The full set of Polarized Deep Inelastic Scattering Structure Functions at NNLO accuracy Ignacio Borsa

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The full set of Polarized Deep Inelastic Scattering Structure Functions at

NNLO accuracy

Ignacio Borsa∗

Universidad de Buenos Aires and IFIBA,

Facultad de Ciencias Exactas y Naturales,

Departamento de F´ısica. Buenos Aires, Argentina.

Daniel de Florian†and Iv´an Pedron‡

International Center for Advanced Studies (ICAS), ICIFI and ECyT-UNSAM,

25 de Mayo y Francia, (1650) Buenos Aires, Argentina

We present the second order contributions to the coeﬃcient functions for the parity vio-

lating polarized structure functions gLand g4, thus completing the O(α2

S) knowledge on DIS

structure functions. We obtain the missing O(α2

S) pieces from the known parity conserving

unpolarized coeﬃcient functions. We also present a phenomenological analysis for the phase

space region the future Electron-Ion Collider is set to explore.

PACS numbers:

I. INTRODUCTION

The deep inelastic scattering (DIS) process plays a fundamental role in the analysis of polarized

QCD phenomena, and ultimately in our understanding of the spin structure of the proton in terms

of polarized parton distributions functions (pPDFs). In addition to the pure QED contributions

due to photon exchange, which have been thoroughly studied in pPDFs global analyses [1–5], the

DIS process also receives contributions associated to the exchange of the weak Zand W±bosons,

which introduce parity violating terms to the cross section. Both neutral and charged current (NC

and CC, respectively) DIS involve quark PDF combinations diﬀerent from those of the pure photon

counterpart, making the polarized DIS data an important source of complementary information

on the proton spin decomposition, since it allows to disentangle the individual quark contributions

from its corresponding antiquark ones [6, 7]. Nonetheless, and contrary to the unpolarized case, the

rather low values of Q2explored by polarized DIS experiments so far, for which weak contributions

are highly suppressed, have made the study of Zand W-mediated processes rather unnecessary.

However, with the construction of the new Electron-Ion-Collider (EIC) on the horizon, there is a

prospect of reaching an unprecedented level of precision in measurements of polarized processes and

to extend the kinematical coverage in terms of the DIS variables xand Q2[8]. Besides the extension

∗Electronic address: iborsa@df.uba.ar

†Electronic address: deﬂo@unsam.edu.ar

‡Electronic address: ipedron@unsam.edu.ar

arXiv:2210.12014v2 [hep-ph] 13 Dec 2022

l, k

l′, k′

γ, Z, W

N, p

FIG. 1: Leading order diagram for the lepton-nucleon deep-inelastic scattering. The exchanged

particle can be either a photon γor a weak boson Z, W ±, and carries momentum q.

of the kinematical range, the EIC will give access to the hadron helicity states independently, and

thus allow to measure the asymmetries with polarized protons and unpolarized leptons for the

ﬁrst time [7]. These new precision measurement need to be matched by correspondingly accurate

theoretical predictions. As it is already the standard for the unpolarized sector in Large-Hadron-

Collider (LHC) computations, next-to-next-to-leading order (NNLO) calculations are becoming

a benchmark for polarized processes, with results already available for inclusive process, such as

Drell-Yan [9] and pure QED DIS [10], the helicity splitting functions [11–13], as well as the recent

addition of exclusive process like jet production in DIS [14, 15], Wboson production in proton-

proton collisions [16] and semi-inclusive DIS (in an approximated form) [17, 18]. In this paper,

we present the O(α2

S) parity violating (longitudinally) polarized structure functions g4and gL, in

order to match the NNLO precision for inclusive NC and CC DIS.

Our paper is organized as follows: in section II we introduce and provide the expressions of the

polarized DIS structure functions at O(α2

S). In section III we analyze the impact of the second

order corrections to the polarized structure functions, for both NC and CC processes and we discuss

the combination of polarized structure functions related to the single-spin cross section. Finally,

in section IV we summarize our work and present our conclusions.

II. POLARIZED STRUCTURE FUNCTIONS

In the following we consider the inclusive lepton-nucleon DIS process, deﬁned as

l(k) + N(p)→l0(k0) + X,

where, for simplicity, we assume the incoming lepton is either an electron or a positron and work

within the lowest order approximation in the EW theory, i.e. assuming single-boson exchange.

Here, kand pare the momenta of the incoming lepton and nucleon, respectively, k0is the momentum

of the outgoing scattered lepton (either an electron/positron in the NC or a neutrino/antineutrino

in the CC case, respectively), and Xrepresents the whole recoiling hadronic ﬁnal state. The

particle exchanged in the process can be any of the electroweak bosons γ,Zor W±, and carries

a momentum q=k−k0determined by the lepton kinematics. The usual variables utilized in the

description of DIS are the boson virtuality Q2, the Bjorken variable xand the inelasticity y, which

are deﬁned by

Q2=−q2, x =Q2

2p·q, y =q·p

k·p.(1)

While in the CC case the kinematics of the outgoing neutrino may not be experimentally accessible,

the values of xand Q2can be reconstructed from the hadronic ﬁnal state using the Jacquet-Blondel

method [6]. The diagram for the lowest order contribution is shown in Fig. 1.

Following the notation from [19], the DIS cross section can be expressed in terms of the product

of a hadronic (W) and a leptonic (L) tensor

d2σ

dxdy =2πyα2

Q4X

ηiLµν

iWi

µν ,(2)

with the summation over iindicating the contributions associated to the diﬀerent gauge bosons.

For NC processes, the cross section receives contributions from the exchange of γ,Zas well as

their interference, so i=γ, Z,γ/Z. For CC processes, i=W. The factors ηiare the ratios of the

corresponding propagators and couplings to the ones of the photon exchange

ηγ= 1, ηγ/Z =GFM2

2√2πα Q2

Q2+M2

Z,

ηZ=η2

γ/Z , ηW=1

2GFM2

4πα

Q2+M2

W2

(3)

The ﬁrst piece of the cross section in Eq. (2) is the leptonic tensor Lµν

i. It corresponds to the

square amplitude of the QED/EW interaction vertex between the boson and the leptons. In terms

of the charge eand helicity λof the incoming massless lepton (with λ2= 1), it has the following

expressions for each boson contribution:

Lµν

γ= 2 −k·k0gµν +kµk0ν+k0µkν−iλµναβkαk0

β,

Lµν

Z= (ge

V+eλge

A)2Lµν

γ,

Lµν

γ/Z = (ge

V+eλge

A)Lµν

γ,

Lµν

W= (1 + eλ)2Lµν

γ,

(4)

where ge

V=−1

2+ 2 sin2θWand gA=−1

2, and e=±1 is the charge of the incoming lepton. In

the case of pure QED, the leptonic tensor has a symmetric and an antisymmetric part, with the

latter being proportional to the lepton helicity. The case of the weak bosons is more involved, since

the axial coupling mixes the helicity dependence in both structures. However, the key point to

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ThefullsetofPolarizedDeepInelasticScatteringStructureFunctionsatNNLOaccuracyIgnacioBorsaUniversidaddeBuenosAiresandIFIBA,FacultaddeCienciasExactasyNaturales,DepartamentodeFsica.BuenosAires,Argentina.DanieldeFlorianyandIvanPedronzInternationalCenterforAdvancedStudies(ICAS),ICIFIandECyT-UNSAM,25de...

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The full set of Polarized Deep Inelastic Scattering Structure Functions at NNLO accuracy Ignacio Borsa

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