# Electromagnetic and two-photon transition form factors of the pseudoscalar
mesons: An algebraic model computation

I. M. Higuera-Angulo<EMAIL_ADDRESS>Instituto de Física y
Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Morelia,
Michoacán 58040, México R. J. Hernández-Pinto<EMAIL_ADDRESS>Facultad de
Ciencias Físico-Matemáticas, Universidad Autónoma de Sinaloa, Ciudad
Universitaria, Culiacán, Sinaloa 80000, México K. Raya<EMAIL_ADDRESS>Department of Integrated Sciences and Center for Advanced Studies in Physics,
Mathematics and Computation, University of Huelva, E-21071 Huelva, Spain A.
Bashir<EMAIL_ADDRESS>Instituto de Física y Matemáticas, Universidad
Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán 58040, México
Department of Integrated Sciences and Center for Advanced Studies in Physics,
Mathematics and Computation, University of Huelva, E-21071 Huelva, Spain

###### Abstract

We compute electromagnetic and two-photon transition form factors of ground-
state pseudoscalar mesons: $\pi,\,K,\,\eta_{c},\,\eta_{b}$. To this end, we
employ an algebraic model based upon the coupled formalism of Schwinger-Dyson
and Bethe-Salpeter equations. Within this approach, the dressed quark
propagator and the relevant Bethe-Salpeter amplitude encode the internal
structure of the corresponding meson. Electromagnetic properties of the meson
are probed via the quark-photon interaction. The algebraic model employed by
us unifies the treatment of all ground-state pseudoscalar mesons. Its
parameters are carefully fitted performing a global analysis of existing
experimental data including the knowledge of the charge radii of the mesons
studied. We then compute and predict electromagnetic and two-photon transition
form factors for a wide range of probing photon momentum-squared which is of
direct relevance to the experimental observations carried out thus far or
planned at different hadron physics facilities such as the Thomas Jefferson
National Accelerator Facility (JLab) and the forthcoming Electron-Ion
Collider. We also present comparisons with other theoretical models and
approaches and lattice quantum chromodynamics.

##  I Introduction

Unraveling the internal structure of hadrons is one of the most intriguing,
challenging and yet highly researched topics within the strong interaction
physics. Although the underlying fundamental field theory, i.e., quantum
chromodynamics (QCD), was proposed on firm grounds some five decades ago, its
most compelling success is predominantly restricted to its perturbative domain
alone. The emergent non-perturbative phenomena of confinement, hadron masses
and structural properties of even the simplest strongly interacting bound
states are notoriously hard to be elucidated from the underlying first
principles. The plain reason is that the well-established and well-trusted
orthodox perturbative methods are no longer suitable for this undertaking
Gross  _et al._ (2023); Accardi  _et al._ (2023). Different non-perturbative
approaches are relied upon, including: lattice QCD, coupled formalism of
Schwinger-Dyson (SDEs) and Bethe-Salpeter equations (BSEs), and other
effective theories and models (see, _e.g._ Constantinou  _et al._ (2021);
Bashir  _et al._ (2012); Aguilar  _et al._ (2019); Lepage  _et al._ (1992);
Brambilla  _et al._ (2014)). From the experimental point of view, unveiling
the properties of hadrons in terms of cross sections is equally challenging
but essential to make progress through direct comparison and contrast with
theory. Ongoing and planned facilities aim at probing the internal structure
of hadrons at different energy scales with an unprecedented resolution and
precision Accardi  _et al._ (2023); Aznauryan  _et al._ (2013); Chávez  _et
al._ (2022); Chen  _et al._ (2020); Huber  _et al._ (2006).

Much like the hydrogen atom for quantum electrodynamics (QED), pions and
kaons, the simplest two-body strongly interacting bound states, serve as a
portal to understand the intricate dynamics of QCD. They hold a unique
relationship with the dynamical mechanism underlying the Emergent Hadronic
Mass (EHM) Arrington  _et al._ (2021); Horn and Roberts (2016). Comparison
with the heavy counterparts of these light mesons, such as $\eta_{c}$ and
$\eta_{b}$, reveals essential differences with the diametrically opposed mass
generating mechanisms, namely, EHM and the mass triggered by the famous Higgs
mechanism. These competing sources expose themselves via different hadron
observables such as form factors and parton distributions. In this article, we
focus on the form factors of ground-state pseudoscalar mesons
($\scriptsize{\hbox{M}}=\pi,\,K,\,\eta_{c},\,\eta_{b}$), including
electromagnetic ($\gamma^{*}\scriptsize{\hbox{M}}\to\scriptsize{\hbox{M}}$)
and transition ($\gamma\gamma^{*}\to\scriptsize{\hbox{M}}$) form factors
(labeled as EFF and TFF, respectively).

EFFs provide valuable information about the internal dynamics of hadrons, such
as the charge and current distribution, and how asymptotic QCD predictions can
be approached for sufficiently large photon momentum squared $Q^{2}$. It’s of
course more and more unlikely to prevent a meson from breaking up at large
$Q^{2}$ and it becomes increasingly hard for the experiment to probe widely
disparate momentum scales in one single experiment. Both the Thomas Jefferson
National Accelerator Facility (JLab) and the planned Electron-Ion Collider
(EIC) will push the observable range of $Q^{2}$ to unprecedentedly large
intervals. On the other hand, TFFs $\gamma\gamma^{*}\to\scriptsize{\hbox{M}}$
can be accessed to much larger $Q^{2}$ with relative ease as they rely on
mesons production mechanism rather than keeping them intact; in this scenario,
the main challenge would be primarily controlling the background produced by
other processes.

Noticeably, pseudoscalar meson’s EFF and two-photon TFF are characterized by a
single form factor, hence facilitating its experimental extraction Horn and
Roberts (2016). Furthermore, these form factors provide a neat platform to
test fundamental predictions of QCD, such as scaling violations and
factorization formulae Lepage and Brodsky (1980); Efremov and Radyushkin
(1980); Lepage and Brodsky (1979). The latter entails that, at sufficiently
large energy scales ($Q^{2}\gg\Lambda_{\text{QCD}}^{2}$), asymptotic QCD
limits are approached:

$\displaystyle Q^{2}F_{\scriptsize{\hbox{M}}}(Q^{2})$ $\displaystyle\sim$
$\displaystyle[f_{\scriptsize{\hbox{M}}}\tilde{\mathnormal{w}}_{\scriptsize{\hbox{M}}}^{q}(Q^{2})]^{2}\alpha_{s}(Q^{2})\;,$
(1) $\displaystyle Q^{2}G_{\scriptsize{\hbox{M}}}(Q^{2})$ $\displaystyle\sim$
$\displaystyle[f_{\scriptsize{\hbox{M}}}\tilde{\mathnormal{w}}_{\scriptsize{\hbox{M}}}^{q}(Q^{2})]^{1}\;,$
(2)

where $F_{\scriptsize{\hbox{M}}}$ and $G_{\scriptsize{\hbox{M}}}$ denote the
EFF and TFF, respectively; $\alpha_{s}(Q^{2})$ is the strong coupling constant
at one loop, $f_{\scriptsize{\hbox{M}}}$ is the pseudoscalar meson leptonic
decay constant and

$\displaystyle\tilde{\mathnormal{w}}_{\scriptsize{\hbox{M}}}^{q}(Q^{2})=\int_{0}^{1}dx\frac{1}{x}\phi_{\scriptsize{\hbox{M}}}^{q}(x,Q^{2})\;.$
(3)

Here $\phi_{\scriptsize{\hbox{M}}}^{q}(x,Q^{2})$ refers to the leading-twist
parton distribution amplitude (PDA) Chang  _et al._ (2013a). It is a close
analogue of the quantum mechanical wave function of the meson whose asymptotic
form adopts the form:

$\displaystyle\phi_{\scriptsize{\hbox{M}}}^{q}(x,Q^{2}\to\infty)\to\phi_{asy}(x)=6x(1-x)\;.$
(4)

Scaling violations are explicit in Eq. (1) due to the presence of
$\alpha_{s}(Q^{2})$; and, although the expression of the TFF, Eq. (2), does
not plainly exhibit such scaling violations, the way in which this asymptotic
result is approached for increasing $Q^{2}$ is implicitly governed by these
scaling violations Raya  _et al._ (2016). The factorization theorems also
reveal that, asymptotically, the form factors would be weighted by the decay
constant, which is a measure of dynamical chiral symmetry breaking Sultan  _et
al._ (2021). For the TFF, this behavior is manifest in the opposite energy
domain where the Abelian chiral anomaly entails Adler (1969); Bell and Jackiw
(1969); Adler (2004):

$\displaystyle 2f_{\pi}^{0}G_{\pi}(Q^{2}=0)=1\;,$ (5)

where $f_{\pi}^{0}=0.092$ GeV is the value of the pion decay constant in the
chiral limit.

Several experimental efforts are aimed at measuring the pion EFF Amendolia
_et al._ (1986a); Ackermann  _et al._ (1978); Volmer  _et al._ (2001); Horn
_et al._ (2006); Huber  _et al._ (2008) as well as TFF Behrend  _et al._
(1991); Gronberg  _et al._ (1998); Aubert  _et al._ (2009); Uehara  _et al._
(2012). Notably, for the pion EFF, all collaborations converge to an excellent
agreement within the available domain of results (i.e., up to
$2.5\,\text{GeV}^{2}$). Notwithstanding, current trend of experimental results
falls short of approaching the asymptotic limit, i.e., Eq. (1). JLab and EIC
planned experiments aim to extend the upper bound of observed $Q^{2}$ to
around 8.5 GeV2 (12 GeV upgrade), 15 GeV2 (potential 22 GeV upgrade), and 35
GeV2 Arrington  _et al._ (2021); Aguilar  _et al._ (2019); Accardi  _et al._
(2023); Horn and Roberts (2016); Huber  _et al._ (2006), respectively.

