# Recent data analysis to revisit the spin structure function of nucleon in
Laplace space

Hoda Nematollahi1 ID<EMAIL_ADDRESS>Abolfazl Mirjalili2 ID
<EMAIL_ADDRESS>Shahin Atashbar Tehrani3 ID<EMAIL_ADDRESS>(1)Faculty
of Physics, Shahid Bahonar University of Kerman, Kerman, Iran
(2)Physics Department, Yazd University, Yazd, Iran
(3)School of Particles and Accelerators, Institute for Research in Fundamental
Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran

(August 27, 2024)

###### Abstract

Considering a fixed-flavor number scheme and based on laplace transdormation,
we perform a leading-order and next-to-leading-order QCD analysis which are
including world data on polarized structure functions $g_{1}$ and $g_{2}$.
During our analysis, taking the DGLAP evolution, we employ the Jacobi
polynomials expansion technique. In our recent analysis we utilize the recent
available data and consequently include more data than what we did in our
previous analysis. we obtain good agreements between our results for the
polarized parton densities and nucleon structure functions with all available
experimental data and some common parametrization models.

###### Contents

  1. I Introduction
  2. II Polarized DGLAP evolution equations in Laplace space
    1. II.1 Leading-order approximation
    2. II.2 Next-Leading-order approximation
  3. III The Jacobi Polynomial Method
  4. IV QCD Analysis & Parametrization of PPDFs
    1. IV.1 Parametrization
    2. IV.2 First Moments of $\Delta u_{v}$ and $\Delta d_{v}$
    3. IV.3 Polarized DGLAP evolution
    4. IV.4 The $g_{2}$ structure function
  5. V Fitting contents in QCD analysis
    1. V.1 Overview of data sets
    2. V.2  $\chi^{2}$ minimization
      1. V.2.1 Gluon and sea quarks
  6. VI The Sum Rules
    1. VI.1 Bjorken sum rule
    2. VI.2 Proton helicity sum rule
    3. VI.3 The twist-3 reduced matrix element $d_{2}$
    4. VI.4 Burkhardt-Cottingham sum rule
  7. VII Conclusion

##  I Introduction

High-energy scattering of polarized leptons by polarized protons, neutrons,
and deuterons provides a measurement of the nucleon spin structure functions.
These structure functions give information on the polarized quark
contributions to the spin of the proton and the neutron and allow tests of the
quark-parton model and quantum chromodynamics (QCD). After precise
consideration of the unpolarized deep inelastic scattering (DIS) experiments,
polarized DIS program has been planed to study the spin structure of the
nucleon using polarized lepton beams (electrons and muons) scattered by
polarized targets. These fixed-target experiments have been used to
characterize the spin structure of the proton and neutron and also to test
fundamental sum rules of QCD and quark-parton model (QPM) [1, 2]. The first
experiments in polarized electron-polarized proton scattering, performed
around 50 years ago, helped to establish the parton structure of the proton.
About two decades later, by performing an experiment with polarized muon and
polarized proton, this reality has been raveled that the QPM sum rule was
violated which seemed to indicate that the quarks do not contribute alone to
the spin of the proton. This “proton-spin crisis” gave birth to a new
generation of experiments at several high-energy physics laboratories around
the world. The new and extensive data sample, collected from these fixed
target experiments, has enabled a careful characterization of the spin-
dependent parton substructure of the nucleon. The results have been used to
test QCD, to find an independent value for $\alpha_{s}(Q^{2})$, to probe the
polarized parton distributions with reasonable precision, and to provide a
first look at the polarized gluon distribution [3].

On this purpose we try to solve the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi
(DGLAP) evolution equations, using the Laplace transformation. This is done at
leading order (LO) and next-to-leading order (NLO) approximations in
SubSec.II.1 and II.2 of Sec.II. We then construct the $xg_{1}$ polarized
structure function using the expansion in terms of Jaccobi polynomials in
Sec.III. The evolution of partons requires some inputs which are in fact the
polarized parton distribution functions (PPDFs) at initial energy scale,
$Q_{0}^{2}$. On this base we need to some parameterizations for the input
PPDFs which is introduced in Sec.IV. To determine the unknown parameters of
the input PPDFs we should take all the recent and available data from
different DIS experiments. We then use them in a fitting process as is
illustrated in Sec.V. Getting the $g_{1}$ structure function, it is possible
to calculate the $g_{2}$ structure function which is done also in this
section. To validate the results from data analysis for $g_{1}$ structure
function, several sum rules are computed. We find them in good agreement with
experimental data and the results which are arising out from theoretical
investigations. This part of work is presented in Sec.VI. The worthiness of
the work which we handle is to employ the last reported data for polarized
targets in DIS experiments. The compatible results with data at different
energy scales and some models confirm authenticity of the utilized theoretical
framework, including QPM and some outputs of QCD which are described in
Sec.VII as the last section of this paper.

##  II Polarized DGLAP evolution equations in Laplace space

###  II.1 Leading-order approximation

In this work we generalize the method of Laplace transformation to employ QCD
evolution equations in polarized case to investigate the polarized parton
distributions of the nucleon. Here we focus on the polarization of singlet and
non-singlet quarks to indicate the efficiency of this method for solving the
DGLAP evolution equations [4, 5, 6, 7]. In order to extract the polarized
parton distribution functions, we review the method in this section briefly.

By introducing the variable $\nu\equiv ln(\frac{1}{x})$ into leading order
coupled DGLAP equations, it is possible to turn them into coupled convolution
equations in $\nu$ space. One can use two Laplace transformations, one from
$\nu$ space to $s$ space and the second one from $\tau$ space to $U$ space,
using new variable
$\tau\equiv\frac{1}{4\pi}\int_{Q_{0}^{2}}^{Q^{2}}\alpha_{s}(Q^{\prime 2})d\ln
Q^{\prime 2}$. The DGLAP evolution equations can be solved using these two
laplace transformations and set of convolution integrals of polarized parton
distributions at the initial scale $Q_{0}^{2}$. Finally by applying two
inverse laplace transforms, we can back to ordinary space (x, $Q^{2}$) [8].

The DGLAP evolution equations for the polarized parton distributions are
written as [4, 5, 6, 7, 9]:

$\frac{4\pi}{\alpha_{s}(Q^{2})}\frac{\partial\Delta F_{NS}}{\partial\ln
Q^{2}}(x,Q^{2})=\Delta F_{NS}\otimes\Delta P_{qq}^{0}(x,Q^{2})\;,$ (1)

$\frac{4\pi}{\alpha_{s}(Q^{2})}\frac{\partial\Delta F_{S}}{\partial\ln
Q^{2}}(x,Q^{2})=\Delta F_{S}\otimes\Delta P_{qq}^{0}+\Delta G\otimes\Delta
P_{qg}^{0}(x,Q^{2})\;,$ (2)

$\frac{4\pi}{\alpha_{s}(Q^{2})}\frac{\partial\Delta G}{\partial\ln
Q^{2}}(x,Q^{2})=\Delta F_{S}\otimes\Delta P_{gq}^{0}+\Delta G\otimes\Delta
P_{gg}^{0}(x,Q^{2})\;,$ (3)

where $\Delta P_{ij}^{0}$’s are the LO polarized splitting functions.

By introducing the variable change $w\equiv\ln(1/z)$, applying two other ones:
$v\equiv\ln(1/x)$ and
$\tau(Q_{0}^{2},Q^{2})\equiv\frac{1}{4\pi}\int_{Q_{0}^{2}}^{Q^{2}}\alpha_{s}(Q^{\prime
2})d\ln Q^{\prime 2}$, introduced before, and using the notation
$\Delta\widehat{F}_{NS}(\nu,\tau)\equiv\Delta F_{NS}(e^{-v},Q^{2})$ ,
$\Delta\widehat{F}_{NS}(w,\tau)\equiv\Delta F_{NS}(e^{-w},\tau)$,
$\Delta\widehat{F}_{s}(\nu,\tau)\equiv\Delta F_{s}(e^{-v},Q^{2})$,
$\Delta\widehat{F}_{s}(w,\tau)\equiv\Delta F_{s}(e^{-w},\tau)$,
$\Delta\widehat{G}(v,\tau)\equiv\Delta G(e^{-v},\tau)$,
$\Delta\widehat{G}_{s}(w,\tau)\equiv\Delta G_{s}(e^{-w},\tau)$, the above
DGLAP equations in terms of the convolution integrals are given as:

$\frac{\partial\Delta\widehat{F}_{NS}}{\partial\tau}(v,\tau)=\int_{0}^{v}\Delta\widehat{F}_{NS}(w,\tau)\Delta\widehat{H}_{qq}^{0}(v-w)dw\;,$
(4)

$\displaystyle\frac{\partial\Delta\widehat{F}_{S}}{\partial\tau}(v,\tau)=\int_{0}^{v}\Delta\widehat{F}_{s}(w,\tau)\Delta\widehat{H}_{qq}^{0}(v-w)dw+$
$\displaystyle\int_{0}^{v}\Delta\widehat{G}_{s}(w,\tau)\Delta\widehat{H}_{qg}^{0}(v-w)dw\;,$
(5)

