# On the feasibility of dynamical analysis of network models of biochemical
regulation

Luis M. Rocha,1,2,3
1Department of Systems Science and Industrial Engineering,
Binghamton University, State University of New York, Binghamton, NY, USA.
2Instituto Gulbenkian de Ciência, 2780-156, Oeiras, Portugal
3Center for Social and Biomedical Complexity,
Luddy School of Informatics, Computing & Engineering,
Indiana University, Bloomington IN, USA
<EMAIL_ADDRESS>
To the Editor: A recent article by Weidner et al. (2021) presents a method to
extract graph properties that are predictive of the dynamical behavior of
multivariate, discrete models of biochemical regulation. In other words, a
method that uses only features from the structure of network interactions to
predict which nodes are most involved in automata network dynamics. However,
the authors claim that dynamical analysis of large automata network models is
“not even feasible.” To make sure that others are not discouraged from working
on this problem, it is important to clarify that effective dynamical analysis
of automata network models, to the contrary, is feasible. Unlike what is
suggested in the article, graph-based analysis of static features is not the
only analytical avenue for large systems biology models of regulation and
signaling dynamics because there are dynamical methods that are, indeed,
scalable111 By scalable we mean that the computational complexity of methods
employed to analyze multivariate dynamical systems in regard to their
dynamical behavior (e.g., controlability, convergence to attractors,
robustness to perturbations, etc.) is manageable. That is, a given method is
scalable if results can be computed in finite (and reasonable) time, with
finite memory, for a system of a reasonable size—ideally for network models in
Systems Biology, up to thousands of nodes. .

There has been much interest recently in predicting multivariate dynamics from
static network structure alone, especially in regard to the controlability of
systems biology models of gene regulation, signalling, and cellular
differentiation (Liu et al., 2011; Nacher and Akutsu, 2013; Fiedler et al.,
2013; Zanudo et al., 2017). These are quite welcome methods because we often
lack information about the underlying causal interaction dynamics. However,
two very popular (and scalable) methods in network science lead to very
erroneous predictions of the subsets of (driver) variables that control
dynamics (Gates and Rocha, 2016). The most accurate of these methods are based
on feedback vertex set theory (Fiedler et al., 2013; Zanudo et al., 2017),
which does not scale well and can only make predictions about the entire
ensemble of dynamical systems that fits the same static interaction graph
(Gates et al., 2021).

Another putative reason for pursuing structure-only methods is that even when
the underlying interaction dynamics of each variable is known, it is not
feasible to compute the dynamical (or attractor) landscape of the entire
multivariate system when the interaction network is sufficiently large. This
prevents us from exhaustively enumerating all possible interventions that can
control the dynamics from one attractor basin to another222 The most important
scalability constraint in the analysis of automata networks is the number of
node variables, $n$, as the dynamical landscape of such systems is comprised
of $s^{n}$ possible configurations of states $s$ (Gates and Rocha, 2016). For
instance, a well-known Boolean network model of intracellular signaling
networks in generic fibroblasts is comprised of 130 nodes (Helikar et al.,
2008), thus it can be in one of $2^{130}$ possible state configurations—a
dynamical landscape that is too large to be exhaustively searched. . Still, it
is also true that enumeration of all possible interventions is infeasible in a
simple graph of sufficient size. The computational complexity of finding all
possible subsets of the set of nodes, or generating the powerset, is at least
$\mathcal{O}(2^{n})$ (Moore, 1971). Therefore, exhaustive search of all
possible interventions in any network is ultimately infeasible for large
graphs whether one uses structure- or dynamics-based methods.

Computing the full dynamics of a multivariate system certainly adds to the
complexity of exhaustively searching all possible interventions. But many
software tools exist—and are collected in repositories such as the CoLoMoTo
Consortium (Naldi et al., 2015)—that allow for the identification of
attractors in automata networks without full enumeration of their dynamical
landscapes, such as PyBoolNet (Klarner et al., 2016) and Boolink (Karanam et
al., 2021). In particular, recent developments in attractor identification
algorithms and code (PyStableMotifs) now allow the dynamical analysis of
thousands of networks, some with over 15,000 nodes (Rozum et al., 2021). From
another angle, a novel update scheme for asynchronous automata networks has
been shown to reduce the complexity of attractor identification, enabling the
modelling and dynamical analysis of genome-scale networks (Paulevé et al.,
2020).

Moreover, various computational approaches have been developed and applied to
extract the key drivers of collective dynamics of biochemical network models
without going through every possible subset of nodes, much less the entire
dynamical landscape, in a brute force manner (Hari et al., 2021; Zañudo and
Albert, 2015; Biane and Delaplace, 2018; Su and Pang, 2020; Rozum et al.,
2021). Indeed, scalable methods exist that remove the redundancy of the
dynamics of each variable (micro-level) to allow for a characterization of the
entire causal macro-level dynamics, in both complete (Marques-Pita and Rocha,
2013) and probabilistic (Gates et al., 2021) manners. These scalable methods,
and the associated software tool CANA (Correia et al., 2018), provide causal
graph representations of automata networks that synthesize both structure and
dynamics. They are exhaustive in the sense that they preserve all effective
interactions of the micro-level dynamics—only redundant interactions are
disregarded. Thus, it is very feasible to analyze systems biology models
without disregarding their dynamics, allowing the precise study of any
putative intervention that controls the dynamics just as easily as structure-
only methods, but with additional accuracy afforded by information about the
dynamics.

## Acknowledgements

The author is indebted to the anonymous reviewers for their most valuable
comments and references which have considerably strengthened this letter.

## References

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# Effect of symmetry breaking of polarized light sea quarks on the nucleon and
nuclear structure functions, and sum rules

Fatemeh Arbabifar1 ID<EMAIL_ADDRESS>Shahin Atashbar Tehrani2 ID
<EMAIL_ADDRESS>Hamzeh Khanpour3,4,2 ID<EMAIL_ADDRESS>(1)Department
of Physics, Nasibeh campus, Farhangian University, P.O.Box 14665-889, Tehran,
Iran
(2)School of Particles and Accelerators, Institute for Research in Fundamental
Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran
(3)AGH University of Science and Technology, Faculty of Physics and Applied
Computer Science, Al. Mickiewicza 30, 30-055 Krakow, Poland
(4)Department of Physics, University of Science and Technology of Mazandaran,
P.O.Box 48518-78195, Behshahr, Iran

(August 31, 2024)

###### Abstract

In this study, we performed calculations and analyses of the structure
functions of polarized nucleons and light nuclei, specifically 3He and 3H,
using second-order Feynman diagrams. Our investigation focused on two main
aspects: Firstly, we examined the symmetry properties of polarized light sea
quarks. Secondly, we conducted a detailed investigation into the impacts of
symmetry breaking on the structure functions of both nucleons and nuclei. To
achieve this, we utilized the existing polarized Parton Distribution Functions
(polarized PDFs) available in the literature. These PDFs were used to
calculate and compare the polarized structure functions $g_{1}$ and $g_{2}$ of
the nuclei. Additionally, we examined and analyzed the Bjorken and Efremov-
Leader-Teryaev sum rules by utilizing the moments of the polarized structure
functions. The Lorentz color force components, namely $F_{E}^{y,n}$ and
$F_{B}^{y,n}$, are determined using the twist-2, twist-3, and twist-4 matrix
elements. When symmetry breaking is applied, it is observed that they have
similar magnitudes but opposite signs. Our theoretical predictions for the
polarized structure functions of nucleons and light nuclei, taking into
account the symmetry breaking of light sea quarks, exhibit better agreement
with experimental data.

