School of particle and accelerator

Department of Theoretical Physics

Nuclear parton distribution functions with uncertainties in a generalmass variable flavor number scheme
Hamzeh Khanpour
In collaboration withMaryam Soleymaninia (IPM), Shahin Atashbar Tehrani (IPM),
Hubert Spiesberger (Mainz) and Vadim Guzey (PNPI)
arXiv:2010.00555 [hep-ph]; Phys. Rev. D 104 (2021) 034010
20th Lomonosov Conferences
on Elementary Particle Physics, 23 August 2021, Moscow, Russia

Outline❖ Short Introduction❖ Highlights from the nuclear DIS experiments❖ A review on available nuclear PDFs analyses❖ Parametrization of KSASG20 nuclear PDFs
❖ Experimental data sets used in KSASG20 nuclear PDFs
❖ 𝜒 2 analysis method and uncertainty determination❖ KSASG20 results of nuclear PDFs
❖ Summary and conclusions

DIS and probing the structure of the proton• Electron-proton scattering provides a powerful tool for probing the structure of the proton. At low
energies, the dominant process is elastic scattering where the proton remains intact. Elastic
scattering is described by the interaction of a virtual photon with the proton as a whole, and thus
provides a probe of the global properties of the proton, such as its charge radius.
• At high energies, the dominant process is deep inelastic scattering, where the proton breaks up.
Here the underlying process is the elastic scattering of the electron from one of the quarks within
the proton. Consequently, DIS provides a probe of the momentum distribution of the quarks.

Inclusive DIS kinematics
The nature of e-p scattering depending on the wavelength of the virtual photon.

The Quark Parton Model• The basic idea of the QPM is that in the DIS process, ep→eX, the virtual proton interacts with oneof the quark constituents of the proton.
• As a result the ep interaction may be written as a sum (of probabilities) of scattering from single
free quarks.

fq(ξ) is the probability of finding the quark q in the proton
carrying a fraction ξ of its momentum
electron-quark cross section

Inclusive DIS structure functions• The structure functions Fa as a physical observables describe the DIS processes.
• For the structure functions Fa, describing the deep inelastic processes ℓ+p → ℓ′+X, thefactorization formula has the following form:

Universal parton densities (of the proton). They cannot becalculated in perturbative QCD, but their Q2 dependence
is calculable using the DGLAP evolution equations

Coefficient functions. They are calculable fromperturbative QCD as a power series in αs, but
are unique to the particular observable, Fa.

DIS experiments on nuclear targets
Nuclear EMC effectHEP Data

This effect was first observed in 1983 at CERN by the European Muon Collaboration, hence the name "EMCeffect". The observed x-dependence of this ratio is in disagreement with existing theoretical predictions.
The ratio of the nucleon structure functions F2(Fe)/F2(D) for iron and deuterium
European Muon Collaboration, Published in Phys. Lett. B123 (1983) 275-278, CERN-EP/83-14

Annu. Rev. Nucl. Part. Sci. 1995.45:337-390

Experimental efforts

Nuclear EMC Effects

x-Dependence

A-Dependence

Ratios of the deep inelastic cross section on nuclear several targets.
Annu. Rev. Nucl. Part. Sci. 1995.45:337-390

Typical nuclear effects seen in the DIS measurement
0.1 < x < 0.3
Quark momentum distributions
in nucleons bound inside anuclei are different from thoseof free nucleons!!!

Quarks in nuclei carry lessmomentum than quarks infree nucleons.

x > 0.7

Nucleon motion
inside the nucleus

x < 0.05-0.1

Rev. Mod. Phys. 89 (2017) 4, 045002

0.3 < x < 0.7

Physics Reports 240 (1994) 301-393

Charged-lepton scattering of the nuclei and nuclear PDFs
❖ Parton densities are less precisely known for nuclei than for nucleons.
❖ The LHC data opens a previously unexplored kinematic region.

JLAB (CLAS), Nature 566, no.7744, 354-358 (2019) [arXiv:2004.12065 [nucl-ex]].

Nominal x and Q2 coverage of nuclear DIS.

❖ These data do not have the power to constrain all of the nuclear PDFs components.

Deeply inelastic scattering (DIS) on nuclear targets❖ The partonic coefficient functions 𝜎ො and the DGLAP evolution of 𝑓𝑖𝐴 are the same as in the caseof free proton scattering. The goal is to carry out very similar program as in the case of freeproton analyses.

Standard definition of nuclear PDFs

Nuclear modifications

Free proton baseline❑ Nuclear modification at low and large x poorly constrained by present data.

Recent determination of nuclear PDFs in 2019/2021➢ nNNPDF [Eur. Phys. J. C 79, no. 6, 471 (2019), [arXiv:1904.00018 [hep-ph]].

➢ TUJU19 [Phys. Rev. D 100, no. 9, 096015 (2019), [arXiv:1908.03355 [hep-ph]].

➢ KSASG20 [Phys. Rev. D 104 (2021) 3, 034010; arXiv:2010.00555 [hep-ph]]➢ nCTEQ15HIX [Phys. Rev. D 103 (2021) 11, 114015, arXiv:2012.11566 [hep-ph]].

