#  Pion inspired by QCD: Nakanishi and Light-Front Integral Representations

R. M. Moita  Instituto Tecnológico de Aeronáutica, DCTA, 12228-900 São José
dos Campos, Brazil  J. P. B. C. de Melo  Laboratório de Física Teórica e
Computacional - LFTC,
Universidade Cruzeiro do Sul and Universidade Cidade de São Paulo (UNICID)
01506-000 São Paulo, Brazil  T. Frederico  Instituto Tecnológico de
Aeronáutica, DCTA, 12228-900 São José dos Campos, Brazil  W. de Paula
Instituto Tecnológico de Aeronáutica, DCTA, 12228-900 São José dos Campos,
Brazil

###### Abstract

The pion structure in Minkowski space is explored using the Nakanishi integral
representation. A general framework is developed for the pion Bethe-Salpeter
amplitude based on the Källen-Lehmann representation of the dressed quarks
with an ansatz for the pseudo-scalar $\pi-q\bar{q}$ vertex fulfilling the
axial Ward-Takahashi identity. The Nakanishi weight functions are derived for
the scalar amplitudes associated with the decomposition of the pion Bethe-
Salpeter amplitude in different operator bases in the Dirac spinor space in
terms of the involved spectral densities. The approach is applied to an
analytical model of the pion Bethe-Salpeter amplitude, which is combined with
Landau gauge lattice QCD results for the quark running mass at space-like
momentum. From the Nakanishi integral representation several pion observables
were calculated, such as the decay constant, the spin decomposition of the
valence probabilities, longitudinal and transverse momentum distributions from
the valence component of the light-front wave function.

##  I Introduction

The intricate infrared (IR) and non-perturbative physics of Quantum
Chromodynamics (QCD), which is expected to describe the hadron phenomenology
from low to high energies, is under intense research both theoretically and
experimentally. Understanding the richness of the strong interaction,
reflected in the possibility of 3D imaging of hadrons in terms of their quark
and gluon content, is among the motivations to built new electron-ion collider
(see e.g. [1]).

The IR physics of QCD evolves the hadron partonic description in terms of the
fundamental degrees of freedom of quarks and gluons to dressed constituents.
While heavy quarks acquires the bulk of their masses from the coupling to the
Higgs field [2], this mechanism is not enough to give mass to the light
flavors and does not explain the nucleon mass. On the other hand, the strong
IR interaction, as shown by lattice QCD (LQCD calculations dresses both light
quarks [3] and gluons [4]), as also found by continuum Euclidean methods using
truncated Dyson-Schwinger [5, 6, 7]. As a result, at low momentum scales the
light quarks become massive, and consequently, the hadrons composed by them
gain their weight. This mechanism corresponds to the dynamical chiral symmetry
breaking (DCSB), where the pion and kaon are identified as the associated
Goldstone bosons [8, 9].

The infrared non-perturbative dynamics of QCD at long distances, responsible
for the color confinement, is also understood as the origin of DCSB with the
light quarks dressed by the strong interaction. They are understood as
effective degrees of freedom associated with constituent quarks, which are
widely used in the literature for modeling both the spectrum and the structure
of hadrons (see e.g. the review of the quark model in [10]). In particular,
the constituent quark model, formulated on the light-front (LF), is a
practical theoretical approach to study non-perturbative problems in hadron
structure and spectrum [11]. For example, this framework has been used as the
conceptual basis for building a confining LF Hamiltonian model for quarks and
gluons, where the hadron eigenstates are found by diagonalization, resorting
to the basis light-front quantization (BLFQ) method [12, 13]. This approach
has been applied to study light meson parton distribution functions [14] and
generalized parton distributions [15].

Each hadron reflects the full complexity of QCD in Minkowski space, e.g., the
LF wave function consists of an infinite sum of Fock components dynamically
coupled by the interaction [16]. However, Haag’s theorem [17, 18] implies that
the formulation of interacting and free quantum field theories can only be
done on inequivalent Hilbert-space representations of the field algebra, which
from the fundamental point of view could turn questionable the construction of
the Fock-space basis from the free LF QCD Hamiltonian to describe hadron
states. This issue has been recently circumvented in [19] by showing the
equivalence of the scattering theory formulated in instant and light-front
quantum field theories. This study was based on a two-Hilbert space
representation, without demanding the existence of a free dynamics on the
Hilbert space and the adopted one particle basis, which also belongs to the
asymptotic states, is built from the interacting theory. On the other hand,
dressed quarks and gluons are confined and rigorously can not be asymptotic
states of QCD, despite of that this work suggests the possibility of using
dressed degrees of freedom as the building blocks of the hadron LF wave
function.

Following this reasoning, we adopt the phenomenological perspective, where the
dressed quark is considered as the effective degree of freedom, which builds,
in particular, the pion state. Such practical point of view has already been
applied to study light mesons starting from Euclidean frameworks (see e.g.
[7]). Our work add to previous studies the exploration of the Minkowski space
structure of the pion, concomitantly with constraints from DCSB. We mention
that models incorporating DCSB have been indeed explored[20, 21], as in [22]
where the pion Bethe-Salpeter amplitude of a chiral model was compared with
that extracted from lattice QCD calculations. More recently, the pion quasi-
distributions were also obtained [23].

The pion model proposed in Ref. [24], is build from a simple and analytical
quark mass function fitted to LQCD results [3] in the space-like momentum
region (see also [25]). At the same time, given the Goldstone boson nature of
the pion, the scalar part of the quark self-energy provides the pseudo-scalar
component of the quark-pion vertex. This is demanded by the axial-vector Ward-
Takahashi identities in the limit of vanishing current quark mass (see e.g.
[5, 8]). It is important to mention that it is not yet known how the Fock
components of the pion LF wave function, beyond the valence one, are
constructed in terms of dressed quarks and gluons [26] in a way to satisfy
Haag’s theorem. However, we note that in the case of the pion the valence
accounts for 70% of the normalization of the wave function [27, 28, 29] found
in a dynamical model where the quark does not have momentum dependent self-
energy and it is an eigenstate of the LF free Hamiltonian. Therefore, it could
be questionable if such a valence content is maintained once the quark is
dressed.

In the present work, the analytical structure of the BS amplitude of the pion
is further explored, using the Nakanishi integral representation (NIR) [30,
31] considering dressed quarks. An essential aspect needed in our work is
Källén-Lehmann (KL) [32] spectral representation of the quark propagator. Here
we derive the Nakanishi weight functions and make an application to the pion
model of Ref. [24]. Furthermore, we obtain the spin decomposition of the
valence probabilities, the longitudinal and transverse momentum distributions,
and the pion decay constant.

As further observations, we mention that integral representations are useful
to expand the applicability of calculations performed in Euclidean space, such
as those from lattice QCD or from continuous methods which solves the
Schwinger-Dyson and Bethe-Salpeter equations, allowing to construct the Bethe-
Salpeter amplitude in Minkowski space, and with this eventually access
partonic distributions, as already done in Ref. [33] for the pion. The KL
spectral representation has also been used to solve non-perturbatively the
Schwinger-Dyson equations for the fermion self-energy in Minkowski space [34,
35, 36, 37, 38]. In addition, the NIR has already been applied to solve the BS
equation in Minkowski space, pioneered in Refs. [39, 40, 41]. Later on, the
combination with the projection onto the LF [41] was also used to solve the BS
equation for the $0^{+}$ bound state of two fermions [42, 43, 44] and to
calculate the excited states of the Wick-Cutkosky model[45]. Particularly, in
this approach the valence LF wave function corresponds to a Stieltjes
transform [46, 47], which will be extensively used in the present study of the
pion.

This work is organized as follows. In Sect. II the general framework is
presented, with the key ingredients, namely the KL representation of the
scalar and vector parts of the quark propagator, the Nakanishi integral
representation of the $\pi-q\bar{q}$ pseudoscalar vertex and the BS model
amplitude. Its decomposition in an non-orthogonal and orthogonal basis in the
Dirac spinor space is also done, preparing the setup for the Nakanishi
integral representation of the BS amplitude, that is presented in Sect. III.
In that section, we introduce an auxiliary functional, which allows to build
the weight of the Nakanishi integral representation for the scalar functions
associated with the non-orthogonal and orthogonal basis in terms of the
spectral densities. In Sect. IV, in regard to the valence LF wave function, we
present its form in terms of the weight functions, its spin decomposition, the
associated probabilities and decay constant. In Sect. V, we made the
quantitative application of the developed framework to the pion Bethe-Salpeter
amplitude model of Ref. [24] with dressed quarks. We obtain the Nakanishi
weight functions, discuss their properties and the normalization, which now
has the contribution from the running quark mass. In Sect. VI, we present the
results for the valence probability and decay constant. In Sect. VII, we
present the results for valence wave function, its spin components, obtained
from the LF projected amplitudes, and the longitudinal and transverse momentum
distributions. In Sect. VIII, the conclusions of our study are presented. One
appendix is provided with details of the necessary steps for the
transformation of the weight functions between the two basis.

##  II General framework

In this work we calculate pion observables using a model for the Bethe-
Salpeter amplitude (BSA). The dressed fermionic propagators are written in
terms of Källen-Lehman integral representation and the BSA is described in
terms of the Nakanishi integral representation [31].

###  II.1 Key ingredients

Our goal is to describe the pion as a dressed quark-antiquark bound state. The
corresponding Bethe-Salpeter amplitude is given by

$\Psi_{\pi}(k;p)=S_{F}\left(k_{q}\right)\,\Gamma_{\pi}(k;p)\,S_{F}\left(k_{\overline{q}}\right),$
(1)

where the quark and antiquark momenta are $k_{q}=k+\frac{p}{2}$ and
$k_{\overline{q}}=k-\frac{p}{2}$, respectively. In this model, the dressing of
the quarks is also incorporated in the quark propagators, besides the vertex
function which carries all the complexity from QCD.

The pion quark-antiquark pseudo-scalar vertex, denoted here by
$\Gamma_{\pi}(k,p)$, is constrained by the pseudoscalar nature of the pion:

$\Gamma_{\pi}(k,p)=\gamma_{5}[\imath E_{\pi}(k,p)+/p\,F_{\pi}(k,p)\\\
+k^{\mu}p_{\mu}\,/k\,G_{\pi}(k,p)+\sigma_{\mu\nu}k^{\mu}p^{\nu}H_{\pi}(k,p)]\,,$
(2)

where $E_{\pi}(k,p)$, $F_{\pi}(k;p)$, $G_{\pi}(k;p)$ and $H_{\pi}(k,p)$ are
four scalar amplitudes.

The dressed quark propagator is written as [8, 24],

$\displaystyle S_{F}(k)=\imath\,S_{v}(k^{2})\,/k\,+\,\imath\,S_{s}(k^{2})$
$\displaystyle=\imath\left(A(k^{2})\,/k-B(k^{2})\right)^{-1}\,.$ (3)

The self-energies can also be written in terms of the scalar functions
introduced for the propagator, namely:

$\displaystyle
A(k^{2})=\frac{S_{v}(k^{2})}{k^{2}S_{v}^{2}(k^{2})-S_{s}^{2}(k^{2})}\,,$ (4)
$\displaystyle
B(k^{2})=\frac{S_{s}(k^{2})}{k^{2}S_{v}^{2}(k^{2})-S_{s}^{2}(k^{2})}\,.$ (5)

The scalar functions $S_{v}(k)$, and $S_{s}(k)$ can be written in terms of the
Källen-Lehmann spectral decomposition [32],

$\displaystyle
S_{v}(k^{2})=\int_{0}^{\infty}d\mu^{2}\frac{\rho_{v}(\mu^{2})}{k^{2}-\mu^{2}+\imath\varepsilon}\,,$
(6) $\displaystyle
S_{s}(k^{2})=\int_{0}^{\infty}d\mu^{2}\frac{\rho_{s}(\mu^{2})}{k^{2}-\mu^{2}+\imath\varepsilon}\,.$
(7)

Considering the chiral limit, $m_{\pi}=0$, we have for the vertex, Eq. (2),
the expression [24]

$E_{\pi}(k;p)=\imath\,B(k^{2})/f^{0}_{\pi}\,.$ (8)

In addition, one can also write the scalar self-energy $B(k^{2})$ in terms of
an integral representation:

$B(k^{2})=\int_{0}^{\infty}d\mu^{2}\frac{\rho_{B}(\mu^{2})}{k^{2}-\mu^{2}+\imath\varepsilon}~{}.$
(9)

We observe that, in principle by taking the imaginary parts of $A(k^{2})$ and
$B(k^{2})$ in Eqs. (4) and (5) one can obtain the spectral densities for
$A(k^{2})$ and $B(k^{2})$ in terms of $\rho_{s}(\mu^{2})$ and
$\rho_{v}(\mu^{2})$, which for our purposes is not necessary to be detailed.