##  I Introduction

Over the past years, the Deep Inelastic Scattering (DIS) of leptons from
nucleons has been a widely used experimental method for investigating the
internal structure of nucleons. Future lepton and hadron colliders, such as
the Large Hadron-electron Collider (LHeC) LHeC:2020van , and the Future
Circular Collider (FCC-he) FCC:2018byv  will also play an important role in
investigating the internal structure of nucleons Hobbs:2020vme ;
Amoroso:2022eow . Experimental groups such as E142, E143, COMPASS, HERMES, and
JLAB have played a significant role in advancing our knowledge of nucleon
structure by publishing a wealth of experimental results from collider
experiments E142:1996thl ; E143:1998hbs ; COMPASS:2015mhb ; HERMES:2004zsh ;
JeffersonLabHallA:2003joy ; JeffersonLabHallA:2016neg . These experiments have
focused on the DIS of polarized electrons by polarized nucleons and light
nuclei, providing valuable insights into the nature of nucleon spin and its
substructure.

The polarized structure functions of nucleons and nuclei provide useful
information about the spin distribution of partons Lin:2017snn ;
Ethier:2020way ; Geesaman:2018ixo ; Deur:2018roz ; Zhou:2022wzm . In the
simple picture of the 3He nuclei, all nucleons are in the $S$ state wherein
two protons with opposite spins exist, hence, their spins in the asymmetry are
completely canceled and the nuclei polarization is determined solely by the
neutron spin. Therefore, the use of 3He targets in DIS experiments of leptons
from polarized targets is common and is considered as an alternative target to
the neutron. The same method applies in the case of 3H, where the neutrons are
replaced with protons. However, in more precise calculations and by
considering other components of the three-particle wave, such as $S^{\prime}$
and $D$ states, the spins of the protons are no longer canceled in the 3He
structure function and must be taken into account.

In most published phenomenological models, the nucleon spin fractions carried
by sea quarks are often assumed to be equal, and symmetry breaking is not
taken into account., i.e. $\delta\bar{u}=\delta\bar{d}=\delta\bar{s}$
AtashbarTehrani:2007odq ; Khorramian:2010qa ; Taghavi-Shahri:2016idw ;
AtashbarTehrani:2013qea ; Khanpour:2017cha ; Salajegheh:2018hfs ;
Nematollahi:2021ynm ; Mirjalili:2022cal ; Sidorov:2006fi ; Leader:2014uua ;
deFlorian:2005mw ; Gluck:2000dy ; Blumlein:2002qeu ; deFlorian:2008mr ;
Hirai:2008aj ; Blumlein:2010rn ; Leader:2011tm ; Leader:2010dx ; Leader:2009tr
; Nocera:2014uea . In some other phenomenological models, both flavor SU(2)
and SU(3) symmetry breaking are taken into consideration, and hence, the
nucleon spin fraction carried by the light sea quarks considered to be unequal
as $\delta\bar{u}\neq\delta\bar{d}\neq\delta\bar{s}$ Khanpour:2017fey ;
Leader:2010rb ; deFlorian:2009vb ; deFlorian:2014yva ; Ball:2013lla ;
Arbabifar:2013tma ; Khorramian:2020gkr ; Lin:2017snn ; Geesaman:2018ixo .

In the present work, both of the mentioned phenomenological methods are
applied, and the polarized structure functions of nucleons and nuclei are
calculated using some selected polarized PDF models available in the
literature, namely NAAMY21 Nematollahi:2021ynm , AKS14 Arbabifar:2013tma ,
DSSV09 deFlorian:2009vb , and BB10 Blumlein:2010rn .

In the most recent analysis, NAAMY21, the polarized Deep Inelastic Scattering
(DIS) data is utilized, and the polarized PDFs of protons, neutrons, and
deuterons are calculated at the next-to-leading order (NLO) approximation,
without taking into account symmetry breaking. In other phenomenological
model, AKS14, the asymmetry data from inclusive and semi-inclusive polarized
Deep Inelastic Scattering (SIDIS) are utilized, and both flavor SU(2) and
SU(3) symmetry breaking are taken into consideration. The QCD-PEGASUS software
package Vogt:2004ns  is employed in both analyses for the DGLAP (Dokshitzer-
Gribov-Lipatov-Altarelli-Parisi) evolution, and a QCD fit is conducted using
the experimental polarized data. Both the NAAMY21 and AKS14 polarized PDFs are
presented at the next-to-leading order (NLO) approximation in perturbative
QCD.

In the present work, after calculating the moments of the polarized structure
functions using the two mentioned polarized PDFs in the Mellin space, the
DGLAP evolution equations Lampe:1998eu  are solved. Then, using the Jacobi
polynomials, the polarized structure functions of nucleons are calculated in
the Bjorken $x$ space. Finally, the polarized structure functions of the light
nuclei Helium-3 (3He) and Tritium (3H) are extracted in the second
approximation of the Feynman diagram. After applying the necessary corrections
to the nuclei structure functions, we compare our results with the available
experimental data for validation and comparison. Having the polarized
structure functions of nucleons and nuclei, along with their moments, one can
calculate various important quantities such as the Bjorken sum rule, Efremov-
Leader-Teryaev sum rule, and the Lorentz color force components.

The organization of this paper is as follows: In Section II, we present the
theoretical background of this study, including the polarized PDFs, the
polarized structure functions, and a comparison with the available
experimental data. In Section III, we discuss the polarized structure
functions of nuclei, the associated corrections, and compare the results with
experimental data. Sections IV and V present our results on the sum rules,
specifically the Bjorken sum rule and the Efremov-Leader-Teryaev sum rule. In
Section VI, we perform the calculation of the Lorentz color force components.
Finally, Section VII provides a summary and conclusion of the work.

##  II Polarized structure functions of nucleons

In this section, we will present the theoretical framework for the calculation
of the polarized structure functions of nucleons in the Mellin space.
Following that, we will review the Jacobi polynomial method employed to
transform the calculated structure functions from the Mellin space to the
Bjorken $x$ space. Additionally, we will provide a brief overview of previous
studies on polarized PDFs, specifically the NAAMY21 and AKS14 models, which
were utilized to compute the structure functions of nucleons and nuclei.

