# QCD analysis of valence structure functions using deep inelastic lepton-
nucleon scattering

Javad Shahrzad<EMAIL_ADDRESS>Ali Khorramian ID
<EMAIL_ADDRESS>Faculty of Physics, Semnan University, P. O. Box
35131-19111, Semnan, Iran

###### Abstract

A new “$\mathtt{SK24}$” non-singlet QCD analysis of the structure functions at
the NNLO approximation is performed, utilizing the global fit of the data from
various charged lepton scattering experiments. We extract the valence parton
distribution functions (PDFs) and provide a parametrization of them, along
with the correlated errors for a wide range of $x$ and $Q^{2}$. We compare
valence PDFs and their uncertainties with those from different PDF sets
provided by various groups. We also obtain valence PDFs and the strong
coupling constant $\alpha_{s}(M_{Z}^{2})$, taking into account the nuclear
correction concerning large $x$ as well as the target mass correction (TMC)
and higher twist (HT) effects at the NNLO. In the large $x$ region, we extract
the higher twist contributions of $xF_{3}(x,Q^{2})$, $F_{2}^{p}(x,Q^{2})$, and
$F_{2}^{d}(x,Q^{2})$. We determine $\alpha_{s}(M_{Z}^{2})$ without and with
considering the TMC and HT corrections and perform a comparison with the world
average of $\alpha_{s}(M_{Z}^{2})$ and other reported results. The extracted
results concerning valence PDFs with their uncertainties and
$\alpha_{s}(M_{Z}^{2})$ value agree with available theoretical models.

Keywords: QCD Analysis, Non-singlet Structure Function, Valence Quark.

##  I Introduction

Deep-inelastic scattering (DIS) of high-energy leptons off hadrons is a
fundamental process for studying the structure of hadrons in terms of their
parton distribution functions (PDFs) Taylor:1991ew ; Kendall:1991np ;
Friedman:1991nq . For both nucleons and nuclei, the data obtained from
electron and neutrino DIS play a significant role in modern determinations of
PDFs, illuminating and decoding the internal structure of hadrons. These kinds
of data have offered detailed insights into parton densities, revealing
valuable information at both small and large values of the parton momentum
fraction $x$ of the nucleon. The investigation will once again take the
spotlight at the future Electron-Ion Collider (EIC) Accardi:2012qut ;
AbdulKhalek:2021gbh , proposing an opportunity to study DIS off nucleons and
various nuclear targets with remarkable precision across a wide kinematic
range, with particular emphasis on the large $x$ region.

Progress in understanding the inner structure of hadrons continues through
ongoing research efforts, with new experiments, improved theoretical models,
and increasingly sophisticated computational methods. Achieving precise
theoretical calculations remains challenging despite having theoretically
motivated expectations for certain aspects of the structure. Quantum
chromodynamics (QCD) global fits to these data are instrumental in
constraining the PDFs. DIS has been highly successful in investigating various
features of QCD, including the study of polarized Khorramian:2020gkr ;
Nematollahi:2023dvj ; Mirjalili:2022cal ; Han:2021dkc ; Salimi-Amiri:2018had ;
Sato:2016tuz ; TaheriMonfared:2014var ; Jimenez-Delgado:2013boa ;
Arbabifar:2013tma ; Khorramian:2010qa ; Leader:2010rb ; Blumlein:2010rn ;
AtashbarTehrani:2007odq ; Khorramian:2004ih  and unpolarized PDFs Yan:2022pzl
; Gao:2022uhg ; Hou:2022onq ; Abdolmaleki:2019tbb ; NNPDF:2021njg ;
Bailey:2020ooq ; H1:2015ubc ; Hou:2019qau  within a hadron.

Furthermore, there has been a growing interest in the large $x$ behavior of
proton PDFs in recent years Courtoy:2020fex ; Ball:2016spl ; Fu:2023rrs . The
importance of large $x$ PDFs is notable in specific high-energy scattering
processes, such as the investigation of intrinsic heavy quarks within the
proton, which is dominant at large $x$ BHPS1 ; BHPS2 ; Brodsky:2015fna ;
Brodsky:2020hgs . In recent years, there have been increasing efforts to
examine the role of intrinsic heavy quarks within a proton Ball:2022qks ;
Ball:2023anf . The results presented in Ref. Azizi:2022nqm  discuss the
implications of the large $x$ PDFs on various high-energy processes.

Determining the valence PDFs with a particular emphasis on the large $x$
region is a topic of significant importance, where the impact of large $x$
PDFs is remarkable. Valence quarks, being among the constituent quarks, play a
crucial role in understanding the internal structure of the proton. Very
recently, new constraints on the $u$-valence PDF are reported in Ref.
Aggarwal:2022cki .

Deep inelastic neutrino-nucleon scattering provides essential information for
extracting $xu_{v}(x,Q^{2})$ and $xd_{v}(x,Q^{2})$ valence quark densities.
The non-singlet structure function, $xF_{3}(x,Q^{2})$, plays an important role
in the QCD global analysis of PDFs, especially at large $x$, where valence
quark distributions are essential. The $xF_{3}$ structure functions of deep
inelastic neutrino-nucleon scattering have been measured by several
experimental groups, including the Chicago-Columbia-Fermilab-Rochester
collaboration (CCFR) ccfr:1977 , Neutrinos at the Tevatron (NuTeV)
Tzanov:2005kr , CERN Hybrid Oscillation Research ApparatUS (CHORUS)
collaboration at CERN Onengut:2005kv , and CERN-Dortmund-Heidelberg-Saclay-
Warsaw collaboration (CDHSW) Berge:1989hr . These experimental data are a
precise foundation for determining the valence quark densities and the strong
coupling constant. Previous non-singlet analyses of CCFR data have been
performed using various approximation methods based on orthogonal polynomial
expansions, such as Jacobi polynomials Kataev:1994rj ; Kataev:1996vu ;
Kataev:1997nc ; Kataev:1997vv ; Alekhin:1998df ; Alekhin:1999af ;
Kataev:1999bp ; Kataev:2001kk ; Kataev:2002wr ; Sidorov:2013aza ;
AtashbarTehrani:2009zz , Bernstein polynomials Santiago:2001mh ;
Khorramian:2006wg , and Laguerre polynomials GhasempourNesheli:2015tva .
Recent non-singlet QCD results based on CCFR, NuTeV, CHORUS, and CDHSW
neutrino-nucleon data to determine $xu_{v}(x,Q^{2})$ and $xd_{v}(x,Q^{2})$,
taking into account nuclear and higher twist corrections at the next-to-
leading order (NLO) and next-to-next-to-leading order (NNLO) are presented in
Ref. Tooran:2019cfz  using the xFitter framework. The analysis covered a wide
range of $x$ and $Q^{2}$, offering valuable insights into the valence quark
distributions in the nucleon. The QCD analysis of the non-singlet $xF_{3}$
structure function, utilizing the Laplace transform approach, is reported in
Ref. MoosaviNejad:2016ebo  without considering any nuclear and higher twist
corrections.

In addition, performing a non-singlet QCD analysis of $F_{2}(x,Q^{2})$
structure function using the world data for charged lepton scattering offers
an opportunity to extract the valence quark parton densities and the
associated correlated errors on a broad range of $x$ and $Q^{2}$. By employing
non-singlet analysis of $F_{2}$, we can also extract the flavor non-singlet
$xu_{v}(x,Q^{2})$ and $xd_{v}(x,Q^{2})$ PDFs from the available $e(\mu)p$ and
$e(\mu)d$ world data of $F_{2}^{p}(x,Q^{2})$ and $F_{2}^{d}(x,Q^{2})$ in the
valence-quark region, as well as from the
$F_{2}^{NS}(x,Q^{2})=2[F_{2}^{p}(x,Q^{2})-F_{2}^{d}(x,Q^{2})]$ structure
functions. Data sets from various experiments, including BCDMS, NMC, SLAC, H1,
and ZEUS, are utilized for this purpose BCDMS:1989ggw ; BCDMS:1989qop ;
BCDMS:1989gtb ; Whitlow:1991uw ; NewMuon:1996fwh ; H1:2000muc ; H1:2003xoe ;
ZEUS:1998agx ; ZEUS:2001mhd . Our previous non-singlet analyses of
electron(muon)-proton and deuteron $F_{2}$ data based on Jacobi polynomials
are reported in Refs. Khorramian:2008yh ; Khorramian:2009xz . Previous results
of non-singlet analyses Blumlein:2006be , presented without using the Jacobi
polynomials approach, extended up to N3LO, and instead, utilized the Mellin
space technique. In fact, this analysis is based on the approach of J.
Blümlein, et al. in Ref. Blumlein:2006be  and has developed it by utilizing
various data and different parameterization in $x$-space by QCDNUM open-source
framework Botje:2010ay .

Our motivation to include $xF_{3}(x,Q^{2})$ and $F_{2}(x,Q^{2})$ data in our
QCD non-singlet analysis is the study of valence PDFs at large $x$, where
these kinds of PDF is not one of the very well known PDFs in the proton,
particularly for $x$ > 0.5. In Figure 1, we compare the NNLO valence PDF
ratios of the NNPDF4.0 NNPDF:2021njg , CT18 Hou:2019qau , and HERAPDF2.0
H1:2015ubc  predictions to the MSHT20 set Bailey:2020ooq . All PDF sets are
imported from LHAPDF6 Buckley:2014ana . This figure shows the sizable
difference between the extracted $u$ and $d$ valence PDF results for large $x$
values. Therefore, in this study, we are motivated to provide a valence PDF
set in the presence of $xF_{3}(x,Q^{2})$ and $F_{2}(x,Q^{2})$ data. We aim to
find the influence of such data on the central value and corresponding
uncertainties of valence PDFs, with a particular focus on large values of $x$
and the strong coupling constant.

The current study explicitly emphasizes a non-singlet analysis to extract
valence PDFs. It involves integrating DIS neutrino-nucleon $xF_{3}(x,Q^{2})$
structure function data with $F_{2}(x,Q^{2})$ structure function measurements
obtained from charged electron (muon)-proton and deuteron collisions from
various experiments. Indeed, this analysis benefits from being free of
assumptions on the gluon PDF, resulting in improved systematic considerations.
The reduced number of parameters needed to describe the parton distributions
and the lack of correlation between the QCD strong coupling constant
$\alpha_{s}$ and the gluon PDF contribute to the enhanced reliability of the
results. Heavy flavor contributions for non-singlet structure function are
investigated in Ref. Blumlein:2006be . As this article has pointed out, these
effects are negligible at NNLO except for the large $Q^{2}$ region
($Q^{2}\simeq 10^{4}$ GeV2), which is not an effective area of the present
paper. Additionally, according to Ref. Blumlein:2021lmf , it is demonstrated
that within the large $Q^{2}$ region, heavy flavor corrections may reach up to
1% for $x\leq 0.4$ at N3LO level of accuracy. Moreover, it is observed that
the relative effects tend to be larger at higher values of $x$. In this
analysis, $F_{2}$ also depends on sea quark corrections in the valence region
to some extent. The $u$ and $d$-valence PDFs at $Q^{2}_{0}=4$ GeV2 are
affected only slightly, as demonstrated in Ref. Blumlein:2012se .