Considering only the dominant component of the vertex function ($\gamma_{5}$),
the Bethe-Salpeter amplitude has the following form:

$\Psi_{\pi}(k,p)=\gamma_{5}~{}\chi_{1}(k,p)+/\\!\\!\\!{k}_{q}\gamma_{5}~{}\chi_{2}(k,p)\\\
+\gamma_{5}~{}/\\!\\!\\!{k}_{\overline{q}}~{}\chi_{3}(k,p)+/\\!\\!\\!{k}_{q}~{}\gamma_{5}~{}/\\!\\!\\!{k}_{\overline{q}}~{}\chi_{4}(k,p)~{}\,,$
(10)

where the scalar amplitudes $\chi_{i}(k,p)$ are defined in terms of
$S_{v}(k^{2})$, $S_{s}(k^{2})$ and $B(k^{2})$, as explicitly shown in what
follows.

###  II.2  Non-orthogonal basis

The scalar amplitudes in the pion BS amplitude in the non-orthogonal basis
decomposition, Eq. (10), are written in terms of an auxiliary functional of
the spectral densities as:

$\displaystyle\chi_{1}(k,p)=\mathcal{F}(k,p;[\rho_{B},\rho_{s},\rho_{s}])\,,$
$\displaystyle\chi_{2}(k,p)=\mathcal{F}(k,p;[\rho_{B},\rho_{v},\rho_{s}])\,,$
$\displaystyle\chi_{3}(k,p)=\mathcal{F}(k,p;[\rho_{B},\rho_{s},\rho_{v}])\,,$
$\displaystyle\chi_{4}(k,p)=\mathcal{F}(k,p;[\rho_{B},\rho_{v},\rho_{v}])\,,$
(11)

where the functional is defined as

$\displaystyle\mathcal{F}(k,p;[o,f,h])={1\over
f_{\pi}^{0}}\int_{0}^{\infty}d\mu^{\prime\prime 2}\frac{o(\mu^{\prime\prime
2})}{k^{2}-\mu^{\prime\prime 2}+\imath\varepsilon}$
$\displaystyle\times\int_{0}^{\infty}d\mu^{\prime 2}\frac{f(\mu^{\prime
2})}{k_{q}^{2}-\mu^{\prime
2}+\imath\epsilon}\int_{0}^{\infty}d{\mu}^{2}\frac{h({\mu}^{2})}{k_{\overline{q}}^{2}-{\mu}^{2}+\imath\epsilon}\,.$

The symmetry properties of the scalar functions

$\displaystyle\chi_{1}(k,p)=\chi_{1}(-k,p)\,,$
$\displaystyle\chi_{2}(k,p)=\chi_{3}(-k,p)\,,$
$\displaystyle\chi_{4}(k,p)=\chi_{4}(-k,p)\,,$ (13)

are reflected in the weight functions of the Nakanishi integral representation
representing the BSA. In the following we present the decomposition of the BSA
in an orthogonal basis used to study the pion with a dynamical model in
Minkowski space [27].

###  II.3 Orthogonal basis

The Bethe-Salpeter amplitude, $\Psi_{\pi}(k,p)$, can be decomposed as:

$\displaystyle\Psi_{\pi}(k,p)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{4}S_{i}(k,p)\phi_{i}(k,p)\,,$ (14)

where the orthogonal basis for the pion $0^{-}$ states is given by:

$\displaystyle S_{1}(k,p)$ $\displaystyle=$ $\displaystyle\gamma^{5},\quad
S_{2}(k,p)=\dfrac{/p}{M_{\pi}}\gamma^{5},$ $\displaystyle S_{3}(k,p)$
$\displaystyle=$ $\displaystyle\dfrac{k\cdot
p}{M_{\pi}^{3}}~{}/p\gamma^{5}~{}-~{}\dfrac{1}{M_{\pi}}/k\gamma^{5},$
$\displaystyle S_{4}(k,p)$ $\displaystyle=$
$\displaystyle\dfrac{\imath}{M^{2}_{\pi}}\sigma^{\mu\nu}p_{\mu}k_{\nu}\gamma^{5}~{},$
(15)

here,
$\sigma^{\mu\nu}=\frac{\imath}{2}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})$.
The set of operators satisfies the orthogonality condition, given by

$\text{Tr}[S_{i}(k,p)~{}S_{j}(k,p)]~{}=\mathcal{N}_{i}\,\delta_{ij}\,.$ (16)

where

$\displaystyle\mathcal{N}_{1}$ $\displaystyle=$
$\displaystyle-\mathcal{N}_{2}=4\,,$ $\displaystyle\mathcal{N}_{3}$
$\displaystyle=$
$\displaystyle\mathcal{N}_{4}=\frac{4}{M_{\pi}^{4}}\left((k\cdot
p)^{2}-M_{\pi}^{2}\,k^{2}\right)\,.$ (17)

The scalar functions $\phi_{i}(k,p)$ depends on the scalars
$(k^{2},p^{2},k\cdot p)$, having well defined properties under the exchange
$k~{}\rightarrow-k$ due to its the pion fermionic nature, namely they have to
be even for $i=1,2,4$ and odd for $i=3$ (see e.g. [42, 43, 44]). From the
orthogonality conditions, they are written as:

$\phi_{i}(k,p)=\frac{1}{\mathcal{N}_{i}}\,\text{Tr}[S_{i}(k,p)\Psi_{\pi}(k,p)]\,.$
(18)

The scalar amplitudes $\phi_{i}(k,p)$ can be written in terms of
$\chi_{i}(k,p)$ by using Eq. (18) with the BSA written in Eq. (10). The result
is:

$\displaystyle\phi_{1}(k,p)$ $\displaystyle=$
$\displaystyle\chi_{1}(k,p)-\left(k^{2}-\frac{M_{\pi}^{2}}{4}\right)\chi_{4}(k,p)\,,$
(19) $\displaystyle\phi_{2}(k,p)$ $\displaystyle=$
$\displaystyle\left(\frac{M_{\pi}}{2}+\frac{p\cdot
k}{M_{\pi}}\right)\chi_{2}(k,p)$ (20) $\displaystyle+$
$\displaystyle\left(\frac{M_{\pi}}{2}-\frac{p\cdot
k}{M_{\pi}}\right)\chi_{3}(k,p)\,,$ $\displaystyle\phi_{3}(k,p)$
$\displaystyle=$ $\displaystyle M_{\pi}~{}(\chi_{3}(k,p)-\chi_{2}(k,p))\,,$
(21) $\displaystyle\phi_{4}(k,p)$ $\displaystyle=$ $\displaystyle
M^{2}_{\pi}\,\chi_{4}(k,p)\,.$ (22)

Note that the symmetry properties of $\phi_{i}(k,p)$ under the transformation
$k\to-k$ are consistent with the ones fulfilled by $\chi_{i}(k,p)$ expressed
by (13), taking into account the above relations.

##  III Integral Representation

The denominator of a generic Feynman diagram contributing to the fermionic
transition amplitudes has the same expression as in the boson case analyzed by
Nakanishi [31]. Based on this fact, we write our amplitudes following the
Nakanishi integral representation:

$\chi_{i}(k;p)=\int_{-1}^{1}dz\int_{-\infty}^{\infty}d\gamma\frac{G_{i}(\gamma,z)}{[k^{2}+z~{}k\cdot
p-\gamma+\imath\epsilon]^{3}},$ (23)

One can also write the scalar functions $\phi_{i}(k,p)$ with NIR as:

$\phi_{i}(k,p)=\int_{-1}^{1}dz^{\prime}\int_{-\infty}^{\infty}d\gamma^{\prime}\frac{g_{i}(\gamma^{\prime},z^{\prime})}{[k^{2}+z^{\prime}~{}k\cdot
p-\gamma^{\prime}+\imath\epsilon]^{3}}\,.$ (24)

###  III.1 Auxiliary functional

In order to write the scalar functions $\chi_{i}(k,p)$ and $\phi_{i}(k,p)$ in
terms of NIR, we have to express the product of the three denominators in
$\mathcal{F}(k,p;[f,h,o])$ through the integral representation, which is
achieved by resorting to the Feynman parametrization as derived in [24], and
written below:

$\int_{-1}^{1}dz\int_{-\infty}^{\infty}d\gamma\,\frac{\imath\,F(\gamma,z;\mu^{\prime\prime},\mu^{\prime},\mu)}{[k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon]^{3}}=\\\ \frac{1}{[(k+p/2)^{2}-\mu^{\prime
2}+\imath\epsilon][k^{2}-\mu^{\prime\prime 2}+\imath\epsilon]}\\\
\times\frac{1}{[(k-p/2)^{2}-{\mu}^{2}+\imath\epsilon]}\,,$ (25)

where the weight function is given by:

$F(\gamma,z;\,\mu^{\prime\prime},\mu^{\prime},\mu)=\\\
-\frac{2\imath\,\theta(z-2~{}\alpha+1)~{}\theta(\alpha-z)~{}\theta(1-\alpha)~{}\theta(\alpha)}{|\mu^{\prime
2}+\mu^{2}-\frac{M_{\pi}^{2}}{2}-2\mu^{\prime\prime 2}|}\,,$ (26)

with

$\alpha=\frac{\gamma-z(\mu^{\prime\prime
2}-\mu^{2}+\frac{M_{\pi}^{2}}{4})-\mu^{\prime\prime 2}}{\mu^{\prime
2}+\mu^{2}-\frac{M_{\pi}^{2}}{2}-2\mu^{\prime\prime 2}}\,.$ (27)

The support of the function
$F(\gamma,z;\,\mu^{\prime\prime},\mu^{\prime},\mu)$ in $\gamma$ is finite and
constrained by the theta functions for each values of the variables $-1\leq
z\leq 1$, $\mu^{\prime\prime}$, $\mu^{\prime}$ and $\mu$ for a given pion
mass. One can check that $\gamma>0$ when $\mu+\mu^{\prime}>M_{\pi}$,
$\mu^{\prime\prime}+\mu^{\prime}>M_{\pi}$ and
$\mu+\mu^{\prime\prime}>M_{\pi}$. In addition, the function $F$ has the
symmetry property, although not immediately obvious:

$F(\gamma,z;\,\mu^{\prime\prime},\mu^{\prime},\mu)=F(\gamma,-z;\,\mu^{\prime\prime},\mu,\mu^{\prime})\,,$
(28)

which corresponds to the invariance of the product of the three denominators
in the right side of Eq. (26) under the transformation $k\to-k$ and
$\mu^{\prime}\leftrightarrow\mu$, associated with the symmetry properties (13)
of the $\chi_{i}$ amplitudes. Another property of the weight function
$F(\gamma,z;\,\mu^{\prime\prime},\mu^{\prime},\mu)$ is that it strictly
vanishes at the end-points $z=\pm 1$, which is easily checked in Eq. (26), due
to the product of the theta functions.

By using the auxiliary integral representation given in Eq. (26), the
functional $\mathcal{F}$ of Eq. (II.2) is written as a NIR form:

$\mathcal{F}(k,p;[o,f,h])=\\\
=\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma\frac{H(\gamma,z;[o,f,h])}{[k^{2}+z^{\prime}~{}k\cdot
p-\gamma+\imath\epsilon]^{3}}\,,$ (29)

where the weight functional is:

$H(\gamma,z;[o,f,h])=\int d\mu^{\prime\prime 2}d\mu^{\prime
2}d\mu^{2}\,F(\gamma,z;\,\mu^{\prime\prime},\mu^{\prime},\mu)\\\ \times
o(\mu^{\prime\prime 2})\,f(\mu^{\prime 2})\,h(\mu^{2})\,,$ (30)

with the integrals having support from 0 to $\infty$.

###  III.2 Weight functions: non-orthogonal basis

The weight functions of the scalar amplitudes $\chi_{i}$ are:

$\displaystyle G_{1}(\gamma,z)=H(\gamma,z;[\rho_{B},\rho_{s},\rho_{s}])\,,$
(31) $\displaystyle
G_{2}(\gamma,z)=H(\gamma,z;[\rho_{B},\rho_{v},\rho_{s}])\,,$ (32)
$\displaystyle G_{3}(\gamma,z)=H(\gamma,z;[\rho_{B},\rho_{s},\rho_{v}])\,,$
(33) $\displaystyle
G_{4}(\gamma,z)=H(\gamma,z;[\rho_{B},\rho_{v},\rho_{v}])\,,$ (34)

and one immediately have the following symmetry properties for the weight
functions taking into account Eq. (28), namely:

$\displaystyle G_{1}(\gamma,z)=G_{1}(\gamma,-z)\,,$ (35) $\displaystyle
G_{2}(\gamma,z)=G_{3}(\gamma,-z)\,,$ (36) $\displaystyle
G_{4}(\gamma,z)=G_{4}(\gamma,-z)\,,$ (37)

which also implies in the symmetry properties expressed in Eq. (13) for the
scalar amplitudes.

