# Nucleon spin structure functions, considering target mass correction and
higher twist effects at the NNLO accuracy and their transverse momentum
dependence

Abolfazl Mirjalili1 ID<EMAIL_ADDRESS>(Corresponding author) Shahin
Atashbar Tehrani2 ID<EMAIL_ADDRESS>(1)Physics Department, Yazd University,
P.O.Box 89195-741, Yazd, Iran
(2)School of Particles and Accelerators, Institute for Research in Fundamental
Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran

(September 3, 2024)

###### Abstract

Using recent and updated world data on polarized structure functions $g_{1}$
and $g_{2}$ we perform QCD analysis at the next-next-to-leading-order (NNLO)
accuracy. We include also target mass correction and higher twist effect to
get more precise results in our fitting procedure. To confirm the validity of
our fitting results several sum rules are examined and we do a comparison for
them with results from other models. In our analysis we employ Jacobi
polynomials approach to obtain analytical solutions of the DGLAP evolution
equations for parton distribution functions (PDFs). Using the extracted PDFs
from our data analysis as input we also compute the $x$\- and
$\mathbf{p}_{T}$-dependence of some transverse momentum dependence (TMD) PDFs
in polarized case, based on covariant parton model. These functions are
naively even time-reversal (T-even) at twist-2 approximation. The results for
TMDs are indicating proper and acceptable behaviour with respect to what are
presented in other literatures.

###### Contents

  1. I Introduction
  2. II Leading twist spin dependence of structure function
  3. III Jacobi polynomials expansion technique
  4. IV Target mass corrections in polarized case
  5. V Twist-3 contribution
  6. VI Fitting contents in QCD analysis
    1. VI.1 Parametrization
    2. VI.2 Overview of data sets
    3. VI.3  $\chi^{2}$ minimization
  7. VII The Sum Rules
    1. VII.1 Bjorken sum rule
    2. VII.2 Proton helicity sum rule
    3. VII.3 The twist-3 reduced matrix element $d_{2}$
    4. VII.4 Burkhardt-Cottingham (BC) sum rule
    5. VII.5 Efremov-Leader-Teryaev (ELT) Sum Rule
  8. VIII Comparison for the spin structure functions
  9. IX Predictions for polarized TMDs
  10. X Conclusions

##  I Introduction

The determination of the nucleon’s spin into its quark and gluon components is
still an important challenge in particle physics. The deep-inelastic
scattering (DIS) experiments performed at DESY, SLAC, CERN, and JLAB have
refined our understanding of the spin distributions and revealed the spin-
dependent structure functions of the nucleon. The polarized structure
functions $g_{1}(x,Q^{2})$ and $g_{2}(x,Q^{2})$ are measured in deep-inelastic
scattering of a longitudinally polarized lepton on polarized nuclear targets.
We do the required analysis on the polarized structure function to extract the
desired parton densities at the initial $Q_{0}$.

In exact consideration of inclusive processes it is required to take into
account the distributions in which the role of transverse momentum is
embedded. These distributions are known as transverse momentum dependent (TMD)
distributions. TMDs are the generalization of PDFs which provide us an
extensive knowledge to investigate the hadron structure function. In a native
parton model in which the effect of transverse momentum of a quark is not
outstanding, there is a proper computational frame which is called infinite
momentum frame (IMF) imf1 ; imf2 . In this frame the target (nucleon) is
moving fast, comparable to speed of light and because of Lorentz contraction
the nucleon seems like a flat disc. In this case one can imagine a transverse
space position of quark inside the disk with respect to the moving direction
of target. This space coordinate is called impact parameter and denoted
usually by $b_{T}$. Corresponding to the impact parameter in coordinate space
we can attribute to a quark inside the target a transverse momentum, $k_{T}$,
that is perpendicular to moving direction of nucleon. This momentum component
is ignorable against the quark longitudinal momentum. This model then gives
oversimplified relations between structure and distribution functions. In an
another model which is called covariant parton model (CPM) Landshoff:1971xb  a
more exact but much more complex relations between structure and distribution
functions are given. The original assumptions of this model is based on
covariance of relations together with a spherically symmetric quark momenta
distribution in the nucleon rest frame where one photon exchange is used in a
charged lepton-quark interaction. The output of this model is such us the
quark transverse momentum is as important as longitudinal one and the
transverse momentum dependence of parton densities are obtained analytically
cpm1 .

The extended PDF is then describing the parton distribution with respect to
both $x$ and $k_{T}$ variables. On the other words quarks can have transverse
momentum with respect to the motion of parent hadron. The transverse momentum
of parton at initial state and inside the parent hadron is called the
intrinsic transverse momentum, denoted by $k_{T}$. In the final state the
transverse momentum of parton with respect to the momentum of produced hadron
is denoted by $p_{T}$. TMDs have outstanding effect on the momentum feature of
produced hadron. They also have crucial role to describe the spin asymmetry in
produced hadron sa  by analysing the semi inclusive DIS (SIDIS) processes sa1
; ours . To achieve the three dimensional (3D) picture of nucleon, some
processes like SIDIS are required in which one can measure the effect of
transverse momentum of partons in created hadron. It is therefore required to
consider the spin dependence of PDFs. Early applications to polarized
structure functions were made by Jackson:1989ph ; Roberts:1996ub ;
Blumlein:1996tp

The PDFs in polarized case are two types. The first one is related to the
longitudinal polarized quark inside longitudinal polarized nucleon, denoted by
$g_{1}(x)$ that is called helicity function. The second one is related to
transverse polarized quark inside the transverse polarized nucleon, denoted by
$h_{1}(x)$ and is called the transversity function. The type of polarization
is determined with respect to moving direction of nucleon. If the parton
transverse momentum as an extra degree of freedom is also considered then
total number of PDFs, involving polarized cases, are arising to eight ones sa2
. In this article the polarized TMDs which are even time reversal functions,
based on covariant parton model, are investigated.

The organization of this paper are as following. In Sec.II an overview on
theoretical aspects of polarized structure function is done. In Sec.III the
theoretical framework of Jacobi polynomials approach is reviewed. Sec.IV is
devoted to discuss the target mass correction for $g_{1}$ and $g_{2}$
structure functions. Additionally in Sec.V higher twist effect is demonstrated
for polarized structure functions. In Sec.VI which includes also some
subsections we illustrate our QCD data analysis which we call it as MA22
analysis. To get more validation of our MA22 results, we examine in Sec.VII
several sum rules. In Sec.VIII our prediction for polarized PDFs and structure
functions are presented. Using the results of our MA22 analysis, some
polarized TMDs can be calculated. We do it in Sec.IX. In the last part that is
Sec.X our conclusions is given.

##  II Leading twist spin dependence of structure function

To achieve the main goal of this article to calculate the polarized TMDs we
first need to analysis DIS structure function in polarized case. For this
purpose linear combination of polarized parton densities and coefficient
functions can be used to express the leading twist spin-dependent proton and
neutron structure functions, $g_{1}^{\rm p}(x,Q^{2})$ at the next-next-to-
leading-order (NNLO) accuracy as it follows Shahri:2016uzl ; Goto:1999by ;
Khanpour:2017cha :

$\displaystyle g_{1}^{\rm p}(x,Q^{2})=\frac{1}{2}\sum_{q}e^{2}_{q}\Delta
q_{v}(x,Q^{2})\otimes$
$\displaystyle\left(1+\frac{\alpha_{s}(Q^{2})}{2\pi}\Delta
C^{(1)}_{q}+\left(\frac{\alpha_{s}(Q^{2})}{2\pi}\right)^{2}\Delta
C^{(2)}_{ns}\right)$ $\displaystyle+e^{2}_{q}(\Delta
q_{s}+\Delta\bar{q_{s}})(x,Q^{2})\otimes$
$\displaystyle\left(1+\frac{\alpha_{s}(Q^{2})}{2\pi}\Delta
C^{(1)}_{q}+\left(\frac{\alpha_{s}(Q^{2})}{2\pi}\right)^{2}\Delta
C^{(2)}_{s}\right)$
$\displaystyle+\frac{2}{9}\left(\frac{\alpha_{s}(Q^{2})}{2\pi}\Delta
C^{(1)}_{g}+\left(\frac{\alpha_{s}(Q^{2})}{2\pi}\right)^{2}\Delta
C^{(2)}_{g}\right)\otimes\Delta g(x,Q^{2})$

Here $\Delta q_{v}$, $\Delta q_{s}$ and $\Delta g$ are the polarized valance,
sea and gluon densities, respectively. The pQCD evolution kernel for PPDFs is
now available at the NNLO accuracy in Ref. Moch:2014sna ; Blumlein:2021ryt ;
Blumlein:2021enk . The $\Delta C^{(1)}_{q}$ and $\Delta C^{(1)}_{g}$ in
Eq.(II) are denoting to the NLO spin-dependent quark and gluon hard scattering
coefficients, calculable in pQCD Lampe:1998eu . We now apply the hard
scattering coefficients, extracted at NNLO approximation. At this order the
Wilson coefficients are different for quarks and antiquarks . They are
presented in Eq.(II) by $\Delta C^{(2)}_{ns}$ and $\Delta C^{(2)}_{s}$
respectively and their analytical relations have been reported in
Zijlstra:1993sh . The symbol $\otimes$ in Eq.(II) is representing typical
convolution in $x$-space.

The neutron structure function, $g_{1}^{\rm n}(x,Q^{2})$, can be obtained from
the proton one by considering isospin symmetry. Hence the deuteron structure
function at leading twist would be available, utilizing the $g_{1}^{\rm p}$
and $g_{1}^{\rm n}$ structure functions such as:

$\displaystyle g_{1}^{\rm\tau 2(d)}(x,Q^{2})=\frac{1}{2}\\{g_{1}^{\rm
p}(x,Q^{2})+g_{1}^{\rm n}(x,Q^{2})\\}\times(1-1.5w_{D})\,,$

where $w_{D}=0.05\pm 0.01$ is the probability to find the deuteron in a
$D-$state Lacombe:1981eg ; Buck:1979ff ; Zuilhof:1980ae . Using the Wandzura
and Wilczek (WW) relation Wandzura:1977qf  the leading twist polarized
structure function of $g_{2}^{\tau_{2}}(x,Q^{2})$ can be fully determined via
$g_{1}^{\tau_{2}}(x,Q^{2})$ structure function:

$\displaystyle g_{2}^{\tau_{2}}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
g_{2}^{WW}(x,Q^{2})=$ (3) $\displaystyle-
g_{1}^{\tau_{2}}(x,Q^{2})+\int_{x}^{1}\frac{dy}{y}g_{1}^{\tau_{2}}(y,Q^{2})\,.$

This relation that is in the leading twist approximation can also be used when
target mass correction (TMC) is included Wandzura:1977qf .

The $g_{1}^{\tau_{2}}(x,Q^{2})$ and $g_{2}^{\tau_{2}}(x,Q^{2})$ structure
functions at the leading twist order have valid definition in the Bjorken
limit, i.e. $Q^{2}\rightarrow\infty,\leavevmode\nobreak\ x=$ fixed. But at the
a moderate low $Q^{2}$ ($\sim 1-5$ GeV2) and $W^{2}($$4$ GeV${}^{2}<W^{2}<10$
GeV2) where $W^{2}$ is the invariant mass of the hadronic system, both TMC
along with higher twist corrections should be considered. We investigate them
in Sec.IV and Sec.V.

Next section is devoted to illustrate the nucleon and deuteron structure
functions, based on Jacobi polynomial approach which yield us these functions
in moment-$n$ space.

