbigg][bracketrightbigg] ln2

p2T

[notdef]2 [parenrightbigg]

[notdef]

1 + O

[parenleftBig]

p2T

T Q

[parenrightBig][bracketrightbigg]

. (5.27)

The remainder is

[integraldisplay]

dz1 z1

fj(x1/z1, [notdef])

fq(x1, [notdef]) CjPqj(z1) ln min [parenleftbigg]

p1 z1[notdef]

2, T p1z1(1 z1)[notdef]2, p2T(1 z1)2[notdef]2

[integraldisplay]

1 dz1 z1

fj(x1/z1, [notdef])

fq(x1, [notdef]) CjPqj(z1) [bracketleftbigg]

p2T

[notdef]2[parenrightbigg] 2 ln(1 z1) [bracketrightbigg]

[integraldisplay]

1 dz1 CF Pqq(z1)

[bracketleftbigg]

p2T

[notdef]2[parenrightbigg] 2 ln(1 z1)

[bracketrightbigg][bracketleftbigg]1 + O

[parenleftBig]

p2T

T Q

[parenrightBig][bracketrightbigg]

[integraldisplay]

1 dz1 z1

fj(x1/z1, [notdef])

fq(x1, [notdef]) CjPqj(z1) [bracketleftbigg]

JHEP02(2015)117

p2T

[notdef]2[parenrightbigg] 2 ln(1 z1) [bracketrightbigg]

+ CF

2 ln2

p1[notdef]

[parenrightbigg]+ 4 ln

p1 [notdef]

T[notdef] [parenrightbigg]+ 2 ln2

T[notdef] [parenrightbigg]

2 ln

p2T

[notdef]2[parenrightbigg][bracketleftbigg] ln

p1[notdef]

[parenrightbigg]+ ln

T[notdef] [parenrightbigg]+ 34[bracketrightbigg][bracketleftbigg]1 + O

[parenleftBig]

p2T

T Q

[parenrightBig][bracketrightbigg]

. (5.28)

We have veried that this agrees with the SCET+ factorization formula in eq. (2.15) expanded to NLO, providing an important check on our e ective theory framework.

We now consider zb > za, i.e. pT < T , which is only allowed by the power counting

in the SCETII region of phase space. In contrast with eq. (5.22), we now only have two regions: 1 z1 zc and zc z1 x1, where

zc = 11 + pT /p1

. (5.29)

This leads to the following correction to the SCET+ result,

(T pT ) [summationdisplay]

[integraldisplay]

dz1 z1

fj(x1/z1, [notdef])

fq(x1, [notdef]) CjPqj(z1) [bracketleftbigg]

T p1 z1(1 z1)[notdef]2

[parenrightbigg]

(p1)2 z21[notdef]2[parenrightbigg][bracketrightbigg]

[integraldisplay]

dz1 z1

fj(x1/z1, [notdef])

fq(x1, [notdef]) CjPqj(z1) [bracketleftbigg]

p2T(1 z1)2[notdef]2

[parenrightbigg]

(p1)2 z21[notdef]2 [parenrightbigg][bracketrightbigg]

= CF (T pT )

[braceleftBigg] [integraldisplay]

za dz1 Pqq(z1)[bracketleftbigg]

[parenleftbigg]

[parenrightbigg]

ln(1 z1)

[bracketrightbigg]

[integraldisplay]

dz1 Pqq(z1)

2 ln

p1[parenrightbigg] 2 ln(1 z1)

[bracketrightbigg][bracerightBigg][bracketleftbigg]1 + O

[parenrightBig][bracketrightbigg]

= 12 CF (T pT ) ln2 [parenleftbigg]

[parenrightBig][bracketrightbigg]

. (5.30)

The rst line erases the earlier contributions from za < z1 < zb and the second line from zb < z1 < zc. This agrees with the FU soft function in eq. (4.10).

We conclude this section by briey commenting on the size of the various power corrections we encountered. In section 5.2, we restricted to pT , T Q, dropping some (but

not all) terms of O(p2T /Q2, T /Q). In our SCETI analysis in this section, we systemat

ically expanded up to corrections of O(T /Q). For SCETII the power corrections were

p2T

T 2

[parenrightbigg][bracketleftbigg]

1 + O

Mode: Scaling (, +, ?)

collinear Q(1, 2r/ , r/ )

collinear-soft Q r , ( 2)r( 2) , ( 1)r( 1)

[parenrightBig]

soft Q( , , )

Table 2. Modes and power counting in SCET+ for the double angularity measurement on a single jet. The power counting parameter is , with e e1/r and 1 > r > / .

O(T /Q p2T /(T Q)), and for SCET+ they were O(p2T /(T Q)) in size. Contrary to our

expectation in section 2.1, we found no O(T 2/p2T ) power corrections at NLO. However, it

is quite possible that this changes at higher orders.

6 Measuring two angularities on one jet

We will now apply our e ective eld theory framework to the measurement of two angularities on one jet. The angularity e of a jet is dened as [32, 65, 102]

e =

Xi2jet

EiEjet

Here, Ei and i denote the energy and angle (with respect to the jet axis) of particle i, and Ejet and R are the jet energy and radius. To avoid the issue of recoil [28, 103105], we use the winner-take-all axis [105, 106]. This ensures that the direction of the collinear radiation coincides with the jet axis.

For the measurement of two angularities e , e (with > ), the phase space is given by e / e e at NLL. The e ective eld theories on the boundaries were discussed

in ref. [34], so we focus on the intermediate regime described by SCET+. The modes of SCET+ are shown in table 2. Their power counting is xed by the requirement that these modes are on-shell, that the collinear mode contributes to e , the soft mode contribute to e and the collinear-soft mode contribute to both. This leads to the following factorization formula

d2i

de de =

(0)iHi(Q2, [notdef])

. (6.1)

JHEP02(2015)117

[integraldisplay]

dec Q decs Q decs Q des Q Ji(ec Q , [notdef]) Si(ecs Q, ecs Q ) Si(es Q, [notdef])

[notdef] (e ecs ec ) (e ec ecs ) , (6.2)

for quark (i = q) and gluon (i = g) jets. Here,

(0)i is the tree-level cross section, and H the hard function describing hard virtual corrections. The jet function J, soft function S and collinear-soft function S capture the e ect of collinear, soft and collinear-soft radiation, respectively. The rst two have been dened in ref. [34] while the third is the analog of eq. (2.17) but for the double angularity measurement. Since we only work up to NLL order, we are allowed to consider a single jet. At higher orders we need to take the rest of the event into account, and eq. (6.2) must accordingly be generalized to e.g. e+e event

shapes. We expect the power corrections to be O(e /e , e / /e ), which blow up at the edges of the phase space, where the boundary theories should be used instead.

Below we collect what is needed for NLL resummation. The RG equation and the anomalous dimension of the hard function are

[notdef] d

d[notdef] Hi(Q2, [notdef]) = iH(Q2, [notdef]) Hi(Q2, [notdef]) ,

iH(Q2, [notdef]) = icusp( s) ln Q2

[notdef]2 + iH( s) . (6.3)

For the jet function we have

[notdef] d

d[notdef] Ji(e Q , [notdef]) = [integraldisplay]

JHEP02(2015)117

0 de[prime] Q iJ(e Q e[prime] Q , [notdef]) Ji(e[prime] Q , [notdef]) ,

iJ(e Q , [notdef]) =

2 1

icusp( s) 1

[notdef] L0

e Q

[notdef]

[parenrightbigg]+ iJ( s) (e Q ) , (6.4)

and the soft function satises

[notdef] d

d[notdef] Si(e Q, [notdef]) =

[integraldisplay]

0 de[prime] Q iS(e Q e[prime] Q, [notdef]) Si(e[prime] Q, [notdef]) ,

iS(e Q, [notdef]) = 2

icusp( s) 1

[notdef]L0

e Q

[notdef]

[parenrightbigg]+ iS( s) (e Q) . (6.5)

The anomalous dimension of the collinear-soft function is constrained by consistency of the cross section in eq. (6.2). These anomalous dimensions involve icusp( s), given in appendix B, and the non-cusp parts

iX( s) =

iX,n

s 4

n+1, (6.6)

with X = H, J, S. At NLL we only need the leading coe cients,

qH,0 = 6CF , gH,0 = 2 0 , iJ,0 = iH,0 , iS,0 = 0 , (6.7)

where 0 = 113 CA 43 TF nf.

