11institutetext:  FMI, Sofia University “St. Kliment Ohridski”, Bulgaria
22institutetext: University of Cambridge, UK 33institutetext: Releva.ai,
Bulgaria 44institutetext: Qatar Computing Research Institute, HBKU, Qatar

# Leaf: Multiple-Choice Question Generation

Kristiyan Vachev 11 Momchil Hardalov 11 Georgi Karadzhov 22 Georgi Georgiev 33
Ivan Koychev 11 Preslav Nakov 44

###### Abstract

Testing with quiz questions has proven to be an effective way to assess and
improve the educational process. However, manually creating quizzes is tedious
and time-consuming. To address this challenge, we present Leaf, a system for
generating multiple-choice questions from factual text. In addition to being
very well suited for the classroom, Leaf could also be used in an industrial
setting, e.g., to facilitate onboarding and knowledge sharing, or as a
component of chatbots, question answering systems, or Massive Open Online
Courses (MOOCs). The code and the demo are available on
GitHub.111https://github.com/KristiyanVachev/Leaf-Question-Generation

###### Keywords:

Multiple-choice questions, education, self-assessment, MOOCs.

##  1 Introduction

Massive Open Online Courses (MOOCs) have revolutionized education by offering
a wide range of educational and professional training. However, an important
issue in such a MOOC setup is to ensure an efficient student examination
setup. Testing with quiz questions has proven to be an effective tool, which
can help both learning and student retention [38]. Yet, preparing such
questions is a tedious and time-consuming task, which can take up to 50% of an
instructor’s time [41], especially when a large number of questions are needed
in order to prevent students from memorizing and/or leaking the answers.

To address this issue, we present an automated multiple-choice question
generation system with focus on educational text. Taking the course text as an
input, the system creates question–answer pairs together with additional
incorrect options (distractors). It is very well suited for a classroom
setting, and the generated questions could also be used for self-assessment
and for knowledge gap detection, thus allowing instructors to adapt their
course material accordingly. It can also be applied in industry, e.g., to
produce questions to enhance the process of onboarding, to enrich the contents
of massive open online courses (MOOCs), or to generate data to train
question–answering systems [10] or chatbots [22].

##  2 Related Work

While Question Generation is not as popular as the related task of Question
Answering, there has been a steady increase in the number of publications in
this area in recent years [1, 18]. Traditionally, rules and templates have
been used to generate questions [29]; however, with the rise in popularity of
deep neural networks, there was a shift towards using recurrent
encored–decoder architectures [2, 8, 9, 33, 40, 46, 47] and large-scale
Transformers [7, 20, 23, 27, 36].

The task is often formulated as one of generating a question given a target
answer and a document as an input. Datasets such as SQuAD1.1 [37] and NewsQA
[44] are most commonly used for training, and the results are typically
evaluated using measures such as BLEU [32], ROUGE [25], and METEOR [21]. Note
that this task formulation requires the target answer to be provided
beforehand, which may not be practical for real-world situations. To get over
this limitation, some systems extract all nouns and named entities from the
input text as target answers, while other systems train a classifier to label
all word $n$-grams from the text and to pick the ones with the highest
probability to be answers [45]. To create context-related wrong options (i.e.,
distractors), typically the RACE dataset [19] has been used along with beam
search [3, 11, 31]. Note that MOOCs pose additional challenges as they often
cover specialized content that goes beyond knowledge found in Wikipedia, and
can be offered in many languages; there are some open datasets that offer such
kinds of questions in English [5, 6, 19, 28, 42] and in other languages [4,
12, 13, 15, 16, 24, 26, 30].

Various practical systems have been developed for question generation.
WebExperimenter [14] generates Cloze-style questions for English proficiency
testing. AnswerQuest [39] generates questions for better use in Question
Answering systems, and SQUASH [17] decomposes larger articles into paragraphs
and generates a text comprehension question for each one; however, both
systems lack the ability to generate distractors. There are also online
services tailored to teachers. For example, Quillionz [35] takes longer
educational texts and generates questions according to a user-selected domain,
while Questgen [34] can work with texts up to 500 words long. While these
systems offer useful question recommendations, they also require paid
licenses. Our Leaf system offers a similar functionality, but is free and
open-source, and can generate high-quality distractors. It is trained on
publicly available data, and we are releasing our training scripts, thus
allowing anybody to adapt the system to their own data.

##  3 System

System architecture: Leaf has three main modules as shown in Figure 1. Using
the Client, an instructor inputs a required number of questions and her
educational text. The text is then passed through a REST API to the Multiple-
Choice Question (MCQ) Generator Module, which performs pre-processing and then
generates and returns the required number of question–answer pairs with
distractors. To achieve higher flexibility and abstraction, the models
implement an interface that allows them to be easily replaced.

Figure 1: The general architecture of Leaf.

Question and Answer Generation: To create the question–answer pairs, we
combined the two tasks into a single multi-task model. We fine-tuned the small
version of the T5 Transformer, which has 220M parameters, and we used the
SQuAD1.1 dataset [37], which includes 100,000 question–answer pairs. We
trained the model to output the question and the answer and to accept the
passage and the answer with a 30% probability for the answer to be replaced by
the [MASK] token. This allows us to generate an answer for the input question
by providing the [MASK] token instead of the target answer. We trained the
model for five epochs, and we achieved the best validation cross-entropy loss
of 1.17 in the fourth epoch. We used a learning rate of 0.0001, a batch size
of 16, and a source and a target maximum token lengths of 300 and 80,
respectively. For question generation, we used the same data split and
evaluation scripts as in [9]. For answer generation, we trained on the
modified SQuAD1.1 Question Answering dataset as proposed in our previous work
[45], achieving an Exact Match of 41.51 and an F1 score of 53.26 on the
development set.

Distractor Generation: To create contextual distractors for the
question–answer pairs, we used the RACE dataset [19] and the small pre-trained
T5 model. We provided the question, the answer, and the context as an input,
and obtained three distractors separated by a [SEP] token as an output. We
trained the model for five epochs, achieving a validation cross-entropy loss
of 2.19. We used a learning rate of 0.0001, a batch size of 16, and a source
and a target maximum token lengths of 512 and 64, respectively. The first, the
second, and the third distractor had BLEU1 scores of 46.37, 32.19, and 34.47,
respectively. We further extended the variety of distractors with context-
independent proposals, using sense2vec [43] to generate words or multi-word
phrases that are semantically similar to the answer.

Figure 2: Screenshot of Leaf showing the generated questions for a passage
from the Wikipedia article on Oxygen. All distractors in Question 1 are
generated by the T5 model, and the last two distractors in Question 2 are
generated by the sense2vec model.

User Interface: Using the user interface shown on Figure 2, the instructor can
input her educational text, together with the desired number of questions to
generate. Then, she can choose some of them, and potentially edit them, before
using them as part of her course.

##  4 Conclusion and Future Work

We presented Leaf, a system to generate multiple-choice questions from text.
The system can be used both in the classroom and in an industrial setting to
detect knowledge gaps or as a self-assessment tool; it could also be
integrated as part of other systems. With the aim to enable a better
educational process, especially in the context of MOOCs, we open-source the
project, including all training scripts and documentation.

In future work, we plan to experiment with a variety of larger pre-trained
Transformers as the underlying model. We further plan to train on additional
data. Given the lack of datasets created specifically for the task of Question
Generation, we plan to produce a new dataset by using Leaf in real university
courses and then collecting and manually curating the question–answer pairs
Leaf generates over time.

## Acknowledgements

This research is partially supported by Project UNITe BG05M2OP001-1.001-0004
funded by the Bulgarian OP “Science and Education for Smart Growth.”

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# Electromagnetic Form Factors and Charge Radii of Pseudoscalar and Scalar
Mesons: A Comprehensive Contact Interaction Analysis

R. J. Hernández-Pinto<EMAIL_ADDRESS>Facultad de Ciencias Físico-
Matemáticas, Universidad Autónoma de Sinaloa, Ciudad Universitaria, Culiacán,
Sinaloa 80000, México  L. X. Gutiérrez-Guerrero<EMAIL_ADDRESS>CONACyT-Mesoamerican Centre for Theoretical Physics, Universidad Autónoma de
Chiapas, Carretera Zapata Km. 4, Real del Bosque (Terán), Tuxtla Gutiérrez,
Chiapas 29040, México  A. Bashir<EMAIL_ADDRESS><EMAIL_ADDRESS>Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de
Hidalgo, Edificio C-3, Ciudad Universitaria, Morelia, Michoacán 58040, México
Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606,
USA  M. A. Bedolla<EMAIL_ADDRESS>Facultad de Ciencias en Física y
Matemáticas, Universidad Autónoma de Chiapas, Carretera Emiliano Zapata Km. 8,
Rancho San Francisco, Ciudad Universitaria Terán, Tuxtla Gutiérrez, Chiapas
29040, México  I. M. Higuera-Angulo<EMAIL_ADDRESS>Instituto de
Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo,
Edificio C-3, Ciudad Universitaria, Morelia, Michoacán 58040, México

###### Abstract

We carry out a comprehensive survey of electromagnetic form factors of all
light, heavy and heavy-light ground-state pseudoscalar and scalar mesons. Our
analysis is based upon a Schwinger-Dyson equations treatment of a vector
$\times$ vector contact interaction. It incorporates confinement and ensures
axial vector and vector Ward-Takahashi identities are satisfied along with the
corresponding corollaries such as the Goldberger-Treiman relations. The
algebraic simplicity of the model allows us to compute the form factors at
arbitrarily large virtualities of the probing photon momentum squared with
relative ease. Wherever possible and insightful, we compare our results for
the electromagnetic form factors and the charge radii with those obtained
earlier through Schwinger-Dyson equations, lattice and with experimental
observations available. We also comment on the scope and shortcomings of the
model.