On the other hand, there remains an existing controversy concerning the two-
photon pion TFF. In this case, while all available experimental data agree at
low to moderate values of $Q^{2}$, the results above $Q^{2}\sim
10\,\text{GeV}^{2}$ of the BaBar collaboration, Aubert  _et al._ (2009),
disagree markedly with those from the Belle collaboration Uehara  _et al._
(2012) and from the perturbartive QCD prediction of Eq. (2) for sufficiently
large values of $Q^{2}$. Moreover, is appears to be at par with all other
extant pion properties. This marked discrepancy has resulted in some analyses
to argue that the BaBar data might not be considered an accurate measure of
the pion TFF Roberts  _et al._ (2010); Brodsky  _et al._ (2011a); Bakulev  _et
al._ (2011); Brodsky  _et al._ (2011b); Stefanis  _et al._ (2013); Bakulev
_et al._ (2012); El-Bennich  _et al._ (2013); Melikhov and Lucha (2014); Raya
_et al._ (2016). Concerning the kaon, empirical information is currently
scarce and limited to the low-$Q^{2}$ regime Amendolia  _et al._ (1986b);
Dally  _et al._ (1980); Carmignotto  _et al._ (2018); but more data, and in a
larger domain, which is crucially required, is under preliminary analysis,
Horn  _et al._ . It is worth recalling that there is available BaBar data for
the $\gamma\gamma^{\ast}\rightarrow\eta_{c}$ transition which extends up to
probing photon momentum squared $Q^{2}>35\,\text{GeV}^{2}$, displaying
satisfactory consistency with Eq. (2), Raya  _et al._ (2017). Note that no
experimental results are yet reported for the transition
$\gamma\gamma^{\ast}\rightarrow\eta_{b}$ for which predictions are available
Raya  _et al._ (2017). Within a set of sophisticated truncations of SDEs/BSEs,
several form factors have been scrutinized, including
$\gamma^{*}\pi^{+}\to\pi^{+}$ and $\gamma^{*}K^{+}\to K^{+}$ EFFs Chang  _et
al._ (2013b); Gao  _et al._ (2017a); Eichmann  _et al._ (2020); Miramontes
_et al._ (2022); Xu  _et al._ (2023); Raya  _et al._ (2022a), TFFs
$\gamma^{*}\gamma^{*}\to\\{\pi^{0},\eta,\,\eta^{\prime},\,\eta_{c},\,\eta_{b}\\}$
Raya  _et al._ (2016, 2017); Ding  _et al._ (2019); Raya  _et al._ (2020);
Eichmann  _et al._ (2017). Time-like EFFs Miramontes and Bashir (2023a, b);
Miramontes  _et al._ (2021) have also begun to emerge which extends the domain
of applicability of the SDE formalism. Then there are much simpler SDE-based
Contact Interaction (CI) results Roberts  _et al._ (2010); Raya  _et al._
(2018); Bedolla  _et al._ (2016); Xing  _et al._ (2021); Wang  _et al._
(2022); Hernández-Pinto  _et al._ (2023); Dang  _et al._ (2023); Xing  _et
al._ (2024), whose reliable applicability is limited to small $Q^{2}$ values.
We work with a recently proposed Algebraic Model (AM) which captures large
$Q^{2}$ properties of QCD fairly well while still preserving the simplicity of
the CI.

This manuscript is organized as follows: in Sec. II we describe the basic
elements of the AM employed throughout this work, and construction of the
quark-photon vertex (QPV); in Sec. III we derive semi-analytical expressions
for the EFF of pseudoscalar mesons, considering the most generic scenario,
_i.e._ , with different flavours of dressed quarks; Sec. IV contains an
analogous derivation for the TFFs of pseudoscalar mesons. Sec. V details the
results of our global fit of EFFs and TFFs within the AM, related to the best
fitted parameters for pion, kaon, $\eta_{c}$ and $\eta_{b}$ mesons. Finally,
Sec. VI is dedicated to the conclusions of this work and perspectives for
future research directions.

##  II The formalism

The internal structure of the pseudoscalar meson is encoded within the so
called Bethe-Salpeter amplitude (BSA), which in turn is related with the fully
dressed quark propagator through the BSE and the axial vector Ward-Takahashi
identity. Important aspects of the structural properties of the meson are
exposed via its electromagnetic interaction with photons, such as those
unravelled by the EFFs and TFFs. Within the present approach, the standard
application of the Feynman rules to the meson-photon interactions, _i.e._ ,
those describing the $\scriptsize{\hbox{M}}\gamma\scriptsize{\hbox{M}}$ and
$\gamma\scriptsize{\hbox{M}}\gamma$ vertices, demands the knowledge of the
QPV. Moreover, the complete evaluation of the form factors requires prior
knowledge of the dressed quark propagator, the meson BSA, and the QPV. Their
construction is discussed below.

###  II.1 The algebraic model

In order to systematically describe the structure of a pseudoscalar meson, M,
a typical starting point is the derivation of the Bethe-Salpeter wave function
(BSWF). This is expressed in terms of the $q$-quark propagator $S_{q}$ and the
corresponding BSA, $\Gamma_{\scriptsize{\hbox{M}}}$, in the following form:

$\displaystyle{\chi}_{\scriptsize{\hbox{M}}}\left(k_{-},P\right)=S_{q}(k)\Gamma_{\scriptsize{\hbox{M}}}\left(k_{-},P\right)S_{\bar{q}^{\prime}}\left(k-P\right)\,,$
(6)

where $q$ and $\bar{q}^{\prime}$ correspond to the valence quark and antiquark
respectively, $k_{-}=k-P/2$ ($k$ is the relative momentum between the two
quarks), and $P^{2}=-m_{\scriptsize{\hbox{M}}}^{2}$ is the squared mass of the
meson. The quark propagator and BSA can be obtained by solving the
corresponding SDE and BSE, which might require arduous work Qin and Roberts
(2020). However, taking advantage of the advances and understanding in this
direction, it is possible to build simple, yet efficient and reliable
Ans$\ddot{a}$tze of these entities that observe key QCD symmetries and mimic
the expected behavior of the numerical solutions, and thus translate the full
numerical approach to an amicable and trustable algebraic analysis in some
measure.

A family of such models have been employed in, among others, Refs. Bedolla
_et al._ (2016); Mezrag  _et al._ (2016); Xu  _et al._ (2018); Raya  _et al._
(2022b); Albino  _et al._ (2022); Almeida-Zamora  _et al._ (2024) for
pseudoscalar mesons and in Ref. Almeida-Zamora  _et al._ (2023) for the vector
meson case. We adopt the phenomenologically motivated AM described in Albino
_et al._ (2022), which is determined once the quark propagator and the
pseudoscalar meson BSA are specified as follows:

$\displaystyle S_{q(\bar{q}^{\prime})}(k)$ $\displaystyle=$
$\displaystyle\left[-i\gamma\cdot
k+m_{q(\bar{q}^{\prime})}\right]\Delta\left(k^{2};m_{q(\bar{q}^{\prime})}^{2}\right)\,,$
(7) $\displaystyle
n_{\scriptsize{\hbox{M}}}\Gamma_{\scriptsize{\hbox{M}}}(k,p)$ $\displaystyle=$
$\displaystyle
i\gamma_{5}\int_{-1}^{1}dw\,\rho_{\scriptsize{\hbox{M}}}(w)\left[\hat{\Delta}\left(k_{w}^{2};\Lambda_{w}^{2}\right)\right]^{\nu},$
(8)

where $\Delta(a;b)=(a+b)^{-1},\;\hat{\Delta}(a;b)=b\Delta(a;b)$ and
$k_{w}=k+(w/2)p$. Herein, $\nu=1+\delta$ is a parameter that controls the
asymptotic behavior of the BSA, with $\delta$ playing the role of an anomalous
dimension; $m_{q(\bar{q}^{\prime})}$ is a mass scale that corresponds to the
dressed mass for a given quark (antiquark) flavor $q(\bar{q}^{\prime})$, and
$n_{\scriptsize{\hbox{M}}}$ is a normalization constant. Moreover,
$\rho_{\scriptsize{\hbox{M}}}(w)$ is identified with a spectral density whose
form dictates the pointwise behavior of the BSA and has significant impact on
a kaleidoscopic array of meson observables111We refer to the Appendix for the
extraction of the spectral density function in the AM.. As explained in Albino
_et al._ (2022) in great detail, the spectral density can be determined if the
PDA is known. This process largely anticipates that an appropriate PDA would
produce sensible form factors, distribution functions, generalized parton
functions, and other related physical observables. Finally,
$\Lambda_{w}^{2}\equiv\Lambda^{2}(w)$ is defined as follows:

$\displaystyle\Lambda_{w}^{2}$ $\displaystyle\equiv$ $\displaystyle
m_{q}^{2}-\frac{1}{4}\left(1-w^{2}\right)m_{\scriptsize{\hbox{M}}}^{2}+\frac{1}{2}\left(1-w\right)\left(m_{\bar{q}^{\prime}}^{2}-m_{q}^{2}\right).$
(9)

It leads to the requirement that
$m_{\scriptsize{\hbox{M}}}<m_{q}+m_{\bar{q}^{\prime}}$ and prevents the
appearance of zeroes in the denominator. The model for the quark propagator
and the BSA do not impose quark confinement. However, the constraint of Eq.
(9) is naturally satisfied by the Nambu-Goldstone bosons (_i.e._ $\pi,\,K$).
Moreover, numerical solutions for the heavier pseudoscalar mesons
($\eta_{c},\,\eta_{b}$),  _e.g._ , Refs. Sultan  _et al._ (2021); Ding  _et
al._ (2019) are also consistent with it. The spectral densities constructed
through this ansatz correspond to a non point-like state, and produce
momentum-dependent form factors in agreement with the computation through
other theoretical tools as well as experiment.