At the next-to-leading order (NLO) approximation, the polarized structure
function of nucleons in Mellin space, denoted as ${\cal{M}}[xg_{1},{\rm
N}](Q^{2})$, can be expressed in terms of the polarized PDFs and the
corresponding Wilson coefficient functions $\Delta C_{i}^{N}$. Therefore, the
polarized structure function is given by the following expression:

$\displaystyle{\cal{M}}[xg_{1}^{p},{\rm N}](Q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum\limits_{q=u,d,s}e_{q}^{2}\left\\{(1+\frac{\alpha_{s}}{2\pi}\Delta
C_{q}^{N})\right.$ (1) $\displaystyle\times$ $\displaystyle[\delta
q(N,Q^{2})+\delta\bar{q}(N,Q^{2})]$ $\displaystyle+$
$\displaystyle\left.\frac{\alpha_{s}}{2\pi}\>2\Delta C_{g}^{N}\delta
g(N,Q^{2})\right\\}\;,$

$\displaystyle{\cal{M}}[xg_{1}^{n},{\rm N}](Q^{2})$ $\displaystyle=$
$\displaystyle{\cal{M}}[xg_{1}^{p},{\rm
N}](Q^{2})-\frac{1}{6}(1+\frac{\alpha_{s}}{2\pi}\Delta C_{q}^{N})$ (2)
$\displaystyle\times$ $\displaystyle(\delta u(N,Q^{2})-\delta d(N,Q^{2}))\;,$

$\displaystyle{\cal{M}}[xg_{1}^{d},{\rm N}](Q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}({\cal{M}}[xg_{1}^{p},{\rm
N}]+{\cal{M}}[xg_{1}^{n},{\rm N}])$ (3) $\displaystyle\times$
$\displaystyle\left(1-\frac{3}{2}\omega_{D}\right)\;,$

for the proton ($p$), neutron ($n$) and deuteron ($d$), respectively.

In the equations above the $\omega_{D}=0.058$ Lacombe:1981eg ; Buck:1979ff ;
Zuilhof:1980ae  and the Wilson coefficient $\Delta C_{q}^{N}$ and $\Delta
C_{g}^{N}$ can be found in Lampe:1998eu , and written as follows,

$\displaystyle\Delta C_{q}^{N}$ $\displaystyle=$
$\displaystyle\frac{4}{3}\>\left\\{-S_{2}(N)+(S_{1}(N))^{2}+\left(\frac{3}{2}-\frac{1}{N(N+1)}\right)\right.$
(4) $\displaystyle\times$
$\displaystyle\left.S_{1}(N)+\frac{1}{N^{2}}+\frac{1}{2N}+\frac{1}{N+1}-\frac{9}{2}\right\\},$

$\displaystyle\Delta C_{g}^{N}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[-\frac{N-1}{N(N+1)}(S_{1}(N)+1)-\frac{1}{N^{2}}+\frac{2}{N(N+1)}\right],$

with $S{}_{1}(n)=\sum_{j=1}^{n}\frac{1}{j}=\psi(n+1)+\gamma_{E}$,
$S{}_{2}(n)=\sum_{j=1}^{n}\frac{1}{j^{2}}=(\frac{\pi^{2}}{6})-\psi^{\prime}(n+1)$,
$\psi(n)=\Gamma^{\prime}(n)/\Gamma(n)$, $\gamma_{E}=0.577216$ and
$\psi^{\prime}(n)=d^{2}\ln\Gamma(n)/dn^{2}$.

It is important to note that the structure function of neutrons, defined
through Eq. 2, takes into account symmetry breaking, meaning that the
contribution of sea quarks is considered. However, if the symmetry of light
sea quarks is neglected, the second term subtracted from the proton structure
function can be replaced by $\delta u_{v}(N,Q^{2})-\delta d_{v}(N,Q^{2})$.

After determining the moments of the structure functions, they can be
transformed to the Bjorken $x$ space using the Jacobi polynomials
Khorramian:2010qa . The transformation is given by the following expression:

$x\,g_{1}(x,Q^{2})=x^{\beta}(1-x)^{\alpha}\ \,\sum_{n=0}^{\rm N_{\rm
max}}a_{n}(Q^{2})\,\Theta_{n}^{\alpha,\beta}(x)\,,$ (6)

where the maximum order of expansion is denoted by $\rm N_{\rm max}$, and the
index $n$ refers to the order of expansion. Two free parameters $\alpha$ and
$\beta$ are chosen to make the fastest convergence for the series in Eq. 6.
The $a_{n}(Q^{2})$ as Jocbi moment discloses the dependence of polarized
structure functions on Q2. By using the Jacobi polynomials, one can factor out
from the structure function a weight function $w^{\alpha,\beta}(x)\equiv
x^{\beta}(1-x)^{\alpha}$. Hence, the Jacobi polynomials
$\Theta_{n}^{\alpha,\beta}(x)$ can be written as the following expansion:

$\Theta_{n}^{\alpha,\beta}(x)=\sum_{j=0}^{n}\,c_{j}^{(n)}(\alpha,\beta)\,x^{j}\,.$
(7)

It can be shown that the expansion coefficient $c_{j}^{(n)}(\alpha,\beta)$ is
given by a combination of Gamma $\Gamma$ function in terms of $n$, $\alpha$
and $\beta$ parameters Parisi:1978jv .

The following orthogonality relation is satisfied by the Jacobi polynomials
with the weight function $w^{\alpha,\beta}(x)$,

$\int_{0}^{1}dx\,x^{\beta}(1-x)^{\alpha}\,\Theta_{n}^{\alpha,\beta}(x)\,\Theta_{l}^{\alpha,\beta}(x)\,=\,\delta_{n,l}\,.$
(8)

Based on the orthogonality condition, the Jacobi moments $a_{n}(Q^{2})$ can be
obtained as Khorramian:2010qa ; Kataev:2001kk ; Kataev:2005ci :

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

where the ${\cal M}[xg_{1},\,j+2](Q^{2})$ is the Mellin transform of the
$xg_{1}(x,Q^{2})$ from Eqs. 1, 2 and 3, which is given by,

$\displaystyle{\cal
M}[xg_{1},\,N](Q^{2})=\int_{0}^{1}dx\,x^{N-2}\,xg_{1}(x,Q^{2})\;,$ (10)

Now by substituting Eq. 9 into Eq. 6 the polarized structure function
$xg_{1}(x,Q^{2})$, based on Jacobi polynomial expansion method, can be
constructed. Therefore the following expression for $xg_{1}(x,Q^{2})$ could be
obtained,

$\displaystyle xg_{1}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
x^{\beta}(1-x)^{\alpha}\,\sum_{n=0}^{\rm
N_{max}}\,\Theta_{n}^{\alpha,\beta}(x)$ (11) $\displaystyle\times$
$\displaystyle\sum_{j=0}^{n}\,c_{j}^{(n)}{(\alpha,\beta)}\,{\cal
M}[xg_{1},j+2](Q^{2})\,.$

where $N_{max}$ is considered to be 9 , and $\alpha$ and $\beta$ are fixed to
3 and 0.5, respectively Khorramian:2010qa ; Khanpour:2017cha ;
Salajegheh:2018hfs ; Khanpour:2017fey .

The polarized PDFs of the NAAMY21 and AKS14 are depicted in Figures 1 and 2,
respectively.