##  I Introduction

A major challenge in strong interaction physics is the description of hadrons
from first principles, i.e., by commencing from the Lagrangian dynamics of
elementary degrees of freedom of quantum chromodynamics (QCD), namely, quarks
and gluons. The arduous task is then to describe hadron properties by sewing
together the Green functions of dressed quarks through relativistic bound
state equations. In close analogy with the hydrogen atom of electrodynamics,
the simplest bound states of QCD are the two-particle systems (mesons)
composed of a quark and an antiquark ($q\bar{q}^{\prime}$). Relativistic
description of such states through the Bethe-Salpeter equation (BSE) was first
formulated in Ref. Salpeter and Bethe (1951). Solutions of this equation
presuppose the knowledge of the dressed quark propagator and the
$q\bar{q}^{\prime}$ scattering kernel. The quark propagator is obtained by
solving the gap equation while the $q\bar{q}^{\prime}$ scattering kernel is
constructed by ensuring the axial vector Ward-Takahashi identity is satisfied.

Several experimental facilities around the globe study electromagnetic
properties of mesons for a gradually increasing interval of momentum squared
($Q^{2}$) transferred to the target by the incident probing photon. It
enhances the possibility of observing a gradual transition from non-
perturbative QCD effects to its perturbative domain, finally settling onto its
asymptotic predictions estimated decades ago, all that in one single
experiment. Resulting elastic or electromagnetic form factors (EFFs) of mesons
thus provide us with an ideal platform to study numerous uncanny facets of
QCD, unfolding the complex structure of these bound states at varying
resolutions scales.

EFFs of pseudoscalar (PS) mesons, pion and kaon in particular but, have been
studied extensively, for example, within the functional approach via
Schwinger-Dyson equations (SDEs) Maris and Tandy (2000); Chang et al. (2013);
Bhagwat and Maris (2008), lattice QCD Shultz et al. (2015); Alexandrou et al.
(2022); Gao et al. (2021); Davies et al. (2018), contact interaction (CI)
Gutierrez-Guerrero et al. (2010a); Wang et al. (2022); Xing and Chang (2022),
other models and formalisms, asymptotic QCD Lepage and Brodsky (1979) and, of
course, experimentally Amendolia et al. (1986a); Horn et al. (2006); Volmer et
al. (2001). $\pi^{+}$, $K^{+}$, $K^{0}$, $D^{+}$ and $D_{0}$ have also been
studied in a hybrid model that combines the generalized Bertlmann-Martin
inequalities with smearing corrections due to relativistic effects Lombard and
Mares (2000). Light and heavy PS mesons in the light-front framework have been
reported in Hwang (2002) while with a QCD potential model there are results
for $D^{+}$, $D^{0}$, $D_{s}^{+}$, $B^{+}$, $B^{0}$,
$B^{0}_{{\color[rgb]{0,0.5,0}s}}$ Das et al. (2016).

PS mesons have additional and important relevance as they contribute to the
hadronic light-by-light (HLbL) piece of the muon anomalous magnetic moment
(AMM), most dominantly through the single exchange of the light mesons such as
$\pi,\eta$ and $\eta^{\prime}$. Furthermore, there are also loops with charged
pions ($\pi^{\pm}$) and kaons ($K^{\pm}$). These contributions have been
computed with desirable accuracy within the SDE formalism Goecke et al.
(2011); Eichmann et al. (2019); Raya et al. (2020); Eichmann et al. (2020);
Miramontes et al. (2022). On the other hand, scalar (S) mesons have been less
studied for technical hindrances and due to the fact their composition is
still debatable. However, similarly to the PS mesons, they contribute to the
AMM of the muon, see the review article Aoyama et al. (2020) and references
therein.

Additional overwhelming interest in studying mesons arises from the fact that
their BSE analysis provides an important first step towards studying baryons
in a quark-diquark picture. It is firmly established that the non-pointlike
diquark correlations play an important role in baryons Barabanov et al.
(2021). As a clear illustration, it has been demonstrated that the quark-
diquark picture of a nucleon produces its mass within 5% of what the Faddeev
equation of a three quark system Eichmann (2011) yields. With this
realisation, it is useful to know that the BSE for diquarks is exactly the
same as that for corresponding mesons up to a color and charge factor. The
chiral partners form a set of particles which transform into each other under
chiral transformation, like $(\sigma,\pi)$ and $(\rho,a_{1})$.
Correspondingly, there are diquark partners ($0^{-},0^{+}$) and
$(1^{+},1^{-})$. The Bethe-Salpeter amplitudes (BSAs) as well as the EFFs for
$\sigma,\pi,\rho$ and $a_{1}$ yield the corresponding description of diquarks
$0^{-},0^{+},1^{+}$ and $1^{-}$. The quark-diquark picture has been
successfully used to calculate EFFs and transition form factors (TFF) of
baryons Cloet et al. (2009); Wilson et al. (2012); Segovia et al. (2014a, b);
Raya et al. (2018a, 2021); Segovia et al. (2015). For comprehensive reviews in
this connection, one can consult Refs. Aznauryan et al. (2013); Bashir et al.
(2012).

We have already mentioned CI in the preceding discussion. It is a symmetry
preserving vector × vector interaction based on a momentum-independent gluon
propagator. It results in four quarks interacting at a point. It was first
proposed in Gutierrez-Guerrero et al. (2010a) to calculate pion EFF.
Subsequently, CI has extensively been employed to study EFFs and TFFs of
mesons in Refs. Roberts et al. (2010, 2011); Chen et al. (2013); Raya et al.
(2018b); Wang et al. (2022) and of baryons in Refs. Wilson et al. (2012);
Segovia et al. (2014a, b); Raya et al. (2018a, 2021). It is a well-known
realization that the EFFs obtained from the CI are harder than the ones
obtained from full QCD predictions. However, the simplicity of the model
allows us to perform algebraic calculations. Moreover, the results obtained
provide a benchmark to compare and contrast with refined QCD-based SDE results
in order to understand the correct pattern of dynamical chiral symmetry
breaking (DCSB) and the large $Q^{2}$ evolution of the EFFs which stems from
asymptotic QCD where $Q^{2}$ is much larger than any other mass scale relevant
to the problem. In this work, we compute EFFs using this momentum-independent
interaction, regularized in a symmetry-preserving manner for a large number of
PS and S mesons composed of light quarks, heavy quarks, and the heavy-light
combinations. We must emphasize that the scalars like $\sigma$ have a
complicated internal structure, possibly including a large component of pion
correlations. The $\sigma$ in our article refers to a quark-antiquark state
alone, parity partner of the pion and approximately twice as heavy as
$\sigma(600)$, Pelaez and Rios (2006).

The article is organized as follows: in Sec. II we collect the basic
ingredients required to carry out the analysis in the CI model: the dressed
quark masses obtained through the gap equation and the general expression for
the BSAs for PS and S mesons. We discuss the generalities of the EFFs for PS
and S mesons in Sec. III, i.e., the quark-photon vertex and the triangle
diagram which are the two building blocks to calculate all the meson EFFs in
our formalism. Sec. IV is dedicated to computing EFFs of the ground state PS
mesons. It allows us to evaluate their charge radii in the limit $Q^{2}\approx
0$, and simultaneously understand the asymptotic behaviour of the meson EFFs
at large $Q^{2}$, i.e., $Q^{2}\rightarrow\infty$. In Sec. V, we repeat our
study for the S mesons. A brief summary and perspectives for future work are
presented in Sec. VI.

##  II The Ingredients

Calculation of the meson EFFs presupposes the knowledge of the dynamically
generated dressed valence quark masses, BSAs of the mesons as well as the
quark-photon interaction vertex at different probing momenta of the incident
photon. In this section, we provide a brief but self contained introduction to
the CI, its essential ingredients and characteristics, namely, the gluon
propagator, the quark-gluon vertex and the set of parameters employed which,
collectively, define the CI. This discussion is followed by the solution of
the gap equation to obtain dynamically generated dressed quark masses. We then
provide the general expressions of the BSAs for PS and S mesons. The
corresponding BSE is set up consistently with the gap equation. The numerical
solution is presented in the respective sections dedicated to the analysis of
these mesons.

###  II.1 The Gap Equation

The starting point for our study is the dressed-quark propagator for a quark
of flavor $f$, which is obtained by solving the gap equation,

$\displaystyle S(p)^{-1}$ $\displaystyle=$ $\displaystyle i\gamma\cdot
p+m_{f}+\Sigma(p)\;,$ $\displaystyle\Sigma(p)$ $\displaystyle=$
$\displaystyle\frac{4}{3}\int\\!\frac{d^{4}q}{(2\pi)^{4}}g^{2}D_{\mu\nu}(p-q)\gamma_{\mu}S(q)\Gamma_{\nu}(q,p),\;$
(1)

where $m_{f}$ is the Lagrangian current-quark mass, $D_{\mu\nu}(p)$ is the
gluon propagator and $\Gamma_{\nu}(q,p)$ is the quark-gluon vertex. It is a
well-established fact by now that the Landau gauge gluon propagator saturates
in the infrared and a large effective mass scale is generated for the gluon,
see for example Boucaud et al. (2012); Ayala et al. (2012); Bashir et al.
(2013); Binosi et al. (2017); Deur et al. (2016); Rodríguez-Quintero et al.
(2018). It also leads to the saturation of the effective strong coupling at
large distances. This modern understanding of infrared QCD forms the defining
ideas of the CI proposed in Gutierrez-Guerrero et al. (2010b). We assume that
the quarks interact, not through massless vector-boson exchange but via a CI.
Thus the gluon propagator no longer runs with a momentum scale but is frozen
into a CI in keeping with the infrared properties of QCD, see Fig. 1.

