#  Proton quark distributions from a light-front Faddeev-Bethe-Salpeter
approach

E. Ydrefors<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  T. Frederico<EMAIL_ADDRESS>Instituto
Tecnológico de Aeronáutica, DCTA, 12228-900 São José dos Campos, Brazil

###### Abstract

The projection onto the Light-Front of a Minkowski space Faddeev-Bethe-
Salpeter equation model truncated at the valence level is applied to study the
proton structure with constituent quarks. The dynamics of the model has built-
in: (i) a bound diquark brought by a contact interaction, and (ii) the
separation by $\sim\Lambda_{QCD}$ of the infrared and ultraviolet interaction
regions. The model parameters are fine tuned to reproduce the proton Dirac
electromagnetic form factor and mass. From that, the non-polarized
longitudinal and transverse momentum distributions were computed. The results
for the evolved non-polarized valence parton distributions suggest that: (i)
the explicit consideration of the spin degree of freedom of both quark and
diquark seems not relevant to it, and (ii) the comparison with the global fit
from the NNPDF4.0 calls for higher Fock components of the wave function beyond
the valence one.

###### keywords:

Faddeev-Bethe-Salpeter equation, Light-Front, proton structure, valence state,
momentum distributions

††journal: Physics Letters B

Introduction. The distinctive characteristic of Quantum Chromodynamics (QCD)
is its ability to dynamically enhance the interaction in the infrared (IR)
region or at long distances. Phenomena like the mass generation of the light
hadrons is outside the known Higgs mechanism, but it is born in the
enhancement of the IR interaction among quarks and gluons, leading to chiral
symmetry breaking and the quark dressing [1, 2, 3, 4]. At the same time gluons
are also dressed and in extreme IR region they behave as massive particles of
about 600 MeV [5] (for a recent discussion of the gluon propagator in the
Landau gauge see [6]). Any successful representation of the QCD dynamics by
means of models has to incorporate the strengthening of the effective
interaction in IR region [7] and dressed degrees of freedom. The interaction
among these dressed quarks is based on the exchange of dressed gluons, which
has a range of about $\sim 0.3$ fm inside the hadron, where the interaction at
larger distances becomes strong, or below momentum of 600 MeV$\sim
2\Lambda_{QCD}$. Therefore, to some extend the constituent quarks have a
short-range interaction at low momentum, which could be parameterized
effectively as a contact one, as happens in the successful phenomenology
provided by Nambu-Jona-Lasinio quark models (see e.g. [8]).

The implementation of effective interactions among the constituent quarks on
the hypersurface $(x^{+}=t+z=0)$ within light-cone quantization [9, 10, 11,
12], allows to find the hadron state as an eigenstate of the light-front (LF)
Hamiltonian. On the other hand such description leads to the partonic picture
of the hadron in the ultraviolet (UV) momentum region. The hadron state
expanded in the LF Fock basis defined with constituent degrees of freedom,
ultimately allows to build the hadron image through the different probability
densities (see e.g. [13]). The associated parton distributions to be studied
in future facilities as the Electron Ion Collider (EIC) [14] and the one in
China (EicC) [15], will provide precise information on the QCD nonperturbative
IR physics.

Among the Fock-components of the hadron wave function, the dominant valence
one can also be obtained from the projection onto the LF of the Minkowski
space Bethe-Salpeter amplitude (see e.g. [16, 17]). Furthermore, the valence
wave function is an eigenstate of the effective LF mass squared operator
reduced to the valence sector, which can be derived from the BS equation using
the quasi-potential approach applied to two bosons [18], two fermions [19] and
to three-particles [20, 21]. The effective interaction contains the infinite
sum over intermediate states in the Fock-space. Alternatively, the eigenvalue
equation for the effective LF mass squared operator can be derived using the
”iterated resolvent method” [9].

