[a]Bruno El-Bennich

# Heavy meson chiral Lagrangians, effective flavored couplings, SU(4) flavor
breaking and their consequences

Fernando E. Serna

###### Abstract

We review heavy quark flavor and spin symmetries, their exploitation in heavy
meson effective theories and the flavored couplings of charmed and light
mesons in the definition of their effective Lagrangians. We point out how
nonperturbative continuum QCD approaches based on Dyson-Schwinger and Bethe-
Salpeter equations can be used to calculate strong and leptonic decays of
open-charm mesons and heavy quarkonia. The strong decay $D^{*}\to D\pi$ serves
as a benchmark, as it is the only physical open-charm observable that can be
related to the effective Lagrangian’s couplings. Nonetheless, a quantitative
comparison of $D^{*}D\pi$, $\rho DD$, $\rho D^{*}D$ and $\rho D^{*}D^{*}$
couplings for a range of off-shell momenta of the $\rho$-meson invalidates
SU(4)F symmetry relations between these couplings. Thus, besides the breaking
of flavor symmetry by mass terms in the Lagrangians, the flavor-symmetry
breaching in couplings and their dependence on the $\rho$-meson virtuality
cannot be ignored. We also take the opportunity to present new results for the
effective $J/\psi DD$ and $J/\psi D^{*}D$ couplings. We conclude this
contribution with a discussion on how the description of pseudoscalar and
vector $D$, $D_{s}$, $B$ and $B_{s}$ meson properties can be drastically
improved with a modest modification of the flavor-dependence in the Bethe-
Salpeter equation.

##  1 Introduction

Heavy flavor physics has been an important and very active field of particle
physics for most of the past three decades and is posed to remain a formidable
source of new discoveries. Most efforts have been dedicated to the study of
weak $B$-meson decays and to establishing experimental evidence for direct
and/or indirect (in oscillation) CP-violation. In the 2000s, the BaBar
experiment at the Stanford Linear Accelerator Center (SLAC) and the Belle
Experiment at the High Energy Accelerator Research Organization (KEK), both
electron-positron colliders with the center of mass energy tuned to the
$\Upsilon(4S)$ resonance, were very successful in observing a large number of
CP-violating processes in $B$ decays. In the following decade dedicated
experiments were pursued by the LHC$b$ experiment with the Large Hadron
Collider (LHC) at CERN. The new generations of detectors at the LHC provided
hitherto unprecedented luminosities, which led to the discovery of CP
violation in $B_{s}$ decays [1] and most recently CP violation was reported in
$D^{0}\to K^{-}K^{+}$ and $D^{0}\to\pi^{-}\pi^{+}$ decays [2].

These important experimental efforts by BaBar and Belle, and nowadays by the
LHC$b$ and Belle II collaborations, have strongly improved our understanding
of CP-violating mechanisms. Much of our current knowledge about the
electroweak sector of the Standard Model will serve as a guideline to
interpret the weak interactions of any new particles to be discovered. On the
other hand, the contributions of the strong force to heavy meson decay
amplitudes are still the major source of uncertainty in this field. That is to
say, theoretical approaches to weak decays based on perturbative Quantum
Chromodynamics (QCD), such as heavy quark effective field theory and
associated factorization theorems, provide the means to systematically
integrate out energy scales in the perturbative domain. This yields
approximations of the decay amplitudes in terms of products of hard and soft
matrix elements valid in the heavy-quark limit. However, a reliable evaluation
of the soft physics, namely hadronic wave functions and form factors, is
notoriously difficult. These difficulties must be overcome, as the precise
knowledge of hadronic contributions to heavy meson decay amplitudes is crucial
to determine strong phases without which CP violation cannot occur [3, 4, 5].

Flavor physics is not merely a fertile playground for indirect precision tests
of the Standard Model, as it also provides a powerful tool to study
nonperturbative aspects of QCD in charm and bottom mesons and in heavy
quarkonia. Charm physics, on its own right, will be the object of much
experimental activity at the Facility for Antiproton and Ion Research (FAIR),
at the Japan Proton Accelerator Research Complex (J-PARC) and at the Beijing
Spectrometer (BES III). Notably, in this context, the masses and quantum
numbers of recently discovered $D$ mesons and their possible radial
excitations are not yet precisely determined (the preferred interpretations
are scalar and vector $D$ and $D_{s}$ mesons in some cases). The discovery of
new types of charmonia, especially the plethora of observed $X,Y,Z$ states, is
exciting as these objects challenge the commonplace view of hadrons as either
$\bar{q}q$ or $qqq$ color-singlet states. Their composition remains
controversial and numerous theoretical descriptions have been proposed. In
case of the $X(3872)$ this ranges from mixtures of pure charmonia with
molecular states to purely molecular bound states and tetraquarks. Molecular
states, which are bound states of mesons, are plausible since the mass
difference between the $X(3872)$ and the $\bar{D}^{0}D^{*0}$ threshold is
tiny.

The effective Lagrangian approaches we discuss in Section 2 treat the observed
decays, for instance $X(3872)\to J/\psi\,\pi^{+}\pi^{-}$ [7, 8, 9, 10], in
terms of mesonic degrees of freedom, where a $J/\psi\,\rho$ state in
$X(3872)\to J/\psi\,\rho\to J/\psi\,\pi^{+}\pi^{-}$ is preceded by a $D$-meson
loop that couples to the $J/\psi$ and $\rho$ [11]. Similarly, the reaction
$D^{(*)}D^{(*)}\to\pi X(3872)$ is effectively described by an intermediate
triangle diagram with pseudoscalar and vector mesons [12]. As we will see, the
effective Lagrangians are expressed in terms of effective flavored couplings
between $D$\- and light mesons which are not known _a priori_.

Figure 1: A set of representative yet not exhaustive diagrams of $DN$
scattering via the exchange of light pseudoscalar and vector mesons. The $\pi
NN$ coupling can be be deduced from the experimental pion-nucleon scattering
lengths of pionic atoms [6] and the $D^{*}D\pi$ coupling can be extracted from
the physical decay width of $D^{*+}\to D^{0}\pi^{+}$. It is, however,
misguided to assume SU(4)F flavor symmetry expressed by relations of the kind
$g_{D^{*}D\pi}=g_{\rho DD}=g_{\rho D^{*}D^{*}}$ in applications of effective
Lagrangians to these scattering amplitudes. At which light-meson virtuality
$q^{2}$ the couplings are evaluated should also be taken into consideration.

Related considerations apply to the prospects of studying in-medium $D$
mesons, which may open the possibility of forming novel charmed nuclear bound
states. Their existence is contentious, yet the study of interactions between
charmed mesons and nuclear matter represents an important component of the
proposed $\overline{\textrm{P}}$ANDA activities at FAIR, where low-momentum
charmonia, such as $J/\psi$ and $\psi$, as well as $D^{(\ast)}$ mesons, will
be produced by antiproton annihilation on nuclei [13]. Effective Lagrangians
have also been employed to calculate $DN$ cross section with the diagrams
depicted in Figure 1. They are typically formulated with couplings between
$D^{(\ast)}$ mesons and light pseudoscalar and vector mesons and are derived
from an SU(4)F extension of light-flavor chirally-symmetric Lagrangians.

At a future Electron-Ion Collider (EIC), the electroproduction of these
charmonia from nuclei at threshold may be exploited to reveal the existence of
hidden-charm pentaquarks [14] and offers prospects for a quantitative estimate
of the QCD trace anomaly [15, 16, 17]. Irrespective of the experiment’s aim,
the extraction of valuable information from cross sections implies the
difficult task of understanding charmonium production and its final-state wave
function [18] whose light-front projections were recently computed in Ref.
[19] and are discussed in another contribution to these proceedings [20].

Thus, while we just swept a wide range of physical motivations and
experimental and theoretical challenges in heavy flavor physics, their strong
appeal to a hadron physicist should be evident. In the following, we will
review various aspects of charm physics and the motivation for heavy flavor
symmetry in effective Lagrangians, while also pointing out a possible
misguidance in assuming universal couplings between heavy and light mesons.
Flavor-symmetry breaking and a nonperturbative description of mesons are
indeed the common thread of this contribution.

##  2 Flavor and spin symmetries in effective theories

In an effective heavy-quark expansion of QCD it is common practice to take the
limit $m_{Q}\to\infty$ as a good approximation for quarks with masses
$m_{Q}\gg\Lambda_{\mathrm{QCD}}$. This leads to heavy-spin and -flavor
symmetries which are _not_ manifest in the QCD Lagrangian but emerge in heavy
quark effective theory (HQET) [21, 22, 23]. In a nutshell, this effective
theory consists in an expansion in powers of $m_{Q}^{-1}$ and $\alpha_{s}$ and
to leading order the HQET Lagrangian reads,

$\mathcal{L}_{\mathrm{HQET}}\ =\ \bar{h}_{\mathrm{v}}\,i\mathrm{v}\cdot
D\,h_{\mathrm{v}}\ +\
\frac{1}{2m_{Q}}\Big{[}\bar{h}_{\mathrm{v}}\left(i\,{\vec{D}}\right)^{\\!2}\\!h_{\mathrm{v}}+c(\mu)\dfrac{g}{2}\,\bar{h}_{\mathrm{v}}\,\sigma_{\mu\nu}\,t^{a}G_{\mu\nu}^{a}\,h_{\mathrm{v}}\Big{]}\
+\ \ \mathcal{O}\left(m_{Q}^{-2},\alpha_{s}^{2}\right)\ ,$ (1)

where $D_{\mu}=\partial_{\mu}+gt^{a}A_{\mu}^{a}$ is the usual covariant
derivative, $A^{a}_{\mu}$ is the gluon field, $G_{\mu\nu}^{a}$ is the gluon
field strength tensor and $t^{a}=\lambda^{a}/2$ are the SU(3) color group
generators. The Wilson coefficient $c(\mu)$ represents higher-order operator
corrections in $\alpha_{s}(\mu)=g^{2}/4\pi$ with a matching condition at tree
level that implies $c(m_{Q})=1+\mathcal{O}[\alpha_{s}(m_{Q}^{2})]$ at the
renormalization point $\mu=m_{Q}$. The heavy quark interacts with the light
quark by exchanging momenta $k$ much smaller than its mass, i.e.
$k\sim\Lambda_{\mathrm{QCD}}\ll m_{Q}$. Hence, being almost on-shell, it moves
with the hadron velocity111 The four-velocity satisfies $\mathrm{v}^{2}=1$
with $\mathrm{v}_{\mu}=(1,\vec{0})$ in the rest frame of the heavy quark.
$\mathrm{v}_{\mu}$ and to a good approximation the heavy-quark momentum can be
written as,

$p_{\mu}=m_{Q}\mathrm{v}_{\mu}+k_{\mu}\ ,$ (2)

close to the momentum $m_{Q}\mathrm{v}_{\mu}$ of the hadron’s motion. In other
words, in the limit $m_{Q}\to\infty$, the heavy-quark velocity becomes a
conserved quantity and the momentum exchange with surrounding light
constituents is predominantly soft. In the Lagrangian (1), $h_{\mathrm{v}}$
denotes the large component of the Dirac spinor field obtained from the
positive-energy projection,

$h_{\mathrm{v}}(x)=e^{im_{Q}\mathrm{v}\cdot x}\
\frac{1+\gamma\cdot\mathrm{v}}{2}\ Q(x)\ ,$ (3)

where $Q(x)$ is the heavy-quark spinor in the QCD Lagrangian,
$\mathcal{L}_{Q}=\bar{Q}(\gamma\cdot D-m_{Q})Q$. The exponential phase factor
serves to subtract $m_{Q}\mathrm{v}_{\mu}$ from the heavy quark momentum, so
that in momentum space a derivative acting on $h_{\mathrm{v}}$(x) produces the
residual momentum $k$. The small component of the quark spinors is suppressed
in $m_{Q}^{-1}$ and integrated out in the effective theory.