In this article, we manage our non-singlet QCD analysis using the QCDNUM open-
source framework Botje:2010ay . Incorporating nuclear corrections, target mass
corrections (TMCs), and higher twist (HT) effects is essential to accommodate
the diverse data sets utilized in this analysis. These corrections are
performed during the fit procedure and play an essential role in our study. We
incorporate these adjustments into our codes, which are not included in the
main QCDNUM package.

The paper is organized as follows: We present a brief overview of the basic
formalism for DIS structure functions and parton distributions in Section II.
Moving on to Section III, we present the inputs for the fit, encompassing both
theoretical and experimental aspects. This section also covers our
parametrization for valence PDFs and the practical data sets utilized in the
present QCD analysis. Section IV is dedicated to QCD corrections, where we
discuss nuclear effects, target mass corrections, and higher twist effects,
ensuring a comprehensive understanding of their impact. Section V presents the
fit results for the valence distribution functions, their evolution,
corresponding errors, and our findings regarding $\alpha_{s}(M_{Z}^{2})$ at
the NNLO. We also compare these results with other theoretical outcomes to
establish their significance and reliability. Finally, we conclude our study
by discussing the results in Section VI.

##  II DIS structure functions and parton distributions

The present flavor non-singlet analysis is based on four complementary data
sets: the non-singlet structure function $xF_{3}(x,Q^{2})$, as well as
$F_{2}^{p}(x,Q^{2})$ and $F_{2}^{d}(x,Q^{2})$ structure functions in the
valence-quark region $x\geq 0.3$, and the combination of
$F_{2}^{p,d}(x,Q^{2})$ is employed to determine
$F_{2}^{NS}(x,Q^{2})=2[F_{2}^{p}(x,Q^{2})-F_{2}^{d}(x,Q^{2})]$. Therefore, we
proceed to introduce these structure functions in this section. Theoretical
results of these structure functions, including anomalous dimensions, the
massless Wilson coefficients, and the massive Wilson coefficients up to three-
loop orders, are described in Refs. Moch:2002sn ; Moch:2004pa ; Vogt:2004mw ;
Blumlein:2021enk ; Politzer:1974fr ; vanNeerven:1999ca ; Vermaseren:2005qc ;
vanNeerven:1991nn ; Zijlstra:1991qc ; Zijlstra:1992kj ; Blumlein:2022gpp ;
Ablinger:2017err ; Ablinger:2012ej . Also, one can refer to a good review of
Ref. Blumlein:2023aso  and references therein.

The charged-current (CC) deep inelastic neutrino (antineutrino)-nucleon
scattering differential cross sections at the leading order of the running
coupling constant $\alpha_{s}$ can be described in terms of structure
functions related to parton distributions Eisele:1986uz ; Diemoz:1986kt . In
the quark parton model (QPM) framework, the structure of neutrinos and
antineutrinos interacting with nucleons is described at the leading order (LO)
regarding their connection to the valence PDFs. Within the QPM, the averaged
nucleon structure for neutrinos and antineutrinos is exclusively governed by
the valence quark distributions. Investigating the DIS neutrino
(antineutrino)-nucleus $xF_{3}$ structure function requires considering the
nuclear effect due to neutrino detection involving heavy nuclear targets
deFlorian:2011fp . Various neutrino experiments employ nuclear targets, such
as CCFR, NuTeV, CDHSW (using an iron target), and CHORUS (using a lead
target). Nuclear PDFs are crucial in obtaining the average of the neutrino and
antineutrino nucleus structure functions. In Ref. Schienbein:2007fs , a study
on the nuclear effects in CC neutrino-iron DIS data is conducted to determine
the iron PDFs. Furthermore, Ref. Tooran:2019cfz  comprehensively analyzes
nuclear effects on heavy neutrino-nucleus scattering. The average of the
neutrino and antineutrino nucleus structure functions is given by

$\displaystyle xF_{3}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
xu_{v}(x,Q^{2})+xd_{v}(x,Q^{2})~{}.$ (1)

In the following sections, we will introduce the nuclear effects for neutrino
DIS structure functions (see, e.g., Tooran:2019cfz , and references therein).

Alternatively, when considering the non-singlet regime and the valence region
with $x\geq 0.3$ for $F_{2}^{p}$ at LO Blumlein:2006be ; Khorramian:2008yh ;
Khorramian:2009xz , the combinations of parton densities can be expressed as

$F_{2}^{p}(x,Q^{2})=\left[\frac{1}{18}\,xq_{{\rm
NS,}8}^{+}+\frac{1}{6}\,xq_{{\rm
NS,}3}^{+}\right](x,Q^{2})+\frac{2}{9}x\Sigma(x,Q^{2})~{}.$ (2)

Considering the neglect of sea PDFs in the region $x\geq 0.3$, we can deduce
the following relations: $xq_{{\rm NS},3}^{+}=xu_{v}-xd_{v}$ represents the
difference between up ($u_{v}$) and down ($d_{v}$) valence quarks. $xq_{{\rm
NS},8}^{+}=xu_{v}+xd_{v}$ represents the sum of up and down valence quarks.
Additionally, since sea PDFs can be neglected in this region, the term
$\Sigma$ represents the sum of up and down valence quarks. Consequently, in
the $x$-space, we obtain $x\Sigma(x)=xu_{v}(x)+xd_{v}(x)$. These insights
allow us to understand the behavior of quarks in the specified region better
and facilitate the simplification of the QCD analysis. Consequently, in the
$x$-space, we obtain the following:

$\displaystyle F_{2}^{p}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\left(\frac{5}{18}\,xq_{{\rm NS,}8}^{+}+\frac{1}{6}\,xq_{{\rm
NS,}3}^{+}\right)(x,Q^{2})$ (3) $\displaystyle=$
$\displaystyle\frac{4}{9}\,xu_{v}(x,Q^{2})+\frac{1}{9}\,xd_{v}(x,Q^{2})~{}.$

Alternatively, in the specified region, the combinations of parton densities
for $F_{2}^{d}$ are also given by Khorramian:2008yh ; Khorramian:2009xz :

$\displaystyle F_{2}^{d}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\left(\frac{5}{18}\,xq_{{\rm NS,}8}^{+}\right)(x,Q^{2})$ (4)
$\displaystyle=$ $\displaystyle\frac{5}{18}\,x(u_{v}+d_{v})(x,Q^{2})~{}.$

In the region $x\leq 0.3$, for the difference between the proton and deuteron
data, we utilize Khorramian:2008yh ; Khorramian:2009xz

$\displaystyle F_{2}^{NS}(x,Q^{2})$ $\displaystyle\equiv$ $\displaystyle
2(F_{2}^{p}-F_{2}^{d})(x,Q^{2})=\frac{1}{3}\,xq_{{\rm NS,}3}^{+}(x,Q^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{3}\,x(u_{v}-d_{v})(x,Q^{2})-\frac{2}{3}\,x(\bar{d}-\bar{u})(x,Q^{2})~{}.$

In the region $x\leq 0.3$, since sea quarks cannot be neglected for smaller
values of $x$, we update the expression for $q_{{\rm NS,}3}^{+}$ as
$u_{v}-d_{v}+2(\bar{u}-\bar{d})$. Besides parameterizing the valence PDFs, the
non-singlet structure function necessitates the inclusion of the distribution
$x(\bar{d}-\bar{u})(x,Q^{2})$ as an input. It is important to note that
directly extracting the $x(\bar{d}-\bar{u})(x,Q^{2})$ distribution from deep
inelastic scattering (DIS) analysis is challenging; however, this distribution
can be determined by utilizing Drell-Yan data. Very recently, strong evidence
has emerged for a flavor asymmetry between the $u$ and $d$ sea PDFs in the
proton, as observed in deep-inelastic scattering and Drell-Yan experiments by
FNAL E906/SeaQuest Collaboration SeaQuest:2022vwp . The reported results of
the $(\bar{d}-\bar{u})(x,Q^{2})$ differences are deduced for the range
$0.13<x<0.45$. Interestingly, the latest SeaQuest data demonstrate that the
$\bar{d}(x)$ distribution continues to exceed the $\bar{u}(x)$ distribution
even at the highest $x$ value ($x=0.45$).

Figure 2 illustrates recent SeaQuest $\bar{d}(x)-\bar{u}(x)$ results
SeaQuest:2022vwp  in comparison with data from NuSea NuSea:1998kqi ;
NuSea:2001idv . Additionally, the figure displays the modern PDF set results
as a function of $x$, which incorporates MSHT20 Bailey:2020ooq , NNPDF4.0
NNPDF:2021njg  and CT18 Hou:2019qau  at Q2=25.5 GeV2.

In our analysis, we choose to utilize the $x(\bar{d}-\bar{u})(x,Q^{2})$
distribution from the MSHT20 Bailey:2020ooq  set as an input, which
incorporates the asymmetry in sea quarks. The MSHT20 PDF set is one of the
most recent and modern PDF sets obtained from global fits in a wide range of
experimental data, including DIS measurements. While this parametrization has
a minor impact on our analysis, it is still essential to assess the influence
of this distribution by comparing it with various asymmetrical sea quark
distributions derived from other studies. We will discuss the outcomes
obtained when incorporating different asymmetry distributions from other
analyses.

The following section will focus on the discussion of input PDF
parametrizations and the experimental measurements corresponding to theories
expressed in this section.

##  III Inputs for the Fit: Theoretical and Experimental Aspects

This section introduces the $xu_{v}$ and $xd_{v}$ PDF parametrizations at the
input scale of $Q_{0}^{2}$. Additionally, we consider $\alpha_{s}(M_{Z}^{2})$
as another fitting parameter in the present QCD analysis. A detailed
discussion of various combinations of data sets will be presented for
neutrino-nucleon DIS data obtained by CCFR ccfr:1977 , NuTeV Tzanov:2005kr ,
CHORUS Onengut:2005kv , and CDHSW Berge:1989hr  experiments, as well as
charged electron (muon)-nucleon DIS data obtained by BCDMS, NMC, SLAC, H1, and
ZEUS experiments BCDMS:1989ggw ; BCDMS:1989qop ; BCDMS:1989gtb ;
Whitlow:1991uw ; NewMuon:1996fwh ; H1:2000muc ; H1:2003xoe ; ZEUS:1998agx ;
ZEUS:2001mhd . These data sets will be used to determine valence $xu_{v}$ and
$xd_{v}$ PDFs, as well as $\alpha_{s}(M_{Z}^{2})$.

###  III.1 Valence PDF parametrization

This section presents the theoretical framework on which our PDF
parametrization is based. In the presence of only $xF_{3}$ DIS data, the
determination of the $xd_{v}(x,Q_{0}^{2})$ distribution relies on the
$xu_{v}(x,Q_{0}^{2})$ distribution Tooran:2019cfz ; Diemoz:1987xu ;
Gluck:1989ze ; Gluck:1998xa . Specifically, in the presence of $xF_{3}$ DIS
data, it is not possible to determine $xu_{v}(x,Q_{0}^{2})$ and
$xd_{v}(x,Q_{0}^{2})$ separately in a QCD non-singlet analysis. In this case,
the $xd_{v}(x,Q_{0}^{2})$ distribution depends on $xu_{v}(x,Q_{0}^{2})$
Diemoz:1987xu ; Gluck:1989ze ; Gluck:1998xa . For example, in pure $xF_{3}$
QCD analysis Khorramian:2006wg ; Tooran:2019cfz , it is assumed that
$xd_{v}(x,Q_{0}^{2})\propto(1-x)^{b_{d}}xu_{v}(x,Q_{0}^{2})$.

