# Residual Mean Field Model of Valence Quarks in the Nucleon

Christopher Leon and Misak Sargsian  Department of Physics, Florida
International University, Miami, Florida 33199, USA

###### Abstract

We develop a non-perturbative model for valence parton distribution functions
(PDFs) based on the mean field interactions of valence quarks in the nucleonic
interior. The main motivation for the model is to obtain a mean field
description of the valence quarks as a baseline to study the short range
quark-quark interactions that generate the high $x$ tail of PDFs. The model is
based on the separation of the valence three-quark cluster and residual system
in the nucleon. Then the nucleon structure function is calculated within the
effective light-front diagrammatic approach introducing nonperturbative light-
front valence quark and residual wave functions. Within the model a new
relation is obtained between the position, $x_{p}$, of the peak of $xq_{V}(x)$
distribution of the valence quark and the effective mass of the residual
system, $m_{R}$, in the form: $x_{p}\approx{1\over 4}(1-{m_{R}\over m_{N}})$
at starting $Q^{2}$. This relation explains the difference in the peak
positions for d- and u- quarks through the expected difference of residual
masses for valence d- and u- quark distributions. The parameters of the model
are fixed by fitting the calculated valence quark distributions to the
phenomenological PDFs. This allowed us to estimate the overall mean field
contribution in baryonic and momentum sum rules for valence d- and u- quarks.
Finally, the evaluated parameters of the non-perturbative wave functions of
valence 3q-cluster and residual system can be used in calculation of other
quantities such as nucleon form factors, generalized partonic and transverse
momentum distributions.

##  I Introduction

Understanding the dynamics of the valence quarks in the nucleon is one of the
most interesting aspects of quantum chromodynamics (QCD). The problem is not
only how the valence PDFs can be calculated from the first principles in QCD,
but also how in general the three relativistic Fermions become bound. The
special feature of valence quarks is that they define the baryonic quantum
number of the nucleon and one expects them to be a bridge between baryonic
spectroscopy and the QCD structure of the nucleon. Since baryonic spectroscopy
adheres reasonably well to the SU(6) spin-flavor symmetry, one expects such a
symmetry to be found also in the valence quark distributions in the nucleon
and the problem is to understand how and in which parts of the valence quark
momentum distribution this symmetry is broken.

Historically, the first attempts to model valence quark structure of nucleon
were based on the non-relativistic picture of constituent quarks in which
SU(6) symmetry was preserved for the constituents carrying approximately one
third of the nucleon massIsgur:1979be ; Close:1979bt . These models were
successful in describing the phenomenologies of baryonic spectroscopy.
However, experimental extraction of the valence PDFs indicates that SU(6) is
apparently violated, especially at large Bjorken $x$. Thus, one of the
unresolved issues is how to reconcile the apparent success of SU(6) in
baryonic spectroscopy and its breaking down in partonic distributions.

Another unique property of valence quarks is that their distribution weighted
by Bjorken $x$ has a well defined peak, even if the shape of the PDF is not an
observable. The peaking of the momentum distribution for the bound system
implies the importance of mean-field dynamics for the individual valence
fermions in the nucleon, while the position of the peak is sensitive to the
dynamical aspects of interaction in the mean field

The history of the modeling of partonic distributions is very rich, ranging
from non-relativisticIsgur:1979be  and relativisticBrodsky:1981jv  constituent
quark models, bag modelsJaffe:1974nj ; Chodos:1974pn ; Miller:1979kg , models
combining partonic and pionic cloud picture of the nucleonThomas:1981vc ;
Schreiber:1991qx ; Miller:2002ig  as well as models based on the di-quark
picture of the nucleonClose:1988br ; Anselmino:1992vg ; Roberts:1994dr ;
Cloet:2005pp ; Cloet:2013jya  following from the spin-dependent part of the
one-gluon exchange in the non-relativistic approximationDeRujula:1975qlm . All
these models are non-perturtbative in their nature and their predictions
varied widely for the general characteristics of the valence quark
distribution such as position of the peak and relative strength of the $u$\-
and $d$-quark distributions. While calculations based on lattice QCD reproduce
the general characteristics of valence PDFs, their complexity does not always
allow an understanding of the dominant mechanisms of interactions.

In this and the following papers we develop a new model for valence quark
distributions of the nucleon based on the multi-quark correlation picture. The
validity of such a model is based on the fact that even though the number of
the quarks is not conserved in the nucleon, the number of valence quarks is
effectively conserved and therefore it is possible to describe them in the
framework used for the description of a bound system of finite number of
fermions. This approach is similar to highly successful multi-nucleon
correlation model of calculation of momentum distribution of nucleons in the
nucleiFrankfurt:2008zv ; Egiyan:2005hs ; Frankfurt:1988nt ; Arrington:2011xs ;
Fomin:2017ydn ; Artiles:2016akj  as well as the calculation of the momentum
distribution in ultra-cold atomic Fermi gas with contact interactions (see
e.g. Tan2008 ; Hen:2014lia ). In our approach we consider three distinct
interaction dynamics of valence quarks which are: mean field three valence
quark (3q) cluster, two-quark and three-quark short range interactions, all of
which are expected to dominate at different momentum fraction ranges of the
valence quarks. We demonstrate that such a framework in the description of the
valence quark dynamics brings a new insight and methodology in analyzing the
partonic distributions in the nucleon.

Why modeling? The advantage of the proposed framework is that it creates a new
ground for making predictions for different QCD processes involving nucleons,
since in this case one can make unique predictions based on whether the
process under study is dominated by the interaction of quarks in the mean
field or in two-/three- quark correlations. As experience from nuclear physics
shows, eventually the ab-initio calculationsSchiavilla:2006xx ;
Wiringa:2013ala  (lattice calculation in the case of QCD) will reproduce the
phenomena observed based on the model description of the
processesPiasetzky:2006ai ; Sargsian:2012sm ; Hen:2014nza . However, ab-initio
calculations, because of their general approach, are not always well
positioned for making experimentally verifiable specific predictions.

The article is organized as follows: in Sec.II we first present the
justification for the mean field, two- and three- quark short range
interaction picture of partonic distributions. We then discuss the
phenomenology of partonic distributions, which are almost universal for the
all recent PDFs extracted from the analysis of different high energy
scattering data of electro-production and weak interaction processes (e.g.
deep inelastic scattering and Drell-Yan). Our focus is on the dynamical
characteristics of the $x$ weighted valence PDFs, such as the position and
height of the peaks as well as the ratio of d- to u- quark distributions. We
examine how the the position and the height of the peak changes due to QCD
evolution while the difference between the peak positions for u- and d- quark
distribution is largely $Q^{2}$ independent. Another observation is the
approximate validity of SU(6) symmetry in the region close to the peak of the
partonic distributions. The latter observation is used as a justification for
the assumption of approximate SU(6) symmetry for mean-field valence quarks. In
Sec.III the mean field model of valence quark distributions is presented.
Based on this model we calculate the $xq_{V}(x,Q^{2})$ in leading order (LO),
which is then fitted to the phenomenological distributions (in Sec.IV) to
evaluate the parameters defining the non-perturbative wave functions. The
latter allowed us to ascertain the expected strength of the two- and three-
quark correlation effects in the normalization and momentum sum rule of
valence d- and u- quark distributions. The evaluated parameters of the wave
functions can also be used for estimation of mean field contributions of the
different QCD processes sensitive to the valence quark dynamics at moderate
Bjorken x $\approx 0.2$. Sec.V summarizes our results, discusses the
predictions that follow from the model and the possibility of their
experimental verification. We also discuss the limitations of the model and
outline the second part of the work in which the mechanism of quark-quark
short range interaction is included to calculate the high momentum component
of valence quark distributions. In Appendix A we summarize the Light-Front
effective diagrammatic rules on which the calculation in the paper is based.
In Appendix B we present the mathematical details of the derivation related to
the integration in the transverse momentum space.

##  II The model of valence quark distributions and phenomenology of PDFs

The uniqueness of the valence quarks as carriers of the baryonic quantum
number of nucleons is that even if the number of quarks in the bound nucleon
is not conserved their “effective” number is conserved. This situation allows
us to introduce a concept of potential energy and apply a rather well known
theoretical framework for the description of a bound system consisting of a
finite number of fermions. In this framework the bulk of the momentum
distribution is defined by the mean field dynamics while, if interaction is
short range, the fermion-fermion short-range correlations are responsible for
the high momentum part of the same distribution. As it was mentioned in the
introduction, this approach has been successfully applied in nuclear physics
and physics of ultra-cold atomic Fermi gases.

###  II.1 Mean Field and Quark Correlation Model of Valence Quark
Distributions in the Nucleon

Any finite size, bound system with a fixed number of fermions with nonzero
interaction length will exhibit a mean field interaction of its constituents
Migdal:1967ab . Hereafter, we call a mean field an approximation in which the
interaction of a given constituent with other constituents in the system is
characterized by a single effective potential representing sum of all possible
non-perturbative interactions between constituents. (Such a condition is
similar to one in which Hartree-Fock approximation is justified.) In quantum
mechanics if the binding of a constituent is due to the minimum of the mean
field potential, the latter can be approximated by a harmonic oscillator
potential near the vicinity of the minimum. Thus the momentum distribution of
the constituent in such a mean field can be described in exponential form with
the coefficient of the exponent characterizing the size of the system (or
orbit). The same mean field dynamics define also the characteristic average
momentum carried by the constituents which can be related to the position of
the peak of the momentum weighted distributions. For the system in which
interaction strength between constituents is not negligible at distances much
smaller than the bulk size of the system (like the contact interaction, see
e.g. Hen:2014lia ), the high momentum part of the momentum distributions is a
result of the short range pair-wise interactions between constituents, often
referred to as short-range correlations.

In the present work we apply such a framework for the calculation of the
light-front momentum distribution of valence quarks in the nucleonLeon:2020cev
. The approach is based on the argument, that the “effective” number of
valence quarks is conserved and the strength of the quark-quark interactions
is sufficiently large in the volume they occupy. It is assumed that the volume
the valence quarks occupy is smaller than the actual size of the nucleon. Thus
we arrive at the picture of a non-perturbative valence quark cluster embedded
in the nucleon. We assume the following interaction dynamics for valence
quarks in the nucleon: the non-perturbative interaction among the three
valence quarks provides the confinement (referred hereafter as mean field
interaction), the short range perturbative quark-quark interactions through
hard gluon exchanges generate the high momentum (high Bjorken x) tail of the
valence quark distribution (hereafter, referred to as quark-quark short range
correlations) and the interaction of three-valence quark cluster with the
residual nucleon system, R, influences the final momentum distribution of
valence quarks in the nucleon.

Such a framework in leading order, is represented by the three light-cone
time-ordered diagrams in Fig.1 and it is assumed that they should describe the
major properties of partonic distribution at Bjorken $x>0.1$. For the smaller
x the valence quark distributions are defined predominantly by Regge dynamics
dominated by $\alpha={1\over 2}$ trajectory, which is beyond the scope of this
paper (see e.g. Ref.Roberts:1990ww ).

Figure 1: The mean field (a), two-valence (b) and three-valence (c) quark
short-range correlation contributions to the deep-inelastic scattering off the
nucleon in the partonic picture.

Before proceeding with the calculations within this framework, one can
evaluate the valence quark distributions parametrically to validate the model
at least on a qualitative level. First, the mean-field dynamics is largely
responsible for the bulk properties of the bound three-quark system such as
its size and average momentum of its constituents. The functional form of the
latter can be described by an exponential function with the exponent being
proportional to the spatial extension of the valence quarks.

Figure 2: The graph shows the comparison of the x weighted valence u-quark
distribution fitted by three analytic functions corresponding to the mean
field, 2q- and 3q- correlations.

To identify the range of Bjorken $x$ that one expects to be generated by mean
field dynamics we note that the phenomenology of PDFs is well consistent with
the average transverse momentum of valence quarks being about $\langle
k_{t}\rangle\sim 200$ MeV/cFeynman:1973xc . Using the approximate spherical
symmetry of momentum distribution, we estimate that the Bjorken $x$ range
relevant to the mean-field to be $x\lesssim{2\langle k_{t}\rangle\over
m_{N}}\sim 0.4$. Thus for massless valence quarks one expects an exponential
type distribution for the range of $0.1\lesssim x\lesssim 0.4$.