It is worth highlighting once again that the AKS14 model also conducted an
additional analysis that included a positive polarized gluon to explore the
impact of SU(2) and SU(3) symmetry breaking, as well as the effect of SIDIS
data on the polarized gluon distribution. In Figs. 3 and 4 we show the
polarized PDFs obtained from AKS14 and NAAMY21 in comparison with experimental
data from COMPASS COMPASS:2010hwr  and HERMES HERMES:2004zsh  experiments. The
results of two other available models, DSSV09 deFlorian:2009vb  and BB10
Blumlein:2010rn , are also shown for comparison. To calculate the $\delta q/q$
in Fig 4, the most recent unpolarized PDFs from CT18 Hou:2019efy  is used.

Considering the results presented in Figures 3 and 4, several remarks can be
made. As shown in these plots, in general, the compatibility of the AKS14
model with the experimental data and their error is better than those of
NAAMY21, more especially for the case of $\delta s$. However, it is important
to note that the compatibility of the data with the NAAMY21 model is also
good.

As can be seen, the DSSV09 results which come from the analysis with symmetry
breaking consideration are mostly close to the AKS14 curves and both are in
agreement with the experimental data. However, due to the wide dispersion of
the related data, the plots might not be able to support them as one may
expect. Also it is obvious that the results of BB10, with no symmetry
breaking, is closer to the NAAMY21. For the case of
$x\delta\bar{u}-x\delta\bar{d}$, both the AKS14 and DSSV09 models show the
same level of compatibility and are in agreement with the COMPASS data. For
the specific parton species, it is observed that both the AKS14 and DSSV09
models demonstrate a better agreement with the COMPASS data in terms of
$x\delta\bar{u}$ and $x\delta\bar{d}$ compared to other groups. Significant
deviations from the COMPASS data are observed for the NAAMY21 and BB10 for the
case of $x\delta\bar{u}$ and $x\delta s$.

Figure 1: The polarized PDFs from NAAMY21 Nematollahi:2021ynm  model at
$Q_{0}^{2}=1\,\textrm{GeV}^{2}$ in NLO approximation, considering
$\delta\bar{q}=\delta\bar{u}=\delta\bar{d}=\delta\bar{s}$.  Figure 2: The
polarized PDFs from AKS14 Arbabifar:2013tma  model at
$Q_{0}^{2}=1\,\textrm{GeV}^{2}$ at NLO approximation, considering
$\delta\bar{u}\neq\delta\bar{d}\neq\delta\bar{s}$. Solid line indicates the
AKS14 model in sign changing gluon scenario, dashed line denotes the AKS14
model in positive gluon scenario.  Figure 3: The NLO polarized PDFs from
AKS14, NAAMY21, DSSV09 deFlorian:2009vb  and BB10 Blumlein:2010rn  in
comparison with COMPASS COMPASS:2010hwr  experimental data at
$Q^{2}=3\,\textrm{GeV}^{2}$.  Figure 4: Same as Fig. 3 but this time in
comparison with HERMES experimental data HERMES:2004zsh  at
$Q^{2}=2.5\,\textrm{GeV}^{2}$.

With the moment expressions for the polarized structure functions of the
proton, neutron, and deuteron presented in Eqs.1, 2, and 3, respectively, and
utilizing the Jacobi polynomials, the structure functions can be calculated in
the Bjorken $x$ space. Figs. 5, 6, and 7 show the comparison of the structure
functions of proton, neutron, and deuteron with E143 experimental data
E143:1998hbs . We show both the absolute distributions (upper panel) and the
data/theory ratios (down panel) for detailed comparison. In order to correctly
assess weather or not the different theoretical results are statistically
different one from the other, and to examine the compatibility with the data,
the theory predictions for different polarized PDFs sets are also displayed
with error bands. The results from DSSV09 and BB10 are also shown and
compared.

In general, as observed from these plots, the nucleon polarized structure
functions generated by the AKS14 model, which takes into account symmetry
breaking, are quite close to the DSSV09 polarized structure functions, where
SU(2) and SU(3) symmetry breaking is also considered.

Focusing on the comparison with the proton polarized structure function shown
in Figure 5, one can observe a larger error band for the AKS14 model. The
theoretical prediction of the AKS14 model and the data are compatible over the
range of medium to small values of $x$. Although, differences are also seen at
large $x$, they are always compatible within uncertainties. As one can see
from the data/theory ratios (down panel), the compatibility of AKS14 theory
prediction and the E143 experimental data are better than those of NAAMY21 at
large $x$. The compatibility of the AKS14 and NAAMY21 theory predictions with
the data is similar for small values of $x$.

In Fig. 6, we present a comparison of the neutron polarized structure function
with data from the E143 Collaboration. Our analysis indicates that both the
AKS14 and NAAMY21 curves demonstrate a good level of agreement with the
central points of the experimental data. Specifically, for small to medium
values of $x$, the theoretical predictions of both the AKS14 and NAAMY21
models exhibit better accordance with the data, considering the uncertainties,
when compared to other models. However, it is important to note that
deviations from the experimental data can be observed for both models,
particularly at large values of $x$.

Finally, focusing on the deuteron polarized structure function in Fig. 7, it
can be observed that the NLO prediction of the structure function calculated
from the AKS14 model is in good agreement with the E143 experimental data
across the entire range of $x$. The DSSV09 polarized structure functions are
also seen to be close to the AKS14 curves, as both models account for SU(2)
and SU(3) symmetry breaking. Comparing with the NAAMY21 model, the data/theory
ratios (bottom panel) indicate that the compatibility of the AKS14 theory
prediction is better. This difference is particularly evident at very large
and small values of $x$, taking into account the uncertainties.

Figure 5: The NLO prediction for the proton polarized structure function
calculated using the AKS14, NAAMY21, DSSV09, BB10 in comparison with E143
experimental data E143:1998hbs  at $Q^{2}=2\,\textrm{GeV}^{2}$.  Figure 6: The
neutron polarized structure function at $Q^{2}=2\,\textrm{GeV}^{2}$, compared
with AKS14, NAAMY21, DSSV09, BB10 and E143 experimental data E143:1998hbs .
Figure 7: The deuteron polarized structure function at
$Q^{2}=2\,\textrm{GeV}^{2}$, compared with AKS14, NAAMY21, DSSV09, BB10 and
E143 experimental data E143:1998hbs .

In Table. 1, we present the value of the moments of nucleon polarized
structure functions at $3\leavevmode\nobreak\ {\textrm{G}eV}^{2}$ for AKS14,
NAAMY21 polarized PDFs. Comparing the values of $\Gamma_{1}^{p}=0.139\pm
0.003\pm 0.009\pm 0.005$, $\Gamma_{1}^{n}=-0.041\pm 0.006\pm 0.011\pm 0.005$,
and $\Gamma_{1}^{d}=0.049\pm 0.003\pm 0.004\pm 0.004$, reported by COMPASS
experimental group COMPASS:2015mhb  with the values reported in Table. 1, it
is observed that the values of moments of neutron and deuteron polarized
structure functions, when symmetry breaking is taken into account in the
AKS14, are more compatible with COMPASS results. For the case of proton, the
value extracted from NAAMY21 is in better agreement with the COMPASS results.

| $\Gamma_{1}^{p}$ | $\Gamma_{1}^{n}$ | $\Gamma_{1}^{d}$
---|---|---|---
AKS14 | $0.1273\pm 0.0003$ | $-0.0374\pm 0.00035$ | $0.0406\pm 0.00032$
NAAMY21 | $0.1322\pm 0.0053$ | $-0.0554\pm 0.0056$ | $0.0350\pm 0.0050$
Table 1: The first momentum of the nucleon polarized structure functions
calculated using AKS14 and NAAMY21 polarized PDFs.