We now evaluate the double cumulative distribution at NLL order by inserting the tree-level expressions

Hi(Q2, [notdef]) = 1 , Ji(e Q , [notdef]) = (e Q ) ,

Si(e Q, e Q ) = (e Q) (e Q ) , Si(e Q, [notdef]) = (e Q) , (6.8)

in eq. (6.2) and evolving them to the collinear-soft scale [notdef]S . This results in

i(e , e ) =

[integraldisplay]

0 de[prime] [integraldisplay]

0 de[prime]

@e[prime] @e[prime]

(0)i

eKiH +KiJ +KiS E iJ E iS

(1 + iJ) (1 + iS)

Q [notdef]H

2 iH

e1/ Q [notdef]J

e Q [notdef]S

iS. (6.9)

The evolution kernels that enter here are

KiH([notdef]H, [notdef]S ) = 2Ki ([notdef]H, [notdef]S ) + K

iH ([notdef]H, [notdef]S ) , iH([notdef]J, [notdef]S ) = i ([notdef]J, [notdef]S ) , (6.10)

KiJ([notdef]J, [notdef]S ) =

2 1

Ki ([notdef]J, [notdef]S ) + K i

J ([notdef]J, [notdef]S ) , iJ([notdef]J, [notdef]S ) =

i ([notdef]J, [notdef]S ) ,

KiS([notdef]S, [notdef]S ) = 2

Ki ([notdef]S, [notdef]S ) , iS([notdef]S, [notdef]S ) =

i ([notdef]S, [notdef]S ) ,

in terms of Ki , i and K i

X dened in eq. (B.1). As starting point for the RG evolution we use the canonical (natural) scales

[notdef]H = Q ,

[notdef]J = e1/ Q = [notdef]J!J ,

[notdef]S = e1 e 1

1/( )Q = [notdef]J!S ,

[notdef]S = e Q = [notdef]S!S . (6.11)

which we identied with the interpolating scales [notdef]J!J, [notdef]J!S and [notdef]S!S of ref. [34] (see

also appendix C of ref. [37]) to simplify the comparison.

This mostly agrees with the conjecture made in ref. [34]

ref. [34]i(e , e ) =

JHEP02(2015)117

eR(e ,e ) E ~R(e ,e )

(1 + ~R(e , e ))

(6.12)

where

R(e , e ) NLL= KiH([notdef]H, [notdef]S ) KiJ([notdef]J, [notdef]S ) KiS([notdef]S, [notdef]S ) ,~R(e , e ) NLL= iJ([notdef]J, [notdef]S ) + iS([notdef]S, [notdef]S ) . (6.13)

The only di erence12 with our result in eq. (6.9) is in the denominator, where we have

(1 + iJ) (1 + iS) instead of (1 + iJ + iS). These factors agree with each other on the boundary, since either iJ or iS vanishes there, but lead to O( 2s ln2) di erences in the bulk.

(An analogous conjecture to eq. (6.12) in Laplace space does agree with our result.13)

According to ref. [34], the leading di erence between their interpolation and the true NLL cross section is expected to be 4s ln4. However, this is based on boundary conditions for the di erential cross section, which do not a ect the logarithmic accuracy of their calculation in the bulk. Specically, their di erential cross section satises the condition at the boundary e = e / through the addition of terms that are power suppressed.

Since these terms are power suppressed in the bulk, they cannot improve the logarithmic accuracy there.

In ref. [101], we will discuss how a more sophisticated scale choice than eq. (6.11) provides a natural way to satisfy the derivative boundary condition. In addition to requiring

12Ignoring di erences beyond NLL order and power suppressed contributions.

13We thank D. Neill for pointing this out.

[notdef]S to merge with [notdef]J or [notdef]S on the respective boundaries, one can also require a continuous derivative,

@e [notdef]J(e , e )[vextendsingle][vextendsingle][vextendsingle]e

=e / =

dde [notdef]J(e , e / ) ,

@e [notdef]J(e , e )[vextendsingle][vextendsingle][vextendsingle]e

=e / = 0 ,

@e [notdef]S (e , e )[vextendsingle][vextendsingle][vextendsingle]e

=e / =

dde [notdef]J(e , e / ) ,

@e [notdef]S (e , e )[vextendsingle][vextendsingle][vextendsingle]e

=e / = 0 ,

@e [notdef]S(e , e )[vextendsingle][vextendsingle][vextendsingle]e

=e / =

dde [notdef]S(e , e / ) ,

@e [notdef]S(e , e )[vextendsingle][vextendsingle][vextendsingle]e

=e / = 0 , (6.14)

and similarly for the boundary at e = e . These equations closely resemble those imposed on R and ~R in ref. [34] and follow from the same steps. Note that there is a redundancy in the constraints in eq. (6.14), as e.g. the second equation on the rst line implies the rst. The scale choice in transitioning to a region where resummation is turned o has been studied for single variables in e.g. refs. [97, 100], and also in ref. [52].

7 Conclusions

In this paper we studied the resummation of double di erential measurements. We focussed on two examples: Drell-Yan production with a (beam-thrust) jet veto where the pT of the lepton pair is measured, and the measurement of two angularities on one jet. Concerning the latter, in ref. [34] resummation on the two phase space boundaries was achieved, and an interpolation was built to smoothly connect them. We go beyond this by identifying the factorization formula needed to achieve resummation in the intermediate regime. This involves additional collinear-soft modes, and the corresponding collinear-soft function was calculated at one loop. The relations between FU PDFs, collinear-soft functions and (FU) soft functions were investigated. The consistency of our factorization theorem was veried by checking that the anomalous dimensions cancel between the various ingredients, and by comparing to an analytic NLO calculation of the cross section. We also showed how to combine the factorization theorems on the boundaries and interior, to achieve NNLL precision throughout. At variance with ref. [34] we found a universal factorization formula that describes the cross section in all three phase space regions up to power corrections. Numerical results, including the matching to xed order, will be presented in ref. [101].

If the hierarchy of scales for the individual variables is not that large, such that the resummation of them is only marginally important, there may be not enough room for a distinct SCET+ region of phase space. (This can be seen in gure 1, where you have to go deeper into the resummation region for SCET+.) Even in this case, one benets from knowing the correct description of the intermediate regime in building the interpolation between boundaries, as illustrated by the O( 2s ln2) di erence between our NLL results and

the interpolation conjectured in ref. [34].

Looking forward, we hope the results presented here will stimulate the development of more realistic analytic resummations and more robust Monte Carlo descriptions of LHC events. The framework presented here has natural generalizations to resummation in more than two variables. Finally, nding a proper description of the terra incognita in gure 1

JHEP02(2015)117

is important for resolving a long-standing issue over double counting between higher-order corrections and double parton scatterings.

Acknowledgments

We thank Thomas Becher and Mathias Ritzmann for discussions. We thank Andrew Larkoski, Ian Moult, Du Neil, Frank Tackmann and Jonathan Walsh for feedback on this manuscript. W.W. thanks the Galileo Galilei Institute for Theoretical Physics for hospitality and the INFN for partial support during the completion of this work. M.P. acknowledges support by the Swiss National Science Foundation. W.W. is supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Program (PIIF-GA-2012-328913). This work is part of the D-ITP consortium, a program of the Netherlands Organization for Scientic Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).

A Plus distributions

The standard plus distribution for some function g(x) can be dened as

(x)g(x) + = lim !0d dx (x ) G(x)

[bracketrightbig]with G(x) =

[integraldisplay]

1 dx[prime] g(x[prime]) , (A.1)

JHEP02(2015)117

satisfying the boundary condition

[integraltext]

10 dx [ (x)g(x)]+ = 0. Two special cases we need are

Ln(x)

(x) lnn x x

+= lim !0

(x ) lnn x

x + (x )lnn+1

n + 1[bracketrightbigg]

(x )

x1 + (x )

x 1

[bracketrightbigg]

. (A.2)

In our calculations, we use the following expansion in plus distributions

(x) x1+[epsilon1] =

L (x) [bracketleftbigg]

(x) x1

+= lim !0

1 [epsilon1] (x) + L0(x) [epsilon1]L1(x) + O([epsilon1]2) . (A.3)

Rescaling and convolution identities for Ln(x) and L (x) can be found in appendix B

of ref. [97].

