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JHEP12(2017)043

Published for SISSA by Springer

Received: September 18, 2017

Revised: November 30, 2017

Accepted: December 2, 2017

Published: December 11, 2017

Transverse vetoes with rapidity cutoﬀ in SCET

Andrew Hornig,

Daekyoung Kang,

a,b

Yiannis Makris

a,c

and Thomas Mehen

Theoretical Division T-2, Los Alamos National Laboratory,

Los Alamos, NM, 87545, U.S.A.

Key Laboratory of Nuclear Physics and Ion-beam Application (MOE)

and Institute of Modern Physics, Fudan University,

Shanghai, 200433, China

Department of Physics, Duke University,

Durham, NC 27708, U.K.

E-mail: andrew.hornig@gmail.com, dkang@fudan.edu.cn, yiannis@lanl.gov,

mehen@phy.duke.edu

Abstract: We consider di-jet production in hadron collisions where a transverse veto is

imposed on radiation for (pseudo-)rapidities in the central region only, where this central

region is deﬁned with rapidity cutoﬀ. For the case where the transverse measurement

(e.g., transverse energy or min p

for jet veto) is parametrically larger relative to the

typical transverse momentum beyond the cutoﬀ, the cross section is insensitive to the cutoﬀ

parameter and is factorized in terms of collinear and soft degrees of freedom. The virtuality

for these degrees of freedom is set by the transverse measurement, as in typical transverse-

momentum dependent observables such as Drell-Yan, Higgs production, and the event

shape broadening. This paper focuses on the other region, where the typical transverse

momentum below and beyond the cutoﬀ is of similar size. In this region the rapidity

cutoﬀ further resolves soft radiation into (u)soft and soft-collinear radiation with diﬀerent

rapidities but identical virtuality. This gives rise to rapidity logarithms of the rapidity

cutoﬀ parameter which we resum using renormalization group methods. We factorize the

cross section in this region in terms of soft and collinear functions in the framework of soft-

collinear eﬀective theory, then further refactorize the soft function as a convolution of the

(u)soft and soft-collinear functions. All these functions are calculated at one-loop order.

As an example, we calculate a diﬀerential cross section for a speciﬁc partonic channel,

→ qq

, for the jet shape angularities and show that the refactorization allows us to

resum the rapidity logarithms and signiﬁcantly reduce theoretical uncertainties in the jet

shape spectrum.

Keywords: Jets

ArXiv ePrint: 1708.08467

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP12(2017)043

JHEP12(2017)043

In experimental studies of exclusive N-jet production, it is common to impose a veto

on the out-of-jet radiation in order to control soft emissions. Additionally, due to detec-

tor limitations these vetoes are imposed within a speciﬁc (pseudo-)rapidity region and the

veto is not imposed outside this region. These constraints can induce large logarithms of

the cutoﬀ parameter, e

−η

cut

, and ratios of the veto parameter, p

cut

, to other scales in the

problem (e.g., the hard scale µ

∼

√

−t, where t is the usual Mandelstam variable). These

logarithmic enhancements could potentially ruin the eﬀectiveness of the ordinary perturba-

tive expansion. In this work we propose a factorization theorem for resumming logarithms

of e

−η

cut

and p

cut

/µ within the framework of soft-collinear eﬀective theory (SCET) [13–16],

where µ is the factorization scale. SCET was extensively used in the past decade for

factorization of observables with sensitivity to soft and collinear radiation, such as jet sub-

structure measurements in hadronic colliders. Some other interesting applications of SCET

include cross sections for event shapes in the collinear limit, jet production rates [17, 18],

and identiﬁed hadrons within jets [19–26].