In the presence of only $F_{2}$ DIS data Khorramian:2008yh ; Blumlein:2006be ,
and also both $xF_{3}$ and $F_{2}$ DIS data, one can consider a separate
parametrization for $xu_{v}(x,Q_{0}^{2})$ and $xd_{v}(x,Q_{0}^{2})$. To
proceed with our previous works on $xF_{3}$ structure function
Khorramian:2006wg ; Tooran:2019cfz , we are interested in linking $u$ and
$d$-valence PDF parametrizations. To fulfill this purpose, we utilize the CJ
formalism, as mentioned in Accardi:2016qay ; Owens:2012bv , to provide greater
flexibility in determining the $d$-valence PDF, especially at large values of
$x$. With this modification, we gain more flexibility in determining the
valence $xd_{v}$ in the large $x$, which is important for further studies
related to this region in the rest of this paper.
Performing a non-singlet QCD analysis based on $xF_{3}$ DIS data and adding a
world data set for $F_{2}$ can significantly improve the precision of the
valence PDF determination. This approach allows us to present a detailed
parametrization of the valence PDF and its correlated errors over a wide range
of $x$ and $Q^{2}$.

The chosen $u$-valence PDF at the input scale of $Q_{0}^{2}$ is expressed as
follows

$\displaystyle xu_{v}(x,Q_{0}^{2})$ $\displaystyle=$
$\displaystyle{N_{u_{v}}}x^{a_{u}}(1-x)^{b_{u}}(1+c_{u}x+d_{u}\sqrt{x})\;,$
(6)

which in the parametrization provided above, we utilize the same
parametrization as employed in our previous analysis for $xu_{v}$
Khorramian:2006wg ; Tooran:2019cfz .

For the $d$-valence PDF, the parametrization is given by

$\displaystyle xd(x,Q_{0}^{2})$ $\displaystyle=$ $\displaystyle
N_{d_{v}}\left(x^{a_{d}}(1-x)^{b_{d}}(1+c_{d}x+d_{d}\sqrt{x})\right.$ (7)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\left.e_{d}x^{f_{d}}~{}xu_{v}(x,Q_{0}^{2})\right)~{}.$

As mentioned, we employ an alternative form for $d$-valence PDF by combining
$u$-valence PDF, following the approach outlined in Ref. Accardi:2016qay .
This combination permits enhanced flexibility within the large $x$ region. The
term $x^{a_{i}}$ control the behavior of valence PDFs in the low $x$ and
$(1-x)^{b_{i}}$ control large $x$ regions. In contrast, other polynomial terms
play a significant role in intermediate values of $x$. As a result of the
modification for $d$-valence PDF parametrization, we have
$xd(x,Q_{0}^{2})/xu(x,Q_{0}^{2})\rightarrow N_{d_{v}}e_{d}~{}$ in the limit of
$x\rightarrow 1$ if $b_{d}>b_{u}$.

Additionally, the normalization constants $N_{u}$ and $N_{d}$ can be derived
from the other parameters by applying the conservation of the fermion number,
which is expressed as

$\int_{0}^{1}u_{v}~{}dx=2\;,$ (8) $\int_{0}^{1}d_{v}~{}dx=1\;.$ (9)

Using above equations, the normalization constants $N_{u}$ and $N_{d}$ are as
follows

$\displaystyle N_{u}$ $\displaystyle=$ $\displaystyle
2/\big{(}(B(a_{u},1+b_{u})+c_{u}~{}B(1+a_{u},1+b_{u})$ (10) $\displaystyle+$
$\displaystyle d_{u}~{}B(1/2+a_{u},1+b_{u})\big{)},$ $\displaystyle N_{d}$
$\displaystyle=$ $\displaystyle
1/\bigg{(}B\left(a_{d},1+b_{d}\right)+c_{d}~{}B\left(1+a_{d},1+b_{d}\right)$
(11) $\displaystyle+$ $\displaystyle d_{d}~{}B\left(1/2+a_{d},1+b_{d}\right)$
$\displaystyle+$ $\displaystyle
N_{u}~{}{e_{d}}~{}\Big{(}B\left(a_{u}+{f_{d}},1+b_{u}\right)$ $\displaystyle+$
$\displaystyle c_{u}~{}B\left(1+a_{u}+{f_{d}},1+b_{u}\right)$ $\displaystyle+$
$\displaystyle
d_{u}~{}B\left(1/2+a_{u}+{f_{d}},1+b_{u}\right)\Big{)}\bigg{)}~{}.$

In the above parametrization, $B(a,b)$ represents the Euler $\beta$ function,
where the normalization constants $N_{u}$ and $N_{d}$ play a crucial role in
determining the unknown parameters through the QCD fitting procedure. They
effectively constrain the parameter space and ensure consistency with the
conservation of the fermion number.

According to the above parametrization, we may have ten free valence
parameters that can be determined through QCD fits. However, in the next
section, we will observe that specific parameters need to be fixed after the
initial minimization, because the available DIS data may not sufficiently
constrain some of the parameters in Eqs. (6) and (7). In particular, the
errors associated with some parameters can be relatively large compared to
their central values. To address this issue, it is common practice to fix
these parameters after the first minimization, as has been done in previous
non-singlet QCD analyses of $F_{2}(x,Q^{2})$. Similar procedures have been
previously conducted in Refs. Khorramian:2009xz ; Khorramian:2008yh ;
Blumlein:2006be . By fixing these parameters, we ensure stability and avoid
overfitting, considering the limitations of the available DIS data and the
associated uncertainties in parameter determination.

In general, the coupling constant $\alpha_{\rm s}(M_{Z}^{2})$ can be extracted
from global QCD fits to various hadronic processes. However, in this non-
singlet QCD analysis, the strong coupling constant at the scale of $M_{Z}^{2}$
is considered a free parameter and can be determined using DIS data. The
strong coupling constant plays a crucial role, exhibiting a significant
correlation with the PDFs. Since the determination of $\alpha_{\rm
s}(M_{Z}^{2})$ is connected to uncertainties in other non-singlet PDF
parameters, assessing the uncertainty associated with this parameter is
essential.

One way to evaluate this uncertainty is by comparing the fitted value of
$\alpha_{\rm s}(M_{Z}^{2})$ with the world average which reported as
${\alpha_{s}(M_{Z}^{2})}=0.1179\pm 0.0009$ in Ref. ParticleDataGroup:2022pth .
This comparison helps validate the consistency of the obtained value with the
experimental measurements and theoretical expectations for the strong coupling
constant $\alpha_{\rm s}(M_{Z}^{2})$ in the context of the non-singlet QCD
analysis.

In this work, we employed the QCDNUM evolution engine Botje:2010ay  to
determine the $Q^{2}$ evolution of PDFs and the coupling constant. QCDNUM is a
widely used tool in high-energy physics, known for performing precise
calculations related to the evolution of PDFs and the running of the strong
coupling constant. By utilizing QCDNUM in our analysis, we could accurately
account for the $Q^{2}$ dependence of the PDFs and the corresponding evolution
of the strong coupling constant within the framework of QCD.

###  III.2 Experimental data sets

In this section, we employ various experimental data sets for our analysis,
which include neutrino-nucleon DIS data from CCFR, NuTeV, CHORUS, and CDHSW
experiments, as well as charged electron (muon)-proton and deuteron DIS data
from BCDMS, NMC, SLAC, H1, and ZEUS experiments. These diverse data sets offer
valuable insights for determining and analyzing the parameters and
distributions. By incorporating four different data sets: $xF_{3}$,
$F_{2}^{p}$, $F_{2}^{d}$, and $F_{2}^{NS}$, a more comprehensive understanding
of the nucleon structure and the behavior of valence quarks can be obtained.
This expanded approach allows for a more accurate description of the
experimental data and enhances the precision of the extraction process for
valence quark densities and the strong coupling constant.

In this analysis, we include the DIS neutrino (antineutrino) $xF_{3}$
measurements from CCFR [$30\leq E_{\nu}$(GeV)$\leq 360$] with an iron target
ccfr:1977 , NuTeV [$30\leq E_{\nu}$(GeV)$\leq 500$] with an iron target
Tzanov:2005kr , CHORUS [$10\leq E_{\nu}$(GeV)$\leq 200$] with a lead target
Onengut:2005kv , and CDHSW [$20\leq E_{\nu}$(GeV)$\leq 212$] with an iron
target Berge:1989hr . Nevertheless, neutrino-nucleon $xF_{3}$ experiments have
employed targets with high atomic number (Z) and mass number (A), such as iron
or lead. Therefore, it is worth considering the nuclear corrections
deFlorian:2011fp  to have a good accuracy.

In addition, for the current QCD analysis, we include the structure function
data obtained from charged electron (muon)-proton and deuteron DIS
experimental data, such as BCDMS BCDMS:1989ggw ; BCDMS:1989qop ; BCDMS:1989gtb
, SLAC Whitlow:1991uw , NMC NewMuon:1996fwh , H1 H1:2000muc ; H1:2003xoe , and
ZEUS ZEUS:1998agx ; ZEUS:2001mhd . These experiments mentioned above provide a
reliable and accurate foundation for determining valence-quark densities and
$\alpha_{s}(M_{Z}^{2})$, thus enhancing the precision of our analysis. Also,
utilizing deuteron data in the high $x$ region requires nuclear correction,
which we will describe in the next section.

Therefore, all the data sets used in this analysis include $xF_{3}(x,Q^{2})$
data, $F_{2}^{p}(x,Q^{2})$, $F_{2}^{d}(x,Q^{2})$ in the valence quark region
$x\geq 0.3$, and finally $F^{NS}_{2}=2(F^{p}_{2}-F^{d}_{2})$ in the region
$x<0.3$. We approximate the PDFs by considering only pure valence quarks
within the valence quark region.

In this analysis, two different fits at the NNLO with the names of
$\mathtt{SK24[QCD]}$ and $\mathtt{SK24[QCD+TMC+HT]}$ are introduced below:

  * •

$\mathtt{SK24[QCD]}$: In the first fit, only data with $Q^{2}\geq 4\text{
GeV}^{2}$ are included, and a cut in the hadronic mass of $W^{2}\geq
12.5\text{ GeV}^{2}$ is applied to eliminate higher twist effects from the
data samples effectively.

  * •

$\mathtt{SK24[QCD+TMC+HT]}$: In the second fit, we apply the same cuts as in
$\mathtt{SK24[QCD]}$, with the exception that we impose $W^{2}\geq 4\text{
GeV}^{2}$. In this scenario, it is necessary to consider target mass
corrections and higher-twist effects.

As we will see, the imposition of cuts on $W^{2}$ in the datasets used in this
analysis substantially enhances the number of available data points,
contributing to a more effective constraint on valence PDFs. In both fits,
some unknown parameters exist for valence PDFs and $\alpha_{s}(M^{2}_{Z})$.
The MINUIT James:1975dr  program is employed for the minimization in the
fitting procedure.

