# Evaluation of the Gottfried sum with use of the truncated moments method

A. Kotlorz<EMAIL_ADDRESS>Opole University of Technology, 45-758 Opole,
Proszkowska 76, Poland  D. Kotlorz<EMAIL_ADDRESS>Opole University of
Technology, 45-758 Opole, Proszkowska 76, Poland  Bogoliubov Laboratory of
Theoretical Physics, JINR, 141980 Dubna, Russia  O. V. Teryaev
<EMAIL_ADDRESS>Bogoliubov Laboratory of Theoretical Physics, JINR,
141980 Dubna, Russia

###### Abstract

We reanalyze the experimental NMC data on the nonsinglet structure function
$F_{2}^{p}-F_{2}^{n}$ and E866 data on the nucleon sea asymmetry
$\bar{d}/\bar{u}$ using the truncated moments approach elaborated in our
previous papers. With help of the special truncated sum one can overcome the
problem of the unavoidable experimental restrictions on the Bjorken $x$ and
effectively study the fundamental sum rules for the parton distributions and
structure functions. Using only the data from the measured region of $x$, we
obtain the Gottfried sum $\int_{0}^{1}F_{2}^{ns}/x\,dx$ and the integrated
nucleon sea asymmetry $\int_{0}^{1}(\bar{d}-\bar{u})\,dx$. We compare our
results with the reported experimental values and with the predictions
obtained for different global parametrizations for the parton distributions.
We also discuss the discrepancy between the NMC and E866 results on
$\int_{0}^{1}(\bar{d}-\bar{u})\,dx$. We demonstrate that this discrepancy can
be resolved by taking into account the higher-twist effects.

###### pacs:

11.55.Hx, 12.38.-t, 12.38.Bx

##  I Introduction

The deep inelastic scattering (DIS) of leptons on hadrons and hadron-hadron
collisions are a gold mine to study the hadron structure and fundamental
particle interactions at high energies. Especially, so-called DIS sum rules
can provide important information on partonic structure of the nucleon and a
good test for the quantum chromodynamics (QCD). Nowadays, there are known a
number of polarized and unpolarized sum rules for structure functions. Some of
them are rigorous theoretical predictions and other are based on model
assumptions which can be verified experimentally. An example of the latter is
the Gottfried sum rule (GSR) Gottfried (1967). Thus, the GSR violation in a
series of experiments Amaudruz et al. (1991); Arneodo et al. (1994); Baldit et
al. (1994); Hawker et al. (1998); Peng et al. (1998); Towell et al. (2001);
Ackerstaff et al. (1998) revealed that, unlikely to the assumed simple
partonic model of the nucleon with the symmetric light sea, the light sea of
the proton was flavor asymmetric, i.e., $\bar{u}(x)\neq\bar{d}(x)$. This
unexpected result has prompted a large interest for many further studies, for
review, see, e.g., Kumano (1998); Garvey and Peng (2001), related to
theoretical explanations of the flavor asymmetry of the nucleon sea.

In our paper, we present a phenomenological analysis of the experimental NMC
data on the nonsinglet structure function $F_{2}^{p}-F_{2}^{n}$ Arneodo et al.
(1994) and E866 data on the nucleon sea asymmetry $\bar{d}/\bar{u}$ Towell et
al. (2001), utilizing a very effective method for determination of the DIS sum
rules in a restricted region of Bjorken $x$ – the so-called truncated Mellin
moments (TMM) approach Kotlorz et al. (2017).

In the next section, we give a brief recapitulation of the Gottfried sum rule
violation problem and discuss some effects modifying the GSR like the
perturbative QCD corrections, higher-twist terms, small-$x$ behavior and
nuclear shadowing. The method of the evaluation of the DIS rules from the
experimental data with help of the truncated Mellin moments approach is
shortly summarized in Section III. In Section IV, we present our numerical
results on the GSR value and compare them to those provided by the NMC and
E866, and also to other determinations based on the global parton distribution
functions (PDFs) fits. Furthermore, we discuss the higher-twist effects as a
possible explanation of the discrepancy between the NMC and E866 results on
the integrated nucleon sea asymmetry $\int_{0}^{1}(\bar{d}-\bar{u})\,dx$.
Finally, we discuss shortly our prediction for the iso-vector quark momentum
fraction $\langle x\rangle_{u-d}$.
In Section V, we give conclusions for this study.

##  II Violation of the Gottfried sum rule

The Gottfried sum rule Gottfried (1967) states that the integral over Bjorken
variable $0<x<1$ of a difference of electron-proton and electron-neutron
structure functions is a constant ($=1/3$) under flavor symmetry in the
nucleon sea ($\bar{u}(x)=\bar{d}(x)$), which is independent of the transferred
four-momentum $q$ Gottfried (1967):

$S_{G}(Q^{2})=\int^{1}_{0}\left[F_{2}^{p}(x,Q^{2})-F_{2}^{n}(x,Q^{2})\right]{dx\over
x}=\frac{1}{3}\,.$ (1)

Here, $x=Q^{2}/(2Pq)$, where $Q^{2}=-q^{2}$, $P^{2}=m^{2}$, and $m$ is the
nucleon mass. This form of the GSR originates from a simple partonic model of
the nucleon structure functions in which the isospin symmetry of the nucleon
(the u-quark distribution in the proton is equal to the d-quark distribution
in the neutron),

$u^{p}_{v}(x)=d^{n}_{v}(x)\equiv u_{v}(x)\,,$ (2)

and, similarly, $d^{p}=u^{n}$ , $\bar{u}^{p}=\bar{d}^{n}$,
$\bar{d}^{p}=\bar{u}^{n}$, etc., and the flavor symmetry of the light sea in
the nucleon,

$\bar{u}(x)=\bar{d}(x)\,,$ (3)

are assumed. Then, the difference between the proton and neutron structure
functions incorporating implicit perturbative QCD $Q^{2}$ corrections to the
parton model is given by

$F_{2}^{p}(x,Q^{2})-F_{2}^{n}(x,Q^{2})=\frac{1}{3}\,x\left[u_{v}(x,Q^{2})-d_{v}(x,Q^{2})\right]+\frac{2}{3}\,x\left[\bar{u}(x,Q^{2})-\bar{d}(x,Q^{2})\right],$
(4)

where the valence-quark distribution $q_{v}$, ($q=u,d$), is defined by
$q_{v}\equiv q-\bar{q}$, with $\bar{q}$ being the sea-quark distribution.
Taking into account the charge conservation law for the nucleon,

$\int^{1}_{0}u_{v}(x,Q^{2})\,dx=2\,,\quad\quad\int^{1}_{0}d_{v}(x,Q^{2})\,dx=1\,,$
(5)

we obtain

$\int^{1}_{0}\left[F_{2}^{p}(x,Q^{2})-F_{2}^{n}(x,Q^{2})\right]{dx\over
x}=\frac{1}{3}+\frac{2}{3}\int^{1}_{0}\left[\bar{u}(x,Q^{2})-\bar{d}(x,Q^{2})\right]dx.$
(6)

If the light sea is flavor symmetric, Eq. (3), the second term in Eq. (6)
vanishes giving the Gottfried sum rule (1).