B Renormalization group evolution

The functions Ki ([notdef]0, [notdef]), i ([notdef]0, [notdef]) and K i

X ([notdef]0, [notdef]) that enter in the RGE solutions are

dened by

Ki ([notdef]0, [notdef]) =

[integraldisplay]

s([notdef])

s([notdef]0)

d s

( s) icusp( s) [integraldisplay]

s([notdef]0)

d [prime]s

( [prime]s) , i ([notdef]0, [notdef]) = [integraldisplay]

s([notdef])

s([notdef]0)

d s

( s) icusp( s) ,

K i

X ([notdef]0, [notdef]) =

[integraldisplay]

s([notdef])

s([notdef]0)

d s

( s) iX( s) . (B.1)

Expanding the beta function and anomalous dimensions in powers of s,

( s) = 2 s

Xn=0 n

s 4

n+1, icusp( s) =

Xn=0 in

s 4

n+1,

iX( s) =

Xn=0 iX,n

s 4

n+1, (B.2)

their explicit expressions at NNLL order are

Ki ([notdef]0, [notdef]) =

4 20

4 s([notdef]0)

1 1r ln r

[parenrightBig]+

i1 i0

1 0

(1 r + ln r) + 12 0 ln2 r

+ s([notdef]0) 4

JHEP02(2015)117

[bracketleftbigg][parenleftbigg]

21 20

2 0

[parenrightbigg][parenleftBig]

1 r2

2 + ln r

[parenrightBig]

1 i1 0 i0

21 20

(1 r + r ln r)

i2 i0 1 i1 0 i0

(1 r)22[bracketrightbigg]

i ([notdef]0, [notdef]) =

2 0

ln r + s([notdef]0) 4

i1 i0

1 0

(r1)

+ 2s([notdef]0) 162

i2 i0

1 i1 0 i0+ 21 20

2 0

r212[bracketrightbigg]

(r 1)

[bracketrightbigg]

. (B.3)

Here, r = s([notdef])/ s([notdef]0) and the running coupling is given by the three-loop expression

1 s([notdef]) =

X s([notdef]0) +

K i

X ([notdef]0, [notdef]) =

iX,0

2 0

ln r + s([notdef]0) 4

iX,1 iX,0

1 0

14 0 ln X +

s([notdef]0)

162

2 0

1 1X

[parenrightbigg]+ 21 20

ln X

X +1X 1[parenrightbigg][bracketrightbigg]

, (B.4)

where X = 1 + s([notdef]0) 0 ln([notdef]/[notdef]0)/(2).

The coe cients of the beta function [107, 108], cusp anomalous dimension [109, 110], non-cusp anomalous dimensions of the hard function and jet function [1, 110117] and non-cusp anomalous dimension of the rapidity resummation [10, 13, 14, 118] are given below in the MS scheme. At this order gi = (CA/CF ) qi, which are therefore not separately shown.

0 = 11

3 CA

43 TF nf , (B.5)

1 = 34

3 C2A [parenleftbigg]

3 CA + 4CF

TF nf ,

2 = 2857

54 C3A + [parenleftbigg]

C2F

205

18 CF CA

1415

54 C2A[parenrightbigg]

2TF nf +

119 CF +7954 CA

4T 2F n2f ,

q0 = 4CF ,

q1 = 4CF

[bracketleftbigg][parenleftbigg]

2 3

CA 209 TF nf

[bracketrightbigg]

q2 = 4CF

[bracketleftbigg][parenleftbigg]

22 3

C2A +[parenleftbigg] 41827 +40227 56 3 3

245

1342

27 +

114

45 +

CA TF nf

3 + 16 3

[parenleftbigg]

CF TF nf 1627 T 2F n2f[bracketrightbigg]

, (B.6)

qH 0 = 6CF , (B.7)

qH 1 = CF [bracketleftbigg][parenleftbigg]

9 52 3

CA + (3 42 + 48 3)CF + [parenleftbigg]659 + 2[parenrightbigg]

0[bracketrightbigg]

qH 2 = 2CF [bracketleftbigg][parenleftbigg]

66167

324

6862

3024

135

782 3

9 +

442 3

9 + 136 5

C2A

1514 20529 2474135 +844 33 +82 33 + 120 5

CF CA

292 + 32 +845 + 68 3 162 33 240 5

C2F

10781

108 +

4462

81 +

4494

270

1166 3 9

JHEP02(2015)117

[parenleftbigg]

CA 0

2953108 13218 7427 + 128 3 9

1 +[parenleftbigg] 2417324 +526 +2 3 3

20[bracketrightbigg]

gH 0 = 2 0 ,

gH 1 =[parenleftbigg]

118

9 + 4 3

C2A +[parenleftbigg] 389 +2 3

CA 0 2 1 ,

60875

162 +

6342

81 +

5 +

1972 3

402 3

9 32 5

gH 2 =

[parenleftbigg]

C3A

764954 +134281 61445 500 3 9

C2A 0 +

46681 +529 28 3 3

CA 20

1819

54 +

3 +

[parenleftbigg]

45 +

152 3 9

CA 1 2 2 , (B.8)

qJ 0 = 6CF ,

qJ 1 = CF

[bracketleftbigg][parenleftbigg]

146

9 80 3

CA + (3 42 + 48 3)CF + [parenleftbigg]1219 +22 3

0[bracketrightbigg]

qJ 2 = 2CF

[bracketleftbigg][parenleftbigg]

52019

162

8412

824

2056 3

9 +

882 3

9 + 232 5

C2A

1514 20529 2474135 +844 33 +82 33 + 120 5

CACF

292 + 32 +845 + 68 3 162 33 240 5

C2F

7739

54 +

325

81 2 +

6174

270

1276 3 9

[parenleftbigg]

CA 0

[parenleftbigg]

3457

324 +

9 +

16 3 3

20 +

116627 829 414135 + 52 3 9

1[bracketrightbigg]

, (B.9)

gJ 0 = 2 0 ,

gJ 1 =

1829 32 3

C2A +

949 22 3

CA 0 + 2 1 ,

gJ 2 =

4937381 944281 1645 4520 39 +1282 39 + 224 5

C3A

6173

3762

81 +

134

5 +

280 3 9

[parenleftbigg]

C2A 0 +[parenleftbigg] 98681 1029 + 56 3 3

CA 20

176527 223 8445 304 3 9

CA 1 + 2 2 , (B.10)

q ,0 = 0 ,

q ,1 = CF

[bracketleftbigg][parenleftbigg]

9 28 3

CA + 32 3 CF + 569 0[bracketrightbigg]

, (B.11)

g ,0 = 0 ,

g ,1 = CA

CA + 569 0[bracketrightbigg]

. (B.12)

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

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

[1] C.F. Berger, C. Marcantonini, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Higgs Production with a Central Jet Veto at NNLL+NNLO, http://dx.doi.org/10.1007/JHEP04(2011)092

Web End =JHEP 04 (2011) 092 [http://arxiv.org/abs/1012.4480

Web End =arXiv:1012.4480 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.4480

Web End =INSPIRE ].

[2] A. Ban, P.F. Monni, G.P. Salam and G. Zanderighi, Higgs and Z-boson production with a jet veto, http://dx.doi.org/10.1103/PhysRevLett.109.202001

Web End =Phys. Rev. Lett. 109 (2012) 202001 [http://arxiv.org/abs/1206.4998

Web End =arXiv:1206.4998 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1206.4998

Web End =INSPIRE ].

[3] X. Liu and F. Petriello, Reducing theoretical uncertainties for exclusive Higgs-boson plus one-jet production at the LHC, http://dx.doi.org/10.1103/PhysRevD.87.094027

Web End =Phys. Rev. D 87 (2013) 094027 [http://arxiv.org/abs/1303.4405

Web End =arXiv:1303.4405 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1303.4405

Web End =INSPIRE ].

[4] T. Becher, M. Neubert and L. Rothen, Factorization and N3LLp+NNLO predictions for the Higgs cross section with a jet veto, http://dx.doi.org/10.1007/JHEP10(2013)125

Web End =JHEP 10 (2013) 125 [http://arxiv.org/abs/1307.0025

Web End =arXiv:1307.0025 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1307.0025

Web End =INSPIRE ].

[5] I.W. Stewart, F.J. Tackmann, J.R. Walsh and S. Zuberi, Jet pT resummation in Higgs production at NNLL[prime] + NNLO, http://dx.doi.org/10.1103/PhysRevD.89.054001

Web End =Phys. Rev. D 89 (2014) 054001 [http://arxiv.org/abs/1307.1808

Web End =arXiv:1307.1808 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1307.1808

Web End =INSPIRE ].

[6] R. Boughezal, X. Liu, F. Petriello, F.J. Tackmann and J.R. Walsh, Combining Resummed Higgs Predictions Across Jet Bins, http://dx.doi.org/10.1103/PhysRevD.89.074044

Web End =Phys. Rev. D 89 (2014) 074044 [http://arxiv.org/abs/1312.4535

Web End =arXiv:1312.4535 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1312.4535

Web End =INSPIRE ].

[7] Y.L. Dokshitzer, D. Diakonov and S.I. Troian, On the Transverse Momentum Distribution of Massive Lepton Pairs, http://dx.doi.org/10.1016/0370-2693(78)90240-X

Web End =Phys. Lett. B 79 (1978) 269 [http://inspirehep.net/search?p=find+J+Phys.Lett.,B79,269

Web End =INSPIRE ].