$\displaystyle\frac{\partial\Delta\widehat{G}}{\partial\tau}(v,\tau)=\int_{0}^{v}\Delta\widehat{F}_{s}(w,\tau)\Delta\widehat{H}_{gq}^{0}(v-w)+$
$\displaystyle\int_{0}^{v}\Delta\widehat{G}_{s}(w,\tau)\Delta\widehat{H}_{gg}^{0}(v-w)dw\;,$
(6)

where

$\displaystyle\Delta\widehat{H}_{qq}^{0}(v)$ $\displaystyle\equiv$
$\displaystyle e^{-v}\Delta P_{qq}^{0}(e^{-v}),$
$\displaystyle\Delta\widehat{H}_{gq}^{0}(v)$ $\displaystyle\equiv$
$\displaystyle e^{-v}\Delta P_{gq}^{0}(e^{-v}),$
$\displaystyle\Delta\widehat{H}_{qg}^{0}(v)$ $\displaystyle\equiv$
$\displaystyle e^{-v}\Delta P_{qg}^{0}(e^{-v}),$
$\displaystyle\Delta\widehat{H}_{gg}^{0}(v)$ $\displaystyle\equiv$
$\displaystyle e^{-v}\Delta P_{gg}^{0}(e^{-v})\;.$ (7)

By considering the following property of laplace transforms:

$\displaystyle{\cal
L}\left[\int_{0}^{v}\Delta\widehat{F}[w]\Delta\widehat{H}[v-w]dw;s\right]$
$\displaystyle={\cal L}[\Delta\widehat{F}_{s}[v];s]\times{\cal
L}\left[\Delta\widehat{H}[v];s\right],$ (8)

the DGLAP equations in Eqs.(4, 5) and Eq.(6) can be converted to three coupled
ordinary first-order differential equations in terms of the variable $\tau$ in
the Laplace s-space with $\tau$-dependent coefficients as following:

$\displaystyle\frac{\partial\Delta f_{NS}}{\partial\tau}(s,\tau)$
$\displaystyle=$ $\displaystyle\Delta\Phi_{ns}^{LO}(s)\Delta f(s,\tau)\;,$
$\displaystyle\frac{\partial\Delta f_{s}}{\partial\tau}(s,\tau)$
$\displaystyle=$ $\displaystyle\Delta\Phi_{f}^{LO}(s)\Delta
f_{s}(s,\tau)+\Delta\Theta_{f}^{LO}(s)\Delta g(s,\tau)\;,$
$\displaystyle\frac{\partial\Delta g}{\partial\tau}(s,\tau)$ $\displaystyle=$
$\displaystyle\Delta\Phi_{g}^{LO}(s)\Delta
g(s,\tau)+\Delta\Theta_{g}^{LO}(s)\Delta f_{s}(s,\tau)\;.$

In the above equations, we utilize the following abbreviations:

$\displaystyle\Delta f_{NS}(s,\tau)$ $\displaystyle\equiv$ $\displaystyle{\cal
L}[\Delta\widehat{F}_{NS}(v,\tau);s],$ $\displaystyle\Delta f_{s}(s,\tau)$
$\displaystyle\equiv$ $\displaystyle{\cal
L}[\Delta\widehat{F}_{s}(v,\tau);s],$ $\displaystyle\Delta g(s,\tau)$
$\displaystyle\equiv$ $\displaystyle{\cal L}[\Delta\widehat{G}(v,\tau);s]\;.$
(10)

Following that we can write:

$\displaystyle{\cal L}\left[\frac{\partial\Delta\widehat{F}_{NS}}{\partial
w}(w,\tau);s\right]$ $\displaystyle=$ $\displaystyle s\Delta
f_{NS}(s,\tau)\;,$ (11) $\displaystyle{\cal
L}\left[\frac{\partial\Delta\widehat{F}_{s}}{\partial w}(w,\tau);s\right]$
$\displaystyle=$ $\displaystyle s\Delta f_{s}(s,\tau)\;,$ $\displaystyle{\cal
L}\left[\frac{\partial\Delta\widehat{G}}{\partial w}(w,\tau);s\right]$
$\displaystyle=$ $\displaystyle s\Delta g(s,\tau)\;.$

On the other hand the LO coefficients $\Delta\Phi^{LO}$ and
$\Delta\Theta^{LO}$ in Laplace s-space are given by:

$\Delta\Phi_{f}^{LO}=4-\frac{8}{3}\left(\frac{1}{s+1}+\frac{1}{s+2}+2(\psi^{(0)}(s+1)+\gamma_{E})\right),$
(12)

$\Delta\Theta_{f}^{LO}=T_{r}\left(\frac{2}{s+2}-\frac{1}{s+1}\right)\;,$ (13)

$\Delta\Theta_{g}^{LO}=C_{f}\left(\frac{2}{s+1}-\frac{1}{2+s}\right)\;,$ (14)

$\displaystyle\Delta\Phi_{g}^{LO}=C_{a}\left(\frac{11}{6}-\frac{f}{9}+\frac{3}{s+1}-\frac{3}{s+2}+\frac{1}{s+3}+\right.$
$\displaystyle\left.\frac{1}{s+4}+\psi^{(0)}(s+1)-\psi^{(0)}\;.(s+5)\right)\;.$

In Eq.( 12)and Eq.( II.1) $\psi^{(0)}(x)$ denotes digamma function and
$\gamma_{E}=0.5772156$ is Euler’s constant. The evolution of DGLAP equations
in Laplace space at the LO approximation for singlet sector and gluon part can
be written as:

$\displaystyle\Delta f_{1}(s,\tau)=\Delta k_{ff_{1}}(s,\tau)f_{0}(s)+\Delta
k_{fg_{1}}(s,\tau)g_{0}(s)\;,$ $\displaystyle\Delta g_{1}(s,\tau)=\Delta
k_{gg_{1}}(s,\tau)g_{0}(s)+\Delta k_{gf_{1}}(s,\tau)f_{0}(s),$ (16)

where the $\Delta k$’s in above equations are given by:

$\displaystyle\Delta k_{ff_{1}}(s,\tau)$ $\displaystyle\equiv
e^{\frac{\tau}{2}(\Delta\Phi_{f}^{LO}(s)+\Delta\Phi_{g}^{LO}(s))}\left[cosh\left(\frac{\tau}{2}R(s)\right)\right.$
$\displaystyle\left.+\frac{2sinh(\frac{\tau}{2}R(s))}{R(s)}(\Phi_{f}^{LO}(s)-\Phi_{g}^{LO}(s))\right],$
$\displaystyle\Delta k_{fg_{1}}(s,\tau)$ $\displaystyle\equiv
e^{\frac{\tau}{2}(\Delta\Phi_{f}^{LO}(s)+\Delta\Phi_{g}^{LO}(s))}\frac{sinh\left(\frac{\tau}{2}R(s)\right)}{R(s)}\Delta\Theta_{f}^{LO}(s),$
$\displaystyle\Delta k_{gg_{1}}(s,\tau)$ $\displaystyle\equiv
e^{\frac{\tau}{2}(\Delta\Phi_{f}^{LO}(s)+\Delta\Phi_{g}^{LO}(s))}\left[cosh\left(\frac{\tau}{2}R(s)\right)\right.$
$\displaystyle\left.-\frac{2sinh(\frac{\tau}{2}R(s))}{R(s)}(\Phi_{f}^{LO}(s)-\Phi_{g}^{LO}(s))\right],$
$\displaystyle\Delta k_{fg_{1}}(s,\tau)$ $\displaystyle\equiv
e^{\frac{\tau}{2}(\Delta\Phi_{f}^{LO}(s)+\Delta\Phi_{g}^{LO}(s))}\frac{sinh\left(\frac{\tau}{2}R(s)\right)}{R(s)}\Delta\Theta_{g}^{LO}(s).$

In Eq.(LABEL:eq:251), $R(s)$ is defined as:

$R(s)=\sqrt{(\Delta\Phi_{f}^{LO}(s)-\Delta\Phi_{g}^{LO}(s))^{2}+4\Delta\Theta_{f}^{LO}(s)\Delta\Theta_{g}^{LO}(s)}\;,$
(18)

For doing the numerical Laplace inversion in $v$ space, one need the Laplace
inverse of the kernels as $K_{FF}(v,\tau)\equiv{\cal L}^{-1}[\Delta
k_{ff}(s,\tau);v]$, $K_{FG}(v,\tau)\equiv{\cal L}^{-1}[\Delta
k_{fg}(s,\tau);v]$, $K_{GF}(v,\tau)\equiv{\cal L}^{-1}[\Delta
k_{gf}(s,\tau);v]$ and $K_{GG}(v,\tau)\equiv{\cal L}^{-1}[\Delta
k_{gg}(s,\tau);v]$ [10, 8, 11]. So that we can write the decoupled solutions
in $(v,Q^{2})$ space, based on the following convolutions:

$\displaystyle\Delta\widehat{F}_{s}(v,Q^{2})$
$\displaystyle\equiv\int_{0}^{v}K_{FF}(v-w,\tau(Q^{2},Q_{0}^{2}))\Delta\widehat{F}_{so}(w)dw$
$\displaystyle+\int^{v}_{0}K_{FG}(v-w,\tau(Q^{2},Q_{0}^{2}))\Delta\widehat{G}_{so}(w)dw\;,$
$\displaystyle\Delta\widehat{G}_{s}(v,Q^{2})$
$\displaystyle\equiv\int_{0}^{v}K_{GG}(v-w,\tau(Q^{2},Q_{0}^{2}))\Delta\widehat{G}_{so}(w)dw$
$\displaystyle+\int^{v}_{0}K_{GF}(v-w,\tau(Q^{2},Q_{0}^{2}))\Delta\widehat{F}_{so}(w)dw;.$
(19)

It is obvious that $\Delta\widehat{F}_{so}(w)$ and $\Delta\widehat{G}_{so}(w)$
are the Laplace inverse of $f_{0}(s)$ and $g_{0}(s)$ in Eq.(16). Reminding
$w\equiv ln(1/z)$ and recalling that $v\equiv ln(1/x)$, we can finally convert
the above solutions into the usual Bjorken-$x$ space.