##  III Polarized structure functions of nuclei

This section emphasizes the calculation of the polarized structure functions
of light nuclei, as well as the impact of the symmetry breaking of polarized
light sea quarks on the agreement between theory and experimental data. The
calculations in the previous section showed that the polarized PDFs, polarized
structure functions of proton, neutron, deuteron, and their first moments
obtained from the AKS14 polarized PDFs, in general, have better compatibility
with the experimental data from COMPASS COMPASS:2010hwr ; COMPASS:2015mhb ,
and E143 E143:1998hbs .

In the following section, we analyze the polarized structure functions of
light nuclei Helium-3 3He and Tritium 3H. These two are trivalent light nuclei
that, respectively, consist of two protons plus one neutron and two neutrons
plus one proton. Despite their wave function at ground state level is in the
$S$ state, the higher states $S^{\prime}$ and $D$ can be also found in a more
realistic definition. At $S^{\prime}$ and $D$ states, the spin contributions
of two protons in Helium-3 or the two neutrons in Tritium are not canceled and
must be considered in the calculations. The calculation of the polarized
structure function can be performed using the contribution of proton and
neutron polarized structure function in addition to the spin-dependent nucleon
light-cone momentum distributions $\Delta f^{p}_{{}^{3}{\mathrm{H}e}}$ and
$\Delta f^{n}_{{}^{3}{\mathrm{H}e}}$ Yazdanpanah:2009zz ;
CiofidegliAtti:1993zs ; Bissey:2000ed ; Bissey:2001cw ; Afnan:2003vh . They
are given by,

$\displaystyle g_{1}^{{}^{3}{\mathrm{H}e}}=$
$\displaystyle\int_{x}^{3}\frac{dy}{y}\Delta
f^{n}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{n}(x/y)$
$\displaystyle+2\int_{x}^{3}\frac{dy}{y}\Delta
f^{p}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{p}(x/y)\,,$

$\displaystyle g_{1}^{{}^{3}{\rm H}}=$
$\displaystyle\int_{x}^{3}\frac{dy}{y}\Delta
f^{n}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{p}(x/y)$
$\displaystyle+2\int_{x}^{3}\frac{dy}{y}\Delta
f^{p}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{n}(x/y)\,.$

It is noticeable that because of isospin symmetry, the light cone momentum
distribution $\Delta f^{p}_{{}^{3}{\mathrm{H}e}}$ and $\Delta
f^{n}_{{}^{3}{\mathrm{H}}}$ are equal in size. In Fig. 8 we show the spin-
dependent nucleon light-cone momentum distributions $\Delta
f^{n}_{{}^{3}{\mathrm{H}e}}$ and $\Delta f^{p}_{{}^{3}{\mathrm{H}e}}$
Yazdanpanah:2009zz  as a function of $y$. For the calculation of $\Delta
f^{n}_{{}^{3}{\mathrm{H}e}}$ and $\Delta f^{p}_{{}^{3}{\mathrm{H}e}}$, we
utilized the PEST nucleon-nucleon (N-N) interaction which is derived from the
Paris N-N potential Lacombe:1980dr .

Figure 8: The Spin-dependent nucleon light-cone momentum distributions $\Delta
f^{p}_{{}^{3}{\mathrm{H}e}}$ and $\Delta f^{n}_{{}^{3}{\mathrm{H}e}}$
Yazdanpanah:2009zz  as a function of $y$.

In addition to the correction discussed above, various calculations show that
the main contribution to the $g_{1}$ should originate from the
$n\rightarrow\Delta^{\circ}$ non-diagonal transition in 3He and from the
$p\rightarrow\Delta^{+}$ non-diagonal transition in 3H case Frankfurt:1996nf ;
Saito:1990aj ; Carlson:1991ju ; Boros:2000af .

$\displaystyle g_{1}^{{}^{3}{\rm He}}(x,Q^{2})=\int_{x}^{3}\frac{dy}{y}\Delta
f^{n}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{n}(x/y,Q^{2})+$ $\displaystyle
2\int_{x}^{3}\frac{dy}{y}\Delta
f^{p}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{p}(x/y,Q^{2})$ $\displaystyle+$
$\displaystyle
4P_{p\rightarrow\Delta^{+}}g_{1}^{p\rightarrow\Delta^{+}}(x,Q^{2})+2P_{n\rightarrow\Delta^{\circ}}g_{1}^{n\rightarrow\Delta^{\circ}}(x,Q^{2})\,,$

$\displaystyle g_{1}^{{}^{3}{\rm H}}(x,Q^{2})=\int_{x}^{3}\frac{dy}{y}\Delta
f^{n}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{p}(x/y,Q^{2})+$ $\displaystyle
2\int_{x}^{3}\frac{dy}{y}\Delta
f^{p}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{n}(x/y,Q^{2})$ $\displaystyle-$
$\displaystyle
2P_{n\rightarrow\Delta^{\circ}}g_{1}^{p\rightarrow\Delta^{+}}(x,Q^{2})-4P_{p\rightarrow\Delta^{+}}g_{1}^{n\rightarrow\Delta^{\circ}}(x,Q^{2})\,,$

As a result, the structure functions $g_{1}^{n\rightarrow\Delta^{\circ}}$ and
$g_{1}^{p\rightarrow\Delta^{+}}$ should be considered in the calculations,
which will lead to a correction for the three particle nuclei structure
function. Regarding to the model-independent equation Bissey:2001cw , we have

$g_{1}^{p\rightarrow\Delta^{+}}(x,Q^{2})=g_{1}^{n\rightarrow\Delta^{\circ}}(x,Q^{2})=\frac{2\sqrt{2}}{5}(g_{1}^{p}-4g_{1}^{n}),$
(16)

and the polarized structure function equation will be written as,

$\displaystyle g_{1}^{{}^{3}{\mathrm{H}e}}=\int_{x}^{3}\frac{dy}{y}\Delta
f^{n}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{n}(x/y)+2\int_{x}^{3}\frac{dy}{y}\Delta
f^{p}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{p}(x/y)$
$\displaystyle-0.014\Big{(}g_{1}^{p}(x)-4g_{1}^{n}(x)\Big{)}\,,$ (17)

and

$\displaystyle g_{1}^{{}^{3}{\rm H}}=\int_{x}^{3}\frac{dy}{y}\Delta
f^{n}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{p}(x/y)+2\int_{x}^{3}\frac{dy}{y}\Delta
f^{p}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{n}(x/y)$
$\displaystyle+0.014\Big{(}g_{1}^{p}(x)-4g_{1}^{n}(x)\Big{)}\,.$ (18)

At high energy levels or at low values of Bjorken $x$, the virtual photon can
interact with multiple nucleons coherently. This behavior is apparent in
nuclear targets. Nuclear shadowing and antishadowing are examples of these
coherent effects Bissey:2001cw ; Ethier:2013hna . According to the shadowing
and antishadowing correction, the Helium-3 and Tritium nuclei polarized
structure functions can be rewritten as,