FIG. 1: Diagrammatic representation of the CI, employing the simplified model
of the gluon propagator in Eq. (2).

Thus

$\displaystyle g^{2}D_{\mu\nu}(k)$ $\displaystyle=$ $\displaystyle
4\pi\hat{\alpha}_{\mathrm{IR}}\delta_{\mu\nu}$ (2)

where $\hat{\alpha}_{\mathrm{IR}}=\alpha_{\mathrm{IR}}/m_{g}^{2}$. The scale
$m_{g}$ is for dimensional reasons and is interpreted as the infrared gluon
mass scale generated dynamically within QCD Bowman et al. (2004); Gutierrez-
Guerrero et al. (2010a); Gutiérrez-Guerrero et al. (2019). We take currently
accepted value $m_{g}=500\,\text{MeV}$ Boucaud et al. (2012); Aguilar et al.
(2018); Binosi and Papavassiliou (2018); Gao et al. (2018). It is clear that
in the CI gap equation, the effective coupling which appears is
$\hat{\alpha}_{\mathrm{IR}}$ instead of $\alpha_{\mathrm{IR}}.$ We choose
$\alpha_{\rm IR}/\pi$ to be $0.36$ so that $\hat{\alpha}_{\mathrm{IR}}$ has
exactly the same value as in all related previous works Gutierrez-Guerrero et
al. (2010a); Gutiérrez-Guerrero et al. (2019, 2021); Yin et al. (2019). The
interaction vertex is bare, i.e., $\Gamma_{\nu}(q,p)=\gamma_{\nu}$.

This constitutes an algebraically simple but useful and predictive rainbow-
ladder truncation of the SDE of the quark propagator whose solution can
readily be written as follows:

$\displaystyle S(q,M_{f})$ $\displaystyle\equiv$ $\displaystyle-i\gamma\cdot
q\;\sigma_{V}(q,M_{f})+\sigma_{S}(q,M_{f})\,,$ (3)

with

$\displaystyle\sigma_{V}(q,M_{f})=\frac{1}{q^{2}+M_{f}^{2}}\,,\sigma_{S}(q,M_{f})=M_{f}\,\sigma_{V}(q,M_{f})\,,$
(4)

where $M_{f}$, for the CI, is momentum-independent dynamically generated
dressed quark mass determined by

$M_{f}=m_{f}+M_{f}\frac{4\hat{\alpha}_{\rm
IR}}{3\pi}\int_{0}^{\infty}\\!ds\,s\,\frac{1}{s+M_{f}^{2}}\,\,.$ (5)

Our regularization procedure follows Ref. Ebert et al. (1996):

$\displaystyle\frac{1}{s+M_{f}^{2}}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\tau\,{\rm
e}^{-\tau(s+M_{f}^{2})}\rightarrow\int_{\tau_{\rm UV}^{2}}^{\tau_{\rm
IR}^{2}}d\tau\,{\rm e}^{-\tau(s+M_{f}^{2})}$ (6) $\displaystyle=$
$\displaystyle\frac{{\rm e}^{-(s+M_{f}^{2})\tau_{\rm
UV}^{2}}-e^{-(s+M_{f}^{2})\tau_{\rm IR}^{2}}}{s+M_{f}^{2}}\,,$

where $\tau_{\rm IR,UV}$ are, respectively, infrared and ultraviolet
regulators. It is apparent from Eq. (6) that a finite value of $\tau_{\rm
IR}\equiv 1/\Lambda_{\rm IR}$ implements confinement by ensuring the absence
of quark production thresholds. Since Eq. (5) does not define a renormalisable
theory, $\Lambda_{\rm UV}\equiv 1/\tau_{\rm UV}$ cannot be removed but instead
plays a dynamical role, setting the scale of all mass dimensioned quantities.
Using Eq. (6), the gap equation becomes

$\displaystyle M_{f}=m_{f}+M_{f}\frac{4\hat{\alpha}_{\rm IR}}{3\pi}{\cal
C}(M_{f}^{2})\;,$ (7)

where

$\displaystyle\frac{{\cal C}(M^{2})}{M^{2}}=\Gamma(-1,M^{2}\tau_{\rm
UV}^{2})-\Gamma(-1,M^{2}\tau_{\rm IR}^{2})\;$ (8)

and $\Gamma(\alpha,x)$ is the incomplete gamma-function.

Table 1:  Ultraviolet regulator and coupling constant for different combinations of quarks in PS mesons. $\hat{\alpha}_{\mathrm{IR}}=\hat{\alpha}_{\mathrm{IRL}}/Z_{H}$, where $\hat{\alpha}_{\mathrm{IRL}}=4.57$ is extracted from the best-fit to data as explained in Ref. Raya et al. (2018b). $\Lambda_{\rm IR}=0.24$ GeV is a fixed parameter. quarks |  $Z_{H}$ |  $\Lambda_{\mathrm{UV}}\,[\text{GeV}]$ |  $\hat{\alpha}_{\mathrm{IR}}$
---|---|---|---
$\,{\color[rgb]{1,0,0}u},{\color[rgb]{0.64,0.0,0.0}d},{\color[rgb]{0,0.5,0}s}$ | 1 | 0.905 |  4.57
$\,{\color[rgb]{0.55,0.0,0.55}c},{\color[rgb]{1,0,0}u},{\color[rgb]{0,0.5,0}s}$ |  3.034 | 1.322 | 1.50
$\,{\color[rgb]{0.55,0.0,0.55}c}$ | 13.122 | 2.305 | 0.35
$\,{\color[rgb]{0,0,1}b},{\color[rgb]{1,0,0}u}$ | 11.273 | 3.222 | 0.41
$\,{\color[rgb]{0,0,1}b},{\color[rgb]{0,0.5,0}s}$ | 17.537 | 3.574 | 0.26
$\,{\color[rgb]{0,0,1}b},{\color[rgb]{0.55,0.0,0.55}c}$ | 30.537 | 3.886 | 0.15
$\,{\color[rgb]{0,0,1}b}$ | 129.513 | 7.159 |  0.035

We report results for PS mesons using the parameter values listed in Tables 1,
2, whose variation with quark mass was dubbed as heavy parameters in Ref.
Gutiérrez-Guerrero et al. (2019). In this approach, the coupling constant and
the ultraviolet regulator vary as a function of the quark mass. This behavior
was first suggested in Ref. Bedolla et al. (2015) and later adopted in several
subsequent works Bedolla et al. (2016); Raya et al. (2018b); Gutiérrez-
Guerrero et al. (2019); Yin et al. (2019, 2021). Table 2 presents the current
quark masses $m_{f}$ used herein and the dynamically generated dressed masses
$M_{f}$ of ${\color[rgb]{1,0,0}u}$, ${\color[rgb]{0,0.5,0}s}$,
${\color[rgb]{0.55,0.0,0.55}c}$ and ${\color[rgb]{0,0,1}b}$ computed from the
gap equation, Eq. (7).

Table 2:  Current ($m_{f}$) and dressed masses ($M_{f}$) for quarks in GeV, required as an input for the BSE and the EFFs. $m_{{\color[rgb]{1,0,0}u}}=0.007$ | $m_{{\color[rgb]{0,0.5,0}s}}=0.17$ | $m_{{\color[rgb]{0.55,0.0,0.55}c}}=1.08$ | $m_{{\color[rgb]{0,0,1}b}}=3.92$
---|---|---|---
$M_{{\color[rgb]{1,0,0}u}}=0.367$ |  $M_{{\color[rgb]{0,0.5,0}s}}=0.53$ |  $M_{{\color[rgb]{0.55,0.0,0.55}c}}=1.52$ |  $M_{{\color[rgb]{0,0,1}b}}=4.75$

A meson can consist of heavy ($Q$) or light ($q$) quarks. We present the study
of all heavy ($Q\bar{Q}$), heavy-light ($Q\bar{q}$) and (review) light
($q\bar{q}$) mesons. We commence by setting up the BSE for mesons by employing
a kernel which is consistent with that of the gap equation to obey axial
vector Ward-Takahashi identity and low energy Goldberger-Treiman relations,
see Ref. Gutierrez-Guerrero et al. (2010a) for details. The PS mesons are
$J^{PC}=0^{-+}$ states while the S mesons are $J^{PC}=0^{++}$ states. The
solution of the BSE yields BSAs whose general form depends not only on the
spin and parity of the meson under consideration but also on the interaction
employed as explained in the next sub-section.

###  II.2 Bethe Salpeter Equation

FIG. 2: Diagrammatic representation of the BSE. Blue (solid) circles represent
dressed quark propagators $S$, red (solid) circle is the meson BSA $\Gamma$
while the blue (solid) rectangle is the dressed-quark-antiquark scattering
kernel ${\mathcal{K}}$.