##  III Jacobi polynomials expansion technique

To achieve the nucleon structure function in momentum $n$-space we resort to a
method that is based on the Jacobi polynomials expansion. Practical aspects of
this method including its major advantages are presented in our previous
studies Shahri:2016uzl ; Khanpour:2017cha ; Khorramian:2010qa ;
Khorramian:2009xz ; MoosaviNejad:2016ebo ; Khanpour:2016uxh ;
AtashbarTehrani:2013qea ; Nematollahi:2021ynm . According to this method, one
can easily expand the polarized structure functions $xg_{1}^{\rm
QCD}(x,Q^{2})$, in terms of the Jacobi polynomials
$\Theta_{n}^{\alpha,\beta}(x)$, as it follows Ayala:2015epa ; Barker:1982rv ;
Barker:1983iy ; Krivokhizhin:1987rz ; Krivokhizhin:1990ct ; Chyla:1986eb ;
Barker:1980wu ; Kataev:1997nc ; Alekhin:1998df ; Kataev:1999bp ; Kataev:2001kk
; Kataev:2005ci ; Leader:1997kw ,

$x\,g_{1}^{\tau 2}(x,Q^{2})=x^{\beta}(1-x)^{\alpha}\ \,\sum_{n=0}^{\rm N_{\rm
max}}a_{n}(Q^{2})\,\Theta_{n}^{\alpha,\beta}(x)\,,$ (4)

where $\rm N_{\rm max}$ is the maximum order of expansion. The parameters
$\alpha$ and $\beta$ are Jacobi polynomials free parameters which normally
fixed on their best values. These parameters have to be chosen so as to
achieve the fastest convergence of the series on the right-hand side of Eq.
(4). In the polynomial fitting procedure, the evolution equation is combined
with the truncated series to perform a direct fit to the structure functions.

The Jacobi moments, $a_{n}(Q^{2})$ are codifying the Q2-dependence of the
polarized structure functions. The $x$-dependence will be provided by the
weight function $w^{\alpha,\beta}(x)\equiv x^{\beta}(1-x)^{\alpha}$ and the
Jacobi polynomials $\Theta_{n}^{\alpha,\beta}(x)$ which can be written as,

$\Theta_{n}^{\alpha,\beta}(x)=\sum_{j=0}^{n}\,c_{j}^{(n)}(\alpha,\beta)\,x^{j}\,,$
(5)

where the coefficients $c_{j}^{(n)}(\alpha,\beta)$ are combinations of Gamma
functions in terms of $n$, $\alpha$ and $\beta$. The above Jacobi polynomials
are satisfying the following orthonormality condition:

$\int_{0}^{1}dx\,x^{\beta}(1-x)^{\alpha}\,\Theta_{n}^{\alpha,\beta}(x)\,\Theta_{l}^{\alpha,\beta}(x)\,=\,\delta_{n,l}\,.$
(6)

Consequently the Jacobi moments, $a_{n}(Q^{2})$, can be obtained, using the
above relation such as,

$\displaystyle a_{n}(Q^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\,xg_{1}^{\tau_{2}}(x,Q^{2})\,\Theta_{n}^{\alpha,\beta}(x)$
(7) $\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\,c_{j}^{(n)}(\alpha,\beta)\,{\cal
M}[xg_{1}^{\tau_{2}},\,j+2](Q^{2})\,,$

where the Mellin transform ${\cal{M}}[xg_{1}^{\tau 2},\rm N]$ is given by,

$\displaystyle{\cal{M}}[xg_{1}^{\tau_{2}},{\rm
N}](Q^{2})\equiv\int_{0}^{1}dx\,x^{\rm N-2}\,xg_{1}^{\tau_{2}}(x,Q^{2})\,.$
(8)

Using the QCD expressions for the Mellin moments,
${\cal{M}}[xg_{1}^{\tau_{2}},{\rm N}](Q^{2})$, the polarized structure
function $xg_{1}^{\tau_{2}}(x,Q^{2})$, can be constructed. Therefore, based on
the method of Jacobi polynomial expansion, the $xg_{1}^{\tau_{2}}(x,Q^{2})$ is
given by:

$\displaystyle xg_{1}^{\tau_{2}}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
x^{\beta}(1-x)^{\alpha}\,\sum_{n=0}^{\rm
N_{max}}\,\Theta_{n}^{\alpha,\beta}(x)$ (9) $\displaystyle\times$
$\displaystyle\sum_{j=0}^{n}\,c_{j}^{(n)}{(\alpha,\beta)}\,{\cal
M}[xg_{1}^{\tau_{2}},j+2](Q^{2})\,.$

By setting $N_{\rm max}$ = 9, $\alpha$ = 3 and $\beta$ = 0.5, as we have shown
in our previous analyses Shahri:2016uzl ; Khanpour:2017cha ; Khorramian:2010qa
; Khorramian:2009xz ; MoosaviNejad:2016ebo ; Khanpour:2016uxh ;
AtashbarTehrani:2013qea ; Nematollahi:2021ynm , it is possible to obtain the
optimal convergence of above expansion through the whole kinematic region that
is constrained by the polarized DIS data.

In next section we improve our analysis of DIS polarized data, considering the
TMC correction to the nucleon structure functions.

##  IV Target mass corrections in polarized case

Power suppressed corrections to the structure functions can have important
contributions in some kinematic regions. Hence nucleon mass correction cannot
be neglected in low $Q^{2}$ region. The TMCs can be calculated via an
expression which is different from higher twist (HT) effects in dynamical
case. In the case of polarized structure function we follow the suggested
method by Blumlein and Tkabladze Blumlein:1998nv  which is in fact the
generalized one that was established by Georgi and Politzer Georgi:1976ve  for
the unpolarized structure function.

Mellin inversion to $x$-space or the integer moments of structure function can
be used to present these corrections. Leading twist-2 expression for $g_{1}$,
that is containing TMC, is given explicitly by Blumlein:1998nv :

$\displaystyle g_{1}^{\rm\tau_{2}+TMCs}(x,Q^{2})$ (10) $\displaystyle=$
$\displaystyle\frac{xg_{1}^{\tau_{2}}(\xi,Q^{2};{\rm M}=0)}{\xi(1+4{\rm
M}^{2}x^{2}/Q^{2})^{3/2}}$ $\displaystyle+$ $\displaystyle\frac{4{\rm
M}^{2}x^{2}}{Q^{2}}\frac{(x+\xi)}{\xi(1+4{\rm
M}^{2}x^{2}/Q^{2})^{2}}\int_{\xi}^{1}\frac{d\xi^{\prime}}{\xi^{\prime}}g_{1}^{\tau_{2}}(\xi^{\prime},Q^{2};{\rm
M}=0)$ $\displaystyle-$ $\displaystyle\frac{4{\rm
M}^{2}x^{2}}{Q^{2}}\frac{(2-4{\rm M}^{2}x^{2}/Q^{2})}{2(1+4{\rm
M}^{2}x^{2}/Q^{2})^{5/2}}$ $\displaystyle\times$
$\displaystyle\int_{\xi}^{1}\frac{d\xi^{\prime}}{\xi^{\prime}}\int_{\xi^{\prime}}^{1}\frac{d\xi^{\prime\prime}}{\xi^{\prime\prime}}g_{1}^{\tau_{2}}(\xi^{\prime\prime},Q^{2};{\rm
M}=0)\,.$

The twist-2 contribution for the $g_{2}$ structure function, including TMC is
similarly presented by Blumlein:1998nv :

$\displaystyle g_{2}^{\tau_{2}+\rm TMCs}(x,Q^{2})$ (11) $\displaystyle=$
$\displaystyle-\frac{xg_{1}^{\tau_{2}}(\xi,Q^{2};{\rm M}=0)}{\xi(1+4{\rm
M}^{2}x^{2}/Q^{2})^{3/2}}$ $\displaystyle+$ $\displaystyle\frac{x(1-4{\rm
M}^{2}x\xi/Q^{2})}{\xi(1+4{\rm
M}^{2}x^{2}/Q^{2})^{2}}\int_{\xi}^{1}\frac{d\xi^{\prime}}{\xi^{\prime}}g_{1}^{\tau_{2}}(\xi^{\prime},Q^{2};{\rm
M}=0)$ $\displaystyle+$ $\displaystyle\frac{3}{2}\frac{4{\rm
M}^{2}x^{2}/Q^{2}}{(1+4{\rm M}^{2}x^{2}/Q^{2})^{5/2}}$ $\displaystyle\times$
$\displaystyle\int_{\xi}^{1}\frac{d\xi^{\prime}}{\xi^{\prime}}\int_{\xi^{\prime}}^{1}\frac{d\xi^{\prime\prime}}{\xi^{\prime\prime}}g_{1}^{\tau_{2}}(\xi^{\prime\prime},Q^{2};{\rm
M}=0)\,,$

Numerical illustrations for the target mass effects in $g_{1}$ and $g_{2}$
have been given in Blumlein:1999rv . In both above equations M is the nucleon
mass and $\xi$ is called Nachtmann variable that is defined by
Nachtmann:1973mr :

$\xi=\frac{2x}{1+\sqrt{1+4{\rm M}^{2}x^{2}/Q^{2}}}\leavevmode\nobreak\ .$ (12)

It can be seen that by choosing the maximum value for the $x$-Bjorken
variable, the maximum kinematic value of $\xi$ variable would be less than
unity. This means that the target mass corrected structure functions at
leading twist in both the polarized and unpolarized cases, as it is expected,
do not vanish at maximum $x=1$ value.

As we referred before, in addition to target mass correction, higher twist
effects would also be dominant at low $Q^{2}$ values and make contribution to
nucleon structure function in related kinematic region. Next section is
devoted to this effects.

##  V Twist-3 contribution

The long-range nonperturbative multiparton correlations which have outstanding
contributions at low values of $Q^{2}$ will lead to higher twist (HT) terms. A
proper analysis of this effect can be found in Blumlein:2010rn . For a
developing phenomenological analysis an advantageous parametrization is made
by the BLMP model Braun:2011aw  for HT terms. Following that HT distributions
are constructed from convolution integrals that are containing light-cone wave
functions. In this connection a simple model based on three valence quark and
one gluon distributions with the total zero angular momentum are assumed.

Accordingly, we utilize the parameterized form, suggested by the BLMP model at
the twist-3 order for $g_{2}$ structure function in an initial scale $Q_{0}$
as it follows Braun:2011aw ; Blumlein:2012se :

$\displaystyle g_{2}^{{{\tau_{3}}}}(x)$ $\displaystyle=$ $\displaystyle
A[ln(x)+(1-x)+\frac{1}{2}(1-x)^{2}]$ (13) $\displaystyle+$
$\displaystyle(1-x)^{3}[B-C(1-x)+D(1-x)^{2}$ $\displaystyle-$ $\displaystyle
E(1-x)^{3}]\ .$

The unknown coefficients in Eq.(13) are extracted by fitting the data. Since
higher twist contributions are important in a region with large-$x$ values, a
nonsinglet evolution equation is employed. The results of this approach can be
compared with exact evolution equations where a gluon-quark-antiquark
correlation is considered Braun:2011aw . It is expecting that these two
results are in good agreement with each other.

The twist-3 part of different spin-dependent structure functions ,
$g_{1}^{\tau_{3}}$ and $g_{2}^{\tau_{3}}$, are related by the following
integral relation Blumlein:1998nv .

$\displaystyle g_{1}^{\tau 3}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\frac{4x^{2}M^{2}}{Q^{2}}[g_{2}^{\tau 3}(x,Q^{2})$ (14)
$\displaystyle-$ $\displaystyle 2\int_{x}^{1}\frac{dy}{y}g_{2}^{\tau
3}(y,Q^{2})]\ ,$

The $Q^{2}$-dependence of the $g_{\rm 2}^{\tau_{3}}$ can be achieved within
nonsinglet perturbative QCD evolution as

$g_{\rm 2}^{\tau 3}(n,Q^{2})={\cal M}^{\rm NS}(n,Q^{2})\,g_{2}^{\tau 3}(n)\,.$
(15)

Finally the spin-dependent structure functions, considering the TMCs and HT
terms are given by,

$\displaystyle xg_{1,2}$ ${}^{\text{Full=pQCD+TMC+HT}}(x,Q^{2})=\,$ (16)
$\displaystyle
xg_{1,2}^{\rm\tau_{2}+TMCs}(x,Q^{2})+xg_{1,2}^{\tau_{3}}(x,Q^{2})\,.$

One of the particular feature of $xg_{1,2}^{\text{Full}}(x,Q^{2})$ function is
that the twist-3 term is not suppressed there by inverse powers of $Q^{2}$.
Consequently to describe this function, this contribution is so important as
the twist-2 contribution.

Since the required theoretical inputs are accessed by us, we can do now the
concerned data analysis which is done in next section

##  VI Fitting contents in QCD analysis

The fitting procedure, including the recent and updated data for polarized
structure functions which we do in our QCD analysis, are containing the
following parts.