KSASG20 nuclear PDFs

KSASG20 nuclear PDFs parameterization❑ We work within the conventional approach which defines the nuclear PDFs for a bound proton ina nucleus with atomic mass number A with respect to those for a free proton through amultiplicative nuclear modification factor

T. J. Hou et al., “New CTEQ global analysis of quantum
chromodynamics with high-precision data from the LHC,”
Phys. Rev. D 103 (2021) 014013.

❑ In order to give further and enough flexibility to our modification factor, we have considered thefit parameters to be A-dependent:❑ Combining the weight function with PDFs and assuming the isospin symmetry, will yield usnuclear PDFs as in what follows:Total list of experimental data in KSASG20

Total list of experimental data in KSASG20

Nuclear DIS

The measurements of the deuteron structure
function and deuteron-proton ratio.

Nuclear DIS

The Drell-Yan cross section ratios measured by FNAL
Collaboration at JLAB experiment.
The charged-current (anti)neutrino-nucleus DIS data.

The structure functions for neutrino andantineutrino scattering

The neutral-current charged lepton DIS experimental
datasets from Jefferson Lab Hall C and CLAS measured
by JLAB during the 6 GeV electron beam operation.

JLAB (CLAS), Nature 566, no.7744, 354-358 (2019)[arXiv:2004.12065 [nucl-ex]].

Target Mass Corrections❑ TMC effects are most pronounced at large-x regions and moderate-to-small values of photon
virtuality Q2,

Nachtmann variable

The target-mass corrected structure functions

Higher Twist Corrections➢ The inclusion of the Higher Twist (HT) corrections in which particular important at high x and low
Q values✓ For our NLO analysis:

CJ15: Phys. Rev. D 93, no.11, 114017 (2016), [arXiv:1602.03154 [hep-ph]]✓ For our NNLO analysis:

M. Goharipour and S. Rostami, Phys. Rev. D 101, no.7, 074015 (2020),[arXiv:2004.03403 [hep-ph]].

2
𝜒 minimization➢ Global QCD extractions of nuclear PDFs are implemented around an effective 𝜒 2 function thatquantifies the goodness of the fit to the data for a given set of theoretical parameters, which
determine the nuclear PDFs at some input scale 𝑄02 .

➢ The 𝜒 2 function is minimized by the CERN program library MINUIT.
F. James and M. Roos, Comput. Phys. Commun. 10, 343 (1975).

➢ Nuclear PDFs uncertainties: Standard Hessian method

KSASG20 nuclear PDFs and uncertainties

The nuclear modification factors defined as ratios of proton PDFs bound in deuterium (D), Beryllium (Be), iron (Fe) and gold(Au) to the corresponding free-proton PDFs of CT18 at NLO (top row) and NNLO (bottom row) accuracy in pQCD and at theinitial scale Q2 = 2.0 GeV.

KSASG20 nuclear PDFs and their uncertainties at Q2 = 10 GeV2 for iron (left) and lead (right).

NLO/NNLO comparison

Higher-order QCD corrections

Our bound proton PDFs at Q2 = 10 GeV2 for lead at NLO and NNLO accuracy.

Comparison with other groups

The nuclear parton distribution functions KSASG20 in lead at NLO accuracy compared to the nuclear PDF sets nCTEQ15,EPPS16 and TUJU19 shown at a higher scale Q2 = 10 GeV2.

Our bound proton PDFs at the scale Q2 = 100 GeV2 for lead at NLO accuracy. The most recent results from nCTEQ15, EPPS16 andTUJU19 are shown for comparison.

Fit quality and comparison of data and theory

Comparison of our NNLO theory predictions for the ratio F2A/F2C and F2A/F2D as a function of x with some selected nuclear
DIS data.

The neutrino(antineutrino) lead DIS data compared with the KSASG20 NNLO theory predictions.

Comparison of our NLO and NNLO theory predictions for the structure functions ratios F2A/F2D with the most recent darafrom Jefferson Lab CLAS as a function of x with some selected nuclear targets.

Comparison of the KSASG20 NLO theory predictions for the Drell-Yan cross section ratios

Summary and Conclusions➢ We have given a very short introduction to nuclear PDFs and reviewed some available global nuclear PDFsanalyses.
➢ We have presented the KSASG20 NLO and NNLO nuclear PDFs and their uncertainties.
➢ We have introduced several improvements aimed to obtained a well-established nuclear PDFs.

➢ Consistent QCD fit for wide variety of nuclear targets and wide kinematic range have been achieved.
➢ We look forward to more high-precision data from LHC proton-lead ( 𝑝 − 𝑝𝑏 ) collisions to better
determination of nuclear gluon PDF (especially at small-x )➢ Future HERA-like measurements for many nuclear targets covering very small range of Bjorken x (FCC-he &LHeC).
➢ Plans are being made to test this hypothesis with the 12 GeV upgrade of Jefferson Lab by including a more
complete set of light nuclei.