[8] G. Parisi and R. Petronzio, Small Transverse Momentum Distributions in Hard Processes, http://dx.doi.org/10.1016/0550-3213(79)90040-3

Web End =Nucl. Phys. B 154 (1979) 427 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B154,427

Web End =INSPIRE ].

[9] J.C. Collins, D.E. Soper and G.F. Sterman, Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production, http://dx.doi.org/10.1016/0550-3213(85)90479-1

Web End =Nucl. Phys. B 250 (1985) 199 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B250,199

Web End =INSPIRE ].

JHEP02(2015)117

[bracketleftbigg][parenleftbigg]

9 + 4 3

[10] C.T.H. Davies, B.R. Webber and W.J. Stirling, Drell-Yan Cross-Sections at Small Transverse Momentum, http://dx.doi.org/10.1016/0550-3213(85)90402-X

Web End =Nucl. Phys. B 256 (1985) 413 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B256,413

Web End =INSPIRE ].

[11] G. Altarelli, R.K. Ellis, M. Greco and G. Martinelli, Vector Boson Production at Colliders: A Theoretical Reappraisal, http://dx.doi.org/10.1016/0550-3213(84)90112-3

Web End =Nucl. Phys. B 246 (1984) 12 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B246,12

Web End =INSPIRE ].

[12] S. Catani, D. de Florian and M. Grazzini, Universality of nonleading logarithmic contributions in transverse momentum distributions, http://dx.doi.org/10.1016/S0550-3213(00)00617-9

Web End =Nucl. Phys. B 596 (2001) 299 [http://arxiv.org/abs/hep-ph/0008184

Web End =hep-ph/0008184 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0008184

Web End =INSPIRE ].

[13] D. de Florian and M. Grazzini, The structure of large logarithmic corrections at small transverse momentum in hadronic collisions, http://dx.doi.org/10.1016/S0550-3213(01)00460-6

Web End =Nucl. Phys. B 616 (2001) 247 [http://arxiv.org/abs/hep-ph/0108273

Web End =hep-ph/0108273 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0108273

Web End =INSPIRE ].

[14] T. Becher and M. Neubert, Drell-Yan production at small qT , transverse parton distributions and the collinear anomaly, http://dx.doi.org/10.1140/epjc/s10052-011-1665-7

Web End =Eur. Phys. J. C 71 (2011) 1665 [http://arxiv.org/abs/1007.4005

Web End =arXiv:1007.4005 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1007.4005

Web End =INSPIRE ].

[15] A. Idilbi, X.-d. Ji and F. Yuan, Transverse momentum distribution through soft-gluon resummation in e ective eld theory, http://dx.doi.org/10.1016/j.physletb.2005.08.038

Web End =Phys. Lett. B 625 (2005) 253 [http://arxiv.org/abs/hep-ph/0507196

Web End =hep-ph/0507196 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0507196

Web End =INSPIRE ].

[16] S. Mantry and F. Petriello, Transverse Momentum Distributions from E ective Field Theory with Numerical Results, http://dx.doi.org/10.1103/PhysRevD.83.053007

Web End =Phys. Rev. D 83 (2011) 053007 [http://arxiv.org/abs/1007.3773

Web End =arXiv:1007.3773 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1007.3773

Web End =INSPIRE ].

[17] M.G. Echevarria, A. Idilbi and I. Scimemi, Factorization Theorem For Drell-Yan At Low qT And Transverse Momentum Distributions On-The-Light-Cone, http://dx.doi.org/10.1007/JHEP07(2012)002

Web End =JHEP 07 (2012) 002 [http://arxiv.org/abs/1111.4996

Web End =arXiv:1111.4996 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1111.4996

Web End =INSPIRE ].

[18] H.-n. Li, Z. Li and C.-P. Yuan, QCD resummation for light-particle jets, http://dx.doi.org/10.1103/PhysRevD.87.074025

Web End =Phys. Rev. D 87 http://dx.doi.org/10.1103/PhysRevD.87.074025

Web End =(2013) 074025 [http://arxiv.org/abs/1206.1344

Web End =arXiv:1206.1344 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1206.1344

Web End =INSPIRE ].

[19] M. Dasgupta, K. Khelifa-Kerfa, S. Marzani and M. Spannowsky, On jet mass distributions in Z+jet and dijet processes at the LHC, http://dx.doi.org/10.1007/JHEP10(2012)126

Web End =JHEP 10 (2012) 126 [http://arxiv.org/abs/1207.1640

Web End =arXiv:1207.1640 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.1640

Web End =INSPIRE ].

[20] Y.-T. Chien, R. Kelley, M.D. Schwartz and H.X. Zhu, Resummation of Jet Mass at Hadron Colliders, http://dx.doi.org/10.1103/PhysRevD.87.014010

Web End =Phys. Rev. D 87 (2013) 014010 [http://arxiv.org/abs/1208.0010

Web End =arXiv:1208.0010 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1208.0010

Web End =INSPIRE ].

[21] T.T. Jouttenus, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Jet mass spectra in Higgs boson plus one jet at next-to-next-to-leading logarithmic order, http://dx.doi.org/10.1103/PhysRevD.88.054031

Web End =Phys. Rev. D 88 http://dx.doi.org/10.1103/PhysRevD.88.054031

Web End =(2013) 054031 [http://arxiv.org/abs/1302.0846

Web End =arXiv:1302.0846 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1302.0846

Web End =INSPIRE ].

[22] A. Abdesselam et al., Boosted objects: A Probe of beyond the Standard Model physics, http://dx.doi.org/10.1140/epjc/s10052-011-1661-y

Web End =Eur. http://dx.doi.org/10.1140/epjc/s10052-011-1661-y

Web End =Phys. J. C 71 (2011) 1661 [http://arxiv.org/abs/1012.5412

Web End =arXiv:1012.5412 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.5412

Web End =INSPIRE ].

[23] A. Altheimer et al., Jet Substructure at the Tevatron and LHC: New results, new tools, new benchmarks, http://dx.doi.org/10.1088/0954-3899/39/6/063001

Web End =J. Phys. G 39 (2012) 063001 [http://arxiv.org/abs/1201.0008

Web End =arXiv:1201.0008 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1201.0008

Web End =INSPIRE ].

[24] A. Altheimer et al., Boosted objects and jet substructure at the LHC. Report of BOOST2012, held at IFIC Valencia, 23rd-27th of July 2012, http://dx.doi.org/10.1140/epjc/s10052-014-2792-8

Web End =Eur. Phys. J. C 74 (2014) http://dx.doi.org/10.1140/epjc/s10052-014-2792-8

Web End =2792 [http://arxiv.org/abs/1311.2708

Web End =arXiv:1311.2708 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1311.2708

Web End =INSPIRE ].

[25] I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, N-Jettiness: An Inclusive Event Shape to Veto Jets, http://dx.doi.org/10.1103/PhysRevLett.105.092002

Web End =Phys. Rev. Lett. 105 (2010) 092002 [http://arxiv.org/abs/1004.2489

Web End =arXiv:1004.2489 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1004.2489

Web End =INSPIRE ].

[26] J. Thaler and K. Van Tilburg, Identifying Boosted Objects with N-subjettiness, http://dx.doi.org/10.1007/JHEP03(2011)015

Web End =JHEP 03 http://dx.doi.org/10.1007/JHEP03(2011)015

Web End =(2011) 015 [http://arxiv.org/abs/1011.2268

Web End =arXiv:1011.2268 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1011.2268

Web End =INSPIRE ].

JHEP02(2015)117

[27] J. Thaler and K. Van Tilburg, Maximizing Boosted Top Identication by Minimizing N-subjettiness, http://dx.doi.org/10.1007/JHEP02(2012)093

Web End =JHEP 02 (2012) 093 [http://arxiv.org/abs/1108.2701

Web End =arXiv:1108.2701 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.2701

Web End =INSPIRE ].

[28] A. Ban, G.P. Salam and G. Zanderighi, Principles of general nal-state resummation and automated implementation, http://dx.doi.org/10.1088/1126-6708/2005/03/073

Web End =JHEP 03 (2005) 073 [http://arxiv.org/abs/hep-ph/0407286

Web End =hep-ph/0407286 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0407286

Web End =INSPIRE ].

[29] A.J. Larkoski, G.P. Salam and J. Thaler, Energy Correlation Functions for Jet Substructure, http://dx.doi.org/10.1007/JHEP06(2013)108

Web End =JHEP 06 (2013) 108 [http://arxiv.org/abs/1305.0007

Web End =arXiv:1305.0007 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1305.0007

Web End =INSPIRE ].

[30] A.J. Larkoski, I. Moult and D. Neill, Power Counting to Better Jet Observables, http://dx.doi.org/10.1007/JHEP12(2014)009

Web End =JHEP 12 http://dx.doi.org/10.1007/JHEP12(2014)009

Web End =(2014) 009 [http://arxiv.org/abs/1409.6298

Web End =arXiv:1409.6298 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1409.6298

Web End =INSPIRE ].