After applying $Q^{2}\geq 4\text{ GeV}^{2}$ and $W^{2}\geq 12.5\text{
GeV}^{2}$ cuts, we are left with 1045 data points: 283 data points for
$xF_{3}$, 322 data points for $F^{p}_{2}$, 232 data points for $F^{d}_{2}$,
and 208 data points for $F^{NS}_{2}$. We should note that three additional
cuts need to be applied for the neutrino-nucleon and lepton-nucleon DIS data.
A $x>0.4$ cut is necessary for CCFR data due to a discrepancy between CCFR and
NuTeV in this region Tooran:2019cfz . Additionally, to exclude a region with
significant correlated systematic errors, a $y_{\mu}>0.3$ cut is necessary for
the BCDMS data Blumlein:1992we . Also, the NMC data requires an extra cut of
$Q^{2}\geq 8\ \text{GeV}^{2}$. For more information on these additional cuts,
please see Refs. Khorramian:2006wg ; Tooran:2019cfz ; Blumlein:1992we . After
these additional cuts, the total number of DIS data points has been reduced
from 1045 to 814. This includes 263 data points for $xF_{3}$, 227 data points
for $F^{p}_{2}$, 159 data points for $F^{d}_{2}$, and 165 data points for
$F^{NS}_{2}$. By considering $W^{2}\geq 4\text{ GeV}^{2}$ on all data, the
total increases to 1395, including 287 data points for $xF_{3}$, 506 data
points for $F^{p}_{2}$, 437 data points for $F^{d}_{2}$, and 165 data points
for $F^{NS}_{2}$.

Table 1 presents different combinations of subsets of non-singlet $xF_{3}$ and
$F_{2}$ data, along with their corresponding $x$ and $Q^{2}$ ranges, used in
the current analyses. The fourth and fifth columns show the number of
individual data points for each data set with $W^{2}\geq$ 12.5 GeV2 and
$W^{2}\geq$ 4 GeV2 cuts applied to the data. The sixth and seventh columns
($\mathtt{SK24[QCD]}$) and the eighth and ninth ($\mathtt{SK24[QCD+TMC+HT]}$)
display the $\chi^{2}$ and $\chi^{2}/$#Data values for each fit applied on
corresponding cuts. The $\chi^{2}$ and total $\chi^{2}/d.o.f.$ are also shown
in this table.

In Figure 3, we present the experimental data utilized in this analysis,
including neutrino-nucleon DIS data extracted from CCFR, NuTeV, CHORUS, and
CDHSW experiments, as well as the electron (muon)-proton and deuteron DIS
experimental data obtained from BCDMS, NMC, SLAC, H1, and ZEUS experiments,
plotted in the $x$ and $Q^{2}$ plane. The data points are selected by imposing
cuts on $W^{2}$ ($W^{2}\geq$ 12.5 GeV2 and $W^{2}\geq$ 4 GeV2) and $Q^{2}\geq$
4 GeV2. As previously mentioned, a discrepancy exists between NuTeV and CCFR
at $x>0.4$, and consequently, the CCFR data in this region is excluded from
the analysis. Please see Ref. Tooran:2019cfz  for more information and
details. As seen from Figure 3, a significant number of data points for both
$F_{2}^{p}$ and $F_{2}^{d}$ are found within the range of $4\leq W^{2}\leq
12.5,\text{GeV}^{2}$. However, there is no available data for $F_{2}^{NS}$
within this specific region.

In the upcoming section, we will introduce corrections related to the data
selection discussed earlier and the theories explained in Sec. II.

##  IV QCD Corrections

In this section, our main focus is on QCD corrections, specifically
emphasizing nuclear effects, target mass corrections, and higher twist
effects. Through a detailed investigation of these components, we aim to study
their impact on the present non-singlet QCD analysis of DIS structure function
data.

###  IV.1 Deuteron nuclear correction

Implementing deuteron nuclear corrections becomes challenging in the high $x$
region due to the essential requirement for deuteron wave function (DWF) and
nuclear off-shell function (OSF), making it impossible to neglect this effect
Alekhin:2017fpf . In the present paper, due to the differences between the
PDFs in deuterons and free nucleons, and also the use of deuteron DIS data in
very high $x$ region ($x\gtrsim 0.7$), we are required to implement deuteron
nuclear correction concerning large $x$ Accardi:2016qay ; Kovarik:2015cma .
Therefore, we follow the approach investigated by S. A. Kulagin and R. Petti
in Ref. Kulagin:2004ie . In this instruction, the deuteron structure function
becomes Alekhin:2017fpf ; Kulagin:2004ie

$\displaystyle\gamma^{2}F_{2}^{d}(x,Q^{2})=\int\frac{\mathrm{d}^{3}\bm{p}}{(2\pi)^{3}}\left|\Psi_{D}(\bm{p})\right|^{2}$
$\displaystyle\left(1+\frac{\gamma
p_{z}}{M}\right)\left({\gamma^{\prime}}^{2}+\frac{6{x^{\prime}}^{2}\bm{p}_{\perp}^{2}}{Q^{2}}\right)F_{2}^{N}(x^{\prime},Q^{2},p^{2})~{}.$
(12)

In the above equation $\gamma^{2}=1+4x^{2}M^{2}/Q^{2}$ and
${\gamma^{\prime}}^{2}=1+4{x^{\prime}}^{2}p^{2}/Q^{2}$ and also
$\Psi_{D}(\bm{p})$ is a deuteron wave function which is depends on the
momentum of bound nucleon $\bm{p}=\left(\bm{p}_{\perp},p_{z}\right)$ and can
archive from various analyses such as AV18 Veerasamy:2011ak , WJC1
Gross:2008ps  and WJC2 Gross:2010qm .

Mass of bound nucleon can be written as $M=\frac{1}{2}(M_{p}+M_{n})$ and
Bjorken scaling as $x^{\prime}=x/\left[1+\left(\varepsilon+\gamma
p_{z}\right)/M\right]$, in this relation
$\varepsilon=\varepsilon_{D}-\bm{p}^{2}/(2M)$ where deuteron binding energy
define as $\varepsilon_{D}=M_{D}-2M$.
$F_{2}^{\smash{N}}=\tfrac{1}{2}(F_{2}^{p}+F_{2}^{n})$ is the structure
function of the bound nucleon which its $p^{2}$-dependence comes from off-
shell dependence of the LT structure function

$\displaystyle F_{2}^{\mathrm{LT}}\left(x,Q^{2},p^{2}\right)$
$\displaystyle=F_{2}^{\mathrm{LT}}\left(x,Q^{2}\right)\left[1+\delta
f\left(x,Q^{2}\right)v\right]~{},$ (13)

here $v=\left(p^{2}-M^{2}\right)/M^{2}$ is nucleon virtuality. For the off-
shell effect $\delta f(x)$ we consider a phenomenological model which is taken
by Refs. Accardi:2016qay ; Kulagin:2004ie

$\displaystyle\delta
f^{N}=C_{N}\left(x-x_{0}\right)\left(x-x_{1}\right)\left(1+x_{0}-x\right)~{},$
(14)

where $C_{N}$ and $x_{0}$ are free parameters participating in fit procedure
and $x_{1}$ claculated analytically to conserve number of valence quarks. For
more details of this instruction, please see Refs. Kulagin:2004ie ;
Alekhin:2017fpf , and also for the application of this instruction at the
partonic level, please look at Ref. Accardi:2016qay
In the present paper, we consider WJC2 Gross:2010qm  for the deuteron wave
function, and we follow the same treatment for the nuclei structure functions
as our previous analysis in Ref. Tooran:2019cfz .

###  IV.2 Target mass corrections

In our non-singlet analysis, another correction that needs to be included is
TMCs. The effects of target masses were first discussed in the context of
operator-product expansion (OPE) in Ref. Georgi:1976ve . Indeed, to ensure the
desired accuracy of theory calculations, it becomes essential to account TMCs
in the present analyses of non-singlet structure functions. One crucial aspect
that must be considered is the target mass correction in twist $\tau=2$
structure functions for nuclei. Based on the data sets mentioned in the
previous section and the wide range of kinematics covered in the present
analysis, it is necessary to consider the impact of TMCs on unpolarized proton
and nuclear targets. Very recently, a detailed derivation of the formalism of
TMCs to structure functions in DIS has been presented in Ref. Ruiz:2023ozv .

An accurate implication of TMCs, which is subleading $1/Q^{2}$ correction on
structure function in the leading twist approximation, is vital to reliable
PDFs determination and also comprehensive data interpretation. The impact of
this correction is considerable in large $x$ and moderate $Q^{2}$ regions.
Higher values of $Q^{2}$ decrease the effect of TMCs due to elimination of the
mass term in Eq. (15) and subsequently, $\xi_{A}\rightarrow x_{A}$ in Eq.
(16). In terms of this notation, $x_{A}$ is Bjorken scaling variable, also
Nachtmann variable $\xi_{A}$ and quantity of $r_{A}$ define as

$r_{A}\ =\ \sqrt{1+4x_{A}^{2}M_{A}^{2}/Q^{2}}\,,$ (15) $\xi_{A}\ =\
\frac{2x_{A}}{1+r_{A}}\ \equiv
R_{M}x_{A},~{}~{}~{}~{}R_{M}=\frac{2}{1+r_{A}}~{}.$ (16)

In the above, $R_{M}$ denotes the factor dependent on the target mass,
connecting the Bjorken scaling variable $x_{A}$ to the Nachtmann scaling
variable $\xi_{A}$.

According to the master formulas for twist $\tau=2$ target mass corrections to
structure functions for nuclei, we have the $Q^{2}$ dependence of TMCs to
structure functions as

$\displaystyle{F}_{2}^{A,TMC}(x_{A})$ $\displaystyle=$
$\displaystyle\left(\frac{x_{A}^{2}}{\xi_{A}^{2}r_{A}^{3}}\right){F}_{2}^{A,(0)}(\xi_{A})$
(17)
$\displaystyle+\left(\frac{6M_{A}^{2}x_{A}^{3}}{Q^{2}r_{A}^{4}}\right)\,{h}_{2}^{A}(\xi_{A})$
$\displaystyle+\left(\frac{12M_{A}^{4}x_{A}^{4}}{Q^{4}r_{A}^{5}}\right)\,{g}_{2}^{A}(\xi_{A})\,,$
$\displaystyle x_{A}{F}_{3}^{A,TMC}(x_{A})$ $\displaystyle=$
$\displaystyle\left(\frac{x_{A}^{2}}{\xi_{A}r_{A}^{2}}\right){F}_{3}^{A,(0)}(\xi_{A})$
(18)
$\displaystyle+\left(\frac{2M_{A}^{2}x_{A}^{3}}{Q^{2}r_{A}^{3}}\right)\,{h}_{3}^{A}(\xi_{A})\,.$

Here, the $Q^{2}$ dependence of the structure functions is omitted for
brevity. In the present analysis, $A=\\{1,2,56,197\\}$ stand for mass number
related to $F_{2}^{p,NS}$, $F_{2}^{d}$, $xF_{3}^{Fe}$ and $xF_{3}^{Pb}$,
respectively. Also, ${F}_{2,3}^{A,(0)}$ are massless nuclei structure
functions when $M^{2}/Q^{2}\rightarrow 0$ and one can find coefficients of
this structure functions, ${h}_{2,3}^{A}$ and ${g}_{2}^{A}$ in Refs.
Kretzer:2003iu .