We would like to point out that with the simple structure required for the
quark propagator and the BSA, Eq. (7) and Eq. (8), the complete BSWF in the AM
would be fully determined by the parameters $m^{2}_{q(\bar{q}^{\prime})}$ and
$\nu$. An additional one, $\alpha_{q}^{(0)}$, arising from the QPV, is
introduced later on. It leaves us with a total of 3-4 parameters to be
specified, depending on whether the quark and the antiquark are of the same
flavor or observe isospin symmetry. Moreover, these parameters have little
maneuverability as we have a good knowledge of the dressed quark masses, meson
masses and the anomalous dimensions. The complete set of parameters shall be
constrained through employing both experimental measurements and our
theoretical understanding of the hadron structure. To carry out the
calculation of these observables, what remains is the reliable knowledge of
the interaction of the electromagnetic probe with the dressed quarks, _i.e._ ,
the QPV; this is what we set out to discuss in the next subsection.

###  II.2 The quark-photon vertex

As stated before, the extraction of EFF and TFF is related with the
$\scriptsize{\hbox{M}}\gamma\scriptsize{\hbox{M}}$ and
$\gamma\scriptsize{\hbox{M}}\gamma$ vertices, respectively. At the fundamental
level, these meson-photon interactions are expressed in terms of the fully-
dressed QPV, $\Gamma_{\mu}^{q}$. Hence, a proper construction of the latter
becomes essential. It turns out that it is convenient to work with the
unamputated version of the vertex, which reads:

$\displaystyle\chi_{\mu}^{q}(k_{f},k_{i})=S_{q}(k_{f})\Gamma_{\mu}^{q}(k_{f},k_{i})S_{q}(k_{i})\,.$
(10)

We thus adopt the Ansatz described in Ref. Raya  _et al._ (2016):

$\displaystyle\chi_{\mu}^{q}(k_{f},k_{i})$ $\displaystyle=$ $\displaystyle
T^{(1)}_{\mu}\Delta_{k^{2}\sigma_{v}}+T^{(2)}_{\mu}\Delta_{\sigma_{v}}+T^{(3)}_{\mu}\Delta_{\sigma_{s}}\,,$
(11) $\displaystyle T^{(1)}_{\mu}$ $\displaystyle=$
$\displaystyle\gamma_{\mu}\,,$ $\displaystyle T^{(2)}_{\mu}$ $\displaystyle=$
$\displaystyle\not\\!{k_{f}}\gamma_{\mu}\not\\!{k_{i}}+\alpha_{q}(\not\\!{k_{f}}\gamma_{\mu}\not\\!{k_{i}}-\not\\!{k_{i}}\gamma_{\mu}\not\\!{k_{f}})\,,$
$\displaystyle T^{(3)}_{\mu}$ $\displaystyle=$ $\displaystyle
i(\not\\!{k_{f}}\gamma_{\mu}+\gamma_{\mu}\not\\!{k_{i}})$ $\displaystyle+$
$\displaystyle
i\alpha_{q}(\not\\!{k_{f}}\gamma_{\mu}+\gamma_{\mu}\not\\!{k_{i}}-\not\\!{k_{i}}\gamma_{\mu}-\gamma_{\mu}\not\\!{k_{f}})\,;$

here $\Delta_{F}=[F(k_{f}^{2})-F(k_{i}^{2})]/[k_{f}^{2}-k_{i}^{2}]$ and
$\sigma_{v,s}$ are the quark propagator dressing functions Sultan  _et al._
(2021):

$\sigma_{s}(p^{2})=M(p^{2})\sigma_{v}(p^{2})=\frac{Z(p^{2})M(p^{2})}{p^{2}+M^{2}(p^{2})}\;;\,$
(12)

here $Z(p^{2}):=1$ as inferred from Eq. (7). The part proportional to
$\alpha_{q}$ is transverse to the external photon momentum $Q_{\mu}$. It is
introduced as a momentum redistribution factor in the TFF, owing to the
impossibility to simultaneously conserve the vector and axial-vector currents,
which would make it prohibitively difficult to satisfy the Abelian anomaly of
Eq. (5). As long as $\alpha_{q}$ is rapidly damped asymptotically, the point
particle limit, $\Gamma_{\mu}\to\gamma_{\mu}$, is properly recovered. With
that in mind, this factor is expressed as

$\alpha_{q}=\alpha^{(0)}_{q}\,\exp[-Q^{2}/(2m_{q}^{2})]\,,$ (13)

where the parameter $\alpha^{(0)}_{q}$ sets the strength of the transverse
pieces of the QPV. The presence of $m_{q}$ as a flavor-dependent mass scale is
natural, given that the transverse structures of the QPV are closely linked to
the dynamical breaking of chiral symmetry Albino  _et al._ (2019). The value
$\alpha^{(0)}_{q}$ plays an essential role in ensuring the correct
normalization of $G_{\scriptsize{\hbox{M}}}(0)$, hence guaranteeing a proper
description of the TFFs near vanishingly small momenta. The way the
$\alpha_{q}$ term impacts the TFF, in fact, is similar to the contribution of
the anomalous magnetic moment of the quark, whose importance for this and the
anomalous $\gamma^{*}\to 3\pi$ process has recently been explained in Refs.
Dang  _et al._ (2023); Xing  _et al._ (2024). Conversely,
$F_{\scriptsize{\hbox{M}}}(Q^{2})$ and the associated charge radius are
completely independent of $\alpha^{(0)}_{q}$; firstly, because the value
$F_{\scriptsize{\hbox{M}}}(Q^{2}=0)=1$ is completely fixed by the longitudinal
part of the QPV and, secondly, due to the fact that only the dominant meson
BSA has been considered, Raya  _et al._ (2016).

It is worth mentioning that the Ansatz for the QPV in Eq. (11) has shown its
robustness in the computation of the pion EFF Chang  _et al._ (2013b) and all
two-photon TFFs involving ground-state neutral pseudoscalar mesons, Raya  _et
al._ (2016, 2017); Ding  _et al._ (2019); Raya  _et al._ (2020). Furthermore,
having been derived from the so-called gauge-technique Delbourgo and West
(1977), the present vertex construction fulfills crucial mathematical
requirements Albino  _et al._ (2019); Bermudez  _et al._ (2017): it is free of
any kinematic singularities, the free-field limit is properly recovered, and
it satisfies its corresponding Ward-Green-Takahashi identity. The latter
identity is crucial in the present approach, since it enables us to express
the QPV dressing functions in terms of those of the quark propagator,
facilitating any related computation. Furthermore, given the simple character
of the quark propagator in the present AM, Eq. (7), the QPV can be
systematically cast in a compact manner:

$\displaystyle\chi_{\mu}^{q}(k_{f},k_{i})=\frac{\sum_{j=1}^{3}T^{(j)}_{\mu}X_{j}}{[k_{f}^{2}+m_{q}^{2}][k_{i}^{2}+m_{q}^{2}]}\,,$
(14)

where $X_{1}=m_{q}^{2}\,,\,\,X_{2}=-1\,,\,\,X_{3}=-m_{q}$.

Finally, it is worth recalling that this construction of the QPV introduces
only one extra parameter to be determined, i.e., $\alpha^{(0)}_{q}$. Next, in
Sec. III and Sec. IV, we proceed to derive the expressions related to the EFF
and TFF, respectively. The corresponding numerical results are discussed in
Sec. V.

##  III Electromagnetic form factors

Figure 1: $\scriptsize{\hbox{M}}\gamma\scriptsize{\hbox{M}}$ vertex,
corresponding to the EFF, in the impulse approximation. Here
$\Gamma_{\scriptsize{\hbox{M}}}$ denotes the M-meson BSA amplitude and
$\Gamma_{\mu}$ the QPV. The edges of the triangle represent the fully dressed
quark propagator.

Considering all the essential ingredients discussed previously, we can now
proceed to compute the EFF and TFF in the AM. As we shall see, it turns out to
be an analytical task to a large extent. The EFF is written down in the
impulse approximation Chang  _et al._ (2013b) which graphically leads to the
triangle diagram shown in Fig. 1. Mathematically, this corresponds to the
following expression:

$\displaystyle K_{\mu}F_{\scriptsize{\hbox{M}}}^{q}(Q^{2})$ $\displaystyle=$
$\displaystyle
N_{c}\,\mbox{tr}\int\frac{d^{4}k}{(2\pi)^{4}}\chi_{\mu}^{q}(k+p_{f},k+p_{i})$
(15) $\displaystyle\times$
$\displaystyle\Gamma_{\scriptsize{\hbox{M}}}(k_{i},p_{i})S_{\bar{q}\,^{\prime}}(k)\Gamma_{\scriptsize{\hbox{M}}}(k_{f},p_{f})\,,$

where $Q$ is the incoming photon momentum and the trace is over spinor
indices. The incoming and outgoing meson momenta are denoted by $p_{f,i}=K\pm
Q/2$, while the relative momenta of the quark-meson-antiquark vertex are
$k_{f,i}=k+p_{f,i}/2$, such that we have $K\cdot Q=0$ and
$p_{f(i)}^{2}=K^{2}+Q^{2}/2=-m_{M}^{2}$. Note that Eq. (15) corresponds to the
case when the photon strikes quark $q$. Naturally, we would also have to
consider the case when the antiquark is hit. Hence, the total EFF of the meson
would be written as:

$\displaystyle
F_{\scriptsize{\hbox{M}}}(Q^{2})=e_{q}F_{\scriptsize{\hbox{M}}}^{q}(Q^{2})+e_{\bar{q}^{\prime}}F_{\scriptsize{\hbox{M}}}^{\bar{q}^{\prime}}(Q^{2})\,,$
(16)

where, $e_{q(\bar{q^{\prime}})}$ is the quark (antiquark) electric charge, in
units of that of the positron222For neutral mesons composed of same flavored
quarks, the total EFF is simply defined as
$F_{\scriptsize{\hbox{M}}}=F_{\scriptsize{\hbox{M}}}^{q}$..