Though the isospin symmetry, Eq. (2), is not exact, and can also contribute to
the GSR violation, usually the experimental results on the GSR breaking are
interpreted as an evidence of the light flavor asymmetry of the nucleon sea,

$\bar{u}(x)\neq\bar{d}(x)\,.$ (7)

The first clear indication of the GSR violation in DIS experiment was provided
by the New Muon Collaboration (NMC) Amaudruz et al. (1991) and from the
reanalyzed NMC data Arneodo et al. (1994). The obtained NMC measurement of
$S_{G}$,

${\rm NMC~{}~{}1994:}\quad\quad S_{G}(Q^{2}=4\,{\rm GeV}^{2})=0.235\pm 0.026$
(8)

implies the integrated antiquark flavor asymmetry, Eq. (7),

$\int^{1}_{0}\left[\bar{d}(x,Q^{2})-\bar{u}(x,Q^{2})\right]dx=0.148\pm 0.039$
(9)

which means that in the proton, $d$-sea is larger than $u$-sea.

Later, the Gottfried sum rule was tested at the Fermilab in E866 Drell-Yan
(DY) experiments which measured $\bar{d}/\bar{u}$ as a function of $x$ over
the kinematic range of $0.015<x<0.35$ at $Q^{2}=54\,{\rm GeV}^{2}$ Towell et
al. (2001). Again, the data suggested a significant deficit in the sum rule
consistent with the DIS results and also with semi-inclusive DIS (SIDIS)
measurements of the HERMES collaboration for $0.020<x<0.30$ and
$1<Q^{2}<20\,{\rm GeV}^{2}$ Ackerstaff et al. (1998):

$\displaystyle{\rm HERMES~{}~{}1998:}\quad\quad$
$\displaystyle\int^{1}_{0}\left[\bar{d}(x,Q^{2})-\bar{u}(x,Q^{2})\right]dx=0.16\pm
0.03$ (10a) $\displaystyle{\rm E866~{}~{}2001:}\quad\quad$
$\displaystyle\int^{1}_{0}\left[\bar{d}(x,Q^{2})-\bar{u}(x,Q^{2})\right]dx=0.118\pm
0.012$ (10b)

The surprisingly large difference between light sea in the nucleon, Eq. (7),
observed in different experiments like DIS, DY and SIDIS, has triggered many
theoretical efforts to understand and accurately describe the experimental
results (for review, see, e.g., Kumano (1998); Garvey and Peng (2001)). While
the perturbative QCD fails in description of the sea asymmetry, the
nonperturbative mechanisms as Pauli-blocking, meson cloud, chiral-quark,
intrinsic sea, soliton seem to be more promising in explanation of the GSR
breaking. Recently, the statistical parton distributions approach was
developed to study the flavor structure of the light quark sea Soffer and
Bourrely (2019). The authors obtained a remarkable agreement of the
statistical model prediction for the ratio $\bar{d}/\bar{u}$ with the E866
data Garvey and Peng (2001); Peng et al. (2014) up to $x=0.2$. Unfortunately,
none of the studies mentioned above predicts correctly the $\bar{d}/\bar{u}$
behavior in the whole $x$ region, i.e. none of them predicts a sign-change for
$\bar{d}(x)-\bar{u}(x)$ at $x\approx 0.3$ as suggested by the E866 data.
Below, we briefly discuss possible effects modifying the GSR like the
perturbative QCD corrections, higher twist-terms, small-$x$ behavior and
nuclear shadowing effects.

###  II.1 pQCD corrections to the GSR

Here, we show that the perturbative QCD corrections to the GSR are too small
to explain the light sea asymmetry Hinchliffe and Kwiatkowski (1996). The
corrections of order $\alpha_{s}^{2}$ to the GSR were obtained in Kataev and
Parente (2003) basing on numerical calculation of the order $\alpha_{s}^{2}$
contribution to the coefficient function.

From the renormalization group equation analysis for $S_{G}(Q^{2})$ Kataev and
Parente Kataev and Parente (2003) obtained for the number of active flavors
$n_{f}$=4 the following QCD corrections to the GSR:

$S_{G}(Q^{2})=\frac{1}{3}\left[1+0.0384\left(\frac{\alpha_{s}}{\pi}\right)-0.822\left(\frac{\alpha_{s}}{\pi}\right)^{2}\right].$
(11)

Using the above formula we find

$\displaystyle S_{G}(Q^{2}=4\,{\rm GeV}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{3}-0.0015=0.3318$ (12a) $\displaystyle
S_{G}(Q^{2}=54\,{\rm GeV}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{3}-0.0003=0.3330\,.$ (12b)

This means that the magnitude of order $\alpha_{s}^{2}$ perturbative QCD
effects turn out to be about $-0.4\%$ at $Q^{2}=4\,{\rm GeV}^{2}$
($\alpha_{s}\approx 0.31$), and $-0.08\%$ at $Q^{2}=54\,{\rm GeV}^{2}$
($\alpha_{s}\approx 0.20$) of the original constant value of the GSR,
$S_{G}=1/3$.
So, it is clearly seen that the perturbative QCD corrections to the Gottfried
sum rule are very small and cannot explain the experimental results of NMC
Arneodo et al. (1994) and E866 Towell et al. (2001) collaborations where the
GSR is broken on the level of $-29.5\%$ and $-23.6\%$, respectively.

###  II.2 Higher-twist effects

In light of the results obtained in Alekhin et al. (2012), also the higher-
twist terms seem not to be much helpful in description of the large
discrepancy between the theoretical prediction of the GSR, Eq. (1), and the
experimental value of Eq. (8). The authors of Alekhin et al. (2012), basing on
the DIS world data, fitted in the NNLO analysis the twist-4 coefficient
$H_{2}^{\tau=4}(x)$ for the nonsinglet function $F_{2}^{p-n}(x)$ and found the
HT corrections marginal in comparison with the leading twist (LT) terms,

$F_{2}^{p-n}(x,Q^{2})=F_{2}^{p}(x,Q^{2})-F_{2}^{n}(x,Q^{2})=\left[F_{2}^{p-n}(x,Q^{2})\right]^{{\rm
LT}}+\frac{H_{2}^{\tau=4}(x)}{Q^{2}}\,.$ (13)

In Fig. 1, we plot the coefficient of the twist-4 term $H_{2}^{\tau=4}(x)$,
Eq. (13), for the nonsinglet structure function $F_{2}^{p-n}(x)$ obtained in
Alekhin et al. (2012) and compare the corresponding HT corrections with the
results of NMC for $F_{2}^{p-n}(x)$ at $Q^{2}=4\,{\rm GeV}^{2}$.

Figure 1:  Left: the central values (solid line) and the error band for the
coefficient of the twist-4 term $H_{2}^{\tau=4}(x)$, Eq. (13), obtained from
the NNLO fit for the nonsinglet structure function $F_{2}^{p-n}(x)$ Alekhin et
al. (2012). Right: comparison of the NMC data for $F_{2}^{p-n}(x)$ at
$Q^{2}=4\,{\rm GeV}^{2}$ with the corresponding HT corrections.