In this paper, we study rapidity cutoﬀ resummation and develop the necessary ingre-

dients for di-jet cross sections with transverse energy, E

, and jet-veto measurements in

hadronic collisions. Our results can easily be extended to zero and one-jet cross sections

as well. The transverse energy is deﬁned as the sum of the scalar transverse momentum of

all the particles that do not belong to a jet and have rapidity, η, in the range |η| < η

cut

i/∈jet

|Θ(η

cut

− |η

|), (1.1)

where the rapidity is measured with respect to the beam axis. The veto is implemented

by imposing the constraint E

< p

cut

. Due to the nature of the observable one expects

that such a measurement is sensitive to the underlying event (UE). Measurements of the

UE activity have been performed by ATLAS and CMS in inclusive charged particle pro-

duction [27], Drell-Yan [28–30], and exclusive dijet events [31].

The eﬀect of UE in transverse energy resummed distributions was studied in

refs. [32, 33] for η

cut

= 4.5 in the case of Higgs and vector-boson production using monte-

carlo simulation. In contrast to the work in this paper, in refs. [32, 33] the rapidity cutoﬀ

was introduced only during the simulation and not in the resummed distribution. However,

as will be discussed below, for the large values of rapidity cutoﬀ (η

cut

& 4.5) the eﬀect of

the cutoﬀ on the resummed distribution is expected to be small. In this work we ignore

eﬀects of multiparton interactions and focus on contributions from initial and ﬁnal state

radiation. In principle, eﬀects from multiparton interactions could be included later on

top of our analysis as factorization breaking corrections but this is beyond the scope of

this work.

Similarly, the jet-veto measurement imposes |p

veto

)| < p

cut

, where p

is the trans-

verse momenta of the i-th jet reconstructed by a jet algorithm and R

veto

is the jet cone size

parameter used during the vetoing process which could be diﬀerent from the hard jet size,

R. Though the jet-veto measurements are less sensitive to UE, they suﬀer from logarithmic

enhancements of R

veto

. Such logarithms are known as “clustering logarithms” and appear

in next-to-next-leading order (NNLO) calculations [34–36], and could make an important

– 2 –

JHEP12(2017)043

contribution to the cross section. At present there is no known method for resummation

of these logarithms but they could be included order-by-order in perturbation theory.

As a preliminary exercise we study di-jet production under the rapidity constraints in

an electron-positron annihilation process. Speciﬁcally we study the eﬀects of the rapidity

constraints in the small transverse energy regime Λ

QCD

 E

⊥

. ωr  ω, where ω =

√

is the center of mass energy and r = e

−η

cut

is the rapidity cut. For this simple example,

the transverse energy E

⊥

as well as the rapidity is measured with respect to the thrust

axis and therefore we use diﬀerent notation (E

⊥

instead of E

) to avoid confusion. The

schematic form of the factorization of the cross section within SCET is,

dσ

⊥

∼ H × S ⊗ J

⊗ J

¯q

, S(E

⊥

) = S

⊗ S

¯n

, (1.2)

where ⊗ denotes convolution over E

⊥

. The hard function, H describes the hard process:

−

→ q¯q, and the soft function, S, describes the soft radiation and cross talk between

collinear sectors. The collinear radiation along the thrust axis is described by the functions

which can be written in terms of the “unmeasured” jet function,

, introduced in

refs. [37], and contributions from out-of-jet radiation, which we denote as ∆J

. For small

values of the transverse momentum, E

⊥

 ωr, the collinear radiation which is emitted

within the cone has parametrically large transverse momenta, compared to E

⊥

, and does

not contribute to the measurement. In this case the function J

reduces to the standard

unmeasured jet function. The corrections from the out-of-jet radiations are necessary to

describe the process for moderate values of E

⊥

. ωr. The collinear-soft function, S

describes the collinear-soft modes which are collinear in the n-direction and therefore can

resolve the jet-cone boundary. The global-soft function, S

, describes the standard u-soft

modes of SCET

which cannot resolve the small jet radius and therefore for the calculation

of S

no rapidity constraints are imposed. In this e

−

example there is no UE and therefore

the factorization is accurate up to higher orders in the eﬀective ﬁeld theory power counting

parameter λ ∼ E

⊥

/ω. This allows us to directly compare our results with simulation data.