###  III.3 Weight functions: orthogonal basis

Using the previous results for the functions $\phi_{1}(k,P)$ written in terms
of the $\chi$’s with Eq. (19) and the NIR from Eq. (23), we get:

$\phi_{1}(k,p)=\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma\,\Bigg{\\{}\frac{G_{1}(\gamma,z)}{\left[k^{2}+z\,k\cdot
p-\gamma^{\prime}+\imath\epsilon\right]^{3}}\\\
+\frac{\left(\frac{M_{\pi}^{2}}{4}-k^{2}\right)G_{4}(\gamma,z)}{\left[k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon\right]^{3}}\Bigg{\\}}\,.$ (38)

The final expression for the weight function is found after the $k^{2}$ factor
is eliminated as shown in the Appendix A, and resorting to the uniqueness of
Nakanishi weight function, one finds that

$g_{1}(\gamma,z)=G_{1}(\gamma,z)+(M_{\pi}^{2}/4-\gamma)G_{4}(\gamma,z)\\\
+\int_{0}^{\gamma}d\gamma^{\prime}\Big{(}G_{4}(\gamma^{\prime},z)-z~{}\partial_{z}[G_{4}(\gamma^{\prime},z)]\Big{)}.$
(39)

The scalar amplitude $\phi_{2}(k,p)$ is written in terms of $\chi$’s as given
by Eq. (20), and their respective NIR’s:

$\phi_{2}(k,p)=\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma\,\Bigg{\\{}\frac{\left(\frac{M_{\pi}}{2}+\frac{k\cdot
p}{M_{\pi}}\right)G_{2}(\gamma,z)}{[k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon]^{3}}\\\ +\frac{\left(\frac{M_{\pi}}{2}-\frac{k\cdot
p}{M_{\pi}}\right)G_{3}(\gamma,z)}{[k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon]^{3}}\Bigg{\\}}\,.$ (40)

The final expression for the function $g_{2}(\gamma,z)$ is found after the
factor $k\cdot p$ in Eq. (40) is eliminated by following the same steps in
deriving Eqs. (103) and (104). The resulting weight function is,

$g_{2}(\gamma,z)=\frac{M_{\pi}}{2}~{}\left[G_{2}(\gamma,z)+G_{3}(\gamma,z)\right]\\\
-\frac{2}{M_{\pi}}\int_{0}^{\gamma}d\gamma^{\prime}~{}\partial_{z}\left[z\left(G_{2}(\gamma^{\prime},z)-G_{3}(\gamma^{\prime},z)\right)\right].$
(41)

The third scalar amplitude $\phi_{3}$ is given in Eq. (21), and is written in
detail as:

$\phi_{3}(k,p)=-\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma\frac{M_{\pi}~{}G_{2}(\gamma,z)}{\left[k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon\right]^{3}}\\\
+\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma\frac{M_{\pi}~{}G_{3}(\gamma,z)}{\left[k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon\right]^{3}}~{},$ (42)

and resorting to the uniqueness of NIR, one finds that:

$g_{3}(\gamma,z)=M_{\pi}~{}[G_{3}(\gamma,z)-G_{2}(\gamma,z)].$ (43)

The fourth scalar amplitude $\phi_{4}(k,p)$ as given by Eq. (22), and NIR is
written as

$\phi_{4}(k,p)=\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma~{}\frac{M_{\pi}^{2}\,G_{4}(\gamma,z)}{(k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon)^{3}}\,,$ (44)

which leads to

$g_{4}(\gamma,z)~{}=~{}M_{\pi}^{2}~{}G_{4}(\gamma,z).$ (45)

The symmetry properties of the $g_{i}$’s follows from their representation in
terms of the $G_{i}$’s, and are given by:

$\displaystyle g_{i}(\gamma,z)=g_{i}(\gamma,-z)\,,\quad(i=1,2,4)\,,$ (46)
$\displaystyle g_{3}(\gamma,z)=-g_{3}(\gamma,-z)\,.$ (47)

To obtain the above relations, the symmetry properties of the $G_{i}$’s from
Eqs. (35)-(37) were used in Eqs. (39), (41), (43) and (45), which give the
explicit expressions of the $g_{i}$’s in terms of linear combinations of
$G_{i}$’s.

###  III.4 Normalization

In order to calculate the observables of the pion, which in our case are the
valence probability, moment distributions and the decay constant, the BS
amplitude must be properly normalized. We will assume that the BS amplitude of
the QCD-inspired pion model is hypothetically a solution of a BS equation
whose interaction kernel depends only on the transferred moment, and the quark
propagators are dressed, so the normalization expression reduces to (see
[48]):

$Tr\Bigg{[}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\partial}{\partial
p^{\prime\mu}}\\{S^{-1}(k-p^{\prime}/2)\bar{\Psi}_{\pi}(k,p)\\\ \times
S^{-1}(k+p^{\prime}/2)\Psi_{\pi}(k,p)\\}\Bigg{]}_{p=p^{\prime}}=-\imath~{}2p_{\mu}\,.$
(48)

We observe that within the ladder approximation of the BS equation, Eq. (48)
also provides the pion charge normalization [28].

The most general expressions for the BSA of a $0^{-}$ system is given in Eq.
(14) for the ortoghonal basis, and the associated conjugate is,

$\displaystyle\bar{\Psi}_{\pi}(k,p)=-S_{1}(k,p)\,\phi_{1}(k,p)+S_{2}(k,p)\,\phi_{2}(k,p)$
(49) $\displaystyle+$ $\displaystyle
S_{3}(k,p)\,\phi_{3}(k,p)+S_{4}(k,p)\,\phi_{4}(k,p)\,,$

where $S_{i}(k,p)$ are the Dirac operators from Eq. (15). We have already
defined these scalar functions, $\phi_{i}$(k,p), which contain the analytic
behavior imposed by the Feynman prescription, namely $+\imath\epsilon$. These
have to obey well-defined properties under the $k\to-k$ exchange, according to
the anticommutation rule for the fermionic fields.

A comment is appropriate here. The normalization formula in Eq. (91), contains
contributions from the LF Fock-space decomposition of the pion wave function
beyond the valence one, that is, it takes into account the infinite sum of
states with the dressed quark-antiquark pair and any number of dressed gluons,
which have the pion quantum numbers. All this complexity appears when the
Bethe-Salpeter equation is projected onto the light-front, and Fock components
beyond the valence one contribute to the kernel of the effective squared mass
operator that acts on the valence component [49]. Also the BS amplitude as
defined in Minkowski space, has the quark and antiquark propagating in
different LF times, and involves from the point of view of the LF Hamiltonian
evolution a propagation in the relative time, and therefore during the virtual
propagation between these two times the quantum system fluctuates on all Fock
components allowed by the pion quantum numbers. Thus, although the
$\pi-q\bar{q}$ BS amplitude is defined in terms of the matrix element of only
two quark operators, evaluated between the vacuum and the pion states, the
quantization of the quark and gluon fields and the non-diagonal form in the
evolution operator in the LF Fock-space end-up allowing the virtual
propagation of the system in all Fock components allowed by the pion quantum
numbers. There remains the difficult question on how the LF evolution operator
can be defined for the dressed degrees of freedom.

##  IV LF valence wave function

The valence wave function is the Fock space component of the hadron state with
the smallest number of constituents that carries its quantum numbers. The
hadron state on the null-plane is decomposed in an infinite sum of Fock-
components with an arbitrary number of dressed constituents, namely quarks and
gluons, and each of these many-body state carrying the hadron quantum number.
All components contribute to the normalization and, in particular, to the
valence probability. In the pion it is associated to the probability to find a
dressed pair of quark and an antiquark. In the most simplest case of
constituent quarks with no structure, the basis in Fock space can be
formulated using the standard methods of quantum field theory on the Light-
Front, where one defines the creation and annihilation operators for particles
and antiparticles in the null plane with arbitrary spin, and the hadron state
is decomposed in Fock-components with a generic number of constituents [11].

On the other hand, it is recognized that the valence wave function comes from
the elimination of the relative LF time between the minimum number of quark
operators that enter the matrix element between the vacuum and the hadron
state, and that defines the BS amplitude [49, 50]. Alternatively, the valence
wave function can be obtained using the quasi-potential expansion method
adapted to perform the projection on the light-front of the BS equation and
associated amplitude [49, 50]. It is worthwhile to mention that even the BS
amplitude with the minimal number of legs contains the information on the full
LF Fock-space composition of the hadron. It should be noted that gauge links
are always required between the quark field operators to define an observable
in order to keep color gauge invariance [51]. Recently, it was suggested [52]
to obtain the LF probability amplitude of the hadron many-body component from
the matrix elements between the vacuum and the hadron state of quark and gluon
operators connected by gauge links, and then eliminate the relative LF time by
projecting these amplitudes onto the null-plane.

###  IV.1 Spin components

In our phenomenological treatment of the pion state, the formulation of the
valence wave function with dressed constituents will follow Ref. [27], which
derived the relation between the valence components with antialigned and
aligned quark spins in terms of the scalar functions associated with the
decomposition of the BSA in the orthogonal Dirac basis:

$\varphi_{2}(\xi,\vec{k_{\perp}},\sigma_{i},J^{\pi},J_{z})=-\frac{\sigma_{2}}{2}\sqrt{N_{c}}\sqrt{1-z^{2}}\\\
\times\int\frac{dk^{-}}{2\pi}\Bigg{\\{}\delta_{\sigma_{2},-\sigma_{1}}\left(\phi_{2}(k,p)+\left(\frac{k^{-}}{2M_{\pi}}+\frac{z}{4}\right)\phi_{3}(k,p)\right)\\\
-\frac{k_{L(R)}\sqrt{2}}{M_{\pi}}~{}\delta_{\sigma_{2},\sigma_{1}}~{}\phi_{4}(k,p)\Bigg{\\}}\,,$
(50)

for $\sigma_{1}=\pm 1$, $\sigma_{2}=\pm 1$ and $N_{c}=3$ is the number of
colors. The momentum fraction is $0<\xi<1$ and the associated relation
$z=2\xi-1$. Although, strictly speaking the derivation in [27] was done with a
structureless constituent quark degrees of freedom, we will assume the
validity of this formula for dressed quarks, noting that Eq. (50) has no trace
of the particular constituent quark mass, which would forbid its applicability
to dressed quarks. Another indication of the utility of Eq. (50) in a general
case comes from the decay constant formula, which can be written in terms of
the antialigned spin component or solely from $\phi_{2}$ [27].

From the above reasoning, we write the valence probability as given by:

$P_{val}=\frac{1}{(2\pi)^{3}}\sum_{\sigma_{1},\sigma_{2}}\int^{1}_{-1}\frac{dz}{1-z^{2}}\int
d^{2}k_{\perp}\\\
\times|\varphi_{2}(\xi,\vec{k_{\perp}},\sigma_{i},J^{\pi},J_{z})|^{2}\,,$ (51)

where we identify the two-spin components of the pion valence wave function
following Ref. [27] and written as:

$\varphi_{2}(\xi,\vec{k_{\perp}},\sigma_{i};M_{\pi},J^{\pi},J_{z})=-\dfrac{\sigma_{2}}{2}\sqrt{N_{c}}\sqrt{1-z^{2}}\\\
\times\left\\{\delta_{\sigma_{2},-\sigma_{1}}~{}\Psi_{\uparrow\downarrow}(\gamma,z)~{}\mp~{}\delta_{\sigma_{2},\sigma_{1}}~{}e^{\mp\imath\theta}~{}\Psi_{\uparrow\uparrow}(\gamma,z)\right\\}\,,$
(52)

where $k_{x}\mp\imath\,k_{y}=\sqrt{\gamma}\,e^{\mp\imath\,\theta}$. In terms
of the LF projected amplitudes, which were shown for the pion QCD inspired
model in Sect. VII, the two spin components of the pion valence wave function
are given by [27]:

$\Psi_{\uparrow\downarrow}(\gamma,z)=\psi_{2}(\gamma,z)+\frac{z}{2}\psi_{3}(\gamma,z)\\\
+\frac{\imath}{M_{\pi}^{3}}\int_{0}^{\infty}d\gamma^{\prime}\frac{\frac{\partial}{\partial
z}[g_{3}(\gamma^{\prime},z)]}{\left(\gamma+\gamma^{\prime}+\frac{z^{2}M_{\pi}^{2}}{4}-\imath\epsilon\right)}\,,$
(53)

for the anti-aligned quark and antiquark spins, and

$\Psi_{\uparrow\uparrow}(\gamma,z)=\frac{\sqrt{\gamma}}{M_{\pi}}~{}\psi_{4}(\gamma,z)$
(54)

for the aligned quark spins.