[31] J. Thaler and L.-T. Wang, Strategies to Identify Boosted Tops, http://dx.doi.org/10.1088/1126-6708/2008/07/092

Web End =JHEP 07 (2008) 092 [http://arxiv.org/abs/0806.0023

Web End =arXiv:0806.0023 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.0023

Web End =INSPIRE ].

[32] L.G. Almeida et al., Substructure of high-pT Jets at the LHC, http://dx.doi.org/10.1103/PhysRevD.79.074017

Web End =Phys. Rev. D 79 (2009) http://dx.doi.org/10.1103/PhysRevD.79.074017

Web End =074017 [http://arxiv.org/abs/0807.0234

Web End =arXiv:0807.0234 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.0234

Web End =INSPIRE ].

[33] A.J. Larkoski and J. Thaler, Unsafe but Calculable: Ratios of Angularities in Perturbative QCD, http://dx.doi.org/10.1007/JHEP09(2013)137

Web End =JHEP 09 (2013) 137 [http://arxiv.org/abs/1307.1699

Web End =arXiv:1307.1699 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1307.1699

Web End =INSPIRE ].

[34] A.J. Larkoski, I. Moult and D. Neill, Toward Multi-Di erential Cross Sections: Measuring Two Angularities on a Single Jet, http://dx.doi.org/10.1007/JHEP09(2014)046

Web End =JHEP 09 (2014) 046 [http://arxiv.org/abs/1401.4458

Web End =arXiv:1401.4458 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1401.4458

Web End =INSPIRE ].

[35] T. Sjstrand, S. Mrenna and P.Z. Skands, PYTHIA 6.4 Physics and Manual, http://dx.doi.org/10.1088/1126-6708/2006/05/026

Web End =JHEP 05 http://dx.doi.org/10.1088/1126-6708/2006/05/026

Web End =(2006) 026 [http://arxiv.org/abs/hep-ph/0603175

Web End =hep-ph/0603175 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0603175

Web End =INSPIRE ].

[36] M. Bahr et al., HERWIG++ Physics and Manual, http://dx.doi.org/10.1140/epjc/s10052-008-0798-9

Web End =Eur. Phys. J. C 58 (2008) 639 [http://arxiv.org/abs/0803.0883

Web End =arXiv:0803.0883 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0803.0883

Web End =INSPIRE ].

[37] A.J. Larkoski, J. Thaler and W.J. Waalewijn, Gaining (Mutual) Information about Quark/Gluon Discrimination, http://dx.doi.org/10.1007/JHEP11(2014)129

Web End =JHEP 11 (2014) 129 [http://arxiv.org/abs/1408.3122

Web End =arXiv:1408.3122 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1408.3122

Web End =INSPIRE ].

[38] S. Frixione and B.R. Webber, Matching NLO QCD computations and parton shower simulations, http://dx.doi.org/10.1088/1126-6708/2002/06/029

Web End =JHEP 06 (2002) 029 [http://arxiv.org/abs/hep-ph/0204244

Web End =hep-ph/0204244 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0204244

Web End =INSPIRE ].

[39] Z. Nagy and D.E. Soper, Matching parton showers to NLO computations, http://dx.doi.org/10.1088/1126-6708/2005/10/024

Web End =JHEP 10 (2005) http://dx.doi.org/10.1088/1126-6708/2005/10/024

Web End =024 [http://arxiv.org/abs/hep-ph/0503053

Web End =hep-ph/0503053 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0503053

Web End =INSPIRE ].

[40] S. Frixione, P. Nason and C. Oleari, Matching NLO QCD computations with Parton Shower simulations: the POWHEG method, http://dx.doi.org/10.1088/1126-6708/2007/11/070

Web End =JHEP 11 (2007) 070 [http://arxiv.org/abs/0709.2092

Web End =arXiv:0709.2092 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0709.2092

Web End =INSPIRE ].

[41] S. Alioli, P. Nason, C. Oleari and E. Re, A general framework for implementing NLO calculations in shower Monte Carlo programs: the POWHEG BOX, http://dx.doi.org/10.1007/JHEP06(2010)043

Web End =JHEP 06 (2010) 043 [http://arxiv.org/abs/1002.2581

Web End =arXiv:1002.2581 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1002.2581

Web End =INSPIRE ].

[42] S. Alioli et al., Combining Higher-Order Resummation with Multiple NLO Calculations and Parton Showers in GENEVA, http://dx.doi.org/10.1007/JHEP09(2013)120

Web End =JHEP 09 (2013) 120 [http://arxiv.org/abs/1211.7049

Web End =arXiv:1211.7049 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1211.7049

Web End =INSPIRE ].

[43] K. Hamilton, P. Nason, E. Re and G. Zanderighi, NNLOPS simulation of Higgs boson production, http://dx.doi.org/10.1007/JHEP10(2013)222

Web End =JHEP 10 (2013) 222 [http://arxiv.org/abs/1309.0017

Web End =arXiv:1309.0017 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1309.0017

Web End =INSPIRE ].

[44] S. Alioli et al., Matching Fully Di erential NNLO Calculations and Parton Showers, http://dx.doi.org/10.1007/JHEP06(2014)089

Web End =JHEP http://dx.doi.org/10.1007/JHEP06(2014)089

Web End =06 (2014) 089 [http://arxiv.org/abs/1311.0286

Web End =arXiv:1311.0286 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1311.0286

Web End =INSPIRE ].

[45] S. Hoeche, Y. Li and S. Prestel, Drell-Yan lepton pair production at NNLO QCD with parton showers, http://arxiv.org/abs/1405.3607

Web End =arXiv:1405.3607 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1405.3607

Web End =INSPIRE ].

[46] I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Factorization at the LHC: From PDFs to Initial State Jets, http://dx.doi.org/10.1103/PhysRevD.81.094035

Web End =Phys. Rev. D 81 (2010) 094035 [http://arxiv.org/abs/0910.0467

Web End =arXiv:0910.0467 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0910.0467

Web End =INSPIRE ].

JHEP02(2015)117

[47] F.J. Tackmann, J.R. Walsh and S. Zuberi, Resummation Properties of Jet Vetoes at the LHC, http://dx.doi.org/10.1103/PhysRevD.86.053011

Web End =Phys. Rev. D 86 (2012) 053011 [http://arxiv.org/abs/1206.4312

Web End =arXiv:1206.4312 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1206.4312

Web End =INSPIRE ].

[48] C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in B ! Xs in e ective eld theory, http://dx.doi.org/10.1103/PhysRevD.63.014006

Web End =Phys. Rev. D 63 (2000) 014006 [http://arxiv.org/abs/hep-ph/0005275

Web End =hep-ph/0005275 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0005275

Web End =INSPIRE ].

[49] C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, An E ective eld theory for collinear and soft gluons: Heavy to light decays, http://dx.doi.org/10.1103/PhysRevD.63.114020

Web End =Phys. Rev. D 63 (2001) 114020 [http://arxiv.org/abs/hep-ph/0011336

Web End =hep-ph/0011336 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0011336

Web End =INSPIRE ].

[50] C.W. Bauer and I.W. Stewart, Invariant operators in collinear e ective theory, http://dx.doi.org/10.1016/S0370-2693(01)00902-9

Web End =Phys. Lett. http://dx.doi.org/10.1016/S0370-2693(01)00902-9

Web End =B 516 (2001) 134 [http://arxiv.org/abs/hep-ph/0107001

Web End =hep-ph/0107001 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0107001

Web End =INSPIRE ].

[51] C.W. Bauer, D. Pirjol and I.W. Stewart, Soft collinear factorization in e ective eld theory, http://dx.doi.org/10.1103/PhysRevD.65.054022

Web End =Phys. Rev. D 65 (2002) 054022 [http://arxiv.org/abs/hep-ph/0109045

Web End =hep-ph/0109045 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0109045

Web End =INSPIRE ].

[52] C.W. Bauer, F.J. Tackmann, J.R. Walsh and S. Zuberi, Factorization and Resummation for Dijet Invariant Mass Spectra, http://dx.doi.org/10.1103/PhysRevD.85.074006

Web End =Phys. Rev. D 85 (2012) 074006 [http://arxiv.org/abs/1106.6047

Web End =arXiv:1106.6047 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1106.6047

Web End =INSPIRE ].

[53] A. Jain, M. Procura and W.J. Waalewijn, Fully-Unintegrated Parton Distribution and Fragmentation Functions at Perturbative kT , http://dx.doi.org/10.1007/JHEP04(2012)132

Web End =JHEP 04 (2012) 132 [http://arxiv.org/abs/1110.0839

Web End =arXiv:1110.0839 ]

[http://inspirehep.net/search?p=find+EPRINT+arXiv:1110.0839

Web End =INSPIRE ].

