Single Inclusive Hadron Production in DIS at Small x Next to Leading Order Corrections Filip Bergabo12and Jamal Jalilian-Marian12y

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Single Inclusive Hadron Production in DIS at Small x: Next to Leading Order

Corrections

Filip Bergabo1, 2, ∗and Jamal Jalilian-Marian1, 2, †

1Department of Natural Sciences, Baruch College, CUNY,

17 Lexington Avenue, New York, NY 10010, USA

2City University of New York Graduate Center, 365 Fifth Avenue, New York, NY 10016, USA

We calculate the one-loop corrections to single inclusive hadron production in Deep Inelastic

Scattering (DIS) at small xin the forward rapidity region using the Color Glass Condensate for-

malism. We show that the divergent parts of the next to leading order (NLO) corrections either

cancel among each other or lead to x(rapidity) evolution of the leading order (LO) dipole cross

section according to the JIMWLK evolution equation and DGLAP evolution of the parton-hadron

fragmentation function. The remaining ﬁnite parts constitute the NLO (αs) corrections to the LO

single inclusive hadron production cross section in DIS at small x.

I. INTRODUCTION

Gluon saturation [1, 2] at small xas encoded in the Color Glass Condensate (CGC) formalism [3–7] has been the

subject of intense theoretical studies and experimental searches. Theoretical work based on leading order (LO) or

leading log (LL) approximations to gluon saturation have successfully described structure functions, suppression of

the single inclusive hadron transverse momentum spectrum and disappearance of the away side peak in dihadron

angular correlations in high energy proton(deuteron)-gold/lead collisions at RHIC and the LHC [8–54]. Nevertheless

ﬁrmly establishing gluon saturation as the QCD dynamics responsible for these experimental observations requires

more precise theoretical calculations. The ongoing work on improving the accuracy of leading order CGC calculations

can be broadly put into three categories; higher order in αscorrections to leading order results [55–80], sub-eikonal

corrections which aim to relax the inﬁnite energy assumption inherent to eikonal approximation [81–92], and inclusion

of intermediate/large xdynamics into CGC in order to generalize CGC to include DGLAP evolution and collinear

factorization and high ptphysics [93–101]. Here we will focus on next to leading order corrections to single inclusive

hadron production in Deep Inelastic Scattering (DIS) at small xin the forward rapidity region [102] (virtual photon

going direction) for the case when the virtual photon is longitudinal. We note that leading order results for single

inclusive hadron production in DIS in the midrapidity region were obtained in [103].

The ideal environment in which to investigate gluon saturation and CGC is DIS experiments at high energy as the

incoming virtual photon does not interact strongly. Single inclusive hadron production in DIS (SIDIS) at small xis

one of the most attractive channels for gluon saturation studies as it is not sensitive to Sudakov eﬀects which can

obscure saturation dynamics in dihadron production and angular correlations. Furthermore, it is more discriminatory

than the total cross section (structure functions) so that it contains more information about the QCD dynamics of

the target. While there exists leading order calculations of single inclusive hadron production in DIS at small xin

the CGC framework [102, 103] it is highly desirable and in fact urgently needed to perform a next to leading order

calculation which can then be used for quantitative studies of the transverse momentum spectra of produced hadrons

in DIS with proton and nuclear targets at the proposed Electron Ion Collider (EIC).

Here we calculate the next to leading order corrections to single inclusive hadron production in DIS at small xin

the forward rapidity region using the Color Glass Condensate formalism. To do so we use our recent results for next

to leading order corrections to dihadron production [80] in DIS and integrate out one of the ﬁnal state partons. As

expected we encounter various divergences which appear when we integrate over the phase space of the ﬁnal state

parton. We show that UV and soft divergences cancel among each other while the collinear divergences associated

with radiation of a massless parton are absorbed into the parton-hadron fragmentation function. We show that all

quadrupole terms appearing in the intermediate steps of the calculation cancel among various terms and one is left

with dipoles (and squared dipoles) only. The rapidity divergences arising from integrating over longitudinal phase

space of the ﬁnal state parton are absorbed into evolution of the dipoles describing the target dynamics and lead

to JIMWLK evolution of the leading order cross section. The remaining terms are ﬁnite and constitute the O(αs)

corrections to leading order single inclusive hadron production in DIS at small x.

