SIDIS-RC EvGen a Monte-Carlo event generator of semi-inclusive deep inelastic scattering with the lowest-order QED radiative corrections

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SIDIS-RC EvGen: a Monte-Carlo event generator

of semi-inclusive deep inelastic scattering with

the lowest-order QED radiative corrections

Duane Byera,b, Vladimir Khachatryana,b, Haiyan Gaoa,b, Igor Akushevicha,

Alexander Ilyichevc,d, Chao Penge, Alexei Prokudinf,g, Stan Srednyaka,b,

Zhiwen Zhaoa,b

aDepartment of Physics, Duke University, Durham, NC 27708, USA

bTriangle Universities Nuclear Laboratory, Durham, NC 27708, USA

cBelarusian State University, Minsk, 220030, Belarus

dInstitute for Nuclear Problems, Belarusian State University, Minsk, 220006, Belarus

ePhysics Division, Argonne National Laboratory, Lemont, IL 60439, USA

fDivision of Science, Penn State Berks, Reading, PA 19610, USA

gTheory Center, Thomas Jeﬀerson National Accelerator Facility, Newport News, VA

23606, USA

Abstract

SIDIS-RC EvGen is a C++ standalone Monte-Carlo event generator for stud-

ies of semi-inclusive deep inelastic scattering (SIDIS) processes at medium

to high lepton beam energies. In particular, the generator contains binary

and library components for generating SIDIS events and calculating cross

sections for unpolarized or longitudinally polarized beam and unpolarized,

longitudinally or transversely polarized target. The structure of the gener-

ator incorporates transverse momentum-dependent parton distribution and

fragmentation functions, whereby we obtain multi-dimensional binned sim-

ulation results, which will facilitate the extraction of important information

about the three-dimensional nucleon structure from SIDIS measurements. In

order to build this software, we have used recent elaborate QED calculations

of the lowest-order radiative eﬀects, applied to the leading order Born cross

section in SIDIS. In this paper, we provide details on the theoretical for-

malism as well as the construction and operation of SIDIS-RC EvGen, e.g.,

∗Corresponding author.

E-mail address: vladimir.khachatryan@duke.edu (V. Khachatryan)

Preprint submitted to Computer Physics Communications March 9, 2023

arXiv:2210.03785v3 [hep-ph] 7 Mar 2023

how we handle the event generation process and perform multi-dimensional

integration. We also provide example programs, ﬂowcharts, and numerical

results on azimuthal transverse single-spin asymmetries.

Keywords: Monte-Carlo event generators for radiative events; FOAM

Monte-Carlo event generator; ROOT, GSL, VEGAS, Cubature packages;

Semi-inclusive deep inelastic scattering; Transverse momentum-dependent

distribution, fragmentation functions; QED radiative corrections.

PROGRAM SUMMARY

Program Title: SIDIS-RC EvGen

Licensing provisions: GNU General Public License Version 3 (GPLv3)

Developer’s repository link: https://github.com/duanebyer/sidis

Maintainers: Duane Byer; Department of Physics, Duke University

Distribution format: GitHub repository

Programming language: C++, Python

Operating System: Linux, macOS and their distributions

External packages: FOAM, ROOT, GSL, VEGAS, Cubature, Cog, WW-SIDIS,

MSTWPDF

Nature of problem: The task is to ﬁrst create a code for calculations of the lead-

ing order Born cross section as well as radiative corrections (RCs) at the next-

to-leading order (NLO) of the cross section of lepton-hadron semi-inclusive deep

inelastic scattering (SIDIS) at medium to high beam energies with incident un-

polarized or longitudinally polarized lepton beam and unpolarized, longitudinally

or transversely polarized target, enabling to compute azimuthal single-target and

double-beam-target spin asymmetries. Afterwards, a Monte-Carlo event generator

based upon this code is developed, where in the coding and simulation processes

multi-dimensional integrals need to be calculated precisely to obtain the exact

NLO RCs to the SIDIS cross section with high precision beyond ultra-relativistic

limit, which means that the lepton mass is taken into account.