In the infinite-heavy quark limit only the leading term survives in the
Lagrangian (1), which exhibits the full $U(2N_{h})$ spin-flavor symmetry
($N_{h}$: heavy flavor number): it is mass-independent and invariant under
rotations in flavor space, and in the absence of Dirac matrices heavy-quark
interactions with gluons leave its spin unchanged while $h_{\mathrm{v}}(x)$ is
invariant under spin rotation. When applying Feynman rules it gives rise to
the well known quark propagator in the heavy-quark limit, which can also be
derived using Eq. (2):

$S(p)\,=\,i\,\frac{\gamma\cdot p+m_{Q}}{p^{2}-m_{Q}^{2}}\ \
\xrightarrow{m_{Q}\to\infty}\ \
i\,\frac{1+\gamma\cdot\mathrm{v}}{2\,\mathrm{v}\cdot k}\ +\
\mathcal{O}\left(\frac{k}{m_{Q}}\right)\ .$ (4)

Since the heavy-quark spin decouples in this limit, the light quarks do not
experience any different interactions with a heavy constituent quark in a
pseudoscalar or in vector meson. This implies symmetry relations between the
decay constants of heavy mesons: $f_{D}=f_{D^{*}}$ and $f_{B}=f_{B^{*}}$. Of
course, in full QCD this is not even the case for the $B$-mesons. The second
term in Eq. (1) is the kinetic energy, ${\vec{p}\,}^{2}\\!/2m_{Q}$, of a heavy
non-relativistic constituent quark and breaks the heavy quark flavor symmetry.
Both, heavy quark spin and flavor symmetries are broken by
$\mathcal{O}(\Lambda_{\mathrm{QCD}}/m_{Q})$ and $\alpha_{s}$ corrections due
the chromomagnetic term in Eq. (1), and so do higher terms in the expansion as
well as nonperturbative contributions. More precisely, gluon exchanges
renormalize coefficients that multiply effective operators. For example, the
renormalization scale dependence of $c(\mu)$ must be cancelled by that of the
chromomagnetic moment operator. However, in evaluating matrix elements with
initial and final hadron states, the operators give rise to form factors which
break flavor symmetry and whose scale dependence is often unknown due to the
model approach employed. To summarize, HQET can schematically be expressed as
the expansion,

$\mathcal{L}_{\mathrm{HQET}}\ =\
\mathcal{L}_{0}+\frac{1}{m_{Q}}\,\mathcal{L}_{1}+\frac{1}{m_{Q}^{2}}\,\mathcal{L}_{2}+\,\ldots\
,$ (5)

in which $\mathcal{L}_{0}$ is manifestly spin and flavor independent, whereas
the $1/m_{Q}$ terms are symmetry breaking corrections.

Spin and flavor symmetries have also guided the construction of an effective
Lagrangian that describes the interactions of the pseudoscalar and vector
$D^{(*)}$ and $B^{(*)}$ mesons with light pseudoscalar mesons [24, 25, 26].
This Lagrangian satisfies C, P, T and Lorentz invariance and at leading order
in the $1/M_{H}$ expansion, where $M_{H}$ is the heavy-meson mass, it imposes
flavor and spin symmetry in the heavy meson sector and chiral
$\mathrm{SU}(3)_{L}\otimes\mathrm{SU}(3)_{R}$ invariance in the light sector:

$\mathcal{L}_{\mathrm{HMChPT}}\ =\
-\operatorname{Tr}\big{[}\bar{H}_{a}\,i\mathrm{v}\cdot
D_{ab}H_{b}\big{]}+i\hat{g}\operatorname{Tr}\big{[}\bar{H}_{a}\gamma_{\mu}\gamma_{5}\mathbf{A}^{ab}_{\mu}H_{b}\big{]}+\frac{f^{2}_{\pi}}{8}\,\partial_{\mu}\xi^{2}_{ab}\partial_{\mu}\xi_{ba}^{2\dagger}\
,$ (6)

where the trace is over Dirac and flavor indices, the latter represented by a
subscript that denotes the light antiquark in the heavy meson:
$H_{a}=Q\bar{q}_{a}$, $a=u,d,s$. Here, $\mathrm{v}$ are the heavy-meson
velocities and a sum over them is implicit. The covariant derivative is
defined as the sum of the partial derivative and the vector current of the
light-meson octet,

$D^{ab}_{\mu}H_{b}=\partial_{\mu}H_{a}-\,\tfrac{1}{2}\left[\xi^{\dagger}\partial_{\mu}\xi+\xi\partial_{\mu}\xi^{\dagger}\right]_{ab}H_{b}\
,$ (7)

and has the right transformation properties under chiral transformations with
the negative parity spin doublets, $D,D^{*}$ and $B,B^{*}$, represented by
Dirac tensor structures as,

$H_{a}(\mathrm{v})=\frac{1+\gamma\cdot\mathrm{v}}{2}\left[P_{\mu}^{*a}(v)\gamma_{\mu}-P^{a}(v)\gamma_{5}\right]\
,\quad\bar{H}_{a}=\gamma_{0}H_{a}^{\dagger}\gamma_{0}\ .$ (8)

The heavy vector and pseudoscalar fields are described by annihilation
operators, $P^{a}$ and $P^{*a}_{\mu}$, which are normalized as:

$\displaystyle\langle 0|P^{a}|H_{a}(0^{-})\rangle$
$\displaystyle=\sqrt{M_{H}}\ ,$ (9) $\displaystyle\langle
0\left|P^{*a}_{\mu}\right|H^{*}_{a}(1^{-})\rangle$
$\displaystyle=\sqrt{M_{H}}\,\bm{\epsilon}^{\lambda}_{\mu}\ .$ (10)

One has by definition $\mathrm{v}\cdot P^{*a}=0$, $\bm{\epsilon}^{\lambda}$ is
the polarization of the vector meson and $M_{P}=M_{P^{*}}=M_{H}$ at this
leading order. The light-meson octet also enters the Lagrangian via the axial
current $\mathbf{A}_{\mu}^{ab}$,

$\mathbf{A}_{\mu}^{ba}=\frac{i}{2}\left[\xi^{\dagger}\partial_{\mu}\xi-\xi\partial_{\mu}\xi^{\dagger}\right]_{ba}\
,$ (11)

which transforms as the adjoint representation of SU(3)V and makes use of the
exponential representation $\xi=\exp\left(i\mathcal{M}/f_{\pi}\right)$, where
$f_{\pi}$ is the weak decay constant of the pion in the chiral limit and
$\mathcal{M}$ is the $3\times 3$ hermitian, traceless matrix of the pseudo-
Goldstone boson octet:

$\mathcal{M}=\left(\begin{array}[]{ccc}\sqrt{\frac{1}{2}}\,\pi^{0}+\sqrt{\frac{1}{6}}\,\eta&\pi^{+}&K^{+}\\\
\pi^{-}&-\sqrt{\frac{1}{2}}\,\pi^{0}+\sqrt{\frac{1}{6}}\,\eta&K^{0}\\\
K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}}\,\eta\end{array}\right)\ .$ (12)

Their coupling to the heavy mesons is determined by a putative _universal
coupling constant_ $\hat{g}$. The last term in Eq. (6) is the non-linear
Lagrangian that describes the light meson self-interactions. The interactions
between heavy and light mesons are obtained by expanding the field $\xi$ and
taking traces. Corrections to the Lagrangian stem from higher terms in the
$1/M_{H}$ expansion and from chiral symmetry breaking; for details we refer to
Ref. [26].

While SU$(3)_{F}$ is a sensible approximation in hadron physics, though broken
at the $\approx 20\%$ level, it is questionable whether this is still
reasonable for SU$(4)_{F}$. At the quark level, we saw that at next-to-leading
order in the effective Lagrangian (1) flavor and spin symmetries are violated
and similarly so for charmed mesons in the Lagrangian of Heavy Meson Chiral
Perturbation Theory (HMChPT) of Eq. (6). However, the charm and $D$ masses,
unlike those of the $b$-quark and $B$-mesons, sit in an uncomfortable energy
region. They are not light, nor are they heavy enough to neglect higher-order
$1/M_{H}$ corrections. Moreover, it is assumed that the coupling $\hat{g}$
between the light and heavy meson fields is universal at leading order in the
Lagrangian (6), i.e. it can be obtained from the strong $B^{*}B\pi$ coupling
with renormalization of $\hat{g}$ by $\mathcal{O}(1/M_{H})$ corrections [26].
However, as we will see in Section 3, it makes a material difference whether
$\hat{g}$ is extracted from a hadronic $g_{D^{*}D\pi}$ or $g_{B^{*}B\pi}$
coupling. The former coupling can be related to the physical decay $D^{*}\to
D\pi$ [27, 28], though the latter is unphysical considering the mass
difference $m_{B^{*}}-m_{B}<m_{\pi}$. Still, one can calculate $g_{B^{*}B\pi}$
in the chiral limit, $m_{\pi}\to 0$.