For the highest end of the Bjorken x range, the phenomenological observation
is that for $x\geq 0.8$ the valence PDF’s behave as $(1-x)^{3}$ which is
consistent with the dynamics of two hard gluon exchanges between three valence
quarksBrodsky:1974vy ; Lepage:1980fj ; Ball:2016spl  . Hereafter we refer them
as three-quark (3q) correlations. The short range nature of such a correlation
will require that all three quarks have momenta exceeding the above mentioned
$\langle k_{t}\rangle$ and the interacting quark balances the other two
spectator quarks. The latter results in $x\gtrsim{4\langle k_{t}\rangle\over
m_{n}}\sim 0.8$.

Establishing the regions where one expects the dominance of mean-field and
3q-correlations our next conjecture is that the transition between these two
regions happens through the two-quark short range interactions which we will
refer to as 2q-correlation. If such two-quark correlations are due to short
range gluon interactions then they are dominated among two quarks with
opposite helicites resulting in a functional form of the partonic distribution
$(1-x/B)^{2}$, where $B$ is a parameter characterizing the total momentum
fraction carried by the center of mass of the 2q-correlated pair.

In Fig.2 we fitted functional forms following from the above discussed
scenarios of mean-field, 2q- and 3q- correlations to the $x$ weighted valence
$u$-quark distribution. As the figure shows the exponential mean field as well
as $(1-{x\over B})^{2}$ and $(1-x)^{3}$ forms for $2q$\- and $3q$\-
correlations reproduce the shape of the valence quark distribution
surprisingly well starting at $x\gtrsim 0.1$. Note that in principle the same
functions should reproduce the $x<0.1$ part of the distributions, however, as
it was mentioned earlier, starting at $x<0.1$ valence quark distributions are
dominated by Regge behavior which is not included in the framework considered
here.

###  II.2 Phenomenological properties of valence quark distributions

One of the most distinguished properties of valence quarks is that while their
distributions do not exhibit any peaking structure, their $x$ weighted
distribution, $h(x,t)\equiv xq_{V}(x,t)$ has a clear peak at around $x\sim
0.2$ which makes them qualitatively different from the sea quark distributions
(Fig. 3). Note that the peaking property is universal for leading and higher
order approximation despite the shape of PDFs not being a physical observable
with certain degree of arbitrariness in their definition. Here we define
$t=\log{Q^{2}}$.

Figure 3: The up and down valence PDFs (left) and x weighted PDFs (right) at
$Q^{2}=4GeV^{2}$. The 68 % confidence level error bands for PDF
parametrization are shown. The PDF sets CJ15nlo and NNPDF3.1-nnlo
Accardi:2016qay ; Ball:2017nwa  are used in the calculation. Similar features
are also observed for other modern PDF parameterizations.

One expects that the $Q^{2}$ dependence of the position and the height of the
peak originate from QCD evolution, in which case with an increase of $Q^{2}$
the gluon radiations of valence quarks shift the peak towards smaller $x$ and
diminishes the height of the maximum. This can be seen in Fig.4 where the
valence $d$\- and $u$\- quarks structure functions are presented for $Q^{2}$
ranging from $5$ to $100$ GeV2.

Figure 4: The $x$ dependence of $xq_{V}(x,Q^{2})$ function for $d-$ and $u-$
valence quarks, calculated for CT14nnlo Dulat:2015mca  PDFs at various
$Q^{2}$.

In the recent work Leon:2020nfb  we found that the correlation between the
height of the maximum of the $h(x_{p},t)$ function and its position, $x_{p}$
has a universal exponential form: $h(x_{p},t)=Ce^{Dx_{p}}$, where constants, C
and D saturate with the increase of the order of approximation in coupling
constant. As it can bee seen from Fig.5 the exponential behavior extends to
practically the entire range of the coupling constant $\alpha$ being covered
in various deep-inelastic processes.

The observed “height-position” correlation results in the following analytic
form of the valence PDFs at the vicinity of $x_{p}$Leon:2020nfb  :

$h(x,t)\equiv xq_{V}(x,t)\approx C+CDex(1-x)^{1-x_{p}(t)\over x_{p}(t)}.$ (1)

It can be checked that for the above relation the $h(x,t)$ function peaks at
$x=x_{p}$. Eq.(1) indicates that the exponent of the $(1-x)$ term,
${1-x_{p}(t)\over x_{p}(t)}$, is defined by the position of the peak $x_{p}$,
which changes continuously with $t$ due to QCD evolution. However as Fig.4
(see also Fig.(7(b) below) shows, already at starting $Q_{0}^{2}=1.69$ GeV2
(before the onset of QCD evolution) the peak-positions $x_{p}$ are different
for valence d and u quarks, resulting in different $x$-dependencies of PDFs at
the vicinity of peaks. The observation that the exponent of $(1-x)$ term is
flavor-dependent indicates a more complex dynamics in the generation of
valence PDFs at $x\sim x_{p}$ than one expects, for example, from the
mechanism of a fixed number gluon exchanges that will result in a same
exponent for valence u- and d- quark PDFs. A flavor-dependent exponent can be
generated by considering an effective nonperturbative potential in Weinberg
type equations for relativistic bound states Weinberg:1966jm .

Figure 5: The peak height and position correlation described by an exponential
function: $h(x_{p},t)=Ce^{Dx_{p}}$, from Ref.Leon:2020nfb . CT14lo and
CT14nnlo corresponds to the CT14 valence PDF parameterization in leading and
next to next to leading order approximation Dulat:2015mca . Other PDF
parameterizations also result in an exponential form of the peak height-
position correlation. The CT14nnlo is the Hessian error at 68% confidence
level, while the CT14lo shows just the central value.

Since the mean-field part of the valence quark wave function will dominate in
the calculations related to the bulk (integrated) characteristics of baryons
(e.g. mass spectrum Isgur:1979be ), then to explain the approximate SU(6)
symmetry of baryonic spectroscopy one should expect the same symmetry to hold
for the mean-field part of the valence PDFs. For SU(6) symmetry one expects
that the ratio of valence $d$\- to $u$\- quark distributions to scale as 0.5.
As Fig.6 shows all phenomenological valence quark distributions produce a
$d_{V}/u_{V}$ ratio close to $0.5$ in the region, $x\sim x_{p}\sim 0.2$, where
one expects the dominance of mean-field dynamics. It is quite intriguing that
some of these distributions such as CJ12Owens:2012bv  and MMHT2014Harland-
Lang:2014zoa  produce an almost constant scaling for $d_{V}/u_{V}$ ratios
close to 0.5 at $x\approx x_{p}$ 111It will be interesting if in future
analyses of the partonic distributions special attention will be given to the
unambiguous verification of the existence of 0.5 scaling in $d_{V}/u_{V}$
ratios in $x\sim x_{p}\sim 0.2$ region.. The figure also shows the decrease of
the $d_{V}/u_{V}$ ratio with an increase of x ($>$0.3), which is consistent
with the increasingly large violation of SU(6) symmetry. One such mechanism
that can result in such a violation is the onset of quark-correlation dynamics
in which case one expects the violation of SU(6) symmetry due to the helicity
selection rule in quark-quark interactions through the hard vector (gluon)
exchanges (see e.g. Ref.Lepage:1980fj ).

Figure 6: The ratio of the $d-$ to $u-$ valence quark distributions for
various PDF sets Accardi:2016qay ; Martin:2009iq ; Dulat:2015mca ;
Owens:2012bv ; Harland-Lang:2014zoa  calculated at the lowest point of
$Q_{0}^{2}=M_{c}^{2}=1.69$ GeV2 for which all the PDFs are defined.

In Fig.7(a) we show the $Q^{2}$ dependence of the $d_{V}/u_{V}$ ratio
estimated at the peak position of the $h(x,t)=x\cdot d_{V}(x,Q^{2})$
distribution. As the figure shows the ratio $\approx 0.5$ is largely $Q^{2}$
independent, which indicates that such a ratio reflects an underlying property
of the quark wave function of the nucleon rather than being due to QCD
evolution. The QCD evolution only slightly modifies this ratio due to the
migration of possible quark correlation effects from large to small x region.

Finally, another phenomenological feature is that for valence d- quarks the
position of the peak of the $h(x,t)$ function is systematically lower than
that of the valence u- quarks (see Fig. 3 (b)). If such a difference in the
peak positions is due to the dynamical property of valence d- and u- quarks in
the nucleon, then one expects that it should be insensitive to QCD evolution
and be largely independent on $Q^{2}$. Such an expectation is justified by the
weak $Q^{2}$ dependence of the ratio of peak-positions of valence u- and
d-quark distributions for CT class of PDF parameterizations shown in Fig.7
(b). The CJ15 parameterization shows $Q^{2}$ independence starting at
$Q^{2}\approx 10$ GeV2 while the MMHT14 PDFs reach $Q^{2}$ independence at
$Q^{2}>10^{3}$ GeV2. Even though the magnitude of the gap between peaks of
$xd_{V}(x,t)$ and $xu_{V}(x,t)$ varies noticeably between different groups of
PDFs its $Q^{2}$ dependence is largely flat, not showing any systematic
$Q^{2}$ dependence.

Figure 7: (a) The $Q^{2}$ dependence of the ratio of d- to u- valence quark
distributions at the peak position of $h(x,t)$ distribution for $d$ quark. (b)
The $Q^{2}$ dependence of the ratio of the peak positions of the d- and u-
valence quark $h(x,t)$ distributions. Different PDF sets are used from
Refs.Accardi:2016qay ; Martin:2009iq ; Dulat:2015mca ; Owens:2012bv ; Harland-
Lang:2014zoa  .

Summarizing this section we would like to emphasize the dedicated study of the
above discussed characteristics of valence quark PDFs, such as exponential
correlation of Fig.5, the approximate $0.5$ scaling of d- to u- valence quark
PDF ratios at $x\sim x_{p}\sim 0.2$ (Figs.6 and 7(a)) as well as the gap
between d- and u- valence quark peaking positions will give new venues in
understanding the valence quark structure of the nucleon. Finally, even though
in above discussion we considered PDF in higher order approximation (nlo and
nnlo), the observed PDF characteristics are largely the same also for leading
order PDFs.

###  II.3 Main assumptions of the model

The above discussed phenomenology of valence PDFs serves us as a ground for
several assumptions for the model outlined below:

  * •

Dynamics: The main assumption of the model is that the mean field, two- and
three- quark short-range correlations define the dynamics of the valence
quarks in the range of $0.1\lesssim x\lesssim 1$. In the considered model,
lepton scattering from the valence quark, in leading order, proceeds according
to the diagrams presented in Fig.1.

  * •

Valence Quarks in the Nucleon: The model assumes an existence of almost
massless three-quark valence cluster, $V$, in the nucleon. The cluster is
compact with the transverse separation between any pairs of quarks
$b_{qq}\lesssim 0.3$ Fm which indicates that the 3q system will occupy a size
of up to 0.6Fm. Such a separation is consistent with both phenomenology and
theoretical evaluations made in all cases in which the 3q core of the nucleon
is considered (see e.g.Brodsky:1985qs ; Cheedket:2002ik ; Weiner:1985ih ;
Islam:2004ke ). In the model the valence quark system defines the baryonic
number, but not necessarily the total isospin of the nucleon. It can have
total isospin $I_{V}={1\over 2}$ or ${3\over 2}$, each of them corresponding
to different excitations or masses of the residual nucleon system. (For the
lowest mass of the recoil system one expects the 3q system to have the same
isospin and its projection that the considered nucleon has).

  * •

Residual Structure: Introducing the residual structure of the nucleon with the
spectrum of mass, $m_{R}$, represents an introduction of spectral function
formalism in the description of the nucleon structure. The model assumes a
certain universality of the residual structure, $R$, entering in all three
mechanisms of generation of valence quark distribution according to Fig.1.
This universality is reflected in the fact that one can fix its main
properties within one of the approaches (say within mean field model) and
apply it in the calculation of $2q$\- and $3q$\- short range correlation
contributions. The mass spectrum of the residual system is continuous and
effectively depends on whether u- or d- valence quarks are probed. For
example, a given valence quark in the proton, can originate from the 3-valence
quark cluster having the total isospin and its projection the same as the
proton but it can also originate from the 3-quark cluster with isospin,
$I_{V}={1\over 2}$ and projection $I_{V}^{3}=-{1\over 2}$ or isospin,
$I_{V}={3\over 2}$, corresponding to the higher mass of the residual system.
Thus in the case of the proton one can expand the residual nucleon mass in the
form:

$\displaystyle m_{R}(u/d)=\alpha_{u/d}\cdot m_{R}(I_{V}={1\over
2},I^{3}_{V}={1\over 2})$ (2) $\displaystyle+\beta_{u/d}\cdot
m_{R}(I_{V}={1\over 2},I^{3}_{V}=-{1\over 2})$
$\displaystyle+\gamma_{(u/d)}\sum\limits_{I^{3}_{V}=-{3\over 2}}^{3\over
2}m_{R}(I_{V}={3\over 2},I^{3}_{V})+\cdots$

where $I^{3}_{V}$ is the isospin projection of the valence quark cluster and
“$\cdots$” accounts for the possible higher mass spectrum of the residual
system. Here the $\alpha$, $\beta$ and $\gamma$ factors are estimating the
probabilities that u- or the d- valence quarks emerge from different isospin
configurations.