In this work, we further study the proton within the framework of the LF
projected Minkowski space Faddeev-Bethe-Salpeter (LF-FBS) equation for three
particles with a contact pairwise interaction [22, 23] (see [24] for the
detailed derivation within the quasi-potential approach). This valence model
has been used to investigate the proton structure with the totally symmetric
momentum component of the valence wave function [25] and applied recently [24]
to study the proton image on the null-plane. The essential dynamical
ingredient is a diquark [26], which is introduced in the model through the
quark-quark amplitude, analogous to formulations of the nucleon with Euclidean
continuum methods [27]. Other recent approaches to the nucleon emphasize
diquark degrees of freedom (see e.g. [28, 29]).

The kernel of the LF-FBS equation applied for the proton is improved to take
into account the separation between the IR and the UV interaction regions in
the three-quark dynamics. This is achieved by introducing a soft cutoff of the
free three-quark LF propagation in states of high virtuality. The deep
unphysical ground state found previously [30] is now naturally removed. In the
present formulation of the LF-FBS model the quark spin degree of freedom is
not considered, as it is our goal to study the spatial non-polarized
distributions of the quarks in the proton valence state, namely, the parton
distribution function (PDF) and transverse momentum distribution (TMD).

LF three-quark model. We consider only the totally symmetric momentum part of
the the colorless three-quark wave function corresponding to the valence
nucleon state, as we are interested for the time being on the investigation of
the properties associated with the momentum distributions and the image of the
nucleon onto the null-plane. In this case, the valence LF wave function is
written as [24]:

$\Psi_{3}(\\{x,\vec{k}_{\perp}\\})=\sum_{i=1}^{3}\frac{\Gamma(x_{i},k_{i\perp})}{\sqrt{x_{1}x_{2}x_{3}}\left(M_{N}^{2}-M^{2}_{0}(\\{x,\vec{k}_{\perp}\\})\right)},$
(1)

$\\{x,\vec{k}_{\perp}\\}\equiv\\{x_{1},\vec{k}_{1\perp},x_{2},\vec{k}_{2\perp},x_{3},\vec{k}_{3\perp}\\}$
with $\Gamma(x_{i},k_{i\perp})$, where $k_{i\perp}=|\vec{k}_{i\perp}|$, being
the Faddeev component of the vertex function for the bound state,
$x_{1}+x_{2}+x_{3}=1$,
$\vec{k}_{1\perp}+\vec{k}_{2\perp}+\vec{k}_{3\perp}=\vec{0}_{\perp}$ and
$M^{2}_{0}(\\{x,\vec{k}_{\perp}\\})=\sum_{i=1}^{3}\frac{\vec{k}_{i\perp}^{2}+m^{2}}{x_{i}}\,,$
is the free three-body squared mass for on-mass-shell constituents. The
factorized form of the valence wave function, namely with a vertex function
depending solely on the bachelor quark LF momenta, is a consequence of the
effective contact interaction between the constituent quarks, which is an
idealized model resembling the successful Nambu-Jona-Lasinio model applied to
model QCD [31]. It should be understood as an effective low-energy model which
is meant to have significance in the IR region where constituent quarks are
massive and bound forming the nucleon.

The bound state homogeneous Faddeev equation for the vertex component of the
nucleon valence LF wave function, with four-point local interaction between
the constituents was described in Refs. [22, 23] and applied to study the
proton in [25, 24]. One physical key ingredient was missing so far in these
previous studies, namely the kernel of the dynamical integral equation has to
take into account the IR enhancement of the QCD interaction between the quarks
[7] and the weakening at large momentum scale. However, our model four-point
local interaction has its action in the UV region, while in IR quarks and
gluons interact strongly according to QCD. In order to represent the physics
of QCD, which undoubtedly distinguish the IR and UV dynamics, we model the
kernel in a way that it is stronger at low momenta and weaker for large ones.
For this aim we introduced a smooth cutoff in the integral equation for the
Faddeev component of the vertex such that:

$\Gamma(x,k_{\perp})=\frac{\mathscr{F}(M^{2}_{12})}{(2\pi)^{3}}\int_{0}^{1-x}\frac{dx^{\prime}}{x^{\prime}(1-x-x^{\prime})}\\\
\times\int_{0}^{\infty}d^{2}k^{\prime}_{\perp}\frac{\Lambda(\widehat{M}^{2}_{0})}{\widehat{M}_{0}^{2}-M_{N}^{2}}\Gamma(x^{\prime},k^{\prime}_{\perp})\,,$
(2)

where $\mathscr{F}(M^{2}_{12})$ is the quark-quark amplitude,
$\widehat{M}_{0}^{2}=M^{2}_{0}(x,\vec{k}_{\perp},x^{\prime},\vec{k}^{\prime}_{\perp},1-x-x^{\prime},-(\vec{k}_{\perp}+\vec{k}^{\prime}_{\perp}))\,.$
Eq. (2) for $\Lambda(\widehat{M}^{2}_{0})=1$ was derived in detail in [24]
resorting to the quasi-potential technique to perform the projection onto the
LF of the three-boson BS equation in Minkowski space. Such LF equation
corresponds to the truncation of the LF Fock-space only at the valence level.
The form factor is introduced to cut the three-quark resolvent at large
virtuality, such that

$\frac{1}{\widehat{M}_{0}^{2}-M_{N}^{2}}-\frac{1}{\widehat{M}^{2}_{0}+\mu^{2}}=\frac{\Lambda(\widehat{M}^{2}_{0})}{\widehat{M}_{0}^{2}-M_{N}^{2}}\,,$
(3)

where
$\Lambda(\widehat{M}^{2}_{0})=(M_{N}^{2}+\mu^{2})/(\widehat{M}^{2}_{0}+\mu^{2})\,,$
which dampens the kernel when $\widehat{M}_{0}^{2}>>\mu^{2}$, without changing
the low momentum region. In the IR region $F(\widehat{M}^{2}_{0})\sim 1$, as
the minimum value of the three-quark free-mass is $3m$, which is about the
nucleon mass. The IR scale of the model is chosen $\mu\sim\Lambda_{QCD}$.
Noteworthy that the IR enhancement of the kernel also should simulate the
relevance of the coupling of the valence component to the higher Fock-
components at large distances. In addition, the choice of the form factor
eliminates the unphysical solution with $M_{N}^{2}<0$ appearing for the bound
diquark case when $\Lambda(\widehat{M}^{2}_{0})=1$ [24], as we are going to
show.

Model |  $m$ [MeV] | $a.m$ | $\mu/m$ |  $M_{dq}$ [MeV]
---|---|---|---|---
(a) | 366 | 2.70 | 1 | 644
(b) | 362 | 3.60 | $\infty$ | 682
(c) | 317 | -1.84 | $\infty$ | -
Table 1: Model parameters: constituent quark mass (2nd column), scattering
length in units of $m^{-1}$ (3rd column), cutoff mass in units of $m$ (4th
column) and diquark mass (5th column). The nucleon mass is 940 MeV.

We observe that the quark-exchange kernel of the LF-FBS equation (2)
regularized at the mass scale $\mu$ can be recognized by the appearance of the
regulated free resolvent given by Eq. (3). The quark-exchange kernel is also
present in the Euclidean four-dimensional three-quark BSE [4] when diquarks
dominate the quark-quark interaction.

The model takes into account the quark-quark amplitude,
$\mathscr{F}(M^{2}_{12})$, which weights the kernel of the LF-FBS equation for
the vertex function, and has the following expression in the limit of an
effective contact interaction between the constituent quarks:

$\mathscr{F}(M^{2}_{12})=\frac{\Theta(-M_{12}^{2})}{\frac{1}{16\pi^{2}y}\log\frac{1+y}{1-y}-\frac{1}{16\pi
ma}}+\frac{\Theta(M_{12}^{2})\,\,\Theta(4m^{2}-M_{12}^{2})}{\frac{1}{8\pi^{2}y^{\prime}}\arctan
y^{\prime}-\frac{1}{16\pi ma}}\,,$ (4)

where the $\Theta(x)$ denotes the Heaviside theta function. and its argument
is the effective off-shell mass of the two-quark subsystem squared, given by