However, when applied to the Gottfried sum rule, these HT corrections though
too small to be responsible for the observed flavor asymmetry, are not
marginal and can accurately explain the relatively large discrepancy between
the two central values of the experimental results: NMC, Eq. (9), and E866,
Eq. (10b).
Namely, the HT effects modify the original GSR giving the contribution on the
level of $-5.4\%$ at $Q^{2}=4\,{\rm GeV}^{2}$ (NMC) and $-0.4\%$ at
$Q^{2}=54\,{\rm GeV}^{2}$ (E866) of the sum $1/3$. Hence, the corresponding
difference between the NMC and E866 results for the flavor asymmetry of the
light sea

$\Delta(Q^{2})\equiv\int^{1}_{0}\left[\bar{d}(x,Q^{2})-\bar{u}(x,Q^{2})\right]dx\,,$
(14)

implied by the HT effects at different scales of $Q^{2}$, is

$\Delta^{HT}(Q^{2}=4\,{\rm GeV}^{2})-\Delta^{HT}(Q^{2}=54\,{\rm
GeV}^{2})\approx 0.025\pm 0.022\,.$ (15)

This is in a very good agreement with the experimental data:

$\Delta_{NMC}-\Delta_{E866}\approx 0.030\,.$ (16)

Taking into account also the perturbative QCD radiative corrections of Eq.
(11), we arrive at the value even closer to the data:

$\Delta^{Rad+HT}(Q^{2}=4\,{\rm GeV}^{2})-\Delta^{Rad+HT}(Q^{2}=54\,{\rm
GeV}^{2})\approx 0.027\,.$ (17)

It is seen that the $Q^{2}$-dependence of the GSR can resolve a discrepancy
between the flavor asymmetry of the light sea in the nucleon measured in
different experiments. Similar suggestion was made by the authors of Ref.
Szczurek and Uleshchenko (2000).

We have found that the QCD-improved parton model including the NNLO radiative
corrections and also the twist-4 contributions predicts for the Gottfried sum
rule at $Q^{2}=4\,{\rm GeV}^{2}$

$S_{G}(Q^{2}=4\,{\rm GeV}^{2})=\frac{1}{3}\quad\underbrace{-0.0015}_{{\rm
pQCD}}\quad\underbrace{-0.0181}_{{\rm HT}}\approx\frac{1}{3}-0.02\,.$ (18)

This means that the large deficit of the GSR observed in the experiments
($S_{G}\approx 1/3-0.1$) comes from another sources than perturbative
mechanisms and HT effects.

###  II.3 Low-$x$ contribution

Experimental verification of the most sum rules faces the difficulty that in
any realistic experiment one cannot reach arbitrarily small values of the
Bjorken $x$. This is a serious obstacle also in the determination of the
Gottfried sum rule which involves the first Mellin moment, i.e. integral of
the nonsinglet structure function $F_{2}^{ns}$ over the whole range of $x$:
$0\leqslant x\leqslant 1$. The lack of low-$x$ data with good accuracy makes
reasonable the idea that a significant contribution to the integral of the GSR
can come just from the small-$x$ region. We illustrate this in Fig. 2 where we
show different low-$x$ behaviors of $F_{2}^{ns}/x\sim x^{a}$ and the
corresponding truncated GSR, $\int_{x}^{1}F_{2}^{ns}(x)\,dx/x$, together with
the NMC data Arneodo et al. (1994). We use three values for $a$: $-0.2$,
$-0.4$ and $-0.6$. It is seen that the experimental uncertainties in the
small-$x$ region are too large to favor any of them.

Figure 2:  Left: possible parametrizations of $F_{2}^{ns}/x$ reflecting
different small-$x$ behavior $\sim x^{a}$: $a\,=\,-0.2$ (dashed), $-0.4$
(dash-dotted) and $-0.6$ (solid) compared to the NMC data Arneodo et al.
(1994). Right: the truncated Gottfried sum rule
$\int_{x}^{1}F_{2}^{ns}(x)\,dx/x$, respectively.

The different $\sim x^{a}$ behaviors predict significant different very
low-$x$ contributions to the GSR, $\int_{0}^{0.004}F_{2}^{ns}(x)\,dx/x$,
namely 0.006 for $a=-0.2$, 0.011 for $a=-0.4$ and 0.022 for $a=-0.6$. This
means that the very low-$x$ contributions to the GSR can vary from $1-7\,\%$
of the sum $1/3$ and cannot resolve the GSR breaking problem. On the other
hand, the NMC data in the small-$x$ region confirm very well the expectations
of the theoretical studies on $F_{2}$ based on the Regge theory. In the Regge
approach, the small-$x$ behavior of $F_{2}^{ns}(x)$ is controlled by the
reggeon $A_{2}$ exchange Kwiecinski (1996):

$F_{2}^{p-n}(x)=F_{2}^{p}(x)-F_{2}^{n}(x)\sim x^{1-\alpha_{A_{2}}}\,,$ (19)

where $\alpha_{A_{2}}\approx 0.5$ is the $A_{2}$ reggeon intercept. Taking
into account the Regge predictions in the NMC data analysis, we can estimate
the small-$x$ contribution to the Gottfried sum
$\int_{0}^{0.004}F_{2}^{ns}(x)\,dx/x$ as $4-7\,\%$ of the total value $1/3$.

###  II.4 Nuclear shadowing

Since there is no fixed target for the neutron, the deuteron is usually used
for measuring the neutron structure function $F_{2}^{n}$. The same method was
used by the NMC for determination of the Gottfried sum rule. In order to
obtain the difference of the structure functions $F_{2}^{p}-F_{2}^{n}$ of free
nucleons which enters into the GSR, the extracted $F_{2}^{n}$ from the
deuteron data has to be corrected by the shadowing effects:

$F_{2}^{d}=\frac{1}{2}\,\left(F_{2}^{p}+F_{2}^{n}\right)-\delta F_{2}^{d}\,,$
(20)

where $\delta F_{2}^{d}\geqslant 0$.

The shadowing effects in the deuteron were investigated in many works (for
review, see, e.g., Kumano (1998)) providing the small negative correction to
the sum. Thus, the shadowing leads to smaller value of $S_{G}(Q^{2})$ than
that determined experimentally assuming no shadowing, and the GSR violation is
even magnified. The nuclear shadowing which is dominated by the vector-meson-
dominance (VMD) mechanism is non-negligible in the region of $x\leqslant 0.1$
and for low- and moderate $Q^{2}$ relevant for the NMC measurements and has to
be taken into account in the data analysis Badelek and Kwiecinski (1994). This
leads to the following expression for the difference between the proton and
neutron structure functions in the integrand of the Gottfried sum, Eq. (1):

$F_{2}^{p}(x,Q^{2})-F_{2}^{n}(x,Q^{2})=\left(F_{2}^{p}(x,Q^{2})-F_{2}^{n}(x,Q^{2})\right)_{NMC}-2\,\delta
F_{2}^{d}(x,Q^{2})\,,$ (21)

where $(F_{2}^{p}(x,Q^{2})-F_{2}^{n}(x,Q^{2}))_{NMC}$ obtained by NMC is
related to the measured $F_{2}^{d}$ and $F_{2}^{n}/F_{2}^{p}$ via

$\left(F_{2}^{p}(x,Q^{2})-F_{2}^{n}(x,Q^{2})\right)_{NMC}=2\,F_{2}^{d}\;\frac{1-\left(\frac{F_{2}^{n}}{F_{2}^{p}}\right)_{NMC}}{1+\left(\frac{F_{2}^{n}}{F_{2}^{p}}\right)_{NMC}}$
(22)

Using the results of Badelek and Kwiecinski (1994) for $\delta F_{2}^{d}$, in
Fig. 3, we compare the NMC data with the corrected one by the nuclear
shadowing effect. We find that the negative shadowing correction to the
experimental result for the Gottfried sum, $S_{G}(0.004,\,0.8,\,Q^{2})=0.221$,
is $0.0265$ ($\approx 12\%$).