From the above representations one expects that the minimal mass in the
residual system comes from the contribution of $I_{V}={1\over
2},I^{3}_{V}={1\over 2}$ term that has the same isospin numbers that the
proton has. As it was mentioned above, for the cases of total isospin
($I_{V}={1\over 2}$, $I_{V}^{3}=-{1\over 2}$) or $I_{V}={3\over 2}$ the
residual system will correspond to higher mass excitations. With this
assumption, we observe that, for the proton having $uud$ configuration, (per
one valence quark) the u- valence quark will be accompanied with lesser
residual mass than that of the d-quark. Thus one expects that for proton in
general:

$m_{R}(u)<m_{R}(d),$ (3)

since for the d- valence quarks one expects relatively larger contribution
from higher residual masses. For example, the second $\beta$ term contributes
more in the d- valence quark case since it has $udd$ configurations. It is
interesting that such a scenario is in qualitative agreement with violation of
Gottfried sum ruleArneodo:1994sh  or the ”SeaQuest” resultDove:2021ejl , since
large $\beta$ factor for d- valence quark will generate an excess of $\bar{d}$
quarks due to $(ddu)(\bar{d}u)$ configuration in the proton.

In the current paper we will estimate the magnitude of $m_{R}$ for u- and d-
quarks by fitting the calculated valence quark distributions to the empirical
PDFs. In principle it could be possible to extract $m_{R}$ experimentally,
with dedicated measurements of tagged structure functions in semi-inclusive
DIS processes.

  * •

Mean Field Model: In the considered model we assume that the mean-field
dynamics (Fig.1(a)) is largely responsible for the position and the peak of
the valence quark structure function, $h(x,t)$. We assume that mean-field
dynamics preserves the SU(6) flavor-spin symmetry of valence quark-cluster for
both the $I_{V}={1\over 2}$ and ${3\over 2}$ cases. Expecting that mean the
field contribution will be relevant for the region of x up to as much as
0.55-0.65 and requiring the produced final mass in DIS is $\geq 2$ GeV, we set
the starting $Q^{2}=4$ GeV2.

  * •

$2q$\- and $3q$\- Correlations: Starting at $x\gtrsim 0.4$ the model assumes
the onset of $2q$\- short range correlations transitioning to $3q$
correlations at $x\gtrsim 0.7$. All such correlations are taken place within
the valence quark cluster. We assume that such short range correlations are
generated by vector exchanges, which in the pQCD limit will correspond to the
hard gluon exchanges. In the calculation of such interactions the finite range
of correlations will be introduced by the momentum transfer cut-off in the
gluon exchanged diagrams. Because of the selection rule, in which gluon
exchanges are dominated for quarks with opposite helicities in the massless
limit, we expect that with the onset of quark correlations the SU(6) symmetry
of the 3q system will break down.

Using the above assumptions in developing the quark correlation model of
valence quark distributions we also bring forward several experimentally
verifiable predictions. These include the prediction for universality of the
recoil system, $R$, for almost entire range of $x>0.1$ as well as the
prediction for the onset of $2q$\- and $3q$\- short range quark correlations
which will be distinguished by specific correlation properties between struck
and recoil valence quarks. These predictions can be quantified once the model
parameters are fixed by comparing the calculated valence quark distributions
with PDFs extracted from DIS measurements.

It is worth mentioning that the presented approach can be applied also for the
studies of QCD processes in the nuclear mediumSargsian:2002wc ; Cosyn:2017ekf
; Cosyn:2010ux , in which case an additional vertex will be added to the
diagrams of Fig.1 describing the transition of nuclear target to the “nucleon-
residual nucleus” intermediate state.

##  III Mean-Field Model of Valence Quark Distributions

Our focus now is on modeling the mean-field dynamics of valence quark
interaction and calculating the valence quark distribution functions that can
be compared with empirical PDFs. The mean field picture that we adopt assumes
that the nucleon core consists of interacting thee-valence quark cluster
embedded in the residual nucleon system consisting of sea quarks, gluons,
pions and other possible hadrons. The main assumptions of the mean field model
are as follows:

  * •

The valence 3q system occupies a space in which the separation of any given
quark pair in the impact parameter space is about $0.3$ Fm. The interactions
among these quarks is described by coupled relativistic three-dimensional
harmonic oscillators, thus satisfying confinement condition.

  * •

Valence quarks are almost massless with the invariant energy of the 3q system
contributing to the nucleon mass.

  * •

The residual system generates an external field for 3q valence system and
occupies a volume less or equal to the nucleon volume.

Figure 8: The nucleon transition into a valence and residual systems. The
diagram is arranged by light-front $\tau=x^{+}=z+t$ ordering. The two dashed
vertical lines represent the two intermediate states where the light-front
wave functions of residual and 3q systems are defined.

To calculate the amplitude of virtual photon interaction with the valence
quark in the above described system we notice that, in the leading order, the
light-front time, $\tau$ evolution of scattering process proceeds according to
the diagram in Fig.8. Here the first vertex (from the left) describes the
interaction of the 3q valence quark-cluster in the field of the residual
system and the second vertex, the interaction among three valence quarks.

The scattering amplitude corresponding to the diagram of Fig.8 can be
calculated by applying effective light-front perturbation theory in which one
introduces effective vertices that can be absorbed into the definition of
light-cone wave functions. The present approach represents the light-front
projection of the covariant amplitude which formally can be calculated within
the effective Feynman diagrammatic approachSargsian:2001ax .

###  III.1 General Considerations: Reference Frame, Kinematics and the Cross
Section of the Reaction.

The calculations here will be done using light-cone (LC) coordinates with the
4-momenta and four-products defined as follows:

$\displaystyle k^{\mu}=(k^{+},k^{-},\mathbf{k}_{\perp}),\ \ k^{\pm}=E\pm
k_{z},\ \ \mathbf{k}_{\perp}=(k_{x},k_{y}),\ \ $ $\displaystyle k_{1}\cdot
k_{2}={k_{1}^{-}k_{2}^{+}\over 2}+{k_{1}^{+}k_{2}^{-}\over 2}-{\bf
k_{1,\perp}\cdot k_{2,\perp}}.$ (4)

We consider a reference frame where the four-momenta of the nucleon,
$p^{\mu}_{N}$ and the virtual photon, $q^{\mu}$ are:

$\displaystyle
p_{N}^{\mu}=(p^{+}_{N},\frac{m_{N}^{2}}{p^{+}_{N}},\mathbf{0}_{\perp}),\ \
q^{\mu}=(0,\frac{2p\cdot q}{p^{+}_{N}},\mathbf{q}_{\perp}),\ \ $
$\displaystyle Q^{2}=-q^{2}=|\mathbf{q}_{\perp}|^{2},$ (5)

where $m_{N}$ is the mass of the nucleon and we use the standard definition of
Bjorken variable: $x_{B}=\frac{Q^{2}}{2p\cdot q}$.

The important kinematical condition of the chosen reference frame is that we
assume:

$p_{N}^{+}\gg m_{N},k_{i}^{-},k_{i,\perp},$ (6)

where $k_{i}^{-}$ and $k_{i,\perp}$ are the “-” and transverse components of
the momenta of constituents in the nucleon.

In calculating the cross section of deep inelastic inclusive scattering from
the nucleon it is convenient to introduce the nucleonic tensor, $W^{\mu\nu}$,
through which in the one-photon exchange approximation the differential cross
section can be presented as follows:

${d\sigma\over dQ^{2}dx}={2\pi\alpha^{2}\over Q^{4}}{y^{2}\over
Q^{2}}m_{N}L_{\mu\nu}W_{N}^{\mu\nu},$ (7)

where $y={p_{N}\cdot q\over p_{N}\cdot k}$ and the spin averaged leptonic
tensor is defined as:

$L_{\mu\nu}=4[(k_{\mu}-{q_{\mu}\over 2})(k_{\nu}-{q_{\nu}\over
2})]+Q^{2}[-g_{\mu\nu}-{q_{\mu}q_{\nu}\over Q^{2}}],$ (8)

where $k_{\mu}$ is the four momentum of initial lepton.

If one introduces the deep-inelastic nuclear transition current as
$J_{N}^{\mu}(p_{X},s_{X},p_{N},s_{N})$ between the initial nucleon and the
final deep inelastic state, X, then the nucleonic tensor can be presented as

$\displaystyle W_{N}^{\mu\nu}$ $\displaystyle={1\over 4\pi
m_{N}}\int\sum\limits_{X}\sum\limits_{s_{X}}J^{\mu,\dagger}(p_{X},s_{X},p_{N},s_{N})$
$\displaystyle\times
J^{\nu}(p_{X},s_{X},p_{N},s_{N})(2\pi)^{4}\delta^{4}(q+p_{N}-p_{X})$
$\displaystyle\times\delta(p_{X}^{2}-M_{X}^{2}){d^{4}p_{X}\over(2\pi)^{3}},$
(9)

where one sums over the all possible finite DIS states, $X$. For the most
general case the nucleonic tensor for unpolarized electron scattering from an
unpolarized target can be expressed through two invariant structure functions

$\displaystyle W_{N}^{\mu\nu}$ $\displaystyle=-{F_{1}(x_{B},Q^{2})\over
m_{N}}(g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^{2}})$
$\displaystyle+\frac{F_{2}(x_{B},Q^{2})}{m_{N}(p_{N}\cdot q)}(p_{N}^{\mu}-\
\frac{p_{N}\cdot q}{q^{2}}q^{\mu})(p_{N}^{\nu}-\frac{p_{N}\cdot
q}{q^{2}}q^{\nu}),$ (10)

where in the reference frame of Eq.(5):

$\displaystyle F_{2}(x,Q^{2})$ $\displaystyle={m_{N}(p_{N}\cdot
q)\over(p_{N}^{+})^{2}}W_{N}^{++}={m_{N}Q^{2}\over
2x(p_{N}^{+})^{2})}W_{N}^{++}$ $\displaystyle F_{1}(x,Q^{2})$
$\displaystyle={F_{2}(x,Q^{2})\over 2x}\left(1+{2m_{N}^{2}x^{2}\over
Q^{2}}\right)$ $\displaystyle-{m_{N}\over 2}W_{N}^{+-},$ (11)

and because of the gauge invariance,

$W_{N}^{+-}={2Q\over q^{-}}W_{N}^{\perp-}.$ (12)

With these structure functions the invariant cross section of the scattering
(7) can be presented as follows:

$\displaystyle{d\sigma\over dQ^{2}dx}={2\pi\alpha^{2}\over
Q^{4}}y^{2}\bigg{(}2F_{1}(x,Q^{2})$ $\displaystyle+{1\over
2xy^{2}}\bigg{[}(2-y)^{2}-y^{2}\bigg{(}1+{4m_{N}^{2}x^{2}\over
Q^{2}}\bigg{)}\bigg{]}F_{2}(x,Q^{2})\bigg{)}.$ (13)

With respect to Eq.(11) it is worth noting that in the limit of
${2m_{N}^{2}x^{2}\over Q^{2}}\ll 1$, if $W^{+-}=0$, one arrives at Callan-
Gross relation:

$F_{2}(x,Q^{2})=2xF_{1}(x,Q^{2}).$ (14)

This, together with Eq.(12) indicates that the Callan-Gross relation is
fulfilled when $W^{\perp-}=0$, which according to Eq.(9) corresponds to the
absence of the transverse currents $J^{\perp}=0$ in the deep-inelastic
scattering process in the chosen reference frame (5).

In the partonic model the structure function $F_{2}(x,Q^{2})$ is directly
related to the partonic distribution functions Feynman:1973xc :

$F_{2}(x,Q^{2})=\sum\limits_{i}e_{i}^{2}xf_{i}(x,Q^{2}),$ (15)

where the $e_{i}$’s are the charges of interacting quarks relative to the
magnitude of electron charge and $f_{i}(x,Q^{2})$ is the PDF of the quark of
type $i$.