###  VI.1 Parametrization

We start the QCD analysis considering the following parametrization at the
initial scale of $Q_{0}^{2}=1$ GeV2 where $q=\\{u_{v},d_{v},\bar{q},g\\}$:

$x\>\Delta q(x,Q_{0}^{2})={\cal
N}_{q}\eta_{q}x^{a_{q}}(1-x)^{b_{q}}(1+c_{q}x)\ .$ (17)

The normalization constant ${\cal N}_{q}$,

${\cal
N}_{q}^{-1}=\left(1+c_{q}\frac{a_{q}}{a_{q}+b_{q}+1}\right)\,B\left(a_{q},b_{q}+1\right)\
,$ (18)

is determined such that $\eta_{q}$ in Eq.(18) is the first moment of the
polarized parton distribution functions (PPDFs). Here $B(a,b)$ is the Euler
beta function. Considering SU(3) flavor symmetry, we assume
$\Delta\overline{q}\equiv\Delta\overline{u}=\Delta\overline{d}=\Delta
s=\Delta\overline{s}$ .

The unknown free parameters can be extracted through a fit which involves a
large degree of flexibility. Some of parameters can be determined via the
existing constrains, as describing in below:

  * •

The weak matrix elements $F$ and $D$ as measured in neutron and hyperon
$\beta$ decays ParticleDataGroup:2018ovx  can be related to the first moments
of the polarized valence quark densities. Considering these constrains, the
numerical values $\eta_{u_{v}}=0.928\pm 0.014$ and $\eta_{d_{v}}=-0.342\pm
0.018$ are obtained.

  * •

Due to the present accuracy of the data, the $c_{\bar{q}}$ and $c_{g}$
parameters are setting to zero. Considering nonzero values for them, there
would not be observed any improvement in the fit.

  * •

The large-$x$ behavior of the polarized sea quarks and gluons are controlled
by $b_{\bar{q}}$ and $b_{g}$ parameters. In a region that is dominated by the
valence distributions, these parameters have large uncertainties.

  * •

Due to higher twist effect to the $g_{2,\\{p,n,d\\}}$ and consequently
$g_{1,\\{p,n,d\\}}$, there are unknown parameters $\\{A,B,C,D,E\\}$, sea
Eq.(13). By a simultaneous fit to the all polarized structure function data of
$g_{1}$ and $g_{2}$, these parameters can be determined.

  * •

The values of some parameters are frozen in the first minimization procedure.
They involve $\\{\eta_{u_{v}},\eta_{d_{v}},c_{\bar{q}},c_{g}\\}$ and finally
the $b$ parameter. As demonstrated in Tables 1 and 2 the
$\\{b_{\bar{q}},b_{g},c_{u_{v}},c_{d_{v}}\\}$ and $\\{A,B,C,D,E\\}$ parameters
are then fixed in the second minimization. Nine unknown parameters, including
$\alpha_{s}(Q_{0}^{2})$, are left which are determined in the fit. They have
enough flexibility to perform a reliable fit.

  * •

The numerical value $\alpha_{s}(M_{Z}^{2})=0.112804\pm 0.001907$ would be
achieved in which we need to change the energy scale to the Z boson mass. It
is while for the present world average, the value
$\alpha_{s}(M_{Z}^{2})=0.1179\pm 8.5\times 10^{-6}$ is reported Zyla:2020zbs .

To extract the unknown parameters, it needs to access to all available
concerned data sets which we describe them in below.

Figure 1: Our MA22 results for the polarized PDFs at Q${}_{0}^{2}=$ 1 GeV2 as a function of $x$ in the NNLO approximation. It is indicated by a solid curve along with their $\Delta\chi^{2}=1$ uncertainty bands which is computed, based on the Hessian approach hes . The recent results of TKAA16 (dashed-dotted) Shahri:2016uzl  is also shown in NNLO approximation without inclusion of HT terms and TMCs. Additionally the KTA17(dashed) Khanpour:2017cha  in NNLO approximation is presented that including the HT terms and TMCs. The KATAO11(dashed-dotted-dotted) in NLO approximation Khorramian:2010qa  is furthermore indicated. Finally the results of NAAMY21(dashed-dashed-dotted) Nematollahi:2021ynm  in NLO approximation is also plotted.  Table 1:  Final parameter values and their statistical errors at the input scale $Q_{0}^{2}=1$ GeV2 determined from two different global analyses. Those marked with (∗) are fixed. Parameters | Full scenario | pQCD scenario
---|---|---
$\delta u_{v}$ | $\eta_{u_{v}}$ | $\leavevmode\nobreak\ 0.928^{*}\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 0.928^{*}\leavevmode\nobreak\ $
| $a_{u_{v}}$ | $0.898\pm 0.022$ | $0.277\pm 0.0072$
| $b_{u_{v}}$ | $3.218\pm 0.035$ | $2.725\pm 0.029$
| $c_{u_{v}}$ | $3.88^{*}$ | $28.95^{*}$
$\delta d_{v}$ | $\eta_{d_{v}}$ | $-0.342^{*}$ | $-0.342^{*}$
| $a_{d_{v}}$ | $0.217\pm 0.027$ | $0.150\pm 0.012$
| $b_{d_{v}}$ | $2.947\pm 1.45$ | $2.591\pm 0.087$
| $c_{d_{v}}$ | $\leavevmode\nobreak\ 9.335^{*}\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 31.75^{*}\leavevmode\nobreak\ $
$\leavevmode\nobreak\ \delta_{\bar{q}}$ | $\leavevmode\nobreak\ \eta_{\bar{q}}$ | $-0.0288\pm 0.002$ | $-0.0356\pm 0.0033$
| $a_{\bar{q}}$ | $1.227\pm 0.068$ | $1.991\pm 0.041$
| $b_{\bar{q}}$ | $3.364^{*}$ | $11.163^{*}$
| $c_{\bar{q}}$ | $\leavevmode\nobreak\ 0.0^{*}\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 0.0^{*}\leavevmode\nobreak\ $
$\delta g$ | $\eta_{g}$ | $0.0921\pm 0.022$ | $0.178\pm 0.014$
| $a_{g}$ | $10.2\pm 1.22$ | $26.33\pm 0.49$
| $b_{g}$ | $46.32^{*}$ | $99.95^{*}$
| $c_{g}$ | $\leavevmode\nobreak\ 0.0^{*}\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ 0.0^{*}\leavevmode\nobreak\ $
$\alpha_{s}(Q_{0}^{2})$ | $0.3362\pm 0.002$ | $0.4688\pm 0.0008$
$\chi^{2}/ndf$ | $1111.789/957=1.161$ | $1580.751/957=1.651$
Table 2:  Parameter values for the coefficients of the twist-3 corrections at $Q^{2}=1$ GeV2 obtained in the full scenario. | A | B | C | D | E
---|---|---|---|---|---
$g_{2,p}^{tw-3}$ | $0.0879$ | $1.0196$ | $-0.8832$ | $-2.3765$ | $2.4234$
$g_{2,n}^{tw-3}$ | $1.0086$ | $0.3009$ | $-0.6583$ | $0.3466$ | $-2.7571$
$g_{2,d}^{tw-3}$ | $0.8878$ | $1.3430$ | $-2.1334$ | $0.1878$ | $2.2293$
Figure 2: Our results, MA22 polarized parton distributions as a function of
$x$ and for some selected value of Q2 = 10, 100 GeV2.

###  VI.2 Overview of data sets

In our recent analysis which we call it MA22 we focus on the polarized DIS
data samples.. The required DIS data for all PPDFs are coming from the
experiments at electron-proton collider and also in fixed-target including
proton, neutron and heavier targets such as deuteron.

Although it is not possible to separate quarks from antiquarks, nonetheless it
is the inclusive DIS data that are included in the fit. Additionally we take
into our MA22 fitting procedure the $g_{2}$ structure function. Due to the
technical difficulty in operating the required transversely polarized target,
these data have been traditionally neglected before.

The data which we use in our recent analysis are up to date and including more
data than we employed in our pervious analysis Khanpour:2017cha . In fact we
use all available $g_{1}^{p}$ data from E143, HERMES98, SMC, EMC, E155,
HERMES06, COMPASS10, COMPASS16, JLAB06 and JLAB17 experiments Abe:1998wq ;
HERM98 ; Adeva:1998vv ; Ashman:1987hv ; E155p ; HERMpd ; COMP1 ;
Adolph:2015saz ; Dharmawardane:2006zd ; Fersch:2017qrq , and $g_{1}^{n}$ data
from HERMES98, E142, E154, HERMES06, Jlab03, Jlab04 and Jlab05 HERM98 ; E142n
; E154n ; Ackerstaff:1997ws ; JLABn2003 ; JLABn2004 ; JLABn2005  and finally
the $g_{1}^{d}$ data from E143, SMC, HERMES06, E155, COMPASS05, COMPASS06 and
COMPASS17 Abe:1998wq ; Adeva:1998vv ; HERMpd ; E155d ; COMP2005 ; COMP2006 ;
Adolph:2016myg . The DIS data for $g_{2}^{p,n,d}$ from E143, E142, Jlab03,
Jlab04, Jlab05, E155, Hermes12 and SMC Abe:1998wq ; E142n ; JLABn2003 ;
JLABn2004 ; JLABn2005 ; E155pdg2 ; hermes2012g2 ; SMCpg2  also are included.
These data sets are summarized in Table 3. The kinematic coverage, the number
of data points for each given target, and the fitted normalization shifts
${\cal{N}}_{i}$ also presented in this Table. Our MA22 analysis algorithm
computes the $Q^{2}$ evolution and extracts the structure function in $x$
space using Jacobi polynomials approach. It is corresponding to the fitting
programs on the market which solve the DGLAP evolution equations in the Mellin
space.

One of the important quantity which is used as a criteria to indicate the
validation of fit, is the chi-square ($\chi^{2}$) test which is assessing the
goodness of fit between observed values and those expected theoretically. We
discuss about it in the following subsection.

Figure 3: The spin-dependent proton, neutron and deuteron structure functions
as a function of $x$ and Q2. Our results, MA22, at the NNLO approximation
(solid curve) are compared with KTA17 at the same approximation (dashed)
Khanpour:2017cha , with THK14 at the NLO approximation (dashed-dotted)
TaheriMonfared:2014var , with TKAA16 at the NNLO approximation (long-dashed
dotted) Shahri:2016uzl ,with KATAO11 at the NLO approximation (dashed-dotted-
dotted) Khorramian:2010qa  and finally with NAAMY21 at the NLO approximation
(dashed-dashed-dotted) Nematollahi:2021ynm .

###  VI.3  $\chi^{2}$ minimization

The $\chi_{\rm global}^{2}(\rm p)$ quantifies the goodness of fit to the data
for a set of $\rm p$ independent parameters. To determine the best fit, it is
needed to minimize the $\chi^{2}_{\rm global}$ function with the free unknown
parameters. We do it for PPDFs at the NNLO approximation which additionally
includes the QCD cut off parameter, $\Lambda_{\rm QCD}$ which finally
specifies the polarized PDFs at Q${}_{0}^{2}$ = 1 GeV2.

This function is presented as it follows:

$\chi_{\rm global}^{2}({\rm p})=\sum_{n=1}^{N_{\rm exp}}w_{n}\chi_{n}^{2}\,.$
(19)

In this equation, ${w_{n}}$ is a weight factor for the $n^{\rm th}$ experiment
and $\chi_{n}^{2}$ is defined by:

$\chi_{n}^{2}(\rm p)=\left(\frac{1-{\cal N}_{n}}{\Delta{\cal
N}_{n}}\right)^{2}+\sum_{i=1}^{N_{n}^{\rm data}}\left(\frac{{\cal
N}_{n}\,g_{(1,2),i}^{\rm Exp}-g_{(1,2),i}^{\rm Theory}(p)}{{\cal
N}_{n}\,\Delta g_{(1,2),i}^{\rm Exp}}\right)^{2}\,.$ (20)

The minimization of the above $\chi_{\mathrm{\rm global}}^{2}(\rm p)$ function
is done using the CERN program library MINUIT James:1994vla . In the above
equation, the main contribution comes from the difference between the model
and the DIS data within the statistical precision. In the $\chi_{n}^{2}$
function, $g^{\rm Theory}$ indicates the theoretical value for the $i^{\rm
th}$ data point and $g^{\rm Exp}$, $\Delta g^{\rm Exp}$ represent the
experimental measurement and the experimental uncertainty (statistical and
systematic combined in quadrature) respectively.

