Projected transverse momentum resummation in top-antitop pair production at LHC

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IPPP/22/73

MCnet-22-20

Projected transverse momentum resummation in

top-antitop pair production at LHC

Wan-Li Ju(a,b), Marek Sch¨onherr(a)

(a)Institute for Particle Physics Phenomenology, Durham University, Durham DH1 3LE, United Kingdom

(b)INFN, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy

Abstract: The transverse momentum distribution of the t¯

tsystem is of both experimental

and theoretical interest. In the presence of azimuthally asymmetric divergences,

pursuing resummation at high logarithmic precision is rather demanding in gen-

eral. In this paper, we propose the projected transverse momentum spectrum

dσt¯

t/dqτ, which is derived from the classical ~qTspectrum by integrating out the

rejection component qτ⊥with respect to a reference unit vector ~τ, to serve as an

alternative solution to remove these asymmetric divergences, in addition to the

azimuthally averaged case dσt¯

t/d|~qT|. In the context of the eﬀective ﬁeld theories,

SCETII and HQET, we will demonstrate that in spite of the qτ⊥integrations, the

leading asymptotic terms of dσt¯

t/dqτstill observe the factorisation pattern in

terms of the hard, beam, and soft functions in the vicinity of qτ= 0 GeV. Then,

with the help of the renormalisation group equation techniques, we carry out the

resummation at NLL+NLO, N2LL+N2LO, and approximate N2LL0+N2LO accu-

racy on three observables of interest, dσt¯

t/dqT,in, dσt¯

t/dqT,out, and dσt¯

t/d∆φt¯

within the domain Mt¯

t≥400 GeV. The ﬁrst two cases are obtained by choosing ~τ

parallel and perpendicular to the top quark transverse momentum, respectively.

The azimuthal de-correlation ∆φt¯

tof the t¯

tpair is evaluated through its kine-

matical connection to qT,out. This is the ﬁrst time the azimuthal spectrum ∆φt¯

is appraised at or beyond the N2LL level including a consistent treatment of both

beam collinear and soft radiation.

arXiv:2210.09272v3 [hep-ph] 11 Feb 2023

Contents

1 Introduction 2

2 Factorisation 4

2.1 Kinematics and the factorised cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Dynamicregions ........................................... 5

2.3 The case of NJ=0.......................................... 7

2.4 The case of NJ≥1.......................................... 13

2.4.1 The NJ=1conﬁguration .................................. 13

2.4.2 The NJ≥2conﬁguration .................................. 16

2.4.3 Summaryanddiscussion................................... 17

2.5 The soft function with the exponential regulator . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Resummation 21

3.1 Asymptoticbehavior......................................... 21

3.2 Evolutionequations ......................................... 23

3.3 Observables.............................................. 25

3.4 Matchingtoﬁxed-orderQCD.................................... 26

4 Numerical Results 27

4.1 Parameters and uncertainty estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Validation............................................... 28

4.3 Resummation improved results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Conclusions 33

A The double diﬀerential transverse momentum distribution d2σt¯

t/d2~qT34

B Evolution kernel for the non-diagonal anomalous dimension 40

C Decomposition of the theoretical uncertainty estimate 41

1 Introduction

Hadroproduction of top-antitop pairs plays a pivotal part in the physics programme of the LHC experiments

due to its role in the precise extractions of fundamental parameters of the Standard Model (SM). It has

thus drawn plenty of theoretical and experimental attention in the recent years. On the experimental

side, the inclusive top-pair production cross section has been measured at the colliding energies √s=

5.02 TeV [1–4], 7 TeV [5–7], 8 TeV [6–9], 13 TeV [10–16] and 13.6 TeV [17], respectively, whilst a large

number of the diﬀerential spectra have been published in the latest analyses [12, 16, 18–21], including the

transverse momentum of the t¯

tsystem qT, the invariant mass of the top quark pair Mt¯

t, and the azimuthal

opening angle of the top and antitop quarks ∆Φt¯

t. Theoretical calculations of these spectra also have since

long attracted a lot of interest in the community. While NLO QCD corrections to top-pair production were

determined already some time ago [22–25], recent advances have reached the NNLO accuracy [26–32]. Top-

quark decay eﬀects were considered in [33–37] and the electroweak (EW) corrections in [30, 38–45]. Along

with the progress made in ﬁxed-order calculations, in a bid to improve the perturbative convergence and

in turn the predictivity of the theoretical results, resummed calculations have also been carried out within

a variety of frameworks and the kinematical limits. Examples include the mechanic threshold [46–56], the

top-quark pair production threshold [57–63], the low transverse momentum domain [64–68], and the narrow

jettiness regime [69]. Very recently, the combination of the ﬁxed-order results with a parton shower has been

discussed in [70, 71].

