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JHEP02(2017)026

Published for SISSA by Springer

Received: August 28, 2016

Revised: January 9, 2017

Accepted: January 23, 2017

Published: February 6, 2017

Two-loop beam and soft functions for

rapidity-dependent jet vetoes

Shireen Gangal,

Jonathan R. Gaunt,

Maximilian Stahlhofen

and

Frank J. Tackmann

Theory Group, Deutsches Elektronen-Synchrotron (DESY),

Notkestraße 85, D-22607 Hamburg, Germany

Nikhef Theory Group and VU University Amsterdam,

De Boelelaan 1081, NL-1081 HV Amsterdam, the Netherlands

PRISMA Cluster of Excellence, Institute of Physics, Johannes Gutenberg University,

Staudingerweg 7, D-55128 Mainz, Germany

E-mail: shireen.gangal@desy.de, jgaunt@nikhef.nl,

mastahlh@uni-mainz.de, frank.tackmann@desy.de

Abstract: Jet vetoes play an important role in many analyses at the LHC. Traditionally,

jet vetoes have been imposed using a restriction on the transverse momentum p

T j

of jets.

Alternatively, one can also consider jet observables for which p

T j

is weighted by a smooth

function of the jet rapidity y

that vanishes as |y

| → ∞. Such observables are useful

as they provide a natural way to impose a tight veto on central jets but a looser one

at forward rapidities. We consider two such rapidity-dependent jet veto observables, T

and T

, and compute the required beam and dijet soft functions for the jet-vetoed color-

singlet production cross section at two loops. At this order, clustering eﬀects from the jet

algorithm become important. The dominant contributions are computed fully analytically

while corrections that are subleading in the limit of small jet radii are expressed in terms

of ﬁnite numerical integrals. Our results enable the full NNLL

resummation and are an

important step towards N

LL resummation for cross sections with a T

or T

jet veto.

Keywords: Jets, NLO Computations

ArXiv ePrint: 1608.01999

Open Access,

 The Authors.

Article funded by SCOAP

doi:10.1007/JHEP02(2017)026

JHEP02(2017)026

The default jet variable by which jets are currently classiﬁed and vetoed is the trans-

verse momentum p

T j

of a jet. However, there are some drawbacks to using this variable. In

harsh pile-up conditions, it can be diﬃcult to identify (and veto) low-p

jets in the forward

region (beyond |η| & 2.5) when a large part or all of the jet lies in a detector region where

no tracking information is available. One way to get around this problem is by introducing

a hard cut on the jet (pseudo)rapidity and only consider jets at |η

| < η

cut

. However, such

a cut also changes the logarithmic structure [1], and none of the extant resummations of

T j

take account of such a rapidity cut. Another way one might try to avoid the problem

is to raise the cut on p

T j

, but then one loses the potential beneﬁts of a tight jet veto (such

as its utility to identify the initial state of heavy resonances [2]).

An alternative approach is to consider a generalized jet-veto variable [3]

= p

T j

f(y

) (1.1)

that is smoothly dependent on the jet rapidity y

, where the function f(y) is decreasing

for increasing |y| such that limiting T

to small values tightly constrains central jets, but

only loosely constrains the forward ones. In addition to the above practical considerations,

given the importance of jet binning it is highly desirable to have several diﬀerent options

for performing jet vetoes, both experimentally and theoretically. This avoids having to rely

exclusively on p

T j

, and provides important complementary information on the pattern of

additional jets produced, for example in Higgs production [4].

In this paper, we consider the two representative jet observables

= |~p

T j

−|y

= min{p

, p

−

}, T

|~p

T j

2 cosh y

−

+ p

−

. (1.2)

We have introduced light-cone coordinates, where an arbitrary four-vector q

is decomposed

as q

= q

−

/2+q

¯n

/2+q

⊥

with n

and ¯n

being light-like vectors (n

= ¯n

= 0, n·¯n = 2)

along the beam directions. T

gives the plus (minus) momentum of the jet j if the jet

lies in the right (left) hemisphere with p

−(+)

> p

+(−)

. That is, it has the same rapidity

weighting as the global beam thrust hadronic event shape [5]. T

has the same rapidity

weighting as the C-parameter event-shape for e

−

→ dijet processes. It becomes equal

to T

at forward rapidities and approaches p

T j

/2 at central rapidities. The T

spectrum

has been measured in Higgs production by ATLAS [4].

In refs. [1, 3], the factorization of the color-singlet production cross sections with a T

or T

veto, involving jet-dependent beam and soft functions, was formulated within soft

collinear eﬀective theory (SCET) [6–11], and the resummation of the two observables was

performed to the NLL

+NLO order [3].

In ref. [3], these deﬁnitions were denoted with an additional subscript “cm” to distinguish them from

the corresponding boost-invariant versions, for which y

is replaced by y

− Y , where Y is some reference

rapidity (e.g. that of the color singlet in color-singlet production). For our purpose of calculating the relevant

soft and beam functions this distinction is irrelevant as the factorization theorems for the corresponding

jet-vetoed cross sections only diﬀer by the arguments of the beam functions. We will therefore drop the cm

subscript for simplicity of notation in this paper.

– 2 –

JHEP02(2017)026

The resummation for the corresponding global beam thrust in color-singlet production

is currently known to NNLL

[5, 12] (with the results of refs. [13, 14]). The resummation

for a jet-algorithm dependent veto using p

T j

is known up to NNLL

[1, 15–19] and has been

applied to a number of color-singlet processes [2, 20–27].