Figure 3:  The nuclear shadowing corrected NMC data for $F_{2}^{ns}$ (open
circles) calculated in Badelek and Kwiecinski (1994). Solid: fit to the
shadowing contribution to the deuteron structure function, $2\delta
F_{2}^{d}$.

##  III TMM method for determination of sum rules

Here, we briefly present an effective method which allows one to determine any
sum rule value from the experimental data in the available restricted
kinematic range of the Bjorken variable $x$. The method was elaborated in
Kotlorz et al. (2017); Strozik-Kotlorz et al. (2017) for the Bjorken sum rule,
and successfully applied to the experimental data at COMPASS, SLAC and JLab
Kotlorz et al. (2017); Strozik-Kotlorz et al. (2017); Kotlorz and Mikhailov
(2019); Kotlorz et al. (2019).

The main philosophy of the method presented in Kotlorz et al. (2017) is
construction of a special truncated sum $\Gamma$ which approaches the limit of
the sum rule value more quickly, i.e. for larger $x$, than the ordinary sum.
In other words, the use of $\Gamma$ “mimics” the extension of the experimental
kinematic region of $x$ to the lower values.
Below, we give useful formulas for determination of the sum rule value in the
TMM approach which in the next section we shall apply to the GSR. The details
on theoretical aspects of the $\Gamma$ construction and the description of
different approximations of the TMM method can be found in Kotlorz et al.
(2017).

Determination of the sum rules involves the integrals of the parton density or
structure function $f(x,Q^{2})$ over the whole range $(0,1)$ of $x$:

$S(0,\,1)=\int^{1}_{0}f(x)\,dx\,,$ (23)

where for clarity we omit the $Q^{2}$ dependence. The experimental
measurements provide data on $f(x)$ only in the limited range of $x$:
$0<x_{min}\equiv x_{1}<x_{2}<\cdots<x_{max}\equiv x_{N}<1$, where
$x_{min}\equiv Q^{2}_{\text{min}}/(2(Pq)_{\text{max}}>0)$. Thus, in fact, the
experiment gives information on the truncated sum

$S(x_{min},\,x_{max})=\int^{x_{max}}_{x_{min}}f(x)\,dx\,.$ (24)

The truncation at the upper limit $x_{max}$ is less important in comparison to
the low-$x$ limit $x_{min}$ because of the rapid decrease of the parton
densities and structure functions as $x\rightarrow 1$.

In particular, if we define $n$th truncated moment of the structure function
$f(x,Q^{2})$ as

$M_{n}(x_{min},\,x_{max},\,Q^{2})=\int^{x_{max}}_{x_{min}}x^{n-2}f(x,Q^{2})\,dx\,,$
(25)

the Gottfried sum rule $S_{G}(Q^{2})$ is the first moment
$M_{1}(0,\,1,\,Q^{2})$ of the nonsinglet function
$F_{2}^{p}(x,\,Q^{2})-F_{2}^{n}(x,\,Q^{2})$.

The special sum $\Gamma$ is constructed based on the ordinary sum $S$ in the
following way:

$\displaystyle\Gamma(x_{1},\,r)$ $\displaystyle=$ $\displaystyle
S(x_{1},1)+A\,\int_{x_{1}}^{x_{1}/r}f(x)\,dx\,,$ (26)

where $x_{1}$ is the smallest value of $x$ accessible in the experiment and
$A$, and $r$ are parameters calculated from the data. In the limit $x_{1}\to
0$, $\Gamma(x_{1},\,r)$ is equal to $S(x_{1},1)$ providing the sum rule value
$S(0,1)$, Eq. (23), whereas for $x>0$ $\Gamma(x_{1},\,r)$ approaches $S(0,1)$
much earlier than $S(x_{1},1)$ itself. This is illustrated in Fig. 4 where we
compare $S(x_{1},1)$ to a bunch of $\Gamma(x_{1},\,r)$, Eq. (26), plotted for
different values of $A$. We use smooth fits to the NMC and E866 data setting
the ratio of two experimental points $r=x_{1}/x_{2}$ equal to $0.5$ for NMC
and $0.7$ for E866, respectively. Here, we would like to emphasize that in our
analysis we shall use the auxiliary fit only for determination of $A_{I}$
while the rest of calculations will be performed with use of the pure data
$f(x)$ from the measured $x$-region.

Figure 4:  $S(x_{1},1)$, Eq. (24), (black solid) and $\Gamma(x_{1},\,r)$, Eq.
(26), for different values of $A$. Upper (blue) solid line corresponds to
$A=A_{0}$ (left panel) and $A=A_{I}$ (right panel), see description in the
text.

In our approach, as described in Kotlorz et al. (2017), we utilize the quasi-
linear regime of $\Gamma(x_{1},\,r)$ which starts already for $x$
significantly larger than the smallest experimental value of $x_{1}$. This
ensures the applicability of the first (or even zero as in the NMC case) order
approximation for estimation of the value of $S(0,1)$ with help of
$\Gamma(x_{1},\,r)$. Thus, requiring the second derivative to vanished,
$\Gamma^{\prime\prime}(x_{1},\,r)=0$, we obtain $A_{I}$ and the sum rule value
can be determined very effectively in the first order of Taylor expansion:

$\displaystyle
S(0,1)=\Gamma(0,\,r)\approx\Gamma(x_{1},\,r)-x_{1}\,\Gamma^{\prime}(x_{1},\,r)$
$\displaystyle=$
$\displaystyle\Gamma(x_{1},\,r)+(A_{I}+1)\,x_{1}\,f(x_{1})-A_{I}\,\frac{x_{1}}{r}f(x_{1}/r)\,,$
(28) $\displaystyle A_{I}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{r^{2}}\,\frac{f^{\prime}(x_{1}/r)}{f^{\prime}(x_{1})}-1\right]^{-1}\,,$
(29)

where $\Gamma(x_{1},\,r)$ is given by Eq. (26) and $f^{\prime}(x)$ denotes the
first order derivative with respect of $x$.

In a special case, where the small-$x$ experimental data can be well described
by a simple form $f(x)=Nx^{a}$, we have

$r^{n}\frac{f^{(n-1)}(x_{1})}{f^{(n-1)}(x_{1}/r)}=r\frac{f(x_{1})}{f(x_{1}/r)}$
(30)

and all derivatives $\Gamma^{(n)}(x_{1},\,r)$ vanish for the same $A=A_{0}$,

$A_{0}=\left[\frac{1}{r}\,\frac{f(x_{1}/r)}{f(x_{1})}-1\right]^{-1}\,.$ (31)

Hence, we arrive at the zero order approximation for the sum rule value which
reads

$S(0,1)=\Gamma(x_{1},\,r)=S(x_{1},1)+A_{0}\,\int_{x_{1}}^{x_{1}/r}f(x)\,dx\,.$
(32)

The method of estimation of the sum rule value based on the special truncated
sum $\Gamma$ is effective for different small-$x$ behavior of the function
$f\sim x^{a}$, also for $a<0$, as in the case of the Gottfried sum rule.

##  IV Data analysis

Below we present our numerical results for the Gottfried sum rule value
$S(0,1)$ based on the experimental NMC Arneodo et al. (1994) and E866 Towell
et al. (2001) data following the approach described in the previous section.