$M^{2}_{12}=(1-x)M^{2}_{N}-\frac{k^{2}_{\perp}+(1-x)m^{2}}{x}\,,\,\,\\\
y^{\prime}=\frac{M_{12}}{\sqrt{4m^{2}-M^{2}_{12}}}\,,\,\,y=\frac{\sqrt{-M^{2}_{12}}}{\sqrt{4m^{2}-M^{2}_{12}}}\,.$
(5)

The scalar diquark is a pole of the quark-quark amplitude, Eq. (4), for
scattering lengths $\pi/(2m)>a>0$ associated with a diquark mass $M_{dq}$,
which is a model parameter. The value of $M_{dq}$ is suggested by the recent
literature [26] to be around 600 MeV. In the case $a$ is negative no physical
two-body bound-state exists and the nucleon is a Borromean state. In both
situations, when $\pi/(2m)>a>0$ and $a<0$, the quark-quark amplitude has a
pole. In the former case the pole appears in the physical complex-energy
sheet, while in the latter in the $2^{\text{nd}}$ one, meaning the virtual
state. The strong diquark correlation is manifested either for bound or
virtual states and it is a consequence of the enhancement of interaction
between the constituent quarks in the IR region [26]. Some choices of
parameters of the model are given in Table 1, with (a) being the new one for
$\mu=m$, (b) and (c) for $\mu=\infty$, which were already studied in [24]. The
choice (a) of model parameters will become clear later on.

Figure 1: Upper panel: Vertex function, $\Gamma(x,k_{\perp})$, in arbitrary
units for model (a). Lower panel: Computed values of the squared three-body
bound state mass, $M^{2}_{3}$, versus the cutoff mass, $\mu$, for $a/m=3.60$
ground (solid line) and first excited state (dot-dashed line), and $a/m=-1.84$
(dashed line). The asymptotic limits of $\mu\to\infty$ correspond to model (b)
(dot-dashed line) and to model (c) (dashed line).

In the upper panel of Fig. 1, the Faddeev component of the vertex,
$\Gamma(x,k_{\perp})$, is shown. The quark mass in model (a) is chosen close
to 350 MeV which is the IR value obtained in a recent LQCD calculation in the
Landau gauge [32]. We note that the diquark mass of $644\,$MeV presents a
difference of 278 MeV with respect to the quark mass, comparable to the gauge
invariant result from the LQCD calculation [33] of 319(1) MeV at the physical
pion mass. For the model (a) with parameters inspired by LQCD results, the
three-quark system has only one bound state, which is identified with the
nucleon. The vertex function is peaked at $x\sim 1/3$ and it spreads out up to
$k_{\perp}\sim\Lambda_{QCD}$, which is about the constituent quark mass and
turns even more reasonable our choice of $\mu$. In particular, model (a)
provides a fit to the Dirac form factor of the proton, as it will be shown.

To be complete, we present in the lower panel of Fig. 1, the computed values
of the squared three-body mass, $M^{2}_{3}$, as a function of the cutoff mass
$\mu$ for the ground and first excited states for $a=3.6/m$, and ground state
in the case of $a=-1.84/m$. By decreasing the value of $\mu$ the unphysical
ground state of the model with $M^{2}_{3}<0$ for $a=3.6/m$ disappears and
turns to a physical one. A similar effect is found for $a=2.7/m$ which for
$\mu=m$ defines the model parametrization (a) given in Table 1.

Figure 2:  $F_{1}(Q^{2})$ for model (a) (solid line), (b) (dot-dashed line)
and (c) (dotted line). The empirical fit (dashed line) obtained in Ref. [34].