Now for non-singlet sector, $\Delta F_{ns}(x,Q^{2})$, as before, using the
variable change $v\equiv ln(1/x)$ and the variable $\tau$ then the valance
part in Eq.( 1) can be written as:

$\frac{\partial\Delta\widehat{F}_{NS}}{\partial\tau}=\int_{0}^{v}\Delta\widehat{F}_{NS}(w,\tau)e^{-(v-w)}\Delta
P_{qq}^{LO,ns}(v-w)dw\;.$ (20)

Employing the Laplace transformation on above equation, we obtain a linear
differential equation in terms of $\tau$ variable for the $\Delta
f_{ns}(s,\tau)$ as the transformed version of
$\Delta\widehat{F}_{NS}(x,Q^{2})$. This differential equation leads to the
following solution:

$\Delta f_{ns}(s,\tau)\equiv e^{\tau\Delta\Phi_{ns}^{LO}}\Delta f_{ns0}(s)\;.$
(21)

Now using the inverse Laplace transform on Eq.(21) we arrive at the following
convolution:

$\Delta\widehat{F}_{ns}(v,\tau)=\int_{0}^{v}K_{ns}(v-w,\tau)\Delta\widehat{F}_{ns0}(w)dw\;.$
(22)

In this equation we take $\Delta K_{ns}(v,\tau)\equiv{\cal
L}^{-1}[e^{\tau\Delta\Phi_{ns}^{LO}(s)};v]$ where $\Delta\widehat{F}_{ns0}(w)$
is the inverse Laplace of $\Delta f_{ns0}(s)$. Finally by variable change
$\nu=ln(\frac{1}{x})$ the results in $(x,Q^{2})$ space is accessible.

More details to extract parton distribution functions at the LO approximation,
based on the Laplace transformation, can be found in [8].

###  II.2 Next-Leading-order approximation

At the NLO approximation for the non-singlet sector of DGLAP evolution
equation, after changing the required variables which have been introduced
before, we arrive at

$\displaystyle{\partial{\hat{\Delta}F_{ns}}\over\partial\tau}(\nu,\tau)=\int_{0}^{\nu}{\Delta\hat{F}}_{ns}(w,\tau)e^{-(\nu-w)}\left(\Delta
P_{qq}^{LO,ns}(\nu-w)\right.$ $\displaystyle\left.+{\alpha_{s}(\tau)\over
4\pi}\Delta P_{qq}^{NLO,ns}(\nu-w)\right)d\,w\;,$

which is in fact the extended version of Eq.(4) for $\hat{\Delta}F_{ns}$.

Taking above equation to Laplace $s$-space, we obtain a linear differential
equation in terms of $\tau$ variable for the transformed $\Delta
f_{ns}(s,\tau)$. This equation has the simple solution as:

$\displaystyle\Delta f_{NS}(s,\tau)=e^{\tau\Delta\Phi_{NS}(s)}\Delta
f^{0}_{NS}(s),$
$\displaystyle\qquad\Delta\Phi_{NS}(s)\equiv\Delta\Phi_{NS}^{LO}(s)+{\tau_{2}\over\tau}\Delta\Phi_{NS}^{NLO}(s),$
(24)

where

$\displaystyle\tau_{2}$ $\displaystyle\equiv$ $\displaystyle{1\over
4\pi}\int_{0}^{\tau}\alpha_{s}(\tau^{\prime})\,d\tau^{\prime}={1\over(4\pi)^{2}}\int_{Q^{2}_{0}}^{Q^{2}}\alpha_{s}^{2}(Q^{\prime
2})\,d\ln Q^{\prime 2},$

and

$\displaystyle\Delta\Phi_{NS}^{LO}(s)\equiv{\cal L}\left[e^{-v}\Delta
P_{qq}^{LO,ns}(e^{-\nu});s\right],$
$\displaystyle\qquad\Delta\Phi_{NS}^{NLO}(s)\equiv{\cal L}\left[e^{-\nu}\Delta
P_{qq}^{NLO,NS}(e^{-\nu});s\right].$ (26)

It should be noted that at the LO approximation
$\Delta\Phi_{NS}(s)=\Delta\Phi_{f}^{LO}(s)$ where $\Delta\Phi_{f}^{LO}(s)$ has
been given explicitly in Eq.(12). The evaluation of $\Delta\Phi_{NS}^{NLO}(s)$
has been presented in Ref.[12].

Figure 1: Our NMA23 results for the polarized PDFs at Q${}_{0}^{2}=$ 1.3 GeV2
with respect to $x$ in LO approximation, which is plotted by a solid curve
along with their $\Delta\chi^{2}/\chi^{2}=2\%$ uncertainty bands computed with
the Hessian approach. We also present the result obtained in earlier global
analyses of BB (dashed) [76], GRSV (dashed-dotted) [77], AAC00 (dashed-dashed-
dotted) [78] in LO approximation.  Figure 2: Our NMA23 results for the spin-
dependent PDFs at Q${}_{0}^{2}=$ 1.3 GeV2 with respect to $x$ in NLO
approximation which is plotted by a solid curve along with their
$\Delta\chi^{2}/\chi^{2}=2\%$ uncertainty bands computed with the Hessian
approach. We also display the result obtained in earlier global analyses of
NNPDF(dashed-dotted-dotted) [79], KATAO (long dashed) [21], BB (dashed) [69],
DSSV (dashed-dotted) [68], AAC09 (dashed-dashed-dotted) [80] in NLO
approximation.  Figure 3: The evolved polarized quark densities as a function
of $x$ in the LO approximation.  Figure 4: The evolved polarized quark
densities as a function of $x$ in the NLO approximation.

We can find any non-singlet solution, $\Delta F_{NS}(x,Q^{2})$, by applying
the non-singlet kernel $K_{ns}(v)\equiv{\cal
L}^{-1}\left[e^{\tau\Delta\Phi_{NS}(s)};v\right]$, and using the Laplace
convolution relation as:

$\displaystyle{\Delta\hat{F}}_{NS}(v,\tau)=\int_{0}^{v}K_{ns}(v-w,\tau)\Delta\hat{F}^{0}_{NS}(w)\,dw\;.$
(27)

Taking the variable change $\nu\equiv\ln(1/x)$, the results are obtained in
$(x,Q^{2})$ space as before with this difference that the splitting functions
in $s$-space include as well the NLO contributions in correspond to the
following expressions:

$\displaystyle\Delta\Phi_{S}(s)\equiv\Delta\Phi_{S}^{LO}(s)+a_{0}\Delta\Phi_{S}^{NLO}(s),$
$\displaystyle\Delta\Phi_{g}(s)\equiv\Delta\Phi_{g}^{LO}(s)+a_{0}\Delta\Phi_{g}^{NLO}(s),$
$\displaystyle\Delta\Theta_{S}(s)\equiv\Delta\Theta_{S}^{LO}(s)+a_{0}\Delta\Theta_{S}^{NLO}(s),$
$\displaystyle\Delta\Theta_{g}(s)\equiv\Delta\Theta_{g}^{LO}(s)+a_{0}\Delta\Theta_{g}^{NLO}(s)\;.$
(28)

The analytical expressions for $\Delta\Phi_{NS}^{NLO}(s)$ ,
$\Delta\Phi_{S}^{NLO}(s)$ , $\Delta\Phi_{g}^{NLO}(s)$,
$\Delta\Theta_{S}^{NLO}(s)$ and $\Delta\Theta_{g}^{NLO}(s)$ in Laplace
transform $s$-space have been presented in Ref.[12].

We indicate the NLO expression for ${\alpha_{s}(\tau)\over 4\pi}$ by
$a(\tau)$. We can numerically show that an excellent approximation to
$a(\tau)\equiv{\alpha_{s}(\tau)\over 4\pi}$, with a precision of a few parts
in $10^{4}$, is given by the expression

$\displaystyle a(\tau)$ $\displaystyle\approx a_{0}+a_{1}e^{-b_{1}\tau},$ (29)

where the unknown parameters $a_{1},b_{1}$ and $a_{0}$ are found by a least
squared fit to $a(\tau)$. It should be pointed that this approximation is
inspired by the fact that at the LO approximation the expression for
$\alpha_{s,LO}(\tau)$ is exactly given by
$\alpha_{s,LO}(Q_{0}^{2})e^{-b\tau}$.

Now for the singlet sector and gluon part of Eq.(II.1) at the NLO
approximation, the results of DGLAP evolution equations in Laplace s-space
could be obtained and are given by:

$\displaystyle\Delta f(s,\tau)$ $\displaystyle=$ $\displaystyle
k_{ff}(a_{1},b_{1},s,\tau)\Delta f^{0}_{S}(s)+k_{fg}(a_{1},b_{1},s,\tau)\Delta
g^{0}(s),$ $\displaystyle\Delta g(s,\tau)$ $\displaystyle=$ $\displaystyle
k_{gg}(a_{1},b_{1},s,\tau)\Delta g^{0}(s)+k_{gf}(a_{1},b_{1},s,\tau)\Delta
f^{o}_{S}(s)\;.$

Here the functions $k_{ij}(a_{1},b_{1},s,\tau)$ are expressed as a power
series in terms of the NLO expansion parameter $a_{1}$ whose coefficients are
analytic functions with respect to $s$ and $\tau$ parameters. These
expressions which are used to extract polarized parton distribution functions
at NLO approximation, can be found in Ref.[12]. Finally, recalling that
$\nu\equiv\ln(1/x)$, we can convert the above solutions to the $(x,Q^{2})$
space. Therefore we should write the NLO decoupled solutions, $\Delta
F_{s}(x,Q^{2})$ and $\Delta G(x,Q^{2})$, with the knowledge requirement of
$\Delta F(x)$ and $\Delta G(x)$ at initial scale $Q_{0}^{2}$. We also use
these analytical solutions for polarized parton distributions in the next
sections to extract polarized structure functions of proton, neutron and
deuteron.