In Eqs. (17, 18), the functions ${h}_{2}^{A}(\xi_{A})$, ${g}_{2}^{A}(\xi_{A})$
and ${h}_{3}^{A}(\xi_{A})$ are expressed as integrals

$\displaystyle{h}_{2}^{A}(\xi_{A},Q^{2})$
$\displaystyle=\int_{\xi_{A}}^{1}du_{A}\
\frac{{F}_{2}^{A,(0)}(u_{A},Q^{2})}{u_{A}^{2}},$ (19)
$\displaystyle{g}_{2}^{A}(\xi_{A},Q^{2})$
$\displaystyle=\int_{\xi_{A}}^{1}du_{A}\ {h}_{2}^{A}(u_{A},Q^{2})~{},$ (20)
$\displaystyle{h}_{3}^{A}(\xi_{A},Q^{2})$
$\displaystyle=\int_{\xi_{A}}^{1}du_{A}\
\frac{{F}_{3}^{A,(0)}(u_{A},Q^{2})}{u_{A}}~{}.$ (21)

By incorporating Eqs. (17, 18) and also (19 $\\!\\!-\\!\\!$ 21), we can
consider the TMC effects at twist $\tau=2$ structure functions for nucleons
and nuclei in the current analysis.

The accuracy of the calculation of the TMCs has a significant effect on our
further study of higher twist correction, this impact can be seen by the
values and uncertainties of its parameters. For reliable implication and
increasing precision of TMCs calculation like integration and interpolation,
we used the MBUTIL package released by QCDNUM Botje:2010ay . This FORTRAN
package involves public libraries CERNLIB and NETLIB and some privately
developed routines.

In the following subsection, we discuss higher twists and their application to
target mass corrections, involving them as free parameters directly in the
fit.

###  IV.3 Higher twist effects

The final correction considered in the present analysis involves HT effects.
These effects significantly contribute to improving the accuracy of the
present analysis. By considering HT effects, a more complete and detailed
understanding of the data can be achieved. Continuing with the analysis, we
further investigate the influence of HT effects on the current QCD analysis of
the $xF_{3}$ and $F_{2}$ structure function measurements. By explicitly
accounting HT effects, we aim to gain insights into the additional
nonperturbative contributions to the structure functions.

In the standard analysis of DIS structure function data, it is essential to
incorporate appropriate cuts on the invariant-mass squared
$W^{2}=Q^{2}(1/x-1)+m_{N}^{2}$ and the virtual photon $Q^{2}$ at the NNLO.
Choosing an appropriate $W^{2}$ cut value for the structure function data is
crucial to disregard the influence of nonperturbative effects. By applying
these cuts, the analysis can focus primarily on the perturbative QCD aspects
of the data.

To address the HT effects in the data, we initially applied standard cuts on
$Q^{2}$ and invariant-mass squared $W^{2}$. Specifically, we implemented cuts
$Q^{2}\geq 4$ GeV2 and $W^{2}\geq$12.5 GeV2 to exclude the influence of HT
effects from the data. By employing these cuts, our objective was to minimize
nonperturbative contributions and concentrate primarily on the perturbative
QCD aspects of the analysis. Following the application of these cuts, we
proceeded with the extraction of the unknown parameters through QCD fits on
the data. Through this fit, we aimed to determine the best-fit values of the
parameters and gain a more precise understanding of the QCD dynamics
underlying the structure function measurements.

In the subsequent step, we investigated the impact of the HT contribution by
utilizing all available data within the $Q^{2}\geq 4$ GeV2 and $W^{2}\geq 4$
GeV2 region. This approach allowed us to incorporate data spanning the DIS
region into our QCD fits, enabling a comprehensive analysis of the HT effects
on the structure function measurements. In this context, notable efforts have
been made in previous works such as Refs. Kataev:1997vv ; Blumlein:2008kz ;
Sidorov:1996wb ; Sidorov:1996if ; Virchaux:1991jc  to address the HT
contributions and investigate their implications.

To account for the HT contribution, the average of DIS structure functions can
be expressed as a combination of the perturbative and nonperturbative
contributions

$\displaystyle
F_{i}(x,Q^{2})=F_{i}{\mathtt{[QCD+TMC]}}(x,Q^{2})\left(1+\frac{C^{i}_{HT}(x)}{Q^{2}}\right)~{}.$
(22)

Here, the $Q^{2}$ dependence of the first term is obtained through
perturbative QCD and TMC, and the coefficient function for higher twists is
parameterized by a polynomial function, expressed as Accardi:2016qay ;
Owens:2012bv

$\displaystyle C^{i}_{HT}(x)=h_{i1}x^{h_{i2}}(1+h_{i3}x)~{},$ (23)

where $i=1,2,3$ present the HT effects for the $F_{2}^{p}$, $F_{2}^{d}$, and
$xF_{3}$ structure functions, respectively. Through a comprehensive fitting
procedure using experimental data, the unknown parameters of $h_{ij}$ with
$j=1,2,3$ and their corresponding uncertainties for the function
$C^{i}_{HT}(x)$ can be simultaneously extracted along with other unknown
parameters, including those associated with the valence PDFs and the strong
coupling constant.

As we discussed, the fitting process allows us to incorporate the
modifications arising from nuclear effects, target mass corrections, and
higher twist effects in our analysis. These modifications are essential for
achieving a more accurate description of the data.

##  V Results

A comprehensive non-singlet QCD analysis is presented in this work, focusing
on the structure functions of both neutrino-nucleon and electron-nucleon deep
inelastic scattering. The analysis encompasses the charged current neutrino-
nucleon DIS data, corresponding NNLO approximations, accounting for nuclear
and higher twist corrections. Therefore, the world data for lepton scattering
is employed for the NNLO analysis of the DIS structure functions
$xF_{3}(x,Q^{2})$, $F_{2}^{p}(x,Q^{2})$, $F_{2}^{d}(x,Q^{2})$, and
$F_{2}^{NS}(x,Q^{2})$. This analysis is performed to determine the parameters
of the valence PDFs together $\alpha_{s}(M_{Z}^{2})$ with the correlated
errors.

This analysis performs two separate fits, considering the cut studies in both
$Q^{2}$ and $W^{2}$. Remarkably, it has been observed that power corrections
are essentially negligible in the kinematic region Q${}^{2}\geq 4$ GeV2 and
$W^{2}>12.5$ GeV2. Consequently, we have extended these cuts to the entire
data set, considering HT corrections. Additionally, we have considered the
TMCs by unfolding them for all the data and nuclear effects for lepton-nucleon
structure functions.

Figures 4 $\\!\\!-\\!\\!$ 8 display the comparison of $xF_{3}(x,Q^{2})$ and
$F_{2}(x,Q^{2})$ structure functions with and without TMC and HT obtained from
the different fits at the NNLO as a function of $Q^{2}$ in various $x$ in the
valence quark region. The vertical dashed line indicates the regions with W2
$\geq$ 12.5 GeV2.

By conducting this analysis, we extract the valence $xu_{v}$ and $xd_{v}$
PDFs, within a wide range of $x$ and $Q^{2}$. The parametrization of these
PDFs is accompanied by their respective correlated errors, providing a
comprehensive understanding of their uncertainties. Table 2 presents a summary
of the fit results for the parameters of $xu_{v}(x,Q_{0}^{2})$ and
$xd_{v}(x,Q_{0}^{2})$ PDFs, and $\alpha_{s}(M_{Z}^{2})$ at the NNLO. Also in
our analysis, we determine off-shell function $\delta f(x)$,
phenomenologically, from Eq. 14 during the fit process. Our result for off-
shell function parameters satisfies the constraints $0<x_{1}<x_{0}<1$ and
$(1+x_{0})>1$ as mentioned in Ref. Kulagin:2004ie , and confirms the result of
this analysis in the shape of the off-shell function. Additionally, the higher
twist parameter values and their uncertainties have been reported in this
table.

In all fits, the covariance matrix is verified to be positive definite. Table
3 represents the values of the covariance matrix for $\mathtt{SK24[QCD]}$ fit,
as an example, corresponding to each free parameter. The previous non-singlet
analyses in the presence of only $F_{2}$ DIS data Blumlein:2006be ;
Khorramian:2008yh  used two free parameters for each valence PDF, and also, in
our previous research in the presence of only $xF_{3}$ DIS data Tooran:2019cfz
; Khorramian:2006wg  just one free parameter used for $xd_{v}$ distribution.
In the present analysis, due to the combination of $xF_{3}$ and $F_{2}$ DIS
data and also the parametrization formalism mentioned in Eqs. (6, 7), we gain
more freedom in valence PDF distributions by using five and six free
parameters for SK24[QCD] and SK24[QCD + TMC + HT] fit procedures,
respectively. Additional flexibility is achieved from the modified form of
$xd_{v}(x,Q^{2})$ at the input scale of $Q_{0}^{2}$.

Due to the effect of deuteron nuclear correction on $xd_{v}$, especially in
large $x$ region, we consider different treatments regarding the form of
parameterization in $\mathtt{SK24[QCD]}$ and $\mathtt{SK24[QCD+TMC+HT]}$. In
the case of $\mathtt{SK24[QCD]}$, the parameter $d_{u}$ is considered free in
the first minimization while $e_{d}$ and $f_{d}$ are fixed on zero. That means
we consider separate parameterization formalism for each valence PDF in
$\mathtt{SK24[QCD]}$ fit procedure. Otherwise, a rise in the $xd_{v}$ PDF
shape in large $x$ region ($x\gtrsim 0.7$) will occur. On the other hand, in
the case of $\mathtt{SK24[QCD+TMC+HT]}$, selecting $d_{u}$ as a free parameter
results in a noticeable rise in the $xd_{v}$ PDF shape in the intermediate $x$
region, influenced by the effective area of this parameter and the dependency
of $xd_{v}$ on $xu_{v}$ in the parametrization space. We choose $d_{u}=0$ in
this fit to maintain consistency, like CJ PDF analysis Accardi:2016qay ;
Owens:2012bv .

We determined the strong coupling constant, $\alpha_{s}(M_{Z}^{2})$, at the
NNLO fits as $0.1154\pm 0.0009$ and $0.1149\pm 0.0014$ in the case of
$\mathtt{SK24[QCD]}$ and $\mathtt{SK24[QCD+TMC+HT]}$ fits, respectively. The
world and DIS averages for ${\alpha_{s}(M_{Z}^{2})}$ is reported as
${\alpha_{s}(M_{Z}^{2})}=0.1179\pm 0.0009$ and
${\alpha_{s}(M_{Z}^{2})}=0.1162\pm 0.0020$ in Ref. ParticleDataGroup:2022pth .