###  IV.1 NMC

The violation of the GSR was first observed by the New Muon Collaboration at
CERN in 1991 Amaudruz et al. (1991). NMC measured the cross section ratio for
deep inelastic scattering of muons from hydrogen and deuterium targets in the
kinematic range extended to the low-$x$ region, $0.004\leqslant x\leqslant
0.8$. The difference of the structure functions was calculated by Eq. (22) and
the ratio $F_{2}^{n}/F_{2}^{p}=2F_{2}^{d}/F_{2}^{p}-1$ was determined by the
NMC experiment, where the deuteron structure function $F_{2}^{d}$ was taken
from a fit to various experimental data. The results were obtained by
interpolation or extrapolation to $Q^{2}=4{\rm\,GeV}^{2}$. In the reanalyzed
data Arneodo et al. (1994), which are under study in this section, NMC used
their own data for $F_{2}^{d}$ and revised $F_{2}^{n}/F_{2}^{p}$ ratios.

The NMC data for $F_{2}^{p-n}$ which form the GSR,

$S_{G}(0,1)=\int^{1}_{0}\left[F_{2}^{p}(x,Q^{2})-F_{2}^{n}(x,Q^{2})\right]{dx\over
x}\,,$ (33)

can be for $x\leqslant 0.4$ well described by the fit function $\sim
0.2\,x^{0.6}$ Arneodo et al. (1994), which agrees with theoretical prediction
of the Regge-like behavior, Eq. (19), Kwiecinski (1996). The corresponding
truncated function $\Gamma(x_{1},\,r)$ saturates to the constant $S_{G}(0,1)$
already at large $x$ (see upper solid line in the left panel of Fig. 4) and we
can estimate $S_{G}(0,1)$ using the zero order formulas, Eqs. (31) and (32),
which take the form

$\displaystyle S_{G}(0,1)$ $\displaystyle=$ $\displaystyle
S_{G}(x_{1},1)^{\rm{NMC}}+A_{0}\,\int_{x_{1}}^{x_{1}/r}F_{2}^{p-n}(x,Q^{2})\frac{dx}{x}\,,$
(34) $\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle\left[\frac{F_{2}^{p-n}(x_{1}/r,Q^{2})}{F_{2}^{p-n}(x_{1},Q^{2})}-1\right]^{-1}.$
(35)

All quantities in Eqs. (34) and (35) are directly provided by the data or can
be calculated from the data without necessity to use of any fit function.
Namely, $S_{G}(x_{1},1)^{\rm{NMC}}$ is the contribution to the GSR from the
measured region of $x$ together with the correction for $x>0.8$, $x_{1}$
denotes the smallest $\langle x\rangle$ in the analysis, and $r=x_{1}/x_{k}$
is a ratio of two experimental points where $x_{k}>x_{1}$. The integral in Eq.
(34) can be calculated as a sum of the partial experimental contributions,
respectively:

$\int_{x_{i}}^{x_{i+1}}F_{2}^{p-n}(x)\,\frac{dx}{x}=\frac{x_{i+1}-x_{i}}{\langle
x_{i}\rangle}\,F_{2}^{p-n}(\langle x_{i}\rangle)\,.$ (36)

In Table 1 we show our estimations for $S_{G}(0,1)$ obtained for two values of
$x_{1}$: $0.007$ and $0.015$ and corresponding $r$ and $A_{0}$, up to
$x_{k}\approx 0.4$. The experimental value of $S_{G}(0,1)$, where the
small-$x$ contribution from the region $x<0.004$ is determined from the fit
described in Arneodo et al. (1994), is displayed in the last row.

Table 1: Estimations of the Gottfried sum rule value $S_{G}(0,1)$ obtained in the zero order approximation, Eqs. (34) and (35), for two values of $x_{1}$: $0.007$ and $0.015$ based on the NMC data at $Q^{2}=4$ GeV2 Arneodo et al. (1994). The ratio $r=x_{1}/x_{1+i}$, where $i=1,\,2,\,...\,8$. The experimental value of $S_{G}(0,1)$ is displayed in the last row. $x_{1}=0.007$ | $~{}~{}~{}~{}~{}~{}x_{1}=0.015$
---|---
$r$ | $~{}~{}~{}A_{0}$ | $~{}~{}~{}S_{G}(0,1)$ | $~{}~{}~{}~{}~{}~{}r$ | $~{}~{}~{}A_{0}$ | $~{}~{}~{}S_{G}(0,1)$
$0.47$ | $~{}~{}~{}2.0$ | $~{}~{}~{}0.236$ | $~{}~{}~{}~{}~{}~{}0.50$ | $~{}~{}~{}1.07$ | $~{}~{}~{}0.223$
$0.23$ | $~{}~{}~{}0.53$ | $~{}~{}~{}0.230$ | $~{}~{}~{}~{}~{}~{}0.30$ | $~{}~{}~{}0.94$ | $~{}~{}~{}0.236$
$0.14$ | $~{}~{}~{}0.48$ | $~{}~{}~{}0.236$ | $~{}~{}~{}~{}~{}~{}0.19$ | $~{}~{}~{}0.52$ | $~{}~{}~{}0.232$
$0.09$ | $~{}~{}~{}0.29$ | $~{}~{}~{}0.234$ | $~{}~{}~{}~{}~{}~{}0.12$ | $~{}~{}~{}0.34$ | $~{}~{}~{}0.232$
$0.06$ | $~{}~{}~{}0.20$ | $~{}~{}~{}0.233$ | $~{}~{}~{}~{}~{}~{}0.09$ | $~{}~{}~{}0.31$ | $~{}~{}~{}0.236$
$0.04$ | $~{}~{}~{}0.19$ | $~{}~{}~{}0.236$ | $~{}~{}~{}~{}~{}~{}0.06$ | $~{}~{}~{}0.22$ | $~{}~{}~{}0.234$
$0.03$ | $~{}~{}~{}0.14$ | $~{}~{}~{}0.235$ | $~{}~{}~{}~{}~{}~{}0.04$ | $~{}~{}~{}0.19$ | $~{}~{}~{}0.235$
$0.02$ | $~{}~{}~{}0.12$ | $~{}~{}~{}0.235$ | $~{}~{}~{}~{}~{}~{}0.03$ | $~{}~{}~{}0.17$ | $~{}~{}~{}0.237$
EXP. NMC $S_{G}(0,1)=0.235\pm 0.026$

We obtain $S_{G}(0,1)=0.234\pm 0.003_{\,\rm SD}$. The low-$x$ contribution
$S_{G}(0,0.004)=0.012\pm 0.004$. Here, we estimate the error for the low-$x$
contribution as an average deviation for the composed errors of
$S_{G}(0,0.004)$ calculated for the data sets summarized in Table 1. Hence, we
find finally $S_{G}(0,1)=0.234\pm 0.022$. The obtained result is in a good
agreement with the value provided by NMC. This means that in the case of the
NMC data, the Gottfried sum can be determined very effectively already in the
zero order approximation of the TMM approach.

In Fig. 5, we compare the NMC data for $F_{2}^{ns}/x$ and
$\int_{x}^{0.8}F_{2}^{ns}dx/x$ with the predictions of two parametrizations
based on the global PDFs fit, CTEQ6 Pumplin et al. (2002) and MSTW08 Martin et
al. (2009), and to our TMM estimation for the GSR. In the left panel we plot
also the low-$x$ fit function $f_{\rm fit}^{\rm NMC}$ for illustration of the
regular Regge behavior of the NMC data up to $x\approx 0.4$. We find

$\frac{1}{x}\left[F_{2}^{p}(x,Q^{2})-F_{2}^{n}(x,Q^{2})\right]_{\rm
NMC}\approx f_{\rm fit}^{\rm NMC}=ax^{b},\quad\quad a=0.198\pm 0.006,\quad
b=-0.410\pm 0.012,$ (37)

which is consistent with the form provided in Arneodo et al. (1994).