$\displaystyle g_{1}^{{}^{3}{\mathrm{H}e}}=\int_{x}^{3}\frac{dy}{y}\Delta
f^{n}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{n}(x/y)+2\int_{x}^{3}\frac{dy}{y}\Delta
f^{p}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{p}(x/y)$
$\displaystyle-0.014\Big{(}g_{1}^{p}(x)-4g_{1}^{n}(x)\Big{)}+a(x)g_{1}^{n}(x)+b(x)g_{1}^{p}(x)\
,$

and

$\displaystyle g_{1}^{{}^{3}{\mathrm{H}}}=\int_{x}^{3}\frac{dy}{y}\Delta
f^{n}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{p}(x/y)+2\int_{x}^{3}\frac{dy}{y}\Delta
f^{p}_{{}^{3}{\mathrm{H}e}}(y)g_{1}^{n}(x/y)$
$\displaystyle+0.014\Big{(}g_{1}^{p}(x)-4g_{1}^{n}(x)\Big{)}+a(x)g_{1}^{n}(x)+b(x)g_{1}^{p}(x)\,,$

where $a(x)$ and $b(x)$ functions describe the shadowing and anti-shadowing
effect as a function of $x$ and $Q^{2}$ Bissey:2001cw . Since the current
experimental data do not cover very small values of $x$ considerably, the
corrections from shadowing $(10^{-4}\leq x\leq 0.03-0.07)$ and $(0.03-0.07\leq
x\leq 0.2)$ anti-shadowing can be completely ignored in the calculations of
polarized nuclei JeffersonLabHallA:2016neg . However, the calculations of
Refs. Bissey:2001cw ; Ethier:2013hna  have shown that these effects are quite
significant and impact the extraction of the nucleon polarized structure
functions at small values of Bjorken $x$.

According to Eqs. III and III, the Helium-3 and Tritium polarized structure
functions are extracted at $5\,\textrm{GeV}^{2}$ and $2.5\,\textrm{GeV}^{2}$
respectively and they are shown in Figs. 9 and 10.

Polarized structure functions of 3He and 3H at NLO and NNLO from NAAMY21 and
KTA17 Khanpour:2017fey  polarized PDFs, both concerning symmetry of light sea
quarks, are compared with AKS14 Arbabifar:2013tma  model at NLO concerning
symmetry breaking of light sea quarks.

In Fig.9, we also include the E142E142:1996thl  and JLAB
JeffersonLabHallA:2004tea  experimental data for comparison. As shown in this
figure, the theory prediction from the AKS14 model, even though it is similar
to that of NAAMY21, exhibits slightly better compatibility with the
experimental results across most of the Bjorken $x$ region covered by the
data. This compatibility is particularly emphasized for larger values of $x$
($x>0.1$) within the uncertainty bands.

In Fig. 10 and for the case of $g_{1}^{{}^{3}\rm{H}}$ polarized structure
functions, one can see that the difference between the curves with and without
considering symmetry breaking is noticeable, especially at low value of $x$.

Fig. 11 shows the comparison of $x^{2}g_{1}^{{}^{3}\rm{He}}$ polarized
structure function in comparison with E142 E142:1996thl , JLAB04
JeffersonLabHallA:2004tea , JLAB03 jlab:WilliamandMary , and JLAB16
JeffersonLabHallA:2016neg  experimental data at different energies in the
range of $1.1<\rm{Q}^{2}<5.89\,\rm{GeV}^{2}$. Based on this plot, it can be
concluded that the AKS14 model exhibits comparable agreement to the other
models within the region of $0.1\leq x\leq 0.4$. For higher values of $x$, the
NAAMY21 and KTA17 models show better compatibility with the experimental data
points. However, it should be noted that both the AKS14 and NAAMY21 models
remain compatible within the given uncertainties throughout the entire range.

Figure 9: The theory prediction for $g_{1}^{{}^{3}\rm{He}}$ polarized
structure functions at NLO and NNLO approximation from NAAMY21, AKS14 and
KTA17 in comparison with experimental data from E142 E142:1996thl  and JLAB
JeffersonLabHallA:2004tea .  Figure 10: Same as Fig. 9 but this time for the
$g_{1}^{{}^{3}\rm{H}}$ polarized structure functions. Figure 11: Same as Fig.
9 but this time for the $x^{2}g_{1}^{{}^{3}\rm{He}}$ polarized structure
functions in comparison with experimental data from E142 E142:1996thl , JLAB04
JeffersonLabHallA:2004tea , JLAB03 jlab:WilliamandMary  and JLAB16
JeffersonLabHallA:2016neg .  Figure 12: Same as Fig. 9 but this time for the
$g_{2}^{{}^{3}\rm{He}}$ polarized structure functions in comparison with E142
experimental data E142:1996thl .  Figure 13: Same as Fig. 9 but this time for
the $g_{2}^{{}^{3}He}$ polarized structure functions in comparison with the
JLAB experimental data JeffersonLabHallA:2004tea .

The Wandzura-Wilczek $g_{2}$ structure function Wandzura:1977qf  can be
calculated as $g_{2}^{WW}(x)=-g_{1}(x)-\int_{x}^{1}$ $\frac{dy}{y}g_{1}(y)$,
and for the case of Helium-3 and Tritium structure functions, they are given
by:

$\displaystyle g_{2}^{{}^{3}{\rm He}}=\int_{x}^{3}\frac{dy}{y}\Delta
f^{n}_{{}^{3}{\rm He}}(y)g_{2}^{n}(x/y)$
$\displaystyle+2\int_{x}^{3}\frac{dy}{y}\Delta f^{p}_{{}^{3}{\rm
He}}(y)g_{2}^{p}(x/y)$
$\displaystyle-0.014\Big{(}g_{2}^{p}(x)-4g_{2}^{n}(x)\Big{)}\,,$ (21)

and

$\displaystyle g_{2}^{{}^{3}{\rm H}}=\int_{x}^{3}\frac{dy}{y}\Delta
f^{n}_{{}^{3}{\rm He}}(y)g_{2}^{p}(x/y)$
$\displaystyle+2\int_{x}^{3}\frac{dy}{y}\Delta f^{p}_{{}^{3}{\rm
He}}(y)g_{2}^{n}(x/y)$
$\displaystyle+0.014\Big{(}g_{2}^{p}(x)-4g_{2}^{n}(x)\Big{)}\,.$ (22)

Figs. 12 and 13 display the polarized structure of $g_{2}^{{}^{3}He}$
extracted from AKS14 and NAAMY21 at NLO, and KTA17 Khanpour:2017fey  at the
NNLO approximation compared to the JLAB04 JeffersonLabHallA:2003joy  and
JLAB16 JeffersonLabHallA:2016neg  experimental data. Although the curves are
not exactly the same, a similar agreement in the behavior of three models in
comparison with the experimental data of JLAB04 and JLAB16 and their error bar
is observed.