To do a proper fit an over normalization factor for the data of experiment $n$
is needed which is denoted by ${\cal N}_{n}$. An uncertainty ${\Delta{\cal
N}_{n}}$ is attributed to this factor which should be considered in the fit.
These factors, considering the uncertainties, quoted by the experiments are
used to relate different experimental data sets. In fact they are taken as a
free parameters which are determined simultaneously with the other parameters
in the fit. They are obtained in the pre-fitting procedure and then fixed at
their best values in further steps. Numerical results for the unknown
parameters, resulted from $\chi^{2}$ minimization, are listed in Table.1 and
2. Different data sets which are used in the fit, is presented in Table.3.

Now we are at stage to do some analytical computations for a more confirmation
of the fitting validation, taken into account the several sun rules as we do
it in the next section.

Table 3: Summary of published polarized DIS experimental data points with measured $x$ and $Q^{2}$ ranges and the number of data points. Experiment | Ref. | [$x_{\rm min},x_{\rm max}$] | Q2 (GeV2) | Num. of data poi. | $\chi^{2}$ | ${\cal N}_{i}$
---|---|---|---|---|---|---
SLAC/E143(p) | Abe:1998wq  | [0.031–0.749] | 1.27–9.52 | 28 | 19.0218 | 0.99705
HERMES(p) | HERM98  | [0.028–0.66] | 1.01–7.36 | 39 | 55.2816 | 0.99982
SMC(p) | Adeva:1998vv  | [0.005–0.480] | 1.30–58.0 | 12 | 8.9328 | 1.00009
EMC(p) | Ashman:1987hv  | [0.015–0.466] | 3.50–29.5 | 10 | 3.8416 | 1.00592
SLAC/E155 | E155p  | [0.015–0.750] | 1.22–34.72 | 24 | 41.7453 | 0.99915
HERMES06(p) | HERMpd  | [0.026–0.731] | 1.12–14.29 | 51 | 21.0559 | 0.99915
COMPASS10(p) | COMP1  | [0.005–0.568] | 1.10–62.10 | 15 | 23.1003 | 1.00073
COMPASS16(p) | Adolph:2015saz  | [0.0035–0.575] | 1.03–96.1 | 54 | 52.6140 | 1.00296
SLAC/E143(p) | Abe:1998wq  | [0.031–0.749] | 2-3-5 | 84 | 122.0060 | 0.99578
HERMES(p) | HERM98  | [0.023–0.66] | 2.5 | 20 | 35.2073 | 0.99726
SMC(p) | Adeva:1998vv  | [0.003–0.4] | 10 | 12 | 14.8138 | 1.00071
Jlab06(p) | Dharmawardane:2006zd  | [0.3771–0.9086] | 3.48–4.96 | 70 | 99.6438 | 1.00127
Jlab17(p) | Fersch:2017qrq  | [0.37696–0.94585] | 3.01503–5.75676 | 82 | 171.5716 | 1.00282
$\bf{g_{1}^{p}}$ |  |  |  | 501 |  |
SLAC/E143(d) | Abe:1998wq  | [0.031–0.749] | 1.27–9.52 | 28 | 38.3735 | 1.00210
SLAC/E155(d) | E155d  | [0.015–0.750] | 1.22–34.79 | 24 | 20.0319 | 1.00228
SMC(d) | Adeva:1998vv  | [0.005–0.479] | 1.30–54.80 | 12 | 18.3574 | 1.00006
HERMES06(d) | HERMpd  | [0.026–0.731] | 1.12–14.29 | 51 | 44.4642 | 1.00654
COMPASS05(d) | COMP2005  | [0.0051–0.4740] | 1.18–47.5 | 11 | 7.3430 | 1.00760
COMPASS06(d) | COMP2006  | [0.0046–0.566] | 1.10–55.3 | 15 | 8.4408 | 1.00052
COMPASS17(d) | Adolph:2016myg  | [0.0045–0.569] | 1.03–74.1 | 43 | 36.2019 | 1.01090
SLAC/E143(d) | Abe:1998wq  | [0.031–0.749] | 2–3–5 | 84 | 127.5502 | 0.99981
$\bf{g_{1}^{d}}$ |  |  |  | 268 |  |
SLAC/E142(n) | E142n  | [0.035–0.466] | 1.10–5.50 | 8 | 8.0235 | 0.99881
HERMES(n) | HERM98  | [0.033–0.464] | 1.22–5.25 | 9 | 2.7585 | 0.99995
E154(n) | E154n  | [0.017–0.564] | 1.20–15.00 | 17 | 14.6888 | 0.99908
HERMES06(n) | Ackerstaff:1997ws  | [0.026–0.731] | 1.12–14.29 | 51 | 18.1873 | 0.99913
Jlab03(n) | JLABn2003  | [0.14–0.22] | 1.09–1.46 | 4 | 1.803e-2 | 0.99950
Jlab04(n) | JLABn2004  | [0.33–0.60] | 2.71–4.8 | 3 | 2.2174 | 1.05642
Jlab05(n) | JLABn2005  | [0.19–0.20] | 1.13–1.34 | 2 | 3.2639 | 0.98666
$\bf{g_{1}^{n}}$ |  |  |  | 94 |  |
E143(p) | Abe:1998wq  | [0.038–0.595] | 1.49–8.85 | 12 | 7.1338 | 1.00074
E155(p) | E155pdg2  | [0.038–0.780] | 1.1–8.4 | 8 | 11.9908 | 0.99886
Hermes12(p) | hermes2012g2  | [0.039–0.678] | 1.09–10.35 | 20 | 22.6010 | 0.99898
SMC(p) | SMCpg2  | [0.010–0.378] | 1.36–17.07 | 6 | 1.6804 | 1.00000
$\boldsymbol{g_{2}^{p}}$ |  |  |  | 46 |  |
E143(d) | Abe:1998wq  | [0.038–0.595] | 1.49–8.86 | 12 | 8.3504 | 1.00010
E155(d) | E155pdg2  | [0.038–0.780] | 1.1–8.2 | 8 | 1.9800 | 1.00296
$\boldsymbol{g_{2}^{d}}$ |  |  |  | 20 |  |
E143(n) | Abe:1998wq  | [0.038–0.595] | 1.49–8.86 | 12 | 8.87903 | 1.00001
E155(n) | E155pdg2  | [0.038–0.780] | 1.1–8.8 | 8 | 6.0324 | 1.01893
E142(n) | E142n  | [0.036–0.466] | 1.1–5.5 | 8 | 3.8955 | 0.99999
Jlab03(n) | JLABn2003  | [0.14–0.22] | 1.09–1.46 | 4 | 0.9362 | 0.99337
Jlab04(n) | JLABn2004  | [0.33–0.60] | 2.71–4.83 | 3 | 3.9915 | 1.10299
Jlab05(n) | JLABn2005  | [0.19–0.20] | 1.13–1.34 | 2 | 15.5600 | 0.98986
$\boldsymbol{g_{2}^{n}}$ |  |  |  | 37 |  |
Total |  966 | 1111.7891 |
Figure 4: The spin-dependent proton, neutron and deuteron structure functions,
$xg_{2}$, as a function of $x$ and Q2. Our results, MA22, at the NNLO
approximation (solid curve) are compared with KTA17 at the same approximation
(dashed) Khanpour:2017cha .  Figure 5: The spin-dependent proton structure
function, $xg_{1}$, as a function of $x$ and Q2. Our result, MA22, at the NNLO
approximation (solid curve) is compared with KTA17 at the same approximation
(dashed) Khanpour:2017cha  and with NAAMY21 at the NLO approximation (dashed-
dotted) Nematollahi:2021ynm .

##  VII The Sum Rules

Sum rules like total momentum fraction carried by partons or the total
contribution of parton spin to the spin of the nucleon are important tools to
investigate some fundamental properties of the nucleon structure. Inclusion of
TMCs and HT terms into the NNLO polarized structure function analysis leads to
an improvement for the precision of PPDF determination as well as QCD sum
rules and we are exploring herein their effects. In what are following by
utilizing available experimental data, we describe some important polarized
sum rules.

###  VII.1 Bjorken sum rule

The polarized Bjorken sum rule expresses the integral over the spin
distributions of quarks inside the nucleon in terms of its axial charge,
$g_{A}$ (as measured in neutron $\beta$ decay), times a coefficient function,
$C_{Bj}[\alpha_{s}(Q^{2})]$ Bjorken:1969mm , and considering higher twist (HT)
corrections, it is given by

$\displaystyle\Gamma_{1}^{\rm NS}(Q^{2})$ $\displaystyle=$
$\displaystyle\Gamma_{1}^{p}(Q^{2})-\Gamma_{1}^{n}(Q^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}[g_{1}^{p}(x,Q^{2})-g_{1}^{{n}}(x,Q^{2})]dx$
$\displaystyle=$ $\displaystyle\frac{1}{6}\leavevmode\nobreak\
|g_{A}|\leavevmode\nobreak\ C_{Bj}[\alpha_{s}(Q^{2})]+\text{HT
corrections}\,.$

Bjorken sum rule potentially provides a very precise handle on the
$\alpha_{s}$ as strong coupling constant. The value of coupling can be
extracted via $C_{Bj}[\alpha_{s}(Q^{2})]$ expression from experimental data.
This function has been calculated in 4-loop pQCD corrections in the massless
Baikov:2010je  and very recently massive cases Blumlein:2016xcy . As
previously reported in Ref. Altarelli:1998nb , determination of $\alpha_{s}$
from the Bjorken sum rule suffers from small-$x$ extrapolation ambiguities.

The $\alpha_{s}$ is also available form accurate methods to compute the width
decay of $\tau$-lepton and the $Z$-boson into hadrons akrami1 ; akrami2 . An
important test of QCD consistency can be offered by comparing these values.

Our results for the Bjorken sum rule can be compared with experimental
measurements such as E143 Abe:1998wq , SMC SMCpg2 , HERMES06 HERMpd  and
COMPASS16 Adolph:2015saz . The comparisons indicate an adequate consistency as
we list them in Table 4.

Table 4:  Comparison our computed MA22 result for the Bjorken sum rule, $\Gamma_{1}^{NS}$, with world data from E143 Abe:1998wq , SMC SMCpg2 , HERMES06 HERMpd  and COMPASS16 Adolph:2015saz . Only HERMES06 HERMpd  results are not extrapolated in full $x$ range (measured in region $0.021\leq x\leq 0.9$).  |  E143 Abe:1998wq  |  SMC SMCpg2  |  HERMES06 HERMpd  |  COMPASS16 Adolph:2015saz  |  KTA17 Khanpour:2017cha  | MA22
---|---|---|---|---|---|---
|  $Q^{2}=5$ GeV2 |  $Q^{2}=5$ GeV2 |  $Q^{2}=5$ GeV2 |  $Q^{2}=3$ GeV2 |  $Q^{2}=5$ GeV2 |  $Q^{2}=5$ GeV2
$\Gamma^{\rm NS}_{1}$ | $0.164\pm 0.021$ | $0.181\pm 0.035$ | $0.148\pm 0.017$ | $0.181\pm 0.008$ | $0.167\pm 0.005$ | $0.171\pm 0.001$
Figure 6: The twist-3 contribution to $xg_{1}^{p}$ at $Q^{2}$=4 GeV2 as a
function of $x$ is compared with the results of LSS Leader:2006xc  and KTA17
Khanpour:2017cha  at NNLO approximation and with JAM Jimenez-Delgado:2013boa
at the NLO approximation.  Figure 7: The twist-3 contribution to $xg_{2}^{p}$
at $Q^{2}$ = 4 GeV2 as a function of $x$. Our result, MA22 (solid curve), is
compared with KTA17 at the NNLO approimation Khanpour:2017cha (dashed-dotted),
JAM Jimenez-Delgado:2013boa (dashed) and BLMP Braun:2011aw  (dashed dashed
dotted) at the NLO approximation. E143 experimental data Abe:1998wq  have also
been added.  Figure 8: The twist-3 contribution of $xg_{2}$ for the proton,
neutron, and deuteron as a function of $x$ and for different values of Q2
according to our result, MA22, at the NNLO analysis.  Figure 9: The twist-3
contribution of $xg_{1}$ for the proton, neutron, and deuteron as a function
of $x$ and for different values of Q2 according to our results, MA22, at the
NNLO analysis.

###  VII.2 Proton helicity sum rule

This sum rule is related to the extrapolation of proton spin among its
constituents that is completing our knowledge in the field of nuclear physics
Leader:2016sli . An accurate picture of the quark and gluon helicity density
are obtained, considering proton’s momentum sum rule that needs a precise
extraction of PPDFs.