Besides HMChPT, other like-minded approaches to interactions between heavy and
light mesons are based on a phenomenological “bottom-up” SU$(4)_{F}$
Lagrangian, not derived from a $1/m_{Q}$ expansion of the heavy-quark spinor
and tantamount to a meson-exchange model. In the limit of SU$(4)_{F}$
invariance, the free Lagrangian for pseudoscalar and vector mesons is given by
[29, 30, 31, 32],

$\mathcal{L}_{0}\,=\,\operatorname{Tr}\big{(}\partial_{\mu}P^{\dagger}\partial_{\mu}P\big{)}-\frac{1}{2}\operatorname{Tr}\left(F_{\mu\nu}^{\dagger}F_{\mu\nu}\right)\
,$ (13)

where $F_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}$, and $P$ and
$V$ denote the $4\times 4$ pseudoscalar and vector meson matrices in
SU$(4)_{F}$ with proper normalization:

$P=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cccc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}+\frac{\eta_{c}}{\sqrt{12}}&\pi^{+}&K^{+}&\bar{D}^{0}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}+\frac{\eta_{c}}{\sqrt{12}}&K^{0}&D^{-}\\\
K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}}\eta+\frac{\eta_{c}}{\sqrt{12}}&D_{s}^{-}\\\
D^{0}&D^{+}&D_{s}^{+}&-\frac{3\eta_{c}}{\sqrt{12}}\end{array}\right)\ ,$ (14)
$V=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega^{\prime}}{\sqrt{6}}+\frac{J/\psi}{\sqrt{12}}&\rho^{+}&K^{*+}&\bar{D}^{*0}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega^{\prime}}{\sqrt{6}}+\frac{J/\psi}{\sqrt{12}}&K^{*0}&D^{*-}\\\
K^{*-}&\bar{K}^{*0}&-\sqrt{\frac{2}{3}}\omega^{\prime}+\frac{J/\psi}{\sqrt{12}}&D_{s}^{*-}\\\
D^{*0}&D^{*+}&D_{s}^{*+}&-\frac{3J/\psi}{\sqrt{12}}\end{array}\right)\ .$ (15)

Next, to introduce interactions between pseudoscalar and vector mesons one
applies the following minimal prescription:

$\displaystyle\partial_{\mu}P$ $\displaystyle\ \rightarrow\
\partial_{\mu}P-\frac{i\hat{g}}{2}\left[V_{\mu},P\right]\ ,$ (16)
$\displaystyle F_{\mu\nu}$ $\displaystyle\ \rightarrow\
\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}-\frac{i\hat{g}}{2}\left[V_{\mu},V_{\nu}\right]\
.$ (17)

Applying these substitutions in the Lagrangian (13) and taking into account
the hermiticity of the matrices, namely $P=P^{\dagger}$ and $V=V^{\dagger}$,
one arrives at the Lagrangian:

$\displaystyle\mathcal{L}\ =$ $\displaystyle\ \
\mathcal{L}_{0}+i\hat{g}\operatorname{Tr}\left(\partial_{\mu}P\left[P,V_{\mu}\right]\right)-\frac{\hat{g}^{2}}{4}\operatorname{Tr}\left(\left[P,V_{\mu}\right]^{2}\right)$
$\displaystyle+\
i\hat{g}\operatorname{Tr}\left(\partial_{\mu}V_{\nu}\left[V_{\mu},V_{\nu}\right]\right)+\frac{\hat{g}^{2}}{8}\operatorname{Tr}\left(\left[V_{\mu},V_{\nu}\right]^{2}\right)\
.$ (18)

In this approach, SU$(4)_{F}$ flavor symmetry is explicitly broken by hadron-
mass terms we here omitted, as they are irrelevant to the discussion, and for
which experimental values are used. However, due to the flavor symmetry
imposed, some exact relations between the couplings exist as a consequence of
group properties, as will be seen shortly. The Lagrangian in Eq. (18) accounts
for the three-point functions $PPV$ and $VVV$. Yet, to include anomalous
parity terms that give rise to $PVV$ vertices, it is customary to resort to
the gauged Wess-Zumino action and derive an anomalous three-meson Lagrangian
[31, 33]:

$\mathcal{L}_{\mathrm{an.}}=\,-\frac{\hat{g}_{a}^{2}N_{c}}{16\pi^{2}f_{\pi}}\,\epsilon_{\mu\nu\alpha\beta}\operatorname{Tr}\left(\partial_{\mu}V_{\nu}\partial_{\alpha}V_{\beta}P\right)\
.$ (19)

For practical applications to specific mesons, the Lagrangians (18) and (19)
are expanded in terms of the pseudoscalar- and vector-meson matrices in Eqs.
(14) and (15), where only the entries for the mesons of interest are kept and
all other matrix elements are set to zero; see Ref. [32] for further details.
Specifying to the case of the $\pi$, $\rho$, $D$ and $J/\psi$ mesons, which is
relevant to a phenomenological description of $J/\psi$ suppression in final-
state interactions of heavy-ion collisions, the relevant interaction
Lagrangians derived from Eq. (18) and (19) are:

$\displaystyle\mathcal{L}_{D^{*}\\!D\pi}$ $\displaystyle=\
ig_{D^{*}Dpi}\,D^{*}_{\mu}\,\vec{\tau}\cdot\left(\bar{D}\partial_{\mu}\vec{\pi}-\partial_{\mu}\bar{D}\vec{\pi}\right)\
,$ (20) $\displaystyle\mathcal{L}_{D^{*}\\!D^{*}\pi}$
$\displaystyle=-g_{D^{*}D^{*}\pi}\,\epsilon_{\mu\nu\alpha\beta}\,\partial_{\mu}D_{\nu}^{*}\vec{\tau}\cdot\vec{\pi}\,\partial_{\alpha}\bar{D}_{\beta}^{*}\
,$ (21) $\displaystyle\mathcal{L}_{\psi DD}$ $\displaystyle=\ ig_{\psi
DD}\,\psi_{\mu}\left(\partial_{\mu}D\bar{D}-D\partial_{\mu}\bar{D}\right)\ ,$
(22) $\displaystyle\mathcal{L}_{\psi D^{*}\\!D}$ $\displaystyle=\ g_{\psi
D^{*}D}\,\epsilon_{\mu\nu\alpha\beta}\,\partial_{\mu}\psi_{\nu}\left(\partial_{\alpha}D_{\beta}^{*}\bar{D}+D\partial_{\alpha}\bar{D}_{\beta}^{*}\right)\
,$ (23) $\displaystyle\mathcal{L}_{\psi D^{*}\\!D^{*}}$ $\displaystyle=\
ig_{\psi
D^{*}D^{*}}\,\left[\psi_{\mu}\left(\partial_{\mu}D^{*}_{\nu}\bar{D}_{\nu}^{*}-D^{*}_{\nu}\partial_{\mu}\bar{D}_{\nu}^{*}\right)+\left(\partial_{\mu}\psi_{\nu}D_{\nu}^{*}-\psi_{\nu}\partial_{\mu}D_{\nu}^{*}\right)\bar{D}^{*}_{\mu}\right.$
$\displaystyle\left.+\
D^{*}_{\mu}\left(\psi_{\nu}\partial_{\mu}\bar{D}_{\nu}^{*}-\partial_{\mu}\psi_{\nu}\bar{D}_{\nu}^{*}\right)\right]\
,$ (24) $\displaystyle\mathcal{L}_{\rho DD}$ $\displaystyle=\ ig_{\rho
DD}\left(D\vec{\tau}\,\partial_{\mu}\bar{D}-\partial_{\mu}D\vec{\tau}\,\bar{D}\right)\cdot\vec{\rho}_{\mu}\
,$ (25) $\displaystyle\mathcal{L}_{\rho D^{*}\\!D}$ $\displaystyle=\ g_{\rho
D^{*}D}\,\epsilon_{\mu\nu\alpha\beta}\,\partial_{\mu}\rho_{\nu}\left(\partial_{\alpha}D_{\beta}^{*}\vec{\tau}\bar{D}+D\vec{\tau}\partial_{\alpha}\bar{D}_{\beta}^{*}\right)\
,$ (26) $\displaystyle\mathcal{L}_{\rho D^{*}\\!D^{*}}$ $\displaystyle=\
ig_{\rho
D^{*}D^{*}}\left[\left(\partial_{\mu}D^{*}_{\nu}\vec{\tau}\,\bar{D}_{\nu}^{*}-D^{*}_{\nu}\vec{\tau}\,\partial_{\mu}\bar{D}_{\nu}^{*}\right)\cdot\vec{\rho}_{\mu}+\left(D^{*}_{\nu}\vec{\tau}\cdot\partial_{\mu}\vec{\rho}_{\nu}-\partial_{\mu}D^{*}_{\nu}\vec{\tau}\cdot\vec{\rho}_{\nu}\right)\bar{D}^{*}_{\mu}\right.$
$\displaystyle\left.+\
D^{*}_{\mu}\left(\vec{\tau}\cdot\vec{\rho}_{\nu}\,\partial_{\mu}\bar{D}_{\nu}^{*}-\vec{\tau}\cdot\partial_{\mu}\vec{\rho}_{\nu}\bar{D}_{\nu}^{*}\right)\right]\
,$ (27)

In the above expressions, $\vec{\pi}$ and $\vec{\rho}$ denote the pion and rho
meson isospin triplets, respectively, and $\vec{\tau}$ are Pauli matrices,
while $D\equiv(D^{0},D^{+})$ and $D^{*}\equiv(D^{*0},D^{*+})$ describe the
pseudoscalar and vector $D$-meson doublets.

The charge-specific coupling constants, $g_{PPV}$, $g_{PVV}$ and $g_{VVV}$,
are functions of the universal SU$(4)_{F}$ couplings $g$ and $g_{a}$ defined
with the Lagrangians in Eqs (18) and (19) and their explicit relations are
given in Table 1 of Ref. [32]. We merely remind that exact SU$(4)_{F}$
symmetry implies certain relationships between the couplings, amongst which we
highlight:

$\displaystyle g_{D^{*}D\pi}$
$\displaystyle=g_{D^{*-}D^{0}\pi^{+}}=g_{D^{*+}D^{0}\pi^{-}}\,=\sqrt{2}\,g_{D^{*\pm}D^{\mp}\pi^{0}}=\sqrt{2}\,g_{D^{*0}D^{0}\pi^{0}}=\tfrac{\hat{g}}{2\sqrt{2}}\
,$ (28) $\displaystyle g_{\psi DD}$ $\displaystyle=g_{\psi D^{0}D^{0}}=g_{\psi
D^{+}D^{-}}\ ,\quad g_{\psi DD}=g_{\psi D^{*}D^{*}}=\tfrac{\hat{g}}{\sqrt{6}}\
,$ (29) $\displaystyle g_{\rho D^{*}D}$
$\displaystyle=g_{\rho^{+}D^{0}D^{*-}}=g_{\rho^{-}\bar{D}^{0}D^{*+}}=g_{\rho^{+}D^{-}D^{*0}}=g_{\rho^{-}D^{+}\bar{D}^{*0}}=\tfrac{1}{2\sqrt{2}}\tfrac{\hat{g}_{a}^{2}N_{c}}{16\pi^{2}f_{\pi}}\
,$ (30) $\displaystyle g_{D^{*}\\!D\pi}$ $\displaystyle=g_{\rho DD}=g_{\rho
D^{*}\\!D}=g_{\rho D^{*}\\!D^{*}}\ .$ (31)

The SU$(3)_{F}$ relation $g_{\rho KK}=g_{\rho\pi\pi}/2$ is preserved and
generalized to:

$g_{\rho DD}=g_{\rho KK}=\frac{g_{\rho\pi\pi}}{2}\ .$ (32)

If one takes SU$(4)_{F}$ at face value, a single known coupling constant, for
example $g_{D^{*+}D^{0}\pi^{+}}$ which has been measured experimentally [27],
suffices to infer all other couplings. However, as we will see, in particular
the last algebraic relation (32) strongly indicates that the SU$(4)_{F}$
breaking between couplings appears to be incompatible with the idea of a
universal coupling. We may thus ask: can $\hat{g}$ be unambiguously related to
all the strong-interaction matrix elements mentioned above, $g_{D^{*}D\pi}$,
$g_{D\rho D}$, etc.? Given that the rationale of building Lagrangians for
heavy mesons interactions with light mesons is nowadays extended to the
SU$(5)_{F}$ flavor group [34], our aim is to emphasize this shortcoming and to
propose effective couplings from quark-degrees of freedom that will account
for the flavor symmetry breaking effects without resorting to additional terms
in the $1/M_{H}$ expansion.