Furthermore, we compare our valence-quark densities results and their
uncertainties with predictions obtained using alternative PDF sets from
different research groups. Figure 9 displays the valence $xu_{v}$ and $xd_{v}$
PDF comparison between present $\mathtt{SK24[QCD]}$ and
$\mathtt{SK24[QCD+TMC+HT]}$ analyses, respectively, at Q2 = 4, 5, 10, 100,
1000 and 5000 GeV2. Also, the middle and right panels of Figure 9 show the
relative uncertainties $\delta xu_{v}(x,Q^{2})/xu_{v}(x,Q^{2})$ and $\delta
xd_{v}(x,Q^{2})/xd_{v}(x,Q^{2})$, respectively.

Figure 10 illustrate our fit results to present the evolution of the valence
quark densities $xu_{v}(x,Q^{2})$ and also $xd_{v}(x,Q^{2})$ from $Q^{2}=4\
\mathrm{GeV}^{2}$ to $Q^{2}=5000\ \mathrm{GeV}^{2}$ in the region
$x\in[0.0001,1]$ at NNLO with correlated errors. Error propagation is carried
out using the evolution equations to calculate the associated error bands.
First and third columns of these figures compare our results with other NNLO
analyses like the results of CT18 Hou:2019qau  and very recent NNPDF4.0
NNPDF:2021njg  and MSHT20 Bailey:2020ooq  analysis. Note that, in the
extracted results, we compare combined non-singlet/singlet analyses, and
deviations for valence PDFs in smaller values of $x$ may arise due to
different assumptions regarding the sea PDFs. Also, as the value of $Q^{2}$
increases, the distributions tend to flatten at large values of $x$ and rise
at low values. Notably, these illustrations include two ratios for comparison
with other models. The second and fourth columns of Figure 10 depict log plots
for the ratios of $xu_{v}(x,Q^{2})/xu_{v}(x,Q^{2}){\text{ref}}$ and
$xd_{v}(x,Q^{2})/xd_{v}(x,Q^{2}){\text{ref}}$, respectively, with respect to
$\mathtt{SK24[QCD]}$.

Figures 11 and 12 display linear plots for relative uncertainties $\delta
xu_{v}(x,Q^{2})/xu_{v}(x,Q^{2})$ and $\delta xd_{v}(x,Q^{2})/xd_{v}(x,Q^{2})$,
respectively for both $\mathtt{SK24[QCD]}$ and $\mathtt{SK24[QCD+TMC+HT]}$
fit. In these figures, we also compare our results with other NNLO analyses,
such as the results of CT18 Hou:2019qau , as well as very recent NNPDF4.0
NNPDF:2021njg  and MSHT20 Bailey:2020ooq  analyses.

Figure 13 present the comparison of our extracted value of
$\alpha_{s}(M_{Z}^{2})$, at the NNLO with other results obtained from A02
Alekhin:2002fv , A06 Alekhin:2006zm , MRST03 Martin:2003tt , ABMP16
Alekhin:2017kpj ; Alekhin:2018pai , BBG06 Blumlein:2006be , JR14 Jimenez-
Delgado:2014twa , ABKM09 Alekhin:2009ni , ABM11 Alekhin:2012ig , KKT
Khorramian:2009xz , CT18 Hou:2019qau , MSHT20 Bailey:2020ooq , NNPDF3.1
Ball:2018iqk , NNPDF4.0 NNPDF:2021njg , AKP22 Azizi:2022nqm , GKA
Tooran:2019cfz , and ZEUS ZEUS:2023zie . Furthermore, the world and DIS
average values of the strong coupling constant $\alpha_{s}(M_{Z}^{2})$
ParticleDataGroup:2022pth  are shown. There are potential sources for
variation of the coupling constant value $\alpha_{s}(M_{Z}^{2})$. Generally,
for this issue, three cases can be mentioned: (a) the theoretical models used
in the analyses, such as different treatments for non-perturbative structure
function, (b) variation and accuracy of used data, and (c) methodology for
implementation of the analysis such as using $x$-space, Mellin-space or usage
of the neutral network for fitting procedure. For applying the matching of
flavor thresholds at $Q^{2}=m_{c}^{2}$ and $Q^{2}=m_{b}^{2}$, we chose
$m_{c}=1.43\,\text{GeV}$ and $m_{b}=4.5\,\text{GeV}$. It is important to note
that other results may choose different values for $m_{c}$ and $m_{b}$.

It is valuable to examine the influence of individual $xF_{3}$ data sets on
$\alpha_{s}(M_{Z}^{2})$, identifying one of the sources for variation of this
quantity in this analysis comparing to others. To address this aim, we first
exclude all four sets of $xF_{3}$ data. Subsequently, we apply the
$\mathtt{SK24[QCD]}$ fitting procedure only on $F_{2}$ data and determine
$\alpha_{s}(M_{Z}^{2})$ as $0.1142\pm 0.0009$. This value is consistent with
previous analyses of the $F_{2}$ structure function found in Refs.
Khorramian:2008yh ; Blumlein:2006be . Expanding this investigation, we
reintroduce each set of $xF_{3}$ data individually. This yields
$\alpha_{s}(M_{Z}^{2})$ values ranging from 0.1152 to 0.1157 for all four
possible combinations. Consequently, we conclude that the inclusion of such
$xF_{3}$ data into the $F_{2}$ structure function dataset leads to an increase
in the $\alpha_{s}(M_{Z}^{2})$ value presented in this analysis compared to
our previous study using Bernstein polynomials in Ref. Khorramian:2006wg .

Another method for comparing our NNLO fit results involves calculating moments
of $xu_{v}(x,Q^{2})$ and $xd_{v}(x,Q^{2})$ distributions, as well as moments
of $xu_{v}(x,Q^{2})-xd_{v}(x,Q^{2})$. The moments of $xq_{v}$ valence PDFs at
the scale of $Q^{2}$ are as follows:

$<x^{n-1}>_{q_{v}}(Q^{2})=\int_{0}^{1}dx~{}x^{n-2}~{}xq_{v}(x,Q^{2})~{},$ (24)

where $n=1,2,3,...$ corresponds to the first, second, third, etc., Mellin
moments, respectively, equivalent to the zeroth, first, second, etc., $x$
moments, and $q_{v}=u_{v},d_{v},u_{v}-d_{v}$. The first Mellin moments are
consistent with the quark number sum rules.

In Table 4, we provide the lowest order Mellin moments ($n=2,3,4$) of these
valence PDFs at $Q^{2}=Q_{0}^{2}=4$ GeV2 in NNLO and compare them to the
corresponding moments obtained from non-singlet GKA19 Tooran:2019cfz , KT08
Khorramian:2008yh , KT07 Khorramian:2006wg , BBG06 Blumlein:2006be  and
combined non-singlet/singlet MSHT20 Bailey:2020ooq , NNPDF4.0 NNPDF:2021njg
and CT18 Hou:2019qau  global fits analysis. The $\mathtt{SK24[QCD]}$ moment
results of the $xu_{v}$ PDF demonstrate a closer agreement with BBG06
Blumlein:2006be , as well as our KT08 analysis Khorramian:2008yh  based on the
Jacobi polynomial approach. In the case of $xd_{v}$ moments, our results are
closer to NNPDF4.0 NNPDF:2021njg , and GKA19 Tooran:2019cfz . The
$\mathtt{SK24[QCD+TMC+HT]}$ moments of the $xu_{v}$ PDF shows well
compatibility with BBG06 Blumlein:2006be  and KT08 Khorramian:2008yh , also,
$xd_{v}$ PDF moments illustrate close agreement with MSHT20 Bailey:2020ooq
and KT07 Khorramian:2006wg . There are still some deviations and, especially
in the case of $xd_{v}(x,Q^{2})$, attributed to variations in data sets,
parametrization, high $x$ deuteron nuclear effect, and combinations of non-
singlet/singlet QCD analyses.

Impact of the coefficient function for higher twists $C^{i}_{HT}(x)$ [GeV2]
from Eq. (23) on large $x$ region for the $F_{2}^{p}$, $F_{2}^{d}$, and
$xF_{3}$ structure functions corresponding to NNLO fit are presented in Figure
14. The higher-twist coefficient function $C^{i}_{HT}(x)$ exhibits growth as
$x$ increases. This growth pattern is consistent with the results of the
earlier analysis Blumlein:2006be ; Virchaux:1991jc .

The effects of the TMC and HT corrections are illustrated in Figure 15, where
the ratios of the $xF_{3}$ and $F_{2}$ structure functions with and without
TMC and HT effects are shown, namely $xF_{3}/xF_{3}[\mathtt{QCD}]$,
$F_{2}^{p}/F_{2}^{p}[\mathtt{QCD}]$, and $F_{2}^{d}/F_{2}^{d}[\mathtt{QCD}]$
for Q2 = 4, 10, 25 and 50 GeV2. As seen in this figure, the ratios of
structure functions considering TMC and HT, compared to those without these
corrections, exceed unity at higher values of $x$. Additionally, with an
increase in the value of $Q^{2}$, the ratio approaches unity. These behaviors
are entirely consistent with the theoretical expectations of TMC and HT
corrections.

##  VI Conclusions

We performed a new QCD analysis of the non-singlet world data to extract the
valence $xu_{v}$ and $xd_{v}$ PDFs, incorporating correlated errors. The
article describes a non-singlet QCD analysis using the QCDNUM Botje:2010ay
open-source framework. Integrating nuclear corrections, TMCs, and HT effects
during the fit procedure becomes essential to account for diverse data sets.
We have combined these adjustments into our codes, which are not part of the
main QCDNUM package.

In addition to parameterizing the valence PDFs, the non-singlet structure
function requires the incorporation of the distribution
$x(\bar{d}-\bar{u})(x,Q^{2})$ as an input. Although this parametrization has a
minor impact on our analysis, it remains crucial to evaluate the influence of
this distribution by taking various asymmetrical sea quark distributions
derived from other analyses as input. We observed no significant impact when
selecting different asymmetry distributions from other studies. Therefore, we
used $x(\bar{d}-\bar{u})(x,Q^{2})$ from the MSHT20 PDF set due to its novelty
and consistency with experimental data.

As mentioned, this analysis is free of assumptions on the gluon to well
understanding behaviors of valence PDFs. We investigated the impact of the
gluon PDF on all structure functions in our QCD analysis. Our findings
indicate that including the gluon contribution does not significantly impact
our results.

Due to different kinematic cuts to the data sets, we applied target mass
corrections and higher twist effects. We found that by choosing the same
functional form for the $xu_{v}(x,Q^{2})$ and $xd_{v}(x,Q^{2})$ PDFs, a
considerable increase is observed in the uncertainty of the coefficient
function for higher twists, $C^{i}_{HT}(x)$, particularly at large values of
$x$ for deuteron structure function. So, we considered an alternative form for
$d$-valence PDF by combining $u$-valence PDF in a broad range of $x$ and
$Q^{2}$.

In this analysis the higher twist coefficient $C_{HT}(x)$ for $F_{2}^{p}$,
$F_{2}^{d}$, and $xF_{3}$ is extracted in the region where $W^{2}\geq$ 4 GeV2.
Finally, we determine the strong coupling constant $\alpha_{s}(M_{Z}^{2})$ for
our different fits. The obtained results are in good agreement with the world
average value and the previous results from different NNLO unpolarized and
polarized DIS analyses for $\alpha_{s}(M_{Z}^{2})$.