In the above expressions for the two spin components of the pion wave
function, the LF time projected amplitudes are [44, 27],

$\psi_{i}(\gamma,z)=\int\frac{dk^{-}}{2\pi}~{}\phi_{i}(k,p),$ (55)

with the notation ${\vec{k}}_{\perp}^{2}=\gamma\geq 0$ and $z=-2k^{+}/p^{+}$,
where the scalar functions are represented by the NIR written in Eq. (24). The
momentum are transformed to the light-front ones: $k^{-}=k^{0}-k^{3}$,
$k^{+}=k^{0}+k^{3}$ and $\vec{k}_{\perp}$. The final expression for
$\psi_{i}(\gamma,z)$ [53],

$\psi_{i}(\gamma,z)=-\frac{\imath}{M_{\pi}}\int_{0}^{\infty}d\gamma^{\prime}~{}\frac{g_{i}(\gamma^{\prime},z)}{\left[\frac{z^{2}M_{\pi}^{2}}{4}+\gamma+\gamma^{\prime}\right]^{2}}\,,$
(56)

where we have used the property of the support of the Nakanishi weight
functions. Therefore, the denominator never vanishes for the pion bound state,
and the $\imath\epsilon$ is irrelevant in this case.

###  IV.2 Valence probability decomposition

The components of the valence wave function with spins of antialigned (53) and
aligned quarks (54) are the probability amplitudes, as long as the full wave
function in Fock space is normalized. The valence probability, defined in Eq.
(51), can be written as:

$P_{\text{val}}=\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma~{}\rho_{val}(\gamma,z)\,,$
(57)

where the probability densities associated with the two spin components take
place:

$\rho_{val}(\gamma,z)=\rho_{\uparrow\downarrow}(\gamma,z)+\rho_{\uparrow\uparrow}(\gamma,z)$
(58)

and are defined as

$\displaystyle\rho_{\uparrow\downarrow}(\gamma,z)=\frac{N_{c}}{16\pi^{2}}|{\psi}_{\uparrow\downarrow}(\gamma,z)|^{2}\,,$
(59)
$\displaystyle\rho_{\uparrow\uparrow}(\gamma,z)=\frac{N_{c}}{16\pi^{2}}\frac{\gamma}{M_{\pi}^{2}}|\psi_{4}(\gamma,z)|^{2}\,,$
(60)

being the moment probability densities for the anti-aligned and aligned spin
components, respectively. The probability for each valence spin component is
therefore given by:

$\displaystyle P_{\uparrow\downarrow}$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma~{}\rho_{\uparrow\downarrow}(\gamma,z)\,,$
(61) $\displaystyle P_{\uparrow\uparrow}$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma~{}\rho_{\uparrow\uparrow}(\gamma,z)\,,$
(62)

where we have explicitly written the two contributions to the valence state.

It is important to note that, according to [27], the configuration of anti-
aligned quark and antiquark spins have probabilities around
$P_{\uparrow\downarrow}\sim 0.55$ and $P_{\uparrow\uparrow}\sim 0.15$,
respectively. Reminding that for the anti-aligned spin configuration the quark
and antiquark state is an eigenstate of the operator $L_{z}$ with eigenvalue
$\ell_{z}=0$. In contrast, the aligned configuration has the quark spins
necessarily coupled in an eigenstate of $L_{z}$ with $\ell_{z}=\pm 1$. We
cannot rule out the aligned spin component, even with its low influence on the
relativistic dynamical regime of the quarks within the pion. A quantitative
study of this component has a key role in understanding the characteristics of
light mesons (and possible relativistic corrections for heavier ones).

###  IV.3 Decay constant

The decay constant is defined in terms of the BSA, as follows

$\imath~{}p^{\mu}f_{\pi}=N_{c}\int\frac{d^{4}k}{(2\pi)^{4}}~{}\mbox{Tr}[\,\gamma^{\mu}\,\gamma^{5}\,\Psi_{\pi}(k,p)]\,.$
(63)

Performing the contraction with $p_{\mu}$, from the right in each member and
using the BS amplitude decomposition, given in Eq. (14), we will have

$\imath\,M^{2}_{\pi}f_{\pi}=-4~{}M_{\pi}~{}N_{c}\int\frac{d^{4}k}{(2\pi)^{4}}~{}\phi_{2}(k,p).$
(64)

Here we use the light front variables, remembering that when integrating
$\phi_{2}$ into the variable that corresponds to energy, $k^{-}$, we obtain
the component $\psi_{2}$, i.e.

$\displaystyle\imath~{}M_{\pi}~{}f_{\pi}=-4~{}N_{c}~{}\frac{1}{2}\int\frac{d^{2}{k_{\perp}}}{(2\pi)^{2}}\int\frac{dk^{+}}{(2\pi)}~{}\psi_{2}(\gamma,z)\,.$
(65)

We must analyze the integrals in the longitudinal momentum, $k^{+}$ and
$\vec{k}_{\perp}$ by writing $dk^{+}=-\frac{M_{\pi}}{2}dz$ and
$d^{2}{k_{\perp}}=d\gamma\frac{d\theta}{2}$, where we use that
$\gamma=|\vec{k_{\perp}}|^{2}$, with $\theta$ in the range $[0,2\pi]$, and
using the definition of $\psi_{2}(\gamma,z)$, which was presented in Eq. (56):

$f_{\pi}=\frac{N_{c}}{8\pi^{2}M_{\pi}}\int_{0}^{\infty}d\gamma\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma^{\prime}\frac{g_{2}(\gamma^{\prime},z)}{\left[\gamma+\gamma^{\prime}+z^{2}\dfrac{M^{2}_{\pi}}{4}\right]^{2}},$
(66)

and finally, we just integrate in $\gamma$ and apply the limits, obtaining:

$f_{\pi}=\frac{N_{c}}{8\pi^{2}~{}M_{\pi}}\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma^{\prime}\frac{g_{2}(\gamma^{\prime},z)}{\gamma^{\prime}+z^{2}\,\dfrac{M^{2}_{\pi}}{4}}.$
(67)

We stress that all observables are calculated after obtaining the
normalization constant $\mathcal{N}$, present in the Nakanishi weight
functions, which is obtained from the normalization condition given in Eq.
(91).

We should also point out that it is possible to express $f_{\pi}$ in terms of
the anti-aligned spin component of the valence wave function, given by Eq.
(53), as discussed in Ref. [27]. The demonstration of this equivalence was
done using the “+” component of the axial current, and we get the expression
below,

$\displaystyle
f_{\pi}=\imath\frac{N_{c}}{8\pi^{2}}\int_{0}^{\infty}d\gamma\int_{-1}^{1}dz~{}\Psi_{\uparrow\downarrow}(\gamma,z)\,,$
(68)

which is equivalent to the final form found in Eq. (67).

##  V Application

The present example extends the investigation of the pion by exploring the
integral representation of the BS amplitude, as developed in Sect. III,
applying the formalism to the Minkowski space QCD-inspired model proposed in
Ref. [24]. In this model the pion Bethe-Salpeter amplitude incorporates the
dressed quark propagator. Even though only the quark momentum-dependent
running mass is considered, the decay constant and the electromagnetic form
factor experimental are reproduced [24].

The pseudo-scalar pion-quark vertex in the QCD-inspired model is proportional
to the running mass of the dressed quark propagator (see Eqs. (8) and (10)).
The analytical form of the running mass contains a single pole in the time-
like region, which is fitted to the Landau gauge lattice QCD calculations in
the Euclidean space [3]. Such parametrization allows to analytically extend
the BS amplitude to the Minkowski space, and therefore a more detailed study
of this model exploring the Nakanishi integral representation [31].

In this section, we derive the explicit form of the NIR weight functions of
the four scalar amplitudes associated with the decomposition of the pion BS
amplitude using two choices of operator basis in the Dirac spinor space: (i) a
non-orthogonal basis (see Sect. III.2), and (ii) an orthogonal basis (see
Sect. III.3) , where the latter one has been used to solve the BS equation in
Minkowski space for the pion [27, 28]. In addition, we present the numerical
results for the weight functions and for the projected scalar amplitudes on
the light-front in the orthogonal basis, which are necessary to build the
valence wave function, as detailed in Sec. VII. The numerical results will be
discussed, emphasizing their symmetry properties, as well as the
characteristics brought by the running dressed quark mass.

###  V.1 Model parametrization

The general form of the dressed quark propagator is written in the form:

$S_{F}(k)=\imath\,Z(k^{2})\left[/k-M(k^{2})+\imath\epsilon\right]^{-1}\,.$
(69)

In the adopted model from Ref. [24], it is used the simplification that
$Z(k^{2})=1/A(k^{2})=1$. Such simplification makes the BS amplitude, in the
infrared region, about twice larger than it should be, considering that
$Z(0)\sim 0.7$ [3]. On the other hand, such enhancement is compensated by the
computed values of $f_{\pi}$, the pion charge radius and the electromagnetic
form factor in close agreement with the experimental data, as shown in [24].

Here, in order to illustrate the general framework we have developed in the
previous sections, we make use of such quark propagator model, which has the
simplified form:

$S_{F}(k)=\imath\frac{/k+M(k^{2})}{k^{2}-M^{2}(k^{2})+\imath\epsilon}.$ (70)

Following the previous work [24], we adopt a quark mass function with a
monopole shape having a single time-like pole added to the current quark mass
(see also [54]), which is fitted to lattice QCD calculations [3], and given
by:

$M(k^{2})=m_{0}-m^{3}\left[k^{2}-\lambda^{2}+\imath\epsilon\right]^{-1},$ (71)

where $m_{0}=0.014$ GeV, $m=0.574$ GeV and $\lambda=0.846$ GeV. The result of
the dressed quark mass employed here is shown in Fig. (1). It is compared to
the lattice QCD results in the Landau gauge from Ref. [3], and also with the
parametrization given in [55].

Figure 1: Quark running mass model (dashed line), with the present parameters,
$m_{0}=0.014$ GeV, $m=0.574$ GeV and $\lambda=0.846$ GeV [24]; compared with
lattice QCD calculations [3] (full circles) and the parametrization (solid
line) given in [55]. Results are presented for space-like momentum.

The introduction of Eq. (71) in the quark propagator presents poles in time-
like region, which are the zeros of:

$k^{2}\left(k^{2}-\lambda^{2}\right)^{2}-\left[m_{0}\left(k^{2}-\lambda^{2}\right)-m^{3}\right]^{2}~{}=0\,,$
(72)

and, with the given parameters, the solutions are such that only
$k^{2}=m_{i}^{2}>0$ are found, with the following values: $m_{1}=323$ MeV,
$m_{2}=645$ MeV, and $m_{3}=954$ MeV.

The time-like poles of the quark propagator can be factorized in the form:

$S_{F}(k)=\imath\,\,\frac{\left(k^{2}-\lambda^{2}\right)^{2}\,(/k+m_{0})-\left(k^{2}-\lambda^{2}\right)\,m^{3}}{{\prod_{i=1,3}}(k^{2}-m^{2}_{i}+\imath\epsilon)}\,,$
(73)

and comparing with Eq. (II.1), we single out its vector and scalar components,
respectively as:

$\displaystyle S_{v}(k^{2})$
$\displaystyle=\sum_{i=1}^{3}\frac{D_{i}}{\left(k^{2}-m_{i}^{2}+\imath\epsilon\right)}\,,$
(74) $\displaystyle S_{s}(k^{2})$
$\displaystyle=\sum_{i=1}^{3}\frac{E_{i}}{\left(k^{2}-m_{i}^{2}+\imath\epsilon\right)}+\,S_{v}(k^{2})m_{0}\,,$

where,

$D_{i}=\dfrac{(\lambda^{2}-m_{i}^{2})^{2}}{(m_{i}^{2}-m_{j}^{2})(m_{i}^{2}-m_{k}^{2})}\,,$
(75)

and

$E_{i}=-\frac{(m_{i}^{2}-\lambda^{2})\,m^{3}}{(m_{i}^{2}-m_{j}^{2})(m_{i}^{2}-m_{k}^{2})}\,.$
(76)

with cyclic permutations of $\\{i,j,k\\}\equiv\\{1,2,3\\}$.

Table 1: Residue of the vector and scalar parts given in columns three and four, respectively, of the dressed quark propagator at each pole of mass $m_{i}$ shown in column two. $i$ |  $m_{i}$ [MeV] | $D_{i}$ |  $C_{i}$ [GeV]
---|---|---|---
1 | 323 | 1.487 | 0.481
2 | 645 | -0.583 | -0.376
3 | 954 | 0.096 | -0.091

The spectral density of the vector component of the quark propagator
$S_{v}(k^{2})$ in the form of Eq. (6) is

$\rho_{v}(\mu^{2})~{}=\sum_{i=1}^{3}D_{i}\,\delta(\mu^{2}-m_{i}^{2})\,,$ (77)

and the spectral density of the scalar part of the quark propagator in the
form of Eq. (7) is written as:

$\rho_{s}(\mu^{2})~{}=~{}\sum_{i=1}^{3}C_{i}\,\delta(\mu^{2}-m_{i}^{2})\,,$
(78)

where $C_{i}=E_{i}+m_{0}\,D_{i}$.