[54] J.C. Collins, T.C. Rogers and A.M. Stasto, Fully unintegrated parton correlation functions and factorization in lowest-order hard scattering, http://dx.doi.org/10.1103/PhysRevD.77.085009

Web End =Phys. Rev. D 77 (2008) 085009 [http://arxiv.org/abs/0708.2833

Web End =arXiv:0708.2833 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0708.2833

Web End =INSPIRE ].

[55] T.C. Rogers, Next-to-Leading Order Hard Scattering Using Fully Unintegrated Parton Distribution Functions, http://dx.doi.org/10.1103/PhysRevD.78.074018

Web End =Phys. Rev. D 78 (2008) 074018 [http://arxiv.org/abs/0807.2430

Web End =arXiv:0807.2430 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.2430

Web End =INSPIRE ].

[56] S. Mantry and F. Petriello, Factorization and Resummation of Higgs Boson Di erential Distributions in Soft-Collinear E ective Theory, http://dx.doi.org/10.1103/PhysRevD.81.093007

Web End =Phys. Rev. D 81 (2010) 093007 [http://arxiv.org/abs/0911.4135

Web End =arXiv:0911.4135 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0911.4135

Web End =INSPIRE ].

[57] ATLAS collaboration, M. Myska, Measurement of the double parton scattering in W + 2 jets production at ps = 7 TeV with the ATLAS detector, http://dx.doi.org/10.1051/epjconf/20136020013

Web End =EPJ Web Conf. 60 (2013) 20013 [ http://inspirehep.net/search?p=find+J+00776,60,20013

Web End =INSPIRE ].

[58] CMS collaboration, Study of double parton scattering using W + 2-jet events in proton-proton collisions at ps = 7 TeV, http://dx.doi.org/10.1007/JHEP03(2014)032

Web End =JHEP 03 (2014) 032 [http://arxiv.org/abs/1312.5729

Web End =arXiv:1312.5729 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1312.5729

Web End =INSPIRE ].

[59] M. Cacciari, G.P. Salam and S. Sapeta, On the characterisation of the underlying event, http://dx.doi.org/10.1007/JHEP04(2010)065

Web End =JHEP 04 (2010) 065 [http://arxiv.org/abs/0912.4926

Web End =arXiv:0912.4926 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0912.4926

Web End =INSPIRE ].

[60] M. Diehl, D. Ostermeier and A. Schafer, Elements of a theory for multiparton interactions in QCD, http://dx.doi.org/10.1007/JHEP03(2012)089

Web End =JHEP 03 (2012) 089 [http://arxiv.org/abs/1111.0910

Web End =arXiv:1111.0910 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1111.0910

Web End =INSPIRE ].

[61] M.G. Ryskin and A.M. Snigirev, A Fresh look at double parton scattering, http://dx.doi.org/10.1103/PhysRevD.83.114047

Web End =Phys. Rev. D 83 http://dx.doi.org/10.1103/PhysRevD.83.114047

Web End =(2011) 114047 [http://arxiv.org/abs/1103.3495

Web End =arXiv:1103.3495 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1103.3495

Web End =INSPIRE ].

[62] A.V. Manohar and W.J. Waalewijn, What is Double Parton Scattering?, http://dx.doi.org/10.1016/j.physletb.2012.05.044

Web End =Phys. Lett. B 713 http://dx.doi.org/10.1016/j.physletb.2012.05.044

Web End =(2012) 196 [http://arxiv.org/abs/1202.5034

Web End =arXiv:1202.5034 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1202.5034

Web End =INSPIRE ].

[63] J.R. Gaunt, Single Perturbative Splitting Diagrams in Double Parton Scattering, http://dx.doi.org/10.1007/JHEP01(2013)042

Web End =JHEP 01 http://dx.doi.org/10.1007/JHEP01(2013)042

Web End =(2013) 042 [http://arxiv.org/abs/1207.0480

Web End =arXiv:1207.0480 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.0480

Web End =INSPIRE ].

[64] B. Blok, Y. Dokshitzer, L. Frankfurt and M. Strikman, Perturbative QCD correlations in multi-parton collisions, http://dx.doi.org/10.1140/epjc/s10052-014-2926-z

Web End =Eur. Phys. J. C 74 (2014) 2926 [http://arxiv.org/abs/1306.3763

Web End =arXiv:1306.3763 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.3763

Web End =INSPIRE ].

JHEP02(2015)117

[65] S.D. Ellis, C.K. Vermilion, J.R. Walsh, A. Hornig and C. Lee, Jet Shapes and Jet Algorithms in SCET, http://dx.doi.org/10.1007/JHEP11(2010)101

Web End =JHEP 11 (2010) 101 [http://arxiv.org/abs/1001.0014

Web End =arXiv:1001.0014 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1001.0014

Web End =INSPIRE ].

[66] T.T. Jouttenus, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, The Soft Function for Exclusive N-Jet Production at Hadron Colliders, http://dx.doi.org/10.1103/PhysRevD.83.114030

Web End =Phys. Rev. D 83 (2011) 114030 [http://arxiv.org/abs/1102.4344

Web End =arXiv:1102.4344 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1102.4344

Web End =INSPIRE ].

[67] M. Dasgupta and G.P. Salam, Resummation of nonglobal QCD observables, http://dx.doi.org/10.1016/S0370-2693(01)00725-0

Web End =Phys. Lett. B http://dx.doi.org/10.1016/S0370-2693(01)00725-0

Web End =512 (2001) 323 [http://arxiv.org/abs/hep-ph/0104277

Web End =hep-ph/0104277 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0104277

Web End =INSPIRE ].

[68] M. Dasgupta and G.P. Salam, Resummed event shape variables in DIS, http://dx.doi.org/10.1088/1126-6708/2002/08/032

Web End =JHEP 08 (2002) http://dx.doi.org/10.1088/1126-6708/2002/08/032

Web End =032 [http://arxiv.org/abs/hep-ph/0208073

Web End =hep-ph/0208073 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0208073

Web End =INSPIRE ].

[69] I.W. Stewart, Lectures on the Soft-Collinear E ective Theory, MIT Open Course Ware, E ective Field Theory (Spring 2013), http://ocw.mit.edu/courses/physics/8-851-effective-field-theory-spring-2013/lecture-notes/MIT8_851S13_scetnotes.pdf

Web End =http://ocw.mit.edu/courses/physics/8-851-e ective http://ocw.mit.edu/courses/physics/8-851-effective-field-theory-spring-2013/lecture-notes/MIT8_851S13_scetnotes.pdf

Web End =eld-theory-spring-2013/lecture-notes/MIT8 851S13 scetnotes.pdf .

[70] T. Becher, A. Broggio and A. Ferroglia, Introduction to Soft-Collinear E ective Theory, http://arxiv.org/abs/1410.1892

Web End =arXiv:1410.1892 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1410.1892

Web End =INSPIRE ].

[71] J.-y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, The Rapidity Renormalization Group, http://dx.doi.org/10.1103/PhysRevLett.108.151601

Web End =Phys. http://dx.doi.org/10.1103/PhysRevLett.108.151601

Web End =Rev. Lett. 108 (2012) 151601 [http://arxiv.org/abs/1104.0881

Web End =arXiv:1104.0881 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1104.0881

Web End =INSPIRE ].

[72] J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory, http://dx.doi.org/10.1007/JHEP05(2012)084

Web End =JHEP 05 (2012) 084 [http://arxiv.org/abs/1202.0814

Web End =arXiv:1202.0814 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1202.0814

Web End =INSPIRE ].

[73] J.-y. Chiu, F. Golf, R. Kelley and A.V. Manohar, Electroweak Corrections in High Energy Processes using E ective Field Theory, http://dx.doi.org/10.1103/PhysRevD.77.053004

Web End =Phys. Rev. D 77 (2008) 053004 [http://arxiv.org/abs/0712.0396

Web End =arXiv:0712.0396 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0712.0396

Web End =INSPIRE ].

[74] A.V. Manohar and I.W. Stewart, The zero-bin and mode factorization in quantum eld theory, http://dx.doi.org/10.1103/PhysRevD.76.074002

Web End =Phys. Rev. D 76 (2007) 074002 [http://arxiv.org/abs/hep-ph/0605001

Web End =hep-ph/0605001 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0605001

Web End =INSPIRE ].

[75] G.T. Bodwin, S.J. Brodsky and G.P. Lepage, Initial State Interactions and the Drell-Yan Process, http://dx.doi.org/10.1103/PhysRevLett.47.1799

Web End =Phys. Rev. Lett. 47 (1981) 1799 [http://inspirehep.net/search?p=find+J+Phys.Rev.Lett.,47,1799

Web End =INSPIRE ].