In order to extract the individual contributions of dressed quarks,
$F_{\scriptsize{\hbox{M}}}^{q}$, we contract Eq. (15) with $K^{\mu}$ and take
the corresponding Dirac trace. With the quark propagator, meson BSA and QPV as
defined in the previous section, one eventually arrives at:

$\displaystyle
F_{\scriptsize{\hbox{M}}}^{q}(Q^{2})=\int_{k}\int_{-1}^{1}\prod_{i=1}^{2}dw_{i}\rho(w_{i})\Lambda_{w_{i}}^{2\nu_{i}}\frac{\mathcal{M}_{q,\bar{q}^{\prime}}(k;K,Q)}{\mathcal{D}_{q,\bar{q}^{\prime}}^{\nu_{1,2}}(k;K,Q)}\,,$
(17)

where the numerator $\mathcal{M}_{q,\bar{q}^{\prime}}$ has the form,

$\displaystyle\mathcal{M}_{q,\bar{q}^{\prime}}(k;K,Q)\equiv\frac{2N_{c}}{K^{2}}(4m_{M}^{2}+Q^{2})[2k^{2}+2m_{q}m_{\bar{q}^{\prime}}]$
$\displaystyle\hskip 14.22636pt-2(K\cdot
k)[2k^{2}+4m_{q}^{2}m_{\bar{q}^{\prime}}^{2}-2m_{q}^{2}-2m_{M}^{2}]\,,$ (18)

and $\mathcal{D}_{q,\bar{q}^{\prime}}^{\nu_{1,2}}(k;K,Q)$ contains the product
of all denominators related with the quark propagators, BSAs and QPV:

$\displaystyle\mathcal{D}_{q,\bar{q}^{\prime}}^{\nu_{1,2}}(k;K,Q)=\Delta(k_{(1+w_{1}),i}^{2},\Lambda_{w_{1}}^{2})^{\nu_{1}}\Delta(k_{(1-w_{2}),f}^{2},\Lambda_{w_{1}}^{2})^{\nu_{2}}$
$\displaystyle\hskip
48.36967pt\times\;\Delta((k+p_{f})^{2},m_{q}^{2})\Delta((k+p_{i})^{2},m_{q}^{2})$
$\displaystyle\hskip
48.36967pt\times\;\Delta(k^{2},m_{\bar{q}^{\prime}}^{2})\,,$ (19)

with, $k_{(1\pm w),i}=k+(1\pm w)p_{i}/2$. Additionally, the simplifying
notation $\int_{k}\equiv\int\frac{d^{4}k}{(2\pi)^{4}}$ has been employed.

To compute the integration on $k$ in Eq. (17), we introduce Feynman
parameterization and perform a suitable change of variables. Consequently, we
find that the denominator can be simply reduced to
$[k^{2}+\Omega^{2}]^{\nu_{1}+\nu_{2}+3}$ where $\Omega^{2}$ has the following
form,

$\displaystyle\Omega^{2}=\frac{1}{4}Q^{2}c_{0}+\frac{1}{2}(m_{q}^{2}-m_{\bar{q}^{\prime}}^{2})c_{1}+\frac{1}{4}m_{\scriptsize{\hbox{M}}}^{2}c_{2}+m_{\bar{q}^{\prime}}^{2}\,,$
(20)

with $c_{0,1,2}$ given in terms of the Feynman parameters $u_{i}$ as follows:

$\displaystyle c_{0}$ $\displaystyle=$
$\displaystyle[u_{1}(w_{1}+1)+2u_{4}][u_{2}(1-w_{2})+2u_{3}]\,,$
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle[u_{1}(w_{1}+1)+u_{2}(w_{2}+1)+2u_{3}+2u_{4}]\,,$ $\displaystyle
c_{2}$ $\displaystyle=$ $\displaystyle
u_{1}^{2}(w_{1}+1)^{2}+u_{2}^{2}(w_{2}-1)^{2}$ (21)
$\displaystyle+2u_{1}(w_{1}+1)[u_{2}(1-w_{2})+2u_{3}+2u_{4}-1]$
$\displaystyle+2u_{2}(1-w_{2})(2u_{3}+2u_{4}-1)$
$\displaystyle+4[u_{3}^{2}+u_{3}(2u_{4}-1)+(u_{4}-1)u_{4}]\,.$

Considering that we are computing an elastic scattering, the initial meson is
the same as the final meson. Thus, without loss of generality, we choose
$\nu=\nu_{1}=\nu_{2}$. Hence, the integral in Eq. (17) becomes

$\displaystyle\int_{k}\frac{\mathcal{M}_{q,\bar{q}^{\prime}}(k,K)}{[t^{2}+\Omega^{2}]^{2\nu+3}}$
$\displaystyle=$
$\displaystyle\int_{k}\frac{k^{2}\mathcal{A}+\mathcal{B}}{[k^{2}+\Omega^{2}]^{2\nu+3}}$
(22) $\displaystyle=$
$\displaystyle\alpha\frac{\mathcal{A}}{[\Omega^{2}]^{2\nu}}+\beta\frac{\mathcal{B}}{[\Omega^{2}]^{2\nu+1}}\,,$

where we have defined:

$\displaystyle\mathcal{A}=3[u_{1}(1+w_{1})+u_{2}(1-w_{2})+2u_{3}+2u_{4}]-8\,,$
$\displaystyle\mathcal{B}=\frac{-1}{2}m_{\scriptsize{\hbox{M}}}^{2}[u_{1}(1+w_{1})+u_{2}(1-w_{2})+2(u_{3}+u_{4}-1)]^{2}$
$\displaystyle\times[u_{1}(1+w_{1})+u_{2}(1-w_{2})+2(u_{3}+u_{4})]$
$\displaystyle+4m_{q}^{2}m_{\bar{q}^{\prime}}^{2}[u_{1}(1+w_{1})+u_{2}(1-w_{2})+2(u_{3}+u_{4}-1)]$
$\displaystyle-\frac{1}{2}Q^{2}[u_{1}(1+w_{1})+2u_{4}][u_{2}(1-w_{2})+2u_{3}]$
$\displaystyle\times[u_{1}(1+w_{1})+u_{2}(1-w_{2})+2(u_{3}+u_{4}-2)]\,,$ (23)

with $\alpha$ and $\beta$ being the following constants

$\displaystyle\alpha=\frac{1}{(4\pi)^{2}}\frac{1}{2\nu(2\nu^{2}+3\nu+1)}\,,\,\beta=\nu\alpha\,.$
(24)

Thus, the EFF can be expressed as follows

$\displaystyle F_{\scriptsize{\hbox{M}}}^{q}(Q^{2})$ $\displaystyle=$
$\displaystyle
2N_{c}\frac{\Gamma[2\nu+3]}{\Gamma[\nu]^{2}}\int_{-1}^{1}\prod_{i=1}^{2}dw_{i}\rho(w_{i})\Lambda_{w_{i}}^{2\nu}$
(25)
$\displaystyle\times\int_{0}^{1}du_{1}\int_{0}^{1-u_{1}}du_{2}\cdots\int_{0}^{1-u_{1}-u_{2}-u_{3}}du_{4}\,(u_{1}\,u_{2})^{\nu-1}$
$\displaystyle\times\bigg{[}\frac{\alpha\mathcal{A}}{\Omega^{2(2\nu)}}+\frac{\beta\mathcal{B}}{\Omega^{2(2\nu+1)}}\bigg{]}\,.$

From the knowledge of the EFF, one can define the flavor contribution to the
so called charge radius:

$(r_{\scriptsize{\hbox{M}}}^{q})^{2}=-\frac{1}{6}\frac{dF_{\scriptsize{\hbox{M}}}^{q}(Q^{2})}{dQ^{2}}\Bigg{|}_{Q^{2}\to
0}\,,$ (26)

where the total charge radius is defined in analogy with Eq. (16). This
quantity is important as it is related to the slope of the EFF at zero
momentum, and in some cases, it is practically the only reliable information
available.

Finally, it is important to recall that the integration on $k$ is performed
analytically. The subsequent evaluation of Eq. (25) is performed numerically
with ease over the entire domain of the Feynman parameters and spectral
density $\rho(w_{i})$. We now proceed to a similar analysis of the TFFs of
pseudoscalar mesons.

##  IV Transition form factors

Complementary to the EFFs under investigation, are the two photon TFFs.
Notably, the $\gamma\scriptsize{\hbox{M}}\gamma$ interaction vertex is also
characterized by a single form factor which is customarily defined as follows:

$\displaystyle\mathcal{T}_{\mu\nu}(k_{1},k_{2})$ $\displaystyle=$
$\displaystyle\mathcal{T}_{\mu\nu}(k_{1},k_{2})+\mathcal{T}_{\nu\mu}(k_{2},k_{1})$
$\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi^{2}}\epsilon_{\mu\nu\alpha\beta}k_{1\alpha}k_{2\beta}G_{\scriptsize{\hbox{M}}}(k_{1}^{2},k_{2}^{2},k_{1}\cdot
k_{2})\,,$

where the momentum of the meson is $P=k_{1}+k_{2}$, with $k_{1}$ and $k_{2}$
the momenta of incoming photons. The case in which one of the photons is on-
shell describes the $\gamma\gamma^{*}\rightarrow\scriptsize{\hbox{M}}$
transition, which has been measured experimentally for the
$\\{\pi,\,\eta,\,\eta^{\prime},\,\eta_{c}\\}$ systems (see _e.g._ Ref.
Danilkin  _et al._ (2019)). Here we focus on the
$\\{\pi^{0},\,\eta_{c},\,\eta_{b}\\}$ mesons, employing related experimental
and phenomenological information to constrain the parameter space of the AM.

Figure 2: Analogous of Fig. 1 for the TFF.