Table 1: 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) | data poi. | $\chi^{2}_{LO}$ | $\chi^{2}_{NLO}$ | ${\cal N}_{i}$
---|---|---|---|---|---|---|---
SLAC/E143(p) | [33] | [0.031–0.749] | 1.27–9.52 | 28 | 25.9418 | 25.5862 | 0.997894385554395
HERMES(p) | [34] | [0.028–0.66] | 1.01–7.36 | 39 | 57.6213 | 54.4579 | 1.00121847577922
SMC(p) | [38] | [0.005–0.480] | 1.30–58.0 | 12 | 5.7338 | 7.807 | 0.999682542198503
EMC(p) | [36] | [0.015–0.466] | 3.50–29.5 | 10 | 5.5308 | 5.2507 | 0.998746870048074
SLAC/E155 | [37] | [0.015–0.750] | 1.22–34.72 | 24 | 30.8939 | 25.8174 | 1.00866491330750
HERMES06(p) | [35] | [0.026–0.731] | 1.12–14.29 | 51 | 25.1976 | 30.7736 | 0.978727363512146
COMPASS10(p) | [39] | [0.005–0.568] | 1.10–62.10 | 15 | 21.1180 | 21.7614 | 0.986930436542322
COMPASS16(p) | [40] | [0.0035–0.575] | 1.03–96.1 | 54 | 38.1731 | 36.7525 | 0.998144618082626
SLAC/E143(p) | [33] | [0.031–0.749] | 2-3-5 | 84 | 108.5918 | 102.0200 | 0.998508244589941
HERMES(p) | [34] | [0.023–0.66] | 2.5 | 20 | 34.3329 | 32.2223 | 1.00223612389713
SMC(p) | [38] | [0.003–0.4] | 10 | 12 | 10.2414 | 10.2829 | 1.00050424220605
Jlab06(p) | [41] | [0.3771–0.9086] | 3.48–4.96 | 70 | 101.8636 | 103.0134 | 0.999973926530214
Jlab17(p) | [42] | [0.37696–0.94585] | 3.01503–5.75676 | 82 | 178.2875 | 186.4491 | 1.00223612389713
$\bf{g_{1}^{p}}$ |  |  |  | 501 |  |  |
SLAC/E143(d) | [33] | [0.031–0.749] | 1.27–9.52 | 28 | 37.8573 | 38.4526 | 1.00123946481403
SLAC/E155(d) | [49] | [0.015–0.750] | 1.22–34.79 | 24 | 19.6385 | 18.3428 | 1.00093216947938
SMC(d) | [38] | [0.005–0.479] | 1.30–54.80 | 12 | 19.1969 | 18.8501 | 1.00004243740921
HERMES06(d) | [35] | [0.026–0.731] | 1.12–14.29 | 51 | 48.9197 | 48.3606 | 1.00287123838967
COMPASS05(d) | [50] | [0.0051–0.4740] | 1.18–47.5 | 11 | 8.0128 | 8.6001 | 1.00103525695298
COMPASS06(d) | [51] | [0.0046–0.566] | 1.10–55.3 | 15 | 5.4623 | 7.6560 | 1.00014044224998
COMPASS17(d) | [52] | [0.0045–0.569] | 1.03–74.1 | 43 | 33.1002 | 31.4721 | 1.00401646687751
SLAC/E143(d) | [33] | [0.031–0.749] | 2–3–5 | 84 | 125.8333 | 125.7379 | 0.999955538651217
$\bf{g_{1}^{d}}$ |  |  |  | 268 |  |  |
SLAC/E142(n) | [43] | [0.035–0.466] | 1.10–5.50 | 8 | 7.9561 | 7.7586 | 0.998697021286838
HERMES(n) | [34] | [0.033–0.464] | 1.22–5.25 | 9 | 2.3999 | 2.5486 | 0.999948762872958
E154(n) | [45] | [0.017–0.564] | 1.20–15.00 | 17 | 24.3141 | 21.4152 | 0.999115473491324
HERMES06(n) | [44] | [0.026–0.731] | 1.12–14.29 | 51 | 18.2773 | 17.3811 | 0.998906625804611
Jlab03(n) | [46] | [0.14–0.22] | 1.09–1.46 | 4 | 5.6144e-2 | 5.7129e-2 | 0.999554786214114
Jlab04(n) | [47] | [0.33–0.60] | 2.71–4.8 | 3 | 14.5393 | 8.8133 | 0.994389736514520
Jlab05(n) | [48] | [0.19–0.20] | 1.13–1.34 | 2 | 7.7029 | 6.7595 | 0.999939246942750
$\bf{g_{1}^{n}}$ |  |  |  | 94 |  |  |
E143(p) | [33] | [0.038–0.595] | 1.49–8.85 | 12 | 11.1698 | 10.9170 | 1.00335621969896
E155(p) | [53] | [0.038–0.780] | 1.1–8.4 | 8 | 12.8132 | 15.6826 | 1.04042312148245
Hermes12(p) | [54] | [0.039–0.678] | 1.09–10.35 | 20 | 25.1271 | 21.5690 | 1.00274614220085
SMC(p) | [55] | [0.010–0.378] | 1.36–17.07 | 6 | 1.8259 | 1.7117 | 1.00001538561406
$\boldsymbol{g_{2}^{p}}$ |  |  |  | 46 |  |  |
E143(d) | [33] | [0.038–0.595] | 1.49–8.86 | 12 | 9.6009 | 9.6132 | 1.00047527921467
E155(d) | [53] | [0.038–0.780] | 1.1–8.2 | 8 | 12.2433 | 12.275 | 1.01388492611179
$\boldsymbol{g_{2}^{d}}$ |  |  |  | 20 |  |  |
E143(n) | [33] | [0.038–0.595] | 1.49–8.86 | 12 | 8.9660 | 9.0283 | 1.00004452228917
E155(n) | [53] | [0.038–0.780] | 1.1–8.8 | 8 | 14.1622 | 13.6977 | 1.03135886789238
E142(n) | [43] | [0.036–0.466] | 1.1–5.5 | 8 | 16.48220 | 3.8883 | 1.00000431789744
Jlab03(n) | [46] | [0.14–0.22] | 1.09–1.46 | 4 | 17.6171 | 13.8173 | 1.03226263480608
Jlab04(n) | [47] | [0.33–0.60] | 2.71–4.83 | 3 | 4.2782 | 4.4079 | 0.900030714490705
Jlab05(n) | [48] | [0.19–0.20] | 1.13–1.34 | 2 | 10.1400 | 8.0260 | 0.981366577296903
$\boldsymbol{g_{2}^{n}}$ |  |  |  | 37 |  |  |
Total |  966 | 1171.6502 | 1128.9857 |

##  III The Jacobi Polynomial Method

We perform a fit in the LO approximation for the polarized parton
distributions using Jacobi polynomials [13, 16, 14, 15] to reconstruct the $x$
dependent quantities from their Laplace moments. The application of Jacobi
polynomials has a number of advantages; especially, it will provide us an
opportunity to factorize out the $x$and $Q^{2}$ dependence which help us to
have an efficient parametrization and to do the evolution of the structure
functions.

For example, we can expand the spin structure function $xg_{1}(x,Q^{2})$, as
[14]:

$xg_{1}(x,Q^{2})=x^{\beta}(1-x)^{\alpha}\ \sum_{n=0}^{N_{max}}a_{n}(Q^{2})\
\Theta_{n}^{\alpha,\beta}(x)\,$ (31)

in which $\Theta_{n}^{\alpha,\beta}(x)$ are Jacobi polynomials of order $n$,
and $N_{max}$ is the maximum order of the expansion. In this case, the
$Q^{2}$-dependence of the polarized structure function is contained in the
Jacobi moments, $a_{n}(Q^{2})$. On the other hand, we can factor out the
essential part of its $x$-dependence into a weight function using the Jacobi
polynomials [17].

For computational purpose, the $x$-dependence of the Jacobi polynomials is
given by the following expansion [15]:

$\Theta_{n}^{\alpha,\beta}(x)=\sum_{j=0}^{n}c_{j}^{(n)}(\alpha,\beta)\ x^{j},$
(32)

where the $c_{j}^{(n)}(\alpha,\beta)$’s are combinations of
$\Gamma$-functions. The Jacobi polynomials satisfy an orthogonality condition
with weight function $x^{\beta}(1-x)^{\alpha}$ such that:

$\int_{0}^{1}dx\;x^{\beta}(1-x)^{\alpha}\Theta_{k}^{\alpha,\beta}(x)\Theta_{l}^{\alpha,\beta}(x)=\Delta_{k,l}\
.$ (33)

Hence, the polarized structure function $xg_{1}(x,Q^{2})$ could be
reconstructed from Eq. (31), by giving the Jacobi moments $a_{n}(Q^{2})$ [18,
19, 20, 21, 22, 23, 24, 25, 26, 12].