∗fbergabo@gradcenter.cuny.edu

†jamal.jalilian-marian@baruch.cuny.edu

arXiv:2210.03208v3 [hep-ph] 20 Dec 2022

II. LEADING ORDER CROSS SECTION

To get the leading order single inclusive hadron production in DIS at small xwe start with the quark antiquark

production cross section in DIS given by

dσγ∗A→q¯qX

d2pd2qdy1dy2

=e2Q2(z1z2)2Nc

(2π)7δ(1 −z1−z2)Zd8x[S122010−S12 −S1020+ 1]

eip·(x0

1−x1)eiq·(x0

2−x2)4z1z2K0(|x12|Q1)K0(|x1020|Q1) +

(z2

1+z2

2)x12 ·x1020

|x12||x1020|K1(|x12|Q1)K1(|x1020|Q1).(1)

where (p, y1) and (q, y2) are the transverse momentum and rapidity of the produced quark and antiquark, respectively,

and Q2is the virtuality of the incoming photon. We have made the following deﬁnitions and short hand notations,

Qi=Qpzi(1 −zi),xij =xi−xj,d8x= d2x1d2x2d2x10d2x20.(2)

We have also deﬁned z1≡p+

l+, z2≡q+

l+as the momentum fractions carried by the ﬁnal state quark and antiquark

relative to the photon’s longitudinal momentum l+. In terms of these momentum fractions the rapidity is related

via dyi=dzi

zi. All the dynamics of the strong interactions and gluon saturation are contained in the dipoles Sij and

quadrupoles Sijkl, normalized correlation functions of two and four Wilson lines

Sij =1

tr DViV†

jE, Sijkl =1

tr DViV†

jVkV†

lE,(3)

where the index irefers to the transverse coordinate xiand the following notation is used for Wilson lines,

Vi=ˆ

Pexp ig Zdx+A−(x+,xi).(4)

The Wilson lines eﬃciently resum the multiple scatterings of the quark and antiquark from the target hadron or

nucleus. The angle brackets in Eq. 3 signify color averaging 1. It is important to keep in mind that as this is a

classical result the cross section has no non-trivial x(or rapidity/energy) dependence. It is also easy to check that if

one integrates over the phase space of the quark and antiquark one recovers the standard expressions for the virtual

photon-target total cross section at small x.

Integrating over the quark’s momentum then sets z1= 1 −z2and x0

1=x1and gives

dσγ∗A→¯qX

d2qdy2

=e2Q2z2

2(1 −z2)Nc

(2π)5Zd6x[S220−S12 −S120+ 1]

eiq·(x0

2−x2)4z2(1 −z2)K0(|x12|Q2)K0(|x120|Q2) +

z2

2+ (1 −z2)2x12 ·x120

|x12||x120|K1(|x12|Q2)K1(|x120|Q2)(5)

where the ﬁrst (second) term inside the big square bracket corresponds to contribution of longitudinal (transverse)

photons. To get the full single inclusive production cross section one must also consider the case when one integrates

out the antiquark. It can however be shown that the two results are identical so that we will only integrate out the

quark and multiple our ﬁnal results by a factor of 2. This can also be shown to be true when we calculate the next

to leading order corrections. Therefore we will consider only the case when the quark is integrated out. Furthermore

and as before we will consider only the case of longitudinal photons in this paper.

1Throughout the paper we assume that these dipoles and quadrupoles are real, nevertheless both can have imaginary parts which however

do not contribute here.

III. ONE-LOOP CORRECTIONS

l−k1

iAa

l−k1

iAa

l−k1

k1−k2

iAa

l−k1

k1−k2

iAa

FIG. 1: The real corrections iAa

1, ..., iAa

4. The arrows on Fermion lines indicate Fermion number ﬂow, all momenta

ﬂow to the right. The thick solid line indicates interaction with the target.

l−k1

k3−p

k2−k1

iA5

l−k1

iA6

k3−q

k2−k1

iA7

l−k1

k3−p

k2−k1

l−k1

iA8

k3−q

k2−k1

l−k1

iA9

k2−p

k2−q

iA10

l−k1

iA11

k2−k1

l−k1

k2−k1

iA12

l−k1

iA13

k2+p

q−k2

l−k1

iA14

k1−k2

l−k2

FIG. 2: The ten virtual NLO diagrams iA5, ..., iA14. All momenta ﬂow to the right, except for gluon momenta.

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SingleInclusiveHadronProductioninDISatSmallx:NexttoLeadingOrderCorrectionsFilipBergabo1,2,andJamalJalilian-Marian1,2,y1DepartmentofNaturalSciences,BaruchCollege,CUNY,17LexingtonAvenue,NewYork,NY10010,USA2CityUniversityofNewYorkGraduateCenter,365FifthAvenue,NewYork,NY10016,USAWecalculatetheone-loopc...

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Single Inclusive Hadron Production in DIS at Small x Next to Leading Order Corrections Filip Bergabo12and Jamal Jalilian-Marian12y

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