In the impulse approximation Raya  _et al._ (2016), this form factor is
obtained from the triangle diagram depicted in Fig. 2. The standard Feynman
rules yield the following mathematical expression:

$\displaystyle\mathcal{T}_{\mu\nu}(k_{1},k_{2})$ $\displaystyle=$
$\displaystyle
e^{2}\mathcal{Q}^{2}_{\scriptsize{\hbox{M}}}\mbox{tr}\int_{l}i\chi_{\mu}^{q}(l,l+k_{1})\Gamma_{\scriptsize{\hbox{M}}}(l+k_{1},l-k_{2})$
(28) $\displaystyle\hskip 8.5359pt\times
S(l-k_{2})i\Gamma_{\nu}^{q}(l-k_{2},l)\,,$

where $\mathcal{Q}^{2}_{\pi,\,\eta_{c},\,\eta_{b}}=\\{1/3,\,4/9,\,1/9\\}$.
Moreover, the momenta squared of the virtual and real photons are,
respectively, $k_{1}^{2}=Q^{2}$, $k_{2}^{2}=0$; thus, the on-shell meson
condition imposes $2k_{1}\cdot k_{2}=-(m_{\scriptsize{\hbox{M}}}^{2}+Q^{2})$.

Plugging in various quantities from the AM, namely, Eqs. (7), (8) and (14),
into the above expression, the TFF is expressed compactly as follows:

$\displaystyle G_{\scriptsize{\hbox{M}}}(Q^{2})$ $\displaystyle=$
$\displaystyle\int_{l}\int_{-1}^{1}dw\rho(w)\Lambda_{w}^{2\nu}\frac{\mathcal{M}^{\prime}(l;k_{1},k_{2})}{\mathcal{D^{\prime}}(l;k_{1},k_{2})}\;,$
(29)

where, as in the EFF case, $\mathcal{D^{\prime}}(l;k_{1},k_{2})$ denotes the
product of denominators from the quark propagator, the BSA and the QPV:

$\displaystyle\mathcal{D^{\prime}}(l;k_{1},k_{2})$ $\displaystyle=$
$\displaystyle\Delta((l-k_{2})^{2},m_{q}^{2})\Delta((l+k_{1})^{2},m_{q}^{2})$
(30) $\displaystyle\times$
$\displaystyle\Delta(l^{2},m_{q}^{2})\Delta(k_{(w-1)}^{2},\Lambda_{w}^{2})^{\nu}\;.$

Naturally, the scalar function $\mathcal{M}^{\prime}(l;k_{1},k_{2})$ refers to
the numerator arising after the Dirac tracing and contraction with a sensible
projector operator
$\mathcal{P}_{\mu\nu}\sim\epsilon_{\mu\nu\lambda\rho}k_{1\lambda}k_{2\rho}$.
Once again, after introducing Feynman parametrization and adequately
simplifying the denominator with a suitable change of variables, the TFF can
be cast in a form similar to Eq. (25), and the final results can be obtained
straightforwardly via numerical integration.

With semi analytical expressions for the EFF and TFF at hand, we can now
proceed to discuss the data-driven method to fix the required model
parameters, and analyse the viability of the approach.

##  V $\gamma^{*}\scriptsize{\hbox{M}}\to\scriptsize{\hbox{M}}$ and
$\gamma\gamma^{*}\to\scriptsize{\hbox{M}}$ Form Factors

In this section we present the collection of results for the EFF
($\gamma^{*}\scriptsize{\hbox{M}}\to\scriptsize{\hbox{M}}$) and TFF
($\gamma\gamma^{*}\to\scriptsize{\hbox{M}}$) of ground-state pseudoscalar
mesons $\scriptsize{\hbox{M}}=\pi,\,K,\,\eta_{c},\,\eta_{b}$. We start by
discussing the data-driven analysis for fixing the 3-4 AM parameters. We then
compute the predictions of our model for a wide range of $Q^{2}$ values.

The standard $\chi^{2}$ statistical test for a set with $N$ number of data
points has the expression:

$\displaystyle\chi^{2}=\sum_{i=1}^{N}\frac{(E_{i}-T_{i})^{2}}{\delta
E_{i}^{2}}\,,$ (31)

where $E_{i}$ represents the $i$-th experimental data to fit, $T_{i}$ is the
theoretically predicted $i$-th value for a given point and $\delta E_{i}$ is
the estimated error associated with the $i$-th fitted data point. We implement
statistical analysis for the experimental data from the references shown in
Table 1 to minimize the $\chi^{2}$.

Meson | EFF | TFF
---|---|---
$\pi$ |  Refs. Amendolia  _et al._ (1986a); Ackermann  _et al._ (1978); Volmer  _et al._ (2001); Horn  _et al._ (2006); Huber  _et al._ (2008) |  Ref. Behrend  _et al._ (1991); Gronberg  _et al._ (1998); Aubert  _et al._ (2009); Uehara  _et al._ (2012)
$K$ |  Ref. Amendolia  _et al._ (1986b); Dally  _et al._ (1980) | —
$\eta_{c}$ | — |  Ref. Lees  _et al._ (2010)
Table 1: Experimental data fitted in our global analysis of EFFs and TFFs of
pseudoscalar mesons.

As can be read from the Table 1, there is more information available for the
pion as compared to other pseudoscalar mesons. For charged kaon, precise data
are only available at very low momentum (and, naturally, there is no two-
photon to kaon process). There exists data for the $\eta_{c}$ TFF, which we
complement with a theoretical computation of the charge radius, Bhagwat and
Maris (2008), based upon SDEs. Finally, there is no experimental data
concerning the $\eta_{b}$. So, we rely on the charge radius obtained in Ref.
Bhagwat  _et al._ (2007), along with the SDE prediction for the TFF from Ref.
Raya  _et al._ (2017).

Figure 3: Pion EFF and TFF. Left panel- Pion EFF. Our pion results are
represented with the light blue band. The purple band corresponds to the
previous AM results from the overlap representation presented in Albino  _et
al._ (2022). Dashed black line is the SDE result for the pion Chang  _et al._
(2013b) and the SDE-driven prediction from the CI is represented by the dotted
dark red line Hernández-Pinto  _et al._ (2023). Diamonds, rectangles,
triangles and circles represent the experimental data from Refs. Amendolia
_et al._ (1986a); Ackermann  _et al._ (1978); Volmer  _et al._ (2001); Horn
_et al._ (2006); Huber  _et al._ (2008). Right panel- Pion TFF. The blue light
band corresponds to our TFFs results of pion. The light red dashed line is the
asymptotic limit, $2f_{\pi}$. The black solid line corresponds to the SDEs
results from Raya  _et al._ (2016). Experimental results are taken from
Behrend  _et al._ (1991); Gronberg  _et al._ (1998); Aubert  _et al._ (2009);
Uehara  _et al._ (2012).

###  V.1 $\pi$ meson

We start our analysis with the lightest mesons, namely, pions. Besides being
responsible for holding protons and neutrons together in the atomic nucleus,
these are the lightest hadrons in nature. Being lightest pseudo-Nambu-
Goldstone bosons makes them different from all other hadrons. Since pions are
copiously produced in the high energy collision environments, we take the
advantage of the largest amount of data points for both soft and hard
processes to constrain the parameters of the AM using available experimental
data. We assume isospin symmetry, i.e., $m_{u}=m_{d}$ (implying
$m_{\pi^{\pm}}=m_{\pi^{0}}$), constraining its value around a typical range of
$0.3\mbox{ GeV}<m_{u}<0.5\mbox{ GeV}$. In order to find the best
phenomenologically fitted parameters, the pion mass is also allowed to vary in
the close vicinity of the value reported in the Particle Data Group (PDG) Ref.
Workman  _et al._ (2022). Experimental data from Refs. Amendolia  _et al._
(1986a); Ackermann  _et al._ (1978); Volmer  _et al._ (2001); Horn  _et al._
(2006); Huber  _et al._ (2008); Behrend  _et al._ (1991); Gronberg  _et al._
(1998); Uehara  _et al._ (2012); Aubert  _et al._ (2009) constrain
dimensionless parameters $\nu_{\pi}$ and $\alpha_{u}^{(0)}$ and the dressed
quark masses. In Tab. 2 we present the obtained best parameters to optimize
the value of $\chi^{2}$.

$m_{u}$ | $m_{\pi^{\pm}}$ | $\nu_{\pi}$ | $\alpha_{u}^{(0)}$
---|---|---|---
0.3135 |  0.1395 | 0.8428 |  0.1964
Table 2: Best fitted values for the pion in the AM. Isospin symmetry is
assumed such that $m_{u}=m_{d}$. Note that we allow minimal variation in
$m_{\pi^{\pm}}$ to obtain the best fit. All masses are given in GeV.

Results of this exploration are presented in Fig. 3. In addition to the
agreement with empirical data, the produced curves are highly compatible with
sophisticated SDE predictions Chang  _et al._ (2013b); Eichmann  _et al._
(2019); Raya  _et al._ (2016), and with a previous exploration of the AM
employing the overlap representation Albino  _et al._ (2022). CI model results
Hernández-Pinto  _et al._ (2023), characterized by momentum independent mass
functions and BSAs, are also displayed for comparison. From the obtained EFF,
one can determine the corresponding charge radius via Eq. (26), producing:

$\displaystyle r^{\rm fit}_{\pi}=0.67\mbox{ fm}\,,$ (32)

which presents a small 1.67 % deviation from the experimental value
$r_{\pi}^{\rm exp}=0.659(4)\mbox{ fm}$ Workman  _et al._ (2022). The error
band in Fig. 3 accounts for a $5\%$ uncertainty of the above value, stemming
from a systematic variation of $m_{u}$.

In Table 3 we report the corresponding $\chi^{2}$ per set of data points for
pions. The global fit has a $\chi^{2}/d.o.f.\sim 1.93$ confirming
quantitatively that the predictions are in agreement with experimental data.
Despite the controversy, both BaBar Aubert  _et al._ (2009) and Belle Uehara
_et al._ (2012) data sets of the pion TFF were considered in the analysis. The
former is mildly disfavored as compared to Belle data; BaBar data set produces
$\chi^{2}/d.o.f.\sim 1.82$, in contrast with the Belle set, from which
$\chi^{2}/d.o.f.\sim 1.45$ is obtained.