This work will investigate the projection of the t¯

tsystem’s transverse momentum with respect to a reference

unit vector ~τ on the azimuthal plane, more explicitly, qτ≡ |~qτk| ≡ |~qT·~τ|. In contrast to the traditional

transverse momentum spectrum dσt¯

t/dqT,qT=|~qT|, where both components of ~qTare measured and thereby

constrained, the present observable dσt¯

t/dqτonly concerns the projected piece ~qτk, leaving the perpendicular

part ~qτ⊥unresolved and, hence, it should be integrated out. As will be demonstrated in this paper, the act

of integrating out the perpendicular components will introduce new and distinguishing features to the qτ

spectrum, particularly in regards to the treatment on the azimuthal asymmetric contributions [66,67,72,73].

Induced by the soft and collinear radiation, the ﬁxed-order calculation of the qτdistribution exhibits substan-

tial higher-order corrections in the small qτregion. This, thus, necessitates a resummation of the dominant

contributions in this regime to all orders to stabilise the perturbative predictions. In order to accomplish

this target, one of the prerequisite conditions is to determine the dynamic regions driving the asymptotic

behaviour in the limit qτ→0. For the classic transverse momentum resummation, this analysis was ﬁrst

presented for Drell-Yan production in [74] by means of inspecting the power laws of a generic conﬁgura-

tion on the pinch singularity surface [75–77]. It was proven that the leading singular behaviour in the

small qTdomain was well captured by the beam-collinear, soft, and hard regions. However, this conclusion

cannot be straightforwardly applied onto the qτresummation in top-pair production, as the deep recoil

conﬁguration |~qτ⊥| ∼ Mt¯

tqτ, which stems from the integral over the perpendicular component, was kine-

matically excluded in [74]. Therefore, for delivering an honest and self-consistent study on dσt¯

t/dqτ, this

work will reappraise the scalings of the relevant conﬁgurations, comprising both the qT-like conﬁguration

Mt¯

t |~qτ⊥| ∼ qτand the asymmetric one |~qτ⊥| ∼ Mt¯

tqτ.

To this end, we will exploit the strategy of expansion by regions [78–81] to motivate the momentum modes

governing the low qτregime, which will cover the beam-collinear, soft, central-jet, and hard regions. Then,

the soft-collinear eﬀective theory (SCET) [82–91] and the heavy quark eﬀective theory (HQET) [92–95]

are used to embody those dynamic modes, thereby calculating the eﬀective amplitudes and the respective

diﬀerential cross sections. After carrying out a multipole expansion, the results constructed by those dynamic

regions all reﬂect the unambiguous scaling behaviors, which can be determined from the power prescriptions

of the relevant eﬀective ﬁelds. From the outcome, we point out that the leading asymptotic behavior of

dσt¯

t/dqτis still resultant of the symmetric conﬁguration Mt¯

t |~qτ⊥| ∼ qτ, which is in practice dictated by

the beam-collinear, soft, and hard momenta, akin to the case of dσt¯

t/dqT, whereas the contributions from

the |~qτ⊥| ∼ Mt¯

tqτpattern are suppressed by at least one power of λτ≡qτ/Mt¯

Upon the identiﬁcation of the leading regions, we make use of the decoupling properties of the soft modes [58,

82] to derive the factorisation formula for dσt¯

t/dqτ. Owing to the integration over ~qτ⊥, the impact space

integrals herein are all reduced from 2D to 1D. Thus, the azimuthal asymmetric contributions, which in

principle contain divergent terms in the asymptotic regime in the general dσt¯

t/d~qTcross section after com-

pleting the inverse Fourier transformations, do not contribute any divergences to the qτspectrum. This

is the second ~qT-based observable free of asymmetric singularities in addition to the azimuthally averaged

spectra dσt¯

t/dqT[64, 65, 68].

To implement the resummation, we employ the renormalisation group equations (RGE) and the rapidity

renormalisation group equations (RaGE) to evolve the intrinsic scales in the respective ingredients and in turn

accomplish the logarithmic exponentiations [96–99]. Alternative approaches can also be found in [100–109].