The relativistic bound-state problem for hadrons characterized by two valence-
quarks may be studied using the homogeneous BSE whose diagrammatic
representation can be seen in Fig. 2. This equation is mathematically
expressed as Salpeter and Bethe (1951),

$[\Gamma(k;P)]_{tu}=\int\\!\frac{d^{4}q}{(2\pi)^{4}}[\chi(q;P)]_{sr}{\mathcal{K}}_{tu}^{rs}(q,k;P)\,,$
(9)

where $[\Gamma(k;P)]_{tu}$ represents the bound-state’s BSA and
$\chi(q;P)=S(q+P)\Gamma S(q)$ is the BS wave-function; $r,s,t,u$ represent
colour, flavor and spinor indices; and ${\mathcal{K}}$ is the relevant quark-
antiquark scattering kernel. This equation possesses solutions on that
discrete set of $P^{2}$-values for which bound-states exist.

A general decomposition of the BSA for the PS and the S mesons
(${\color[rgb]{0,0,1}f_{1}}{\color[rgb]{0,0,1}\bar{f_{2}}}$) in the CI has the
following form

$\displaystyle\Gamma_{{\color[rgb]{0,0,1}{PS}}}(P)$ $\displaystyle=$
$\displaystyle
i\gamma_{5}\,E_{{\color[rgb]{0,0,1}{PS}}}(P)+\frac{1}{2M_{R}}\gamma_{5}\gamma\cdot
P\,F_{{\color[rgb]{0,0,1}{PS}}}(P)\,,$
$\displaystyle\Gamma_{{\color[rgb]{0,0,1}{S}}}(P)$ $\displaystyle=$
$\displaystyle I_{D}\,E_{{\color[rgb]{0,0,1}{S}}}(P)\,.$ (10)

Note that $E_{i}(P)$ and $F_{i}(P)$ with $i\in\\{PS,S\\}$ are known as the
BSAs of the meson under consideration, $P$ is its total momentum, $I_{D}$ is
the identity matrix and
$M_{R}=M_{{\color[rgb]{0,0,1}f_{1}}}M_{{\color[rgb]{0,0,1}\bar{f_{2}}}}/[M_{{\color[rgb]{0,0,1}f_{1}}}+M_{{\color[rgb]{0,0,1}\bar{f_{2}}}}]$
is the reduced mass of the system. Eq. (9) has a solution when
$P^{2}=-M_{M}^{2}$ with $M_{M}$ being the meson mass. After this initial and
required set up of the gap equation and the BSE, we now turn our attention to
the description of the EFFs of mesons.

##  III  Electromagnetic Form Factors

The EFFs provide crucial information on the internal structure of mesons. At
low momenta, EFFs allow us to unravel the complexities of non-perturbative
QCD, i.e., confinement, DCSB and the fully dressed quarks. At high energies,
we expect to confirm the validity of asymptotic QCD for its realistic models
while at intermediate energies, we observe a smooth transition from one facet
of strong interactions to the other, all in one single experiment if we are
able to chart out a wide range of momentum transfer squared $Q^{2}$ without
breaking up the mesons under study. While there are plenty of studies on the
pion EFFs, only a few are found about heavy-quarkonia and practically none on
heavy-light mesons. The process involves an incident photon which probes
mesons, interacting with the electrically charged quarks making up these two-
particles bound states. Therefore, it is natural to start this section by
looking at the the structure of the quark-photon vertex within the CI.

###  III.1 The Quark-Photon Vertex

The quark-photon vertex, denoted by
$\Gamma_{\mu}^{\gamma}(k_{+},k_{-},M_{{\color[rgb]{0,0,1}f_{1}}})$, is related
to the quark propagator through the following vector Ward-Takahashi identity:

$\displaystyle
iP_{\mu}\Gamma_{\mu}^{\gamma}(k_{+},k_{-},M_{{\color[rgb]{0,0,1}f_{1}}})=S^{-1}(k_{+},M_{{\color[rgb]{0,0,1}f_{1}}})-S^{-1}(k_{-},M_{{\color[rgb]{0,0,1}f_{1}}})\,.$
(11)

This identity is crucial for a sensible study of a bound-state’s EFF. It is
determined through the following inhomogeneous BSE,

$\displaystyle\Gamma_{\mu}^{\gamma}(Q,M_{{\color[rgb]{0,0,1}f_{1}}})=$
$\displaystyle\gamma_{\mu}-\frac{16\pi\hat{\alpha}_{\rm
IR}}{3}\int\frac{d^{4}q}{(2\pi)^{4}}\gamma_{\alpha}\chi_{\mu}(q_{+},q,M_{{\color[rgb]{0,0,1}f_{1}}})\gamma_{\alpha}\,,$
(12)

where
$\chi_{\mu}(q_{+},q,M_{{\color[rgb]{0,0,1}f_{1}}})=S(q+P,M_{{\color[rgb]{0,0,1}f_{1}}})\Gamma_{\mu}(Q)S(q,M_{{\color[rgb]{0,0,1}f_{1}}})$.
Owing to the momentum-independent nature of the interaction kernel, the
general form of the solution is

$\displaystyle\Gamma_{\mu}^{\gamma}(Q,M_{{\color[rgb]{0,0,1}f_{1}}})=\gamma_{\mu}^{L}(Q)P_{L}(Q^{2},M_{{\color[rgb]{0,0,1}f_{1}}})+\gamma_{\mu}^{T}(Q)P_{T}(Q^{2},M_{{\color[rgb]{0,0,1}f_{1}}}),$

where $\gamma_{\mu}^{L}+\gamma_{\mu}^{T}=\gamma_{\mu}$ and

$\displaystyle\gamma_{\mu}^{T}(Q)=\gamma_{\mu}-\frac{\gamma\cdot
Q\;}{Q^{2}}\,Q_{\mu}\,.$ (14)

Inserting this general form into Eq. (12), one readily obtains (on simplifying
notation)

$\displaystyle P_{L}=1\,,\quad
P_{T}=\frac{1}{1+K_{\gamma}(Q^{2},M_{{\color[rgb]{0,0,1}f_{1}}})},$ (15)

FIG. 3: Dressing function of the transverse quark-photon vertex,
$P_{T}(Q^{2})$, in Eq. (15).

with

$\displaystyle K_{\gamma}(Q^{2},$ $\displaystyle
M_{{\color[rgb]{0,0,1}f_{1}}})=\frac{4\hat{\alpha}_{\rm
IR}}{3\pi}\int_{0}^{1}d\alpha\,\alpha(1-\alpha)Q^{2}\,\bar{\mathcal{C}}_{1}(\omega)\,,$
(16)

where

$\displaystyle\bar{\cal C}_{1}(z)=-\frac{d}{dz}{\cal
C}(z)=\Gamma(0,z\,\tau_{\rm UV}^{2})-\Gamma(0,z\,\tau_{\rm IR}^{2})\,$ (17)

and

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\omega(M_{{\color[rgb]{0,0,1}f_{1}}}^{2},\alpha,Q^{2})=M_{{\color[rgb]{0,0,1}f_{1}}}^{2}+\alpha(1-\alpha)Q^{2}\,.$
(18)

One can clearly observe from Fig. 3 that $P_{T}(Q^{2})\rightarrow 1$ when
$Q^{2}\rightarrow\infty$, yielding the perturbative bare vertex $\gamma_{\mu}$
as expected. This quark-photon vertex provides us with the required
electromagnetic interaction capable of probing the EFFs of mesons through a
triangle diagram which keeps the identity of the meson bound state intact.

###  III.2 The Triangle Diagram

Let us start from the general considerations for the electromagnetic
interactions of mesons. In the impulse approximation, the $M\gamma M$ vertex,
which describes the interaction between a meson
(${\color[rgb]{0,0,1}f_{1}}{\color[rgb]{0,0,1}\bar{f_{2}}}$) and a photon,
reads

$\displaystyle\Lambda^{M,{\color[rgb]{0,0,1}f_{1}}}$ $\displaystyle=$
$\displaystyle N_{c}\int\frac{d^{4}\ell}{(2\pi)^{4}}{\rm
Tr}\;\mathcal{G}^{M,{\color[rgb]{0,0,1}f_{1}}}\,,$ (19)

where

$\displaystyle\mathcal{G}^{M,{\color[rgb]{0,0,1}f_{1}}}$ $\displaystyle=$
$\displaystyle\,i\Gamma_{M}(k_{f})\,S(\ell+k_{i},M_{{\color[rgb]{0,0,1}f_{1}}})\,i\Gamma_{\lambda}(Q,M_{{\color[rgb]{0,0,1}f_{1}}})$
$\displaystyle\times$
$\displaystyle\,S(\ell+k_{f},M_{{\color[rgb]{0,0,1}f_{1}}})i\bar{\Gamma}_{M}(-k_{i})\,S(\ell,M_{{\color[rgb]{0,0,1}\bar{f_{2}}}})\,.$

The notation assumes that it is the quark ${\color[rgb]{0,0,1}f_{1}}$ which
interacts with the photon while the antiquark
${\color[rgb]{0,0,1}\bar{f_{2}}}$ remains a spectator. We define
$\Lambda^{M,{\color[rgb]{0,0,1}\bar{f_{2}}}}$ similarly. Furthermore, we
denote the incoming photon momentum by $Q$ while the incoming and outgoing
momenta of $M$ by: $k_{i}=k-Q/2$ and $k_{f}=k+Q/2$, respectively. The
assignments of momenta are shown in the triangle diagram of Fig. 4.