The NNLO grid output for $\mathtt{SK24[QCD]}$ and $\mathtt{SK24[QCD+TMC+HT]}$
analyses are available by contacting the authors via email.

##  VII Acknowledgments

The authors wish to thank the referees for their valuable comments and
suggestions, which have significantly enhanced the quality of this paper. We
appreciate F. Olness for helpful discussion and valuable comments and
suggestions. We acknowledge A. Shabanpour for their guidance on computational
programs. We thank F. Gross and W. N. Polyzou for providing the deuteron wave
function and their insights during implementation. A.K. is also grateful to
the CERN TH-PH division for the hospitality, where a portion of this work was
performed.

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|  |  |  #Data | $\mathtt{SK24[QCD]}$ | $\mathtt{SK24[QCD+TMC+HT]}$
---|---|---|---|---|---
Data set | $x$ |  $Q^{2}$(GeV2) |  $W^{2}\geq$ 12.5 GeV2 |  $W^{2}\geq$ 4 GeV2 | $\chi^{2}$ |  $\chi^{2}/$#Data | $\chi^{2}$ |  $\chi^{2}/$#Data
$\bf xF_{3}$ |  |  |  |  |  |  |  |
CCFR | 0.0125 - 0.35 | 5.00 - 125.9 | 67(87-20) | 67 | 22.614 | 0.337 | 26.995 | 0.402
CDHSW | 0.0150 - 0.65 | 4.53 - 196.3 | 96 | 107 | 58.750 | 0.611 | 39.246 | 0.366
NuTeV | 0.0150 - 0.75 | 5.01 - 125.89 | 59 | 65 | 85.943 | 1.456 | 64.141 | 0.986
CHORUS | 0.0450 - 0.65 | 5.14 - 81.55 | 41 | 48 | 48.443 | 1.181 | 19.651 | 0.409
F$\bf{}_{2}^{p}$ |  |  |  |  |  |  |  |
BCDMS100 | 0.35 - 0.75 | 11.75 - 75 | 29 | 39 | 36.045 | 1.242 | 60.403 | 1.548
BCDMS120 | 0.35 - 0.75 | 13.25 - 57 | 32 | 36 | 50.913 | 1.591 | 49.821 | 1.383
BCDMS200 | 0.35 - 0.75 | 32.50 - 99 | 28 | 28 | 63.344 | 2.262 | 37.020 | 1.322
BCDMS280 | 0.35 - 0.75 | 43.00 - 115.5 | 26 | 26 | 29.949 | 1.151 | 23.490 | 0.903
NMC | 0.35 - 0.50 | 4.500 - 65.0 | 14 | 20 | 10.123 | 0.723 | 13.007 | 0.650
SLAC | 0.30 - 0.90 | 4.000 - 31.31 | 57 | 316 | 43.981 | 0.771 | 219.19 | 0.693
H1 | 0.40 - 0.65 | 200.0 - 20000 | 26 | 26 | 37.372 | 1.437 | 36.770 | 1.414
ZEUS | 0.40 - 0.65 | 650.0 - 30000 | 15 | 15 | 2.8309 | 0.188 | 2.7382 | 0.182
F$\bf{}_{2}^{d}$ |  |  |  |  |  |  |  |
BCDMS120 | 0.35 - 0.75 | 13.25 - 57 | 32 | 36 | 55.084 | 1.721 | 70.657 | 1.962
BCDMS200 | 0.35 - 0.75 | 32.50 - 99 | 28 | 28 | 41.659 | 1.487 | 26.418 | 0.943
BCDMS280 | 0.35 - 0.75 | 43.00 - 115.5 | 26 | 26 | 28.072 | 1.079 | 15.393 | 0.592
NMC | 0.35 - 0.50 | 4.500 - 65.0 | 14 | 20 | 21.724 | 1.551 | 22.128 | 1.106
SLAC | 0.30 - 0.89 | 4.010 - 30.2 | 59 | 327 | 71.655 | 1.214 | 227.67 | 0.696
F$\bf{}_{2}^{\bf NS}$ |  |  |  |  |  |  |  |
BCDMS120 | 0.0700 - 0.275 | 8.75 - 43 | 30 | 30 | 36.210 | 1.207 | 33.948 | 1.131
BCDMS200 | 0.0700 - 0.275 | 17.0 - 75 | 28 | 28 | 20.120 | 0.718 | 20.358 | 0.727
BCDMS280 | 0.1000 - 0.275 | 32.5 - 115.5 | 26 | 26 | 12.946 | 0.497 | 12.579 | 0.483
NMC | 0.0125 - 0.275 | 4.50 - 65 | 53 | 53 | 12.263 | 0.231 | 11.392 | 0.214
SLAC | 0.1530 - 0.293 | 4.18 - 8.22 | 28 | 28 | 24.138 | 0.862 | 21.323 | 0.761
Total Data |  |  | 814 | 1395 |  |  |  |
Total $\mathbf{\chi^{2}}$ |  |  |  |  | 814.188 | 1054.358
Total $\mathbf{\chi^{2}/d.o.f.}$ |  |  |  |  | 814.188/806 = 1.010 | 1054.358/1377 = 0.765

Table 1: Various combinations of the different subsets of $xF_{3}$ and non-singlet $F_{2}$ data, along with their corresponding $x$ and $Q^{2}$ ranges. The fourth and fifth columns present the number of individual data points for each data set considering $W^{2}\geq$12.5 GeV2 and $W^{2}\geq$ 4 GeV2 cuts on the data, respectively. Furthermore, the reduction in the number of CCFR data points due to additional cuts applied to this data set (i.e., $x>0.4$) accounts for the discrepancy between CCFR and NuTeV in this region. The sixth and seventh columns ($\mathtt{SK24[QCD]}$) and the eighth and ninth ($\mathtt{SK24[QCD+TMC+HT]}$) contain the $\chi^{2}$ and $\chi^{2}/$#Data values for each fit applied on corresponding cuts. The total $\chi^{2}$ and $\chi^{2}/d.o.f.$ are also shown.  Parameters |  $\mathtt{SK24[QCD]}$ | $\mathtt{SK24[QCD+TMC+HT]}$
---|---|---
$N_{u}$ |  $0.8426$ | $1.8777$
$a_{u}$ |  $0.4484~{}\pm~{}0.0265$ | $0.5626~{}\pm~{}0.0324$
$b_{u}$ |  $3.9178~{}\pm~{}0.0183$ | $4.2169~{}\pm~{}0.0185$
$c_{u}$ |  $9.3388~{}\pm~{}1.9361$ | $6.8458~{}\pm~{}1.1963$
$d_{u}$ |  $2.6550$ | $0.0$
$N_{d}$ |  $1.5705$ | $4.9200$
$a_{d}$ |  $0.5499~{}\pm~{}0.0149$ | $0.7715~{}\pm~{}0.0201$
$b_{d}$ |  $6.4429~{}\pm~{}0.1535$ | $7.0218~{}\pm~{}0.1803$
$c_{d}$ |  $8.9695$ | $6.7446$
$d_{d}$ |  $-2.072$ | $-2.957$
$e_{d}$ |  $0.0$ | $0.4955~{}\pm~{}0.0731$
$f_{d}$ |  $0.0$ | $6.3712$
$\alpha_{s}(M_{Z}^{2})$ |  $0.1154~{}\pm~{}0.0009$ | $0.1149~{}\pm~{}0.0014$
$h_{11}$ |  … | $-11.71\hskip 0.7pt\pm~{}1.5808$
$h_{12}$ |  … | $~{}~{}~{}\\!3.296\hskip 0.5pt\pm~{}0.2311$
$h_{13}$ |  … | $-1.868\hskip 0.9pt\pm~{}0.1114$
$h_{21}$ |  … | $-2.005\hskip 0.9pt\pm~{}0.7045$
$h_{22}$ |  … | $~{}~{}~{}\\!2.584\hskip 0.5pt\pm~{}0.2172$
$h_{23}$ |  … | $-7.203\hskip 0.9pt\pm~{}2.2586$
$h_{31}$ |  … | $-12.69\hskip 0.9pt\pm~{}3.4884$
$h_{32}$ |  … | $~{}~{}~{}\\!1.641\hskip 0.5pt\pm~{}0.2168$
$h_{33}$ |  … | $-1.952\hskip 0.9pt\pm~{}0.1035$
$C_{N}$ |  $0.4468\hskip 3.0pt\pm~{}0.0949$ | $0.4938\hskip 3.0pt\pm~{}0.1096$
$x_{0}$ |  $0.8142\hskip 3.0pt\pm~{}0.0696$ | $0.5491\hskip 3.0pt\pm~{}0.0451$
$x_{1}$ |  $0.0986$ | $0.0856$
Table 2: The parameter values of the $u$\- and $d$-valence quark densities in
Eqs. (6, 7) at the input scale of $Q^{2}=4.0$ GeV2, obtained from the best fit
considering QCD and nuclear corrections ($\mathtt{SK24[QCD]}$), as well as TMC
and HT corrections ($\mathtt{SK24[QCD+TMC+HT]}$) at NNLO. The parameter values
without error have been fixed after the first minimization, due to the fact
that the data do not constrain some parameters well enough.

NNLO | $a_{u}$ | $b_{u}$ | $c_{u}$ | $a_{d}$ | $b_{d}$ | $\alpha_{s}$ | $C_{N}$ | $x_{0}$
---|---|---|---|---|---|---|---|---
$a_{u}$ | 0.704E-03 |  |  |  |  |  |  |
$b_{u}$ |  $-0.282$E-$03$ | 0.338E-03 |  |  |  |  |  |
$c_{u}$ |  $-0.508$E-$01$ |  $0.233$E-$01$ | 0.375E+01 |  |  |  |  |
$a_{d}$ |  $-0.160$E-$03$ |  $0.384$E-$05$ |  $0.123$E-$01$ | 0.223E-03 |  |  |  |
$b_{d}$ |  $-0.202$E-$02$ |  $0.616$E-$03$ |  $0.157$E+$00$ |  $0.194$E-$02$ | 0.236E-01 |  |  |
$\alpha_{s}$ |  $0.972$E-$05$ |  $-0.603$E-$05$ |  $-0.584$E-$03$ |  $0.357$E-$05$ |  $-0.241$E-$05$ | 0.844E-06 |  |
$C_{N}$ |  $0.324$E-$03$ |  $-0.552$E-$03$ |  $-0.323$E-$01$ |  $-0.365$E-$03$ |  $-0.245$E-$02$ |  $-0.108$E-$04$ | 0.901E-02 |
$x_{0}$ |  $-0.448$E-$03$ |  $0.496$E-$03$ |  $0.389$E-$01$ |  $0.379$E-$03$ |  $0.431$E-$02$ |  $0.664$E-$06$ |  $-0.617$E-$02$ | 0.485E-02

Table 3: NNLO covariance matrix of $\mathtt{SK24[QCD]}$ fit at Q${}_{0}^{2}=4$
GeV2.