Dirac form factor. The present model accounts only for the valence state of
the nucleon wave function, and it allows to obtain the Dirac form factor as
discussed in detail in [24]. Resorting to the Drell-Yan condition where the
plus component of the momentum transfer vanishes $(q^{+}=0)$, the form factor
computed with the valence component of the wave function is given by:

$F_{\text{1}}(Q^{2})=\int\\{dx\,d^{2}k_{\perp}\\}\,\Psi_{3}^{\dagger}(\\{x,\vec{k}^{\text{f}}_{\perp}\\})\Psi_{3}(\\{x,\vec{k}^{\text{i}}_{\perp}\\}),$
(6)

where the phase-space integral is

$\int\\{dx\,d^{2}k_{\perp}\\}=\prod_{i=1}^{2}\int\frac{d^{2}k_{i\perp}}{(2\pi)^{2}}\int^{1}_{0}dx_{i}\,\Theta\big{(}1-x_{1}-x_{2}\big{)}\,,$
(7)

with $\sum_{i=1}^{3}\vec{k}_{i\perp}=0$, $\sum_{i=1}^{3}x_{i}=1$ and
$Q^{2}=\vec{q}_{\perp}\cdot\vec{q}_{\perp}$. Furthermore, choosing the Breit
frame the momenta of the quarks in Eq. (6) are:

$\vec{k}^{\text{f(i)}}_{i\perp}=\vec{k}_{i\perp}\pm\frac{\vec{q}_{\perp}}{2}x_{i}\,\,\,(i=1,2)\quad\text{and}\quad\\\
\vec{k}^{\text{f(i)}}_{3\perp}=\pm\frac{\vec{q}_{\perp}}{2}(x_{3}-1)-\vec{k}_{1\perp}-\vec{k}_{2\perp}\,,$
(8)

with -(+) for f(i).

In Fig. 2, we present the results for the proton Dirac form factor,
$F_{1}(Q^{2})$, with model (a), which was fine tuned to be consistent with the
global fit to experimental data by Ye et al [34] up to 10 GeV2. In addition,
we present the previous results [24] obtained with model (b) having a bound
diquark, and with model (c) presenting a virtual diquark, both calculations
have $\mu=\infty$. The IR dynamics is not privileged by models (b) and (c) and
the calculated Dirac form factor is not able to reproduce the experimental
fit, which is now possible. The enhancement of the interaction kernel of Eq.
(2) in the IR with respect to the UV region, was achieved with the
introduction of the regularization scale $\mu\sim\Lambda_{QCD}$ in Eq. (3),
which finally lead the model to the experimental Dirac form factor.

Figure 3:  Valence PDF computed with model (a): $I_{11}$ (solid thin line),
$I_{22}+I_{33}$ (dashed line), $I_{12}+I_{13}$ (dot-dashed line), $I_{23}$
(dotted line) and the total (solid thick line) normalized to 1. Results for
the total PDF for model (b) (full squares) and (c) (full circles) are
connected by the dashed lines.

Valence quark momentum distribution. The valence PDF should be given by the
sum over all Fock components of the proton light-front wave function, namely:

$\displaystyle f_{1}(x)=$
$\displaystyle\sum_{n=3}^{\infty}\,\Bigg{\\{}\prod_{i}^{n}\int\frac{d^{2}k_{i\perp}}{(2\pi)^{2}}\int^{1}_{0}dx_{i}\Bigg{\\}}\,$
(9)
$\displaystyle\times\delta\left(x-x_{1}\right)\delta\left(1-\sum_{i=1}^{n}x_{i}\right)\,\delta\left(\sum_{i=1}^{n}\vec{k}_{i\perp}\right)$
$\displaystyle\times\big{|}\Psi_{n}(x_{1},\vec{k}_{1\perp},x_{2},\vec{k}_{2\perp},...)\big{|}^{2}\,,$

where $n$ indicates the number of partons in each Fock contribution to the
probability amplitude. However, in the present model, we consider only the
valence component of the proton LF wave function, namely the truncation takes
into account only $n=3$, which presumably is the dominant one. It has been
verified for the pion state [35] where the valence contribution accounts for
70% of the LF wave function described in terms of constituent quark degrees of
freedom.