[76] J.C. Collins, D.E. Soper and G.F. Sterman, Soft Gluons and Factorization, http://dx.doi.org/10.1016/0550-3213(88)90130-7

Web End =Nucl. Phys. B http://dx.doi.org/10.1016/0550-3213(88)90130-7

Web End =308 (1988) 833 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B308,833

Web End =INSPIRE ].

[77] J.R. Gaunt, Glauber Gluons and Multiple Parton Interactions, http://dx.doi.org/10.1007/JHEP07(2014)110

Web End =JHEP 07 (2014) 110 [http://arxiv.org/abs/1405.2080

Web End =arXiv:1405.2080 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1405.2080

Web End =INSPIRE ].

[78] E. Laenen, G.F. Sterman and W. Vogelsang, Recoil and threshold corrections in short distance cross-sections, http://dx.doi.org/10.1103/PhysRevD.63.114018

Web End =Phys. Rev. D 63 (2001) 114018 [http://arxiv.org/abs/hep-ph/0010080

Web End =hep-ph/0010080 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0010080

Web End =INSPIRE ].

[79] I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, The Quark Beam Function at NNLL, http://dx.doi.org/10.1007/JHEP09(2010)005

Web End =JHEP 09 (2010) 005 [http://arxiv.org/abs/1002.2213

Web End =arXiv:1002.2213 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1002.2213

Web End =INSPIRE ].

[80] R. Kelley, M.D. Schwartz, R.M. Schabinger and H.X. Zhu, The two-loop hemisphere soft function, http://dx.doi.org/10.1103/PhysRevD.84.045022

Web End =Phys. Rev. D 84 (2011) 045022 [http://arxiv.org/abs/1105.3676

Web End =arXiv:1105.3676 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1105.3676

Web End =INSPIRE ].

[81] A. Hornig, C. Lee, I.W. Stewart, J.R. Walsh and S. Zuberi, Non-global Structure of the O( 2s) Dijet Soft Function, http://dx.doi.org/10.1007/JHEP08(2011)054

Web End =JHEP 08 (2011) 054 [http://arxiv.org/abs/1105.4628

Web End =arXiv:1105.4628 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1105.4628

Web End =INSPIRE ].

[82] A. Hornig, C. Lee, J.R. Walsh and S. Zuberi, Double Non-Global Logarithms In-N-Out of Jets, http://dx.doi.org/10.1007/JHEP01(2012)149

Web End =JHEP 01 (2012) 149 [http://arxiv.org/abs/1110.0004

Web End =arXiv:1110.0004 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1110.0004

Web End =INSPIRE ].

[83] A.V. Manohar, Deep inelastic scattering as x ! 1 using soft collinear e ective theory, http://dx.doi.org/10.1103/PhysRevD.68.114019

Web End =Phys.

http://dx.doi.org/10.1103/PhysRevD.68.114019

Web End =Rev. D 68 (2003) 114019 [http://arxiv.org/abs/hep-ph/0309176

Web End =hep-ph/0309176 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0309176

Web End =INSPIRE ].

JHEP02(2015)117

[84] C.W. Bauer, C. Lee, A.V. Manohar and M.B. Wise, Enhanced nonperturbative e ects in Z decays to hadrons, http://dx.doi.org/10.1103/PhysRevD.70.034014

Web End =Phys. Rev. D 70 (2004) 034014 [http://arxiv.org/abs/hep-ph/0309278

Web End =hep-ph/0309278 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0309278

Web End =INSPIRE ].

[85] J.R. Gaunt and M. Stahlhofen, The Fully-Di erential Quark Beam Function at NNLO, http://dx.doi.org/10.1007/JHEP12(2014)146

Web End =JHEP 12 (2014) 146 [http://arxiv.org/abs/1409.8281

Web End =arXiv:1409.8281 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1409.8281

Web End =INSPIRE ].

[86] J.C. Collins and D.E. Soper, Parton Distribution and Decay Functions, http://dx.doi.org/10.1016/0550-3213(82)90021-9

Web End =Nucl. Phys. B 194 http://dx.doi.org/10.1016/0550-3213(82)90021-9

Web End =(1982) 445 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B194,445

Web End =INSPIRE ].

[87] M. Ritzmann and W.J. Waalewijn, Fragmentation in Jets at NNLO, http://dx.doi.org/10.1103/PhysRevD.90.054029

Web End =Phys. Rev. D 90 http://dx.doi.org/10.1103/PhysRevD.90.054029

Web End =(2014) 054029 [http://arxiv.org/abs/1407.3272

Web End =arXiv:1407.3272 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1407.3272

Web End =INSPIRE ].

[88] J. Collins, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology. Book 32: Foundations of perturbative QCD, Cambridge University Press, Cambridge U.K. (2011).

[89] M.D. Schwartz, Resummation and NLO matching of event shapes with e ective eld theory, http://dx.doi.org/10.1103/PhysRevD.77.014026

Web End =Phys. Rev. D 77 (2008) 014026 [http://arxiv.org/abs/0709.2709

Web End =arXiv:0709.2709 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0709.2709

Web End =INSPIRE ].

[90] S. Fleming, A.H. Hoang, S. Mantry and I.W. Stewart, Top Jets in the Peak Region: Factorization Analysis with NLL Resummation, http://dx.doi.org/10.1103/PhysRevD.77.114003

Web End =Phys. Rev. D 77 (2008) 114003 [http://arxiv.org/abs/0711.2079

Web End =arXiv:0711.2079 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0711.2079

Web End =INSPIRE ].

[91] G. Parisi, Summing Large Perturbative Corrections in QCD, http://dx.doi.org/10.1016/0370-2693(80)90746-7

Web End =Phys. Lett. B 90 (1980) 295 [ http://inspirehep.net/search?p=find+J+Phys.Lett.,B90,295

Web End =INSPIRE ].

[92] G.F. Sterman, Summation of Large Corrections to Short Distance Hadronic Cross-Sections, http://dx.doi.org/10.1016/0550-3213(87)90258-6

Web End =Nucl. Phys. B 281 (1987) 310 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B281,310

Web End =INSPIRE ].

[93] L. Magnea and G.F. Sterman, Analytic continuation of the Sudakov form-factor in QCD, http://dx.doi.org/10.1103/PhysRevD.42.4222

Web End =Phys. Rev. D 42 (1990) 4222 [http://inspirehep.net/search?p=find+J+Phys.Rev.,D42,4222

Web End =INSPIRE ].

[94] T.O. Eynck, E. Laenen and L. Magnea, Exponentiation of the Drell-Yan cross-section near partonic threshold in the DIS and MS-bar schemes, http://dx.doi.org/10.1088/1126-6708/2003/06/057

Web End =JHEP 06 (2003) 057 [http://arxiv.org/abs/hep-ph/0305179

Web End =hep-ph/0305179 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0305179

Web End =INSPIRE ].

[95] C. Balzereit, T. Mannel and W. Kilian, Evolution of the light cone distribution function for a heavy quark, http://dx.doi.org/10.1103/PhysRevD.58.114029

Web End =Phys. Rev. D 58 (1998) 114029 [http://arxiv.org/abs/hep-ph/9805297

Web End =hep-ph/9805297 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/9805297

Web End =INSPIRE ].

[96] M. Neubert, Renormalization-group improved calculation of the B ! Xs branching ratio,

http://dx.doi.org/10.1140/epjc/s2005-02141-1

Web End =Eur. Phys. J. C 40 (2005) 165 [http://arxiv.org/abs/hep-ph/0408179

Web End =hep-ph/0408179 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0408179

Web End =INSPIRE ].

[97] Z. Ligeti, I.W. Stewart and F.J. Tackmann, Treating the b quark distribution function with reliable uncertainties, http://dx.doi.org/10.1103/PhysRevD.78.114014

Web End =Phys. Rev. D 78 (2008) 114014 [http://arxiv.org/abs/0807.1926

Web End =arXiv:0807.1926 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.1926

Web End =INSPIRE ].

[98] A.V. Manohar and W.J. Waalewijn, A QCD Analysis of Double Parton Scattering: Color Correlations, Interference E ects and Evolution, http://dx.doi.org/10.1103/PhysRevD.85.114009

Web End =Phys. Rev. D 85 (2012) 114009 [http://arxiv.org/abs/1202.3794

Web End =arXiv:1202.3794 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1202.3794

Web End =INSPIRE ].

[99] L.G. Almeida et al., Comparing and counting logs in direct and e ective methods of QCD resummation, http://dx.doi.org/10.1007/JHEP04(2014)174

Web End =JHEP 04 (2014) 174 [http://arxiv.org/abs/1401.4460

Web End =arXiv:1401.4460 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1401.4460

Web End =INSPIRE ].