From Eqs.(11) and (15) it follows that to calculate the PDFs one needs to
calculate the $W^{++}$ component of the nucleonic tensor, $W^{\mu\nu}$.

###  III.2 Calculation of the Scattering Amplitude

The transition current, $J^{\mu}$ in Eq.(9) in our approach is identified with
the scattering amplitude $A^{\mu}$, which corresponds to the process described
by the diagram of Fig.8. To calculate this amplitude we apply the effective
light-front diagrammatic rules (summarized in Appendix A) which results in:

$\displaystyle
A^{\mu}=\sum_{h_{V},h_{1}}\frac{1}{k_{V}^{+}}\frac{1}{k_{1}^{+}}\bar{u}(k_{1}^{\prime},h_{1}^{\prime})(ie_{1}\gamma^{\mu})u(k_{1},h_{1})$
$\displaystyle\times{\prod_{i=1}^{3}\bar{u}(k_{i},h_{i})\Gamma^{V\rightarrow
3q}\chi_{V}\over\mathcal{D}_{2}}{\chi^{\dagger}_{V}\chi^{\dagger}_{R}\Gamma^{B\rightarrow
VR}u(p_{N},h_{N})\over\mathcal{D}_{1}}$ (16)

where $\Gamma^{N\rightarrow VR}$ is the effective vertex for the nucleon
transition to the valence quark cluster $V$ and residual $R$ systems, the
$\Gamma^{V\rightarrow 3q}$ vertex describes the transition of the state V into
three individual valence quarks. (These vertices are non-perturbative objects
and can not be calculated with any finite order diagrammatic approach. In our
framework they will be associated with respective wave functions which will be
modeled and evaluated by comparing with experiment.) All the $h$’s denote
helicities, the $\chi_{V}$ and $\chi_{R}$ denote the respective spin functions
for the valence and residual systems 222For example, a spinor for a spin 1/2
configuration, a polarization vector for spin 1, etc.. The light front energy
denominators are $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$, which characterize
two intermediate states marked by dashed vertical lines in Fig.8. Here we
label the struck quark with $i=1$. The color indices are omitted since they
are not involved in electromagnetic interaction. The sequence of $N\rightarrow
VR$ followed by $V\rightarrow 3q$ transitions as presented in Fig.8 is
justified based on the assumption that the $VR$ system is characterized by
smaller internal momenta than the $3q$ system. The assumption of small
relative momentum in the $VR$ system allows us to approximate the first
intermediate state as on-shell. For the 3q system such an approximation is not
valid and in principle for the valence quark propagator one will have an on-
shell and instantaneous terms due to relation:
$\not{k}+m=\sum_{h}u(k,h)\bar{u}(k,h)+\frac{\gamma^{+}(k^{2}-m^{2})}{k^{+}}$.
However, since we are interested in the calculation of the $A^{+}$ component
of the scattering amplitude which enters in $W^{++}$, the instantaneous term
drops out since it couples with the extra $\gamma^{+}$ factor in the $\gamma
qq$ vertex and $\gamma^{+}\gamma^{+}=0$. Thus, this allows us to represent the
second intermediate state by on-shell quark spinors only.

Applying the LF diagrammatic rules (Appendix A) for $\mathcal{D}_{1}$ and
$\mathcal{D}_{2}$ denominators one obtains:

$\displaystyle\mathcal{D}_{1}$ $\displaystyle=P^{-}-k_{R}^{-}-k_{V}^{-}$
$\displaystyle=\frac{m_{N}^{2}}{p_{N}^{+}}-\frac{k_{R,\perp}^{2}+m_{R}^{2}}{k_{R}^{+}}-\frac{k_{V,\perp}^{2}+m_{V}^{2}}{k_{V}^{+}})$
$\displaystyle=\frac{1}{p_{N}^{+}}\left(m_{N}^{2}-\frac{k_{R,\perp}^{2}+m_{R}^{2}}{x_{R}}-\frac{k_{V,\perp}^{2}+m_{V}^{2}}{x_{V}}\right)$
$\displaystyle\mathcal{D}_{2}$
$\displaystyle=\frac{1}{k_{V}^{+}}\big{(}m_{V}^{2}-\sum_{i=1}^{3}\frac{k_{i,\perp}^{2}+m_{i}^{2}}{\beta_{i}}),$
(17)

where $k_{R}$ and $k_{V}$ are the four momenta of the residual and valence
systems respectively, $m_{R}$ and $m_{V}$ are their masses and, the summation
is over the valence quarks, $i$, with $\beta_{i}=\frac{x_{i}}{x_{V}}$.

Substituting Eq.(17) into Eq.(16) results in:

$\displaystyle A^{\mu}$
$\displaystyle=\sum_{h_{1},h_{V}}\frac{1}{x_{V}}\frac{1}{\beta_{1}}\bar{u}(k_{1}^{\prime},h_{1}^{\prime})(ie_{1}\gamma^{\mu})u(k_{1},h_{1})$
$\displaystyle\times\frac{\prod_{i=1}^{3}\bar{u}(k_{i},h_{i})\Gamma^{V\rightarrow
3q}\chi_{V}}{m_{V}^{2}-\sum_{i=1}^{3}\frac{k_{i,\perp}^{2}+m_{i}^{2}}{\beta_{i}}}$
$\displaystyle\times\frac{\bar{\chi_{V}}\bar{\chi}_{R}\Gamma^{B\rightarrow
VR}u(p_{N},h_{N})}{m_{N}^{2}-\frac{k_{V,\perp}^{2}+m_{V}^{2}}{x_{V}}-\frac{k_{R,\perp}^{2}+m_{R}^{2}}{x_{R}}}.$
(18)

To proceed, we introduce light-front wave functions for VR and 3q systems as
follows:

$\displaystyle\psi_{VR}(x_{V},\mathbf{k}_{R,\perp},x_{R},\mathbf{k}_{V,\perp})$
$\displaystyle=\frac{\bar{\chi_{V}}\bar{\chi_{R}}\Gamma^{B\rightarrow
VR}u(p_{N},h_{N})}{m_{N}^{2}-\frac{k_{V,\perp}^{2}+m_{V}^{2}}{x_{V}}-\frac{k_{R,\perp}^{2}+m_{R}^{2}}{x_{R}}}$
$\displaystyle\psi_{3q}(\\{\beta_{i},\mathbf{k}_{i,\perp},h_{i}\\}_{i=1}^{3})=\frac{\prod_{i=1}^{3}\bar{u}(k_{i},h_{i})\Gamma^{V\rightarrow
3q}\chi_{V}}{m_{V}^{2}-\sum_{i=1}^{3}\frac{k_{i,\perp}^{2}+m_{i}^{2}}{\beta_{i}}},$
(19)

where $\\{\beta_{i},\mathbf{k}_{i,\perp},h_{i}\\}_{i=1}^{3}$ denotes the LC
momenta and helicities of the three valence quarks in the wave function.

Using the above definitions of LF wave functions the scattering amplitude can
be presented in the following form:

$\displaystyle A^{\mu}$
$\displaystyle=\sum_{h_{1},h_{V}}\bar{u}(k_{1},h_{1})(ie_{1}\gamma^{\mu})u(k_{1},h_{1})$
$\displaystyle\times\frac{\psi_{VR}(x_{V},\mathbf{k}_{R,\perp},x_{R},\mathbf{k}_{V,\perp})}{x_{V}}\frac{\psi_{3q}(\\{\beta_{i},\mathbf{k}_{i,\perp},h_{i}\\}_{i=1}^{3})}{\beta_{1}}.$
(20)

###  III.3 Calculation of the Valence PDF

Figure 9: Hadronic tensor expressed as a cut diagram.

The hadronic tensor, $W_{N}^{\mu\nu}$ of Eq.(9) for the considered scenario of
scattering in which the final state consists of three outgoing valence quarks
and a residual nucleon system, in the leading order, corresponds to the cut
diagram of Fig.9 for which one obtains:

$\displaystyle W_{N}^{\mu\nu}(x,Q^{2})=\frac{1}{4\pi
m_{N}}\sum_{q,h_{i}}\int\delta(k_{R}^{2}-m_{R}^{2})\frac{d^{4}k_{R}}{(2\pi)^{4}}$
$\displaystyle\times\delta(k^{\prime,2}_{1}-m_{1}^{2})\frac{d^{4}k^{\prime}_{1}}{(2\pi)^{4}}\prod_{i=2}^{3}\delta(k_{i}^{2}-m_{i}^{2})\frac{d^{4}k_{i}}{(2\pi)^{4}}$
$\displaystyle\times\delta^{(4)}(P+q-k^{\prime}_{1}+\sum_{i=2}^{3}k_{i}-k_{R})A^{\mu\dagger}A^{\nu},$
(21)

where the sum $\sum\limits_{q,h_{i}}$ is over all the spins/flavors of the
outgoing valence quarks and the residual system. The Lorentz invariant phase
space of all final particles can be presented as follows:

$\displaystyle\delta(k_{j}^{2}-m_{j}^{2})d^{4}k$
$\displaystyle=\frac{1}{2}\delta(k_{j}^{+}k_{j}^{-}-k_{j,\perp}^{2}-m_{j}^{2})dk_{j}^{-}dk_{j}^{+}d^{2}\mathbf{k}_{j,\perp}$
$\displaystyle=\frac{dx_{j}d^{2}\mathbf{k}_{j,\perp}}{2x_{j}}|_{k_{j}^{-}=\frac{k_{j,\perp}^{2}+m_{j}^{2}}{x_{j}p_{N}^{+}}},$
(22)

where $x_{j}=\frac{k_{j}^{+}}{p_{N}^{+}}$ is the light-cone momentum fraction
of outgoing particles, $j=1^{\prime},2,3$ and $R$. To evaluate the energy-
momentum conservation condition, first we express the $\delta^{4}()$ function
through the LC components:

$\displaystyle\delta^{(4)}(p_{N}+q-k^{\prime}_{1}-\sum_{i=2}^{3}k_{i}-k_{R})$
$\displaystyle=2\delta(p_{N}^{+}+q^{+}-k^{\prime,+}_{1}-\sum_{i=2}^{3}k_{i}^{+}-k_{R}^{+})$
$\displaystyle\times\delta(p_{N}^{-}+q^{-}-k^{\prime,-}_{1}-\sum_{i=2}^{3}k_{i}^{-}-k_{R}^{-})$
$\displaystyle\times\delta^{(2)}(\mathbf{p}_{N,\perp}+\mathbf{q}_{\perp}-k^{\prime}_{1,\perp}-\sum_{i=2}^{3}\mathbf{k}_{i,\perp}-\mathbf{k}_{R,\perp}).$
(23)

Using the reference frame of Eq.(5) in which $q^{+}=0$, one observes that
$k^{+}_{1}=k^{\prime,+}_{1}$ and using the definition of LC momentum fractions
$x_{j}=\frac{k_{j}^{+}}{p_{N}^{+}}$ one obtains:

$\delta(p_{N}^{+}+q^{+}-k^{\prime+}_{1}-\sum_{i=2}^{3}k_{i}^{+}-k_{R}^{+})=\frac{1}{p_{N}^{+}}\delta(1-\sum\limits_{i=1}^{3}x_{i}-x_{R}).$
(24)

For the “-” component we use $q^{-}={2p_{N}\cdot q\over p_{N}^{+}}$ (Eq.(5)),
the on-mass shell condition for outgoing particles (see Eq.(22)), and
conditions of high energy scattering of Eq.(6). In the large $Q^{2}$ limit for
which $k^{\prime}_{1\,\perp}=k_{1,\perp}+q_{\perp}\gg k_{1,\perp}$ and
$k^{\prime 2}_{1,\perp}\approx Q^{2}$ one obtains:

$\displaystyle\delta(p_{N}^{-}+q^{-}-k^{\prime-}_{1}-\sum_{i=2}^{3}k_{i}^{-}-k_{R}^{-})$
$\displaystyle=\delta\bigg{(}\frac{m_{N}^{2}}{p_{N}^{+}}-+\frac{2p_{N}\cdot
q}{p_{N}^{+}}-\frac{k^{\prime 2}_{1,\perp}+m^{2}}{x^{\prime}_{1}p_{N}^{+}}$
$\displaystyle-\sum_{i=2}^{3}\frac{k_{i,\perp}^{2}+m^{2}}{x_{i}p_{N}^{+}}-\frac{k_{R,\perp}^{2}+m_{R}^{2}}{x_{R}p_{N}^{+}}\bigg{)}$
$\displaystyle\approx\delta\left(\frac{2p_{N}\cdot
q}{p_{N}^{+}}-\frac{Q^{2}}{x_{1}^{\prime}p_{N}^{+}+}\right)$
$\displaystyle=\frac{x_{1}p_{N}^{+}}{2p_{N}\cdot
q}\delta(x_{1}-x_{B})|_{x_{B}=\frac{Q^{2}}{2p_{N}\cdot q}}.$ (25)