$<x^{n-1}>_{q_{v}}$ | $\mathtt{SK24[QCD]}$ | $\mathtt{SK24[QCD+TMC+HT]}$ | $\mathtt{GKA19}$ | $\mathtt{KT08}$ | $\mathtt{KT07}$ | $\mathtt{BBG06}$ | $\mathtt{MSHT20}$ | $\mathtt{NNPDF4.0}$ | $\mathtt{CT18}$
---|---|---|---|---|---|---|---|---|---
|  |  | (Jacobi poly.) | (Bernstein poly.) |  |  |  |  |
$<x^{1}>_{u_{v}}$ | $0.3009\pm 0.0063$ | $0.3011\pm 0.0136$ | $0.3112$ | $0.3056$ | $0.2934$ | $0.2986$ | $0.2855$ | $0.2845$ | $0.2902$
$<x^{2}>_{u_{v}}$ | $0.0887\pm 0.0018$ | $0.0876\pm 0.0041$ | $0.0890$ | $0.0871$ | $0.0825$ | $0.0871$ | $0.0834$ | $0.0820$ | $0.0845$
$<x^{3}>_{u_{v}}$ | $0.0344\pm 0.0007$ | $0.0335\pm 0.0017$ | $0.0340$ | $0.0330$ | $0.0311$ | $0.0333$ | $0.0324$ | $0.0313$ | $0.0327$
$<x^{1}>_{d_{v}}$ | $0.1024\pm 0.0068$ | $0.1131\pm 0.0135$ | $0.1019$ | $0.1235$ | $0.1143$ | $0.1239$ | $0.1147$ | $0.1099$ | $0.1200$
$<x^{2}>_{d_{v}}$ | $0.0231\pm 0.0020$ | $0.0274\pm 0.0045$ | $0.0207$ | $0.0298$ | $0.0262$ | $0.0315$ | $0.0289$ | $0.0267$ | $0.0300$
$<x^{3}>_{d_{v}}$ | $0.0071\pm 0.0008$ | $0.0094\pm 0.0020$ | $0.0058$ | $0.0098$ | $0.0083$ | $0.0105$ | $0.0097$ | $0.0086$ | $0.0103$
$<x^{1}>_{u_{v}-d_{v}}$ | $0.1985\pm 0.0104$ | $0.1880\pm 0.0186$ | $0.2093$ | $0.1821$ | $0.1791$ | $0.1747$ | $0.1708$ | $0.1746$ | $0.1701$
$<x^{2}>_{u_{v}-d_{v}}$ | $0.0655\pm 0.0030$ | $0.0602\pm 0.0060$ | $0.0683$ | $0.0573$ | $0.0563$ | $0.0556$ | $0.0545$ | $0.0553$ | $0.0545$
$<x^{3}>_{u_{v}-d_{v}}$ | $0.0273\pm 0.0012$ | $0.0240\pm 0.0026$ | $0.0282$ | $0.0232$ | $0.0228$ | $0.0228$ | $0.0227$ | $0.0227$ | $0.0224$

Table 4: Low order moments ($n=2,3,4$) of
$<x^{n-1}>_{u_{v},d_{v},u_{v}-d_{v}}$, at $Q^{2}$= 4 GeV2 for our QCD non-
singlet fits ($\mathtt{SK24[QCD]}$) with comparison of the NNLO analysis,
GKA19 Tooran:2019cfz , KT08 Khorramian:2008yh , KT07 Khorramian:2006wg , BBG06
Blumlein:2006be , MSHT20 Bailey:2020ooq , NNPDF4.0 NNPDF:2021njg  and CT18
Hou:2019qau .

Figure 1: Valence PDF ratios of the NNLO NNPDF4.0 NNPDF:2021njg , CT18
Hou:2019qau , and HERAPDF H1:2015ubc  predictions to the MSHT20 set
Bailey:2020ooq , $xu_{v}/xu_{v}$[MSHT20]$(x,Q^{2})$, as a function of $x$ at
Q${}^{2}=4$ GeV2.

Figure 2:  Comparison of recent SeaQuest $\bar{d}(x)-\bar{u}(x)$ results
SeaQuest:2022vwp  and data from NuSea NuSea:1998kqi ; NuSea:2001idv  with the
NNLO analysis from MSHT20 Bailey:2020ooq , NNPDF4.0 NNPDF:2021njg , and CT18
Hou:2019qau  as a function of $x$ at Q2=25.5 GeV2.

Figure 3: The $x$ and $Q^{2}$ plane of various experiments of DIS lepton-
nucleon data for $xF_{3}(x,Q^{2})$, $F_{2}^{p}(x,Q^{2})$, $F_{2}^{d}(x,Q^{2})$
and $F_{2}^{NS}(x,Q^{2})$. The dotted lines represent the kinematic cuts
applied in the analysis, which require $Q^{2}\geq 4$ GeV2, $W^{2}\geq 12.5$
GeV2 and $W^{2}\geq 4$ GeV2. Data points above the $W^{2}=12.5$ GeV2 line are
used for QCD fits ($\mathtt{SK24[QCD]}$), while data points above the
$W^{2}=4$ GeV2 line are used for QCD+TMC+HT fits ($\mathtt{SK24[QCD+TMC+HT]}$)
in the current analysis. Figure 4: The comparison of structure function
$xF_{3}$ (Iron target) obtained from the $\mathtt{SK24[QCD]}$ and
$\mathtt{SK24[QCD+TMC+HT]}$ fits as a function of $Q^{2}$ in the various $x$,
at the NNLO approximation. The vertical dashed line indicates the regions with
W2 $\geq$ 12.5 GeV2. Figure 5: The comparison of structure function $xF_{3}$
(Lead target) obtained from the $\mathtt{SK24[QCD]}$ and
$\mathtt{SK24[QCD+TMC+HT]}$ fits as a function of $Q^{2}$ in the various $x$,
at the NNLO approximation. The vertical dashed line indicates the regions with
W2 $\geq$ 12.5 GeV2. Figure 6: The comparison of structure function
$F_{2}^{p}$ obtained from the $\mathtt{SK24[QCD]}$ and
$\mathtt{SK24[QCD+TMC+HT]}$ fits as a function of $Q^{2}$ in the various $x$,
at the NNLO approximation. The vertical dashed line indicates the regions with
W2 $\geq$ 12.5 GeV2.  Figure 7: The comparison of structure function
$F_{2}^{d}$ obtained from the $\mathtt{SK24[QCD]}$ and
$\mathtt{SK24[QCD+TMC+HT]}$ fits as a function of $Q^{2}$ in the various $x$,
at the NNLO approximation. The vertical dashed line indicates the regions with
W2 $\geq$ 12.5 GeV2.  Figure 8: The comparison of structure function
$F_{2}^{NS}$ obtained from the $\mathtt{SK24[QCD]}$ and
$\mathtt{SK24[QCD+TMC+HT]}$ fits as a function of $Q^{2}$ in the various $x$,
at the NNLO approximation.

Figure 9: The NNLO comparison of $xu_{v}$ and $xd_{v}$ obtained from the
$\mathtt{SK24[QCD]}$ and $\mathtt{SK24[QCD+TMC+HT]}$ at $Q^{2}$=4, 5, 10, 100,
1000, 5000 GeV2 as a function of $x$. In the left panels, our results for
$\mathtt{SK24[QCD]}$ and $\mathtt{SK24[QCD+TMC+HT]}$ are presented, while the
middle and right panels display the relative uncertainties $\delta
xu_{v}(x,Q^{2})/xu_{v}(x,Q^{2})$ and $\delta xd_{v}(x,Q^{2})/xd_{v}(x,Q^{2})$,
respectively.

Figure 10: The NNLO comparison of $xu_{v}(x,Q^{2})$ and $xd_{v}(x,Q^{2})$
obtained from the $\mathtt{SK24[QCD]}$ fit compared with MSHT20 Bailey:2020ooq
, NNPDF4.0 NNPDF:2021njg  and CT18 Hou:2019qau  non-singlet/singlet
predictions at $Q^{2}$=4, 5, 10, 100, 1000, 5000 GeV2 as a function of $x$. In
the first and third columns our results for $\mathtt{SK24[QCD]}$ are
presented, and the second and fourth columns show log plots for ratios of
$xu_{v}(x,Q^{2})/xu_{v}(x,Q^{2})_{ref}$ and
$xd_{v}(x,Q^{2})/xd_{v}(x,Q^{2})_{ref}$, with respect to $\mathtt{SK24[QCD]}$.

Figure 11: The NNLO comparison of relative uncertainties $\delta
xu_{v}(x,Q^{2})/xu_{v}(x,Q^{2})$ obtained from the $\mathtt{SK24[QCD]}$ and
$\mathtt{SK24[QCD+TMC+HT]}$ fits as a function of $x$ at $Q^{2}$=4, 5, 10,
100, 1000, 5000 GeV2 compared with MSHT20 Bailey:2020ooq , NNPDF4.0
NNPDF:2021njg  and CT18 Hou:2019qau  non-singlet/singlet predictions.

Figure 12: The NNLO comparison of relative uncertainties $\delta
xd_{v}(x,Q^{2})/xd_{v}(x,Q^{2})$ obtained from the $\mathtt{SK24[QCD]}$ and
$\mathtt{SK24[QCD+TMC+HT]}$ fits as a function of $x$ at $Q^{2}$=4, 5, 10,
100, 1000, 5000 GeV2 compared with MSHT20 Bailey:2020ooq , NNPDF4.0
NNPDF:2021njg  and CT18 Hou:2019qau  non-singlet/singlet predictions. Figure
13: The obtained values and uncertainties of $\alpha_{s}(M_{Z}^{2})$ in
present non-singlet $\mathtt{SK24[QCD]}$ and $\mathtt{SK24[QCD+TMC+HT]}$
analysis comparing with available results A02 Alekhin:2002fv , A06
Alekhin:2006zm , MRST03 Martin:2003tt , ABMP16 Alekhin:2017kpj ;
Alekhin:2018pai , BBG06 Blumlein:2006be , JR14 Jimenez-Delgado:2014twa ,
ABKM09 Alekhin:2009ni , ABM11 Alekhin:2012ig , KKT Khorramian:2009xz , CT18
Hou:2019qau , MSHT20 Bailey:2020ooq , NNPDF3.1 Ball:2018iqk , NNPDF4.0
NNPDF:2021njg , AKP22 Azizi:2022nqm , GKA Tooran:2019cfz  and ZEUS
ZEUS:2023zie  at the NNLO approximation. Furthermore, the world and DIS
average values of the strong coupling constant $\alpha_{s}(M_{Z}^{2})$
ParticleDataGroup:2022pth  illustrated for well comparison. Figure 14: Impact
of higher twist function ${C_{HT}^{i}(x)[GeV^{2}]}$ from Eq. (23) on large $x$
region for $i=1,2,3$ corresponding to the HT effects for the $F_{2}^{p}$,
$F_{2}^{d}$, and $xF_{3}$ structure functions, respectively (see text).

Figure 15: Ratio of the $xF_{3}/xF_{3}[\mathtt{QCD}]$,
$F_{2}^{p}/F_{2}^{p}[\mathtt{QCD}]$, and $F_{2}^{d}/F_{2}^{d}[\mathtt{QCD}]$
as a function of $x$ for Q2 = 4, 10, 25 and 50 GeV2.