In Table 1, we present the residue of the poles in the scalar and vector
components of the dressed quark propagator, which are weights of the Dirac-
deltas in the spectral densities. It was shown in [24] that this model
violates the spectral density positivity relations [32], which are not in
principle required to be satisfied, as the quark is not a colorless asymptotic
state of QCD.

###  V.2 Bethe-Salpeter amplitude model

The pion Bethe-Salpeter amplitude is built in terms of the quark propagator,
Eq. (73), which has the scalar and vector components given by Eq. (74) with
the corresponding spectral densities written in Eqs. (77) and (78),
respectively. It reads

$\Psi_{\pi}(k;p)=-\imath\left[S_{v}(k_{q}^{2})\,/\\!\\!\\!{k}_{q}+S_{s}(k_{q}^{2})\right]\,\\\
\times\frac{\mathcal{N}\,\gamma_{5}~{}m^{3}}{k^{2}-\lambda^{2}+\imath\epsilon}\left[S_{v}(k_{\overline{q}}^{2})\,/\\!\\!\\!{k}_{\overline{q}}+S_{s}(k_{\overline{q}}^{2})\right]~{},$
(79)

where $k_{q}=(k+p/2)$, and $k_{\overline{q}}=(k-p/2)$. The above BS amplitude
fits into the form adopted in Eq. (10), albeit with spectral densities
associated with a finite number of time-like poles in the self-energy and
associated quark propagator. We should clarify that in our example, as
suggested by the model of Ref. [24], the pion vertex is obtained from the
running mass term in the chiral limit when $m_{0}\to 0$, while the quark
propagators in the pion BS amplitude kept the finite value of $m_{0}$ in
$M(k^{2})$ as written in Eq. (71).

The spectral density of the vertex function is

$\rho_{B}(\mu^{2})=\mathcal{N}m^{3}\delta(\mu^{2}-\lambda^{2})\,,$ (80)

and with this formula we have all the ingredients to derive the Nakanishi
integral representation of the four scalar amplitudes appearing in the
decomposition of the BS amplitude in the Dirac spinor space.

Figure 2: $G_{1}(\gamma,z)$ as a function of $z$ with $\gamma=$0.75 GeV2
(solid line) and 0.45 GeV2 (dashed line). For the plot we use in Eq. (81) the
arbitrary value of $\mathcal{N}=100$, drop out the factor of $i$ and all
parameters are in units of GeV.

In what follows, we will obtain the weight functions $G_{i}(\gamma,z)$,
associated with the NIR of the scalar amplitudes, Eq. (23), appearing in the
decomposition of the BS amplitude in the non-orthogonal basis. We also compute
the weights, $g_{i}(\gamma,z)$, associated with the NIR of the scalar
amplitudes, Eq. (24), corresponding to the orthogonal basis decomposition of
the BS amplitude.

###  V.3  NIR: non-orthogonal basis

Having at hand the spectral densities of the vector and scalar parts of the
dressed quark propagator and the one for the vertex function, Eqs. (77), (78)
and (80), respectively, the NIR weight functions can be derived. The
$G_{i}(\gamma,z)$ $(i=1-4)$ are obtained using the general form given in Sect.
III.2 and written in Eqs. (31)-(34).

We remind that the auxiliary function $F(\gamma,z;\lambda,\mu^{\prime},\mu)$,
defined in Eq. (26), is essential to build the weight functions for the scalar
amplitudes $\chi_{i}(k,p)$ in Eq. (23). The support of the function
$F(\gamma,z;\lambda,\mu^{\prime},\mu)$ in $\gamma$ with $-1\leq z\leq 1$, for
$\lambda=$ 0.846 GeV, $\mu^{\prime}=m_{i}$ and $\mu=m_{j}$ corresponding to
the time-like poles of the dressed quark propagator, are limited by the
theta’s in Eq. (26), which strictly imposes $\gamma>0$ for each pair
$(m_{i},m_{j})$. Interesting to observe that $\gamma$ has an upper bound of
1.394 GeV2, meaning that the weight functions incorporate a scale of about 1
GeV brought by the the running quark mass function parameter, $\lambda$, which
was obtained by a fitting to LQCD results.

The weight function $G_{1}(\gamma,z)$ is calculated with Eq. (31) by taking
into account the spectral densities of the dressed quark propagator vector and
scalar parts, and the spectral density of the pion vertex function. Therefore,
by substituting Eqs. (77), (78) and (80) in Eq. (31), it is found that:

$G_{1}(\gamma,z)=w\sum\limits_{1\leq i,j\leq
3}C_{i}C_{j}F(\gamma,z;\lambda,m_{i},m_{j}),$ (81)

where $w=-\mathcal{N}m^{3}$ gives the proper normalization to the residue of
the scalar part of the quark propagator. In Fig. (2), we present the weight
function $G_{1}(\gamma,z)$ for two values of $\gamma$ equal to 0.45 and 0.75
GeV2, we confirm the even property under the transformation $z\to-z$
corresponding to Eq. (35). Furthermore, one observes the characteristic
plateaus in the weight function from the theta functions in
$F(\gamma,z;\lambda,m_{i},m_{j})$.

The weight function $G_{2}(\gamma,z)$ is derived following the same steps that
leads to $G_{1}(\gamma,z)$, namely we substitute the spectral densities from
Eqs. (77), (78) and (80) in Eq. (32), resulting in:

$G_{2}(\gamma,z)=w\sum\limits_{1\leq i,j\leq
3}D_{i}C_{j}F(\gamma,z;\lambda,m_{i},m_{j}).$ (82)

The transformation property of the original form of $G_{2}$ and $G_{3}$ from
(32) and (33) turns into the symmetry property in Eq. (36) gives that
$G_{3}(\gamma,z)=G_{2}(\gamma,-z)$. Motivated by this property, we present in
Fig. (3), the even and odd combinations $G_{3}(\gamma,z)+G_{2}(\gamma,z)$ and
$G_{3}(\gamma,z)-G_{2}(\gamma,z)$ for two values of $\gamma$ equal to 0.45 and
0.75 GeV2. The plateaus from the theta functions in
$F(\gamma,z;\lambda,m_{i},m_{j})$ are clearly seen in the figure. Comparing to
$G_{1}$ the two combinations have a factor of about $\sim 3$ smaller, which
comes from the residue $D_{i}$ compared to $E_{i}+m_{0}D_{i}$ as shown in
Table (1). The model also presents $G_{3}(\gamma,z)+G_{2}(\gamma,z)\sim
3G_{1}(\gamma,z)$. The $z$-odd function $G_{3}(\gamma,z)-G_{2}(\gamma,z)$ is
also shown in the lower panel of the figure. One observe that the signs of the
plateaus alternate by changing $\gamma$, as a consequence of the alternate
signs of the residue (see Table 1).

Figure 3: Top panel: $G_{2}(\gamma,z)+G_{3}(\gamma,z)$ as a function of $z$
with $\gamma=$ 0.75 GeV2 (dashed line) and 0.45 GeV2 (solid line). Bottom
panel: $G_{3}(\gamma,z)-G_{2}(\gamma,z)$ as a function of $z$ with $\gamma=$
0.75 GeV2 (solid line) and 0.45 GeV2 (dashed line). For the plot we use in Eq.
(82) the arbitrary value of $\mathcal{N}=100$, drop out the factor of $i$ and
all parameters are in units of GeV.  Figure 4:  $G_{4}(\gamma,z)$ as a
function of $z$ with $\gamma=$ 0.75 GeV2 (solid line) and 0.45 GeV2 (dashed
line). For the plot we use in Eq. (83) the arbitrary value of
$\mathcal{N}=100$, drop out the factor of $i$ and all parameters are in units
of GeV.

The weight function $G_{4}(\gamma,z)$ is derived by introducing the spectral
densities (77), (78) and (80) in Eq. (34), which results in:

$G_{4}(\gamma,z)=w\sum\limits_{1\leq i,j\leq
3}D_{i}D_{j}F(\gamma,z;\lambda,m_{i},m_{j})\,.$ (83)

In Fig. (4) the dependence of $G_{4}(\gamma,z)$ with $z$ is shown for $\gamma$
with values of 0.45 and 0.75 GeV2. Its even property with the transformation
$z\to-z$ is clearly seen, as given by the general form of the model, Eq. (37),
and discussed in Sect. III.2. Such symmetry property, although not trivially
apparent when one inspects the weight function
$F(\gamma,z;\lambda,m_{i},m_{j})$ written in Eq. (26), leads to the symmetry
properties we have seen so far in this example, is traced back to Eqs. (13),
and followed by our construction of the Nakanishi weight functions.

As an observation, the dependence of the weight functions $G_{i}(\gamma,z)$ in
$\gamma$ for a given $z$ also exhibit the characteristic plateaus from the
thetas in the weight function $F(\gamma,z;\lambda,m_{i},m_{j})$. Due to the
proximity between the masses $m_{2}$ and $m_{3}$ given in Table 1 is not
always apparent the several plateaus and superposition of them in the
$G_{i}$’s, which result in less structures than one could think. On the other
side, it is expected that the spectral densities will be smooth functions, and
not having a discrete support, as in the model used here. In this case the
plateaus would be smoothed out giving place, eventually, to oscillations, as a
function of $\gamma$ and $z$. Another strict property of these weight
functions, namely, the vanishing at the end-points due to $F(\gamma,\pm
1;\lambda,m_{i},m_{j})=0$ is of course shown in the figures 2-4, which leads
to the sharp ending peaks close to $z=\pm 1$ for some $\gamma$ values.

Figure 5:  Weight function $g_{1}(\gamma,z)$ as a function of $z$ for
$\gamma=0.45$ GeV2 coming from the covariant model with fixed constituent mass
(dashed line) and with the QCD pion inspired model (the single constituent
mass pole model has $g_{3}=0$). For the plot we use the arbitrary value of
$\mathcal{N}=100$, drop out the factor of $i$ and all parameters are in units
of GeV.

###  V.4 NIR: orthogonal basis

The Nakanishi weight functions of the scalar amplitudes for the pion BS
amplitude decomposed in the orthogonal basis are derived below using Eqs.
(39)-(45), and having $G_{i}(\gamma,z)$ from Eqs. (81)-(83). The result is:

$\displaystyle g_{k}(\gamma,z)=$ $\displaystyle\sum\limits_{1\leq i,j\leq
3}\int_{0}^{\infty}d\gamma^{\prime}\,{\widehat{Y}}^{(k)}_{ij}(\gamma,\gamma^{\prime},z)$
(84) $\displaystyle\times$
$\displaystyle\,F(\gamma^{\prime},z;\lambda,m_{i},m_{j})\,,$

where $k=$1, 2, 3 or 4. We have also introduced the functional
${\widehat{Y}}^{(k)}_{ij}(\gamma,\gamma^{\prime},z)$ to simplify the notation.
Note that at the end-points $g_{k}(\gamma,\pm 1)=0$, as follows from
$F(\gamma,\pm 1;\lambda,m_{i},m_{j})=0$.

The functional associated with the weight function $g_{1}$ is given by:

${\widehat{Y}}^{(1)}_{ij}(\gamma,\gamma^{\prime},z)=wD_{i}D_{j}\,\theta(\gamma-\gamma^{\prime})\,(1-z\partial_{z})\\\
+\Bigg{[}C_{i}C_{j}+D_{i}D_{j}\left(\frac{M^{2}_{\pi}}{4}-\gamma^{\prime}\right)\Bigg{]}\delta(\gamma-\gamma^{\prime})\,,$
(85)

which follows from Eqs. (39), (81) and (83). It has the symmetry property,

${\widehat{Y}}^{(1)}_{ij}(\gamma,\gamma^{\prime},z)={\widehat{Y}}^{(1)}_{ji}(\gamma,\gamma^{\prime},z)\,,$
(86)

which leads to $g_{1}(\gamma,-z)=g_{1}(\gamma,z)$, and we observe that the
contribution from $z\partial_{z}$ keeps the symmetry property under the
transformation $z\to-z$ of the weight function.

Figure 6:  Weight function $g_{2}(\gamma,z)$ as a function of $z$ for
$\gamma=0.45$ GeV2 coming from the covariant model with fixed constituent mass
(dashed line) and with the QCD pion inspired model (continuous line). For the
plot we use the arbitrary value of $\mathcal{N}=100$, drop out the factor of
$i$ and all parameters are in units of GeV.

In Fig. 5, we present the results for $g_{1}(\gamma,z)$ for $\gamma=0.45$ GeV2
as a function of $z$, the characteristic plateaus are visible, as well as the
vanishing at the end points. For comparison, we plot the weight function for
the case where the quark propagator has only a single pole at the constituent
mass of $m_{1}=0.323$ GeV, while we kept the values of $\lambda$, $D_{1}$ and
$C_{1}$ in Table 1. The single pole model presents just one plateau for
$g_{1}$, a much simpler structure compared to the full model, which has three
poles, with specific residua that makes vanish the weight function in the
region where $g_{1}$, for the single pole model, is nonzero. The
characteristic oscillations of the plateaus of $g_{1}$ represents the
interference between the residua with opposite signs and the overlapping of
support of the weight function $F(\gamma,z;\lambda,m_{i},m_{j})$, while the
model with a constant constituent quark mass, has a much simple structure.