$\Lambda^{M,{\color[rgb]{0,0,1}f}}$ corresponds to the EFFs of different
mesons under study. The contribution from the interaction of the photon with
quark ${\color[rgb]{0,0,1}f_{1}}$ can be represented as
$F^{M,{\color[rgb]{0,0,1}f_{1}}}(Q^{2})$ (stemming from
$\Lambda^{M,{\color[rgb]{0,0,1}f_{1}}}$) while the contribution arising from
its interaction with quark ${\color[rgb]{0,0,1}\bar{f_{2}}}$ can be
represented as $F^{M,{\color[rgb]{0,0,1}\bar{f_{2}}}}(Q^{2})$ (coming from
$\Lambda^{M,{\color[rgb]{0,0,1}\bar{f_{2}}}}$). The total form factor
$F^{M}(Q^{2})$ is defined as follows Hutauruk et al. (2016):

$F^{M}(Q^{2})=e_{{\color[rgb]{0,0,1}f_{1}}}F^{M,{\color[rgb]{0,0,1}f_{1}}}(Q^{2})+e_{{\color[rgb]{0,0,1}\bar{f_{2}}}}F^{M,{\color[rgb]{0,0,1}\bar{f_{2}}}}(Q^{2})\,,$
(20)

where $e_{{\color[rgb]{0,0,1}f_{1}}}$ and
$e_{{\color[rgb]{0,0,1}\bar{f_{2}}}}$ are the quark and the antiquark electric
charges, respectively 111For neutral mesons composed of same flavored quarks,
the total EFF is simply $F^{M}=F^{M,{\color[rgb]{0,0,1}f_{1}}}$.. Both for PS
and S mesons, $F^{M,f_{1}}$ is straightforwardly related to
$\Lambda^{M,{\color[rgb]{0,0,1}f_{1}}}$:

$\displaystyle\Lambda^{{\color[rgb]{0,0,1}{S}},{\color[rgb]{0,0,1}f_{1}}}=-2k_{\lambda}F^{{\color[rgb]{0,0,1}{S}},{\color[rgb]{0,0,1}f_{1}}}\,,\,\Lambda^{{\color[rgb]{0,0,1}{PS}},{\color[rgb]{0,0,1}f_{1}}}=-2k_{\lambda}F^{{\color[rgb]{0,0,1}{PS}},{\color[rgb]{0,0,1}f_{1}}}\,.$
(21)

FIG. 4:  The triangle diagram for the impulse approximation to the $M\gamma M$
vertex.

All information necessary for the calculation of the EFFs is now complete. We
can employ numerical values of the parameters listed in Tables 1 and 2 and
proceed to compute the EFFs. Our evaluated analytical expressions and
numerical results for PS and S mesons occupy the details of the next two
sections. Keeping in mind that the pairs of (PS, S) mesons can be considered
as parity partners, we embark upon their treatment in the following sections
in that order.

##  IV  Pseudoscalar Mesons

Table 3:  Calculated values for the BSAs and masses for PS mesons in CI model using the parameters in Tables 1 and 2 (compare the parameters with the ones in Ref. Gutiérrez-Guerrero et al. (2019)). | Mass[GeV] | $E_{{\color[rgb]{0,0,1}{PS}}}$ | $F_{{\color[rgb]{0,0,1}{PS}}}$ |  $m_{PS}^{\rm exp}$[GeV] | error [%]
---|---|---|---|---|---
${\color[rgb]{1,0,0}u}\bar{\color[rgb]{0.64,0.0,0.0}d}$ | 0.139 | 3.59 | 0.47 | 0.139 | 0.008
${\color[rgb]{1,0,0}u}\bar{\color[rgb]{0,0.5,0}s}$ | 0.499 | 3.81 | 0.59 | 0.493 | 1.162
${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0.5,0}s}}$ | 0.701 | 4.04 | 0.75 | — | —
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{1,0,0}u}}$ | 1.855 | 3.03 | 0.37 | 1.864 | 0.494
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0.5,0}s}}$ | 1.945 | 3.24 | 0.51 | 1.986 | 1.183
${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0,1}b}}$ | 5.082 | 3.72 | 0.21 | 5.279 | 3.735
${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0,1}b}}$ | 5.281 | 2.85 | 0.21 | 5.366 | 1.586
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0,1}b}}$ | 6.138 | 2.58 | 0.39 | 6.274 | 2.166
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0.55,0.0,0.55}c}}$ | 2.952 | 2.15 | 0.40 | 2.983 | 1.053
${\color[rgb]{0,0,1}b}\bar{{\color[rgb]{0,0,1}b}}$ | 9.280 | 2.04 | 0.39 | 9.398 | 1.262
---
FIG. 5:  EFFs of PS mesons in a CI model. Left panel: electrically charged
mesons composed of quarks of different flavors. Central panel: quarkonia
including a hypothetical ground state strangeonium
(${\color[rgb]{0,0.5,0}s}\bar{\color[rgb]{0,0.5,0}s}$). Right panel:
electrically neutral mesons composed of quarks of different flavors.

We start with a detailed discussion and results on the ground state PS mesons.
These are negative parity, zero angular momentum $0^{-+}$ states and occupy a
special role in hadron physics. Simultaneously these are the simplest bound
states of a quark and antiquark and also emerge as Goldstone bosons associated
with DCSB. Pions are the lightest hadrons and are produced copiously in
collider machines at all energies. The pion cloud effect substantially
contributes to several static and dynamical hadron properties. Therefore,
understanding their internal structure has been of great interest both for
experimenters and theoreticians. The study of PS mesons is crucial in
understanding the capabilities and limitations of the CI model employed in
this work to reproduce and predict phenomenological results. Being the
Goldstone bosons associated with DCSB, their analysis requires care in
treating the associated subtleties. From Eqs. (II.2), we can see that the BSA
of PS mesons is the only one to be composed of two terms, necessary to ensure
the axial vector Ward-Takahashi identity and the Goldberger-Treiman relations
are exactly satisfied. In this article, we extend and expand the work
presented in Gutierrez-Guerrero et al. (2010a); Bedolla et al. (2016); Raya et
al. (2018b) and compute the EFFs of a larger number of PS mesons composed of
$qq,qQ$ and $QQ$ quarks. With straightforward algebraic manipulations:

$\displaystyle
F^{{\color[rgb]{0,0,1}{PS}},{\color[rgb]{0,0,1}f_{1}}}=P_{T}(Q^{2})\bigg{[}E_{{\color[rgb]{0,0,1}{PS}}}^{2}T^{{\color[rgb]{0,0,1}{PS}}}_{EE}(Q^{2})+E_{{\color[rgb]{0,0,1}{PS}}}F_{{\color[rgb]{0,0,1}{PS}}}T^{{\color[rgb]{0,0,1}{PS}}}_{EF}(Q^{2})$
$\displaystyle\hskip
18.77882pt+F_{{\color[rgb]{0,0,1}{PS}}}^{2}T^{{\color[rgb]{0,0,1}{PS}}}_{FF}(Q^{2})\bigg{]}\,,$
(22)

where

$\displaystyle T^{{\color[rgb]{0,0,1}{PS}}}_{EE}(Q^{2})$
$\displaystyle=\frac{3}{4\pi^{2}}\bigg{[}\int_{0}^{1}d\alpha\,\overline{\mathcal{C}}_{1}(\omega_{1})$
$\displaystyle+2\int_{0}^{1}d\alpha\,d\beta\,\alpha\,\mathcal{A}^{{\color[rgb]{0,0,1}{PS}}}_{EE}\,\overline{\mathcal{C}}_{2}(\omega_{2})\bigg{]}\,,$
$\displaystyle T^{{\color[rgb]{0,0,1}{PS}}}_{EF}(Q^{2})$
$\displaystyle=-\frac{3}{2\pi^{2}}\frac{1}{M_{R}}\int_{0}^{1}d\alpha\,d\beta\,\alpha\bigg{[}\mathcal{A}_{EF}^{(1)}\,\overline{\mathcal{C}}_{1}(\omega_{2})$
$\displaystyle+(\mathcal{A}_{EF}^{(2)}-\omega_{2}\mathcal{A}_{EF}^{(1)})\,\overline{\mathcal{C}}_{2}(\omega_{2})\bigg{]}\,,$
$\displaystyle T^{{\color[rgb]{0,0,1}{PS}}}_{FF}(Q^{2})$
$\displaystyle=\frac{3}{4\pi^{2}}\frac{1}{M_{R}^{2}}\int_{0}^{1}d\alpha\,d\beta\,\alpha\bigg{[}\mathcal{A}_{FF}^{(1)}\,\overline{\mathcal{C}}_{1}(\omega_{2})$
$\displaystyle+(\mathcal{A}_{FF}^{(2)}-\omega_{2}\mathcal{A}_{FF}^{(1)})\,\overline{\mathcal{C}}_{2}(\omega_{2})\bigg{]}\,,$

and

$\displaystyle\omega_{1}$
$\displaystyle=\omega_{1}(M_{{\color[rgb]{0,0,1}f_{1}}},\alpha,Q^{2})=M_{{\color[rgb]{0,0,1}f_{1}}}^{2}+\alpha\,Q^{2}(1-\alpha)\,,$
$\displaystyle\omega_{2}$
$\displaystyle=\omega_{2}(M_{{\color[rgb]{0,0,1}f_{1}}},M_{{\color[rgb]{0,0,1}\bar{f_{2}}}},\alpha,\beta,M_{M})=\alpha\,M_{{\color[rgb]{0,0,1}f_{1}}}^{2}+(1-\alpha)M_{{\color[rgb]{0,0,1}\bar{f_{2}}}}^{2}$
$\displaystyle-\alpha(1-\alpha)\,M^{2}_{M}+\alpha^{2}\,\beta\,(1-\beta)\,Q^{2}\,,$
$\displaystyle\bar{\mathcal{C}}_{2}(z)$ $\displaystyle=(\exp(-z\,\tau_{\rm
uv})-\exp(-z\,\tau_{\rm ir}))/(2z)\,.$ (23)