Figure 5:  $F_{2}^{ns}(x,Q^{2})/x$ (left) and
$\int_{x}^{0.8}F_{2}^{ns}(x,Q^{2})\,dx/x$ (right) at $Q^{2}=4\,{\rm GeV^{2}}$.
A comparison of the NMC data to CTEQ6 Pumplin et al. (2002) and MSTW08 Martin
et al. (2009) predictions and also to the TMM estimation for the Gottfried sum
rule. The low-$x$ fit function $f_{\rm fit}^{\rm NMC}$ in the left panel has
the form Eq. (37).

In Table 2, we collect the contributions to the Gottfried sum rule, $\int
F_{2}^{ns}(x,Q^{2})\,dx/x$, obtained for different $x$ ranges at
$Q^{2}=4\,{\rm GeV}^{2}$. A comparison of the NMC data to TMM, CTEQ6 and
MSTW08 predictions is shown.

Table 2: The contributions to the Gottfried sum rule, $\int F_{2}^{ns}(x,Q^{2})\,dx/x$, integrated over different $x$ ranges at $Q^{2}=4\,{\rm GeV}^{2}$. Compared are the NMC data to TMM, CTEQ6 Pumplin et al. (2002) and MSTW08 Martin et al. (2009) predictions. (∗) The result contains a fit to the unmeasured region of small-$x$. $~{}~{}~{}~{}~{}x$ range |  NMC |  TMM |  CTEQ6 |  MSTW08
---|---|---|---|---
$0<x<1$ | $0.235\pm 0.026^{*}$ | $0.234\pm 0.022$ | $0.255$ | $0.274$
$0.004<x<0.8$ | $~{}~{}~{}0.221\pm 0.021~{}~{}~{}$ |  | $0.231$ | $0.218$
$0<x<0.004$ | $0.013\pm 0.005^{*}$ | $0.012\pm 0.004$ | $0.023$ | $0.055$
$0.8<x<1$ | $0.001\pm 0.001$ |  | $0.001$ | $0.001$

The small-$x$ contribution to the GSR in the unmeasured region $x<0.004$ was
determined by NMC with use of the fit to the data. Our method, which is
totally based on the experimental data in the measured region of $x$, provides
almost the same result. The CTEQ6 prediction is slightly above the NMC
estimation for $S_{G}(0,0.004)$ while agreeing for the total GSR value and
also for the contribution from the measured region. In turn, the MSTW08
parametrization supports the experimental measurements but its predictions for
$S_{G}(0,0.004)$ and $S_{G}(0,1)$ are much larger than both the experimental
and our estimations. A reason for this discrepancy is a small-$x$ behavior of
$\bar{d}-\bar{u}$ assumed by MSTW08 which implies a decrease of
$\int(\bar{d}-\bar{u})\,dx$ and hence the increase of $\int F_{2}^{ns}/x\,dx$
in this region in comparison to other global PDF fits. We shall discuss it
also in the next subsection which is devoted to the E866 experiment.

###  IV.2 E866

Fermilab experiment E866 Towell et al. (2001) was a fixed target experiment
that has measured the light sea quark asymmetry in the nucleon using Drell-Yan
process of di-muon production in 800 GeV proton interactions with hydrogen and
deuterium targets. From the data, the ratio $\bar{d}/\bar{u}$ was determined
over a wide range in Bjorken-$x$. The obtained results confirmed previous
measurements by E866/NuSea Hawker et al. (1998), which were the first
demonstration of a strong $x$-dependence of the $\bar{d}/\bar{u}$ ratio, and
extended them to lower-$x$. To obtained the antiquark asymmetry
$\bar{d}-\bar{u}$ and also the integrated asymmetry
$\int(\bar{d}-\bar{u})\,dx$, E866 used their data for $\bar{d}/\bar{u}$ and
the PDF parametrization CTEQ5M Lai et al. (2000) for $\bar{d}+\bar{u}$. In
order to estimate the contribution from the unmeasured region $0<x<0.015$,
MRST Martin et al. (1998) and CTEQ5M fits were used. Moreover, it was assumed
that the contribution for $x>0.35$ was negligible.
In our TMM analysis, presented below, we use the experimental data only from
the measured region $0.015<x<0.35$. We compare our results with the E866 ones
and also with the predictions of the updated global parametrizations – CTEQ6
Pumplin et al. (2002) and MSTW08 Martin et al. (2009).

To determine the light sea quark asymmetry from the E866 data,

$\Delta(0,1)=\int^{1}_{0}\left[\bar{d}(x,Q^{2})-\bar{u}(x,Q^{2})\right]dx\,,$
(38)

we apply the universal first order approximation of the special truncated sum
$\Gamma$ method given by Eqs. (26)-(29). In the terms of the experimental data
they read

$\displaystyle\Delta(0,1)=\Delta(x_{1},1)^{\rm{E866}}+$ $\displaystyle A_{I}$
$\displaystyle\,\int_{x_{1}}^{x_{1}/r}[\bar{d}-\bar{u}](x)\,dx+(A_{I}+1)\,x_{1}\,[\bar{d}-\bar{u}](x_{1})-A_{I}\,\frac{x_{1}}{r}[\bar{d}-\bar{u}](x_{1}/r)\,,$
(39) $\displaystyle A_{I}$
$\displaystyle=\left[\frac{1}{r^{2}}\,\frac{f_{fit}^{\prime}(x_{1}/r)}{f_{fit}^{\prime}(x_{1})}-1\right]^{-1}\,,$
(40)

where the prime denotes a derivative with respect to $x$ of the fit function
$f_{\rm fit}^{\rm E866}$ to the E866 data on $\bar{d}-\bar{u}$:

$\displaystyle\left[\bar{d}(x,Q^{2})-\bar{u}(x,Q^{2})\right]_{\rm E866}\approx
f_{\rm fit}^{\rm E866}$ $\displaystyle=$ $\displaystyle
ax^{b}(1-x)^{c}(1+d\,x),$ $\displaystyle a=0.55\pm 0.02,\quad b=-0.19\pm
0.02,\quad c$ $\displaystyle=$ $\displaystyle 2.8\pm 0.3,\quad d=-3.7\pm
0.1\,.$ (41)

The fit function, which we use only for calculation of $A_{I}$ in Eq. (40), is
shown as a dotted line in the left panel of Fig. 6. All other quantities in
Eqs. (39) and (40) are directly provided by the data. Again, as in the NMC
analysis, $x_{1}$ denotes the smallest $\langle x\rangle$, and the ratio
$r=x_{1}/x_{k}$ is determined from the kinematics.
$\Delta(x_{1},1)^{\rm{E866}}$ denotes the contribution to the light sea
asymmetry from the measured region of $x$.