The results for the valence parton distribution (PDF) at the hadron scale are
shown in Fig. 3 for model (a), (b) and (c). The PDF is obtained from the
integrand of Eq. (6) for the Dirac form factor at $Q^{2}=0$ or equivalently
from Eq. (9) truncated at the valence state:

${f}_{1}(x)=\sum_{3\geq j\geq i\geq 1}(2-\delta_{ij})\,I_{ij}(x)\,,$ (10)

where

$I_{ij}(x)=\int\\{dx\,d^{2}k_{\perp}\\}\frac{\delta(x-x_{1})}{x_{1}x_{2}x_{3}}\frac{\Gamma(x_{i},\vec{k}_{i\perp})\Gamma(x_{j},\vec{k}_{j\perp})}{\big{(}M_{N}^{2}-M^{2}_{0}(\\{x,\vec{k}_{\perp}\\})\big{)}^{2}}\,,$
(11)

with $\sum_{i=1}^{3}\vec{k}_{i\perp}=0$ and $\sum_{i=1}^{3}x_{i}=1$. The
contributions to the PDF indicated in the figure are identified by $I_{ij}$
defined in Eq. (11). Due to the symmetry of the nucleon wave function under
the exchange of quarks 2 and 3, it follows that $I_{22}=I_{33}$ and
$I_{12}=I_{13}$, these relations are taken into account in Eq. (10).
Noteworthy to find that all the contributions have about the same size, and
are peaked around 1/3, despite our choice of quark 1 to obtain the PDF. This
property can be traced back to the denominator appearing in the valence wave
function in Eq. (1), which is the three-quark resolvent and has its maximum
value at the smallest virtuality of the three-quark system. The contribution
from $I_{11}$ corresponds to the situation where the quark 1 is picked up
while the pair of quarks 2 and 3 are in a diquark correlation, which is a
small fraction of the total PDF, showing that the symmetrization of the
valence wave function is relevant for building the proton PDF.

All contributions to the PDF are indeed similar in magnitude, and no one is
dominant, as we have already shown previously in the study of models (b) and
(c) in Ref. [24]. Interesting to observe that models (a) and (b), which both
present a bound diquark and provide results close to the experimental proton
Dirac form factor, as seen in Fig. 2, also have very similar PDF’s. This
suggests that for these observables, form factor and PDF, the formation of the
diquark is a dominant feature, quite independent on the cutoff mass. However,
model (b) presents an unphysical ground state, which is eliminated by the
introduction of the cutoff, as we have discussed.

Figure 4:  Valence $u$ and $d$ quark PDFs evolved to $Q=3.097\,$GeV.
Comparison with DSE [27]. The shaded areas correspond to model (a) for
$Q_{0}=0.33\pm 0.03\,$GeV. NNPDF 4.0 global fit [36] (dotted lines).

Evolved valence PDFs. We present results for model (a) at $Q=3.097\,$GeV for
the $u$ and $d$ valence quarks in Fig. 4. We compare with the recent results
obtained with the Dyson-Schwinger Equation (DSE) approach [27]. In this work
we adopted the same method as the one used in [37] for the evolution of the
pion PDF. Namely, the DGLAP equation with lowest-order splitting function was
used together with the effective charge from [38].

We also took into account an uncertainty with respect to the initial proton
scale: $Q_{0}=0.33\pm 0.03\,$GeV [38], and compared our calculations with the
results from the NNPDF 4.0 global fit111Shown in the Fig. 4 are the central
values of the NNPDF 4.0 global fit available in the PDF database at
https://lhapdf.hepforge.org/pdfsets.html and the data was extracted by using
the LHAPDF software [39]. [36]. The model (a) produces results quite
consistent with the DSE approach, which privileges the strong diquark
correlation, only at large $x$ the discrepancy becomes visible with a softer
behavior from the DSE approach. The present model has a strong UV damping due
to the cutoff, being not asymptotically free at large momentum, which is
associated with a harder behavior at the end-point. Furthermore, it has only
the contribution for the PDF from the valence light-front wave function, which
is strongly peaked around 1/3. It is expected that the higher Fock-components
contribution to the PDF moves the peak towards smaller $x$, as the proton
longitudinal momentum is shared between more than three quarks. Therefore, the
value of $\langle x_{q}\rangle$ should be somewhat decreased, as we observe
from the comparison of model (a) with NNPDF4.0 in Table 2 for the first Mellin
moments.