##  IV Bjorken sum rule

To quantify the nuclear corrections as expressed in Eqs. III and III we use
the ratio known as $\eta$. Bjorken sum rule evaluates the difference between
the first moment between Helium and Tritium polarized structure functions. It
is given by,

$\int_{0}^{3}[g_{1}^{{}^{3}{\mathrm{H}}}(x,Q^{2})-g_{1}^{{}^{3}\mathrm{He}}(x,Q^{2})]dx=\frac{1}{6}g_{A}|_{triton}[1+{\cal
O}(\frac{\alpha_{s}}{\pi})]\,,$ (23)

where $g_{A}|_{triton}=1.211\pm 0.002$ Budick:1991zb . According to the
Bjorken sum rule Bjorken:1966jh  for proton and neutron without nuclei
constraint we have

$\int_{o}^{1}[g_{1}^{p}(x,Q^{2})-g_{1}^{n}(x,Q^{2})]dx=\frac{1}{6}g_{A}[1+{\cal
O}(\frac{\alpha_{s}}{\pi})]\,,$ (24)

where $g_{A}=1.2670\pm 0.0035$ ParticleDataGroup:2016lqr . Finally the ratio
$\eta$ can be written as,

$\displaystyle\eta$ $\displaystyle\equiv$
$\displaystyle\frac{g_{A}|_{triton}}{g_{A}}=$
$\displaystyle\frac{\int_{0}^{3}[g_{1}^{{}^{3}{\mathrm{H}}}(x,Q^{2})-g_{1}^{{}^{3}\mathrm{He}}(x,Q^{2})]dx}{\int_{0}^{1}[g_{1}^{p}(x,Q^{2})-g_{1}^{n}(x,Q^{2})]dx}=0.956\pm
0.004\,.$

In the calculations, we obtained a value of $0.936\pm 0.002$ for NAAMY21 model
and $0.954\pm 0.003$ for AKS14. This result indicates that the analysis taking
into account symmetry breaking produces a value of the ratio $\eta$ that is
closer to the Bjorken sum rule compared to the analysis that neglects symmetry
breaking.

##  V Efremov-Leader-Teryaev (ELT) Sum Rule

The Efremov-Leader-Teryaev (ELT) sum rule can be derived by integrating the
valence part of the $g_{1}$ and $g_{2}$ structure functions over the Bjorken
$x$ variable in the limit $m_{q}\rightarrow 0$ blum:1997 . The ELT sum rule is
expressed as follows:

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

where $g_{1(2)}^{V}$ denotes the valence quark contributions to the
$g_{1(2)}$. When considering the symmetry of light sea quarks and assuming
they carry an equal fraction of the spin in protons and neutrons, the Efremov-
Leader-Teryaev (ELT) sum rule can be written as follows:

$\displaystyle\int_{0}^{1}x[g_{1}^{p}(x)-g_{1}^{n}+2(g_{2}^{p}(x)-g_{2}^{n}(x)]dx=0\leavevmode\nobreak\
.$ (27)

The values obtained $0.01017\pm 0.00004$ and $-0.030763\pm 0.0004071$ from
NAAMY21 and AKS14 respectively. It seems the consideration of the light sea
quarks symmetry breaking, the ELT sum rule is derived directly from Eq. 26.
The value of the left hand side of above equation is obtained $-0.011\pm
0.008$ from E155 E155:2003  analysis at Q${}^{2}=5\leavevmode\nobreak\
\rm{GeV}^{2}$. The symmetry breaking makes the results negative, however are
of a similar magnitude. Also it is concluded that disregarding the symmetry of
$\delta\bar{u}$, $\delta\bar{d}$ and $\delta\bar{s}$ can be effective on the
magnitude of ELT sum rule to be far from zero.

##  VI Lorentz color force components

The force exerted by the probing quark on the struck quark in the DIS process,
perpendicular to the direction of motion, is referred to as the Lorentz color
force. It is divided into electric and magnetic components Burkardt:2008ps

$F_{E}^{y,n}=-\frac{M_{n}^{2}}{6}(2d_{2}^{n}+f_{2}^{n})\,,$ (28)

$F_{B}^{y,n}=-\frac{M_{n}^{2}}{6}(4d_{2}^{n}-f_{2}^{n})\,.$ (29)

Here $d_{2}^{n}$ is twist-3 matrix element and can be evaluated via

$d_{2}(Q^{2})=\int_{o}^{1}x^{2}[2g_{1}(x,Q^{2})+3g_{2}(x,Q^{2})]dx\,,$ (30)

and $f_{2}^{n}$ is twist-4 matrix element and can be extracted from the first
moment equation of $g_{1}(x,Q^{2})$ Kataev:1994gd

$\displaystyle\eta_{1}$ $\displaystyle\equiv$
$\displaystyle\int_{o}^{1}g_{1}(x,Q^{2})dx$ (31) $\displaystyle=$
$\displaystyle\mu_{2}+\frac{M^{2}}{9Q^{2}}(a_{2}+4d_{2}+4f_{2})+{\cal
O}\left(\frac{1}{Q^{4}}\right)\,.$ (32)

The quantity $\mu_{2}$ is known as twist-2 contribution

$\displaystyle\mu_{2}(Q^{2})=C_{ns}(Q^{2})\left(-\frac{1}{12}g_{A}+\frac{1}{36}a_{8}\right)+C_{s}(Q^{2})\Delta\Sigma,$

in which $C_{ns}$ and $C_{s}$ are the nonsinglet and singlet Wilson
coefficients Larin:1997qq . $g_{A}$ and $a_{8}$ are respectively the flavor-
triplet and the octet axial charge and $\Delta\Sigma$ denotes the singlet
axial current Osipenko:2004xg ; Meziani:2004ne . The quantity $a_{2}$ is the
third moment of $g_{1}$ known as $a_{2}=\int x^{2}g_{1}dx$.

In Table. 2, we present the calculated values for the $d_{2}^{n}$, $f_{2}^{n}$
and $a_{2}^{n}$ using two models of AKS14 and NAAMY21. The corresponding value
from JLAB experiment JeffersonLabHallA:2016neg  also has been shown as well.

Table. 3 presents the calculated magnetic and electric Lorentz color force
components calculated from AKS14 and NAAMY21 models. The measured values from
JLAB for Lorentz color force components JeffersonLabHallA:2016neg  also
presented in Table. 3 for comparison.