For our analysis we use MadGraph [38] + Pythia 8 [39, 40]. Our calculations are in very

good agreement with Monte Carlo for most values of E

⊥

The refactorization of the soft function into global-soft and collinear-soft terms intro-

duces rapidity divergences which we regulate using the rapidity regulator of refs. [41, 42].

The rapidity scale dependence allow us to derive rapidity renormalization group (RRG)

equations which we solve to resum global logarithms of r up to next-to-leading logarithmic

(NLL) accuracy. This process closely follows the analysis in refs. [43, 44] where resumma-

tion of jet size parameter is performed in the context of electron-positron annihilation and

in refs. [45, 46] for proton-proton collisions. Non-global logarithms (NGLs) [47–54] of r

appear at NNLO calculations. Their resummation is particularly challenging since they do

not have the same pattern at each order in perturbative expansion. NGLs can be included

order by order in α

when their contribution is not large. Otherwise, resummation strate-

We use the terminology of ref. [37] and we refer to jets for which no substructure observable is measured

as unmeasured jets.

– 3 –

JHEP12(2017)043

gies developed in perturbative QCD [47, 48, 55–60] or recent approaches in the framework

of SCET [43, 61–64] should be adopted.

Other logarithms of ratios of widely separated scales also appear in the factorized cross

section (e.g. ratios of µ

= ω, µ

= ωr, and µ

= E

⊥

) are resummed by using the standard

RG evolution within the eﬀective theory. We summarize the RG evolution properties for

all relevant terms in the sections 2.2, 3.2, and appendix A.3.

In hadronic collisions the beam direction plays the role that thrust axis plays in

electron-positron collisions and the collinear radiation along the n

-direction is described

by the beam functions [65]. The corrections to the beam function from out-of-beam radi-

ation will contribute to the transverse energy (or jet-veto) measurement in a similar way

as corrections to the jet function in electron-positron annihilation. For jet production,

in addition to the n

-collinear-soft and n

-collinear modes, we also have corresponding

modes along each jet direction. The contribution from n

-collinear-soft modes is considered

through further refactorization of the soft function including the n

-collinear-soft function,

n,J

. The factorization theorem for N -jet production in hadronic collisions is,

dσ

∼ Tr

ab→1,2...,N

unmeas

⊗ B

a/P

⊗ B

b/P



i=1



, (1.3)

where

ab→1,2...,N

unmeas

= S

ab→1,2...,N

⊗ S

(a)

n,B

⊗ S

(b)

n,B

⊗ S

(1)

n,J

··· ⊗ S

(N)

n,J

, (1.4)

and S

is the global N-jet soft function, S

(a)

n,B

, and S

(i)

n,J

are the collinear soft functions along

the beam and jet directions respectively. The superscripts (a) and (i) denote the partons

associated with these functions and it should be noted that S

(a)

n,B

and S

(i)

n,J

are diﬀerent

functions even for the same parton a = i because the veto for both functions is always

applied respect to the beam direction, not individual jet and beam directions. The parton

dependence of the soft functions will be suppressed for the rest of the text for simplicity

of notation. The contributions from the n

-collinear modes to the transverse energy are

suppressed and therefore the jet functions J

do not participate in the convolutions over

, while the beam functions B

i/P

do through E

/ωr terms in the power corrections. This

is discussed in appendix C. It should be noted that in this paper we focus in the region

∼ ωr where these corrections are important and NGLs of the form ln(E

/ωr) ∼ 1 and

therefore resummation is not needed.

The factorization formula in eq. (1.3) assumes the jets are in the central rapidity re-

gion, hence the factorization is invalid for jets in the large rapidity region, for which pure

t-channel forward scattering dominates. This process was extensively studied in the frame-

work of High Energy Jets (HEJ) [66–68] developed to resum logarithms of the rapidity

diﬀerence between jets and in the context of factorization violation by Glauber-gluon ex-

change [69]. The formula in eq. (1.3) can be used to calculate exclusive N-jet production

in the absence of UE and to understand its eﬀect by comparing to ATLAS and CMS

measurements [28–31].

– 4 –

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