##  3 Effective couplings and flavor SU(4)${}_{\textit{F}}$ breaking

Soon after the introduction of effective SU(4)F Lagrangians, such as the ones
discussed in Section 2, the study of interactions between open-charm mesons
and nuclei to explore the possibility of charmed nuclear bound states became
the object of intense activity [35, 36, 37, 38, 39, 40]. Their application has
gone beyond that field and include the reaction $D^{(*)}D^{(*)}\to\pi X(3872)$
which involves an intermediate triangle diagram with pseudoscalar and vector
mesons [12], the decay of a highly excited $D$ meson [41], the $J/\psi$
regeneration in the hadronic gas phase following the cooling of a quark-gluon
plasma [42], the production of the $\psi(3770)$ resonance in
$\bar{p}p\to\bar{D}D$ [43] or the thermal properties of pseudoscalar and
vector charm mesons [44], amongst others. At the same time, the effective
couplings, quite naturally, have also been the object of interest [23, 45, 46,
47, 48, 50, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66, 67, 68, 69, 70, 71, 72, 73, 75, 74].

Figure 2: Diagrammatic representation of the impulse approximation to the
$D^{*+}\to D^{0}\pi^{+}$ decay in Eq. (34). The internal lines represent
dressed quark propagators and the filled ellipses are the meson’s BSAs. In
case of the soft-pion $B^{*+}\to B^{0}\pi^{+}$ amplitude the following
substitutions are in order: $c\to\bar{b}$, $\bar{d}\to u$ and $u\to\bar{d}$.

We here summarize earlier results for the $D^{*}D\pi$, $D_{s}^{*}DK$,
$B^{*}B\pi$, $B_{s}^{*}BK$, $\rho\pi\pi$, $\rho KK$, $\rho DD$, $\rho D^{*}D$
and $\rho D^{*}D^{*}$ couplings calculated in _impulse approximation_ , where
the dressed heavy- and light- quark propagators were algebraic approximations
to numerical solutions of the quark’s Dyson-Schwinger equation (DSE) [76], and
the Bethe-Salpeter amplitude (BSA) for the $D$ and $B$ were modeled in absence
of available solutions for asymmetric heavy-light bound states at that time
[23, 46, 47, 48]. As already mentioned in Section 2, the simplest coupling
that gives access to $\hat{g}$ is found in the definition of the $D^{*+}\to
D^{0}\pi^{+}$ decay amplitude:

$\langle D(p_{2})\pi(q)|D^{*}(p_{1},\lambda)\rangle\ :=\
g_{D^{*}D\pi}\,\bm{\epsilon}^{\lambda_{D^{*}}}(p_{1})\cdot q\ .$ (33)

While here illustrated for a strong $D^{*}$ decay, this matrix element
generally defines the dimensionless coupling of a heavy-light vector meson
$H^{*}$, characterized by a polarization state $\lambda$, and a heavy-light
pseudoscalar meson, $H$, to a pion with momentum $q=p_{1}-p_{2}$. Hence, one
may also use this amplitude for the unphysical process $B^{*}\to B\pi$ in the
chiral $m_{\pi}\to 0$ limit which defines $g_{B^{*}B\pi}$. A consistent
calculation within the same framework allows therefore for a quantitative
estimate of the degree to which notions of heavy-quark symmetry are sensible
in the charm sector. The diagrammatic representation of such a decay is
depicted in Figure 2. The decay $H^{*}\to H\pi$ amplitude in impulse
approximation is given by,

$g_{H^{*}\\!H\pi}\ \bm{\epsilon}^{\lambda_{H^{*}}}\\!\\!\cdot q\ =\
\operatorname{tr}\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}}\
\bm{\epsilon}^{\lambda_{H^{*}}}\\!\cdot\Gamma_{H^{*}}(k;p_{1})\,S_{Q}(k_{Q})\,\bar{\Gamma}_{H}(k;-p_{2})\,S_{u}(k_{u})\,\bar{\Gamma}_{\pi}(k;-q)\,S_{d}(k_{d})\
,$ (34)

where the trace is over Dirac spinor and color indices,
$\bm{\epsilon}^{\lambda_{H^{*}}}$ is the vector-meson polarization and the
momenta are $k_{Q}=k+\eta p_{1}$, $k_{d}=k-\bar{\eta}p_{1}$ and $k_{u}=k+\eta
p_{1}-p_{2}$ with the momentum partition parameters $\eta+\bar{\eta}=1$.
Moreover, $S(k)$ and $\Gamma(k,p)$ are dressed quark propagators [77] and
BSAs, respectively, discussed in Section 4.

The dimensionless coupling $g_{H^{*}H\pi}$ is related at leading order in the
$1/M_{H}$ expansion to the strong meson coupling $\hat{g}$ [26]:

$g_{H^{*}\\!H\pi}\ =\ 2\,\frac{\sqrt{M_{H}M_{H^{*}}}}{f_{\pi}}\ \hat{g}\,.$
(35)

Using a constituent charm quark propagator and Eqs. (34) and (35) the
difference in extracting $\hat{g}$ from either $g_{D^{*}\\!D\pi}$ or
$g_{B^{*}\\!B\pi}$ is material [46],

$g_{D^{*}\\!D\pi}=15.8^{+2.1}_{-1.0}\ \Longrightarrow\
\hat{g}_{c}=0.53^{+0.07}_{-0.03}\ ,\qquad g_{B^{*}\\!B\pi}=30.0^{+3.2}_{-1.4}\
\Longrightarrow\ \hat{g}_{b}=0.37^{+0.04}_{-0.02}\ ,$ (36)

where in order to distinguish between the extraction of the universal coupling
from either $H^{*}H\pi$ vertex we label the quark flavor: $\hat{g}_{c}$ and
$\hat{g}_{b}$. One can also employ a confining heavy-quark propagator of the
algebraic form,

$S_{Q}(k)\ =\ \frac{-i\gamma\cdot k+m_{Q}}{m_{Q}^{2}}\
\mathcal{F}(k^{2}/m_{Q}^{2})\ ,$ (37)

with the entire function ${\cal F}(x)=[1-\exp(-x)]/x$. Unlike the constituent-
quark propagator $S_{Q}(k)=[i\gamma\cdot k+m_{Q}]^{-1}$, Eq. (37) implements
confinement yet still produces a momentum independent heavy-quark mass-
function. With this substitution for the heavy-quark propagator and with
$m_{c}=1.32$ GeV and $m_{b}=4.65$ GeV one obtains [23]:

$g_{D^{*}\\!D\pi}=18.7^{+2.5}_{-1.4}\ \Longrightarrow\
\hat{g}_{c}=0.63^{+0.08}_{-0.05}\ ,\qquad g_{B^{*}\\!B\pi}=31.8^{+4.1}_{-2.8}\
\Longrightarrow\ \hat{g}_{b}=0.39^{+0.05}_{-0.03}\ .$ (38)

Our value for $g_{D^{*}\\!D\pi}$ is in good agreement with the coupling
extracted by CLEO from the $D^{*}\to D\pi$ decay width:
$g_{D^{*}\\!D\pi}^{\mathrm{exp.}}=17.9\pm 1.9$ [27]. A more recent analysis of
the $D^{*}(2010)^{+}$ decay width by the BABAR collaboration [28] yields a
somewhat smaller coupling, $g_{D^{*}\\!D\pi}^{\mathrm{exp.}}=16.9\pm 0.14$,
which still agrees with most theoretical calculations. In Table 1 and 2 we
collect a set of representative values for the couplings obtained with our
approach, QCD sum rules (QCDSR) and Lattice QCD.

|  CLEO [27] |  BABAR [28] |  DSE [23] |  QCDSR [32] |  Lattice [68] |  Lattice [69]
---|---|---|---|---|---|---
$g_{D^{*}D\pi}$ |  $17.9\pm 1.92$ |  $16.9\pm 0.14$ |  $18.7_{-1.4}^{+2.5}$ |  $17.5\pm 1.5$ |  $20.0\pm 2.0$ |  $15.8\pm 0.8$
$\hat{g}_{c}$ |  $0.60\pm 0.06$ |  $0.57\pm 0.006$ |  $0.63^{+0.08}_{-0.05}$ |  $0.59\pm 0.05$ |  $0.67\pm 0.07$ |  $0.53\pm 0.03$
Table 1: Comparison of theoretical values for $g_{D^{*}D\pi}$ and $\hat{g}_{c}$ with those extracted from the experimental $D^{*}\to D\pi$ decay widths. Note that $\hat{g}_{c}$ is extracted at leading order via Eq. (35).  |  DSE [23] |  QCDSR [45] |  Lattice [67] |  Lattice [70] |  Lattice [71] |  Lattice [72]
---|---|---|---|---|---|---
$g_{B^{*}B\pi}$ |  $31.8^{+4.1}_{-2.8}$ |  $29.0\pm 3.0$ |  — |  — |  — |  —
$\hat{g}_{b}$ |  $0.39^{+0.05}_{-0.03}$ |  $0.36\pm 0.04$ |  $0.44\pm 0.03^{+0.07}_{-0.0}$ |  $0.56\pm 0.08$ |  $0.49\pm 0.03$ |  $0.45\pm 0.05$
Table 2: Comparison of theoretical values for $g_{B^{*}B\pi}$ and
$\hat{g}_{b}$ in the soft-pion limit.

Turning our attention now to the $B^{*}B\pi$ coupling, we observe that
$\hat{g}_{b}$ obtained with lattice QCD is uniformly above $0.4$, yet the
result of Ref. [70] stands out by providing the sole value in agreement with
the $D^{*}D\pi$ coupling. This is not generally the case and whether $\hat{g}$
is extracted from the $D^{*}D\pi$ or $B^{*}B\pi$ vertex in the DSE-BSE
approach has significant consequences. Indeed, it is often interpreted as a
sign that the leading term of a $\Lambda_{\mathrm{QCD}}/m_{c}$ expansion may
not be reliable. On the other hand, the numerical values obtained for either
couplings in lattice QCD do not allow for a clear picture, i.e. whether
$\hat{g}_{c}>\hat{g}_{b}$.