In the present LF-FBS model the proton has 100% probability to be in the
valence state, as we have considered only $n=3$ in the Fock space
decomposition of Eq. (9). We verified that the sum rule $\langle
x_{u}\rangle+\langle x_{d}\rangle=1$ is saturated at the proton initial scale
by adopting the standard normalization:

$\int_{0}^{1}dx\,f_{u}(x)=2\quad\text{and}\quad\int_{0}^{1}dx\,f_{d}(x)=1\,,$
(12)

where $f_{q}(x)=(2-\delta_{q,d})f_{1}(x)$ with $f_{1}(x)$ normalized to 1 and
obtained from Eq. (10). The assumption of only valence state in the proton at
the initial scale is not fully sustained by the comparison with NNPDF4.0
global fit as shown in Table 2 at $Q=3.097\,$GeV, as the model Mellin moments
$\langle x_{u}\rangle$ and $\langle x_{d}\rangle$ are clearly over estimated.
On the other hand the comparison with the continuum DSE results from Ref.
[27], which takes into account the detailed spin structure of the quarks and
diquarks, suggests that the spin contribution is averaged out in the non-
polarized PDF, as shown in both Fig. 4 and in Table 2 for the first few
moments at $Q=3.097\,$GeV.

Noteworthy, the present model represents the truncation of the FBS equation at
the valence level, and its full representation on the LF has to consider the
contribution of an induced three-body interaction from the coupling of the
valence with higher Fock-states [40, 30]. In principle, this four-dimensional
FB equation model only takes into account the quark sea, and despite of that a
quite relevant effect in the binding energy was found in [30]. Here, with the
introduction of the soft cutoff, this effect would be somewhat reduced.
However, gluons are not taken into account in this model which presumably are
the main degrees of freedom to share the quark longitudinal momentum.

$q$ | $u$ | $d$
---|---|---
Model (a) |  |
$\langle x_{q}\rangle$ | $0.296\pm 0.025$ | $0.148\pm 0.012$
$\langle x_{q}^{2}\rangle$ | $0.071\pm 0.009$ | $0.036\pm 0.005$
$\langle x_{q}^{3}\rangle$ | $0.021\pm 0.004$ | $0.011\pm 0.002$
$\langle x_{q}^{4}\rangle$ | $0.007\pm 0.002$ | $0.004\pm 0.001$
DSE [27] |  |
$\langle x_{q}\rangle$ | $0.303$ | $0.137$
$\langle x_{q}^{2}\rangle$ | $0.077$ | $0.032$
$\langle x_{q}^{3}\rangle$ | $0.032$ | $0.009$
$\langle x_{q}^{4}\rangle$ | $0.010$ | $0.003$
NNPDF4.0 |  |
$\langle x_{q}\rangle$ | $0.261$ | $0.101$
$\langle x_{q}^{2}\rangle$ | $0.072$ | $0.023$
$\langle x_{q}^{3}\rangle$ | $0.027$ | $0.007$
$\langle x_{q}^{4}\rangle$ | $0.012$ | $0.003$
Table 2: Mellin moments, $\langle x^{n}_{q}\rangle$, of the valence quark PDF
$(q=u,d)$ at $Q=3.097\,$GeV for model (a) compared with the DSE [27] and the
NNPDF 4.0 global fit [36], obtained by integrating the PDFs shown in Fig. 4.