The coefficients ${\cal A}_{i}$ are given by the following expressions:

$\displaystyle\mathcal{A}^{{\color[rgb]{0,0,1}{PS}}}_{EE}=\alpha(M_{{\color[rgb]{0,0,1}f_{1}}}^{2}+M^{2}_{M})+2(1-\alpha)M_{{\color[rgb]{0,0,1}f_{1}}}M_{{\color[rgb]{0,0,1}\bar{f_{2}}}}+(\alpha-2)M_{{\color[rgb]{0,0,1}\bar{f_{2}}}}^{2}\,,$
$\displaystyle\mathcal{A}_{EF}^{(1)}=M_{{\color[rgb]{0,0,1}f_{1}}}+M_{{\color[rgb]{0,0,1}\bar{f_{2}}}}\,,$
$\displaystyle\mathcal{A}_{EF}^{(2)}=2M_{{\color[rgb]{0,0,1}f_{1}}}^{2}M_{{\color[rgb]{0,0,1}\bar{f_{2}}}}-\alpha
M_{{\color[rgb]{0,0,1}f_{1}}}(4(\alpha-1)M_{M}^{2}+\alpha Q^{2})$
$\displaystyle\hskip
11.95013pt+M_{{\color[rgb]{0,0,1}\bar{f_{2}}}}(2(\alpha-1)^{2}M_{M}^{2}+\alpha
Q^{2}(2\alpha(\beta-1)\beta+\alpha-1))\,,$
$\displaystyle\mathcal{A}_{FF}^{(1)}=(3\alpha-2)M_{M}^{2}+\alpha Q^{2}\,,$
$\displaystyle\mathcal{A}_{FF}^{(2)}=2\alpha((\alpha-1)^{2}M_{M}^{4}$
$\displaystyle\hskip 11.95013pt+\,\alpha
M_{M}^{2}Q^{2}(3\alpha\beta^{2}-3\alpha\beta+\alpha-2\beta^{2}+2\beta-1))$
$\displaystyle\hskip 11.95013pt+2\alpha
M_{M}^{2}M_{{\color[rgb]{0,0,1}f_{1}}}^{2}-2M_{{\color[rgb]{0,0,1}f_{1}}}M_{{\color[rgb]{0,0,1}\bar{f_{2}}}}(2(\alpha-1)M_{M}^{2}+\alpha
Q^{2})\,.$

The resulting EFFs for charged as well as neutral mesons are shown in Fig. 5.
For the practical utility and intuitive understanding of their low and large
$Q^{2}$ behavior, we perform an interpolation for PS mesons EFF in the range
$Q^{2}\in[0,8M_{M}^{2}]$. We adopt the following functional form:

$\displaystyle
F^{PS}(Q^{2})=\frac{e_{M}+a_{PS}\,Q^{2}+b_{PS}\,Q^{4}}{1+c_{PS}\,Q^{2}+d_{PS}\,Q^{4}}\,,$
(24)

where $e_{M}:=F^{PS}(Q^{2}=0)$ is the electric charge of the meson and
$a_{PS},b_{PS},c_{PS},d_{PS}$ are the fitted coefficients. The best fit
corresponds to the values listed in Table 4. The fit of Eq. (24) resonates
with our observation that the EFFs of PS mesons tend to constant values for
large $Q^{2}$ when it becomes by far the largest energy scale in the problem.
It is a straightforward consequence of CI treatment, and it is characteristic
of a point-like interaction which leads to harder EFF. However, the heavy as
well as heavy-light mesons approach a constant value much slower than the
light ones. This comparative large $Q^{2}$ behavior of EFFs owes itself to the
fact that $Q^{2}$ becomes larger than all other energy scales at much higher
values.

Table 4: Parameters from the fit of Eq.(24) for PS mesons | $a_{PS}$ | $b_{PS}$ | $c_{PS}$ | $d_{PS}$
---|---|---|---|---
${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0.64,0.0,0.0}d}}$ |  0.330 |  0.029 |  1.190 |  0.068
${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0.5,0}s}}$ | 0.335 | 0.029 | 1.092 | 0.065
${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0.5,0}s}}$ | 0.328 | 0.040 | 0.874 | 0.092
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{1,0,0}u}}$ | 0.616 |  $-$0.001 | 1.370 | 0.109
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0.5,0}s}}$ | 0.615 | 0.028 | 0.897 | 0.111
${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0,1}b}}$ | 1.143 | 0.033 | 1.921 | 0.146
${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0,1}b}}$ | 0.218 | 0.000 | 0.840 | 0.009
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0,1}b}}$ | 0.333 | 0.003 | 0.493 | 0.021
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0.55,0.0,0.55}c}}$ | 1.778 | 0.057 | 1.994 | 0.334
${\color[rgb]{0,0,1}b}\bar{{\color[rgb]{0,0,1}b}}$ | 0.099 | 0.000 | 0.127 | 0.002

The behavior of the form factors at the other extreme, $Q^{2}\simeq 0$ allows
us to extract charge radii:

$r_{M}^{2}=-6\left.\frac{\mathrm{d}F_{M}(Q^{2})}{\mathrm{d}Q^{2}}\right|_{Q^{2}=0}\,.$
(25)

For ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{1,0,0}u}}$ and
${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0,1}b}}$ states, which are
normalized to $F^{M}(0)=0$, we define $r_{M}^{2}$ with a positive sign in the
above equation. The charge radii set the trend for the subsequent evolution of
the form factors as a function of $Q^{2}$, specially for its small and
intermediate values. Fig. 6 depicts charge radii for all the PS mesons
studied, allowing for a 5% variation around the central value. With this
permitted spread in the charge radii, one can obtain a band for the $Q^{2}$
evolution of the EFFs. To avoid over-crowding, we have avoided depicting such
a band for each EFF. However, Fig. 7 shows a representative plot for the pion
permitting a 5% variation in its charge radius in conjunction with the
available experimental results.

Finally we list the central values of all ground state PS mesons charge radii
in Table 5, along with a direct comparison with available experimental
observations, lattice results and the SDE findings. Moreover, we also report
the transition charge radii of light PS mesons and flavorless neutral heavy PS
mesons to two photons invoking the following analytical parameter fit Ding et
al. (2019):

$\displaystyle r^{t}_{M}=\frac{r_{0}}{1+(M_{M}/m_{t})\,{\rm
ln}(1+M_{M}/m_{t})}\,,$ (26)

where $r_{0}=0.67$ fm and $m_{t}=1.01$ GeV. It also yields reasonable results
for the $\pi$ point (mass = 0.139 GeV) and the $K$ point (mass = 0.493 GeV) as
they are made of light quarks. But we cannot expect it to serve exactly as it
is for mesons with vastly off-balanced quark masses. However, if CI results
were to follow this formula, we would only need to assign $r_{0}=0.458$ fm.
The last row of Table 5 lists the resulting values which we denote as
$r^{t}_{M}(CI)$. Let us now summarize our findings and make explicit
comparisons with related works:

FIG. 6: Charge radii of ground state PS mesons in the CI.

FIG. 7: EFF for $\pi$-meson. The central curve is obtained using the
$\tau_{\rm UV}$ value from the Table 1. The filled band allows for a $5\%$
variation in the charge radius. Dots represent the experimental data from
Refs. Amendolia et al. (1986b); Volmer et al. (2001); Horn et al. (2006).

FIG. 8: EFF for $K$-meson. The central curve of the (blue) band is obtained by
using the $\Lambda_{\rm UV}$ value from Table 1. The filled (blue) band allows
for a $5\%$ variation in the charge radius. The experimental data is from Ref.
Amendolia et al. (1986b).

FIG. 9: EFF for $\eta_{c}$-meson. The lower (green) solid curve is the lattice result, Ref. Dudek et al. (2007). The central curve of the (blue) band is obtained using the $\Lambda_{\rm UV}$ value from Table 1. The filled (blue) band allows for a $5\%$ variation in the charge radius. Table 5:  The charge radii of PS meson systems, calculated with the CI model, refined SDE studies, lattice QCD and extraction from data, in a hybrid model (HM), light-front framework (LFF) and a QCD potential model (PM). In the two rows after the experimental results, we also provide the best fit results for the transition charge radii of PS mesons to $\gamma\gamma^{*}$, Eq. (26) and the same fit adapted to the CI. All results are presented in fm. | ${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0.64,0.0,0.0}d}}$ | ${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0.5,0}s}}$ | ${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0.5,0}s}}$ | ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{1,0,0}u}}$ | ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0.5,0}s}}$ | ${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0,1}b}}$ | ${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0,1}b}}$ | ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0,1}b}}$ | ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0.55,0.0,0.55}c}}$ | ${\color[rgb]{0,0,1}b}\bar{{\color[rgb]{0,0,1}b}}$
---|---|---|---|---|---|---|---|---|---|---
Our Result | 0.45 | 0.42 |  0.36 |  0.36 |  0.26 |  0.34 |  0.24 |  0.17 |  0.20 |  0.07
SDE Miramontes et al. (2022); Bhagwat and Maris (2008) | $0.676\pm 0.002$ | $0.593\pm 0.002$ | - | - | - | - | - | - |  0.24 |  0.09
Lattice Gao et al. (2021); Dudek et al. (2006); Davies et al. (2018) |  $0.648\pm 0.141$ |  0.566 (extracted) | - | - | - | - | - | - |  0.25 | -
Exp. Zyla et al. (2020) | $0.659\pm 0.004$ | $0.560\pm 0.031$ | - | - | - | - | - | - | - | -
$r^{t}_{M}$ Ding et al. (2019) | $0.658$ | $0.568$ | - | - | - | - | - | - |  0.13 |  0.03
$r^{t}_{M}({\rm CI})$ | $0.45$ | $0.38$ |  0.33 | - | - | - | - | - |  0.09 |  0.02
HM Lombard and Mares (2000) | 0.66 | 0.65 | - | 0.47 | 0.50 | - | - | - | - | -
LFF Hwang (2002) | 0.66 | 0.58 | - | 0.55 | 0.35 | 0.61 | 0.34 | 0.20 | - | -
PM Das et al. (2016) | - | - | - | 0.67 | 0.46 | 0.73 | 0.46 | - | - | -

  * •

As desired, pion EFF and its charge radius agree with the first results
employing the CI Gutierrez-Guerrero et al. (2010a). As an add-on, in this
article we allow for a 5% variation of the pion charge radius to see its
effect on the evolution of the EFF as a function of $Q^{2}$, Fig. 7. A small
variation of the initial slope of the curve $Q^{2}\simeq 0$ opens a noticeable
spread for large $Q^{2}$ but keeps the qualitative and quantitative behaviour
fully intact.