The spin of the nucleon are carried by its constituents that is generally
represented by

$\frac{1}{2}=\frac{1}{2}\Delta\Sigma(Q^{2})+\Delta{\mathrm{G}}(Q^{2})+{\mathrm{L}}(Q^{2}).$
(22)

Here $\Delta\Sigma(Q^{2})=\sum_{i}\int_{0}^{1}dx\leavevmode\nobreak\ (\Delta
q(x,Q^{2})+\Delta\bar{q}(x,Q^{2}))$ denotes spin contribution of the singlet
flavour, $\Delta{\rm G(Q^{2})}=\int_{0}^{1}dx\leavevmode\nobreak\ \Delta
g(x,Q^{2})$ is interpreted as the gluon spin contribution and finally
${\mathrm{L}}(Q^{2})$ represents the total contribution from quark and gluon
orbital angular momentum. Each individual term in Eq.(22) is a function of
$Q^{2}$ but the sum is not. Finding a way to measure them is a real challenge.
Describing the measurement methods is the beyond the scope of this paper.

In Table 5 the amount of first moment for the singlet-quark and gluon are
listed at Q2=10 GeV2. Our results are compared to those from the NNPDFpol1.0
Ball:2013lla , NNPDFpol1.1 Nocera:2012hx  and DSSV08 deFlorian:2008mr  at both
truncated and full $x$ region.

In Table 6 our results, MA22, are presented and compared with the results of
DSSV08 deFlorian:2008mr , BB10 Blumlein:2010rn , LSS10 Leader:2010rb  ,
NNPDFpol1.0 Ball:2013lla  and KTA17 Khanpour:2017cha  at Q2=4 GeV2.

As can be seen from the Table 5 and Table 6 for the $\Delta\Sigma$, our MA22
results are consistent within uncertainties with that of other groups. It is
back to this reason that the first moment of polarized densities are mainly
fixed by semileptonic decays. Very different values are reported by various
groups when we turn to the gluon. Considering their large uncertainty are
avoiding us to reach a firm conclusion about the full first moment of gluon.

Table 5:  Results for the full and truncated first moments of the polarized singlet-quark $\Delta\Sigma(Q^{2})=\sum_{i}\int_{0}^{1}dx[\Delta q_{i}(x)+\Delta\bar{q}_{i}(x)]$ and gluon distributions at the scale Q2=10 GeV2 in the $\overline{{\rm MS}}$–scheme. Also shown are the recent polarized global analysis of NNPDFpol1.0 Ball:2013lla , NNPDFpol1.1 Nocera:2012hx  and DSSV08 deFlorian:2008mr .  |  DSSV08 deFlorian:2008mr  |  NNPDFpol1.0 Ball:2013lla  |  NNPDFpol1.1 Nocera:2012hx  |  KTA17 Khanpour:2017cha  | MA22
---|---|---|---|---|---
Full $x$ region $[0,1]$ |  |  |  |  |
$\Delta\Sigma{\rm(Q^{2})}$ | $0.242$ | $+0.16\pm 0.30$ | $+0.18\pm 0.21$ | $0.2587\pm 0.044$ | $0.2445\pm 0.0048$
$\Delta{\rm G(Q^{2})}$ | $-0.084$ | $-0.95\pm 3.87$ | $0.03\pm 3.24$ | $0.2104\pm 0.034$ | $0.1205\pm 0.03$
Truncated $x$ region [$10^{-3},1$] |  |  |  |  |
$\Delta\Sigma{\rm(Q^{2})}$ | $0.366\pm 0.017$ | $+0.23\pm 0.15$ | $+0.25\pm 0.10$ | $0.2661\pm 0.038$ | $0.2551\pm 0.0066$
$\Delta{\rm G(Q^{2})}$ | $0.013\pm 0.182$ | $-0.06\pm 1.12$ | $0.49\pm 0.75$ | $0.2104\pm 0.034$ | $0.1205\pm 0.03$
Table 6: Same as Table 5, but only for the full first moments of the polarized singlet-quark and gluon distributions at the scale Q${}_{0}^{2}$ = 4 GeV2 in the $\overline{{\rm MS}}$–scheme. Those of DSSV08 deFlorian:2008mr , BB10 Blumlein:2010rn , LSS10 Leader:2010rb  and NNPDFpol1.0 Ball:2013lla  are presented for comparison.  |  DSSV08 deFlorian:2008mr  |  BB10 Blumlein:2010rn  |  LSS10 Leader:2010rb  |  NNPDFpol1.0 Ball:2013lla  |  KTA17 Khanpour:2017cha  | MA22
---|---|---|---|---|---|---
$\Delta\Sigma{\rm(Q^{2})}$ | $0.245$ | $0.193\pm 0.075$ | $0.207\pm 0.034$ | $0.18\pm 0.20$ | $0.1774\pm 0.029$ | $0.2607\pm 0.0065$
$\Delta{\rm G(Q^{2})}$ | $-0.096$ | $0.462\pm 0.430$ | $0.316\pm 0.190$ | $-0.9\pm 4.2$ | $0.1882\pm 0.0294$ | $0.1095\pm 0.027$

Based on the extracted values presented in Table 6 we can finally discuss the
proton spin sum rule. Hence the amount of quark and gluon orbital angular
momentum to the spin of the proton would be:

${\mathrm{L}}(Q^{2}=4\leavevmode\nobreak\ \rm GeV^{2})=0.3591\pm 0.0779\,.$
(23)

A definite conclusion about the contribution of the total orbital angular
momentum to the spin of the proton can not be done because of the large
uncertainty that is mainly originating from the gluons. To obtain a precise
determination of each individual contribution, it is required to improve the
current level of experimental accuracy.

###  VII.3 The twist-3 reduced matrix element $d_{2}$

One of the quantity which is not considered as a sum rule but its numerical
evaluation is remarkable to invastigate the higher twist effect is the twist-3
reduced matrix element and is denoted by $d_{2}$. Detailed of higher twist
analyses for $g_{1}$ polarized structure function have been performed in
Blumlein:2010rn . In operator product expansion (OPE) theorem ope  the effect
of quark-gluon correlations can be studied through the moments of $g_{1}$ and
$g_{2}$ structure functions. These moments lead to definition of reduced
matrix element, $d_{2}(Q^{2})$, as it follows

$\displaystyle d_{2}(Q^{2})$ $\displaystyle=$ $\displaystyle
3\int_{0}^{1}x^{2}\bar{g_{2}}(x,Q^{2})\leavevmode\nobreak\ dx$ (24)
$\displaystyle=$
$\displaystyle\int_{0}^{1}x^{2}[3g_{2}(x,Q^{2})+2g_{1}(x,Q^{2})]\leavevmode\nobreak\
dx.$

In this equation $\bar{g}_{2}=g_{2}-g_{2}^{WW}$ where $g_{2}^{WW}$ is given by
Wandzura and Wilczek (WW) relation as in Eq.(3). The $d_{2}(Q^{2})$ that is in
fact the twist-3 reduced matrix element of spin dependent operators in
nucleon, can be used to measure the deviation of $g_{2}$ from
$g_{2}^{\tau_{2}}$. Due to the $x^{2}$ weighting factor in Eq.(24), this
matrix element is specially sensitive to the large-$x$ behaviour of
$\bar{g_{2}}$. Some insights into the size of the multi-parton correlation
terms can be obtained by extracting the $d_{2}$ which indicates its important.

The significance of higher twist terms in QCD analyses is revealed by having
non-zero value for $d_{2}$. To achieve precise information on the higher twist
operators and to improve model prediction, a much more accurate experimental
measurement for $d_{2}$ is required. In Table 7 we present our results for
$d_{2}$ which are compared with the other theoretical predictions and also
experimental values.

Table 7:  $d_{2}$ moments of the proton, neutron and deuteron polarized structure functions from the SLAC E155x Kuhn:2008sy , E01-012 Solvignon:2013yun , E06-014 Flay:2016wie , Lattice QCD Gockeler:2005vw , CM bag model Song:1996ea , JAM15 Sato:2016tuz , JAM13 Jimenez-Delgado:2013boa , KTA17 Khanpour:2017cha  compared with MA22 results.  | Ref. |  $Q^{2}$ [GeV2] | $10^{2}d^{p}_{2}$ | $10^{5}d^{n}_{2}$ | $10^{3}d^{d}_{2}$
---|---|---|---|---|---
MA22 |  | $5$ | $1.0929\pm 0.0106$ | $209.095\pm 3.96$ | $7.206\pm 0.078$
KTA17 | Khanpour:2017cha  | $5$ | $0.718\pm 0.01$ | $105.36\pm 74.58$ | $5.16\pm 0.02$
E06-014 | Flay:2016wie  | 3.21 |  | $-421.0\pm 79.0\pm 82.0\pm 8.0$ | -
E06-014 | Flay:2016wie  | 4.32 |  | $-35.0\pm 83.0\pm 69.0\pm 7.0$ | -
E01-012 | Solvignon:2013yun  | 3 | - | $-117\pm 88\pm 138$ | -
E155x | E155pdg2  | $5$ | $0.32\pm 0.17$ | $790\pm 480$ | -
E143 | Abe:1998wq  | $5$ | $0.58\pm 0.50$ | $500\pm 2100$ | $5.1\pm 9.2$
Lattice QCD | Gockeler:2005vw  | 5 | 0.4(5) | -100(-300) | -
CM bag model | Song:1996ea  | $5$ | $1.74$ | $-253$ | $6.79$
JAM15 | Sato:2016tuz  | $1$ | $0.5\pm 0.2$ | $-100\pm 100$ | -
JAM13 | Jimenez-Delgado:2013boa  | $5$ | $1.1\pm 0.2$ | $200\pm 300$ | -

###  VII.4 Burkhardt-Cottingham (BC) sum rule

Considering dispersion relations for virtual Compton scattering in all Q2,
Burkhardt and Cottingham predicted that the zeroth moment of $g_{2}$ goes to
zero Burkhardt:1970ti  such as:

$\Gamma_{2}=\int_{0}^{1}dx\,g_{2}(x,Q^{2})=0\leavevmode\nobreak\ .$ (25)

This relation is called Burkhardt-Cottingham (BC) sum rule and is trivial
consequence of the WW relation for $g_{2}^{\tau_{2}}$ (see Eq.(3)). It should
be noted that zeroth moment of structure function does not exist in the light
cone expansion and hence can not be described by local operator product
expansion Blumlein:1996vs . Even if the target mass corrected structure
function is used, this sum rule is still established Blumlein:1998nv .
Consequently any violation of the BC sum rule is an evidence for the presence
of HT contributions hermes2012g2 .

Our MA22 results for $\Gamma_{2}$ together with data from E143 Abe:1998wq ,
E155 E155pdg2 , HERMES2012 hermes2012g2 , RSS Slifer:2008xu , E01012
Solvignon:2013yun  groups for proton, deuteron and neutron are listed in table
8. The low-$x$ behaviour of $g_{2}$ which is not yet precisely measured, has
considerable effect on any conclusion which we might be get.

The BC sum rule can be obtained analytically from the covariant parton model
as it is discussed in cpm1-1 .

