Factorization at subleading power in deep inelastic scattering in the x1limit

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Factorization at subleading power in deep inelastic scattering in the x→1limit

Michael Luke,∗Jyotirmoy Roy,†and Aris Spourdalakis‡

Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada

We examine the endpoint region of inclusive deep inelastic scattering at next-to-leading power

(NLP). Using a soft-collinear eﬀective theory approach with no explicit soft or collinear modes,

we discuss the factorization of the cross section at NLP and show that the overlap subtraction

procedure introduced to eliminate double counting of degrees of freedom at leading power ensures

that spurious endpoint divergences in the rate cancel at NLP at one loop. For this cancellation to

occur at all renormalization scales a nontrivial relation between the anomalous dimensions of the

leading and subleading operators is required, which is demonstrated to hold at one loop.

I. INTRODUCTION

Soft-collinear eﬀective theory (SCET) [1–7] is a well-

established tool for studying hard scattering processes

in QCD as an expansion in inverse powers of the hard

scattering scale Q. While there has been much work on

applications of SCET at leading power (LP), power cor-

rections suppressed by inverse powers of Q2have proven

more involved, in part due to the appearance of addi-

tional divergences which spoil naive factorization. Recent

work studying power corrections to various processes in

SCET include beam thrust [8], Drell-Yan production near

threshold [9, 10] and at small qT[11], threshold Higgs

production from gluon fusion [12], Higgs production and

decay [13], the energy-energy correlator in N= 4 Su-

persymmetric Yang-Mills [14], Higgs to diphoton decay

[15–17] oﬀ-diagonal deep inelastic scattering [18], gluon

thrust [19] and muon-electron backward scattering [20].

Power corrections have also been studied using non-EFT

QCD techniques [21–30].

SCET has complications not found in more familiar

eﬀective ﬁeld theories (EFT’s) such as four-Fermi the-

ory or heavy quark eﬀective theory because it simulta-

neously describes particles with parametrically diﬀerent

momentum scaling. In its most familiar formulations

the relevant modes contributing to a given process (soft,

collinear, ultrasoft, hard-collinear as well as others, de-

pending on the scales of interest) are described by sep-

arate ﬁelds. In a more recent formalism [31] it was ar-

gued that SCET is more simply written as a theory of

separate sectors, deﬁned such that the invariant mass of

pairs of particles in diﬀerent sectors is of order Q2, but

the invariant mass of pairs of particles in the same sector

is parametrically smaller than Q2. Particles in diﬀer-

ent sectors are described by diﬀerent ﬁelds, but modes

of a given particle in a single sector are described by

the same ﬁeld, as in QCD. In every formulation, how-

ever, spurious divergences arise in individual graphs in

SCET because loop- and phase-space integrals integrate

over all momenta, including momenta which violate the

∗luke@physics.utoronto.ca

†jro1@physics.utoronto.ca

‡aspourda@physics.utoronto.ca

power counting of the corresponding ﬁeld, giving unphys-

ical contributions. As was demonstrated many years ago

[32], matrix elements in SCET are only well-deﬁned if an

appropriate subtraction procedure has been implemented

to remove this double counting between diﬀerent modes

or sectors. Since the EFT by construction must repro-

duce the physics of QCD these divergences must cancel

in physical observables once the appropriate subtractions

have been made; however, this is not always simple to

demonstrate.

Endpoint divergences in particular are unphysical di-

vergences arising from convolutions of Wilson coeﬃcients

and operators in SCET, and lead to an apparent viola-

tion of factorization. The appearance of these endpoint

divergences is a common feature at NLP and has been

recently studied in context of various processes [15–19].

While individual terms in the factorization formula are

divergent, it was demonstrated in these works that the

factorized physical quantities remain ﬁnite since the di-

vergences cancel between diﬀerent terms in the endpoint

region. This property was exploited to rearrange and

rewrite the individual terms in a “refactorized” form.

In this work, we examine next-to-leading power (NLP)

corrections to deep inelastic scattering (DIS) [33, 34] in

the endpoint limit using the formalism introduced in

[31]. DIS in this limit was one of the ﬁrst and sim-

plest processes studied in SCET [35–37], and provides

a simple example of a process with endpoint divergences

at NLP. The cross section is a function of the invari-

ant mass −q2≡Q2Λ2

QCD of the oﬀ shell photon

and the dimensionless variable x≡−q2

2P·q, where Pµis

the four-momentum of the incoming proton and qµis

the four-momentum of the photon. The cross section is

well known to factorize into a hard scattering amplitude,

depending on Q2, and nonperturbative parton distribu-

tion functions (PDFs), which depend on the details of

low-energy QCD. Large logarithms of Q2/µ2in the cross

section, where µ∼ΛQCD, may be resummed by evolving

the PDFs using the DGLAP equations. In the endpoint

region where x→1, additional large logarithmic correc-

tions appear in the perturbative expansion of the hard

scattering amplitude which must be resummed to obtain

a reliable calculation of the cross section. These arise be-

cause in this limit the invariant mass of the ﬁnal state,

F∼Q2(1 −x), is parametrically smaller than the hard

arXiv:2210.02529v2 [hep-ph] 20 Apr 2023

scattering scale Q2.