Table 3: Estimations of the integrated quark asymmetry $\Delta(0,1)$ obtained in the first order approximation, Eqs. (39) and (40), for two values of $x_{1}$: $0.026$ and $0.038$ based on the E866 data at $Q^{2}=54$ GeV2, Towell et al. (2001). The ratio $r=x_{1}/x_{1+i}$ where $i=1,\,2,\,3,\,...\,8$. The experimental value of $\Delta(0,1)$ is displayed in the last row. $x_{1}=0.026$ | $~{}~{}~{}~{}~{}~{}x_{1}=0.038$
---|---
$r$ | $~{}~{}~{}A_{I}$ | $~{}~{}~{}\Delta(0,1)$ | $~{}~{}~{}~{}~{}~{}r$ | $~{}~{}~{}A_{I}$ | $~{}~{}~{}\Delta(0,1)$
$0.68$ | $~{}~{}~{}1.79$ | $~{}~{}~{}0.098$ | $~{}~{}~{}~{}~{}~{}0.73$ | $~{}~{}~{}2.19$ | $~{}~{}~{}0.097$
$0.50$ | $~{}~{}~{}0.79$ | $~{}~{}~{}0.098$ | $~{}~{}~{}~{}~{}~{}0.57$ | $~{}~{}~{}1.03$ | $~{}~{}~{}0.105$
$0.39$ | $~{}~{}~{}0.48$ | $~{}~{}~{}0.101$ | $~{}~{}~{}~{}~{}~{}0.46$ | $~{}~{}~{}0.67$ | $~{}~{}~{}0.101$
$0.32$ | $~{}~{}~{}0.35$ | $~{}~{}~{}0.099$ | $~{}~{}~{}~{}~{}~{}0.39$ | $~{}~{}~{}0.50$ | $~{}~{}~{}0.103$
$0.27$ | $~{}~{}~{}0.27$ | $~{}~{}~{}0.101$ | $~{}~{}~{}~{}~{}~{}0.34$ | $~{}~{}~{}0.40$ | $~{}~{}~{}0.104$
$0.23$ | $~{}~{}~{}0.22$ | $~{}~{}~{}0.101$ | $~{}~{}~{}~{}~{}~{}0.30$ | $~{}~{}~{}0.33$ | $~{}~{}~{}0.101$
$0.20$ | $~{}~{}~{}0.19$ | $~{}~{}~{}0.100$ | $~{}~{}~{}~{}~{}~{}0.27$ | $~{}~{}~{}0.29$ | $~{}~{}~{}0.103$
$0.18$ | $~{}~{}~{}0.17$ | $~{}~{}~{}0.101$ | $~{}~{}~{}~{}~{}~{}0.24$ | $~{}~{}~{}0.26$ | $~{}~{}~{}0.105$
EXP. E866 $\Delta(0,1)=0.118\pm 0.012$

In Table 3 we show our estimations for $\Delta(0,1)$ obtained for two values
of $x_{1}$: $0.026$ and $0.038$ and corresponding $r$ and $A_{I}$. To minimize
a possible error implied by the decrease of $r$, Kotlorz et al. (2017), we
proceed our analysis up to $x_{k}\approx 0.15$.
In Fig. 6, we compare the E866 data for $[\bar{d}-\bar{u}](x,Q^{2})$ and
$\int_{x}^{0.35}[\bar{d}-\bar{u}](x,Q^{2})\,dx$ with the predictions of two
parametrizations based on the global PDFs fits, CTEQ6 Pumplin et al. (2002)
and MSTW08 Martin et al. (2009), and to the TMM results for
$\int_{0}^{1}[\bar{d}-\bar{u}](x,Q^{2})\,dx$. In Table 4, we present a
comparison of the light sea quark asymmetry $\Delta$ integrated over different
$x$ ranges for the E866 data, TMM approach and CTEQ6 and MSTW08 predictions.
The TMM results are shown together with the average deviation of the composed
errors calculated from Eqs. (39) and (40) for the set data from Table 3.

Figure 6:  $[\bar{d}-\bar{u}](x,Q^{2})$ (left) and $\int_{x}^{0.35}[\bar{d}-\bar{u}](x,Q^{2})\,dx$ (right) at $Q^{2}=54\,{\rm GeV^{2}}$. A comparison of the E866 data to CTEQ6 Pumplin et al. (2002) and MSTW08 Martin et al. (2009) predictions and also to the TMM estimation for the integrated asymmetry of the light sea quarks $\int_{0}^{1}[\bar{d}-\bar{u}](x,Q^{2})\,dx$. The fit function $f_{\rm fit}^{\rm E866}$ in the left panel has the form Eq. (41). Table 4: Integrated light sea quark asymmetry $\Delta$ over different $x$ ranges at $Q^{2}=54$ ${\rm GeV^{2}}$ obtained in the TMM approach. A comparison to the E866 data and CTEQ6 Pumplin et al. (2002) and MSTW08 Martin et al. (2009) predictions. (∗) The result contains a fit to the unmeasured region of small-$x$. $~{}~{}~{}~{}~{}x$ range |  E866 |  TMM |  CTEQ6 |  MSTW08
---|---|---|---|---
$0<x<1$ | $0.118\pm 0.012^{*}$ | $0.101\pm 0.016$ | $0.119$ | $0.089$
$0.015<x<0.35$ | $~{}~{}~{}0.0803\pm 0.011~{}~{}~{}$ |  | $0.083$ | $0.077$
$0<x<0.015$ | $0.038\pm 0.004^{*}$ | $0.021\pm 0.012$ | $0.037$ | $0.014$
$0.35<x<1$ | $0$ |  | $\\!\\!\\!-0.001$ | $\\!\\!\\!-0.002$

The low-$x$ contribution $\Delta(0,0.015)=0.021\pm 0.012$ obtained in our
analysis is essentially smaller than the E866 estimation obtained with use of
the combined fits MRST98 and CTEQ5M. It is also smaller than the CTEQ6
prediction but larger than the more recent global fit prediction of MSTW08.
Since the NMC and E866 data were used in the global fit analysis, the CTEQ6
and MSTW08 predictions are in a good agreement with these experimental data
from the measured $x$-region. The problem is in determination of the GSR and
$\Delta$ contributions coming from the unmeasured regions, especially from the
small-$x$ region. While all reasonable fits to the data assume
$\bar{u}=\bar{d}$ as $x\rightarrow 0$, it is achieved differently for
different parametrizations. This is shown in the left panel of Fig. 6 where we
compare the E866 data on $\bar{d}(x)-\bar{u}(x)$ with CTEQ6 and MSTW08 NLO
fits at $Q^{2}=54\,{\rm GeV^{2}}$. Our result for the small-$x$ contribution
$\Delta(0,0.015)$ lies between the values of the CTEQ6 and MSTW08 predictions.
MSTW08 parametrization of $\bar{d}(x)-\bar{u}(x)$ at $Q_{0}^{2}=1\,{\rm
GeV^{2}}$ goes to zero as $x^{0.8}$ at small-$x$ and this behavior is not
excluded by the E866 data.