Table 8:  Comparison of the result of BC sum rule for $\Gamma_{2}^{p}$, $\Gamma_{2}^{d}$ and $\Gamma_{2}^{n}$ with world data from E143 Abe:1998wq , E155 E155pdg2 , HERMES2012 hermes2012g2 , RSS Slifer:2008xu , E01012 Solvignon:2013yun .  |  E143 Abe:1998wq  |  E155 E155pdg2  |  HERMES2012 hermes2012g2  |  RSS Slifer:2008xu  |  E01012 Solvignon:2013yun  |  KTA17 Khanpour:2017cha  | MA22
---|---|---|---|---|---|---|---
| $0.03\leq x\leq 1$ | $0.02\leq x\leq 0.8$ | $0.023\leq x\leq 0.9$ | $0.316<x<0.823$ | $0\leq x\leq 1$ | $0.03\leq x\leq 1$ | $0.03\leq x\leq 1$
|  $Q^{2}=5$ GeV2 |  $Q^{2}=5$ GeV2 |  $Q^{2}=5$ GeV2 |  $Q^{2}=1.28$ GeV2 |  $Q^{2}=3$ GeV2 |  $Q^{2}=5$ GeV2 |  $Q^{2}=5$ GeV2
$\Gamma_{2}^{p}$ | $-0.014\pm 0.028$ | $-0.044\pm 0.008$ | $0.006\pm 0.029$ | $-0.0006\pm 0.0022$ | … | $-0.0196\pm 0.0011$ | $-0.01554\pm 0.00033$
$\Gamma_{2}^{d}$ | $-0.034\pm 0.082$ | $-0.008\pm 0.012$ | - | $-0.0090\pm 0.0026$ | … | $-0.0036\pm 0.0005$ | $-0.00401\pm 0.00006$
$\Gamma_{2}^{n}$ | - | - | - | $-0.0092\pm 0.0035$ | $0.00015\pm 0.00113$ | $\leavevmode\nobreak\ 0.0060\pm 0.0001$ | $\leavevmode\nobreak\ 0.00721\pm 0.00033$

###  VII.5 Efremov-Leader-Teryaev (ELT) Sum Rule

Considering the valence part of $g_{1}$ and $g_{2}$ structure functions and
integrating them over $x$ variable the Efremov-Leader-Teryaev (ELT) sum rule
is obtained. The ELT sum rule is derived like the Bjorken sum rule since the
sea quarks are assumed to be identical in protons and neutrons. Hence it
appears as:

$\displaystyle\int_{0}^{1}dx\leavevmode\nobreak\
x[g_{1}^{V}(x)+2g_{2}^{V}(x)]=$
$\displaystyle\int_{0}^{1}dx\leavevmode\nobreak\
x[g_{1}^{p}(x)-g_{1}^{n}(x)+2(g_{2}^{p}(x)-g_{2}^{n}(x))]=0.$ (26)

This sum rule is only valid in the case of massless quarks and receives
corrections from the quark mass but under presence of target mass corrections
is preserved Blumlein:1996vs . Like the BC sum rule, the ELT sum rule can be
obtained by analytical considerations of CPM. More details can be found in
cpm1-1 .

By combining the data of E143 Abe:1998wq  and E155 E155pdg2  the numerical
value for this sum rule at Q2=5 GeV2 is $-0.011\pm 0.008$ and what we obtain
at the same energy scale would be $0.01017\pm 0.00004$.

##  VIII Comparison for the spin structure functions

Since our QCD analysis has been validated by extracting the PPDFs via the
fitting processes and also obtaining their evolved outputs and in continuation
by considering several sum rules, we are now at the position to investigate
the polarized structure functions. In this regard, we first back to what we
got before. Our results, MA22 PPDFs, as a function of $x$ at Q${}_{0}^{2}=$ 1
GeV2 along with the corresponding uncertainty bounds, is presented in Fig. 1.

The evolution of MA22 polarized parton distributions for a selection of
$Q^{2}$ values indicates in Fig.2 while for comparison various
parameterizations of KTA17 Khanpour:2017cha ,KATAO11 Khorramian:2010qa ,TKAA16
Shahri:2016uzl ,NAAMY21 Nematollahi:2021ynm  at the NLO approximation are
illustrated there. It is seen that by increasing $Q^{2}$, except for the gluon
density, the evolution of all distributions tends to flatten out the peak.

Now for the structure functions, we see that in different panels of Fig.3, our
MA22 predictions for the polarized structure functions of the proton
$xg_{1}^{p}(x,Q^{2})$, neutron $xg_{1}^{n}(x,Q^{2})$ and deuteron
$xg_{1}^{d}(x,Q^{2})$ are compared with respect to the fixed-target DIS
experimental data from E143. As we mentioned, MA22 refers to ‘pQCD+TMC+HT’
scenario. The results from KATAO11 analysis in NLO approximation
Khorramian:2010qa , TKAA16 analysis in NNLO approximation Shahri:2016uzl ,
KTA17 analysis in NNLO approximation Khanpour:2017cha , THK14 analysis in NLO
approximation TaheriMonfared:2014var  and finally NAAMY21 analysis in NLO
approximation Nematollahi:2021ynm  are also depicted there. We find our
results are in good agreement with the experimental data and in accord with
other determinations over the entire range of $x$ at $Q^{2}$=5 GeV 2.

Further illustrations of the fit quality are presented in different panels of
Fig.4, for the $xg_{2}^{i=p,n,d}(x,Q^{2})$ polarized structure functions,
obtained from Eq. (16). In comparison with the $g_{1}$ data, the $g_{2}$ data
have generally larger uncertainties which indicates the lack of knowledge for
the $g_{2}$ structure function. At the current level of accuracy, MA22 is in
agreement with data within their uncertainties. We need to a large number of
data with higher precision to get a precise quantitative extraction of the
$xg_{2}(x,Q^{2})$. In fact we concentrate on the general characteristic of the
$xg_{2}(x,Q^{2})$ structure function.

Fig. 5 is presenting our MA22 prediction for the polarized structure functions
of the proton $xg_{1}^{p}(x,Q^{2})$ while a comparison with the fixed-target
DIS experimental data from JLAB17 Fersch:2017qrq  is done there.

Fig. 6 represents our $xg_{1}^{\tau 3}(x,Q^{2})$ with the results from LSS
Leader:2006xc  and JAM Jimenez-Delgado:2013boa  groups. Analysis of the LSS
group is based on splitting the measured $x$ region into seven bins to
determine the HT correction to $g_{1}$. The HT contribution has been extracted
by LSS group in a model-independent way while its scale dependence is ignored.
On the other side an analytical form for the twist-3 part of $g_{2}$ is
parameterized by the JAM group where using integral relation of Eq.(14) they
calculated $g_{1}^{\tau 3}$ at the NLO accuracy in a global fit.

E143 collaboration at SLAC reported the twist-3 contribution to proton spin
structure function $xg_{2}^{p}$ structure function with relatively large
errors Abe:1998wq . We employ them and present our MA22 results for twist-3
part of $g_{2}$ in Fig. 7 which are accompanied with those of JAM Jimenez-
Delgado:2013boa  and BLMP Braun:2011aw  groups.

However, within experimental precision the $g_{2}$ data are well described by
the twist-2 contribution but the precision of the current data is not
sufficient enough to distinguish model precision. Hence we compute twist-3
part of $g_{2}$ for different targets and depict them in Fig.8 which has
significant contribution even at large Q2 values.

In continuation to have a comparison, we compute the $xg_{\rm 1}^{\tau_{3}}$
and indicate them in Fig.9. We find that these functions vanish rapidly at
$Q^{2}>5$ GeV2 where in the limit of Q${}^{2}\rightarrow,\infty$, the $xg_{\rm
2}^{\tau_{3}}$ remains nonzero.

Up here we focused on longitudinal polarized parton densities and structure
functions. In next section we utilize our MA22 analysis which we have done
before to illustrate the transversal case which are including the polarized
TMDs.

Figure 10: The TMD $g_{1}^{q}(x,\mathbf{p_{T}})$ for $u$\- (upper panel) and
$d$-quarks (lower panel). Left panel: $g_{1}^{q}(x,\mathbf{p_{T}})$ as
function of $x$ for $p_{T}/M=0.10$ (dashed), 0.13 (dotted), 0.20 (dash-dotted
line). Right panel: $g_{1}^{q}(x,\mathbf{p_{T}})$ as function of $p_{T}/M$ for
$x=0.15$ (solid), 0.18 (dashed), 0.22 (dotted), 0.30 (dash-dotted line).
Figure 11: $h_{1}^{q}(x,\mathbf{p_{T}})$, for $u$\- and $d$-quarks.  Left
panel: The TMDs as functions of $x$ for $p_{T}/M=0.10$ (dashed), 0.13
(dotted), 0.20(dash-dotted lines). Right panel: The TMDs as functions of
$p_{T}/M$ for $x=0.15$ (solid), 0.22 (dotted), 0.30 (dash-dotted lines).
Figure 12: $g_{1T}^{\bot q}(x,\mathbf{p_{T}})$, for $u$\- and $d$-quarks.
Left panel: The TMDs as functions of $x$ for $p_{T}/M=0.10$ (dashed), 0.13
(dotted), 0.20(dash-dotted lines). Right panel: The TMDs as functions of
$p_{T}/M$ for $x=0.15$ (solid), 0.22 (dotted), 0.30 (dash-dotted lines).
Figure 13: $h_{1T}^{\perp q}(x,\mathbf{p_{T}})$ for $u$\- and $d$-quarks.
Left panel: The TMDs as functions of $x$ for $p_{T}/M=0.10$ (dashed), 0.13
(dotted), 0.20(dash-dotted lines). Right panel: The TMDs as functions of
$p_{T}/M$ for $x=0.15$ (solid), 0.18 (dashed), 0.22 (dotted), 0.30 (dash-
dotted lines).

##  IX Predictions for polarized TMDs

Since we achieved to sufficient information on longitudinal polarized parton
distributions and structure function, we are now at a situation to utilize the
covariant parton model cpm1 ; cpm2  and extract the transverse momentum
dependent (TMD) distributions in polarized case. Indeed TMDs provide us new
insight toward a more complete understanding of the quark-gluon structure in a
nucleon Collins:2003fm ; Collins:2007ph ; Collins:1999dz ; Collins:2000gd ;
Hautmann:2007uw ; Mulders:1995dh ; Bacchetta:2006tn . Without a more accurate
and realistic picture in three dimensions of the nucleon which includes
naturally transverse motion, it would be hard to explain some experimental
observations. In fact TMDs provide such pictures and their necessities feel
more and more in nucleon investigations.

The first and simplest example of quark TMD is $f_{1}^{q}(x,k_{T})$. It arises
when an unpolarized beam scatters off an unpolarized target hadron, and
therefore does not carry quark/hadron spin information. The function
$f_{1}^{q}(x,k_{T})$ provides the probability that a beam particle strikes a
target quark of momentum fraction $x$ and transverse momentum $k_{T}$. It is
related to the traditional DIS PDF $f_{1}^{q}(x)$ by $\int
d^{2}k_{T}\leavevmode\nobreak\ f_{1}^{q}(x,k_{T})=f_{1}^{q}(x)$.

Similarly to $f_{1}^{q}(x,k_{T})$, we get the $g_{1}^{q}(x,k_{T})$ as
longitudinal polarized TMD and $h_{1}^{q}(x,k_{T})$ as transverse polarized
TMD, whose integrals are denoted respectively by $g_{1}^{q}(x)$ (presented
before by $\Delta{q_{i}}(x)$) and $h_{1}^{q}(x)$ that we know them as quark
longitudinal polarized (helicity) distribution and the quark transversity
distribution.

In addition to the three above TMDs for quarks which are direct extension of
the DIS PDFs, there are five other quark TMDs which depend not only on the
magnitude of $k_{T}$ , but also on its direction. Therefore these TMDs vanish
if simply integrated over $k_{T}$, and do not directly connect to DIS PDFs.
They are:

1- The Sivers distribution $f_{1T}^{\bot,q}$ which expresses, in a
transversely polarized hadron, the asymmetric distribution of the quark
transverse momentum, $p_{z}$, around the center of the $p_{x}$ and $p_{y}$
plane sivers . The appearance of azimuthal asymmetric quark distribution in
the transverse momentum space is often called the “Sivers effect”. This TMD
has opposite signs in semi-inclusive DIS (SIDIS) with respect to Drell-Yan
processes and it is therefore an odd time reversal function(T-odd function).

2- The Boer-Mulders function $h_{1}^{\bot,q}(x,k_{T})$ characterizes the
distribution of longitudinal polarized quarks in an unpolarized hadron B-M1 .
It is also a T-odd function, like $f_{1T}^{\bot,q}$. The rest tree TMDs are:

3-Function $h_{1T}^{\bot,q}(x,k_{T})$ which is describing a transverse
polarized quark inside a transverse polarized nucleon while its direction is
perpendicular to a polarized nucleon. It is called Pretzelosity function.

4-Function $g_{1T}^{\bot,q}(x,k_{T})$ that is describing the longitudinal
polarized quark inside a transverse polarized nucleon and is named as Worm-
gear-I function. And finally:

5- Worm-gear-II function, denoted by $h_{1L}^{\bot,q}(x,k_{T})$ and is
describing the transverse polarized quark inside a longitudinal polarized
nucleon,

Similarly to quark TMDs, gluon TMDs allow access to the gluonic orbital
angular momentum, another possibly important contribution to the nucleon spin.
Just as there are eight TMDs for quarks, there are eight gluon TMDs me . Gluon
TMDs were first proposed in 2001 ge .