  * •

In Fig. 8, we draw kaon EFF over the range of $Q^{2}$ values where (relatively
poor) experimental observations are available. Although large error bars
prevent us from commenting decisively on the validity of the CI but we expect
it will yield harder results as compared to precise experimental measurements
whenever these results will become available. Our reported value of its charge
radius is an indication of this behavior.

  * •

As depicted in Table 5, pion and kaon charge radii Zyla et al. (2020); Gao et
al. (2021); Dudek et al. (2006); Davies et al. (2018); Miramontes et al.
(2022); Bhagwat and Maris (2008) are known experimentally and through lattice
and SDE studies. As CI EFFs come out to be harder than full QCD predictions,
we expect our PS mesons charge radii to undershoot the exact results. This is
precisely what we observe for the pion and the kaon. The percentage relative
difference between the experimental value and our calculation for the pion
charge radius is approximately $32\%$, while for the kaon charge radius is
slightly less, $25\%$. Similar difference between the SDE and the CI results
for heavy quarkonia is observed: For $\eta_{c}$, it is $20\%$ while for
$\eta_{b}$ is is $22\%$, not too dissimilar. This comparatively analogous
behavior augments our expectation that we will be in the same ballpark for the
PS mesons whose charge radii are neither known experimentally as yet nor
lattice offers any results.

  * •

There are no experimental or lattice (to the best of our knowledge) results
available for ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{1,0,0}u}}$,
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0.5,0}s}}$,
${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0,1}b}}$,
${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0,1}b}}$ and
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0,1}b}}$ mesons for
comparison. However, the general trend of decreasing charge radii with
increasing constituent quark mass seems reassuring, e.g., the following
hierarchies are noticeable:

$\displaystyle
r_{{\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0.64,0.0,0.0}d}}}>r_{{\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0.5,0}s}}}>r_{{\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{1,0,0}u}}}>r_{{\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0,1}b}}}\,,$
$\displaystyle
r_{{\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0.5,0}s}}}>r_{{\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0.5,0}s}}}>r_{{\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0.5,0}s}}}>r_{{\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0,1}b}}}\,,$
$\displaystyle
r_{{\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{1,0,0}u}}}>r_{{\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0.5,0}s}}}>r_{{\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0.55,0.0,0.55}c}}}>r_{{\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0,1}b}}}\,,$
$\displaystyle
r_{{\color[rgb]{1,0,0}u}\bar{{\color[rgb]{1,0,0}u}}}>r_{{\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0.5,0}s}}}>r_{{\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0.55,0.0,0.55}c}}}>r_{{\color[rgb]{0,0,1}b}\bar{{\color[rgb]{0,0,1}b}}}\,.$

We must emphasize that the CI is only a simple model. Refined QCD calculations
are required to confirm or refute these findings.

This concludes our detailed analysis of all the ground state PS heavy
($Q\bar{Q}$), heavy-light ($Q\bar{q}$) as well as light ($q\bar{q}$) mesons.
We now turn our attention to a similar analysis of the scalar mesons.

##  V  Scalar mesons

Recall that an S meson is a $0^{++}$ state. It can be considered as the chiral
partner of the PS meson Fig. 10. We work under the assumption that all states
are purely quark-antiquark states. Then, for example, the states $\pi$ and
$\sigma$ get transformed into each other through the following chiral
transformation:

$\displaystyle q\rightarrow{\rm
e}^{-i\gamma_{5}\frac{\bm{\tau}}{2}\cdot{\bm{\theta}}}q\,.$ (27)

‘

FIG. 10: The S meson, e.g., $\sigma$ is viewed as the parity partner of the
pion $\pi$. Note that the scalars in this article only refer to their quark-
antiquark content.

The explicit expression for the EFFs for S mesons with mass $M_{M}$
constituted from a quark ${\color[rgb]{0,0,1}f_{1}}$ and an antiquark
$\bar{\color[rgb]{0,0,1}f_{2}}$ is given by Eq. (20) with

$\displaystyle
F^{{\color[rgb]{0,0,1}{S}},{\color[rgb]{0,0,1}f_{1}}}=P_{T}(Q^{2})E_{{\color[rgb]{0,0,1}{S}}}^{2}T^{{\color[rgb]{0,0,1}{S}}}_{EE}(Q^{2})\,,$
(28)

where

$\displaystyle T^{{\color[rgb]{0,0,1}{S}}}_{EE}(Q^{2})$
$\displaystyle=-\frac{3}{4\pi^{2}}\bigg{[}\int_{0}^{1}d\alpha\,\overline{\mathcal{C}}_{1}(\omega_{1})$
$\displaystyle+2\int_{0}^{1}d\alpha\,d\beta\,\alpha\,\mathcal{A}^{{\color[rgb]{0,0,1}{S}}}_{EE}\,\overline{\mathcal{C}}_{2}(\omega_{2})\bigg{]}\,,$
(29)

Table 6:  Ultraviolet regulator and the coupling constant for different combinations of quarks in S mesons. As before, $\hat{\alpha}_{\mathrm{IR}}=\hat{\alpha}_{\mathrm{IRL}}/Z_{H}$, where $\hat{\alpha}_{\mathrm{IRL}}=4.57$ is extracted from a best-fit to data as explained in Ref. Raya et al. (2018b). $\Lambda_{\rm IR}=0.24$ GeV. quarks | $Z_{H}$ |  $\Lambda_{\mathrm{UV}}\,[\text{GeV}]\,$ | $\hat{\alpha}_{\mathrm{IR}}$
---|---|---|---
$\,{\color[rgb]{1,0,0}u},{\color[rgb]{0.64,0.0,0.0}d},{\color[rgb]{0,0.5,0}s}$ | 1 | 0.905 | 4.57
$\,{\color[rgb]{0.55,0.0,0.55}c},{\color[rgb]{1,0,0}u}$ | 3.034 | 1.322 | 1.50
$\,{\color[rgb]{0.55,0.0,0.55}c},{\color[rgb]{0,0.5,0}s}$ | 3.034 | 2.222 | 1.50
$\,{\color[rgb]{0.55,0.0,0.55}c}$ | 13.122 | 2.305 | 0.35
$\,{\color[rgb]{0,0,1}b},{\color[rgb]{1,0,0}u}$ | 18.473 | 10.670 | 0.25
$\,{\color[rgb]{0,0,1}b},{\color[rgb]{0,0.5,0}s}$ | 29.537 | 11.064 | 0.15
$\,{\color[rgb]{0,0,1}b},{\color[rgb]{0.55,0.0,0.55}c}$ | 34.216 | 14.328 | 0.13
$\,{\color[rgb]{0,0,1}b}$ |  127.013 | 26.873 |  0.036
Table 7:  Computed values of the S mesons masses and BSAs in the CI model, see Gutiérrez-Guerrero et al. (2019) for comparison, using the parameters listed in Tables 2 and 6. |  Mass [GeV] |  $E_{{\color[rgb]{0,0,1}{S}}}$ |  $m_{S}^{\rm exp}$ [GeV] |  error [%]
---|---|---|---|---
${\color[rgb]{1,0,0}u}\bar{\color[rgb]{0.64,0.0,0.0}d}$ | 1.22 | 0.66 | — | —
${\color[rgb]{1,0,0}u}\bar{\color[rgb]{0,0.5,0}s}$ | 1.38 | 0.65 | — | —
${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0.5,0}s}}$ | 1.46 | 0.64 | — | —
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{1,0,0}u}}$ | 2.31 | 0.39 | 2.30 | 0.19
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0.5,0}s}}$ | 2.42 | 0.42 | 2.32 | 3.54
${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0,1}b}}$ | 5.30 | 1.53 | — | —
${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0,1}b}}$ | 5.64 | 0.26 | — | —
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0,1}b}}$ | 6.36 | 1.23 | 6.71 | 5.26
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0.55,0.0,0.55}c}}$ | 3.33 | 0.16 | 3.42 | 2.73
${\color[rgb]{0,0,1}b}\bar{{\color[rgb]{0,0,1}b}}$ | 9.57 | 0.69 | 9.86 | 2.95
---
FIG. 11: EFFs for S mesons in the CI model. Left panel: electrically charges
mesons composed of quarks of different flavors. Central panel: quarkonia
including a hypothetical ground state strangeonium
(${\color[rgb]{0,0.5,0}s}\bar{\color[rgb]{0,0.5,0}s}$). Right panel:
electrically neutral mesons composed of quarks of different flavors. EFFs of
electrically neutral but flavored mesons have been normalized to $F^{S}(0)=0$.