Let us finally comment the discrepancy between the NMC and E866 data. To this
aim we shall use the TMM results which provide even larger discrepancy than
the results reported by NMC and E866. Namely, we compare $\Delta(0,1)$
calculated from the GSR value for the NMC data analysis with that obtained for
the E866 data. We have $\Delta(0,1,Q^{2}=4\,{\rm GeV^{2}})=0.149\pm 0.033$ vs
$\Delta(0,1,Q^{2}=54\,{\rm GeV^{2}})=0.101\pm 0.016$. The both results are
still in agreement with each other and the difference between their central
values can be attributed to the higher-twist effects. As it was described in
Section II.2, using the results obtained for the twist-4 coefficient
$H_{2}^{\tau=4}(x)$ for the nonsinglet function $F_{2}^{p-n}(x)$ Alekhin et
al. (2012), the difference for $\Delta(0,1)$ at $Q^{2}=4$ and $54\,{\rm
GeV^{2}}$ implied by the HT terms is $0.025\pm 0.022$. Using its central value
and taking into account also the perturbative QCD radiative corrections, Eqs.
(12a) and (12b), we are able to reduce the difference
$\Delta(0,1,Q^{2}=4\,{\rm GeV^{2}})-\Delta(0,1,Q^{2}=54\,{\rm
GeV^{2}})=0.048\pm 0.049$ by about $60\%$ .

###  IV.3 Second moment of $F_{2}^{p-n}$

The main aim of our paper is to study the Gottfried sum rule within the TMM
approach, nevertheless, finally, we would like also to discuss shortly our
predictions for the second moment of the structure function $F_{2}^{p-n}$,

$\int^{1}_{0}\left[F_{2}^{p}(x,Q^{2})-F_{2}^{n}(x,Q^{2})\right]\,dx=\frac{1}{3}\left(\langle
x\rangle_{u-d}+\langle x\rangle_{\bar{u}-\bar{d}}\right),$ (42)

where

$\langle
x\rangle_{u-d}\equiv\int^{1}_{0}x\left[u(x,Q^{2})-d(x,Q^{2})\right]dx\,.$ (43)

The latter, $\langle x\rangle_{u-d}$, being the iso-vector quark momentum
fraction, is recently of large interest for the analyses based on the lattice
QCD. This interest, which has triggered many theoretical and phenomenological
investigations, is mainly motivated by a discrepancy of over $25\%$ between
the lattice predictions, $\langle x\rangle_{u-d}>0.2$, and the values obtained
from phenomenological fits to the experimental data, $0.15-0.17$ Bali et al.
(2014).

Below, we present our results for $\langle x\rangle_{u-d}$ at $Q^{2}=4\,{\rm
GeV^{2}}$ obtained within the TMM approach. Since the NMC data provide
knowledge only for the sum $\langle x\rangle_{u-d}+\langle
x\rangle_{\bar{u}-\bar{d}}$, Eq. (42), we use combined results based on the
NMC and E866 data. We take also into account the $Q^{2}$ evolution effects for
the E866 data provided for $Q^{2}=54\,{\rm GeV^{2}}$. To this aim, we correct
the value of $\langle x\rangle_{\bar{u}-\bar{d}}$ calculated from the E866
data by a mean difference $\langle x\rangle_{\bar{u}-\bar{d}}$ obtained for
the two $Q^{2}$ values $4$ and $54$ ${\rm GeV^{2}}$ from the CTEQ6 and MSTW08
fits. Thus, finally, we obtain $\langle x\rangle_{u-d}=0.165\pm 0.007$ and
$\langle x\rangle_{\bar{u}-\bar{d}}=0.007\pm 0.001$.
In Table 5, we compare our TMM results for $\langle x\rangle_{u-d}$ at
$Q^{2}=4\,{\rm GeV^{2}}$ with the predictions of the world-wide fits CTEQ6 and
MSTW08, and also with the recent lattice result Abdel-Rehim et al. (2015).

Table 5: The iso-vector quark momentum fraction $\langle x\rangle_{u-d}$ at $Q^{2}=4$ ${\rm GeV^{2}}$ obtained in TMM approach from the combined NMC and E866 data. A comparison to the global fit predictions CTEQ6 and MSTW08, and to the recent lattice result Abdel-Rehim et al. (2015). TMM |  CTEQ6 |  MSTW08 |  LATTICE
---|---|---|---
$0.165\pm 0.007$ | $0.158$ | $0.161$ | $0.208\pm 0.024$

For comparison, the recent analysis of the DIS data from fixed-target
experiments on the structure function $F_{2}$ performed in the valence-quark
approximation at the NNLO approximation, and incorporating the NMC result on
the Gottfried sum rule, provides $\langle x\rangle_{\bar{u}-\bar{d}}=0.187\pm
0.021$ Kotikov et al. (2018).

##  V Conclusions

In this paper, based on the experimental NMC data on the nonsinglet structure
function $F_{2}^{p}-F_{2}^{n}$ at $Q^{2}=4\,{\rm GeV^{2}}$ Arneodo et al.
(1994), and E866 data on the $\bar{d}/\bar{u}$ asymmetry in the nucleon sea at
$Q^{2}=54\,{\rm GeV^{2}}$ Towell et al. (2001), we have reevaluated the
Gottfried sum rule Gottfried (1967). In our analysis, we used the truncated
moments approach in which, with help of the special truncated sum, one can
overcome in a study of the fundamental integral characteristics of the parton
distributions the problem of the unavoidable kinematic restrictions on the
Bjorken variable $x$ Kotlorz et al. (2017).

Using only the data from the measured region of $x$, we obtained for the
Gottfried sum $\int_{0}^{1}F_{2}^{ns}/x\,dx=0.234\pm 0.022$ which is in a very
good agreement with the value reported by NMC, and $0.101\pm 0.016$ for the
integrated nucleon sea asymmetry $\int_{0}^{1}(\bar{d}-\bar{u})\,dx$. The
latter, though still consistent with the E866 result $0.118\pm 0.012$, is
clearly smaller in its central value. This disagreement can be attributed to
the estimation of the contribution from the unmeasured region $0<x<0.015$.
Namely, our analysis of the data suggests less steep small-$x$ behavior of the
$(\bar{d}-\bar{u})\sim x^{-0.2}$ than the MRST and CTEQ5M parametrizations
used by E866 for the determination of the
$\int_{0}^{0.015}(\bar{d}-\bar{u})\,dx$. For a comparison, the more recent
global fit MSTW08, incorporating also the E866 data, assumes the small-$x$
behavior of the $(\bar{d}-\bar{u})\sim x^{0.8}$ and provides
$\int_{0}^{1}(\bar{d}-\bar{u})\,dx=0.09$ which better agrees with our
estimation.

We have also discussed the well-known discrepancy between the NMC and E866
results on $\int_{0}^{1}(\bar{d}-\bar{u})\,dx$. We demonstrated that this
discrepancy can be understood after taking into account the higher-twist
effects which become important in the case of the NMC data with a relatively
low $Q^{2}=4\,{\rm GeV^{2}}$. Using the results obtained for the twist-4
coefficient $H_{2}^{\tau=4}(x)$ for the nonsinglet function $F_{2}^{p-n}(x)$
Alekhin et al. (2012), we found that the HT effects can be responsible for the
difference of $0.025\pm 0.022$ between the two experimental results obtained
at the different $Q^{2}$ scales.

In the last point of our paper, we obtained in the TMM analysis the iso-vector
quark momentum fraction $\langle x\rangle_{u-d}=0.165\pm 0.007$, which agrees
well with the global fit predictions. We compared it also with the recent
lattice result.

Finally, we note that the presented analysis can be directly applied to
studies of the violation of the Callan-Gross relation and the quark-hadron
duality Christy and Melnitchouk (2011).

###### Acknowledgements.

D. K. thanks A. L. Kataev for useful comments. This work is supported by the
Bogoliubov–Infeld Program. D. K. and O. V. T. acknowledge the support of the
Collaboration Program JINR–Bulgaria.

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