We take advantage of this discussion to remind that the DSE-BSE calculations
in Refs. [46, 23, 47, 48] employ algebraic expressions for the quark
propagators of the light quarks based on DSE solutions and a simple leading
covariant BSA model for the $D$, $D^{*}$ $B$ and $B^{*}$ mesons. These
simplifications were mainly imposed by the lack of proper BSAs for heavy-light
mesons at that time [78]. The shortcomings of a too simplistic ladder-
truncation of the BSE kernel [79, 80, 81, 82] have meanwhile been overcome
with distinct ansätze for the quark-gluon vertex in the heavy and light sector
[83, 84, 19, 85, 86], and a much improved, fully Poincaré-covariant
calculation of $g_{D^{*}\\!D\pi}$ and $g_{B^{*}\\!B\pi}$ is underway [87].

Of course, as in the case of the soft-pion $B^{*}B\to\pi$ decay, one may
calculate other _unphysical_ strong decay amplitudes, for instance
$B^{*}_{s}\to BK$ and $D^{*}_{s}\to DK$ from which the effective couplings
$g_{D_{s}^{*}DK}$ and $g_{B_{s}^{*}BK}$ can be extracted. This is because we
can use Eq. (34) irrespective of the $q^{2}$-value and without resorting to
extrapolations from spacelike to timelike momenta or of the current-quark
mass. The couplings obtained in Ref. [23] are:

$g_{D_{s}^{*}DK}=24.1^{+2.5}_{-1.6}\ \ ,\qquad
g_{B_{s}^{*}BK}=33.3_{-3.7}^{+4.0}\ .$ (39)

It is notable that the ratios, $g_{D_{s}^{*}DK}/g_{D^{*}\\!D\pi}$ and
$g_{B_{s}^{*}BK}/g_{B^{*}\\!B\pi}$ are of comparable magnitude as the decay-
constant ratios $f_{D_{s}}/f_{D}$ and $f_{B_{s}}/f_{B}$, respectively,
indicating a typical $SU(3)_{F}$ flavor breaking pattern of the order of 20%.

Figure 3: Ratios of the couplings $g_{\rho KK}/g_{\rho DD}$ (solid curve) and
$g_{\rho KK}/g_{\rho\pi\pi}$ (dashed curve). In case of exact flavor SU(4)F
symmetry, these ratios take the values $1$ (dot-dashed line) and $1/2$ (dotted
line), respectively. The vertical dotted line marks the on-shell point of the
$\rho$-meson and in Euclidean metric $q^{2}>0$ is spacelike. Figure taken from
Ref. [47].

Further ratios that provide a measure for $SU(3)_{F}$ and $SU(4)_{F}$ breaking
are based on the $\rho\pi\pi$, $\rho KK$, $\rho DD$, $\rho D^{*}D$ and $\rho
D^{*}D^{*}$ couplings. The latter three couplings are those we came across in
the effective Lagrangians (25)–(27), whereas the first coupling relates
directly to the strong decay $\rho\to\pi\pi$. While no physical processes are
associated with the $\rho KK$, $\rho DD$, $\rho D^{*}D$ and $\rho D^{*}D^{*}$
couplings, they are commonly employed in defining $\rho$-mediated exchange
interactions between a nucleon and kaons or $D$-mesons [37, 39]. In these
applications, the $\rho$-meson momentum is necessarily off-shell and
spacelike, and couplings and form factors may be defined once one settles on a
definition of the off-shell $\rho$-meson. At leading-order in a systematic,
symmetry-preserving truncation scheme, in analogy with the physical decay
amplitude in Eq. (34), we can express the $\rho DD$ matrix element as:

$g_{\rho DD}\ \bm{\epsilon}^{\lambda_{\rho}}\\!\cdot p_{1}\ =\
\operatorname{tr}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\
\bar{\Gamma}_{D}(k,-p_{1})\,S_{c}(k_{c})\,\bar{\Gamma}_{D}(k,-p_{2})\,S_{f}(k_{f}^{\prime})\,\bm{\epsilon}^{\lambda_{\rho}}\\!\cdot\Gamma_{\rho}(k,q)\,S_{f}(k_{f})\
,$ (40)

where we work in the isospin symmetric limit, so $f=u=d$. Momentum
conservation requires $q=p_{1}+p_{2}$, $k_{f}=k+\eta q$,
$k_{f}^{\prime}=k-\bar{\eta}q$ and $k_{c}=k+\eta q-p_{1}$, where the relative-
momentum partitioning parameters satisfy $\eta+\bar{\eta}=1$ and
$p_{1}^{2}=p_{2}^{2}=-M_{D}^{2}$. The integral expressions for $g_{\rho
D^{*}D}$ and $g_{\rho D^{*}D^{*}}$, respectively Eqs. (41) and (42), are
obtained with the substitutions
$\Gamma_{D}(k;p)\,\to\,\bm{\epsilon}^{\lambda_{D^{*}}}\\!\cdot\Gamma_{D^{*}}(k;p)$.

As we highlighted in Eq. (32), SU(4)F symmetry implies some stringent
relations between the couplings. As can be read from Figure 3, the equality
$g_{\rho KK}=g_{\rho\pi\pi}/2$ provides a fair approximation on the domain
$P^{2}\in[-m_{\rho}^{2},m_{\rho}^{2}]$ where the deviation ranges from $-10\%$
to 40% and describes again the typical SU(3)F breaking pattern. When it comes
to the relation $g_{\rho DD}=g_{\rho KK}$, the striking discrepancy between
what flavor symmetry dictates and what is found in a covariant calculation
based on quark degrees of freedom is of the order of 360% to 440%. We note
that a lattice QCD calculation [57] finds $g_{\rho DD}(0)=4.84(34)$ and
$g_{\rho D^{*}D^{*}}(0)=5.94(56)$, values that compare well with our coupling
in Figure 4 at zero-recoil momentum: $g_{\rho DD}(0)\simeq 6.3$.

As we continue with the couplings that involve a more complicated spin
structure, we start with the amplitude that describes the transition of on-
shell $D^{*}$ to $D$ mesons emitting an off-shell $\rho$-meson or
equivalently, the unphysical process of an off-shell $\rho$-meson decaying
into a $D^{*}D$ pair. This defines the $g_{\rho D^{*}D}$ couplings as follows:

$g_{\rho
D^{*}D}\,\frac{1}{M_{D^{*}}}\,\varepsilon_{\alpha\beta\mu\nu}\,\bm{\epsilon}_{\alpha}^{\lambda_{D^{*}}}\bm{\epsilon}_{\beta}^{\lambda_{\rho}}\
p_{1\mu}\,p_{2\nu}\ :=\ \langle
D^{*}(p_{2},\lambda_{D^{*}})|\,\rho(q,\lambda_{\rho})\,|D(p_{1})\rangle\,.$
(41)

The three vector-meson vertex $\rho D^{*}D^{*}$, with two mesons on-shell,
introduces additional complexity and requires a minimal set of three
independent couplings analogous to the electromagnetic form factors of vector
mesons [88, 89]:

$\displaystyle\langle
D^{*}(p_{2}\,,\lambda_{D^{*}})|\,\rho(q,\lambda_{\rho})\,|D^{*}(p_{1}\,,\lambda_{D^{*}})\rangle$
$\displaystyle=\
-\sum_{i=1}^{3}T_{\mu\rho\sigma}^{\,i}(p,q)\;g^{i}_{D^{*}\\!\rho
D^{*}}(q^{2})\,\bm{\epsilon}_{\mu}^{\lambda_{\rho}}\bm{\epsilon}_{\rho}^{\lambda_{D^{*}}}\bm{\epsilon}_{\sigma}^{\lambda_{D^{*}}}$
$\displaystyle=\
\Lambda_{\mu\rho\sigma}(p,q)\,\bm{\epsilon}_{\mu}^{\lambda_{\rho}}\bm{\epsilon}_{\rho}^{\lambda_{D^{*}}}\bm{\epsilon}_{\sigma}^{\lambda_{D^{*}}}\
,$ (42)

Figure 4:  _Left panel_ : The dimensionless couplings $g_{D\rho D}$ and
$g_{D^{*}\\!\rho D}$ as a function of the $\rho$-meson’s four-momentum
squared, with the $D$ and $D^{*}$ mesons on-shell. Note that $g_{D^{*}\\!\rho
D}$ rather than $g_{D^{*}\\!\rho D}/m_{D^{*}}$ is plotted and $q^{2}>0$ is
spacelike in Euclidean metric. _Right panel_ : The three-vector couplings
defined in Eq. (42) with the abbreviation $g_{D^{*}\\!\rho D^{*}}^{i}\equiv
g_{i},\,i=1,2,3$, where the $D^{*}$-mesons are on-shell. In both panels the
dashed vertical line denotes the $\rho$-meson on-shell point. Figures taken
from Ref. [48].

with the tensor structures:

$\displaystyle T_{\mu\rho\sigma}^{\,1}(p,q)$ $\displaystyle=$ $\displaystyle
2\,p_{\mu}\,\mathcal{P}_{\rho\gamma}^{T}(p_{1})\,T_{\gamma\sigma}(p_{2})\ ,$
(43) $\displaystyle T_{\mu\rho\sigma}^{\,2}(p,q)$ $\displaystyle=$
$\displaystyle-\left[q_{\rho}-p_{1\rho}\,\frac{q^{2}}{2\,m_{D^{*}}^{2}}\right]T_{\mu\sigma}(p_{2})+\left[q_{\sigma}+p_{2\sigma}\,\frac{q^{2}}{2\,M_{D^{*}}^{2}}\right]T_{\mu\rho}(p_{1})\
,$ (44) $\displaystyle T_{\mu\rho\sigma}^{\,3}(p,q)$ $\displaystyle=$
$\displaystyle\frac{p_{\mu}}{M_{D^{*}}^{2}}\,\left[q_{\rho}-p_{1\rho}\,\frac{q^{2}}{2\,M_{D^{*}}^{2}}\right]\left[q_{\sigma}+p_{2\sigma}\,\frac{q^{2}}{2\,M_{D^{*}}^{2}}\right]\
.$ (45)

In Eqs. (43) to (45) we introduce the four-momentum $p$ via
$p_{1}=p-\tfrac{1}{2}q$ and $p_{2}=p+\tfrac{1}{2}q$,
$p_{1}^{2}=p_{2}^{2}=-M_{D^{(*)}}^{2}\,$ and
$T_{\alpha\beta}(p)=\delta_{\alpha\beta}-p_{\alpha}p_{\beta}/p^{2}$ is the
transverse projection operator. With this decomposition of the tensor
structures all couplings are positive, $g^{i}_{D^{*}\\!\rho D^{*}}(q^{2})\geq
0,\ i=1,2,3$, and the matrix element $\langle
D^{*}(p_{2}\,,\lambda_{D^{*}})|\,\rho(q,\lambda_{\rho})\,|D^{*}(p_{1}\,,\lambda_{D^{*}})\rangle$
satisfies transversality:

$\displaystyle p_{2\rho}\,\Lambda_{\mu\rho\sigma}(p,q)$ $\displaystyle=$
$\displaystyle 0\ ,$ (46) $\displaystyle
p_{1\sigma}\,\Lambda_{\mu\rho\sigma}(p,q)$ $\displaystyle=$ $\displaystyle 0\
,$ (47) $\displaystyle q_{\mu}\,\Lambda_{\mu\rho\sigma}(p,q)$ $\displaystyle=$
$\displaystyle 0\ .$ (48)

The corresponding couplings were obtained in Ref. [48] and are plotted as
functions of the off-shell $\rho$-momentum in Figure 4. The dimensionless
couplings $g_{\rho DD}$ and $g_{\rho D^{*}D}$ are smooth and monotonically
decreasing as $q^{2}$ increases away from the on-shell point
$q^{2}=-m_{\rho}^{2}$. On the average, in the domain
$q^{2}\in[-m_{\rho}^{2},m_{\rho}^{2}]$, one observes $g_{\rho D^{*}D}\gtrsim
3\,g_{\rho DD}$, which can be attributed to differences in the BSA
normalizations of the $D$ and $D^{*}$ and therefore indirectly to
$f_{D^{\ast}}>f_{D}$. This yields the hierarchy relation:

$g_{\rho DD}(0)\approx 6.4<g_{D^{*}\\!D\pi}\approx 17<g_{\rho
D^{*}\\!D}(0)\approx 23\ .$ (49)

The $\rho D^{*}\\!D^{*}$ couplings are also smooth and monotonically
decreasing functions of the $\rho$-momentum. However, there are remarkable
quantitative differences between their magnitudes and damping rates. Averaging
over the interval $q^{2}\in[-m_{\rho}^{2},m_{\rho}^{2}]$, one finds:

$\bar{g}_{2}\big{(}q^{2}\big{)}\ \approx 3\bar{g}_{3}\big{(}q^{2}\big{)}\
\approx\ 5\bar{g}_{1}\big{(}q^{2}\big{)}\ .$ (50)

Such relative strengths are of the same magnitude as those found in the
$\rho$-meson elastic form factor. We also note that $\bar{g}_{3}(s)\approx
0.7\,\bar{g}_{\rho D^{*}\\!D}(s)$, in other words one of the $\rho
D^{*}\\!D^{*}$ couplings is of similar strength than the $\rho D^{*}\\!D$
coupling.

Figure 5: The couplings $g_{J/\psi DD}$ and $g_{J/\psi D^{*}\\!D}$ as
functions of the off-shell momenta square, $p_{D}^{2}=p_{D^{*}}^{2}=q^{2}$,
and with the $J/\psi$ meson on-shell.

In concluding this section, we extend the calculations of charmed couplings to
the $J/\psi$ charmonium and present novel results within the same framework.
In Figure 5, the $J/\psi D\bar{D}$ and $J/\psi D^{*}\\!\bar{D}$ couplings,
defined by $\langle D(p_{1})D(p_{2})|J/\psi(p,\lambda_{J/\psi})\rangle$ and
$\langle
D^{*}(p_{1},\lambda_{D^{*}})D(p_{2})|J/\psi(p,\lambda_{J/\psi})\rangle$, are
plotted, where the $J/\psi$ is on-shell whereas the $D$ mesons are _both_
symmetrically off-shell: $p_{1}^{2}=p^{2}_{2}=q^{2}$. Note that this differs
from the convention in Ref. [32] where either the $J/\psi$ or one of the $D$
mesons are off-shell. This choice is motivated by the observation that the
couplings are commonly required for space-like momenta of the $D$ and $D^{*}$
mesons in loop diagrams that describe $J/\psi\to D^{(*)}\\!\bar{D}\to J/\psi$.
Due to the simplified ansatz for the mesons’s BSA based on Gaussian-type
functions [48] and the use of two charm propagators defined in Eq. (37), these
form factors grow rapidly for larger time-like momenta and we only plot them
up to $q^{2}=-2$ GeV2, which is below the on-shell point of the $D$ meson.
These couplings are often quoted at $q^{2}=0$ GeV2, where we find,

$g_{J/\psi DD}(0)\simeq 2.2\ \ ,\qquad g_{J/\psi D^{*}\\!D}(0)\simeq
0.7~{}\mathrm{GeV}^{-1}\ ,$ (51)

remembering that both $D$-mesons are evaluated off-shell at zero-momentum. As
for the $\rho DD$ couplings, we expect a more realistic behavior as a function
of $q^{2}$ employing the full Poincaré covariant computation of the BSA for
the $D$, $D^{*}$ and $J/\psi$ mesons [87] rather than model wave functions.

##  4 Poincaré covariance in bound-state description of heavy mesons

In the light-meson sector it is crucial to satisfy chiral symmetry and its
breaking pattern to ensures the pion is massless in the chiral limit. The
axialvector Ward-Green-Takahashi identity (axWTI) describes the properties of
the divergence of the vertex and expresses chiral symmetry and its breaking
pattern:

$P_{\mu}\Gamma_{5\mu}^{fg}(k;P)=S_{f}^{-1}\left(k_{\eta}\right)i\gamma_{5}+i\gamma_{5}S_{g}^{-1}\left(k_{\bar{\eta}}\right)-i\left[m_{f}+m_{g}\right]\Gamma_{5}^{fg}(k;P)\
.$ (52)

In this axWTI, $\Gamma_{5\mu}^{fg}(k;P)$ and $\Gamma_{5}^{fg}(k;P)$ are the
color-singlet axialvector and pseudoscalar vertices for two quark flavors, $f$
and $g$, in Eqs. (58) and (59). They are solutions of a BSE with
$\gamma_{5}\gamma_{\mu}$ or $\gamma_{5}$ inhomogeneity, respectively. The
total four-momentum of the meson satisfies $P^{2}=-M_{P}^{2}$ and the quark
momenta are defined as, $k_{\eta}=k+\eta P$ and
$k_{\bar{\eta}}=k-\bar{\eta}P$. The inverse of the dressed quark propagator,
$S_{f}^{-1}=\left[i\gamma\cdot p\,A_{f}(p^{2})+B_{f}(p^{2})\right]^{-1}$, is
fully described by vector and scalar pieces whose respective dressing
functions, $A_{f}(p^{2})$ and $B_{f}(p^{2})$, are solutions of a DSE for a
given flavor $f$ whose derivation is discussed in detail in Ref. [90], for
example. We remind that the DSE is nothing else but the nonperturbative
equation of motion of a particle in a relativistic quantum field theory. In
QCD, this DSE is given by the integral equation,

$S^{-1}_{f}(p)=\,Z_{2}^{f}\,i\,\gamma\cdot p+Z_{4}^{f}\,m_{f}(\mu)\ +\
Z_{1}^{f}g^{2}\\!\int^{\Lambda}\\!\frac{d^{4}k}{(2\pi)^{4}}\,D^{ab}_{\mu\nu}(q)\frac{\lambda^{a}}{2}\gamma_{\mu}S_{f}(k)\,\Gamma^{b}_{\nu,f}(k,p)\
,$ (53)

where the integral describes the infinite tower of the quark’s gluon dressing
and $\Lambda\gg\mu$ is a Poincaré-invariant regularization scale that can be
taken to infinity. In Eq. (53), $m_{f}(\mu)$ is the renormalized current-quark
mass which is related to the bare mass in the QCD Lagrangian via
$Z_{4}^{f}(\mu,\Lambda)\,m_{f}(\mu)=Z_{2}^{f}(\mu,\Lambda)\,m_{f}^{\rm
bm}(\Lambda)$, while $Z_{1}^{f}(\mu,\Lambda)$, $Z_{2}^{f}(\mu,\Lambda)$ and
$Z_{4}^{f}(\mu,\Lambda)$ are respectively flavor-dependent vertex, wave
function and mass renormalization constants. Moreover, $D_{\mu\nu}(q)$ is the
dressed-gluon propagator and
$\Gamma^{a}_{\mu,f}(k,p)=\frac{1}{2}\,\lambda^{a}\Gamma_{\mu,f}(k,p)$ is the
quark-gluon vertex, where $\lambda^{a}$ are the SU(3) color matrices and
$q=k-p$. The most general covariant form of the quark propagator is written in
terms of scalar and vector contributions with $Z_{f}(p^{2})=1/A_{f}(p^{2})$
and $M_{f}(p^{2})=B_{f}(p^{2})/A_{f}(p^{2})$:

$S_{f}(p)\ =\ \frac{1}{i\gamma\cdot p\,A_{f}(p^{2})+B_{f}(p^{2})}\ =\
\frac{Z_{f}(p^{2})}{i\gamma\cdot p+M_{f}(p^{2})}\ .$ (54)

Suppose we take a timid step beyond the abundantly employed rainbow-ladder
(RL) truncation of the DSE and BSE, namely
$\Gamma_{\mu,f}(k,p)\equiv\gamma_{\mu}$, motivated by the observation that the
Abelian WTI it implies for the quark-gluon vertex222 Instead of the correct
Slavnov-Taylor identity.,

$iq\cdot\gamma\ =\
ik\cdot\gamma\,A_{f}(k^{2})-\,ip\cdot\gamma\,A_{f}(p^{2})+\,B_{f}(k^{2})-\,B_{f}(p^{2})\
,$ (55)

is only valid when $B_{f}(k^{2})\simeq B_{f}(p^{2})$ and $A_{f}(k^{2})\simeq
A_{f}(p^{2})\simeq 1$ over a large domain, $k^{2},p^{2}>0$, of momentum
squared. This is only true in the infinitely-heavy quark limit, but does not
apply to the charm quark for which dressing effects are important [23].
Clearly, the vertex must exhibit some sort of flavor dependence. Let us
therefore slightly modify the RL kernel of the DSE:

$Z_{1}^{f}g^{2}D_{\mu\nu}(q)\,\Gamma_{\nu,f}(k,p)\ =\
\left(Z^{f}_{2}\right)^{\\!2}\mathcal{G}_{f}(q^{2})D_{\mu\nu}^{\mathrm{free}}(q)\frac{\lambda^{a}}{2}\gamma_{\nu}\
.$ (56)

In here, the free gluon propagator in Landau gauge,

$D_{\mu\nu}^{\mathrm{free}}(q)\ =\
\delta^{ab}\left(\delta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)\\!\frac{1}{q^{2}}\
,$ (57)

and bare vertex $\gamma_{\nu}$ are multiplied by an effective interaction
model $\mathcal{G}_{f}(q^{2})$ for the product of the gluon and vertex
dressing which is now _flavor dependent_ , unlike in common RL models [79,
82].