Experiment |  # of data points in fit | $\chi^{2}$
---|---|---
NA7 Amendolia  _et al._ (1986a) | 45 | 48.42
DESY Ackermann  _et al._ (1978) | 2 | 2.50
JLab F$\pi$-1 Volmer  _et al._ (2001) | 4 | 1.16
JLab F$\pi$-2 Horn  _et al._ (2006); Huber  _et al._ (2008) | 4 | 2.56
CELLO Behrend  _et al._ (1991) | 5 | 83.55
CLEO Gronberg  _et al._ (1998) | 15 | 15.44
BaBar Aubert  _et al._ (2009) | 17 | 30.95
Belle Uehara  _et al._ (2012) | 15 | 21.72
TOTAL: | 107 |  206.3
Table 3: Data sets used in our global analysis for pions, the individual
$\chi^{2}$ values, and the total $\chi^{2}$ of the fit.

It is important to note that the phenomenological agreement is achieved when
$\nu_{\pi}$ tends to unity from below. In this context, we recall that QCD
prescriptions establish a $1/k^{2}$ falloff for the dominant BSA of
pseudoscalar mesons, which is modified by the corresponding anomalous
dimension Roberts and Williams (1994). The effects of the latter become
apparent in the large momentum regime, but the precise domain in which those
manifest strongly depends on the mass sector. Within the AM, such behavior can
be emulated by adopting a value of $\nu_{\scriptsize{\hbox{M}}}$ close to (but
different from) 1. Therefore, this trend is not only acceptable but expected.
In contrast with involved numerical evaluations,_e.g._ Raya  _et al._ (2016),
a small value for $\alpha^{(0)}_{u}$ is also favored by data. This means that
the correction to the QPV from the transverse terms, given in Eq. (13),
although necessary for a proper description of the TFF, is rather mild.
Notably, it has been seen that when the quark anomalous magnetic moment is
included in a formal manner, Dang  _et al._ (2023), its contribution to the
TFF at the $Q^{2}=0$ limit is commensurate with $\alpha^{(0)}_{u}$.

We can now analyze the expected theoretical predictions of the AM for EFF of
pions for expected center of mass energies of the EIC and JLab. Fig. 4
displays the EFF in a wide range of $Q^{2}$ up to $40\text{GeV}^{2}$. Across
the momentum squared domain covered, there is considerable agreement with SDE
prediction from Ref. Chang  _et al._ (2013b). The noticeable contrast of both
the calculations with a monopole type fit, supports the fact that scaling
violations are already discernible at momentum scales of about
$Q^{2}=4-5\,\text{GeV}^{2}$, Yao  _et al._ (2024). We expect this observation
to be confirmed in a convincing manner by the new generation of electron-ion
colliders Arrington  _et al._ (2021); Aguilar  _et al._ (2019); Accardi  _et
al._ (2023); Horn and Roberts (2016); Huber  _et al._ (2006).

Figure 4: Prediction of the AM for the pion EFF at $Q^{2}$ up to 40 GeV2. A
direct comparison with a monopole fit as well as recent lattice results Ding
_et al._ (2024) and earlier SDE-based computation from Ref. Chang  _et al._
(2013b) is included. Projected EIC and JLab $Q^{2}$ range Arrington  _et al._
(2021); Aguilar  _et al._ (2019); Accardi  _et al._ (2023); Horn and Roberts
(2016); Huber  _et al._ (2006) is also depicted.

Figure 5: Kaon EFF. Left panel- Kaon $F_{K}(Q^{2})$. The purple band
corresponds to the AM previous results from the overlap representation
presented in Albino  _et al._ (2022). For comparison, we have included the
lower gray band that corresponds to the SDE result for the kaon Eichmann  _et
al._ (2020); Gao  _et al._ (2017b) as well as SDE-driven predictions in the CI
model (dotted dark red line) Hernández-Pinto  _et al._ (2023). Right panel-
$Q^{2}F_{K}(Q^{2})$ of Kaon. SDE results taken from Gao  _et al._ (2017b) are
represented with the dashed light red line. Recent lattice results Ding  _et
al._ (2024) are also depicted. In both graphs our results are represented with
the light blue band. And, the circles, diamonds, rectangles and triangles
represent the experimental data from Refs. Amendolia  _et al._ (1986b); Dally
_et al._ (1980); Carmignotto  _et al._ (2018); Horn  _et al._ .

###  V.2 $K$ meson

We now turn to the study of kaon pseudoscalar mesons. The understanding of
kaons, just like that of pions, is of paramount importance. Careful scrutiny
of the similarities and differences between these two systems is expected to
reveal important information regarding the hadronic structure and its
connection to the origin of mass in the Standard Model. As explained before,
within the present approach, the parameters of the $u/d$-quarks, given in Tab.
2, are constrained by the electromagnetic properties of the pion. Similarly,
the $s$-quark parameters are fixed by the related empirical data on the kaon.
The resulting parameters are shown in Tab. 4.

$m_{s}$ | $m_{K}$ | $\nu_{K}$ |  $\alpha^{(0)}_{s}$
---|---|---|---
0.5274 |  0.4936 |  0.913 |  –
Table 4: Best fitted values for kaon mesons in the AM. The $u$-quark related
values are taken from Tab. 2. Masses are given in GeV. The kaon mass gets
fitted to the optimal value given by the PDG Workman  _et al._ (2022).

Large uncertainties on the determination of the experimental data allow us to
have $\nu_{K}=0.913$ close to (but below) 1. The lack of determination of
$\alpha_{s}^{(0)}$ is due to the fact that its value does not affect the EFF.
Therefore, it should be fixed from the two-photon TFF process, which does not
exist in this case.

The result for the EFF arising from the fit of the kaon parameters in the AM
is displayed in Fig. 5. The left-hand side of the figure compares our result
with that obtained within the SDE approach Eichmann  _et al._ (2020), a
previous AM determination via the overlap representation Albino  _et al._
(2022), and the expected harder results of the CI model Hernández-Pinto  _et
al._ (2023). The comparison with the data at small $Q^{2}$, Refs. Amendolia
_et al._ (1986b); Dally  _et al._ (1980), is also shown. The EFF has been
plotted on a larger span of photon momentum on the right-hand side of Fig. 5.
This plot includes comparisons with SDE results Gao  _et al._ (2017a) and
different sets of data Amendolia  _et al._ (1986b); Dally  _et al._ (1980);
Carmignotto  _et al._ (2018); Horn  _et al._ . Notably, although it was not
used in any way to set the parameters, excellent agreement is obtained with
the preliminary JLab data Horn  _et al._ , which increase the range of $Q^{2}$
up to 6 GeV2. Just like the case of the pion, the error bands allow for a
$5\%$ variation in the kaon charge radius 333In this case, the variation in
$m_{u}$ produces the band, whereas the ratio $m_{s}/m_{u}$ is kept fixed.,

$\displaystyle r_{K}^{\rm fit}=0.64\mbox{ fm}\,,$ (33)

which does not substantially differ from the reported experimental result
$r_{K}\sim 0.56\mbox{ fm}$ Workman  _et al._ (2022) and the one obtained in
the SDE approach Miramontes  _et al._ (2022), $r_{K}\sim 0.60\mbox{ fm}$. In
order to be able to quantify the quality of the fit and its predictability, in
Tab. 5, we report the $\chi^{2}$ corresponding to each set of experimental
data on the kaon EFF, producing an overall $\chi^{2}/d.o.f.\sim$ 0.705 and a
rather small $\chi^{2}/d.o.f.\sim 0.141$ for the preliminary JLab set (not
used for the fit).

experiment |  # data in fit | $\chi^{2}$
---|---|---
Amendolia Amendolia  _et al._ (1986b) | 15 | 5.2822
Dally Dally  _et al._ (1980) | 10 | 17.8976
Carmignotto JLab Carmignotto  _et al._ (2018) | 5 | 1.3620
JLab E12-09-011 Horn  _et al._ | 6 | 0.8444
TOTAL: | 36 | 25.3864
Table 5: Produced $\chi^{2}$ values for different available data sets, and its
overall value. The first two rows were considered for the fitting procedure of
the AM; the next two show the degree of predictability of the model.

###  V.3 $\eta_{c}$ meson

We now proceed with the analysis of the heavy quarkonia. Note that these
systems are characterized by having large dressed quark masses which owe
primarily to weak mass generation mechanism of the Higgs field. Including
their study is likely to provide productive comparison with their light
counterparts, namely pions and kaons. We begin with considering the $\eta_{c}$
meson. In this case, we have the charm mass $(m_{c})$, the $\eta_{c}$ mass
($m_{\eta_{c}}$), and the dimensionless parameters $\nu_{\eta_{c}}$ and
$\alpha^{(0)}_{c}$ to be determined from the phenomenological analysis.

Due to the fact that the $\eta_{c}$ is a $q\bar{q}$ system, EFF of
$F_{\eta_{c}}$ is strictly zero (see Eq. (16)) and no experimental input
should be expected in this case. However, $F_{\scriptsize{\hbox{M}}}^{q}$ can
still be computed theoretically by considering the interaction of the photon
only with the quark (or the antiquark). The $\gamma\gamma^{*}\to\eta_{c}$
transition, on the other hand, has been measured in BaBar Lees  _et al._
(2010). We thus employ this phenomenological input for the parameter setting
of the AM in this sector. Supplemented by the SDE estimation of the charge
radius, $r_{\eta_{c}}=0.219$ Bhagwat  _et al._ (2007), we proceed as before
and perform the corresponding $\chi^{2}$ analysis. The optimal parameters are
presented in Tab. 6. We find similar deviations of $\nu_{\eta_{c}}$ from unity
as in the $\pi-K$ scenario and, in addition, the preferred mass of $\eta_{c}$
meson deviates from the PDG value Workman  _et al._ (2022) by about 4.9% to
obtain best results within our AM.

$m_{c}$ | $m_{\eta_{c}}$ | $\nu_{\eta_{c}}$ | $\alpha^{(0)}_{c}$
---|---|---|---
1.7364 |  3.1307 |  0.8021 |  0.2669
Table 6: Best fitted values for $\eta_{c}$ mesons in the AM. Masses are given
in GeV.