Figure 7:  Weight function $g_{3}(\gamma,z)$ as a function of $\gamma$
computed with the QCD pion inspired model for $z=0.5$ (continuous line) and
for $z=-0.5$ (dashed line). For the plot we use the arbitrary value of
$\mathcal{N}=100$, drop out the factor of $i$ and all parameters are in units
of GeV.

The functional that provides the Nakanishi weight function $g_{2}$ in Eq. (84)
is given by:

${\widehat{Y}}^{(2)}_{ij}(\gamma,\gamma^{\prime},z)=\frac{M_{\pi}}{2}~{}w\left[C_{i}D_{j}+C_{j}D_{i}\right]\delta(\gamma-\gamma^{\prime})\\\
-\frac{2}{M_{\pi}}~{}w\left[D_{i}C_{j}-D_{j}C_{i}\right]\frac{\theta(\gamma-\gamma^{\prime})}{M_{\pi}}~{}\partial_{z}z\,.$
(87)

This functional is derived from Eqs. (41) and (82), and it also leads to the
symmetry property $g_{2}(\gamma,-z)=g_{2}(\gamma,z)$. In Fig. 6, we show the
dependence on $z$ of $g_{2}(\gamma,z)$ for $\gamma=0.45$ GeV2 considering the
two models, the complete one with three time-like poles of the quark
propagator, and the simplified one taking into account only the constituent
mass pole corresponding to $m_{1}$. The general structure is similar to what
we have observed for $g_{1}$ in Fig. 5, while the simplified model has only
one plateau, as it is expected from the theta functions in
$F(\gamma,z;\lambda,m_{1},m_{1})$, with the support for $m_{1}=0.323$ GeV
being $0<\gamma\lesssim 1$ GeV2.

The functional that builds the Nakanishi weight function $g_{3}$ is given by:

${\widehat{Y}}^{(3)}_{ij}(\gamma,\gamma^{\prime},z)=M_{\pi}w~{}[C_{i}D_{j}-C_{j}D_{i}]\,\delta(\gamma-\gamma^{\prime})\,,$
(88)

which follows from Eqs. (43) and (82). It is proportional to $G_{3}-G_{2}$,
which provides the odd property under the transformation $z\to-z$. Therefore,
the plot of $g_{3}(\gamma,z)$ for $z=0.5$ as a function of $\gamma$ in Fig. 7
complements the results shown for $G_{3}-G_{2}$ presented in the right panel
of Fig. 3, and also provides one example of the $\gamma$ dependence in the
Nakanishi weight function. Fig. 7 clearly reflects the scales involved in the
model, namely the positions of the quark mass poles between 0.3 and 1 GeV (see
Table 1) as well as the pseudo-scalar vertex scale parameter $\lambda$ close
to 1 GeV. Furthermore, the characteristic oscillations observed in the figure
comes from the different signs of the residua, also as a consequence of the
violation of the positivity constraints of the spectral densities of the
scalar and vector parts of the quark propagator. Note that the single pole
model results $g_{3}=0$. This indeed is a nice indication of the distinctive
effect of the running quark mass model contrasting with the fixed constituent
quark mass one.

Figure 8: Weight function $g_{4}(\gamma,z)$ as a function of $z$ for
$\gamma=0.45$ GeV2 coming from the covariant model with fixed constituent mass
(dashed line) and with the QCD pion inspired model (continuous line). For the
plot we use the arbitrary value of $\mathcal{N}=100$, drop out the factor of
$i$ and all parameters are in units of GeV.

The functional associated with the Nakanishi weight function $g_{4}$ is
derived through Eqs. (45) and (83) is given by:

${\widehat{Y}}^{(4)}_{ij}(\gamma,\gamma^{\prime},z)=~{}M_{\pi}^{2}\,w\,D_{i}\,D_{j}\,\delta(\gamma-\gamma^{\prime})\,.$
(89)

The weight function $g_{4}$ is proportional to $G_{4}$, which provides the
even property under the transformation $z\to-z$. In this case, we plot in Fig.
(8), $g_{4}(\gamma,z)$ as a function of $z$ for $\gamma=0.45$ GeV2 to compare
with the result for the single quark constituent mass pole model. The single
plateau of the latter is contrasted with the structure observed for the result
obtained with the QCD inspired model.

###  V.5 Normalization of the BS amplitude

The normalization of the pion BS amplitude starts with Eq. (48) contracted
with $p_{\mu}$. After performing the Dirac’s algebra, one gets that:

$iN_{c}\int\frac{d^{4}k}{(2\pi)^{4}}\Big{\\{}\phi_{1}\phi_{1}+\phi_{2}\phi_{2}+\mathcal{B}(k^{2})\Big{[}\phi_{3}\phi_{3}+\phi_{4}\phi_{4}\\\
-4\phi_{1}\phi_{4}\Big{]}-\frac{2}{M_{\pi}}\Big{[}M(k_{\bar{q}}^{2})+M(k_{q}^{2})\Big{]}\phi_{2}\phi_{1}\Big{\\}}=1,$
(90)

where $\mathcal{B}(k^{2})=\Big{[}(k\cdot
p)^{2}-k^{2}M_{\pi}^{2}\Big{]}/M_{\pi}^{4}$, $k_{q}=(k+p/2)$ and
$k_{\bar{q}}=(k-p/2)$.

The derivatives over the running masses cancel out in the normalization due to
the opposite signs of $p$ carried by the quark and antiquark momentum in their
arguments. After substituting the NIR of the scalar amplitudes $\phi_{i}$, Eq.
(24), in the normalization formula, the analytic integration over the loop
momentum is easily performed using standard Feynman parametrization. Finally,
one finds that:

$\displaystyle 1=$
$\displaystyle-~{}\frac{3N_{c}}{32\pi^{2}}\int_{-1}^{+1}dz^{\prime}\int_{0}^{\infty}d\gamma^{\prime}\int_{-1}^{+1}dz\int_{0}^{\infty}d\gamma\int^{1}_{0}dv\left\\{~{}v^{2}(1-v)^{2}~{}\left[\frac{\mathcal{G}_{11}+\mathcal{G}_{22}-4\frac{m_{0}}{M_{\pi}}\mathcal{G}_{21}}{\left[\frac{M_{\pi}^{2}}{4}\zeta^{\prime
2}+\gamma^{\prime}v+\gamma(1-v)\right]^{4}}\right.\right.$ (91)
$\displaystyle\left.\left.+\frac{\mathcal{G}_{33}+\mathcal{G}_{44}-4~{}\mathcal{G}_{14}}{2M_{\pi}^{2}\left[\frac{M_{\pi}^{2}}{4}\zeta^{\prime
2}+\gamma^{\prime}v+\gamma(1-v)\right]^{3}}\right]-\frac{16\,m^{3}}{M_{\pi}}\int_{0}^{1}du~{}\frac{\mathcal{G}_{21}~{}(1-v-u)^{2}~{}u^{2}~{}\Theta(1-v-u)}{\left[\frac{M^{2}_{\pi}}{4}\zeta^{\prime\prime
2}-\frac{M^{2}_{\pi}}{4}+\lambda^{2}+t\gamma+u\gamma^{\prime}\right]^{5}}\right\\},$

where $\mathcal{G}_{ij}=g_{i}(\gamma^{\prime},z^{\prime})\,g_{j}(\gamma,z)$,
$t=(1-v-u)$, $\zeta^{\prime}=\big{(}vz^{\prime}+(1-v)z\big{)}$ and
$\zeta^{\prime\prime}=\big{(}(1-v-u)z+uz^{\prime}+v\big{)}\,.$

It is noteworthy to observe that the normalization condition of the BS
amplitude, Eq. (91), contains all the Nakanishi weight functions, contrary to
the valence wave function which does not have $g_{1}$. This naive observation
already suggests that the normalization condition contains contributions from
components beyond the valence one. If $m^{3}=0$ the normalization reduces to
the one found in [27] for fixed constituent quark mass. The presence of the
last term is associated with the contribution of the running quark mass of
this model to the BS normalization, curiously this term comes with the product
of $\mathcal{G}_{21}$, which carries physics beyond the valence state, as
$g_{1}$ is absent in this component of the wave function.

##  VI  Valence probability and decay constant

Once the BS amplitude is properly normalized, we obtain the two spin
components of the valence wave function, namely the spin antialigned one, Eq.
(53), and the spin aligned one, Eq. (54). From the two spin components of the
wave function, it is calculated the associated valence probability
$P_{\text{val}}$, Eq. (57), the corresponding spin decomposition
$P_{\uparrow\downarrow}$ and $P_{\uparrow\uparrow}$, from Eqs. (61) and (62),
respectively. These quantities are not yet obtained for the QCD inspired model
of the pion. The decay constant, Eq. (68) are also computed, as a cross check
of our formalism, giving that it was obtained in [24] using a complete
independent method.

The calculations of these quantities are performed for different models, set
I-IV, with and without running quark mass. Set I corresponds to the QCD
inspired pion model, with the BS amplitude given by Eq. (79). The sets II, III
and IV have fixed constituent quark mass corresponding to each of the poles of
the dressed quark propagator, while keeping the same pseudoscalar vertex form.
The normalization of the BS amplitude in these single pole models of the quark
propagator is done with Eq. (91), disregarding the last term and substituting
$m_{0}$ by the constituent quark mass.

Table 2: Static pion properties ($m_{\pi}=140$MeV) for some sets of input parameters. The second column contains the constituent quark mass. The results for the following quantities are shown: the valence probability $P_{val}$ (third column), and its decomposition in the antialigned spins state, $P_{\uparrow\downarrow}$ (fourth column), and aligned spins state, $P_{\uparrow\uparrow}$ (fifth column); the pion decay constant (sixth column).  Set | Quark mass | $P_{\text{val}}$ | $P_{\uparrow\downarrow}$ | $P_{\uparrow\uparrow}$ |  $f_{\pi}$(MeV)
---|---|---|---|---|---
(I) |  $M(k^{2})$ [Eq. (71)] | 0.70 | 0.58 | 0.12 | 130.1
(II) |  $m_{1}=323$ MeV | 0.52 | 0.41 | 0.11 | 88
(III) |  $m_{2}=645$ MeV | 0.65 | 0.52 | 0.13 | 159
(IV) |  $m_{3}=954$ MeV | 0.86 | 0.69 | 0.17 | 247
Ref. [27] |  $m=255$ MeV | 0.70 | 0.57 | 0.13 | 130

Our results are shown in Table 2. The set I (pion model inspired by QCD)
reproduces the experimental value of the pion’s decay constant,
$f_{\pi^{\pm}}^{exp.}=130.50(1)(3)(13)$ MeV [10], as well as the average value
from LQCD calculations, $f_{\pi^{\pm}}^{LCQD}=130.2(1)(0.8)$ MeV [56].
Furthermore, our result for $f_{\pi}$ agrees with Ref. [24], where it was
computed via integration of the LF momenta, which crosscheck the consistency
of our treatment of the BS amplitude in terms of the NIR. The valence
probabilities in this model, as well as their decomposition in terms of the
spin contributions were obtained. The values found are 0.7 for the total
valence probability, 0.58 for the antialigned spin configuration and 0.12 for
the aligned one.

We also present results for the models with momentum-independent mass,
corresponding to the three quark mass values of 323 MeV (set II), 645 MeV (set
III) and 954 MeV (set IV). The valence probability $P_{val}$ and $f_{\pi}$ are
in the range $[0.52-0.86]$ and $[98-247]$ MeV, respectively. This variation
could be smaller if $f_{\pi}$ fits the experimental value by tuning the
pseudoscalar vertex parameter.

It is interesting to compare with the results obtained by the solution of the
BS equation in Minkowski space of Ref. [27], which includes one gluon exchange
with mass of 637 MeV and a monopole quark-gluon vertex form factor with mass
parameter of 306 MeV. From Table 2, one sees that its results have values for
the valence probability and its spin decomposition quite close to the pion
model inspired by QCD. The valence probabilities, for the anti-aligned and
aligned spin, as well as in the dynamical model from [27], suggest the
expected prevalence of the anti-aligned spins component of the pion wave
function, a feature also exhibited in the single quark mass pole models. Note
that the contribution to the valence probability of aligned spins component of
the pion wave function is exclusively relativistic in nature, although it is
by no means negligible, as the relative weight in the valence state amounts up
to 24%. From the point of view of the expansion of the pion wave function on
the LF Fock space, we have that around 30% of the normalization comes from
higher Fock components, and therefore they are quite relevant in the
description of the pion. Naively speaking, we should add that the remaining
probability, $1-P_{val}$, presumably is distributed between states of dressed
degrees of freedom with few gluons and quark-antiquark pairs beyond the
valence state. However, we should convey that it is still unclear how to build
a basis in the LF Fock-space associated with the dressed quark quasi-particle.