with

$\displaystyle\mathcal{A}_{EE}^{{\color[rgb]{0,0,1}{S}}}$
$\displaystyle=\alpha
M_{{\color[rgb]{0,0,1}f_{1}}}-2(1-\alpha)M_{{\color[rgb]{0,0,1}f_{1}}}M_{{\color[rgb]{0,0,1}\bar{f_{2}}}}$
$\displaystyle+(\alpha-2)M_{{\color[rgb]{0,0,1}\bar{f_{2}}}}^{2}+\alpha
M_{M}^{2}\,.$ (30)

Note the close resemblance between $T_{EE}^{{\color[rgb]{0,0,1}{S}}}$ and
$T_{EE}^{{\color[rgb]{0,0,1}{PS}}}$. As expected, there are only sign
differences between the two due to the presence, or absence, of the
$\gamma^{5}$ matrix. In Table 6, we present the parameters used for S mesons
in order to compute the masses, amplitudes and charge radii. We enlist the
masses and BSAs of S mesons in Table 7 while the EFFs are depicted in Fig. 11.
On the right and central panels we present the results for neutral mesons
while the left panel displays the EFFs of charged mesons. We emphasize that
for electrically neutral but flavored S mesons, we normalize the EFFs to zero
at $Q^{2}=0$, while for flavorless mesons, the normalization is $F^{S}(0)=1$
to be consistent with the definition employed in Eq. (20). We again perform a
fit in the range $Q^{2}\in[0,8M_{M}^{2}]$, where $M_{M}$ is the mass of the S
meson. All the curves are faithfully reproduced by the following choice:

Table 8:  The charge radius for S mesons. All quantities are given in fm. | ${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0.64,0.0,0.0}d}}$ | ${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0.5,0}s}}$ | ${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0.5,0}s}}$ | ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{1,0,0}u}}$ | ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0.5,0}s}}$ | ${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0,1}b}}$ | ${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0,1}b}}$ | ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0,1}b}}$ | ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0.55,0.0,0.55}c}}$ | ${\color[rgb]{0,0,1}b}\bar{{\color[rgb]{0,0,1}b}}$
---|---|---|---|---|---|---|---|---|---|---
$r_{\rm S}$ | 0.55 | 0.54 | 0.50 | 0.46 | 0.42 | 0.21 | 0.41 | 0.53 | 0.43 | 0.43

$\displaystyle
F^{S}(Q^{2})=\frac{e_{M}+a_{S}\,Q^{2}+b_{S}\,Q^{4}}{1+c_{S}\,Q^{2}+d_{S}\,Q^{4}}\,,$
(31)

Table 9: Parameters for the fit in Eq. (31) for S mesons. | $a_{S}$ | $b_{S}$ | $c_{S}$ | $d_{S}$
---|---|---|---|---
${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0.64,0.0,0.0}d}}$ |  0.286 |  0.003 |  1.543 |  0.617
${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0.5,0}s}}$ | 0.266 | 0.002 | 1.486 | 0.629
${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0.5,0}s}}$ | 0.217 | 0.001 | 1.271 | 0.542
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{1,0,0}u}}$ | 0.759 |  $-$0.005 | 0.680 | 0.641
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0.5,0}s}}$ | 0.004 | 0.001 | 0.783 | 0.047
${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0,1}b}}$ | 0.984 | 0.001 | 1.619 | 0.087
${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0,1}b}}$ | 0.210 | 0.001 | 0.175 | 0.115
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0,1}b}}$ | 0.289 | 0.001 | 0.743 | 0.026
${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0.55,0.0,0.55}c}}$ | 0.217 | 0.001 | 0.860 | 0.673
${\color[rgb]{0,0,1}b}\bar{{\color[rgb]{0,0,1}b}}$ | 0.269 | 0.000 | 1.607 | 0.020

where $e_{M}:=F^{S}(Q^{2}=0)$ is the electric charge of the meson and
$a_{S},b_{S},c_{S},d_{S}$ are the parameters of the fit. These values for S
mesons are listed in Table 9. Based on these numbers, we can immediately infer
the large $Q^{2}$ behavior of these EFFs. The coefficient $b_{S}\approx 0$ for
all S mesons under consideration. Therefore, the EFFs for S mesons fall as
$1/Q^{2}$ for large $Q^{2}$.

We present the numerical values of the charge radii for S mesons in Table 10.
We must reiterate that for the S mesons there are no reported measurements of
their charge radii. Theoretical results are also scarce for any direct and
meaningful comparison. It is worth mentioning again that the internal
structure of scalar mesons is not well-established. Our results are based on
considering them as effective quark-antiquark states.

Table 10:  The charge radii for S mesons. All quantities are reported in fm. | ${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0.64,0.0,0.0}d}}$ | ${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0.5,0}s}}$ | ${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0.5,0}s}}$ | ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{1,0,0}u}}$ | ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0.5,0}s}}$ | ${\color[rgb]{1,0,0}u}\bar{{\color[rgb]{0,0,1}b}}$ | ${\color[rgb]{0,0.5,0}s}\bar{{\color[rgb]{0,0,1}b}}$ | ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0,0,1}b}}$ | ${\color[rgb]{0.55,0.0,0.55}c}\bar{{\color[rgb]{0.55,0.0,0.55}c}}$ | ${\color[rgb]{0,0,1}b}\bar{{\color[rgb]{0,0,1}b}}$
---|---|---|---|---|---|---|---|---|---|---
Our Result |  $0.55$ |  $0.54$ |  $0.50$ |  $0.47$ |  $0.44$ |  $0.42$ |  $0.41$ |  $0.40$ |  $0.43$ |  $0.39$

FIG. 12:  EFF for $\sigma$-meson. The central curve is obtained using the
$\Lambda_{\rm UV}$ value from Table 6 while the band represents a $5\%$
variation in the charge radius.

We would like to remind the reader that we again allow for a 5% variation in
the charge radii of S mesons. However, in Fig. 11, we present the EFFs only
for their central values for visual clarity, refraining from showing the
corresponding band to avoid possible overlapping. However, in Fig. 12, we
depict a representative plot with a 5% variation in the charge radius for the
lightest scalar meson, $\sigma$, alone. Other mesons have similar bands.
Finally, in Fig. 13, we plot the charge radii, extracted from the EFFs, as a
function of the S meson mass. In general, the charge radii decrease when the S
meson masses increase just as we observed for the PS mesons.

FIG. 13: Charge radii of S mesons.

##  VI Conclusions

In this work we present an exhaustive computation of EFFs employing the CI
model for twenty ground state PS and S mesons. Note that the CI findings for
light mesons and heavy quarkonia are already found in the literature as
mentioned before Gutierrez-Guerrero et al. (2010a); Bedolla et al. (2015,
2016); Raya et al. (2018b). We include these results for the sake of
completeness and as a guide to pin down the best parameters in order to
explore heavy-light systems. We thus report first results on the latter mesons
within this model/formalism. We expect these new EFFs to be harder than the
exact QCD predictions, especially for the PS mesons due to the necessary
inclusion of the $F$-amplitude. We also anticipate the charge radii to be in
the ballpark of a (20-25)% error in light of the results where comparison with
realistic studies and/or experiment has been possible.

Furthermore, we analyze the sensitivity of the evolution of the EFFs by a
change in appropriate parameters to allow for a 5% variation in the charge
radii of the corresponding mesons. The evolution band has been shown
explicitly for $\pi,K,\sigma$ and $\eta_{c}$ alone to avoid over-crowding in
other collective plots. However, it is worth mentioning that the corresponding
bands in other EFFs are almost identical. Interpolations have also been
provided in Eqs. (24, 31) and Tables 4 and 9 which allow for a convenient
algebraic analysis of the behavior of the EFFs in the momentum range that we
mentioned above and for any application the reader may deem useful. We plan to
recalculate these EFFs for vector and axial vector mesons followed by the same
computation within a more realistic algebraic model and in the long run for
truncations more akin to full non-perturbative QCD. It is also straightforward
to generalise our analysis to study diquarks EFFs which are crucial in the
subsequent computation of baryons EFFs such as the ones reported recently in
Raya et al. (2021). All this is for future.

###### Acknowledgements.

L. X. Gutiérrez-Guerrero wishes to thank the support from Cátedras CONACyT
program of Mexico. The work of R. J. Hernández-Pinto is supported by CONACyT
(Mexico) Project No. 320856 (Paradigmas y Controversias de la Ciencia 2022),
Ciencia de Frontera 2021-2042 and Sistema Nacional de Investigadores as well
as by PROFAPI 2022 Grant No. PRO_A1_024 (Universidad Autónoma de Sinaloa). The
work of A. Bashir is supported in part by the US Department of Energy (DOE)
Contract No. DE-AC05-06OR23177, under which Jefferson Science Associates, LLC
operates Jefferson Lab. A. Bashir also acknowledges Coordinación de la
Investigación Científica of the Universidad Michoacana de San Nicolás de
Hidalgo grant 4.10 and the Fulbright-García Robles scholarship for his stay as
a visiting scientist at the Thomas Jefferson National Accelerator Facility,
Newport News, Virginia, USA. We thank Jozef Dudek and Christine Davies for
helpful communication on lattice results on EFFs and charge radii.

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