Figure 6: $\eta_{c}$ EFF and TFF. Left panel- $\eta_{c}$ EFF. The blue band
represents our results with the parameters of Table 6. For comparison, we have
included the following results: the light purple band that represents the
$\eta_{c}$ previous AM results from the overlap representation discussed in
Albino  _et al._ (2022), the dashed black line that corresponds to the lattice
QCD results from Refs. Dudek  _et al._ (2006, 2007), and the SDE-driven
predictions from the the CI model (dotted dark red line) Hernández-Pinto  _et
al._ (2023). Right panel- $\eta_{c}$ TFF. The blue light band corresponds to
our TFFs results. The SDEs results Raya  _et al._ (2017) are represented by a
black solid line, whereas the green dotted curve depicts the NNLO nrQCD
predictions Wang  _et al._ (2018). The experimental data corresponds to BaBar
collaboration from Lees  _et al._ (2010).

In Tab. 7, we present the obtained $\chi^{2}$ of our analysis of $\eta_{c}$
pseudoscalar mesons. It is noticeable that in this case $\chi^{2}/d.o.f\sim
0.36$ which renders the fit almost perfect. With the values tabulated in Tab.
7, we predict a charge radius for $\eta_{c}$ mesons, $r^{\rm fit}_{\eta_{c}}$,
of

$\displaystyle r_{\eta_{c}}^{\rm fit}=1.04\,r_{\eta_{c}}=0.228\,\text{fm}\,.$
(34)

This value is also quite compatible with the one computed in lattice QCD,
$r_{\eta_{c}}=0.255(2)~{}\text{fm}$ Dudek  _et al._ (2006, 2007).

experiment |  # data in fit | $\chi^{2}$
---|---|---
$r_{\eta_{c}}$Bhagwat  _et al._ (2007) | 1 | 0.67
BaBar Lees  _et al._ (2010) | 11 | 3.97
TOTAL: | 12 |  4.64
Table 7: Data sets used in our global analysis for $\eta_{c}$, the individual
$\chi^{2}$ values, and the total $\chi^{2}$ of the fit.

In Fig. 6, we depict the results obtained for the EFF and TFF for the
$\eta_{c}$. On the left hand side, we plot our findings for the $\eta_{c}$ EFF
and compare them with the computed results in lattice QCD Dudek  _et al._
(2006, 2007), the CI model Hernández-Pinto  _et al._ (2023), and the results
calculated using the AM within the overlap representation Albino  _et al._
(2022). Just like the pions and the kaons, we again report error bands
representing 5% variation in $r_{\eta_{c}}$; this same variation is also
reported for the TFF depicted on the right-hand side. For the case of the TFF,
in addition to the data of the BaBar collaboration, Lees  _et al._ (2010), we
also include the SDE prediction Raya  _et al._ (2017) and the next-to-next-to
leading order (NNLO) non-relativistic QCD (nrQCD) result from Wang  _et al._
(2018). As expected, we observe that our results clearly agree with the BaBar
measurements and with the theoretical predictions. Once the parameters of the
model are carefully fitted, our predictions are readily available for $Q^{2}$
values as large as experiments can access in the future or lattice QCD as well
as other theoretical efforts might be able to produce.

###  V.4 $\eta_{b}$ meson

The heaviest quark found to date in nature which is able to form hadrons is
the $b$-quark. It indicates the strongest Yukawa coupling to the Higgs boson,
thus providing the $b$-quarks with the largest mass through explicit chiral
symmetry breaking in the Lagrangian. In this section, we study the $b$-quark
and the dynamics of the lightest meson which is composed of it and its
antiquark through computing the EFF and the TFF of $\eta_{b}$.

Figure 7: $\eta_{b}$ EFF and TFF. Left panel- $\eta_{b}$ EFF. The blue band
represents our results with the parameters of Table 8. For comparison, we have
included the following results: the light purple band that represents the
$\eta_{b}$ previous AM results from the overlap representation Albino  _et
al._ (2022); and the SDE-driven predictions proceeding from the CI model
(dotted dark red line) Hernández-Pinto  _et al._ (2023). Right panel-
$\eta_{b}$ TFF; the blue light band corresponds to our TFFs results; the black
dashed line represents the SDEs results Raya  _et al._ (2017); the green band
corresponds to NNLO nrQCD result for $\eta_{b}$ (the band width expresses the
sensitivity to the factorisation scale) from Feng  _et al._ (2015).

Figure 8: $\eta_{c}$ and $\eta_{b}$ EFF at $Q^{2}$ up to 10 GeV2. Left panel-
$\eta_{c}$ EFF. Right panel- $\eta_{b}$ EFF. The light blue band corresponds
to our EFFs results allowing for a 5% variation in the charge radius. For
comparison, we include the black dashed line on the left plot which
corresponds to the $\eta_{c}$ lattice QCD results from Refs. Dudek  _et al._
(2006, 2007). No such computation is available for the $\eta_{b}$ meson.

Contrary to the previous cases, there is no analogous experimental information
to perform a data-driven global fit for $\eta_{b}$. Therefore, in order to
compute the EFF and the TFF, we proceed with theory-driven phenomenological
inputs. We take advantage of the available results for the $\eta_{b}$ TFF
calculated using refined SDE truncations Raya  _et al._ (2017). As all
$\gamma^{*}\gamma\to\\{\pi^{0},\,\eta,\,\eta^{\prime},\,\eta_{c},\,\eta_{b}\\}$
TFFs have been computed within the same reliable and unified formalism (see
Ref. Raya  _et al._ (2020)), we obtain agreement with available experiments
and QCD constraints. Thereby, the SDE prediction for the $\eta_{b}$ is
expected to be robust. To complement this information, we also use the
$\eta_{b}$ charge radius $r_{\eta_{b}}=0.086$ fm, which has been reported in
Ref. Bhagwat  _et al._ (2007).

Owing to the lack of data points as such, and instead having a continuous and
smooth TFF curve over a large momentum domain (up to
$Q^{2}=60\,\text{GeV}^{2}$), we construct a random set of points for 12 values
of $Q_{i}^{2}$, allowing for possible error of $5\%$ with respect to their
central values $G_{\scriptsize{\hbox{M}}}(Q^{2}_{i})$. Therefore, we can
proceed analogously to the other cases. Neither the selection of points nor
the size of the error affect the quality of the fit, as long as the selected
set covers the entire domain of interest. The optimal values that minimize the
corresponding $\chi^{2}$ function are collected in Tab. 8. We find a tendency
of $\nu_{\eta_{b}}$ to deviate further from unity, around 27%, and almost no
deviation for $m_{\eta_{b}}$ as compared to the PDG value,
$m_{\eta_{b}}=9.3987(20)\,\text{GeV}$ Workman  _et al._ (2022). The global
analysis in this cases produces a high degree of accuracy, as can be
appreciated from the values tabulated in Tab. 9. These numbers produce a
pseudo-$\chi^{2}/d.o.f.\sim 10^{-3}$.

$m_{b}$ | $m_{\eta_{b}}$ | $\nu_{\eta_{b}}$ | $\alpha^{(0)}_{b}$
---|---|---|---
5.3443 |  9.3886 |  1.2743 |  0.1004
Table 8: Best fitted values for $\eta_{b}$ meson in the AM. Masses are in GeV. experiment |  # data in fit | $\chi^{2}$
---|---|---
$r_{\eta_{b}}$Bhagwat  _et al._ (2007) | 1 | 0.006
SDEs Raya  _et al._ (2017) | 12 |  0.004
TOTAL: | 11 |  0.01
Table 9: Data sets used in our global analysis for $\eta_{b}$, the individual
$\chi^{2}$ values, and the total $\chi^{2}$ of the fit.

Therefore, it is not surprising that there is almost perfect agreement with
the model inputs. This level of compatibility is displayed in Fig. 7, where
EFF and TFF of $\eta_{b}$ are plotted. We reiterate that the reported bands
correspond to a variation of $r_{\eta_{b}}$ of around 5%. In the case of EFF,
we find that

$\displaystyle r_{\eta_{b}}^{\rm fit}=0.99\,r_{\eta_{b}}\,=0.085\,\text{fm}.$
(35)

To provide other points of comparison for the resulting EFF, we have included
results from the AM in the overlap representation Albino  _et al._ (2022), as
well as those obtained from the CI model Hernández-Pinto  _et al._ (2023).
Regarding the TFF, displayed on the right-hand side of Fig. 7, SDEs results
Raya  _et al._ (2017) and NNLO nrQCD projections Feng  _et al._ (2015) are
shown. We must emphasize again that our model has the natural advantage that
once the small number of model parameters are fixed carefully through the
global analysis of experimental data and/or reliable theoretical predictions,
our model is designed to provide predictions for arbitrarily large values of
the probing photon momentum squared $Q^{2}$ which is the very aim of every
experimental facility and each corresponding theoretical endeavor. Fig. 8
displays our results for the $\eta_{c}$ and $\eta_{b}$ EFFs in a larger
$Q^{2}$ domain.

##  VI Summary and conclusions

In this work, we present a detailed calculation of the EFFs and TFFs of the
ground-state pseudoscalar mesons $\pi,\,K,\,\eta_{c},$ and $\eta_{b}$, by
considering the interaction vertices
$\scriptsize{\hbox{M}}\gamma\scriptsize{\hbox{M}}$ and
$\gamma\scriptsize{\hbox{M}}\gamma^{*}$, respectively, in the impulse
approximation. The computation of these form factors is carried out by
evaluating the related triangle diagrams which are completely determined by
the knowledge of the quark propagator, the meson BSA and the QPV. The results
are obtained in an AM developed recently in close connection with QCD.