Solution method: The project for building the SIDIS-RC EvGen software is divided

into a library component called libsidis, for the purpose of calculating the inelas-

tic tail to the SIDIS total cross section, and a binary component called sidisgen,

for generating events. The Monte-Carlo event generation is performed through the

usage of the FOAM library, by applying a spatial partitioning method with hyper-

cubical “foam of cells”. The multi-dimensional numerical integration is carried out

by the GSL, VEGAS, and Cubature packages. The SIDIS structure functions that

describe the SIDIS scattering process, and which are given in terms of transverse

momentum-dependent parton distribution and fragmentation functions, are calcu-

lated in Gaussian and Wandzura–Wilczek-type approximations and implemented

in the generator accordingly.

Additional comments including restrictions and unusual features: The restrictions

depend on the complexity of problems, limited by CPU time. As a consequence,

the program running time is of order of several minutes to hours. SIDIS-RC EvGen

does not comprise yet the contribution coming from the radiative tail of exclu-

sive lepton-hadron reactions, which is a separate contribution involving exclusive

structure functions that are currently not well known.

Contents

1 Introduction and physics motivation 4

2 Some basics of the theoretical framework for studying 3D

momentum structure of the nucleon with SIDIS 8

2.1 SIDIS process and functions describing the nucleon structure 8

2.1.1 Kinematics and cross section of the SIDIS process . . 8

2.1.2 TMD PDFs, TMD FFs, and SIDIS structure functions 12

2.2 Lowest-order QED radiative eﬀects in SIDIS . . . . . . . . . . 15

2.2.1 Leading order cross section . . . . . . . . . . . . . . . 15

2.2.2 Lowest-order radiative corrections to the SIDIS cross

section........................... 18

3 MC event generator SIDIS-RC EvGen for studying SIDIS

processes with lowest-order radiative eﬀects 23

3.1 Dividing the SIDIS inelastic cross section into soft and hard

parts................................ 23

3.2 Library component . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Flexiblemode ...................... 27

3.2.2 Streamlined mode . . . . . . . . . . . . . . . . . . . . 29

3.2.3 Structure function implementation . . . . . . . . . . . 29

3.2.4 Leptonic/Hadronic coeﬃcients . . . . . . . . . . . . . 31

3.3 Binary (generator) component . . . . . . . . . . . . . . . . . 33

3.3.1 MC generation with the FOAM library . . . . . . . . 34

3.3.2 Improvement of the foam eﬃciency . . . . . . . . . . . 35

4 Numerical results obtained from SIDIS-RC EvGen 36

4.1 Charged-hadron Collins and Sivers azimuthal transverse single-

spinasymmetries ......................... 37

4.2 Prospects on charged-hadron multiplicity and asymmetry data

comparisons............................ 43

5 Summary and outlook 44

1. Introduction and physics motivation

A considerable amount of our knowledge on the nucleon’s quark-gluon

composition has been obtained via experimental and theoretical studies of the

collinear parton distribution functions (PDFs) [1] and fragmentation func-

tions (FFs) [2]. The precise knowledge of PDFs is essential for understanding

of Quantum Chromodynamics (QCD), and for making quantitative predic-

tions for the observables related to the strong interaction sector of the Stan-

dard Model. In turn, the search of beyond Standard Model physics also

requires precise understanding of QCD. Within the so-called collinear fac-

torization of deep inelastic scattering (DIS), leading-twist (one-dimensional)

PDFs can be interpreted at the leading order as probability densities for ﬁnd-

ing an unpolarized/longitudinally polarized parton in a fast moving unpolar-

ized/longitudinally polarized nucleon1. They depend on a fraction xBj of the

nucleon momentum carried by the parton and on the “resolution scale” that

is associated with the hard scale Q2of a lepton-nucleon scattering process.