On the other hand, for the BS amplitude models (II)-(IV) we have that the
constituent quark mass determines the binding energy ($B$), and larger values
of $B=2m_{q}-M_{\pi}$ corresponds to a more compact configuration for the
pion. Therefore, the values of $f_{\pi}$ increases by increasing the quark
mass, as we observe in Table 2 an approximate proportionality between this two
quantities (see also [57]). Furthermore, it is expected that the valence
probability increases with the constituent mass, as the system tends to be
non-relativistic and dominated by the valence configuration, which is clearly
shown in the table II.

##  VII Valence wave function

###  VII.1 LF projected amplitudes

We start by studying quantitatively the light-front amplitudes
$\psi_{i}(\gamma,z)$, Eq. (56), resulting from the projection of the scalar
functions $\phi_{i}(k,p)$ from the orthogonal basis decomposition of the pion
QCD inspired model BS amplitude. For convenience it is repeated below:

$\psi_{i}(\gamma,z)=-\frac{\imath}{M_{\pi}}\int_{0}^{\infty}d\gamma^{\prime}~{}\frac{g_{i}(\gamma^{\prime},z)}{\left[\frac{z^{2}M_{\pi}^{2}}{4}+\gamma+\gamma^{\prime}\right]^{2}}\,.$

In the model, the weight functions $g_{i}$ have an upper-bound in $\gamma$,
which leads to the asymptotic behavior of $\psi_{i}(\gamma,z)$ by taking the
limit of $\gamma=|\vec{k}_{\perp}|^{2}\to\infty$:

$\psi_{i}(\gamma,z)\sim-\frac{\imath}{\gamma^{2}}\int_{0}^{\infty}\frac{d\gamma^{\prime}}{M_{\pi}}g_{i}(\gamma^{\prime},z)\,,$
(92)

which for the QCD inspired model should mean in principle $\gamma\gg 1$ GeV2
given the model parameters. However, as we are going to see, in practice it is
enough that $\gamma\gg\Lambda_{QCD}^{2}$, with $\Lambda_{QCD}\sim 0.3$ GeV
[58], to have the asymptotic behavior. Indeed, such scale is not immediately
obvious from the model parameters, it becomes apparent when we study the fall-
off of the pion LF amplitudes and valence wave function with the transverse
momentum. This point will be further explored when discussing the valence wave
functions.

Figure 9: Top panel: $\psi_{1}(\gamma,z)$ as a function of $z$ for
$\gamma\,=\,1.0$ GeV2 (solid line) and 1.5 GeV2 (dashed line). Bottom panel:
$\psi_{1}(\gamma,z)$ as a function of $\gamma$ [GeV2] for $z\,=\,0.5$ (solid
line) and 0.75 (dash-dotted line).

Figure 10:  Top panel: $\psi_{2}(\gamma,z)$ as a function of $z$ for
$\gamma\,=\,0.5$ GeV2 (solid line), 0.75 GeV2 (dashed line) and 1 GeV2 (dotted
line). Bottom panel: $\psi_{2}(\gamma,z)$ as a function of $\gamma$ [GeV2] for
$z\,=\,0.5$ (solid line) and 0.75 (dash-dotted line).

Figure 11: Top panel: $\psi_{3}(\gamma,z)$ as a function of $z$ for
$\gamma=0.5$ GeV2 (solid line), 0.75 GeV2 (dashed line) and 1 GeV2 (dotted
line). Bottom panel: $\psi_{3}(\gamma,z)$ as a function of $\gamma$ [GeV2] for
$z\,=\,0.5$ (solid line) and 0.75 (dash-dotted line).

Figure 12: Top panel: $\psi_{4}(\gamma,z)$ as a function of $z$ for $\gamma=1$
GeV2 (solid line), 1.5 GeV2 (dashed line) and 2 GeV2 (dotted line). Bottom
panel: $\psi_{2}(\gamma,z)$ as a function of $\gamma$ [GeV2] for $z=0.5$
(solid line) and 0.75 (dash-dotted line).

We present in Figs. 9 to 12 the numerical results for the light-front project
amplitudes $\psi_{i}(\gamma,z)$ for the QCD inspired model. The normalization
of the weight functions are arbitrarily chosen as $\mathcal{N}$=1.

In Fig. 9 the dependencies of the amplitude $\psi_{1}(\gamma,z)$ in $z$ and
$\gamma$ are shown in the top and bottom panels of the figure, respectively.
The symmetry around $z=0$ reflects the charge conjugation symmetry of the pion
state. The behavior close to the end-points are shown in the figure, and also
the fast damping by increasing $\gamma$, as anticipated in Eq. (92) with the
asymptotic form $\sim\gamma^{-2}$. Another clear feature is the scale of
$\Lambda^{2}_{QCD}$ relevant for $\psi_{1}$ that drives the asymptotic
behaviour with $\gamma\gg\Lambda^{2}_{QCD}$. Such asymptotic behavior is also
verified in $\psi_{j}(\gamma,z)$ $(j=2,3,4)$ in figures 10-12.

In Fig. 10, the LF amplitude $\psi_{2}(\gamma,z)$ is plotted as a function of
$z$ and $\gamma$ in the top and bottom panels of the figure, respectively. The
symmetry property $z\to-z$ can be checked in the figure, as well as the
behavior close to the end-points.

In Fig. 11 the dependencies of the amplitude $\psi_{3}(\gamma,z)$ in $z$ and
$\gamma$ are shown in the top and bottom panels of the figure, respectively.
The odd property of this amplitude with respect to the transformation $z\to-z$
is seen in the top panel of the figure, where results are presented for
$\gamma=0.5$, 0.75 and 1 GeV2.

In Fig. 12, the amplitude $\psi_{4}(\gamma,z)$ is shown and its dependencies
in $z$ and $\gamma$ are illustrated in the top and bottom panels of the
figure, respectively. The discontinuity of the derivative of
$\partial_{z}\psi_{4}(\gamma,z)$ at $z=0$ is visible in the figure, while this
feature of $\partial_{z}\psi_{i}(\gamma,z)|_{z=0}$ is smoothed out in
$\psi_{i}(\gamma,z=0)$ with $i=(1,2,3)$. Such discontinuity is traced back to
the discrete support of the vector and scalar spectral densities of the quark
propagator, which is expected to be washed out in a more realistic model.

###  VII.2 Spin components

The two spin components of the pion valence wave function, namely the anti-
aligned and aligned ones are obtained from the LF projected amplitudes using
Eqs. (53) and (54), respectively. These LF amplitudes for the QCD inspired
pion model were obtained in the previous section, where we made use of the LF
projected scalar amplitudes $\phi_{i}(k,p)$ as written in Eq. (56). In
particular the LF projected amplitudes $\psi_{i}(\gamma,z)$ for $i=2$, 3 and
4, are the ones needed to compute the valence wave function. Note that the
component $\psi_{1}$ does not directly contribute to these spin components,
but we remind that in the BS amplitude normalization condition (91) all the
scalar functions contribute.

In Figs. 13 and 14, we present the results for the valence wave function in
the anti-aligned and aligned quark spin configurations, respectively. In the
top panel of both figures, the behavior with respect to $z$ near the end-
points are visibly different for the two components, where the anti-aligned
spin one presents a faster falloff towards $z\to\pm 1$ in comparison to the
aligned one. Associated with this property, we notice a narrower distribution
of the former in relation to the latter. These characteristics have been
already observed for the pion wave function obtained within a dynamical model
from the solution of the Minkowski space BS equation in the ladder
approximation [27]. The physical reason for the difference between the end-
point behavior is the relativistic origin of the aligned spin component. Due
to that the quarks in this spin configuration have to explore the short-
distance or UV region more frequently than when they are the in the anti-
aligned spin state.

Figure 13: Top panel: antialigned quark spin component of the valence wave
function, $\Psi_{\uparrow\downarrow}(\gamma,z)$, as a function of $z$, for
$\gamma=0.5$ GeV2 (dotted line), 0.75 GeV2 (dashed line) and 1 GeV2 (solid
line). Bottom panel: dependence on $\gamma$ [GeV2] for $z=0.5$ (solid line)
and 0.75 (dot-dashed line). Arbitrary normalization.

Figure 14: Top panel: aligned quark spin component of the valence wave
function, $\Psi_{\uparrow\uparrow}(\gamma,z)$, as a function of $z$, for
$\gamma=0.5$ GeV2 (dotted line), 0.75 GeV2 (dashed line) and 1 GeV2 (solid
line). Bottom panel: dependence on $\gamma$ [GeV2] for $z=0.5$ (solid line)
and 0.75 (dot-dashed line). Arbitrary normalization.

The derivatives $\partial_{z}\psi_{\uparrow\downarrow}(\gamma,z)$ and
$\partial_{z}\psi_{\uparrow\uparrow}(\gamma,z)$ are discontinuous at $z=0$ for
the QCD inspired model, as seen in the top panels of Figs. 13 and 14,
respectively. Such feature is smoothed for the antialigned spin component,
while it is evident for the aligned spin component of the wave function. As we
have already discussed in the previous subsection, such property is due to the
theta functions appearing in Nakanishi weight functions. The discontinuity in
the derivative is expect to disappear for a spectral densities with a
continuous support and not a discrete one, as in the present model.

Another aspect found in the analysis of the LF amplitudes, $\psi_{i}$, and
discussed in the previous subsection, is the emergent IR scale of
$\Lambda_{QCD}^{2}$ in the $\gamma$ dependence of $\psi_{\uparrow\downarrow}$
and $\psi_{\uparrow\uparrow}$. This IR scale corresponds to confinement
distances of the order of 1 fm, which indeed is corroborated by the value of
0.67 fm for the charge radius of the pion in this model obtained in Ref. [24].

Figure 15:  Pion valence longitudinal-momentum distributions as a function of
the momentum fraction for the QCD inspired model (solid line) and for the
solution of the BS equation in Minkowski space [27] (dashed line). Results
presented for the antialigned spin configuration (left panel), aligned spin
configuration (middle panel) and total (right panel).

The analytical form of the asymptotic behaviour of the spin components of the
wave function are easily found by inspecting Eqs. (53) and (54), considering
the finite support of $g_{i}(\gamma,z)$ in $\gamma$ and Eq. (92):

$\displaystyle\Psi_{\uparrow\downarrow}(\gamma,z)$
$\displaystyle\sim-\frac{\imath}{\gamma}\int_{0}^{\infty}\frac{d\gamma^{\prime}}{M_{\pi}^{3}}\frac{\partial}{\partial
z}[g_{3}(\gamma^{\prime},z)]\,,$ (93)
$\displaystyle\Psi_{\uparrow\uparrow}(\gamma,z)$
$\displaystyle\sim-\frac{\imath}{\gamma^{\frac{3}{2}}}\int_{0}^{\infty}\frac{d\gamma^{\prime}}{M_{\pi}}g_{4}(\gamma^{\prime},z)\,.$

The asymptotic behavior of the spin components of the valence LF wave function
of the model, are consistent with the general form drived for QCD in Ref.
[59], namely $~{}1/\gamma$ for the antialigned component and
$1/\gamma^{\frac{3}{2}}$ for the aligned one. Note in the last case the factor
$\sqrt{\gamma}$ has been considered in the asymptotic behaviour in comparison
with Ref. [59]. By inspecting Figs. 13 and 14 for $\gamma\gg\Lambda_{QCD}^{2}$
the asymptotic behavior of Eq. (93) is confirmed. We should observe that the
presence of the running mass was essential to make $g_{3}$ non vanishing. In
the fixed constituent quark mass case, where $g_{3}=0$, the large transverse
momentum fall-off of the antialigned spin component of the wave function would
be $\sim 1/\gamma^{2}$ in disagreement with QCD.

Figure 16:  Pion valence transverse-momentum distributions as a function of
$\gamma=k^{2}_{\perp}$ [GeV2] for the QCD inspired model (solid line) and for
the solution of the BS equation in Minkowski space [27] (dashed line). Results
presented for the antialigned spin configuration (left panel), aligned spin
configuration (middle panel) and total (right panel).

###  VII.3 Momentum distributions

The longitudinal and transverse momentum distributions of the pion valence LF
wave function are found through the proper integration of the probability
density, $\rho_{val}(\gamma,z)$ defined in Eq. (58), and its spin
contributions $\rho_{\uparrow\downarrow}(\gamma,z)$ and
$\rho_{\uparrow\uparrow}(\gamma,z)$ given in Eqs. (59) and (60), respectively.