Furthermore, introducing the initial transverse momentum of the struck quark
$\mathbf{k}_{1\perp}=\mathbf{k}^{\prime}_{1,\perp}-\mathbf{q}_{\perp}$ for the
conservation of transverse momentum one obtains:

$\displaystyle\delta^{(2)}(\mathbf{p}_{N,\perp}+\mathbf{q}_{\perp}-{\bf
k}^{\prime}_{1,\perp}-\sum_{i=2}^{3}\mathbf{k}_{i,\perp}-\mathbf{k}_{R,\perp})$
$\displaystyle=\delta(\sum_{i=1}^{3}\mathbf{k}_{i,\perp}+\mathbf{k}_{R,\perp})$
(26)

Combining Eqs.(24,25) and (26) in Eq.(23) and substituting it in Eq.(21) one
obtains:

$W_{N}^{\mu\nu}(x,Q^{2})=\frac{1}{4\pi
m_{N}}\sum_{h_{i},q}\int[dx][d^{2}\mathbf{k}_{\perp}]\delta(x_{1}-x_{B})A^{\mu\dagger}A^{\nu},$
(27)

where the integration factors are defined as:

$[dx]=\delta(1-\sum_{i=1}^{3}x_{i}-x_{R})\frac{dx_{R}}{x_{R}}\prod_{i=1}^{3}\frac{dx_{i}}{x_{i}}$
(28)
$[d^{2}\mathbf{k}_{\perp}]=16\pi^{3}\delta^{(2)}(\sum_{i=1}^{3}\mathbf{k}_{i,\perp}+\mathbf{k}_{R,\perp})\frac{d^{2}\mathbf{k}_{R,\perp}}{16\pi^{3}}\prod_{i=1}^{3}\frac{d^{2}\mathbf{k}_{i,\perp}}{16\pi^{3}}.$
(29)

Substituting now the “+” component of the scattering amplitude from Eq.(20) to
Eq.(27) and using Eq.(11) one obtains for the $F_{2}$ structure function the
following expression:

$\displaystyle
F_{2}(x_{B})=\sum_{q}\sum_{h_{i}}\int[dx][d^{2}\mathbf{k}_{\perp}]e_{q}^{2}x_{1}\delta(x_{1}-x_{B})$
$\displaystyle\times|\psi_{3q}(\\{\beta_{i},\mathbf{k}_{i,\perp},h_{i}\\}_{i=1}^{3})|^{2}|\psi_{V}(x_{V},\mathbf{k}_{R,\perp},x_{R},\mathbf{k}_{V,\perp})|^{2},$
(30)

which together with Eq.(15) allows us to obtain the expression for the valence
PDF in the form:

$\displaystyle
f_{q}(x_{B})=\sum_{h_{i}}\int[dx][d^{2}\mathbf{k}_{\perp}]\delta(x_{1}-x_{B})$
$\displaystyle\times|\psi_{3q}(\\{\beta_{i},\mathbf{k}_{i,\perp},h_{i}\\}_{i=1}^{3})|^{2}|\psi_{V}(x_{V},\mathbf{k}_{R,\perp},x_{R},\mathbf{k}_{V,\perp})|^{2}.$
(31)

###  III.4 Modeling the Wave Functions

####  III.4.1 An Approach for Modeling LF Wave Functions

There have been many approaches in modeling light-front valence quark wave
functions of hadrons (see e.g. Lepage:1980fj ; Brodsky:1981jv ;
Pasquini:2008ax ). Quantum mechanically, for a bound system the expansion of
the potential close to the stability point results in a harmonic oscillator
(HO) type interaction potential, thus justifying the use of the HO
eigenfunctions as a basis for modeling nonperturbative wave function of
hadrons. From a practical point of view, the advantage of using HO wave
functions was that they naturally contain the effect of confinement and the
Gaussian form was convenient for analytic calculations. The HO approach was
used both in constituent (e.g. Ref. Isgur:1979be ) and current (e.g.
Ref.Lepage:1980fj ) quark descriptions of hadrons. For the case of current
quark description, one important issue was a proper relativistic
generalization of the wave functions. For two valence quark systems such as
pions, one straightforward step in the relativistic generalization was
introducing the relative momentum of two valence quarks in the center of mass
frame according to $k^{2}={m_{q}^{2}+k^{2}_{\perp}\over x(1-x)}-m_{q}^{2}$,
which allowed to represent the wave function in the boost-invariant form along
the momentum transfer direction $\bf q$ Lepage:1980fj ; Brodsky:1981jv .

In our approach we again use the HO basis for the LF wave functions. In the
relativistic generalization of such wave functions we introduce the relative
momentum similar as it was defined above for two-valence quark system.
However, we also introduce the phase space factor for the residual (spectator)
system that allows us to recover the non-relativistic normalization of the
wave function in the lab frame where the wave function represents the
eigenstate of the HO Hamiltonian. This phase factor appears also if we relate
Weinberg (or Bethe-Salpeter) type equations in the nonrelativistic limit to
the Lippman-Schwinger equation for bound systems in the lab frame, where the
potential of the interaction is defined (see e.g. Sargsian:2001ax ). For
example, in considering a generic two-body bound system, the HO based LF wave
function is represented in the following form:

$\psi_{2}(\alpha_{1},\mathbf{p}_{1\perp},\alpha_{2},\mathbf{p}_{2\perp})=\sqrt{2(2\pi)^{3}M_{T}}\Psi_{0}(k_{cm})\sqrt{\alpha_{2}},$
(32)

where $\Psi_{0}$ is the ground state wave function of the non-relativistic HO
Hamiltonian in the lab frame of the target with mass $M_{T}$, normalized
conventionally as:

$\int|\Psi_{0}(p)|^{2}d^{3}p=1.$ (33)

In Eq.(32) the residual (spectator) particle “2” is treated as on-shell with
$\alpha_{1}=1-\alpha_{2}$, ${\bf p}_{1,\perp}=-{\bf p}_{2,\perp}$, and
$\alpha_{2}={E^{lab}_{2}+p^{lab}_{2z}\over M_{T}}={E^{cm}_{2}+k^{cm}_{z}\over
E_{2,cm}+E_{1,cm}}$, where

$k_{cm}^{2}=\frac{(s-(m_{1}-m_{2})^{2})(s-(m_{1}+m_{2})^{2})}{4s},$ (34)

with

$s=(p_{1}+p_{2})^{2}=\frac{p_{1,\perp}^{2}+m^{2}}{\alpha_{1}}+\frac{p_{2,\perp}^{2}+m^{2}}{\alpha_{2}}.$
(35)

The extra factor of $\sqrt{\alpha_{2}}$ in Eq.(32) is to account for the phase
space of the residual nucleon that allows to relate the relativistic
normalization of the LF wave function to the nonrelativistic normalization
(33) in the lab frame. Namely starting with the relativistic normalization:

$\int|\psi_{2}(\alpha_{1},\mathbf{p}_{1,\perp},\alpha_{2},\mathbf{p}_{2,\perp})|^{2}\frac{d\alpha_{2}}{\alpha_{2}}\frac{d^{2}p_{2,\perp}}{16\pi^{3}}=1,$
(36)

and using Eq.(32) one obtains:

$\displaystyle
M_{T}16\pi^{3}\int|\Psi_{0}(k_{cm})|^{2}\alpha_{2}\frac{d\alpha_{2}}{\alpha_{2}}\frac{d^{2}p_{2,\perp}}{16\pi^{3}}$
$\displaystyle\approx M_{T}16\pi^{3}\int|\Psi_{0}(k_{cm})|^{2}{dp_{2,z}\over
M_{T}}\frac{d^{2}p_{2,\perp}}{16\pi^{3}}$
$\displaystyle=\int|\Psi(p_{2})|^{2}d^{3}p_{2}=1,$ (37)

where in the above derivation we used the nonrelativistic limit for $k_{cm}\to
p_{2}$ and
$\alpha_{2}=\frac{E_{2}+p_{2,z}}{M_{T}}\approx\frac{m_{2}+p_{2,z}}{M_{T}}$
with $\rightarrow d\alpha_{2}=\frac{dp_{2,z}}{M_{T}}$.

It is worth mentioning that the above described prescription is not unique to
HO wave functions and can be applied to any quantum mechanical wave function.

####  III.4.2 Wave Function of Three Valence Quark System

Figure 10: Three valence quarks coupled in pairwise HO internation.

We model the three valence quark system as mutually coupled oscillators
(Fig.10) for which the ground state wave function within the above described
prescription can be presented in the following form:

$\displaystyle\psi_{3q}(\\{x_{i},\mathbf{k}_{i,\perp}\\}_{i=1}^{3})$
$\displaystyle=16\pi^{3}m_{N}A_{V}\exp{\left[-\frac{B_{V}}{2}(k_{12,cm}^{2}+k_{23,cm}^{2}+k_{31,cm}^{2})\right]}$
$\displaystyle\times\sqrt{x_{2}x_{3}},$ (38)

where $A_{V}$ and $B_{V}$ are parameters and $x_{i},k_{i,\perp}$, ($i\neq
j=1,2,3$) are LC momentum fractions and transverse momenta of each valence
quark in the reference frame defined in Eq.(5). The $k_{ij,cm}^{2}$s, ($i\neq
j=1,2,3$) in the exponent of the wave function represent the relative three
momenta in the CM system of $i,j$ pairs defined as follows:

$k_{ij,cm}^{2}=\frac{(s_{ij}-(m_{i}-m_{j})^{2})(s_{ij}-(m_{i}+m_{j})^{2})}{4s_{ij}},$
(39)

where the invariant energy of the $i,j$ pair is:

$\displaystyle s_{ij}$
$\displaystyle=(k_{i}+k_{j})^{+}(k_{i}+k_{j})^{-}-(\mathbf{k}_{i,\perp}+\mathbf{k}_{j,\perp})^{2}$
$\displaystyle=(x_{i}+x_{j})\left(\frac{k_{i,\perp}^{2}+m_{i}^{2}}{x_{i}}+\frac{k_{j,\perp}^{2}+m_{j}^{2}}{x_{j}}\right)$
$\displaystyle-(\mathbf{k}_{i,\perp}+\mathbf{k}_{j,\perp})^{2}.$ (40)

Since the momentum fractions defined as $x_{i}={k^{+}_{i}\over p^{+}_{N}}$,
one can introduce total momentum fraction carried by three-valence quark
system as:

$x_{V}=x_{1}+x_{2}+x_{3}.$ (41)

Next we introduce the valence quark transverse momenta defined with respect to
the total momentum of 3q system $\bf k_{V}$:

$\mathbf{\tilde{k}}_{i,\perp}=\mathbf{k}_{i,\perp}-\frac{x_{i}}{x_{V}}\mathbf{k}_{V,\perp},\
\ \mbox{(i = 1,2,3)},$ (42)

where

$\tilde{k}_{1,\perp}+\tilde{k}_{2,\perp}+\tilde{k}_{2,\perp}=0.$ (43)

Using Eqs.(42) and (43) and assuming same masses for valence quarks one
obtains:

$s_{12}+s_{23}+s_{31}=\sum_{i=1}^{3}x_{V}\frac{\tilde{k}_{i,\perp}^{2}+m^{2}}{x_{i}}+3m^{2},$
(44)

and

$k_{12,cm}^{2}+k_{23,cm}^{2}+k_{31,cm}^{2}=\frac{1}{4}\big{(}\sum_{i=1}^{3}x_{V}\frac{\tilde{k}_{i,\perp}^{2}+m^{2}}{x_{i}}-9m^{2}\big{)}.$
(45)

Inserting the above relation into Eq.(38) the wave function of 3q- state can
be expressed as follows:

$\displaystyle\psi_{3q}(\\{x_{i},\mathbf{k}_{i,\perp}\\}_{i=1}^{3})$
$\displaystyle=16\pi^{3}m_{N}A_{V}\exp\left[-\frac{B_{V}}{8}\left(\sum_{i=1}^{3}x_{V}\frac{\tilde{k}_{i,\perp}^{2}+m^{2}}{x_{i}}-9m^{2}\right)\right]$
$\displaystyle\times\sqrt{x_{2}x_{3}}.$ (46)

####  III.4.3 Wave Function of Recoil System

Using the same approach (Sec.III.4.1) we model the wave function of the recoil
system in the following form:

$\psi_{R}(x_{R},\mathbf{p}_{R,\perp})=\sqrt{16\pi^{3}m_{N}}A_{R}e^{-\frac{B_{R}}{2}p_{R}^{2}}\sqrt{x_{R}}$
(47)

where $x_{R}$ is the light-cone momentum fraction of the nucleon carried by
the recoil system and $p_{R}^{2}=k_{R,\perp}^{2}+p_{R,z}^{2}$ is its three
momentum in the CM of the nucleon. As a first approximation that allows to
perform calculations analytically, we use a non-relativistic kinematics for
the recoil system. In this case in the CM frame of the nucleon one can
approximate $x_{R}\approx{m_{R}+p_{R,z}\over m_{N}}$ and substitute

$p_{R,z}=(x_{R}m_{N}-m_{R}).$ (48)

In numerical evaluations we will treat $B_{R}$ and $m_{R}$ as free parameters
characterizing the recoil system and evaluate $A_{R}$ from the normalization
condition similar to Eq.(36).