[100] R. Abbate, M. Fickinger, A.H. Hoang, V. Mateu and I.W. Stewart, Thrust at N3LL with Power Corrections and a Precision Global Fit for s(mZ), http://dx.doi.org/10.1103/PhysRevD.83.074021

Web End =Phys. Rev. D 83 (2011) 074021 [http://arxiv.org/abs/1006.3080

Web End =arXiv:1006.3080 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1006.3080

Web End =INSPIRE ].

[101] M. Procura, W.J. Waalewijn and L. Zeune, in preparation.

[102] C.F. Berger, T. Kucs and G.F. Sterman, Event shape/energy ow correlations, http://dx.doi.org/10.1103/PhysRevD.68.014012

Web End =Phys. Rev. http://dx.doi.org/10.1103/PhysRevD.68.014012

Web End =D 68 (2003) 014012 [http://arxiv.org/abs/hep-ph/0303051

Web End =hep-ph/0303051 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0303051

Web End =INSPIRE ].

JHEP02(2015)117

[103] S. Catani, G. Turnock and B.R. Webber, Jet broadening measures in e+e annihilation,http://dx.doi.org/10.1016/0370-2693(92)91565-Q

Web End =Phys. Lett. B 295 (1992) 269 [http://inspirehep.net/search?p=find+J+Phys.Lett.,B295,269

Web End =INSPIRE ].

[104] Y.L. Dokshitzer, A. Lucenti, G. Marchesini and G.P. Salam, On the QCD analysis of jet broadening, http://dx.doi.org/10.1088/1126-6708/1998/01/011

Web End =JHEP 01 (1998) 011 [http://arxiv.org/abs/hep-ph/9801324

Web End =hep-ph/9801324 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/9801324

Web End =INSPIRE ].

[105] A.J. Larkoski, D. Neill and J. Thaler, Jet Shapes with the Broadening Axis, http://dx.doi.org/10.1007/JHEP04(2014)017

Web End =JHEP 04 http://dx.doi.org/10.1007/JHEP04(2014)017

Web End =(2014) 017 [http://arxiv.org/abs/1401.2158

Web End =arXiv:1401.2158 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1401.2158

Web End =INSPIRE ].

[106] D. Bertolini, T. Chan and J. Thaler, Jet Observables Without Jet Algorithms, http://dx.doi.org/10.1007/JHEP04(2014)013

Web End =JHEP 04 http://dx.doi.org/10.1007/JHEP04(2014)013

Web End =(2014) 013 [http://arxiv.org/abs/1310.7584

Web End =arXiv:1310.7584 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1310.7584

Web End =INSPIRE ].

[107] O.V. Tarasov, A.A. Vladimirov and A.Y. Zharkov, The Gell-Mann-Low Function of QCD in the Three Loop Approximation, http://dx.doi.org/10.1016/0370-2693(80)90358-5

Web End =Phys. Lett. B 93 (1980) 429 [http://inspirehep.net/search?p=find+J+Phys.Lett.,B93,429

Web End =INSPIRE ].

[108] S.A. Larin and J.A.M. Vermaseren, The Three loop QCD -function and anomalous dimensions, http://dx.doi.org/10.1016/0370-2693(93)91441-O

Web End =Phys. Lett. B 303 (1993) 334 [http://arxiv.org/abs/hep-ph/9302208

Web End =hep-ph/9302208 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/9302208

Web End =INSPIRE ].

[109] G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond theLeading Order, http://dx.doi.org/10.1016/0550-3213(87)90277-X

Web End =Nucl. Phys. B 283 (1987) 342 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B283,342

Web End =INSPIRE ].

[110] S. Moch, J.A.M. Vermaseren and A. Vogt, The Three loop splitting functions in QCD: TheNonsinglet case, http://dx.doi.org/10.1016/j.nuclphysb.2004.03.030

Web End =Nucl. Phys. B 688 (2004) 101 [http://arxiv.org/abs/hep-ph/0403192

Web End =hep-ph/0403192 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0403192

Web End =INSPIRE ].

[111] S. Moch, J.A.M. Vermaseren and A. Vogt, The Quark form-factor at higher orders, http://dx.doi.org/10.1088/1126-6708/2005/08/049

Web End =JHEP http://dx.doi.org/10.1088/1126-6708/2005/08/049

Web End =08 (2005) 049 [http://arxiv.org/abs/hep-ph/0507039

Web End =hep-ph/0507039 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0507039

Web End =INSPIRE ].

[112] S. Moch, J.A.M. Vermaseren and A. Vogt, Three-loop results for quark and gluon form-factors, http://dx.doi.org/10.1016/j.physletb.2005.08.067

Web End =Phys. Lett. B 625 (2005) 245 [http://arxiv.org/abs/hep-ph/0508055

Web End =hep-ph/0508055 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0508055

Web End =INSPIRE ].

[113] A. Vogt, S. Moch and J.A.M. Vermaseren, The Three-loop splitting functions in QCD: TheSinglet case, http://dx.doi.org/10.1016/j.nuclphysb.2004.04.024

Web End =Nucl. Phys. B 691 (2004) 129 [http://arxiv.org/abs/hep-ph/0404111

Web End =hep-ph/0404111 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0404111

Web End =INSPIRE ].

[114] A. Idilbi, X.-d. Ji, J.-P. Ma and F. Yuan, Threshold resummation for Higgs production in e ective eld theory, http://dx.doi.org/10.1103/PhysRevD.73.077501

Web End =Phys. Rev. D 73 (2006) 077501 [http://arxiv.org/abs/hep-ph/0509294

Web End =hep-ph/0509294 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0509294

Web End =INSPIRE ].

[115] A. Idilbi, X.-d. Ji and F. Yuan, Resummation of threshold logarithms in e ective eld theory for DIS, Drell-Yan and Higgs production, http://dx.doi.org/10.1016/j.nuclphysb.2006.07.002

Web End =Nucl. Phys. B 753 (2006) 42 [http://arxiv.org/abs/hep-ph/0605068

Web End =hep-ph/0605068 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0605068

Web End =INSPIRE ].

[116] T. Becher, M. Neubert and B.D. Pecjak, Factorization and Momentum-Space Resummation in Deep-Inelastic Scattering, http://dx.doi.org/10.1088/1126-6708/2007/01/076

Web End =JHEP 01 (2007) 076 [http://arxiv.org/abs/hep-ph/0607228

Web End =hep-ph/0607228 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0607228

Web End =INSPIRE ].

[117] T. Becher and M.D. Schwartz, Direct photon production with e ective eld theory, http://dx.doi.org/10.1007/JHEP02(2010)040

Web End =JHEP http://dx.doi.org/10.1007/JHEP02(2010)040

Web End =02 (2010) 040 [http://arxiv.org/abs/0911.0681

Web End =arXiv:0911.0681 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0911.0681

Web End =INSPIRE ].

[118] T. Gehrmann, T. Luebbert and L.L. Yang, Calculation of the transverse parton distribution functions at next-to-next-to-leading order, http://dx.doi.org/10.1007/JHEP06(2014)155

Web End =JHEP 06 (2014) 155 [http://arxiv.org/abs/1403.6451

Web End =arXiv:1403.6451 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1403.6451

Web End =INSPIRE ].

JHEP02(2015)117

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SISSA, Trieste, Italy 2015

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(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Abstract

LHC measurements involve cuts on several observables, but resummed calculations are mostly restricted to single variables. We show how the resummation of a class of double-differential measurements can be achieved through an extension of Soft-Collinear Effective Theory (SCET). A prototypical application is pp [arrow right] Z + 0 jets, where the jet veto is imposed through the beam thrust event shape ......, and the transverse momentum p ^sub T^ of the Z boson is measured. A standard SCET analysis suffices for ...... and p ^sub T^ ......, but additional collinear-soft modes are needed in the intermediate regime. We show how to match the factorization theorems that describe these three different regions of phase space, and discuss the corresponding relations between fully-unintegrated parton distribution functions, soft functions and the newly defined collinear-soft functions. The missing ingredients needed at NNLL/NLO accuracy are calculated, providing a check of our formalism. We also revisit the calculation of the measurement of two angularities on a single jet in JHEP 1409 (2014) 046, finding a correction to their conjecture for the NLL cross section at .......

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Title

Resummation of double-differential cross sections and fully-unintegrated parton distribution functions

Author

Procura, Massimiliano; Waalewijn, Wouter J; Zeune, Lisa

Pages

1-43

Publication year

2015

Publication date

Feb 2015

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP02(2015)117

ProQuest document ID

1708026296

SISSA, Trieste, Italy 2015

Resummation of double-differential cross sections and fully-unintegrated parton distribution functions

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