We can obtain the Jacobi moments $a_{n}(Q^{2})$, taking the orthogonality
condition on Eq. (31) which finally lead us to:

$\displaystyle a_{n}(Q^{2})$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{n}c_{j}^{(n)}(\alpha,\beta)\ {\cal
L}[xg_{1},s=j+1]\leavevmode\nobreak\ \ .$ (34)

In deriving Eq. (34), we utilize the Laplace transform of $xg_{1}(x,Q^{2})$ as
it follows:

$\displaystyle{\cal{L}}[xg_{1},s]$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{\infty}dv\ e^{-sv}\ xg_{1}(x,Q^{2})\ .$ (35)

We can now relate the polarized structure function, $xg_{1}(x,Q^{2})$, with
its moments in Laplace s-space as following [18, 19, 20, 21, 22, 23, 24, 25,
26, 12]

$\displaystyle xg_{1}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
x^{\beta}(1-x)^{\alpha}\sum_{n=0}^{N_{max}}\Theta_{n}^{\alpha,\beta}(x)$ (36)
$\displaystyle\times$ $\displaystyle\sum_{j=0}^{n}c_{j}^{(n)}{(\alpha,\beta)}\
{\cal{L}}[xg_{1},s=j+1]\ .$

By regarding Eq. (36) for $xg_{1}(x,Q^{2})$, we choose the set
$\\{N_{max},\alpha,\beta\\}$ to reach optimal convergence of this series
throughout the kinematic region constrained by the data. In practice, we find
the following numerical values for above parameters: $N_{max}=9$,
$\alpha=3.0$, and $\beta=0.5$ to be sufficient. We should note that in Mellin
space when we intend to calculate the first moment for quark distribution or
structure function, $g_{1}$, we choose $n=1$ but in Laplace s-space to get the
first moment, we should consider $s=0$ instead of.

##  IV QCD Analysis & Parametrization of PPDFs

The required analysis of PPDFs within the QCD content, are including the
following parts.

###  IV.1 Parametrization

We consider a proton consisted of massless partons which carry momentum
fraction $x$ with helicity distributions $q_{\pm}(x,Q^{2})$ at characteristic
scale $Q^{2}$. The difference $\Delta
q(x,Q^{2})=q_{+}(x,Q^{2})-q_{-}(x,Q^{2})$ measures how much the parton of
flavor $q$ remembers its parent’s proton polarization. On the other words we
can say that it represents the probability of finding a polarized parton with
fraction $x$ of parent hadron momentum and spin align/anti-align to hadron’s
spin. It measures the net helicity of partons in a longitudinally polarized
hadron.

In parametrization process, we consider the following form for the polarized
PDFs at initial scale $Q_{0}^{2}=1.3$ GeV2:

$x\>\Delta q(x,Q_{0}^{2})={\cal
N}_{q}\eta_{q}x^{a_{q}}(1-x)^{b_{q}}(1+c_{q}x)\ ,$ (37)

in which the polarized PDFs are determined by parameters
$\\{\eta_{q},a_{q},b_{q},c_{q}\\}$, and the generic label
$q=\\{u_{v},d_{v},\bar{q},g\\}$ indicates the partonic flavors up-valence,
down-valence, sea, and gluon, respectively. ${\cal N}_{q}$ is the
normalization constant given by:

$\frac{1}{{\cal
N}_{q}}=\left(1+c_{q}\frac{a_{q}}{a_{q}+b_{q}+1}\right)\,B\left(a_{q},b_{q}+1\right)\
,$ (38)

and chosen such that $\eta_{q}$ in Eq.(37) is the first moments of $\Delta
q(x,Q_{0}^{2})$ where $B(a,b)$ is the Euler beta function.

The total up and down quark distributions are a sum of the valence plus sea
distributions: $\Delta u=\Delta u_{v}+\Delta\bar{q}$ and $\Delta d=\Delta
d_{v}+\Delta\bar{q}$. We consider an $SU(3)$ flavor symmetry as
$\Delta\overline{q}\equiv\Delta\overline{u}=\Delta\overline{d}=\Delta
s=\Delta\overline{s}$. Nevertheless we could allow for an $SU(3)$ symmetry
violation term by introducing $\kappa$ such that $\Delta
s=\Delta\overline{s}=\kappa\Delta\bar{q}$. Since the strange quark
distribution is poorly constrained, the results would be insensitive to the
particular choice of $\kappa$.

From Eq. (37), it is obvious that each of four polarized parton densities
$q=\\{u_{v},d_{v},\bar{q},g\\}$ contain four parameters
$\\{\eta_{q},a_{q},b_{q},c_{q}\\}$ which gives a total of 16 parameters that
should be determined. We illustrate that some of these parameters can be
eliminated while maintaining sufficient flexibility to obtain a good fit.

Figure 5: The polarized proton structure functions with respect to $x$ at
$Q^{2}=2\;GeV^{2}$. The results of Jacobi expansion technique in NLO (solid
curve) and LO (dotted) approximations are compared with parametrization models
such as NAAMY (long-dashed) [25], BB (dashed-dotted) [69], GRSV (dashed-
dotted-dotted) [77], LSS05 (dashed-dashed-dotted) [81] and KATAO (dashed)
[21].  Figure 6: As in Fig.5 but at $Q^{2}=3.5\;GeV^{2}$ and in a shorter
range of $x$, corresponding to the available data.  Figure 7: As in Fig.5 but
at $Q^{2}=4.4\;GeV^{2}$ and in a shorter range of $x$, corresponding to the
available data.  Figure 8: The spin dependent neutron structure functions with
respect to $x$ at $Q^{2}$ $=1.5\;GeV^{2}$ and $5\;GeV^{2}$. The results of
Jacobi expansion technique in NLO (solid curve) and LO (dotted) approximations
are compared with parametrization models like NAAMY (long-dashed) [25], BB
(dashed-dotted) [69], GRSV (dashed-dotted-dotted) [77], LSS05 (dashed-dashed-
dotted) [81] and KATAO (dashed) [21].  Figure 9: The spin dependent structure
functions of deuteron as a function of $x$ at $Q^{2}$ $=2\;GeV^{2}$ and
$13\;GeV^{2}$. The results of Jacobi expansion technique in NLO (solid curve)
and LO (dotted) approximations are compared with parametrization models such
as NAAMY (long-dashed) [25], BB (dashed-dotted) [69], GRSV (dashed-dotted-
dotted) [77], LSS05 (dashed-dashed-dotted) [81] and KATAO (dashed) [21].

###  IV.2 First Moments of $\Delta u_{v}$ and $\Delta d_{v}$

The parameters $\eta_{u_{v}}$ and $\eta_{d_{v}}$ are the first moments of the
polarized valence up and down quark densities, denoted by $\Delta u_{v}$ and
$\Delta d_{v}$. These densities can be related to $F$ and $D$ quantities as
the weak matrix elements which are measured in neutron and hyperon
$\beta$–decays. Hence one can write [27]:

$\displaystyle a_{3}$ $\displaystyle=$ $\displaystyle\int_{0}^{1}dx\>\Delta
q_{3}=\eta_{u_{v}}-\eta_{d_{v}}=F+D\ ,$ (39) $\displaystyle a_{8}$
$\displaystyle=$ $\displaystyle\int_{0}^{1}dx\>\Delta
q_{8}=\eta_{u_{v}}+\eta_{d_{v}}=3F-D\ ,$ (40)

where $a_{3}$ and $a_{8}$ denote the non-singlet combinations of the first
moments of the polarized quark densities corresponding to

$\displaystyle q_{3}$ $\displaystyle=$ $\displaystyle(\Delta
u+\Delta\overline{u})-(\Delta d+\Delta\overline{d})\ ,$ (41) $\displaystyle
q_{8}$ $\displaystyle=$ $\displaystyle(\Delta u+\Delta\overline{u})+(\Delta
d+\Delta\overline{d})-2(\Delta s+\Delta\overline{s})\ .$ (42)

A reanalysis of $F$ and $D$ with updated $\beta$-decay constants leads to the
following results: $F=0.464\pm 0.008$ and $D=0.806\pm 0.008$ [27]. With these
values we obtain:

$\displaystyle\eta_{u_{v}}$ $\displaystyle=$ $\displaystyle+0.928\pm 0.014\ ,$
(43) $\displaystyle\eta_{d_{v}}$ $\displaystyle=$ $\displaystyle-0.342\pm
0.018\ .$ (44)

Utilizing the above numerical values of $\eta_{u_{v}}$ and $\eta_{d_{v}}$ will
end to reduce two parameters during the fitting processes.

###  IV.3 Polarized DGLAP evolution

The polarized DGLAP evolution equations can be solved in the Laplace space
using the Jacobi polynomial approach. The Laplace transformation of the parton
densities $\Delta q$ are defined analogous to that of Eq. (35) as:

$\displaystyle{\cal{L}}$ $\displaystyle[\Delta q(x=e^{-v},Q_{0}^{2}),s]\equiv$
$\displaystyle\Delta q(s,Q_{0}^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}e^{-sv}\>\Delta q(x=e^{-v},Q_{0}^{2})\>dv$ (45)
$\displaystyle=$
$\displaystyle{\cal{N}}_{q}\eta_{q}\left(1+c_{q}\>\frac{s+a_{q}}{s+a_{q}+b_{q}+1}\right)\
$ $\displaystyle\times$ $\displaystyle B(s+a_{q},b_{q}+1)\ ,$

where $q=\\{u_{v},d_{v},\overline{q},g\\}$, and $B$ denotes the Euler beta
function.