The purpose of the present paper is to provide the full two-loop corrections to the

-dependent and T

-dependent beam and soft functions that are required to bring the

resummation for T

and T

to the NNLL

level. These corrections also provide the

necessary ﬁxed-order boundary conditions for the N

LL resummation, with the remaining

missing ingredients being the three-loop clustering correction to the noncusp and the four-

loop correction to the cusp anomalous dimensions.

We compute, partly analytically and partly numerically, the beam and dijet soft func-

tions for both rapidity-dependent jet veto observables in eq. (1.2), for all possible color and

parton channels. To be precise, for the beam functions we compute the two-loop perturba-

tive matching coeﬃcients between the beam functions and the standard parton distribution

functions (PDFs). This is the ﬁrst explicit calculation of the full set of two-loop singular

matching corrections for a jet-algorithm dependent jet veto. This includes the complete

set of corrections arising from the clustering of two independent emissions. (In ref. [19],

the full two-loop soft function for p

T j

was calculated, while the two-loop beam functions

required for the NNLL

resummation was extracted numerically from ﬁxed-order codes.)

Our results for the two-loop beam functions can also be used in computations of N-jet

cross sections with a veto on further jets being imposed via T

or T

This paper is structured as follows: in section 2, we deﬁne precisely the soft and beam

functions for T

and T

vetoes, and give an outline of their calculation at two loops.

Additional technical details are relegated to the appendices. In section 3, we give the

obtained results for the two-loop beam and soft functions, and we conclude in section 4.

2 Calculation

The measurement function for a jet veto using a generic variable T

is given by

jet

cut

) =

j∈J(R)

θ(T

< T

cut

) . (2.1)

This constraints all jets to have T

< T

cut

and thus vetoes any jets with T

> T

cut

. The

jets J(R) in eq. (2.2) are identiﬁed with a speciﬁc jet algorithm with jet radius R. At

NNLO, we can have at most two emissions, and the precise form of the jet algorithm is

irrelevant. Our results are valid for any jet algorithm that clusters the emissions together

if they are closer in ∆R

= ∆φ

+∆y

than R

, where ∆y is the separation in rapidity, ∆φ

is the separation in azimuthal angle and R is the jet radius. In experimental analyses at

the LHC typically R = 0.4 or 0.5. For two real emissions in the ﬁnal state with momenta

and k

, eq. (2.1) reduces to

jet

cut

) = θ(∆R < R) θ(T

< T

cut

) + θ(∆R > R) θ(T

< T

cut

) θ(T

< T

cut

) . (2.2)

– 3 –

JHEP02(2017)026

That is, for ∆R < R the two emissions are clustered into a jet and T

is computed from

the sum p

= k

+ k

. For ∆R > R, each emissions forms its own jet and T

and T

are

computed with p

= k

and p

= k

, respectively.

The measurement function M

jet

cut

) is inserted into the usual SCET operator matrix

elements deﬁning the beam and soft functions. In this case, the jets J(R) are obtained

purely from the collinear or soft radiation within each sector. For details we refer to

refs. [1, 3]. The practical implementation of such a measurement in beam and soft function

calculations is discussed below.

The factorization of the cross section with a T

or T

veto in refs. [1, 3] is strictly

speaking valid only to lowest order in an expansion in R. The possibility of clustering inde-

pendent soft and collinear emissions into the same jet breaks the soft-collinear factorization

of the measurement function with the corresponding corrections starting at O(R

) [1]. Since

this only aﬀects the measurement itself but not the soft-collinear factorization of the ampli-

tudes and SCET Lagrangian, these soft-collinear clustering corrections can be computed in

the eﬀective theory and are included in our results. We separate out the corresponding con-

tributions in the two-loop beam and soft functions that are associated with the clustering

of independent emissions. They are denoted with the subscript ‘indep’ and together with

the corrections from soft-collinear clustering reproduce the two-loop clustering behaviour

of independent emissions in full QCD (in the singular limit). In section 3, we give two

prescriptions as to how this collection of terms can be treated in the NNLL

resummation.

In addition, at O(R

) (potentially) factorization breaking eﬀects due to Glauber inter-

actions can play a role [28]. At the perturbative level, they ﬁrst appear in a nonlogarithmic

O(α

) diagram, implying that the factorization breaking eﬀects ﬁrst appear at the N

order [29, 30]. They will not be discussed further here.

Our calculation of the beam and soft functions is organized in an expansion in R as well.

We will give terms in this expansion up to orders high enough for all practical purposes.

This expansion only involves even powers of R (up to few exceptional terms at lower orders).

We ﬁnd the R

expansion to converge very quickly, suggesting that the relevant expansion

parameter is (R/R

)

with R

' 2. (Similar observations have been made recently also in

other contexts involving small-R expansions, see e.g. [31, 32].) As pointed out in ref. [1], in

the small-R limit one should also consider resumming the corresponding logarithms ln R

appearing in the jet-vetoed cross section. The dominant contribution beyond O(α

) was

obtained in ref. [33]. Their resummation at the LL was obtained in refs. [26, 34], and very

recently methods have been developed [35, 36] that could allow one to systematically carry

out their resummation to higher orders.

To perform the computation of the jet-algorithm dependent soft and beam functions we

follow the same strategy used in refs. [1, 15, 19] by computing the diﬀerence to a reference

soft or beam function deﬁned with a global (jet-algorithm independent) measurement, with

the reference functions having been computed elsewhere in the literature. A key property of

the global reference measurements we use is that they coincide with the jet-dependent mea-

surements for the case of one real emission. Then when we compute the diﬀerences, we only

need to consider the double-real emission amplitudes. Since the measurement functions are

diﬀerent for T

and T

, we must perform a separate computation of the soft function for

– 4 –

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