Here we only consider the Quark TMDs that are twist-2 naively and time-
reversal even (T-even) functions. They have been extracted via covariant
parton model (CPM) which is based on the Lorentz invariance and the assumption
of a rotationally symmetric distribution of parton momenta in the nucleon rest
frame Efremov:2009ze .

As a result of CPM, T-even polarized TMDs can be obtained at the leading twist
approximation, in terms of a single “generating function”
$K^{q}(x,\mathbf{p}_{T})$. They are given by Efremov:2011ye ; Bastami:2020rxn

$\begin{array}[c]{rcrcl}g_{1}^{q}(x,\mathbf{p}_{T})&=&\displaystyle\frac{1}{2x}\left(\left(x+\frac{m}{M}\right)^{2}-\frac{\mathbf{p}_{T}^{2}}{M^{2}}\right)&\times&K^{q}(x,\mathbf{p}_{T})\;,\\\
h_{1}^{q}(x,\mathbf{p}_{T})&=&\displaystyle\frac{1}{2x}\left(x+\frac{m}{M}\right)^{2}&\times&K^{q}(x,\mathbf{p}_{T})\;,\\\
g_{1T}^{\perp,q}(x,\mathbf{p}_{T})&=&\displaystyle\frac{1}{x}\left(x+\frac{m}{M}\right)&\times&K^{q}(x,\mathbf{p}_{T})\;,\\\
h_{1L}^{\perp,q}(x,\mathbf{p}_{T})&=&\displaystyle-\,\frac{1}{x}\left(x+\frac{m}{M}\right)&\times&K^{q}(x,\mathbf{p}_{T})\;,\\\
h_{1T}^{\perp,q}(x,\mathbf{p}_{T})&=&\displaystyle-\frac{1}{x}&\times&K^{q}(x,\mathbf{p}_{T})\;.\end{array}$
(27)

According to Efremov:2009ze ) $K^{q}(x,\mathbf{p}_{T})$ as generating function
is defined in compact notation by

$\displaystyle K^{q}(x,\mathbf{p}_{T})$ $\displaystyle=$ $\displaystyle
M^{2}x\int\mathrm{d}\\{p^{1}\\}$ (28) $\displaystyle\mathrm{d}\\{p^{1}\\}$
$\displaystyle\equiv$
$\displaystyle\frac{\mathrm{d}p^{1}}{p^{0}}\;\frac{H^{q}(p^{0})}{p^{0}+m}\;\delta\left(\frac{p^{0}+p^{1}}{M}-x\right)\,.$
(29)

It can be shown that due to rotational symmetry the following relations hold
Efremov:2011ye :

$K^{q}(x,\mathbf{p}_{T})=M^{2}\frac{H^{q}(\bar{p}^{0})}{\bar{p}^{0}+m},\qquad\bar{p}^{0}=\frac{1}{2}\,xM\,\left(1+\frac{\mathbf{p}_{T}^{2}+m^{2}}{x^{2}M^{2}}\right),$
(30) $\pi
x^{2}M^{3}H^{q}\\!\left(\frac{M}{2}x\right)=2\int_{x}^{1}\frac{\mathrm{d}y}{y}\;g_{1}^{q}(y)+3\,g_{1}^{q}(x)-x\;\frac{\mathrm{d}g_{1}^{q}(x)}{\mathrm{d}x},$
(31)

In deriving Eq.(31) the limit $m\rightarrow 0$ has been taken. Consequently
the following result in that limit would be obtained for the generating
function Efremov:2011ye :

$K^{q}(x,\mathbf{p}_{T})=\frac{H^{q}(\frac{M}{2}\xi)}{\,\frac{M}{2}\xi}=\frac{2}{\pi\xi^{3}M^{4}}\left(2\int_{\xi}^{1}\frac{\mathrm{d}y}{y}\;g_{1}^{q}(y)+3\,g_{1}^{q}(\xi)-x\;\frac{\mathrm{d}g_{1}^{q}(\xi)}{\mathrm{d}\xi}\right),\quad\xi=\,x\,\left(1+\frac{\mathbf{p}_{T}^{2}}{x^{2}M^{2}}\right).$
(32)

Substituting the above relations in Eq.(27), the following result for the
$g_{1}^{q}(x,\mathbf{p}_{T})$ would be obtained:

$g_{1}^{q}(x,\mathbf{p}_{T})=\frac{2x-\xi}{\pi\xi^{3}M^{3}}\left(2\int_{\xi}^{1}\frac{\mathrm{d}y}{y}\;g_{1}^{q}(y)+3\,g_{1}^{q}(\xi)-\xi\;\frac{\mathrm{d}g_{1}^{q}(\xi)}{\mathrm{d}\xi}\right).$
(33)

Based on above relation and using the MA22 analysis which we did in this paper
for $g_{1}^{q}(x)$ at $4\,\mathrm{GeV}^{2}$ in the NNLO approximation, we
could obtain the result for $g_{1}^{q}(x,\mathbf{p}_{T})$ which has been shown
in Fig.[10] for $u$ and $d$ quarks.

Using Eq.(27) and in the limit $m\to 0$ the other TMDs can be obtained. They
are presented in below which which are different by simple $x$-dependent
prefactors Efremov:2011ye :

$\displaystyle
h_{1}^{q}(x,\mathbf{p}_{T})=\frac{x}{2}K^{q}(x,\mathbf{p}_{T}),$
$\displaystyle\;\;g_{1T}^{\perp,q}(x,\mathbf{p}_{T})=K^{q}(x,\mathbf{p}_{T}),\;\;$
$\displaystyle
h_{1T}^{\perp,q}(x,\mathbf{p}_{T})=-\frac{1}{x}K^{q}(x,\mathbf{p}_{T}).$ (34)

The result for $h_{1}^{q}(x,\mathbf{p}_{T})$ is depicted in Fig. 11. In Fig.
12 the result for $g_{1T}^{\perp q}$ with respect to $x$ and $p_{T}/M$ is
shown. It does not need to plot $h_{1L}^{\perp q}$ since in the used approach
this TMD is equal to $-g_{1T}^{\perp q}$ Efremov:2009ze . As can be seen from
Fig.10, $g_{1}^{q}(x,\mathbf{p}_{T})$ is the only TMD which has positive and
negative values. The other TMDs in other figures do not change sign which
follows from Eqs.(27, 34).

We should note that among all TMDs, as we see from Fig.13, $h_{1T}^{\perp
q}(x,\mathbf{p}_{T})$ as pretzelosity function has largest absolute value
which is due to the prefactor $1/x$. This function has its own worth since in
some quark models She:2009jq ; Avakian:2010br , including the utilized
approach in Efremov:2010cy ; Avakian:2010nz , this function is related to
quark orbital angular momentum.

##  X Conclusions

Determining the nucleon spin structure functions $g_{1}(x,Q^{2})$ and
$g_{2}(x,Q^{2})$ and their moments is the main goal of our present MA22
analysis. They are essential to test QCD sum rules and to evaluate the TMDs.
We provided a unified and consistent PPDF through an achievement, containing
an excellent description of the fitted data while we employed TMC and HT
effects in our analysis. Within the known very large uncertainties arising
from the lack of constraining data, our helicity distributions are in good
consistency with other extractions. Here the TMCs and HT effects, which are
relevant in the region of low $Q^{2}$, have also been studied for the several
sum rules at the NNLO approximation. Our results for the reduced matrix
element $d_{2}$ at the NNLO approximation have also been presented. We also
studied Burkhardt-Cottinghan and Efremov-Leader-Teryaev sum rules. To scrutiny
them more accurate data are needed.

Finally we studied the behavior of the TMD structure functions which are time-
reversal even with respect to $x$ and $p_{T}/M$ variables at the NNLO
approximation, based on the covariant parton model. Our MA22 results,
containing analysis of up to date and last data on nucleon spin structure
functions, with respect to what we did in Khanpour:2017cha , can be compared
with the results from Efremov:2011ye  which indicated adequate and acceptable
behaviours.

This study can be extended to include other TMDS while higher twist effect is
employed. We hope to report on this issue as our further research task.

## Acknowledgments

The Authors are indebted P. Zavda for reading the manuscript and providing
useful comments. We are grateful to O. V. Treyaev for his useful comments and
suggestion. We are appreciated P. Schweitzer to read our manuscript and give
us his opinion about it. S. A. T. is thankful from the School of Particles and
Accelerators, Institute for Research in Fundamental Sciences (IPM) to make the
required facilities to do this project. A. M acknowledges the Yazd university
for the provided facility to do this project.