The power corrections studied here correspond to

terms suppressed by a single power of 1 −xin the cross

section relative to the leading terms. In [11], it was

shown that the overlap subtraction procedure introduced

to correctly reproduce the infrared behavior of QCD in

Drell-Yan (DY) scattering led automatically to the can-

cellation of rapidity divergences in the EFT, but that at

NLP the cancellations typically involved linear combina-

tions of multiple operators, including power suppressed

subtractions of the leading-order operator. In the DY

process the corresponding divergences were rapidity di-

vergences, which require an additional regulator [14, 38–

41]. Here we show that the same overlap subtraction also

ensures that the DIS rate obtained from SCET is free of

endpoint divergences, but in this case the divergences

are regulated in dimensional regularization. Again, the

required cancellation occurs between separate operators,

giving a nontrivial relation between the anomalous di-

mension of the leading and subleading operators which

is demonstrated to hold at one loop. We defer an all-

orders proof to a future work.

In Sec. II, we review the scattering operators in SCET

for DIS topology up to O1/Q2. We present the one-

loop matrix elements of the product of operators required

for inclusive DIS rate at NLP in Sec. III and show how

the overlap procedure removes the endpoint singularities.

In Sec. IV we show that this cancellation of endpoint di-

vergence holds at all scales and thus puts constraints on

the anomalous dimension of subleading operators. Fi-

nally we present our conclusions in Sec. V.

II. CURRENT TO O(1/Q2)

The incoming state in DIS consists of low invariant

mass partons, p2

I∼Λ2

QCD, while the outgoing state con-

sists of partons with invariant mass p2

F∼Q2(1−x), with

pI·pF∼Q2. Thus, near x= 1 they are described by

diﬀerent sectors in SCET which we denote by the light-

like vectors nµand ¯nµ, where n2= ¯n2= 0 and n·¯n= 2.

Partons in the incoming state are described by the ¯n

sector, and particles in the outgoing state by the nsec-

tor. Particles in the nsector have momenta pn·nQ,

while pn·¯ncan be of order Q(and vice versa for the

¯nsector). For simplicity, we work in a reference frame

where qµhas no components perpendicular to nand ¯n:

qµ= ¯n·qnµ

2+n·q¯nµ

For simplicity we consider a single ﬂavor of quark ψ.

DIS is then mediated by the quark electromagnetic cur-

rent

Jµ=¯

ψγµψ. (1)

At the hard scale µ∼Q, degrees of freedom with in-

variant mass of order Qare integrated out of the theory,

and QCD matrix elements are expanded in inverse pow-

ers of Qand matched onto SCET. In the formalism used

here [31], each low-invariant mass sector of the theory is

described by a diﬀerent copy of QCD, with interactions

between sectors occurring via Wilson lines in the external

current.

Using the operator basis deﬁned in [11, 31, 42], the

current in the EFT is

Jµ(x) = X

Q[i]C(i)

2(µ)O(i)µ

2(x, µ)(2)

where the C(i)

2’s are Wilson coeﬃcients and the O(i)

2’s

are operators in the EFT. Operators in SCET are conve-

niently expressed in terms of the gauge invariant building

blocks [43]

¯χn(x) = ¯

ψn(x)Wn(x)P¯n

χ¯n(x) = P¯nW†

¯n(x)ψ¯n(x)

Bµ1···µN

n(x) = W†

n(x)iDµ1

n(x)···iDµN

n(x)Wn(x)

B†µ1···µN

¯n(x)=(−1)NW†

¯n(x)i←−

Dµ1

¯n(x)···i←−

DµN

¯n(x)W¯n(x)

(3)

where iDµ

n(x) = i∂µ+gAµ

n(x) and i←−

Dµ

¯n(x) = i←−

∂µ−

gAµ

¯n(x) are the usual covariant derivatives in each sector,

and Pn=/

¯n/4, P¯n=/

¯n/

n/4 are projection operators

acting on the four-component spinors ψ. The outgoing

Wilson lines in the ¯nsector and the incoming Wilson

lines in the nsector are deﬁned respectively as

W†

¯n(x) = Pexp igsZ∞

ds n ·Aa

¯n(x+ns)Tae−s0+

Wn(x) = Pexp igsZ0

−∞

ds ¯n·Aa

n(x+ ¯ns)Taes0+(4)

Note that the subscript on a Wilson line corresponds to

the sector with which it interacts rather than its direc-

tion.

The operators in the eﬀective theory for DIS topol-

ogy have already been derived in [31] up to O(1/Q) us-

ing spinor-helicity techniques, and related operators rel-

evant for Drell-Yan scattering and dijet production up to

O(1/Q2) in [11, 42]. Since we will not be using the he-

licity basis, it is instructive to write down the expanded

amplitude and the corresponding SCET operators up to

O(1/Q2) in terms of the usual Dirac matrices.

FIG. 1: One-gluon matrix element of Jµin QCD.

The DIS amplitude to produce an additional gluon in

the ﬁnal state in Fig. 1 is reproduced in SCET by diﬀer-

ent operators depending on whether the outgoing quark

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Factorizationatsubleadingpowerindeepinelasticscatteringinthex!1limitMichaelLuke,JyotirmoyRoy,yandArisSpourdalakiszDepartmentofPhysics,UniversityofToronto,Toronto,OntarioM5S1A7,CanadaWeexaminetheendpointregionofinclusivedeepinelasticscatteringatnext-to-leadingpower(NLP).Usingasoft-collineareectivet...

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Factorization at subleading power in deep inelastic scattering in the x1limit

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