These PDFs are well-known, and over the course of more than two decades

the frontier of exploration has been extended to the three-dimensional (3D)

nucleon structure [3].

The 3D nucleon structure in the momentum space is encoded in the so-

called transverse momentum dependent distribution and fragmentation func-

tions, TMD PDFs and TMD FFs [2, 4–10]. These partonic functions are gen-

eralizations of the aforementioned collinear distribution and fragmentation

functions appearing in the standard collinear factorization [11]. Both of them

depend on two independent variables: xBj and k⊥in the case of TMD PDFs,

as well as zhand p⊥in the case of FFs. In the latter case, zhis the fraction of

the quark momentum transferred to the produced (ﬁnal-state) hadron, and

1Collinear FFs in turn describe how quarks and gluons transform into color-neutral

particles such as hadrons.

p⊥is the transverse momentum of the same ﬁnal-state hadron but with re-

spect to the direction of the fragmenting quark. QCD factorization theorems

have been proven for processes with two distinct measured scales, Q1Q2,

such as the semi-inclusive deep inelastic scattering (SIDIS) [6, 10], Drell-Yan

process [12], and production of two hadrons in e+e−annihilation [2]. In case

of SIDIS, the observed transverse momentum PhT of the produced charged

hadron with respect to the virtual photon momentum plays the role of the

small scale Q1, while the virtual photon virtuality Qplays the role of the

large scale Q2. In the TMD description, the observed transverse momentum

PhT is related to the active parton (quark) intrinsic transverse momentum

k⊥and fragmenting quark p⊥as well as to the eﬀects of the gluon radia-

tion encoded in the evolution equations [9]. By utilizing juxtapositions of

the lepton beam and nucleon target polarizations in SIDIS experiments, we

will be capable of extracting pivotal information on various TMDs, whereby

one can quantify the quark transverse motion inside the nucleon and observe

spin-orbit correlations, as well as obtain quantitative insight on the quark or-

bital angular momentum (OAM) contribution to the proton spin. Note that

the results coming from advances in target polarization techniques, like those

employed by HERMES, COMPASS, LHCb and JLab experiments [13–17],

are crucial in studies of TMDs together with the additional information that

can be accessed from the Drell-Yan process [12] and e+e−annihilation [2].

However, in SIDIS experiments devoted to data extractions on TMDs,

such as HERMES [18–22], COMPASS [23–27], and JLab [28, 29], it is in-

dispensable to have a very good control over systematic uncertainties. In

this regard, one of the dominant sources of the systematic uncertainties in

these experiments, with and without polarization of the lepton beam and

the nucleon target, are the radiative corrections (RC) due to the radiation

of photons oﬀ leptons. Radiative corrections are calculated based on Quan-

tum Electrodynamics (QED). Exact analytical formulas for the lowest-order

model-independent part of QED RCs to the SIDIS cross section, with a lon-

gitudinally polarized lepton beam and arbitrarily polarized nucleon target,

beyond ultra-relativistic approximation2, are calculated in Ref. [30]. The

methodology developed in that paper is anchored upon the covariant ap-

proach for RC calculations, so that the obtained formulas can be directly

applied to any coordinate system. Besides, the Bardin-Shumeiko approach

2It means that the incident beam lepton mass is taken into account in all calculations.

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SIDIS-RCEvGen:aMonte-Carloeventgeneratorofsemi-inclusivedeepinelasticscatteringwiththelowest-orderQEDradiativecorrectionsDuaneByera,b,VladimirKhachatryana,b,HaiyanGaoa,b,IgorAkushevicha,AlexanderIlyichevc,d,ChaoPenge,AlexeiProkudinf,g,StanSrednyaka,b,ZhiwenZhaoa,baDepartmentofPhysics,DukeUniversity,...

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SIDIS-RC EvGen a Monte-Carlo event generator of semi-inclusive deep inelastic scattering with the lowest-order QED radiative corrections

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