Transverse momentum distribution at the proton scale. The single quark
transverse momentum distribution in the forward limit [41] is associated with
the probability density to find a quark with momentum $k_{\perp}$ and $x$,
when truncated to the valence component is:

$\displaystyle\tilde{f}_{1}(k_{\perp},x)$
$\displaystyle=\int_{0}^{1}dx_{1}\delta(x-x_{1})\int\frac{dk_{1\perp}}{(2\pi)^{2}}\delta(k_{\perp}-k_{1\perp})$
(13)
$\displaystyle\times\int_{0}^{2\pi}d\theta_{1}\int\frac{d^{2}k_{2\perp}}{(2\pi)^{2}}\int_{0}^{1-x}dx_{2}\,|\Psi_{3}(\\{x,\vec{k}_{\perp}\\})|^{2}\,,$

where only the dependence on $k_{\perp}=|\vec{k}_{\perp}|$ remains due to the
symmetry of the wave function under rotations in the transverse plane.

The PDF is the integrated TMD on the transverse momentum:

$f_{1}(x)=\int dk_{\perp}k_{\perp}\tilde{f}_{1}(k_{\perp},x)$ (14)

and the integrated TMD in the longitudinal momentum is

$L_{1}(k_{\perp})=k_{\perp}\int_{0}^{1}dx\tilde{f}(k_{\perp},x)\,,$ (15)

which represents the probability density of a single quark with transverse
momentum $k_{\perp}$.

In the upper panel of Fig. 5 the valence TMD given by Eq. (13) is presented
for model (a), and in the lower panel the result for the single quark
transverse momentum density from Eq. (15), for models (a), (b) and (c) are
shown. It is interesting to notice that the momentum scale that dominates the
TMD and the integrated one is about $0.15\,$GeV, deep in the IR region, which
is associated with the proton size in the transverse direction of $\sim
1.3\,$fm. As seen in the lower panel of the figure, the models (a) and (b)
with a bound diquark present quite similar result, and model (c) with the
virtual diquark peaks at considerably lower transverse momentum, reflecting
the lower binding energy (see Table 1). The TMD and the integrated one reflect
and are a source of information on the IR dynamics of QCD, which in part is
reflected in the binding energy of the present model.

Figure 5: Transverse momentum distribution at the proton scale. Upper panel:
transverse momentum distribution, $\tilde{f}_{1}(k_{\perp},x)$ in GeV-2 for
model (a). Lower panel: integrated transverse momentum density vs $k_{\perp}$
for the (a), (b) and (c) models.

Summary. We further develop the LF effective three-quark model for the proton,
based on the notion of the dynamics dominated by the formation of diquarks. We
have disregarded the momentum structure of quark-diquark vertex function,
where the quark-quark amplitude is obtained from a contact interaction. The
improved version of the proton model includes a soft cutoff in the kernel of
the LF Faddeev equation for the vertex function [24]. This soft cutoff allows
to separate the IR and UV interaction regions, with the physical effect of
damping the contributions from three-quark configurations at large
virtualities. This new development eliminated the unphysical ground state for
bound diquarks achieved in previous calculations [30]. The present model was
tuned to reproduce the Dirac proton form factor with a reasonable set of
parameters: a quark mass of $366\,$MeV, a diquark mass of $644\,$MeV and a
cutoff of $366\,$MeV $(\sim\Lambda_{QCD})$, which was enough to give the
proton mass. From this parameter set we explored the proton non-polarized
quark longitudinal and transverse momentum distributions obtained from the
valence wave function. We found that the explicit consideration of the spin
degree of freedom of both quark and diquark is not relevant for the evolved
non-polarized valence PDF. However, the comparison with the global fit from
the NNPDF 4.0 at $Q=3.097\,$GeV suggests that the higher Fock-components,
missed in the model wave function, could be relevant to improve the valence
PDF.

Future challenges for the advance of the present nucleon effective model: the
treatment Bethe-Salpeter amplitude in the four-dimensional Minkowski space
[42, 43], the spin degree of freedom for polarized PDFs, and quark dressing,
which will lead to further insights into the nucleon structure.

This work is a part of the project INCT-FNA #464898/2014-5. This study was
financed in part by Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) under the grant 308486/2015-3 (TF). E.Y. thanks for the
financial support of the grants #2016/25143-7 and #2018/21758-2 from FAPESP.
We thank the FAPESP Thematic grants #2017/05660-0 and #2019/07767-1.

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