Within this approach, the structure of the pseudoscalar meson is fully encoded
in the BSWF, defined in terms of the quark propagator and the meson BSA. The
internal structure of the meson is probed through the electromagnetic QPV. The
quark propagator acquires a rather simple form, Eq. (7), in which the mass
function is demoted to play the role a constant dressed quark mass. However,
this lack of QCD governed momentum dependence is compensated by providing a
suitable Ansatz for the BSA, expressed as an integral representation, Eq. (8).
Its point-wise behavior is dictated by the so called spectral density. One of
the strengths of this approach is that the latter can be extracted
systematically and analytically from the knowledge of the meson PDA Albino
_et al._ (2022). Therefore, well-tested realistic understanding of the PDAs is
available in the literature which is taken advantage of. This set of
distributions and the extraction process is discussed in Appendix A. In
addition, through symmetry requirements, a robust but compact expression for
the QPV is proposed, Eq. (14).

The AM allows different components of the triangle diagrams to be represented
in an algebraically convenient way, which leads to writing expressions for the
EFF and TFF in a largely analytical manner. In particular, numerical
integration over 4-momenta is avoided altogether. Only the integration over
compact domains is required, which can be carried out straightforwardly. A
comprehensive global analysis is subsequently conducted. Using available
experimental data and other phenomenological contributions when needed, we
resort to the minimization of the $\chi^{2}$ function to obtain our best
parameters: $m_{q}$, $m_{q^{\prime}}$, $m_{\scriptsize{\hbox{M}}}$,
$\nu_{\scriptsize{\hbox{M}}}$ and $\alpha_{q}^{(0)}$. The final results are
presented with added error bands, obtained by allowing a 5% variation in the
charge radii of EFF.

For all the mesons we obtain a small $\chi^{2}/d.o.f.$, the largest being 1.93
corresponding to the pion due to the large amount of data available for this
meson including the deviation of experimental measurements from the TFFs at
large $Q^{2}$. Besides, in the case of the pion, we observe that our
predictions of the EFFs at large energy scales are in agreement with the
results of the SDEs and in turn, the AM is capable of mimicking the scaling
violations expected for these form factors. This behavior is expected to be
corroborated by the projected EIC and JLab experiments. On the other hand, in
the case of the kaon, we have that our predictions are again in good agreement
with those obtained by the SDEs even at a higher energy scale than those
normally reported (unlike the CI model, which produces harder form factors).
In turn, they are in almost perfect agreement with the new JLab experimental
data which go up to a $Q^{2}$ of 5.5 GeV2 Horn  _et al._ .

Now, regarding heavy quarkonia, the only experimental information available is
that related to the $\gamma\gamma^{*}\to\eta_{c}$ process, provided by BaBar.
Our findings show that such data can be accurately described by the AM.
Concerning other theoretical approaches, a measurable difference with the
results produced in the CI is found. Notwithstanding, our results are in a
perfect agreement with sophisticated SDE treatments and nrQCD. This confluence
of results, and the ability to reproduce experimental results, would be a
measure of the robustness of the present approach. For the $\eta_{b}$ meson
there is no experimental data available to compare with. However, our
theoretical predictions are in complete agreement with those phenomenological
predictions obtained within the SDE and nrQCD approaches, as well as with
previous determinations of the AM via the overlap representation.

Finally, we conclude that the results generated by the AM compare favorably
with all available data, as well as with different
theoretical/phenomenological frameworks (including lattice QCD, nrQCD and
SDEs). Furthermore, it allows us to extract EFFs and TFFs of pseudoscalar
mesons in a large domain of $Q^{2}$ which will be probed in the upcoming EIC
and potential upgrade of the JLab.

###### Acknowledgements.

I. M. H-A. thanks Consejo Nacional de Humanidades, Ciencias y Tecnologías
(CONAHCyT), Mexico, for the scholarship received to carry out her postgraduate
studies. The work of R. J. H-P is also funded by CONAHCyT through Project No.
320856 (Paradigmas y Controversias de la Ciencia 2022), Ciencia de Frontera
2021-2042 and Sistema Nacional de Investigadores. A. B. acknowledges
Coordinación de la Investigación Científica, Universidad Michoacana de San
Nicolás de Hidalgo grant 4.10. and Fulbright-García Robles scholarship for his
stay as a visiting scientist at the Thomas Jefferson National Accelerator
Facility, Newport News, Virginia, USA, where part of this work was carried
out. A. B. also thanks Beatriz-Galindo support during his current stay at the
University of Huelva, Huelva, Spain. The work of K. R. is supported by the
Spanish MICINN grant PID2022-140440NB-C22, and regional Andalusian project
P18-FR-5057. We are grateful to G. M. Huber for private communication on the
pion and kaon EFF.

##  Appendix A Spectral density differential equation

The leading-twist PDA of the pseudoscalar meson,
$\phi_{\scriptsize{\hbox{M}}}^{q}(x)$, can be derived through the light-cone
projection of the BSWF Chang  _et al._ (2013a). Thereby, in the context of the
present AM, and models that employ integral representations for the BSWF in
general Mezrag  _et al._ (2016); Xu  _et al._ (2018); Raya  _et al._ (2022b);
Albino  _et al._ (2022), the PDA is expressed in terms of a meson spectral
density $\rho_{\scriptsize{\hbox{M}}}(y)$. In our particular case, as detailed
in Ref. Albino  _et al._ (2022), the relationship between
$\rho_{\scriptsize{\hbox{M}}}(y)$ and $\phi_{\scriptsize{\hbox{M}}}^{q}(x)$
turns out to be well-defined. It is characterized by the following
differential equation:

$\displaystyle\eta_{N}\rho_{\scriptsize{\hbox{M}}}(w)$ $\displaystyle=$
$\displaystyle\lambda_{\nu}^{(2)}(w)\varphi^{\prime\prime}(w)+\lambda_{\nu}^{(1)}(w)\varphi^{\prime}(w)$
(36) $\displaystyle+$ $\displaystyle\lambda_{\nu}^{(0)}(w)\varphi(w)\,,$

where $\eta_{N}$ is the normalization factor for the spectral density, such
that $\int_{-1}^{1}\rho_{\scriptsize{\hbox{M}}}(w)dw=1$. Moreover, we have
defined $\varphi(w)\equiv\phi_{\scriptsize{\hbox{M}}}^{q}(\frac{1}{2}(1-w))$.
The rest of all the stated quantities are given by the following expressions:

$\displaystyle\lambda_{\nu}^{(2)}(w)$ $\displaystyle=$
$\displaystyle-\frac{1-w^{2}}{\chi_{+}}\,,$ (37)
$\displaystyle\lambda_{\nu}^{(1)}(w)$ $\displaystyle=$ $\displaystyle
2\frac{\nu
w}{\chi_{+}}-2\frac{\chi_{-}}{\chi_{+}^{2}}+\frac{\nu\chi_{-}}{\Lambda_{w}^{2}}\,,$
(38) $\displaystyle\lambda_{\nu}^{(0)}(w)$ $\displaystyle=$
$\displaystyle\big{\\{}2\nu\chi_{+}^{2}\Lambda_{w}^{2}\left(\chi_{+}^{2}-2\left(1+(1-\nu)w^{2}+\nu\right)\Lambda_{w}^{2}\right)$
(39)
$\displaystyle+4w(1-\nu)\left(2\Lambda_{w}^{2}-\nu\chi_{+}^{2}\right)\Lambda_{w}^{2}\chi_{+}\chi_{-}$
$\displaystyle-\left(\nu(1-\nu)\chi_{+}^{4}+2\nu\chi_{+}^{2}\Lambda_{w}^{2}-8\Lambda_{w}^{4}\right)\chi_{-}^{2}\big{\\}}/\Theta_{w}\,,$

where we identify $\chi_{\pm}\equiv(1-w)M_{\bar{q^{\prime}}}\pm(1+w)M_{q}$ and
$\Theta_{w}\equiv-4\left(1-w^{2}\right)\chi_{+}^{3}\Lambda_{w}^{4}$. Thus Eq.
(36) entails that the prior knowledge of the PDA determines the spectral
density (and vice versa). Now consider the chiral limit
($m_{\scriptsize{\hbox{M}}}=0$, $m_{q}=m_{\bar{q}^{\prime}}$), and $\nu=1$
(which produces the expected falloff of the BSA in the absence of anomalous
dimensions). We then have:

$\displaystyle\lambda_{1}^{(2)}=-\frac{(1-y^{2})}{2M_{q}}\;,\;\lambda_{1}^{(1)}=\lambda_{1}^{(0)}=0\;,$
(40)

which ensures that our AM recovers the known result in Chang  _et al._
(2013a). In other words, taking

$\displaystyle\phi_{asy}(x)=6x(1-x)\;,$ (41)

we produce

$\displaystyle\rho_{\scriptsize{\hbox{M}}}(w)=\rho_{asy}(w):=\frac{3}{4}(1-w^{2})\;.$
(42)

For the present study, which focuses on the analysis of $\pi$, $K$, $\eta_{c}$
and $\eta_{b}$, the PDAs, we employ the following convenient parameterizations
of the PDAs ($\bar{x}=1-x$), reported in Refs. Ding  _et al._ (2016).:

$\displaystyle\phi_{\pi}^{u}(x)$ $\displaystyle=$ $\displaystyle
20.226\,x\bar{x}\,[1-2.509\sqrt{x\bar{x}}+2.025x\bar{x}]\;,\vspace{0.2cm}$
$\displaystyle\phi_{K}^{u}(x)$ $\displaystyle=$ $\displaystyle
18.04\,x\bar{x}\,[1+5x^{0.032}\bar{x}^{0.024}-5.97x^{0.064}\bar{x}^{0.048}]\;,$
$\displaystyle\phi_{\eta_{c}}^{c}(x)$ $\displaystyle=$ $\displaystyle
9.222\,x\bar{x}\,\text{exp}\,[-2.89(1-4x\bar{x})]\;,$
$\displaystyle\phi_{\eta_{b}}^{b}(x)$ $\displaystyle=$ $\displaystyle
12.264\,x\bar{x}\,\text{exp}[-6.25(1-4x\bar{x})]\,.$ (43)

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