## References

  * (1) C. Thorn, Phys. Rev. D 19, 693 (1979).
  * (2) C. Bouchiat, P. Fayet and P. Meyer, Nucl. Phys. B 34, 157 (1971).
  * (3) P. V. Landshoff and J. C. Polkinghorne, Phys. Rept. 5, 1-55 (1972).
  * (4) P. Zavada, Eur.Phys.J.C 52, 121 (2007).
  * (5) M. Anselmino, M. Boglione and F. Murgia, Phys. Lett. B 362, 164 (1995).
  * (6) A. Kotzinian, Nucl. Phys. B 441, 234 (1995).
  * (7) H. Mahdizadeh Saffar, A. Mirjalili, M. M. Yazdanpanah, S. Atashbar Tehrani,  Int.J. Mod.Phys.A 32, 1750175 (2017).
  * (8) J. D. Jackson, G. G. Ross and R. G. Roberts, Phys. Lett. B 226, 159-166 (1989).
  * (9) R. G. Roberts and G. G. Ross, Phys. Lett. B 373, 235-245 (1996).
  * (10) J. Blumlein and N. Kochelev, Phys. Lett. B 381, 296-304 (1996).
  * (11) S. J. Brodsky, D. S. Hwang and I. Schmidt, Phys. Lett. B 530, 99 (2002).
  * (12) F. Taghavi-Shahri, H. Khanpour, S. Atashbar Tehrani and Z. Alizadeh Yazdi, Phys. Rev. D 93, no. 11, 114024 (2016).
  * (13) H. Khanpour, S. T. Monfared and S. Atashbar Tehrani, Phys. Rev. D 95, no. 7, 074006 (2017).
  * (14) Y. Goto et al. [Asymmetry Analysis Collaboration], Phys. Rev. D 62, 034017 (2000).
  * (15) S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys. B 889, 351 (2014).
  * (16) J. Blümlein, P. Marquard, C. Schneider and K. Schönwald, JHEP 01, 193 (2022).
  * (17) J. Blümlein, P. Marquard, C. Schneider and K. Schönwald, Nucl. Phys. B 971, 115542 (2021).
  * (18) B. Lampe and E. Reya, Phys.Rept. 332, 1 (2000).
  * (19) E. B. Zijlstra and W. L. van Neerven, Nucl. Phys. B 417, 61 (1994). [Nucl. Phys. B 426, 245 (1994)] [Nucl. Phys. B 773, 105 (2007)].
  * (20) M. Lacombe, B. Loiseau, R. Vinh Mau, J. Cote, P. Pires and R. de Tourreil, Phys. Lett. B 101, 139 (1981).
  * (21) W. W. Buck and F. Gross, Phys. Rev. D 20, 2361 (1979).
  * (22) M. J. Zuilhof and J. A. Tjon, Phys. Rev. C 22, 2369 (1980).
  * (23) S. Wandzura and F. Wilczek, Phys. Lett. B 72, 195 (1977).
  * (24) A. N. Khorramian, S. Atashbar Tehrani, S. Taheri Monfared, F. Arbabifar and F. I. Olness, Phys. Rev. D 83, 054017 (2011).
  * (25) A. N. Khorramian, H. Khanpour and S. A. Tehrani, Phys. Rev. D 81, 014013 (2010).
  * (26) S. M. Moosavi Nejad, H. Khanpour, S. Atashbar Tehrani and M. Mahdavi, Phys. Rev. C 94, 045201 (2016).
  * (27) H. Khanpour, A. Mirjalili and S. Atashbar Tehrani, Phys. Rev. C 95, 035201 (2017).
  * (28) S. Atashbar Tehrani, F. Taghavi-Shahri, A. Mirjalili and M. M. Yazdanpanah, Phys. Rev. D 87, no.11, 114012 (2013). [erratum: Phys. Rev. D 88, no.3, 039902 (2013)]
  * (29) H. Nematollahi, P. Abolhadi, S. Atashbar, A. Mirjalili and M. M. Yazdanpanah, Eur. Phys. J. C 81, 18 (2021).
  * (30) C. Ayala and S. V. Mikhailov, Phys. Rev. D 92, no. 1, 014028 (2015).
  * (31) I. S. Barker, B. R. Martin and G. Shaw, Z. Phys. C 19, 147 (1983).
  * (32) I. S. Barker and B. R. Martin, Z. Phys. C 24, 255 (1984).
  * (33) V. G. Krivokhizhin, S. P. Kurlovich, V. V. Sanadze, I. A. Savin, A. V. Sidorov and N. B. Skachkov, Z. Phys. C 36, 51 (1987).
  * (34) V. G. Krivokhizhin, S. P. Kurlovich, R. Lednicky, S. Nemecek, V. V. Sanadze, I. A. Savin, A. V. Sidorov and N. B. Skachkov, Z. Phys. C 48, 347 (1990).
  * (35) J. Chyla and J. Rames, Z. Phys. C 31, 151 (1986).
  * (36) I. S. Barker, C. S. Langensiepen and G. Shaw, Nucl. Phys. B 186, 61 (1981).
  * (37) A. L. Kataev, A. V. Kotikov, G. Parente and A. V. Sidorov, Phys. Lett. B 417, 374 (1998).
  * (38) S. I. Alekhin and A. L. Kataev, Phys. Lett. B 452, 402 (1999).
  * (39) A. L. Kataev, G. Parente and A. V. Sidorov, Nucl. Phys. B 573, 405 (2000).
  * (40) A. L. Kataev, G. Parente and A. V. Sidorov, Phys. Part. Nucl. 34, 20 (2003). [Fiz. Elem. Chast. Atom. Yadra 34, 43 (2003)] [Phys. Part. Nucl. 38, no. 6, 827 (2007)].
  * (41) A. L. Kataev, JETP Lett. 81, 608 (2005). [Pisma Zh. Eksp. Teor. Fiz. 81, 744 (2005)].
  * (42) E. Leader, A. V. Sidorov and D. B. Stamenov, Int. J. Mod. Phys. A 13, 5573 (1998).
  * (43) J. Blumlein and A. Tkabladze, Nucl. Phys. B 553, 427 (1999).
  * (44) H. Georgi and H. D. Politzer, Phys. Rev. D 14, 1829 (1976).
  * (45) J. Blumlein and A. Tkabladze, Nucl. Phys. B Proc. Suppl. 79, 541-544 (1999).
  * (46) O. Nachtmann, Nucl. Phys. B 63, 237 (1973).
  * (47) J. Blumlein and H. Bottcher, Nucl. Phys. B 841, 205 (2010).
  * (48) V. M. Braun, T. Lautenschlager, A. N. Manashov and B. Pirnay, Phys. Rev. D 83, 094023 (2011).
  * (49) J. Blumlein and H. Bottcher, [arXiv:1207.3170 [hep-ph]].
  * (50) M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98, 030001 (2018).
  * (51) P. A. Zyla et al. [Particle Data Group], PTEP 2020, 083C01 (2020).
  * (52) J. Pumplin, D. Stump, R. Brock, D. Casey, J. Huston, J. Kalk, H. L. Lai and W. K. Tung, Phys. Rev. D 65, 014013 (2001).
  * (53) K. Abe et al. [E143 Collaboration], Phys. Rev. D 58, 112003 (1998).
  * (54) A. Airapetian et al. [HERMES Collaboration], Phys. Lett. B 442, 484 (1998).
  * (55) A. Airapetian et al. [HERMES Collaboration], Phys. Rev. D 75, 012007 (2007).
  * (56) J. Ashman et al. [European Muon Collaboration], Phys. Lett. B 206, 364 (1988).
  * (57) P. L. Anthony et al. [E155 Collaboration], Phys. Lett. B 493, 19 (2000).
  * (58) B. Adeva et al. [Spin Muon Collaboration], Phys. Rev. D 58, 112001 (1998).
  * (59) M. G. Alekseev et al. [COMPASS Collaboration], Phys. Lett. B 690, 466 (2010). V. Y. Alexakhin et al. [COMPASS Collaboration], Phys. Lett. B 647, 8 (2007).
  * (60) C. Adolph et al. [COMPASS Collaboration], Phys. Lett. B 753, 18 (2016).
  * (61) K. V. Dharmawardane et al. [CLAS Collaboration], Phys. Lett. B 641, 11 (2006).
  * (62) R. Fersch et al. [CLAS Collaboration], Phys. Rev. C 96, 065208 (2017).
  * (63) P. L. Anthony et al. [E142 Collaboration], Phys. Rev. D 54, 6620 (1996).
  * (64) K. Ackerstaff et al. [HERMES Collaboration], Phys. Lett. B 404, 383 (1997).
  * (65) K. Abe et al. [E154 Collaboration], Phys. Rev. Lett. 79, 26 (1997).
  * (66) K. M. Kramer [Jefferson Lab E97-103 Collaboration], AIP Conf. Proc. 675, 615 (2003).
  * (67) X. Zheng et al. [Jefferson Lab Hall A Collaboration], Phys. Rev. C 70, 065207 (2004).
  * (68) K. Kramer, D. S. Armstrong, T. D. Averett, W. Bertozzi, S. Binet, C. Butuceanu, A. Camsonne and G. D. Cates et al., Phys. Rev. Lett. 95, 142002 (2005).
  * (69) P. L. Anthony et al. [E155 Collaboration], Phys. Lett. B 463, 339 (1999).
  * (70) E. S. Ageev et al. [COMPASS Collaboration], Phys. Lett. B 612, 154 (2005).
  * (71) V. Y. .Alexakhin et al. [COMPASS Collaboration], Phys. Lett. B 647, 8 (2007).
  * (72) C. Adolph et al. [COMPASS Collaboration], Phys. Lett. B 769, 34 (2017).
  * (73) P. L. Anthony et al. [E155 Collaboration], Phys. Lett. B 553, 18 (2003).
  * (74) A. Airapetian, N. Akopov, Z. Akopov, E. C. Aschenauer, W. Augustyniak, R. Avakian, A. Avetissian and E. Avetisyan et al., Eur. Phys. J. C 72, 1921 (2012).
  * (75) D. Adams et al. [Spin Muon (SMC) Collaboration], Phys. Rev. D 56, 5330 (1997).
  * (76) S. Taheri Monfared, Z. Haddadi and A. N. Khorramian, Phys. Rev. D 89, 074052 (2014). [erratum: Phys. Rev. D 89, no.11, 119901 (2014)]
  * (77) F. James and M. Roos, Comput. Phys. Commun. 10, 343 (1975).
  * (78) J. D. Bjorken, Phys. Rev. D 1, 1376 (1970).
  * (79) P. A. Baikov, K. G. Chetyrkin and J. H. Kuhn, Phys. Rev. Lett. 104, 132004 (2010).
  * (80) J. Blümlein, G. Falcioni and A. De Freitas, Nucl. Phys. B 910, 568 (2016).
  * (81) G. Altarelli, R. D. Ball, S. Forte and G. Ridolfi, Acta Phys. Polon. B 29, 1145 (1998)
  * (82) M. Akrami and A. Mirjalili Phys. ReV. D 99, 074023 (2019).
  * (83) M. Akrami and A. Mirjalili Phys. ReV. D 101, 034007 (2020).
  * (84) E. Leader, arXiv:1604.00305 [hep-ph]. arXiv:1604.00305 [hep-ph].
  * (85) R. D. Ball et al. [NNPDF Collaboration], Nucl. Phys. B 874, 36 (2013).
  * (86) E. R. Nocera, S. Forte, G. Ridolfi and J. Rojo, arXiv:1206.0201 [hep-ph].
  * (87) D. de Florian, R. Sassot, M. Stratmann and W. Vogelsang, Phys. Rev. Lett. 101, 072001 (2008).
  * (88) E. Leader, A. V. Sidorov and D. B. Stamenov, Phys. Rev. D 82, 114018 (2010).
  * (89) J.C.Collins, Renormalization: an introduction to renormalization, the renormalization group and the operator-product expansion Cambridge university press (1984).
  * (90) H. Burkhardt and W. N. Cottingham, Annals Phys. 56, 453 (1970).
  * (91) J. Blumlein and N. Kochelev, Nucl. Phys. B 498, 285 (1997).
  * (92) K. Slifer et al. [Resonance Spin Structure Collaboration], Phys. Rev. Lett. 105, 101601 (2010).
  * (93) P. Solvignon et al. [E01-012 Collaboration], Phys. Rev. C 92, no. 1, 015208 (2015).
  * (94) P. Zavada, Phys. Rev. D 65, 054040 (2002)
  * (95) S. E. Kuhn, J. P. Chen and E. Leader, Prog. Part. Nucl. Phys. 63, 1 (2009).
  * (96) D. Flay et al., Phys. Rev. D 94, no. 5, 052003 (2016).
  * (97) M. Gockeler, R. Horsley, D. Pleiter, P. E. L. Rakow, A. Schafer, G. Schierholz, H. Stuben and J. M. Zanotti, Phys. Rev. D 72, 054507 (2005).
  * (98) X. Song, Phys. Rev. D 54, 1955 (1996).
  * (99) N. Sato et al. [Jefferson Lab Angular Momentum Collaboration], Phys. Rev. D 93, no. 7, 074005 (2016).
  * (100) P. Jimenez-Delgado, A. Accardi and W. Melnitchouk, Phys. Rev. D 89, no. 3, 034025 (2014).
  * (101) E. Leader, A. V. Sidorov and D. B. Stamenov, Phys. Rev. D 75, 074027 (2007).
  * (102) A.V. Efremov, P. Schweitzer, O.V. Teryaev, P. Zavada, Phys. Rev. D 83, 054025 (2011).
  * (103) J. C. Collins, Acta Phys. Polon. B 34, 3103 (2003) [arXiv:hep-ph/0304122 [hep-ph]].
  * (104) J. C. Collins, T. C. Rogers and A. M. Stasto, Phys. Rev. D 77, 085009 (2008).
  * (105) J. C. Collins and F. Hautmann, Phys. Lett. B 472, 129-134 (2000).
  * (106) J. C. Collins and F. Hautmann, JHEP 03, 016 (2001).
  * (107) F. Hautmann, Phys. Lett. B 655, 26-31 (2007).
  * (108) P. J. Mulders and R. D. Tangerman, Nucl. Phys. B 461, 197-237 (1996). [erratum: Nucl. Phys. B 484, 538-540 (1997)] [arXiv:hep-ph/9510301 [hep-ph]].
  * (109) A. Bacchetta, M. Diehl, K. Goeke, A. Metz, P. J. Mulders and M. Schlegel, JHEP 02, 093 (2007).
  * (110) D. Sivers, Phys. Rev. D 41, 83 (1990).
  * (111) D. Boer, P. J. Mulders, Phys. Rev. D 57, 5780 (1998).
  * (112) S. Meissner, A. Metz, K. Goeke, Phys. Rev. D 76, 034002 (2007).
  * (113) S. Meissner, P. J. Mulders, J. Rodrigues, Phys. Rev. D 63, 094021 (2007).
  * (114) A. V. Efremov, P. Schweitzer, O. V. Teryaev and P. Zavada, Phys. Rev. D 80, 014021 (2009).
  * (115) A. V. Efremov, P. Schweitzer, O. V. Teryaev and P. Zavada, J. Phys. Conf. Ser. 295, 012052 (2011).
  * (116) S. Bastami, A. V. Efremov, P. Schweitzer, O. V. Teryaev and P. Zavada, Phys. Rev. D 103, 014024 (2021).
  * (117) J. She, J. Zhu and B. Q. Ma, Phys. Rev. D 79, 054008 (2009).
  * (118) H. Avakian, A. V. Efremov, P. Schweitzer and F. Yuan, Phys. Rev. D 81, 074035 (2010).
  * (119) A. V. Efremov, P. Schweitzer, O. V. Teryaev, P. Zavada, PoS DIS2010, 253 (2010) [arXiv:1008.3827 [hep-ph]];
  * (120) H. Avakian, A. V. Efremov, P. Schweitzer, O. V. Teryaev and P. Zavada [arXiv:1008.1921 [hep-ph]].