###  III.5 Completing the Calculation of Valence PDF

Substituting the wave functions of 3q (Eq.(46)) and the recoil (Eq.(47))
systems in Eq.(31) for the valence quark partonic distribution one obtains:

$\displaystyle
f_{q}(x_{B})=\left(16\pi^{3}m_{N}\right)^{3}|A|^{2}\int^{Q^{2}}16\pi^{3}\delta(x_{1}-x_{B})$
$\displaystyle\delta(1-\sum\limits_{1}^{3}x_{i}-x_{R})\delta^{2}(\sum\limits_{1}^{3}k_{i,\perp}+k_{R,\perp})$
$\displaystyle\times\exp\bigg{[}-{B_{V}\over
4}\sum_{i=1}^{3}x_{V}\frac{\tilde{k}_{i,\perp}^{2}+m_{i}^{2}}{x_{i}}$
$\displaystyle-
B_{R}\left((m_{N}x_{R}-m_{R})^{2}+k_{R,\perp}^{2}\right)\bigg{]}x_{2}x_{3}x_{R}$
$\displaystyle\times\prod\limits_{i=1}^{3}{dx_{i}\over
x_{i}}{d^{2}k_{i,\perp}\over 16\pi^{3}}{dx_{R}\over
x_{R}}{d^{2}k_{R,\perp}\over 16\pi^{3}},$ (49)

where $A=A_{R}A_{V}e^{{9\over 4}B_{V}m_{q}^{2}}$. Note that the transverse
momenta ${\bf k}_{i,\perp}$ and ${\bf\tilde{k}}_{i,\perp}$ are defined in the
CM of nucleon and 3q- system respectively and related to each other according
to Eq.(42). Using Eq.(42) and relation ${\bf k}_{V,\perp}=-{\bf k}_{R,\perp}$
and $d^{2}\mathbf{k}_{i,\perp}=d^{2}\mathbf{\tilde{k}}_{i,\perp}$ allows us to
factorize integrations by $\tilde{k}_{i,\perp}$ and $k_{R,\perp}$ resulting
in:

$\displaystyle
f_{q}(x_{B},Q^{2})=m_{N}^{3}|A|^{2}\int\prod\limits_{i=1}^{3}{dx_{i}\over
x_{i}}\delta(x_{B}-x_{1})$
$\displaystyle\times\exp\left[-\frac{B_{V}}{4}\sum_{i=1}^{3}x_{V}\frac{m_{i}^{2}}{x_{i}}-B_{R}m_{N}^{2}(x_{R}-\frac{m_{R}}{m_{N}})^{2}\right]$
$\displaystyle\times
x_{2}x_{3}\int^{Q^{2}}\prod\limits_{i=1}^{3}d^{2}\tilde{k}_{i,\perp}\delta^{2}(\sum\limits_{i=1}^{3}{\bf\tilde{k}}_{i,\perp})$
$\displaystyle\times\exp{\left[-\frac{B_{V}}{4}\sum_{i=1}^{3}x_{V}\frac{\tilde{k}_{i,\perp}^{2}}{x_{i}}\right]}\int^{Q^{2}}d^{2}k_{R,\perp}\exp{\left[-B_{R}k_{R,\perp}^{2}\right]}.$
(50)

Furthermore we evaluate the following integrals in the above equation (see
Appendix B):

$\displaystyle\int^{Q^{2}_{R}}\exp{\left[-B_{R}k_{R,\perp}^{2}\right]}d^{2}p_{R,\perp}={\pi\over
B_{R}}(1-e^{-B_{R}Q^{2}_{R,Max}});$
$\displaystyle\int^{Q^{2}}\prod\limits_{i=1}^{3}d^{2}\tilde{k}_{i,\perp}\delta^{2}(\sum\limits_{i=1}^{3}{\bf\tilde{k}}_{i,\perp})\exp{\left[-\frac{B_{V}}{4}\sum_{i=1}^{3}x_{V}\frac{\tilde{k}_{i,\perp}^{2}}{x_{i}}\right]}$
$\displaystyle={16\pi^{2}\over B_{V}^{2}}{x_{1}x_{2}x_{3}\over x_{V}^{3}}$
$\displaystyle\times(1-e^{-a_{cm}Q^{2}_{cm,Max}})(1-e^{-a_{rel}Q^{2}_{rel,Max}}),$
(51)

where $a_{cm}={B_{V}x_{V}\over 4}{x_{V}\over x_{3}(x_{1}+x_{2})}$ and
$a_{rel}={B_{V}x_{V}\over 4}{x_{1}+x_{2}\over x_{1}x_{2}}$. Here
$Q^{2}_{R,Max}$ is the maximal transverse momentum square of the recoil
system. Also in the derivation (see Appendix B) we defined
${\bf\tilde{k}}_{12,\perp}^{cm}={\bf\tilde{k}}_{1,\perp}+{\bf\tilde{k}}_{2,\perp}$
and
${\bf\tilde{k}}_{12,\perp}^{rel}={x_{2}{\bf\tilde{k}}_{1,\perp}-x_{1}{\bf\tilde{k}}_{2,\perp}\over
x_{1}+x_{2}}$ and $Q^{2}_{cm,Max}$ and $Q^{2}_{rel,Max}$ represent the maximal
transverse momentum squares of the center of mass and relative transverse
momenta of “1” and “2” quark system. Inserting the expressions of Eq.(51) into
Eq.(50) and taking the integral over $dx_{1}$ using $\delta(x_{B}-x_{1})$ (see
Appendix B for details) one arrives at:

$\displaystyle
f_{q}(x_{B},Q^{2})=\mathcal{N}\displaystyle\int_{0}^{1-x_{B}}dx_{2}\displaystyle\int_{0}^{1-x_{B}-x_{2}}dx_{3}$
$\displaystyle\times\exp\left[-\frac{B_{V}x_{V}}{4}\sum_{i=1}^{3}\frac{m_{i}^{2}}{x_{i}}-B_{R}M_{N}^{2}(x_{V}-(1-\frac{M_{R}}{M_{N}}))^{2}\right]$
$\displaystyle\times\frac{x_{2}x_{3}}{x_{V}^{3}}\left(1-e^{-a_{cm}Q_{cm}^{max2}}\right)\left(1-e^{-a_{rel}Q_{rel}^{max2}}\right)$
$\displaystyle\times\left(1-e^{-B_{R}Q^{2}}\right),$ (52)

where $x_{1}=x_{B}$ and $x_{V}=x_{B}+x_{2}+x_{3}$ and the normalization
constant

$\mathcal{N}={16\pi^{3}A_{V}^{2}A_{R}^{2}m_{N}^{3}\over
B_{R}B_{V}^{2}}e^{{9\over 4}B_{V}m_{q}^{2}}.$ (53)

The above equation simplifies further when $Q^{2}$ is large so that terms with
it in the exponential are negligible in Eq.III.5:

$\displaystyle
f_{q}(x_{B},Q^{2})=\mathcal{N}\int_{0}^{1-x_{B}}dx_{2}\int_{0}^{1-x_{B}-x_{2}}dx_{3}\frac{x_{2}x_{3}}{x_{V}^{3}}$
$\displaystyle\times\exp\left[-\frac{B_{V}x_{V}}{4}\sum_{i=1}^{3}\frac{m_{i}^{2}}{x_{i}}-B_{R}M_{N}^{2}(x_{V}-(1-\frac{m_{R}}{m_{N}}))^{2}\right]$
(54)

We further simplify this expression by considering the massless limit of
valence quarks and changing the integration variable from $x_{3}$ to $x_{V}$.
This allows us to evaluate the $dx_{2}$ integration analytically resulting in
the final expression for the valence quark distribution in the following form:

$\displaystyle f_{q}(x_{B},Q^{2})={{\cal N}\over
6}\int\limits_{x_{B}}^{1}dx_{V}{(x_{V}-x_{B})^{3}\over x_{V}^{3}}$
$\displaystyle\times\exp\left[-B_{R}m_{N}^{2}\left(x_{V}-(1-\frac{m_{R}}{m_{N}})\right)^{2}\right].$
(55)

Note that as it follows from the above equation, in the massless limit, the
parameter $B_{V}$ enters in the structure function only through the
normalization factor $\cal N$.

###  III.6 Qualitative Features of the Model

One important feature of the model can be observed if we consider Eq.(55) at
$x_{B}\sim x_{p}$, when $x_{V}\sim 1$. In this case the qualitative behavior
of $f_{q}(x_{B},Q^{2})$ can be obtained by evaluating Eq.(55) at the maximum
of the exponent; $x_{V}=1-\frac{m_{R}}{m_{N}}$. This results in
$f_{q}(x_{B},Q^{2})\sim(1-x_{B}-{m_{R}\over m_{N}})^{3}$ and for the valence
quark structure function one obtains:

$h(x_{b},t)=x_{B}f_{q}(x_{B},Q^{2})\sim x_{B}\left(1-x_{B}-{m_{R}\over
m_{N}}\right)^{3}.$ (56)

From the above relation it follows that $h(x,t)$ function peaks at

$x_{p}\approx{1\over 4}(1-{m_{R}\over m_{N}}).$ (57)

Thus one observes that within the considered model the peak of the $h(x,t)$
distribution for valence quarks (see Fig.3(b)) is related to the mass of the
residual system in the nucleon.

Now if we use the characteristic value for the peak position $x_{p}\simeq 0.2$
for valence quarks at starting $Q^{2}=4$ GeV2, from Eq.(57) one observes that
it will correspond to $m_{R}\approx m_{\pi}$. Thus the present model mimics a
pion-cloud like picture for the residual system.

It is interesting that in Ref. Close:1988br  it was observed that within
diquark model the magnitude of $x_{p}$ is related to the mass of the recoil
diquark in the form:

$x_{p}\approx(1-{m_{d}\over m_{N}}),$ (58)

where $m_{d}$ is the mass of the diquark. This and Eq.(58) however are
different, reflecting different dynamics of valence quark generation in the
nucleon.

It is worth mentioning that with an increase of $Q^{2}$, because of the
radiation of valence quarks the magnitude of $x_{V}$ diminishes and due to the
${(x_{V}-x_{B})^{3}\over x_{V}^{3}}$ factor in Eq.(55) the $x_{p}~{}-~{}m_{R}$
relation become more complicated than Eq.(57).

Next we evaluate valence quark PDFs at $x_{B}\rightarrow 1$ limit. For this we
substitute $x_{B}=1-\epsilon$ in Eq.(55) and in the $\epsilon\rightarrow 0$
limit evaluate the integral which results in

$f_{q}(x_{B},Q^{2})\mid_{x_{B}\rightarrow 1}={{\cal N}\over
24}e^{-B_{R}m_{R}^{2}}(1-x_{B})^{4}.$ (59)

This result indicates that the considered mean field model predicts a fall off
of the valence quark PDF faster than the one predicted within perturbative
QCDLepage:1980fj ; $(1-x_{B})^{3}$. Thus the observation of the
$(1-x_{B})^{4}$ behavior at $x_{B}\rightarrow 1$ limit will indicate the
dominance of the mean field dynamics.