The twist-2 contributions to the spin dependent structure function
$g_{1}(s,Q^{2})$ can be expressed in terms of the polarized parton densities,
in the Laplace space as:

$\displaystyle{\cal L}[g_{1}^{p},s]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum\limits_{q}e_{q}^{2}\times\left[\left(1+\frac{\tau}{4\pi}\Delta
C_{q}(s)\right)[\Delta q(s,Q^{2})+\right.$ (46)
$\displaystyle\left.\Delta\bar{q}(s,Q^{2})]+\frac{2}{3}\frac{\tau}{4\pi}\Delta
C_{g}(s)\Delta g(s,Q^{2})\right]\;,$ $\displaystyle{\cal L}[g_{1}^{n},s]$
$\displaystyle=$ $\displaystyle{\cal L}[g_{1}^{p},s]-\frac{1}{6}({\cal
L}[u_{v},s]-$ (47) $\displaystyle{\cal
L}[d_{v},s])\times\left(1+\frac{\tau}{4\pi}\Delta C_{q}(s)\right)\;.$

Here, the summation is over $u,d,s$ quark flavors. $\delta q,\delta\bar{q}$
and $\delta g$ are the polarized quark, anti-quark, and gluon distributions,
respectively. The $\Delta C_{g}$ and $\Delta C_{q}$ denote the spin dependent
Wilson coefficients in Laplace transform $s$ space respectively which are
written as:

$\displaystyle\Delta C_{g}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{2}{s+1}-\frac{2}{s+2}+\frac{\psi^{(0)}(s+1)+\gamma_{E}}{s+1}-\right.$
(48)
$\displaystyle\left.\frac{2\left(\psi^{(0)}(s+2)+\gamma_{E}\right)}{s+2}\right)$
$\displaystyle\Delta C_{q}$ $\displaystyle=$
$\displaystyle\frac{8}{3(s+1)}+\frac{4}{3(s+2)}+\frac{4\left(\psi^{(0)}(s+2)+\gamma_{E}\right)}{3(s+1)}+$
(49)
$\displaystyle\frac{4\left(\psi^{(0)}(s+3)+\gamma_{E}\right)}{3(s+2)}+\frac{4\psi^{(1)}(s+1)}{3}+\frac{4\psi^{(1)}(s+3)}{3}$
$\displaystyle-\frac{4}{3}\left(\frac{9}{2}+\frac{\pi^{2}}{3}\right)$

Employing the inverse Laplace transform on Eqs.(46,47), the $g_{1}^{p}$ and
$g_{1}^{n}$ can be obtained in Bjorken $x$-space. Considering the
parametrization for PPDFs there are finally 9 unknown parameters which should
be determined during the fitting processes, taking the available experimental
data for polarized structure functions.

In addition to proton and neutron structure functions we can also do the
required computations for the deuteron which is in fact a nucleus consists of
one proton and one neutron. The deuteron structure function is given by
$xg_{i}^{d}=\frac{xg_{i}^{p}+xg_{i}^{n}}{2}\times(1-1.5\omega_{D})\;;i=1,2$ .
Here $\omega_{D}=0.05\pm 0.01$ denotes the probability to find the deuteron in
a $D-$state [28, 29, 30]. The available data for deuteron structure function
are also used during the fitting process.

###  IV.4 The $g_{2}$ structure function

Now by accessing to the $g_{1}$ structure function, one can calculate $g_{2}$
via Wandzura–Wilczek [31, 32] relation as in the following

$g_{2}(x,Q^{2})=-g_{1}^{p}(x,Q^{2})+\int_{x}^{1}\frac{dy}{y}g_{1}^{p}(y,Q^{2})\leavevmode\nobreak\
.$ (50)

We are now at the position to investigate the fits to spin dependent structure
functions, as we do it in next section, to extract the PPDFs from the
available data.

##  V Fitting contents in QCD analysis

###  V.1 Overview of data sets

Figure 10: The polarized proton structure functions $xg_{2}^{p}$ as a function
of $x$ at $Q^{2}$ $=5\;GeV^{2}$ . The results of Jacobi expansion technique in
NLO (solid curve) and LO (dashed) approximations are compared with
parametrization models like KTA17 (dashed-dotted) [19], MA22 (dashed-dotted-
dotted) [26].  Figure 11: The spin dependent deuteron structure functions
$xg_{2}^{d}$ with respect to $x$ at $Q^{2}$ $=3\;GeV^{2}$. The result of
Jacobi expansion technique in NLO (solid curve) and LO (dashed) approximations
are compared with parametrization models like KTA17 (dashed-dotted) [19], MA22
(dashed-dotted-dotted) [26].

In our recent analysis which we call it NMA23 we focus on the polarized DIS
data samples. The needed DIS data for all PPDFs are coming from the
experiments at electron-proton collider and also in fixed-target situation
including proton, neutron and heavier targets such as deuteron.

Although separating quarks from antiquarks is not possible, nonetheless it is
the inclusive DIS data that is included in the fit. Additionally we take into
our NMA23 fitting procedure the $g_{2}$ structure function. Due to the
technical difficulty in operating the needed 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 [12]. In fact we use all
available data of $g_{1}^{p}$ from E143, HERMES98, SMC, EMC, E155, HERMES06,
COMPASS10, COMPASS16, JLAB06 and JLAB17 experiments [33, 34, 38, 36, 37, 35,
39, 40, 41, 42], and $g_{1}^{n}$ data from HERMES98, E142, E154, HERMES06,
Jlab03, Jlab04 and Jlab05 [34, 43, 45, 44, 46, 47, 48] and finally the data of
$g_{1}^{d}$ from E143, SMC, HERMES06, E155, COMPASS05, COMPASS06 and COMPASS17
[33, 38, 35, 49, 50, 51, 52]. The DIS data for $g_{2}^{p,n,d}$ from E143,
E142, Jlab03, Jlab04, Jlab05, E155, Hermes12 and SMC [33, 43, 46, 47, 48, 53,
54, 55] are also included. These data sets are listed in Table 1. We also
present the kinematic coverage, the number of data points for each given
target, and the fitted normalization shifts ${\cal{N}}_{i}$ in this Table. Our
NMA23 analysis algorithm calculates the $Q^{2}$ evolution and extracts the
polarized structure function in $x$ space using Jacobi polynomials approach.
It is corresponding to the fitting programs on the market that solve the
polarized DGLAP evolution equations in the Laplace space.

One of the important quantities used as a criteria to indicate the validation
of fit process, is the chi-square ($\chi^{2}$) test which is assessing the
goodness of fit between observed values and those expected theoretically. In
next subsection we deal with about it in more details.

###  V.2  $\chi^{2}$ minimization

The goodness of fit to the data for a set of $\rm p$ independent parameters,
is quantified by the $\chi_{\rm global}^{2}(\rm p)$ . To determine the best
fit, we need to minimize the $\chi^{2}_{\rm global}$ function with the free
unknown parameters. We perform it for PPDFs at the LO and NLO approximations
that additionally include the QCD cut off parameter, $\Lambda_{\rm QCD}$ which
finally yield us the polarized PDFs at Q${}_{0}^{2}$ = 1.3 GeV2.

This function is presented as it follows:

$\chi_{\rm global}^{2}({\rm p})=\sum_{n=1}^{N_{\rm exp}}w_{n}\chi_{n}^{2}\,.$
(51)

In above equation, ${w_{n}}$ denotes a weight factor for the $n^{\rm th}$
experiment. However this factor in principle can have different values for
various data sets but since all of the experimental data sets have identical
worthiness, we take the related weight factor equal to one in our analyses
[56, 57, 58]. Following that the $\chi_{n}^{2}$ in Eq.(51) is defined as:

$\chi_{n}^{2}(\rm p)=\left(\frac{1-{\cal N}_{n}}{\Delta{\cal
N}_{n}}\right)^{2}+\sum_{i=1}^{N_{n}^{\rm data}}\left(\frac{{\cal
N}_{n}\,g_{(1,2),i}^{\rm Exp}-g_{(1,2),i}^{\rm Theory}(p)}{{\cal
N}_{n}\,\Delta g_{(1,2),i}^{\rm Exp}}\right)^{2}\,.$ (52)

The minimization of the $\chi_{\mathrm{\rm global}}^{2}(\rm p)$ function is
done applying the CERN program library MINUIT [59]. In the above equation, the
essential contribution originates 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 $g^{\rm Exp}$, $\Delta g^{\rm Exp}$ denote the experimental
measurement and the experimental uncertainty, respectively that is coming from
statistical and systematic uncertainties, combined in quadrature.

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

We should remind that the results of fitting process for PPDFs at initial
$Q_{0}^{2}$ energy scale is based on Hessian approach [56] such that
$\Delta\chi^{2}/\chi^{2}=0.2\%$ that is corresponding to bigger confidence
level (C.L) than the inconvenience choice $\Delta\chi^{2}=1$ with 68$\%$ C.L
[60]. The plots for polarized parton densities are depicted in Fig.1 and Fig.2
on this base.

####  V.2.1 Gluon and sea quarks

We find the factor $(1+c_{q}x)$ in Eq. (37), provides the flexibility to
obtain a good description of the data, especially for the polarized valence
quark distributions $\Delta u_{v},\Delta d_{v}$. Thus we will make use of the
$c_{q}$ coefficients for the up-valence and down-valence quark distribution
functions; in contrast, we are able to set the values of $c_{\bar{q}}$ and
$c_{g}$ to zero, $(c_{\bar{q}}=c_{g}=0)$, while preserving a good fit and
eliminating two free parameters. We find the fit improves if we use non-zero
values for the $c_{u_{v}},c_{d_{v}}$ parameters, but as these are relatively
flat directions in $\chi$-space we shall fix the values as detailed in Table
2.

Having fixed $\eta_{u_{v}},\eta_{d_{v}}$ and $b_{\bar{q}},b_{g}$ parameters in
preliminary minimization and to take $c_{\bar{q}}=c_{g}=0$ which we referred
before, we then set the $b_{\bar{q}},b_{g},c_{u_{v}},c_{d_{v}}$ parameters as
indicated in Table 2; this gives us a total of 9 unknown parameters, in
addition to $\alpha_{s}(Q_{0}^{2})$.