Whatever modifications we apply to the DSE kernel must have repercussions in
the BSE. To this end, the effect of the ansatz (56) that preserves the axWTI
(52) can be studied by inserting the DSE (53) as well as the axialvector and
pseudoscalar vertices given by,

$\displaystyle\Gamma_{5\mu}^{fg}(k;P)$ $\displaystyle=$ $\displaystyle
Z_{2}^{f}\gamma_{5}\gamma_{\mu}+\int^{\Lambda}\\!\\!\frac{d^{4}q}{(2\pi)^{4}}\,K_{fg}(q,k;P)\,S_{f}(q_{\eta})\Gamma_{5\mu}^{fg}(q;P)S_{g}(q_{\bar{\eta}})\,,$
(58) $\displaystyle\Gamma_{5}^{fg}(k;P)$ $\displaystyle=$ $\displaystyle
Z_{4}^{f}\gamma_{5}+\int^{\Lambda}\\!\\!\frac{d^{4}q}{(2\pi)^{4}}\,K_{fg}(q,k;P)\,S_{f}(q_{\eta})\Gamma_{5}^{fg}(q;P)S_{g}(q_{\bar{\eta}})\,,$
(59)

into the axWTI. (52) which leads to the relation ($l=k-q$),

$\displaystyle\int^{\Lambda}\\!\frac{d^{4}q}{(2\pi)^{4}}\,K_{fg}(q,k;P)\big{[}S_{f}(q_{\eta})\gamma_{5}+\gamma_{5}S_{g}(q_{\bar{\eta}})\big{]}=$
$\displaystyle-\ $
$\displaystyle\int^{\Lambda}\\!\frac{d^{4}q}{(2\pi)^{4}}\,\gamma_{\mu}\big{[}\Delta_{\mu\nu}^{f}(l)S_{f}(q_{\eta})\gamma_{5}+\gamma_{5}\Delta_{\mu\nu}^{g}(l)S_{g}(q_{\bar{\eta}})\big{]}\gamma_{\nu}\,,$
(60)

In Eqs. (58) and (59), $K_{fg}(q,k;P)$ is the fully-amputated quark-anti-quark
scattering kernel suppressing Dirac and color indices, $q_{\eta}=q+\eta P$ and
$q_{\bar{\eta}}=q-\bar{\eta}P$ are the quark and antiquark momenta and we have
defined:

$\Delta_{\mu\nu}^{f}(l)=\frac{4}{3}\,\big{(}Z_{2}^{f}\big{)}^{\\!2}\,\mathcal{G}_{f}(l^{2})\left(\delta_{\mu\nu}-\frac{l_{\mu}l_{\nu}}{l^{2}}\right)\frac{1}{l^{2}}\
.$ (61)

Comparing both sides of Eq. (4) one readily acknowledges that for
$\Delta_{\mu\nu}^{f}(l)=\Delta_{\mu\nu}^{g}(l)$ the identity Eq. (4) is
satisfied by the usual RL kernel,

$K(q,k;P)\ =\
-Z_{2}^{2}\,\mathcal{G}\left(l^{2}\right)D_{\mu\nu}^{\mathrm{free}}(l)\,\gamma_{\mu}\frac{\lambda^{a}}{2}\gamma_{\nu}\frac{\lambda^{a}}{2}\
.$ (62)

On the other hand, with a flavor-dependent interaction
$\mathcal{G}_{f}(l^{2})$ the kernel $K_{fg}(q,k;P)$ on the left-hand side of
Eq. (4) must somehow produce a flavored average of interactions.

Indeed, a consistent ansatz [91] for $K_{fg}(q,k;P)$ that satisfies Eq. (4)
behaves for large momenta $q^{2}$ as,

$K_{fg}\ \sim\
-\,\gamma_{\mu}\left(\frac{\Delta_{\mu\nu}^{f}+\Delta_{\mu\nu}^{g}}{2}\right)\gamma_{\nu}\
,$ (63)

whereas in the infrared limit this becomes,

$K_{fg}\ \sim\
-\,\gamma_{\mu}\left(\frac{\Delta_{\mu\nu}^{f}\sigma_{s}^{f}(0)+\Delta_{\mu\nu}^{g}\sigma_{s}^{g}(0)}{\sigma_{s}^{f}(0)+\sigma_{s}^{g}(0)}\right)\gamma_{\nu}\,.$
(64)

Both limits describe and average of interaction functions, in the latter case
weighted with flavored quark-dressing functions.

Given these considerations, we modified the RL truncation by introducing a
flavor dependence in the effective DSE and BSE vertices [19] which leads to a
different treatment of the light and heavy quarks. For the former, the vertex
dressing is of significant magnitude, while for the latter it amounts to
describing the quark-gluon interaction with all but a bare vertex; see Fig. 1
in Ref. [19] for a comparison of the interaction strength of light ($u,d,s$)
and heavy ($c,b$) quarks. The BSE kernel is thus written as,

$K_{fg}(k,q;P)\ =\ -\mathcal{Z}_{2}^{2}\
\frac{\mathcal{G}_{fg}(l^{2})}{l^{2}}\,\frac{\lambda^{a}}{2}\gamma_{\nu}\frac{\lambda^{a}}{2}\gamma_{\nu}\
,$ (65)

where the wave-function renormalization constants obtained for both quarks
with the DSE (53) is combined into a single one:
$\mathcal{Z}_{2}(\mu,\Lambda)=\sqrt{Z_{2}^{f}Z_{2}^{g}}$. For the averaged
interaction we employ the ansatz,

$\frac{\mathcal{G}_{fg}(l^{2})}{l^{2}}\ =\
\mathcal{G}_{fg}^{\mathrm{IR}}(l^{2})+4\pi\tilde{\alpha}_{\mathrm{PT}}(l^{2})\
,$ (66)

in which the low-momentum domain is described by the Gaussian, infrared-finite
support,

$\mathcal{G}_{fg}^{\mathrm{IR}}(l^{2})\ =\
\frac{8\pi^{2}}{(\omega_{f}\omega_{g})^{2}}\sqrt{D_{f}\,D_{g}}\
e^{-l^{2}/(\omega_{f}\omega_{g})}\ ,$ (67)

and the perturbative tail is given by the usual expression [76],

$4\pi\tilde{\alpha}_{\mathrm{PT}}(q^{2})\ =\
\frac{8\pi^{2}\gamma_{m}\mathcal{F}(q^{2})}{\ln\left[\tau+\left(1+q^{2}/\Lambda^{2}_{\textrm{\tiny
QCD}}\right)^{\\!2}\right]}\ ,$ (68)

with $\gamma_{m}=12/(33-2N_{f})$ being the anomalous dimension, $N_{f}$ is the
active flavor number, $\Lambda_{\textrm{\tiny QCD}}=0.234$ GeV,
$\tau=e^{2}-1$, $\mathcal{F}(q^{2})=[1-\exp(-q^{2}/4m^{2}_{t})]/q^{2}$ and
$m_{t}=0.5$ GeV.

With this ansatz in the homogeneous BSE for a meson $M=P,V$ becomes:

$\Gamma^{fg}_{M}(k,P)\\!=\\!\int^{\Lambda}\\!\\!\\!\frac{d^{4}q}{(2\pi)^{4}}\,K_{fg}(k,q;P)S_{f}(q_{\eta})\,\Gamma^{fg}_{M}(q,P)\,S_{g}(q_{\bar{\eta}})\
.$ (69)

The solutions of the BSA for a pseudoscalar meson with quantum numbers
$J^{PC}=0^{-+}$ can most generally be decomposed into four Lorentz covariants
made from the Dirac matrices $\gamma_{\mu}$, the relative momentum $k_{\mu}$
and the total momentum $P_{\mu}$. For the pseudoscalar mesons, we employ the
tensor structures that are not orthogonal with respect to the Dirac trace,

$\Gamma^{fg}_{P}(k,P)=\ \gamma_{5}\Big{[}iE^{fg}_{P}(k,P)+\gamma\cdot
P\,F^{fg}_{P}(k,P)\ +\ \gamma\cdot k\;k\cdot
P\,G^{fg}_{P}(k,P)+\sigma_{\mu\nu}k_{\mu}P_{\nu}\,H^{fg}_{P}(k,P)\Big{]}\ ,$
(70)

where $E^{fg}_{P}(k,P)$, $F^{fg}_{P}(k,P)$, $G^{fg}_{P}(k,P)$ and
$H^{fg}_{P}(k,P)$ are Lorentz-invariant amplitudes.

Likewise, the most general Poincaré-invariant form of the BSE solution for the
vector vertex $\Gamma_{V\mu}$ in the $J^{PC}=1^{--}$ vector channel is
decomposed into eight Lorentz covariants:

$\displaystyle\Gamma^{fg}_{V\mu}(k,P)\ =\
\sum^{8}_{\alpha=1}\,T^{\alpha}_{\mu}(k,P)\,\mathcal{F}_{\alpha}^{fg}(k,P)\ .$
(71)

In Eq. (71), $\mathcal{F}_{\alpha}(k,P)$ are Lorentz invariant amplitudes and
$T^{\alpha}_{\mu}(k,P)$ is the orthogonal basis with respect to the Dirac
trace:

$\displaystyle T^{1}_{\mu}(k,P)$ $\displaystyle=$ $\displaystyle
i\gamma^{T}_{\mu}\ ,$ (72) $\displaystyle T^{2}_{\mu}(k,P)$ $\displaystyle=$
$\displaystyle i[3k^{T}_{\mu}(\gamma\cdot k^{T})-\gamma^{T}_{\mu}(k^{T})^{2}]\
,$ (73) $\displaystyle T^{3}_{\mu}(k,P)$ $\displaystyle=$ $\displaystyle
i(k\cdot P)k^{T}_{\mu}\gamma\cdot P\ ,$ (74) $\displaystyle T^{4}_{\mu}(k,P)$
$\displaystyle=$ $\displaystyle i[\gamma^{T}_{\mu}\gamma\cdot P(\gamma\cdot
k^{T})+k^{T}_{\mu}\gamma\cdot P]\ ,$ (75) $\displaystyle T^{5}_{\mu}(k,P)$
$\displaystyle=$ $\displaystyle k^{T}_{\mu}\ ,$ (76) $\displaystyle
T^{6}_{\mu}(k,P)$ $\displaystyle=$ $\displaystyle(k\cdot
P)[\gamma^{T}_{\mu}(\gamma\cdot k^{T})-(\gamma\cdot k^{T})\gamma^{T}_{\mu}]\
,$ (77) $\displaystyle T^{7}_{\mu}(k,P)$ $\displaystyle=$
$\displaystyle\gamma^{T}_{\mu}\gamma\cdot P-\gamma\cdot
P\gamma^{T}_{\nu}-2T_{\mu}^{8}(k,P)\ ,$ (78) $\displaystyle T_{\mu}^{8}(k,P)$
$\displaystyle=$
$\displaystyle\hat{k}^{T}_{\mu}(\gamma\cdot\hat{k}^{T})\gamma\cdot P\ .$ (79)

The transverse projections are $V^{T}_{\mu}=V_{\mu}-P_{\mu}(P\cdot V)/P^{2}$
with $P\cdot V^{T}=0$ for any four-vector $V_{\mu}$ and
$\hat{k}^{T}\cdot\hat{k}^{T}=1$. Moreover, the BSA of both, pseudoscalar and
vector mesons, also depend on the angle $z_{k}=k\cdot P/|k||P|$ which is
commonly exploited to expand the eigenfunctions
$\mathcal{F}_{\alpha}^{fg}(k,z_{k},P)$ into Chebyshev polynomials. Such an
expansion allows for a faster convergence in numerical computations and for an
angular analysis of the BSA.