##  IV Numerical Estimates of Valence quark distributions

###  IV.1 Strategy of choosing the parameters of the model

We use Eq.(55) as a baseline formula for fitting to the phenomenological PDFs
evaluated in leading order. In general the model has five parameters $A_{V}$
$B_{V}$, $A_{R}$, $B_{R}$ and $m_{R}$.

For the valence quarks we assume that characteristic separations in the 3q
system in the impact parameter space is $\langle b^{2}_{i,j}\rangle\sim$
(0.3Fm)2. This allows us, based on Eq.(46), to evaluate $B_{V}=4\langle
b^{2}_{i,j}\rangle{x_{i}\over x_{V}}\approx{4\over 3}\langle
b^{2}_{i,j}\rangle\approx 3.08$ GeV-2.

For the recoil system we relate $A_{R}=\left({B_{R}\over\pi}\right)^{3\over
4}$. Finally, the remaining parameter $A_{V}$ is fixed through the
normalization factor, ${\cal N}$ and Eq.(53), yielding:

$A_{V}=\sqrt{B_{R}{\cal N}\over 16\pi^{3}m_{N}^{3}}{B_{V}\over A_{R}}$ (60)

In the fitting procedure the parameters ${\cal N}$, $m_{R}$ and $B_{R}$ are
evaluated by fitting Eq.(55) to the height, position and the width of the
$h(x,t)$ distribution in leading order approximation.

Figure 11: The h(x) distribution at $Q^{2}=4$ GeV2. Left panel for valence d-
quark distribution. Right panel for valence u-quark distributions.

As a starting $Q^{2}$ we choose $Q^{2}=4$ GeV2 such that it provides large
enough produced mass $W\geq 2$ GeV in the range of up to $x\sim 0.65$ for
which we expect that mean-field dynamics have non-neggligible contribution.
This $Q^{2}$ is also large enough that it resolves the $3q$ valence cluster in
the nucleon, for which one requires $Q^{2}\gg{1\over B_{V}}$. In Fig.11 we
show the current status of the valence PDFs in LO for d- and u- quarks
evaluated by using modern PDF parameterizationsAccardi:2016qay ; Martin:2009iq
; Dulat:2015mca ; Owens:2012bv ; Harland-Lang:2014zoa  at $Q^{2}=4$ GeV2. As
the figure shows the height and the peak-position of the distributions vary
between different parameterization on the level of 15-20%. Thus by fitting to
these distributions we evaluate the range of parameters as they change from
one to other parameterizations. We fit using maximum likelihood approximation
without estimating the errors of parameters for specific PDF parameterization.
In such way our estimates can be considered qualitative (see also discussion
in Sec.V).

###  IV.2 Fitting results and estimation of the magnitude of high momentum
component of the valence quark distributions

The results for parameters $\cal{N}$, $B_{R}$ and $m_{R}$ for considered PDF
parameterizations are given in Table 1. As these parameters show despite their
variations, there are common features shared by different parameterizations.
The one is that the residual mass for the case of valence $u$-quarks
($0.04\leq m_{R}^{u}\leq 0.07$ GeV) is less than the corresponding mass for
the $d$-quark ($0.17\leq m_{R}^{d}\leq 0.32$ GeV). Additionally,
$m_{R}^{u}\leq m_{pion}$ and $m_{R}^{d}>m_{pion}$. These evaluations predict
that in the case of scattering from valence d- quark in the proton there is a
larger probability to detect recoil pions than in the case of scattering from
the u- quark.

Table 1: Fitting parameters for valence d- and u- quarks. d-quark | Nd |  B${}_{R}^{d}$(GeV-2) |  m${}_{R}^{d}$ (GeV) |  u-quark |  Nu |  B${}_{R}^{u}$ (GeV-2) |  m${}_{R}^{u}$ (GeV)
---|---|---|---|---|---|---|---
CT14LL0 | 64 |  30 |  0.26 |  | 174 |  42 |  0.07
CT14L0 | 63 |  29 |  0.24 |  | 183 | 42 |  0.06
CJ15lo | 47 |  6 |  0.32 |  | 208 |  50 |  0.05
MMHT2014lo | 76 |  50 |  0.16 |  | 228 |  60 |  0.04
MSTW2008lo | 71 |  50 |  0.17 |  | 235 |  65 |  0.04

Another characteristics of the parameters is that the slope factor for the
residual system in the case of scattering from valence u-quarks ($42\leq
B_{R}^{u}\leq 65$ GeV-2) is systematically larger than for the case of
d-quarks ($29\leq B_{R}^{d}\leq 50$ GeV-2) (not included is the CJ15
parametrization for the d-quarks which yields $B_{R}^{d}\leq 6$ GeV-2.). This
indicates a more compact state for the case of scattering from d quarks which
can be checked by measuring the attenuation of the residual system in the
nucleus depending whether scattering took place on valence d- or u- quark.

In Fig.12 we present a comparison of the model with the CT14nnlo PDFs at
$Q^{2}=4$ GeV2333Similar picture is obtained for other parameterizations,
except for the valence $d$-quark distribution in the case of CJ15 which shows
wider distribution with lower position of the peak, As the figure shows, a
better description is achieved for the valence d-quark than for the u-quark
distribution. This indicates that larger high momentum component is needed for
valence u-quarks than for the d-quark.

To evaluate the expected contribution from the high momentum component of
valence quark distribution, we used the fitting parameters to evaluate
normalization and momentum sum rules for both valence d- and u- quarks. For
the normalization one obtains for the d-quark $N_{d}=0.64$ and for the u-quark
$N_{u}=1.37$. It is interesting that these normalizations result in a
${N_{d}\over N_{u}}\sim 0.5$. These results also indicates that one expects
that Regge mechanism at $x<x_{p}$ together with $qq$-correlations to
contribute $\sim$ 36% and $\sim$32% of total normalizations for d- and u-
quarks respectively.

Figure 12: Comparison of the results of the fitting of mean-field model to the
$xq_{V}(x)$ distribution for d- (a) and u- (b) quarks. Data points corresponds
to the CT14nnlo parameterization for central PDF sets and the shaded areas
shows the uncertainties of PDFs estimated through the PDF error sets (for
details see Ref.Dulat:2015mca ).

The evaluation of the momentum sum rule ($P_{q}=\int xq_{V}(x)dx$) yields
$P_{d}=0.095$ and $P_{u}=0.24$ which should be compared with $P_{d}=0.1$ and
$P_{u}=0.268$ obtained from the CT14llo distribution. Expecting that the Regge
mechanism, which dominates at $x<x_{p}$, has a smaller contribution to the
momentum sum rule the above estimates allow us to evaluate better the
contribution from $qq$ correlations. As the mentioned numbers show, the
expected contribution from $qq$ correlations in the momentum sum rule for the
d-quark is $\sim$ 5% and for the u-quark is $\sim$ 10%. These evaluations
however can be somewhat of an underestimation, since in fitting the model to
the PDFs we assumed that all the strength of PDF at $x=x_{p}$ is due to the
mean-field mechanism. The larger contribution for quark- correlation mechanism
may require a renormalization of the mean field distribution. This problem can
be addressed after modeling high-x tail of valence PDFs and combining it with
the current mean-field model. Overall our current result indicates an expected
larger contribution of high momentum tail to the valence u- quark distribution
compared to that of d-quarks.

Another interesting observable is the ratio of the d- to u- quarks at $x\to 1$
limit. As it follows from Eq.(59) the model predicts

${f_{d}(x)\over f_{u}(x)}\mid_{x_{B}\rightarrow 1}={N_{d}\over
N_{u}}e^{\left(B_{R}^{u}(m_{R}^{u})^{2}-B_{R}^{d}(m_{R}^{d})^{2}\right)}.$
(61)

The interesting feature of the model is that the properties of valence PDFs
(the mass of the residual system, the slope factor and normalization constant)
at the vicinity of the peak define the magnitude of the $d/u$ ratio at x=1.
The mechanism that results in the decrease of ${d_{V}\over u_{V}}$ ratio with
an increase of x, is due to the inequality of Eq.(3) because of which the
recoil wave function for the case of the d- quark decreases faster than that
of the u- quark. Thus the value of the ${d_{V}\over u_{V}}$ ratio at $x\to 1$
in the present model is related to the difference in the peak positions for
$h(x,t)$ distribution for d- and u- quarks.

Using parameters of Table 1 one evaluates

$0.06\leq{d_{V}\over u_{V}}\mid_{x\to 1}\leq 0.1(0.14)$ (62)

where $(0.14)$ is the estimate for CJ15 parameterization. This is the
prediction of the current mean-field model if quark-quark correlations will be
found to be negligible at large $x$.

In Fig.13 we present the $x$ dependence of the ${d_{V}\over u_{V}}$ ratio for
the same parameters used in Fig.12.

Figure 13: The $x$ dependence of ${d_{V}\over u_{V}}$ ratio in valence quark
region, solid curve. The calculation is compared with the one following from
CT14nnlo parametrization. The area takes into account the spread of the PDFs
for d- and u- quark distributionsDulat:2015mca .

If, however, the current estimates of $d/u$ ratio from DIS processes off $A=3$
nuclei Abrams:2021xum ; Segarra:2021exb  is confirmed, it will indicate the
need of $qq$ short range interactions to enhance the $d/u$ ratio at $x=1$, to
the values, $\sim 0.2$.

##  V Summary and Outlook

We have developed a model which describes the mean-field dynamics of valence
quarks in the nucleon by separating the nucleon into a valence 3q- cluster and
residual system. Within this model based on effective light-front diagrammatic
approach we calculated the valence quark distributions in the nucleon in
leading order approximation. The parameters entering the model are ones that
describe the wave functions of nonperturbative 3q and residual systems.

###  V.1 Summary of results and predictions of the model

An important result of the model is the prediction of a new relation for the
position of the peak in the $xq_{V}(x)$ distribution and the mass of the
residual system according to Eq.(57) for moderately large $Q^{2}$. This
relation naturally describes the fact that the valence d- quarks peak at lower
$x_{p}$ than u- quarks following from the expectations, that in the proton,
the residual system for the d-quark in general has larger mass than for the
u-quarks (Eq.(• ‣ II.3)).

The model predicts $(1-x_{B})^{4}$ behavior for valence quark PDFs at
$x_{B}\rightarrow 1$, which is stronger than one predicted in pQCD,
$(1-x_{B})^{3}$. This indicates the need of inclusion of 2q and 3q
correlations to describe high x tail of valence PDFs.

To evaluate the parameters of the model we fitted the calculated valence quark
distributions to the several leading order phenomenological PDFs with starting
$Q^{2}=4$ GeV2. The obtained parameters depend on the particular choice of the
phenomenological PDF. However they demonstrate the following common features

  * •

The mass of the residual system for probed valence u- quarks is systematically
lower than that of the valence d-quarks and it is less than the pion mass

  * •

The slope factor of the residual system in the proton is larger for the
u-quarks than for the d- quark indicating that d-quark- residual system is
more compact

  * •

The magnitude of the parameters indicate the need for larger (by factor of
two) high momentum component for the valence u-quarks than for the d-quark.

The above features allow us to make specific predictions that can be checked
experimentally. One is that there should be a correlation between the
multiplicity of pion productions in the target fragmentation region with the
flavor of the knock-out quark in the forward direction. Specifically one
expects more pions to be produced in the target fragmentation region of the
proton if the forward scattering takes place on the valence d-quark. Another
prediction of the model is that for asymmetric nuclei, attenuation of the
residual system is correlated with the flavor of struck quark in the forward
direction. This prediction is based on the observation that the recoil system
is more compact (larger $B_{R}$) in the case of interaction with the valence
d-quark in the proton.

###  V.2 Possible improvements and fitting to beyond the leading order PDFs

To be able to calculate the nucleon structure function analytically we treated
the kinematics of recoil system non-relativistically (Eq. (48)). However the
fitting results of $m_{R}$ indicate the need of relativistic treatment of the
recoil system. This will be done in the process of fitting the model
parameters to the next to leading order PDFs. For the latter case since all
modern NLO PDFs adopt $\overline{MS}$ factorization scheme the relation
between structure function $F_{2}$ and PDF, $f_{i}(x,t)$ is a convolution
integral which covers region above the probed Bjorken x kinematics. Thus,
fitting to the NLO PDFs will require the modeling of the contribution from
$qq$ correlations presented in Fig.1(b) and (c). These calculations are in
progress and will be published in follow-up paper.

Finally, the reliability of the obtained parameters for the 3q-cluster and
residual system wave functions can be checked by calculating the mean-field
effects at $x\sim 0.1-0.4$, in many different quantities in QCD, such as form-
factors, generalized partonic distribution functions or transverse momentum
distributions for different DIS processes.