LO
---
$\Delta u_{v}$ | $\eta_{u_{v}}$ |  $0.928$(Fixed) | $\Delta\overline{q}$ | $\eta_{\overline{q}}$ | $-0.076068\pm 0.0017283$
| a${}_{u_{v}}$ | $0.2906\pm 0.01061$ |  | a${}_{\overline{q}}$ | $0.42486\pm 0.03115$
| b${}_{u_{v}}$ | $2.0498\pm 0.010617$ |  | b${}_{\overline{q}}$ |  $2.7562$(Fixed)
| c${}_{u_{v}}$ |  $16.4977$(Fixed) |  | c${}_{\overline{q}}$ | $0$
$\Delta d_{v}$ | $\eta_{d_{v}}$ |  $-0.342$(Fixed) | $\Delta g$ | $\eta_{g}$ | $1.1543\pm 0.1792$
| a${}_{d_{v}}$ | $0.1274\pm 0.003728$ |  | ag | $2.4164\pm 0.3716$
| b${}_{d_{v}}$ | $1.8621\pm 0.044186$ |  | bg |  $1.7430$(Fixed)
| c${}_{d_{v}}$ |  $35.7909$(Fixed) |  | cg | $0$
$\Lambda=0.2007\pm 0.05004GeV$
$\alpha(M_{z}^{2})=0.12812\pm 0.0038$
$\chi^{2}/D.O.F=1171.65/957=1.224$
NLO
$\Delta u_{v}$ | $\eta_{u_{v}}$ |  $0.928$(Fixed) | $\Delta\overline{q}$ | $\eta_{\overline{q}}$ | $-0.076272\pm 0.001742$
| a${}_{u_{v}}$ | $0.33889\pm 0.01121$ |  | a${}_{\overline{q}}$ | $0.4844\pm 0.029784$
| b${}_{u_{v}}$ | $2.1075\pm 0.052585$ |  | b${}_{\overline{q}}$ |  $3.3948$(Fixed)
| c${}_{u_{v}}$ |  $15.3475$(Fixed) |  | c${}_{\overline{q}}$ | $0$
$\Delta d_{v}$ | $\eta_{d_{v}}$ |  $-0.342$(Fixed) | $\Delta g$ | $\eta_{g}$ | $0.2526\pm 0.05528$
| a${}_{d_{v}}$ | $0.1294\pm 0.004110$ |  | ag | $2.0898\pm 0.4409$
| b${}_{d_{v}}$ | $1.8654\pm 0.043007$ |  | bg |  $1.0174$(Fixed)
| c${}_{d_{v}}$ |  $41.9067$(Fixed) |  | cg | $0$
$a_{1}=0.0250$(Fixed)
$b_{1}=10.70$(Fixed)
$a_{0}=0.2399$(Fixed)
$\Lambda=0.2156\pm 0.04989GeV$
$\alpha(M_{z}^{2})=0.115457\pm 0.00341$
$\chi^{2}/D.O.F=1128.98/957=1.179$
Table 2: Final parameter values and their statistical errors in the
$\overline{{\rm MS}}$–scheme at the input scale $Q_{0}^{2}=1.3$ GeV2.

Now in order to validate the results of the fitting, we consider and calculate
some sum rules as we do it in next section.

##  VI The Sum Rules

Some fundamental properties of the nucleon structure can be inspected by
considering QCD sum rules like total momentum fraction carried by partons and
also the total contribution of parton spin to the spin of the nucleon. In what
are following by utilizing available experimental data, we analysis some
important polarized sum rules.

###  VI.1 Bjorken sum rule

Integral over the spin distributions of quarks inside the nucleon yields to
the polarized Bjorken sum. It can be written in terms of multiplication of
nucleon axial charge, $g_{A}$ (as measured in neutron $\beta$ decay) with a
coefficient function, $C_{Bj}[\alpha_{s}(Q^{2})]$. Taking into account the
corrections of higher twist (HT), this sum rule is given by [61]:

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

A very precise determination on the $\alpha_{s}$ as strong coupling constant
can be provided by Bjorken sum rule. Using $C_{Bj}[\alpha_{s}(Q^{2})]$
expression the value of coupling can be extracted from experimental data while
the present world average value is $\alpha_{s}(M_{Z}^{2})=0.1179\pm 8.5\times
10^{-6}$ [62]. At 4-loop corrections of perturbative QCD (pQCD) this function
has been calculated in both massless [63] and massive cases [64]. Due to
ambiguities from small-$x$ extrapolation, determining $\alpha_{s}$ from the
Bjorken sum rule is suffering [65]. Nevertheless in our computations the
numerical value for coupling constant at $Z$-boson mass scale can be found
during the fitting process to find the unknown parameters of polarized parton
densities at initial energy scale $Q_{0}$. Outputted results for coupling
constant at LO and NLO analysis are presented in Table. 2 which are in good
agreement with reported world average value of this quantity.

In Table 3 we list our results for the Bjorken sum rule. Experimental
measurements such as E143 [33], SMC [55], HERMES06 [35] and COMPASS16 [40] are
added to this table. An adequate consistency can be seen between them.

Table 3:  Our computed LO and NLO results for the Bjorken sum rule, $\Gamma_{1}^{NS}$, in comparison with world data from E143 [33], SMC [55], HERMES06 [35] and COMPASS16 [40]. Only HERMES06 [35] results are not extrapolated in full $x$ range (measured in region $0.021\leq x\leq 0.9$).  |  E143 [33] |  SMC [55] |  HERMES06 [35] |  COMPASS16 [40] | LO | NLO
---|---|---|---|---|---|---
|  $Q^{2}=5$ GeV2 |  $Q^{2}=5$ GeV2 |  $Q^{2}=5$ GeV2 |  $Q^{2}=3$ GeV2 |  $Q^{2}=5$ GeV2 |  $Q^{2}=5$ GeV2
$\Gamma^{\rm NS}_{1}$ | $0.164\pm 0.021$ | $0.181\pm 0.035$ | $0.148\pm 0.017$ | $0.181\pm 0.008$ | $0.15632\pm 0.0062$ | $0.15350\pm 0.00081$

###  VI.2 Proton helicity sum rule

In order to complete our knowledge in the field of nuclear physics an
extrapolation of proton spin among its constituents can be done and
consequently new sum rule as proton helicity sum rule is achieved [66].
Considering this sum rule, by a precise extraction of PPDFs, one can obtain an
accurate picture of the quark and gluon helicity densities.

Since each constituent of a nucleon is carrying part of nucleon spin, the
total spin of nucleon can be written as:

$\frac{1}{2}=\frac{1}{2}\Delta\Sigma(Q^{2})+\Delta{\mathrm{G}}(Q^{2})+{\mathrm{L}}(Q^{2}).$
(54)

In this equation
$\Delta\Sigma(Q^{2})=\sum_{i}\int_{0}^{1}dx\leavevmode\nobreak\ (\Delta
q(x,Q^{2})+\Delta\bar{q}(x,Q^{2}))$ represents spin contribution of the
singlet flavour, $\Delta{\rm G(Q^{2})}=\int_{0}^{1}dx\leavevmode\nobreak\
\Delta g(x,Q^{2})$ denotes the gluon spin contribution and finally
${\mathrm{L}}(Q^{2})$ is interpreted as the total contribution from quark and
gluon orbital angular momentum. In Eq.(54) each term depends on $Q^{2}$ but
the sum of them does not. The measuring processes of them can not be done
easily and it is beyond the scope of this paper to describe their measurement
methods.

Numerical values of first moments of the singlet-quark and gluon at Q2=10 GeV2
are listed in Table 4. Our results at both truncated and full $x$ region are
compared to those from the NNPDFpol1.1 [67] and DSSV14 [68].

As can be seen from Table 4 for the $\Delta\Sigma$, our NMA23 results are
consistent, within uncertainty, with those of other groups. This is occurred
because in semileptonic decays the first moment of polarized densities are
mainly fixed. Very different values are reported by various groups when the
gluon contribution is considered. Due to their large uncertainty we are
avoided to get a stiffen result for the full first moment of gluon.