For assessing the resummation accuracy, we take the logarithmic counting rule LM∼α−1

s∼λ−1

Lthroughout

and organise the perturbative corrections to the relevant sectors in line with the following prescription,

dσt¯

dqτ∼σBorn

t¯

texp LMf0(αsLM)

| {z }

(LL)

+f1(αsLM)

| {z }

(NLL,NLL0)

+αsf2(αsLM)

| {z }

(N2LL,N2LL0)

+α2

sf3(αsLM)

| {z }

(N3LL,N3LL0)

+. . . 

×1(LL,NLL) + αs(NLL0,N2LL) + α2

s(N2LL0,N3LL) + α3

s(N3LL0,N4LL) + . . . .

(1.1)

Therein, the desired precisions of the anomalous dimensions are speciﬁed between the square brackets in the

exponent as for a given logarithmic accuracy, while the according requirements on the ﬁxed-order elements

are presented within the curly brackets. In this work, we will evaluate and compare the resummed qτspectra

on the next-to-leading-logarithmic (NLL), N2LL, and approximate N2LL0(aN2LL0) levels.

The paper is structured as follows. In Sec. 2, we will utilise the strategy of expansion by dynamic re-

gions [78–81] and eﬀective ﬁeld theories, i.e. SCETII [89–91] and HQET [92–95], to derive the factorisation

⃗

P⊥

⃗

P⊥

∆Φt¯

⃗qT

⃗τ

⃗qτ⊥

⃗qτ∥

Figure 1: The kinematics on the transverse plane in the laboratory reference frame. ~

P⊥

t(¯

t)stands for the

transverse momentum of the (anti-)top quark. ∆Φt¯

tis the azimuthal opening angle between the

top and anti-top quarks. ~τ is a unit reference vector in the transverse plane.

formula governing the leading asymptotic behaviour of dσt¯

t/dqτin the limit qτ→0. Then, the (rapidity)

renormalisation group equations will be solved in Sec. 3 for the respective sectors participating into the

factorisation formula, from which we exponentiate the characteristic logarithmic constituents in the impact

space and thereby accomplish the resummation of the singular terms in the momentum space. Sec. 4 will be

devoted to the numeric evaluations on the spectra qτ. Therein, we will at ﬁrst validate the approximations of

our factorisation formula up to N2LO, and then present the resummation improved diﬀerential distributions

for three particular observables, dσt¯

t/qT,in, dσt¯

t/qT,out, and dσt¯

t/∆φt¯

t.qT,in(out) is a special case of qτon

the choice of ~τ parallel (perpendicular) to the top quark transverse momentum, while ∆φt¯

trepresents the

azimuthal de-correlation of the t¯

tpair and can be extracted through its kinematical connection to qT,out.

Finally, we will oﬀer some concluding remarks in Sec. 5.

2 Factorisation

2.1 Kinematics and the factorised cross section

We start this section with the elaboration on the kinematics. As illustrated in Fig. 1, the main concern of

this work is on the interplay between a reference unit vector ~τ and the transverse momentum ~qTof the t¯

system. By means of the reference vector ~τ ,~qTcan be decomposed into two parts, the projection component

~qτkand the rejection one ~qτ⊥, i.e.,

~qT=~qτ⊥+~qτk=qτ⊥~τ ×~n +qτk~τ , (2.1)

where ~n is another unit vector pointing to one of beam directions in the laboratory reference frame. In the

numeric implementation presented in this paper, the magnitude of the projection ~qτkis of primary interest,

which will hereafter be dubbed qτ≡ |~qτk|.

The ﬁxed-order calculation on the qτspectrum can be realised using the QCD factorisation theorem of [110],

that is,

d5σt¯

dM2

t¯

td2~

P⊥

tdYt¯

tdqτ

sign[Pz

16s(2π)6Zd2~qTΘkin δhqτ− |~qT·~τ|iΣt¯

Mt¯

T|Pz

t|,(2.2)

where Mt¯

tdenotes the invariant mass of the t¯

tsystem, and sis the colliding energy. In this work we will

concentrate on √s= 13 TeV throughout. Yt¯

tand Mt¯

Tare for the pseudorapidity and the transverse mass of

the t¯

tpair in the laboratory frame (LF), respectively. Pz

trepresents the longitudinal components of the top

quark momentum measured from the z-direction rest frame (zRF) of the t¯

tpair. The zRF can be obtained

through boosting the LF along one of the beam directions until the longitudinal momentum of the t¯

tpair

has been eliminated.