With the addition of the two- and three- quark short-range interactions to the
present model we hope to have a framework that will allow us to study the
dynamics of the valence quarks in the all range of $x>0.1$. The parameters
obtained in this work, for the wave functions of mean field 3q-cluster (Eqs.
(46)) and residual system (Eq. (47)) will allow us to use them in the
calculation of the correlation diagrams of Fig.(1)(b) and (c). Note that in
this work we did not use polarized PDF’s to fix the parameters of the model
since one of the assumptions was that the mean-field dynamics adheres SU(6)
symmetry. However once two- and three- quark short-range interactions are
added they will break the SU(6) symmetry because of vector exchange nature of
interactions. Then by fitting calculated distributions to polarized PDFs we
will be able to calibrate contribution from short range interaction at large
$x$.

###### Acknowledgements.

This work is supported by United States Department of Energy grant under
contract DE-FG02-01ER41172.

##  Appendix A Light-Front Diagrammatic Rules

The light-front diagrammatic rules are given for the processes ordered in
light-cone time, $\tau=x^{+}=t+z$ with the possibility of an instantaneous
interaction.

  1. 1.

Draw topologically distinct $\tau$-ordered diagrams identifying effective
interaction vertices. In addition to the usual advanced and retarded
propagation between two events one needs to include a possibility in which the
two events connected by an internal fermion or photon lines that interact at
the same LC $\tau-$time, commonly referred as an instantaneous term.

  2. 2.

For each particle line assign four-momentum $p^{\mu}$ and spin $s$ (or
helicity, $\lambda$) with the on-mass-shell conditions $p^{2}=m^{2}$.

  3. 3.

Incoming fermions/anti-fermions get $u(p,h)/\bar{v}(p,h)$ . Outgoing
fermions/anti-fermions have $\bar{u}(p,h)/v(p,h)$.

  4. 4.

Incoming photons or gluons with polarization $\lambda$ are assigned with
polarization vector $\epsilon^{\mu}_{\lambda}(k)$

  5. 5.

For each internal particle line assign a factor $\frac{\theta(p^{+})}{p^{+}}$

  6. 6.

Intermediate states get a light front energy denominator of:

${1\over\mathcal{D}}=\frac{1}{\sum_{init.}p^{-}-\sum_{interm.}p^{-}+i\epsilon}$

where sums go over the initial and intermediate states.

  7. 7.

Assign to the each vertex in the diagram the effective transition factor
$\Gamma$. For bare particle elementary interactions these factors will
correspond to the fundamental vertices with coupling constants.

  8. 8.

For the transition of the composite particle, A with momentum $p_{A}$ and
helicity $h_{A}$ to its n-constituents one introduces LF wave function in the
form:

$\psi(\\{x_{i},k_{i,\perp},h_{i}\\}_{i}^{n})={\prod\limits_{i=1}^{n}\chi_{fi}(x_{i},k_{i,\perp},h_{i})\Gamma\chi_{A}(p_{A},h_{A})\over
p_{A}^{+}\mathcal{D}},$ (63)

where $x_{i}$ and $k_{i,\perp}$ are LC momentum fractions and transverse
momenta of constituent particles with helicity $h_{i}$. The spin wave
functions of outgoing particles are described by
$\chi_{fi}(x_{i},k_{i,\perp},h_{i})$ and $\Gamma$ is the effective vertex of
the transition of particle A to n-constituents.

  9. 9.

Each closed loop is integrated by the LC space factor

$\int\frac{dp^{+}d^{2}\mathbf{p_{\perp}}}{16\pi^{3}}$

##  Appendix B Integrations in the transverse momentum space

Now, if we introduce the relative transverse momentum
$\mathbf{k}_{i,\perp}=\mathbf{\tilde{k}}_{i,\perp}-\frac{x_{i}}{x_{V}}\mathbf{k}_{V,\perp}$
and using the fact that

$\delta^{(2)}(\sum_{i=1}^{3}\mathbf{k}_{i,\perp}+\mathbf{k}_{R,\perp})=\delta^{(2)}(\sum_{i=1}^{3}\mathbf{\tilde{k}}_{i,\perp}),$
(64)

one factorizes the $d^{2}\tilde{k}_{R,\perp}$ and $d^{2}k_{i,\perp}$
integrations resulting in a following expression for the transverse integrals
of Eq.(49) :

$\displaystyle\int^{Q^{2}}[d^{2}\mathbf{k}_{\perp}]\exp\left[-\frac{B_{V}x_{V}}{4}\sum_{i=1}^{3}\frac{k_{i,\perp}^{2}}{x_{i}}-B_{R}k_{R,\perp}^{2}\right]$
$\displaystyle=\int^{Q^{2}}\prod_{i=1}^{3}\frac{d^{2}\mathbf{\tilde{k}}_{i,\perp}}{16\pi^{3}}16\pi^{3}\delta^{(2)}\left(\sum_{i=1}^{3}\mathbf{\tilde{k}}_{i,\perp}\right)$
$\displaystyle\times\exp\left[-\frac{B_{V}x_{V}}{4}\sum_{i=1}^{3}\frac{\tilde{k}_{i,\perp}^{2}}{x_{i}}\right]\int^{Q^{2}}\frac{d^{2}\mathbf{k}_{R,\perp}}{16\pi^{3}}e^{-B_{R}k_{R,\perp}^{2}}.$
(65)

Furthermore, we take the $d^{2}\mathbf{\tilde{k}}_{3,\perp}$ integral using
the $\delta^{2}()$ function and evaluate the $d^{2}\mathbf{k}_{R,\perp}$
integral analytically, arriving at:

$\displaystyle\frac{1}{(16\pi^{3})^{2}}\int^{Q^{2}}d^{2}\mathbf{\tilde{k}}_{1,\perp}d^{2}\mathbf{\tilde{k}}_{2,\perp}$
$\displaystyle\times\exp{\left[-\frac{B_{V}x_{V}}{4}\left(\frac{\tilde{k}_{1,\perp}^{2}}{x_{1}}+\frac{\tilde{k}_{2,\perp}^{2}}{x_{2}}+\frac{(\mathbf{\tilde{k}}_{1,\perp}+\mathbf{\tilde{k}}_{2,\perp})^{2}}{x_{3}}\right)\right]}$
$\displaystyle\times\frac{\pi}{16\pi^{3}}\int_{0}^{Q^{2}}dk_{R,\perp}^{2}e^{-B_{R}k_{R,\perp}^{2}}$
$\displaystyle=\frac{1}{(16\pi^{3})^{2}}\int^{Q^{2}}d^{2}\mathbf{\tilde{k}}_{1,\perp}d^{2}\mathbf{\tilde{k}}_{2,\perp}$
$\displaystyle\times\exp{\left[-\frac{B_{V}x_{V}}{4}\left(\frac{\tilde{k}_{1,\perp}^{2}}{x_{1}}+\frac{\tilde{k}_{2,\perp}^{2}}{x_{2}}+\frac{(\mathbf{\tilde{k}}_{1,\perp}+\mathbf{\tilde{k}}_{2,\perp})^{2}}{x_{3}}\right)\right]}$
$\displaystyle\times\frac{\pi(1-e^{-B_{R}Q^{2}})}{16\pi^{3}B_{R}}.$ (66)

Next we introduce CM and relative transverse momenta for “1” and “2” quarks:

$\displaystyle\mathbf{k}_{12}^{cm}$ $\displaystyle=$
$\displaystyle\mathbf{\tilde{k}}_{1,\perp}+\mathbf{\tilde{k}}_{2,\perp},\quad\mathbf{\tilde{k}}_{12}^{rel}=\frac{x_{2}\mathbf{\tilde{k}}_{1,\perp}-x_{1}\mathbf{\tilde{k}}_{2,\perp}}{x_{1}+x_{2}},$
(67)

which allows to decouple the momentum dependent term in Eq.(66) as follows:

$\displaystyle\frac{k_{1,\perp}^{2}}{x_{1}}+\frac{k_{2,\perp}^{2}}{x_{2}}+\frac{(\mathbf{k}_{1,\perp}+\mathbf{k}_{2,\perp})^{2}}{x_{3}}$
$\displaystyle=(k_{12}^{rel})^{2}\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}\right)+(k_{12}^{cm})^{2}\left(\frac{x_{V}}{x_{3}(x_{1}+x_{2})}\right).$
(68)

Using this and the relation
$d^{2}\mathbf{\tilde{k}}_{1,\perp}d^{2}\mathbf{\tilde{k}}_{2,\perp}=d^{2}\mathbf{k}_{12}^{cm}d^{2}\mathbf{k}_{12}^{rel}$
for Eq.(65) one obtains:

$\displaystyle\int^{Q^{2}}[d^{2}\mathbf{k}_{\perp}]\exp\left[-\frac{B_{V}x_{V}}{4}\sum_{i=1}^{3}\frac{k_{i,\perp}^{2}}{x_{i}}-B_{R}k_{R,\perp}^{2}\right]$
$\displaystyle=\int^{Q^{2}}d^{2}\mathbf{k}_{12}^{cm}d^{2}\mathbf{k}_{12}^{rel}$
$\displaystyle\times\exp\bigg{[}-\frac{B_{V}x_{V}}{4}\bigg{(}(k_{12}^{rel})^{2}\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}\right)$
$\displaystyle+(k_{12}^{cm})^{2}\frac{x_{V}}{x_{3}(x_{1}+x_{2})}\bigg{)}\bigg{]}$
$\displaystyle\times\frac{1}{(16\pi^{3})^{2}}\frac{\pi(1-e^{-B_{R}Q^{2}})}{(16\pi^{3})B_{R}}.$
(69)

Introducing
$a_{rel}=\frac{B_{V}x_{V}}{4}\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}\right)$ and
$a_{cm}=\frac{B_{V}x_{V}}{4}\frac{x_{V}}{x_{3}(x_{1}+x_{2})}$, the remaining
part of the integration in Eq.(69) can be evaluated analytically resulting in:

$\displaystyle\int^{Q^{2}}[d^{2}\mathbf{k}_{\perp}]\exp\big{[}-B_{V}\sum_{i=1}^{3}x_{V}\frac{k_{i,\perp}^{2}}{x_{i}}-B_{R}k_{R,\perp}^{2}\big{]}$
$\displaystyle=\frac{1}{(16\pi^{3})^{2}}\frac{\pi(1-e^{-a_{cm}Q_{cm}^{max2}})}{16\pi^{3}a_{cm}}\frac{\pi(1-e^{-a_{rel}Q_{rel}^{max2}})}{16\pi^{3}a_{rel}}$
$\displaystyle\times\frac{\pi(1-e^{-B_{R}Q^{2}})}{16\pi^{3}B_{R}}$
$\displaystyle=\frac{x_{1}x_{2}x_{3}}{(16\pi^{3})^{4}x_{V}^{3}B_{V}^{2}B_{R}}(1-e^{-a_{cm}Q_{cm}^{max2}})(1-e^{-a_{rel}Q_{rel}^{max2}})$
$\displaystyle\times(1-e^{-B_{R}Q^{2}}),$ (70)

where $Q_{cm}^{max}$ and $Q_{rel}^{max}$ denote the max values of
$k_{12}^{cm}$ and $k_{12}^{rel}$, respectively. Substituting the above
expression into Eq.(49) one obtains:

$\displaystyle f_{q}(x_{B},Q^{2})=\mathcal{N}\int[dx]\delta(x_{B}-x_{1})$
$\displaystyle\times\exp\left[-\frac{B_{V}x_{V}}{4}\sum_{i=1}^{3}\frac{m_{i}^{2}}{x_{i}}-B_{R}M_{N}^{2}(x_{R}-\frac{M_{R}}{M_{N}})^{2}\right]$
$\displaystyle\times\frac{x_{1}x_{2}^{2}x_{3}^{2}x_{R}}{x_{V}^{3}}\left(1-e^{-a_{cm}Q_{cm}^{max2}}\right)\left(1-e^{-a_{rel}Q_{rel}^{max2}}\right)$
$\displaystyle\times\left(1-e^{-B_{R}Q^{2}}\right),$ (71)

where $\mathcal{N}={16\pi^{3}A_{V}^{2}A_{R}^{2}m_{N}^{3}\over
B_{R}B_{V}^{2}}e^{{9\over 4}B_{V}m_{q}^{2}}$. The two remaining Dirac delta
functions can be used to evaluate the $dx_{1}$ and $dx_{R}$ integrals.

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