As one can see, the magnitudes of $F_{B}^{y,n}$ and $F_{E}^{y,n}$ extracted
from the AKS14 QCD analysis, which takes into account the symmetry breaking of
light sea quarks, exhibit good agreement with the measured values from the
JLAB experiment. In addition, the calculated Lorentz color forces denoted by
$F_{B}^{y,n}$ and $F_{E}^{y,n}$ are in same size with a different signs as
expected from QCD prediction Burkardt:2008ps . The magnitudes of these two
forces, extracted from NAAMY21 QCD analysis with no symmetry breaking, are one
order of magnitude smaller than JLAB measured values and the AKS14 prediction
as well. We should mention here that the agreement in the order of magnitude
between the experimental data and the AKS14 model arises from a combination of
ingredients in which the experimental data and the AKS14 model differ
significantly.

| $<Q^{2}>\left[\textrm{GeV}^{2}\right]$ |  JLAB16 JeffersonLabHallA:2016neg  |  NAAMY21 Nematollahi:2021ynm  |  AKS14 Arbabifar:2013tma
---|---|---|---|---
$d_{2}^{n}\left[\times 10^{-5}\right]$ | 3.21 | $-421.0\pm 79.0\pm 82.0\pm 8.0$ | $-125.566\pm 0.1712$ | $-89.488\pm 2.333$
$d_{2}^{n}\left[\times 10^{-5}\right]$ | 4.32 | $-35.0\pm 83.0\pm 69.0\pm 7.0$ | $-124.783\pm 4.809$ | $-86.225\pm 1.806$
$f_{2}^{n}\left[\times 10^{-3}\right]$ | 3.21 | $53.41\pm 0.79\pm 25.55$ | $9.972\pm 2.166$ | $148.330\pm 25.851$
$f_{2}^{n}\left[\times 10^{-3}\right]$ | 4.32 | $49.66\pm 0.83\pm 25.99$ | $30.732\pm 28.246$ | $219.473\pm 35.044$
$a_{2}^{n}\left[\times 10^{-4}\right]$ | 3.21 | $8.552\pm 1.761\pm 6.125$ | $1.174\pm 0.01$ | $3.953\pm 1.811$
$a_{2}^{n}\left[\times 10^{-4}\right]$ | 4.32 | $5.044\pm 2.270\pm 6.042$ | $1.632\pm 0.0003$ | $3.754\pm 1.712$
Table 2: The values of $d^{n}_{2}$, $a_{2}^{n}$ and $f_{2}^{n}$ from JLAB JeffersonLabHallA:2016neg  experimental data with statistical and systematic uncertainties. The values calculated by AKS14 and NAAMY21 polarized PDFs are also shown as well. | $<Q^{2}>\left[\textrm{GeV}^{2}\right]$ | $F_{E}^{y,n}\left[\textrm{MeV/fm}\right]$ | $F_{B}^{y,n}\left[\textrm{MeV/fm}\right]$
---|---|---|---
JLAB16 JeffersonLabHallA:2016neg  | $3.21$ | $-33.53\pm 1.32\pm 19.07$ | $52.35\pm 2.43\pm 19.18$
JLAB16 JeffersonLabHallA:2016neg  | $4.32$ | $-36.48\pm 1.38\pm 19.38$ | $38.04\pm 2.55\pm 19.46$
NAAMY21 | $3.21$ | $-3.992\pm 3.264$ | $2.207\pm 6.049$
NAAMY21 | $4.32$ | $-4.154\pm 6.332$ | $5.256\pm 6.290$
AKS14 | $3.21$ | $-21.560\pm 3.796$ | $22.350\pm 3.817$
AKS14 | $4.32$ | $-32.037\pm 5.150$ | $32.798\pm 5.166$
Table 3: Measured magnetic and electric Lorentz color force components from
JLAB JeffersonLabHallA:2016neg  experiment along with statistical and
systematic uncertainty. The calculated quantities using AKS14 and NAAMY21
polarized PDFs with statistical uncertainty are also shown for comparison.

##  VII Conclusions

The following conclusions can be drawn from the present study. In this
article, we have conducted an investigation into the impact of symmetry
breaking in polarized light sea quarks on the extraction of polarized PDFs. We
analyze their effects on the polarized structure functions and their moments
for both nucleons and nuclei, providing a detailed discussion. Additionally,
we explore and present the implications of symmetry breaking on the sum rules
and the components of the Lorentz color force. Furthermore, it is important to
note that in our analysis, the polarized structure functions of nuclei are
computed while considering the effective nuclear modifications. This ensures a
more accurate description of the nuclear effects on the polarized structure
functions.

To investigate the impact of symmetry breaking in polarized light sea quarks,
we compare the polarized PDFs of the AKS14, NAAMY21, DSSV09, and BB10 models
with experimental data obtained from HERMES and COMPASS experiments. This
comparison allows us to assess the agreement between the theoretical models
considering symmetry breaking and the experimental data. The primary outcome
of this comparison indicates slightly better agreement between the AKS14
model, which incorporates symmetry breaking, and the experimental data. This
agreement is particularly notable for distributions involving strange and sea
quarks. It suggests that considering symmetry breaking in the modeling of
polarized light sea quarks leads to improved agreement with experimental
observations.

In addition, we conduct a comparison of the polarized structure functions of
the proton, neutron, and deuteron using calculations from the AKS14, NAAMY21,
DSSV09, and BB10 models. This comparison is made with experimental data
obtained from the E143 collaboration. The results obtained from the
comparisons between data and theory indicate that when symmetry breaking is
considered, the extracted predictions exhibit improved agreement with the
experimental data, taking into account the associated uncertainties.

To investigate the impact of symmetry breaking on the nuclei structure
function, we calculate the theoretical predictions of the polarized structure
function $g_{1}^{{}^{3}\textrm{He}}$ using the NAAMY21, AKS14, and KTA17
models at both NLO and NNLO accuracy in perturbative QCD.

These theoretical predictions are subsequently compared with experimental data
from the E142 and JLAB experiments. The findings of this comparison indicate
that the polarized nuclei structure function calculated using the AKS14
polarized PDFs, which consider the symmetry breaking of light sea quarks,
exhibits better agreement with the experimental data compared to other models.
This agreement is particularly notable in the $0.1\leq x\leq 0.4$ regions.

The same findings also apply to the cases of $g_{2}^{{}^{3}\textrm{He}}$ and
$x^{2}g_{1}^{{}^{3}\textrm{He}}$ structure functions. While the extracted
polarized structure functions of nuclei from the AKS14 model demonstrate
agreement with the experimental data, we should clarify that there are
currently no available experimental data for $g_{1}^{{}^{3}\textrm{H}}$.
Furthermore, considering the uncertainties associated with the measurements of
$g_{2}^{{}^{3}\textrm{He}}$ and $x^{2}g_{1}^{{}^{3}\textrm{He}}$, it does not
allow one to strongly prefer the AKS14 model over other models based solely on
these measurements.

Our calculations demonstrate that the ratio $\eta$ of the Bjorken sum rule and
the extracted Lorentz color force components $F_{B}^{y,n}$ and $F_{E}^{y,n}$
from the AKS14 model are all in good agreement with the measured values as
well.

In conclusion, the presented results provide evidence that a comprehensive
understanding of the spin structure of nucleons, nuclei, sum rules, and
Lorentz color force components of polarized structure functions can often be
achieved when considering the breaking of both flavors SU(2) and SU(3)
symmetry in a QCD analysis. By incorporating these symmetry-breaking effects,
a more accurate description of the observed data and a better agreement
between theory and experiment can be achieved.

## Acknowledgments

The authors acknowledge the financial support from the Iran National Science
Foundation (INSF) under grant number 4013570. S. A. T. and H. K. are also
thankful to the School of Particles and Accelerators, Institute for Research
in Fundamental Sciences (IPM). F. A. acknowledges the Farhangian University
for providing support to conduct this research. The authors are grateful to
Abolfazl Mirjalili for several useful comments. They would also like to
express their gratitude to the anonymous referee for their valuable comments
and suggestions, which have greatly contributed to enhancing the quality of
this paper.

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