To perform the integral of ~qTin Eq. (2.2), it is of essence to establish suitable kinematical boundaries to

fulﬁll energy-momentum conservation condition. To this end, we introduce the function Θkin to impose the

following constraints,

Θkin = Θh√s−Mt¯

T− |~qT|iΘhMt¯

T−mt

T−m¯

TiΘ"sinh−1 s(M2

t¯

t+s)2

4s Mt¯

2−1!− |Yt¯

t|#,(2.3)

where Θ[. . . ] is the usual Heaviside function. Therein, mt

Tand m¯

Tare the transverse masses of the top and

anti-top quarks in the LF, respectively. Finally, Eq. (2.2) also entails Σt¯

t, encoding the contributions from

all partonic processes, deﬁned as,

Σt¯

t=X

i,j Z1

dxn

dx¯n

x¯n

fi/N (xn)fj/ ¯

N(x¯n)X

rZr

dΦkm(2π)4δ4 pi+pj−pt−p¯

t−X

km!

×X

hel,col |M(i+j→t+¯

t+X)|2.

(2.4)

Here fi/N (x) is the parton distribution function (PDF) for the parton iwithin the proton Ncarrying the

momentum fraction x. dΦkmcharacterises the phase space integral of the m-th emitted parton, that is,

dΦkm≡d4km

(2π)3δ(k2

m) Θ(k0

m) = 1

dym

2π

d2~

k⊥

(2π)2,(2.5)

where ymand ~

k⊥

mindicate the rapidity and transverse momentum of the occurring emission, respectively.

|M|2is the squared transition amplitude for the partonic processes of the indices indicated. Substituting

Eq. (2.4) into Eq. (2.2), it is ready to perform the ﬁxed-order calculations of the spectra of qτ. In the

vicinity of qτ= 0 GeV, however, this perturbative expansion fails to converge, and an asymptotic expansion

of dσt¯

t/dqτcan be carried out in the small parameter λτ≡qτ/Mt¯

dσt¯

dqτ∼σt¯

m,n αs(Mt¯

4πm



c(0)

m,n

lnn(λτ)

λτ

| {z }

+c(1)

m,n lnn(λτ)

| {z }

NLP

+c(2)

m,n λτlnn(λτ)

| {z }

N2LP

+. . . 



,(2.6)

indicating the leading, next-to-leading and next-to-next-to-leading power terms in λτ, labelled LP, NLP, and

N2LP, respectively. Therein, σt¯

Bis the Born level total cross section of the process pp →t¯

t+X,αsdenotes

the strong coupling constant, and c(k)

m,n represents the coeﬃcient for the asymptotic constituent with the

superscript kspecifying the occurring power. Thus, conventionally, the leading power terms c(0)

m,n lnn(λτ)/λτ

are associated with the most singular behaviors in the low qτdomain and also the main concern of this work.

It is important to note, however, that also the next-to-leading power terms, c(1)

m,n lnn(λτ), are divergent as

λτ→0.

2.2 Dynamic regions

Based on the strategy of expansion of dynamic regions [78–81], the asymptotic series of Eq. (2.6) can

be interpreted with the aid of a set of regions from the phase space and loop integrals. This work, in

particular, will choose the formalism of [81]. We base the deﬁnition of our regions on the criterion of domain

completeness, i.e. the existence of a set of non-intersecting dynamic regions that cover the whole integration

domain. This criterion plays an essential role in consistently extrapolating the expanded integrands from

their own convergent domains to the entire integration ranges. Other constraints are also imposed therein,

including the regularisation of the expanded integrands and the (at least partial) commutativity amongst

the asymptotic expansions. The former case can be fulﬁlled by introducing the rapidity regulator [96–99,

107, 109, 111] in implementing the SCETII formalism [89–91]. However, for the latter criterion, we assume

that all the non-commutative dynamic regions, such as the collinear-plane modes [81], will cancel out in

the eventual qτspectra. It merits noting that, this ansatz, together with the proposal of [81], has been

scrutinised only within the one-loop integrals in the various kinematical limits. We regard their eﬀectiveness

on t¯

thadroproduction as the primary hypothesis in this work. Recent developments on the criteria to

implement the region analysis can be found in [112–114].

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IPPP/22/73MCnet-22-20Projectedtransversemomentumresummationintop-antitoppairproductionatLHCWan-LiJu(a;b),MarekSchonherr(a)(a)InstituteforParticlePhysicsPhenomenology,DurhamUniversity,DurhamDH13LE,UnitedKingdom(b)INFN,SezionediMilano,ViaCeloria16,20133Milano,ItalyAbstract:Thetransversemomentumdistri...

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Projected transverse momentum resummation in top-antitop pair production at LHC

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