Table 4:  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. The recent polarized global analysis of NNPDFpol1.1 [67] and DSSV14 [68] are also presented.  |  DSSV14 [68] |  NNPDFpol1.1 [67] | LO | NLO
---|---|---|---|---
Full $x$ region $[0,1]$ |  |  |  |
$\Delta\Sigma{\rm(Q^{2})}$ | $0.291799$ | $+0.18\pm 0.21$ | $0.12959\pm 0.00922$ | $0.148546\pm 0.0194$
$\Delta{\rm G(Q^{2})}$ | $0.37109$ | $0.03\pm 3.24$ | $1.4693\pm 1.049$ | $0.8626\pm 0.3054$
Truncated $x$ region [$10^{-3},1$] |  |  |  |
$\Delta\Sigma{\rm(Q^{2})}$ | $0.36645$ | $+0.25\pm 0.10$ | $0.03644\pm 0.0406$ | $0.04848\pm 0.0187$
$\Delta{\rm G(Q^{2})}$ | $0.3636$ | $0.49\pm 0.75$ | $1.3848\pm 0.9829$ | $0.7827\pm 0.2854$
Table 5:  $d_{2}$ moments of the proton, neutron and deuteron polarized structure functions from the SLAC E155x [53], E01-012 [75], E06-014 [82], Lattice QCD [83], CM bag model [84], JAM15 [85], JAM13 [86] compared with LO and NLO results.  | Ref. |  $Q^{2}$ [GeV2] | $10^{2}d^{p}_{2}$ | $10^{5}d^{n}_{2}$ | $10^{3}d^{d}_{2}$
---|---|---|---|---|---
LO |  | $5$ | $0.2994\pm 0.00035$ | $127.68\pm 11.0073$ | $1.2534\pm 0.02719$
NLO |  | $5$ | $0.2855\pm 0.00196$ | $23.4919\pm 0.6654$ | $1.3464\pm 0.0743$
E06-014 | [82] | 3.21 |  | $-421.0\pm 79.0\pm 82.0\pm 8.0$ | -
E06-014 | [82] | 4.32 |  | $-35.0\pm 83.0\pm 69.0\pm 7.0$ | -
E01-012 | [75] | 3 | - | $-117\pm 88\pm 138$ | -
E155x | [53] | $5$ | $0.32\pm 0.17$ | $790\pm 480$ | -
E143 | [33] | $5$ | $0.58\pm 0.50$ | $500\pm 2100$ | $5.1\pm 9.2$
Lattice QCD | [83] | 5 | 0.4(5) | -100(-300) | -
CM bag model | [84] | $5$ | $1.74$ | $-253$ | $6.79$
JAM15 | [85] | $1$ | $0.5\pm 0.2$ | $-100\pm 100$ | -
JAM13 | [86] | $5$ | $1.1\pm 0.2$ | $200\pm 300$ | -
Table 6:  The result of BC sum rule for $\Gamma_{2}^{p}$, $\Gamma_{2}^{d}$ and $\Gamma_{2}^{n}$ in comparison with world data from E143 [33], E155 [53], HERMES2012 [54], RSS [74], E01012 [75].  |  E143 [33] |  E155 [53] |  HERMES2012 [54] |  RSS [74] |  E01012 [75] | LO | NLO
---|---|---|---|---|---|---|---
| $0.03\leq x\leq 1$ | $0.02\leq x\leq 0.8$ | $0.023\leq x\leq 0.9$ | $0.316<x<0.823$ | $0\leq x\leq 1$ | $0.03\leq x\leq 1$ | $0.03\leq x\leq 1$
|  $Q^{2}=5$ 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
$\Gamma_{2}^{p}$ | $-0.014\pm 0.028$ | $-0.044\pm 0.008$ | $0.006\pm 0.029$ | $-0.0006\pm 0.0022$ | … | $-0.01911\pm 0.0199$ | $-0.01929\pm 0.00038$
$\Gamma_{2}^{d}$ | $-0.034\pm 0.082$ | $-0.008\pm 0.012$ | - | $-0.0090\pm 0.0026$ | … | $-0.001687\pm 0.000016$ | $-0.0028986\pm 0.00053$
$\Gamma_{2}^{n}$ | - | - | - | $-0.0092\pm 0.0035$ | $0.00015\pm 0.00113$ | $\leavevmode\nobreak\ 0.007824\pm 0.00056$ | $\leavevmode\nobreak\ 0.0034500\pm 0.000016$

The proton spin sum rule can be finally calculated, considering the extracted
values which are listed in Table 4. Accordingly numerical value of quark and
gluon orbital angular momentum, attributed to the spin of the proton, is
obtained as:

${\mathrm{L}}(Q^{2}=10\leavevmode\nobreak\ \rm GeV^{2})=-0.436873\pm
0.334587\,.$ (55)

The contribution of total orbital angular momentum to the spin of the proton
can not be determined tightly and it is due to the large uncertainty which is
mostly coming out from the gluons. By improving the current level of
experimental accuracy, precise determination of each individual contribution
to the nucleon spin can be obtained.

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

Twist-3 reduced matrix element, denoted by $d_{2}$, is not considered as a sum
rule but to investigate the higher twist effect, the numerical evaluation of
this quantity is important. One can find in [69] the detailed analyses of
higher twist, related to the $g_{1}$ polarized structure function. Through the
moments of $g_{1}$ and $g_{2}$ structure functions, considering the operator
product expansion (OPE) theorem [70], the effect of quark-gluon correlations
can be studied. These considerations for the moments will conclude the
following definition for $d_{2}(Q^{2})$ as reduced matrix element:

$\displaystyle d_{2}(Q^{2})$ $\displaystyle=$ $\displaystyle
3\int_{0}^{1}x^{2}\bar{g_{2}}(x,Q^{2})\leavevmode\nobreak\ dx$ (56)
$\displaystyle=$
$\displaystyle\int_{0}^{1}x^{2}[3g_{2}(x,Q^{2})+2g_{1}(x,Q^{2})]\leavevmode\nobreak\
dx.$

In above equation we have $\bar{g}_{2}=g_{2}-g_{2}^{WW}$ where $g_{2}^{WW}$,
corresponding to Eq.(50), is given by Wandzura and Wilczek (WW). The deviation
of $g_{2}$ from $g_{2}^{\tau_{2}}$ which is polarized structure function at
leading twist order can be measured, using the $d_{2}(Q^{2})$ as the twist-3
reduced matrix element of spin dependent operators in nucleon. This matrix
element because of the $x^{2}$ weighting factor in Eq.(56) has remarkable
sensitive to the behaviour of $\bar{g_{2}}$ at large-$x$ values. By extracting
the $d_{2}$ term, valuable intuition about the size of the multi-parton
correlation terms can be achieved which denotes the importance of this
quantity.

Having non-zero value for $d_{2}$ reveals us the importance of higher twist
terms in QCD analyses. To improve model prediction, more information on the
higher twist operators are required and this can be done by precise
measurement of $d_{2}$ term. Our results for $d_{2}$, compared with
experimental values and also some theoretical predictions, are presented in
Table 5.

###  VI.4 Burkhardt-Cottingham sum rule

The zeroth moment of $g_{2}$ structure function, considering dispersion
relations for virtual Compton scattering at all $Q^{2}$ values, is predicted
to get zero value and consequently Burkhardt and Cottingham (BC) sum rule is
obtained such as [71]:

$\Gamma_{2}=\int_{0}^{1}dx\,g_{2}(x,Q^{2})=0\leavevmode\nobreak\ .$ (57)

BC sum rule is insignificant result, arising out from WW relation which is
given by Eq.(50). In light cone expansion, one can not obtained the zeroth
moment of structure function and hence local operator product expansion [72]
can not describe this moment. This sum rule can be used even if the structure
function involves target mass correction [73]. Finally it should be said that
the presence of HT contribution is denoting to violation of the BC sum rule
[54].

In Table. 6 we list our results for $\Gamma_{2}$ at LO and NLO approximations
where data from E143 [33], E155 [53], HERMES2012 [54], RSS [74], E01012 [75]
groups for proton, deuteron and neutron are also added there. The behaviour of
$g_{2}$ structure function at low-$x$ values has not yet measured accurately
but it has significant effect on any possible conclusion which we get.

##  VII Conclusion

For about three decades, studies of the internal spin structure of the proton
have advanced steadily using the technology of polarized beams and polarized
targets. The demand for higher-energy experiments to access the deepest
regions inside the proton and to extract the theoretically cleanest results
continues.

Polarized deep inelastic scattering remains one of the cleanest tests for
studying the internal spin structure of the proton and neutron. Pioneering
experiments at SLAC, scattering polarized electrons off polarized protons,
helped to establish the quark structure of the proton with the observation of
large spin-dependent asymmetries. An exciting follow-up experiment (CERN EMC)
at higher energies uncovered a violation of a quark parton model sum rule,
implying that the quarks accounted for only a small fraction of the proton
spin and giving birth to the proton-spin crisis. Since about three decades ago
a large sample of data from polarized fixed-target experiments at SLAC, CERN,
and DESY have resulted in a substantial perturbative-QCD analysis of the
nucleon spin structure.

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 NMA23
analysis. They are essential to test some QCD sum rules .We provided a unified
and consistent PPDF through an achievement, containing an appropriate
description of the fitted data. Within the known very large uncertainties
arising from the lack of constraining data, our helicity distributions are in
good consistency with other extractions. We studied Bjorken sum rule, proton
helicity and Burkhardt-Cottinghan sum rules. Our results for the reduced
matrix element d2 at the NLO approximation have also been presented. To
investigate them precisely, more accurate data are needed and in this work we
considered all the recent and available data, relating to polarized targets
which are listed in Table.1.

Using Jacobi polynomial technique a fit to the polarized lepton DIS data on
nucleon have been presented at the NLO approximation. We found good agreement
with the experimental data and our results have been in correspond with
determinations from some parametrization models. In general we could
demonstrate that an acceptable progressive has been achieved to describe the
spin structure of the nucleon.

The available data which we use in our recent analysis are up to date and
including more data than we employed in our pervious analysis [12]. These data
sets are summarized in Table 1. The kinematic coverage, the number of data
points for each given target, and the fitted normalization shifts $N_{i}$ also
presented in this table. Our NMA23 analysis algorithm computed the $Q^{2}$
evolution and extracted the spin structure functions in $x$ space using Jacobi
polynomials approach. It corresponded to the fitting programs of other groups
which solve the DGLAP evolution equations in the Mellin space. Results for
PPDFs at different energy scales and $g_{1}^{p,n}$ structure functions
together with deuteron nucleus have been presented in Fig.1 to Fig.11 which
confirm the validity of computations during the fitting process to all updated
and recent related data.

The current analysis can be extended to include transverse polarized targets,
using the Jacobbi polynomial expansions in Laplace $s$-space or other
polynomial expansion which can be done as our further research task.

## Acknowledgments

H. N is indebted to Shaid Bahonar university of Kerman and A. M acknowledges
Yazd university for the provided facilities to do this project. S. A. T. is
grateful to the School of Particles and Accelerators, Institute for Research
in Fundamental Sciences (IPM) to make the required support to do this project.

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