The pion longitudinal momentum distribution decomposed in the individual
contributions from the two spin configurations, is written as

$\phi(\xi)=\phi_{\uparrow\downarrow}(\xi)+\phi_{\uparrow\uparrow}(\xi)\,.$
(94)

These quantities are the probability densities, integrated over the transverse
momentum:

$\displaystyle\phi_{\uparrow\downarrow}(\xi)=\int^{\infty}_{0}d\gamma~{}\rho_{\uparrow\downarrow}(\gamma,z)\,,$
(95)
$\displaystyle\phi_{\uparrow\uparrow}(\xi)=\int^{\infty}_{0}d\gamma~{}\rho_{\uparrow\uparrow}(\gamma,z)\,,$
(96)

where we remind that $\xi=(1+z)/2$ is the quark longitudinal momentum
fraction.

The results for the longitudinal momentum distributions are presented in Fig.
15, which are compared to the results obtained with the solution of the ladder
BS equation in Minkowski space that fits $f_{\pi}$ (model VIII in [27]) and
the experimental pion electromagnetic form factor [28]. For this comparison,
these distributions are normalized to 1, taking into account the respective
probabilities given in Table 2. It should be noted that the distributions of
the QCD-inspired model are narrower than those obtained with the BS solution
from Ref. [27]. This suggests the relevance of the dynamics of the BS equation
in Minkowski space for that observable and on the other side the quark
dressing in the QCD inspired model. In the last case at the end-points
$\phi(\zeta)\sim\xi^{\eta}$ and $\sim(1-\xi)^{\eta}$, with $\eta\simeq 2$,
while $\eta=1.4$ for the solution of the BS equation in Ref. [27].

Another property observed for the longitudinal distribution of valence in Fig.
15 was the expected wider distribution for the aligned spin configuration with
respect to the antialigned one. As we have already discussed such behavior
comes as a consequence of the genuinely relativistic nature of the aligned
spin component of the valence wave function. However, its overall impact on
the total valence distribution is small due to the low probability (see Table
2) associated with this quark spin state.

The pion transverse momentum distribution is written in terms of the
contributions from individual spin configurations as:

$P(\gamma)=P_{\uparrow\downarrow}(\gamma)+P_{\uparrow\uparrow}(\gamma)\,,$
(97)

where they are defined by integrating over $z$, the spin antialigned and
aligned momentum densities written in Eqs. (59) and (60), respectively.

In this way, the transverse momentum distributions, which complements our
study of the valence state and its spin decomposition, are given by:

$\displaystyle
P_{\uparrow\downarrow}(\gamma)=\int^{1}_{-1}dz~{}\rho_{\uparrow\downarrow}(\gamma,z)\,,$
(98) $\displaystyle
P_{\uparrow\uparrow}(\gamma)=\int^{1}_{-1}dz~{}\rho_{\downarrow\downarrow}(\gamma,z).$
(99)

We analyze in Fig. 16 the distribution of the transverse momentum $P(\gamma)$
given in Eqs. (97), and its decomposition into the spin configurations of the
quarks written in Eq. (98) and (99). For comparison purposes with the QCD-
inspired model, we present in the figure the results obtained by solving the
BS equation in Minkowski space in the ladder approximation [27]. In general
the transverse momentum behavior of the two models is similar, with a falloff
scale of the order of $\Lambda_{QCD}$. On overall, the aligned spin transverse
probability density in the region of $\gamma$ up to 0.6 GeV2 exhibits a slower
falloff with respect to the antialigned one, which reflects the sensitivity to
the UV region, or short distances, this property is model independent.
Independently of that, the contribution of the spin aligned state to the total
is about 15% of the total valence probability, as shown in Table 2.

##  VIII Conclusion

We summarize below the main results and discussions to settle down a
theoretical framework for the Nakanishi integral representation of the pion BS
amplitude within a chiral perspective, as well as the application of the
developed tools in practice to an example of a pion model inspired by QCD
[24], where its LF valence wave function and momentum distributions were
studied in detail. We expect that the developed framework can be an useful
addition to further studies of the pion, where the self-energies are to be
obtained dynamically from the solution of the Dyson-Schwinger equation in
Minkowski space.

  1. i)

A general framework was developed to build the Nakanishi integral
representation of the Bethe-Salpeter amplitude of a model that incorporates
the physics of spontaneous chiral symmetry breaking. Technically, we take into
account the Källen-Lehmann spectral decomposition of the quark propagator, and
the integral representation of the pseudo-scalar vertex function. Starting
from these basic ingredients then the Nakanishi integral representation [30]
was build for the scalar amplitudes obtained in the decomposition of the BS
amplitude in two basis of Dirac matrices, namely, one orthogonal, and the
other one not. In particular, the first one was used in the solution of the BS
equation for the pion in Minkowski space [27]. The subtle transformation
between the two sets of Nakanishi weight functions associated with the scalar
amplitudes from the decomposition of the BS amplitude in the non-orthogonal
and orthogonal basis was derived, which made possible our study.

  2. ii)

We also introduced an auxiliary functional which given the spectral densities
of the vector and scalar components of the quark propagator, and also the
weight function of the pseudo-scalar vertex, allows to derived the weight
functions of associated to the BS amplitude. This formulation was restricted
to the case where the density of the integral representation of the pseudo-
scalar vertex function of the pion was chosen, for simplicity, as having
dependence only on a non-compact variable, which represents a scalar function
of the relative quark-antiquark momentum. This form is suggested by the chiral
limit where the axial-vector Ward identities imposes that the dominant
component of the vertex function, the pseudo-scalar one, is proportional to
the quark scalar self-energy [5, 8].

  3. iii)

From the point of view of the LF Fock-space decomposition of the pion wave
function, the BS amplitude normalization receives contributions beyond the
valence one taking into account an infinite sum of over the dressed quark-
antiquark pairs with any number of dressed gluons, which have the pion quantum
numbers. This is the counterpart of the well known LF quantization framework
relying on the LF Fock-space basis built with eigenstates of the free LF
Hamiltonian, where the normalization of the hadron state is represented by the
infinite sum over the probability of each Fock-state component of the wave
function [11]. All this conceptual complexity that populates the public
imagination, can be technically rephrased by the projection onto the light-
front of the Minkowski space Bethe-Salpeter equation [49, 50]. Although the
$\pi-q\bar{q}$ BS amplitude is defined in terms of the matrix element of only
two quark operators, it contains the virtual propagation of the system in all
LF Fock components with the pion quantum numbers. On this picture, it remains
the challenging task on how to build a LF Fock-space basis with quark and
gluons dressed degrees of freedom. Despite of that, in our practical
application of the developed formalism to the pion state, we have made the
initial step and studied its valence LF wave function with the dressed quark
and antiquark degrees of freedom using the integral representation method
developed within the described framework.

  4. iv)

In the practical application of the tools developed in the present formal
framework, we have study the valence LF wave function properties in detail
within the pion model inspired by QCD. In this model the Bethe-Salpeter
amplitude incorporates the quark self-energy, where the mass function is
momentum dependent, and this in turn determines the quark-pion pseudoscalar
vertex at the chiral limit (zero quark current mass), assuming the theoretical
validity of the chiral Ward-Takahashi identities. The model has a satisfactory
phenomenology reproducing the pion decay constant and electromagnetic form
factor [24]. The quark propagator has three time-like poles at 323, 645 and
954 MeV corresponding to spectral densities of the vector and scalar
components of the propagator with a discrete support, which also happens for
the form factor of the pseudo-scalar $\pi-q\bar{q}$ vertex that has a time-
like pole at 846 MeV. Here, we have derived the analytic form of the Nakanishi
weight functions, which has as an essential ingredient the auxiliary
functional built in the development of our general framework. We found that
many of the properties of the weight functions are due to the particular form
of the auxiliary functional, like the vanishing at the end-points, and a rich
structure, due to the alternate signs of the residue associate with the
spectral densities at the time-like poles of the dressed quark propagator
model. Choosing a fixed constituent quark mass, such rich structure is missed
in the Nakanishi weight functions.

  5. v)

We computed from the integral representation of the Bethe-Salpeter amplitude
the pion light-front valence wave function decomposed in terms of quark
helicities coupled in antialigned and aligned spin states. As a cross-check of
the developed formalism, after the normalization of the Bethe-Salpeter
amplitude, we computed the pion decay constant, and it reproduced the original
model results obtained in [24], with a completely independent method. We have
computed the probabilities of the quark spin aligned state and antialigned
spin one in the valence wave function. We found quite unexpected that the
valence probability of 0.70, divided in 0.58 for the antialigned spin state
and 0.12 for the aligned one. Curiously this result agrees with the
probabilities found from the solution of the pion Minkowski space Bethe-
Salpeter equation in the ladder approximation, where the quark-gluon vertex
scale parameter, the quark and gluon masses have values suggested by lattice
QCD calculations and tuned to fit the decay constant.

  6. vi)

We studied the structure of the pion valence LF wave function, by means of its
spin decomposition, namely their dependence on the transverse and longitudinal
momentum fraction. The typical scale for the falloff of the wave functions is
found to be of the order of $\Lambda_{QCD}$ although not immediately visible
from the parameters of the model. Furthermore, we found that the two spin
components fulfill the asymptotic power-law falloff as $1/k_{\perp}^{2}$ and
$1/k_{\perp}^{3}$, for the spin antialigned and aligned wave functions,
respectively, in agreement with the counting rule for hadronic light-cone wave
functions derived for QCD [59].

  7. vii)

The dressing of the quarks kept the property that the valence longitudinal
momentum distribution in the aligned spin state is wider than the antialigned
one, due to the relativistic origin of the former. Such property was also
found from the solution of the BS amplitude in Minkowski space [27]. The
behavior at the end-points, namely $\xi^{\eta}$ and $(1-\xi)^{\eta}$, for
$\xi\to 0$ and $\xi\to 1$, respectively, has the exponent $\eta\simeq 2$ at
the pion scale. In addition, the valence transverse momentum distribution was
studied.

Acknowledgements: TF is grateful to Giovanni Salmè for helpful discussions on
the Haag’s theorem implications. This work was partially supported by
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) under the
grant 88881.309870/2018-01 (WP), Fundação de Amparo à Pesquisa do Estado de
São Paulo (FAPESP) grants 2017/05660-0 and 2019/07767-1 (TF), 2019/02923-5
(JPBCM), and by Conselho Nacional de Desenvolvimento Científico e Tecnológico
(CNPq) grants 308486/2015-3 (TF), 307131/2020-3 (JPBCM), 313030/2021-9 (WP),
and 464898/2014-5 (INCT-FNA).

##  Appendix A Elimination of $k^{2}$ for $g_{1}(\gamma^{\prime},z^{\prime})$

The Lorentz invariant $k^{2}$ in the numerator of Eq. (38) for $\phi_{1}(k,p)$
has to be manipulated before applying the principle of uniqueness of the
Nakanishi integral representation in order to extract
$g_{1}(\gamma^{\prime},z^{\prime})$. For this aim, we use the following
identity:

$k^{2}=(k^{2}+z\,k\cdot p-\gamma)-z\,k\cdot p+\gamma\,.$ (100)

This allows to write:

$\displaystyle\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma\frac{k^{2}\,G_{4}(\gamma,z)}{\left[k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon\right]^{3}}=$
$\displaystyle\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma\frac{G_{4}(\gamma,z)}{\left[k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon\right]^{2}}$ $\displaystyle-(k\cdot
p)\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma\frac{z\,G_{4}(\gamma,z)}{\left[k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon\right]^{3}}$
$\displaystyle+\int_{-1}^{1}dz\int_{0}^{\infty}d\gamma\frac{\gamma\,G_{4}(\gamma,z)}{\left[k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon\right]^{3}}\,.$ (101)

We integrate by parts in $\gamma$ the first term of the right-hand side of Eq.
(A) in order to write the NIR form with power 3 in the denominator, as given
below:

$2\int^{1}_{-1}dz\int_{0}^{\infty}d\gamma\frac{\int_{0}^{\gamma}d\gamma^{\prime}\,G_{4}(\gamma^{\prime},z)}{(k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon)^{3}}\,.$ (102)

For the second term in the right-hand side of Eq. (A), we integrate by parts
in $z$, which results in:

$-\frac{1}{2}\int_{0}^{\infty}d\gamma\int_{-1}^{1}dz\frac{\partial_{z}[z\,G_{4}(\gamma,z)]}{[k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon]^{2}}\,,$ (103)

where the boundary term is set to zero. Then, in order to have power 3 in the
denominator of Eq. (103) we use the same integration by parts as we have done
to derive Eq. (102):

$-\int_{0}^{\infty}d\gamma\int_{-1}^{1}dz\frac{\int_{0}^{\gamma}d\gamma^{\prime}\,\partial_{z}[z\,G_{4}(\gamma^{\prime},z)]}{[k^{2}+z\,k\cdot
p-\gamma+\imath\epsilon]^{3}}\,.$ (104)

Finally, using Eqs. (A)-(104) one arrives at Eq. (39).

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