| $M_{P}$ | $M^{\mathrm{exp}}_{P}$ |  $\epsilon_{M_{P}}$ [%] | $f_{P}$ | $f^{\mathrm{exp/lQCD}}_{P}$ |  $\epsilon_{f_{P}}$ [%]
---|---|---|---|---|---|---
$\pi(u\bar{d})$ | 0.140 | 0.138 | 1.45 | $0.094^{+0.001}_{-0.001}$ | 0.092(1) | 2.17
$K(u\bar{s})$ | 0.494 | 0.494 | 0 | $0.110^{+0.001}_{-0.001}$ | 0.110(2) | 0
$D(c\bar{d})$ | $1.867^{+0.008}_{-0.004}$ | 1.864 | 0.11 | $0.144^{+0.001}_{-0.001}$ | 0.150 (0.5) | 4.00
$D_{s}(c\bar{s})$ | $2.015^{+0.021}_{-0.018}$ | 1.968 | 2.39 | $0.179^{+0.004}_{-0.003}$ | 0.177(0.4) | 1.13
$\eta_{c}(c\bar{c})$ | $3.012^{+0.003}_{-0.039}$ | 2.984 | 0.94 | $0.270^{+0.002}_{-0.005}$ | 0.279(17) | 3.23
$\eta_{b}(b\bar{b})$ | $9.392^{+0.005}_{-0.004}$ | 9.398 | 0.06 | $0.491^{+0.009}_{-0.009}$ | 0.472(4) | 4.03
$B(u\bar{b})$ | $5.277^{+0.008}_{-0.005}$ | 5.279 | 0.04 | $0.132^{+0.004}_{-0.002}$ | 0.134(1) | 4.35
$B_{s}(s\bar{b})$ | $5.383^{+0.037}_{-0.039}$ | 5.367 | 0.30 | $0.128^{+0.002}_{-0.003}$ | 0.162(1) | 20.5
$B_{c}(c\bar{b})$ | $6.282^{+0.020}_{-0.024}$ | 6.274 | 0.13 | $0.280^{+0.005}_{-0.002}$ | 0.302(2) | 10.17
Table 3: Masses and decay constants [in GeV] of pseudoscalar mesons. All
experimental masses and the pion and kaon weak decay constants are averaged
values by the Particle Data Group [93]. The leptonic decay constants of the
$D_{d}$, $D_{s}$, $B_{u}$ and $B_{s}$ mesons are FLAG 2019 averages [94] and
those of the $B_{c}$, $\eta_{c}$ and $\eta_{b}$ mesons are from Ref. [95]. The
relative deviations from experimental values, $v^{\textrm{exp.}}$, are given
by
$\epsilon_{v}=100\%\,|v^{\textrm{exp.}}-v^{\textrm{th.}}|/v^{\textrm{exp.}}$.

We normalize the BSA with the Nakanishi condition [92] which involves the
eigenvalue trajectory $\lambda(P^{2})$ of the BSE solutions :

$\displaystyle\left(\frac{\partial\ln(\lambda)}{\partial
P^{2}}\right)^{-1}\\!=\ \int\frac{d^{4}k}{(2\pi)^{4}}\
\operatorname{tr_{CD}}\left[\bar{\Gamma}^{fg}_{M}(k;-P)\,S_{f}(k_{\eta})\,\Gamma^{fg}_{M}(k;P)S_{g}(k_{\bar{\eta}})\right]\
.$ (80)

The normalization at the mass pole, $P^{2}=-M_{P}^{2}$, is required for the
BSA in the calculation of decay constants and other form factors or transition
amplitudes. The weak decay constant of a pseudoscalar meson is defined by,

$f_{P}P_{\mu}\ =\ \langle
0|\bar{q}_{g}\gamma_{5}\gamma_{\mu}q_{f}|P(P)\rangle\ ,$ (81)

and can be expressed by the integral:

$f_{P}P_{\mu}\ =\
\frac{\mathcal{Z}_{2}N_{c}}{\sqrt{2}}\int^{\Lambda}\\!\frac{d^{4}k}{(2\pi)^{4}}\,\operatorname{Tr_{D}}\left[i\gamma_{5}\gamma_{\mu}S_{f}(k_{\eta})\,\Gamma_{P}^{fg}(k,P)\,S_{g}(k_{\bar{\eta}})\right]\,.$
(82)

As already noted, the quark momenta, $k_{\eta}$ and $k_{\bar{\eta}}$, define
momentum-fraction parameters $\eta+\bar{\eta}=1$. Neither the decay constant
nor any other physical observables can depend on them owing to Poincaré
covariance. The leptonic decay constant of a vector meson is defined by the
amplitude,

$f_{V}M_{V}\bm{\epsilon}^{\lambda}_{\mu}\ =\ \langle
0|\bar{q}_{g}\gamma_{\mu}q_{f}|V(P,\lambda)\rangle\ ,$ (83)

where $\bm{\epsilon}^{\lambda}_{\mu}(P)$ is the polarization vector of the
transverse vector meson which satisfies $\bm{\epsilon}^{\lambda}\cdot P=0$ and
is normalized as
${\bm{\epsilon}^{\lambda}}^{*}\\!\cdot\bm{\epsilon}^{\lambda}=3$. This can
again be expressed by a loop integral:

$f_{V}M_{V}\ =\
\frac{\mathcal{Z}_{2}N_{c}}{3\sqrt{2}}\int^{\Lambda}\\!\frac{d^{4}k}{(2\pi)^{4}}\,\operatorname{Tr_{D}}\left[\gamma_{\mu}S_{f}(k_{\eta})\,\Gamma_{V\mu}^{fg}(k,P)\,S_{g}(k_{\bar{\eta}})\right]\
.$ (84)

| $M_{V}$ | $M^{\mathrm{exp}}_{V}$ |  $\epsilon_{M_{V}}$ [%] | $f_{V}$ | $f^{\mathrm{exp/lQCD}}_{V}$ |  $\epsilon_{f_{V}}$ [%]
---|---|---|---|---|---|---
$\rho(u\bar{u})$ | 0.730 | 0.775 | 5.81 | 0.145 | 0.153(1) | 5.23
$\phi(s\bar{s})$ | 1.070 | 1.019 | 5.20 | 0.187 | 0.168(1) | 11.31
$K^{*}(u\bar{s})$ | 0.942 | 0.896 | 5.13 | 0.177 | 0.159(1) | 11.32
$D^{*}(c\bar{d})$ | 2.021 | 2.009 | 0.60 | 0.165 | 0.158(6) | 4.43
$D^{*}_{s}(c\bar{s})$ | 2.169 | 2.112 | 2.70 | 0.205 | 0.190(5) | 7.90
$J/\psi(c\bar{c})$ | 3.124 | 3.097 | 0.87 | 0.277 | 0.294(5) | 5.78
$\Upsilon(b\bar{b})$ | 9.411 | 9.460 | 0.52 | 0.594 | 0.505(4) | 17.62
Table 4:  Masses and decay constants [in GeV] of ground-state vector mesons
(preliminary without error estimate). The experimental masses are values
listed by the Particle Data Group [93] and the leptonic decay constants for
the $\rho$, $K^{*}$, $\phi$, $J/\psi$ and $\Upsilon$ mesons are extracted from
their experimental decay width [93] via
$f_{V}^{2}=\frac{3m_{V}}{4\pi\alpha^{2}Q^{2}}\,\Gamma_{V\rightarrow
e^{+}e^{-}}$. The reference weak decay constants of the $D^{*}$ and
$D^{*}_{s}$ mesons are those of the ETM collaboration [96]. Relative
deviations as in Table 3.

Our results for the pseudoscalar and vector meson masses and leptonic decay
constants are tabulated in Tables 3 and 4. In the pseudoscalar channel the
experimental masses are reproduced within 1%, while our calculated decay
constants compare very well with experimental or theoretical reference values.
We only note some discrepancy in the case of $f_{B_{s}}$ and $f_{B_{c}}$. We
point out that the pion and kaon set the scale for the light- and strange-
quark masses, respectively, and serve to fix the parameter combinations,
$\omega_{u}D_{u}$ and $\omega_{s}D_{s}$. Therefore, no error estimate is give
for these mesons, whereas the theoretical errors for the remaining meson
masses are due to a certain insensitivity of the computed pion and kaon masses
with respect to $\omega_{u}$ and $\omega_{s}$. The origins of theoretical
errors and the choice of current-quark masses at the renormalization scale
$\mu$ in the DSE (53) are discussed in detail in Ref. [19]. The experimental
$\rho$, $K^{*}$ and $\phi$ masses are also reasonably well reproduced and we
observe very good agreement for the charmed mesons and heavy quarkonia.

Finally, let us place emphasis on the fact that our results for the heavy-
light mesons hinge on our choice of a distinct flavor dependence in the
interaction function (67). Indeed, the strong asymmetry in the momentum
distribution of the heavy and light quarks within these mesons requires a
distinct treatment of their interaction with a gluon. Moreover, this ansatz
also facilitates the calculation of the quark propagators in the complex
momentum plane in Euclidean space [19], where previous studies [79] were
plagued with cuts and singularities in the deep time-like domain.

##  5 Conclusive remarks

We have surveyed a selection of recent developments and outstanding issues
over the past decade that have touched upon hadronic aspects of flavor
physics. The common thread of the different topics we addressed, be it
effective Lagrangians and their effective couplings, flavor symmetry and its
breaking patterns or the properties of flavored mesons and quarkonia, is
nonperturbative QCD. Indeed, in computing couplings and form factors we
necessarily deal with flavored antiquark-quark bound states. The
nonperturbative aspects of bound states reveal themselves at different mass
scales, where they must be considered individually as well as in conjunction
with the other scales of a given physical problem.

We also explored two symmetries, none of which is apparent in the QCD
Lagrangian. Heavy-quark spin and flavor symmetries emerge only when the heavy
sector of QCD is expanded in terms of inverse powers of the heavy-quark mass.
We saw that beyond-leading orders can already significantly contribute to the
breaking of these symmetries. Indeed, nonperturbative calculations of
effective Lagrangian’s couplings in a symmetry-preserving truncation of the
DSE and BSE reveal that SU(3)F flavor symmetry is breached at the order of
20%. While this may be acceptable for calculations with effective hadron
degrees of freedom in the strange sector, the same cannot be said about SU(4)F
as similar calculations demonstrate an order of magnitude larger symmetry-
breaking effect. Along with the observation that the charm quark is not a
heavy enough quark, and therefore cannot be employed in a reliable expansion
of HQET, we conclude that effective Lagrangians based on charmed meson degrees
of freedom must be used judiciously. In particular, flavor-symmetry breaking
in the effective couplings should be accounted for.

At last, we discussed how reliable calculations of heavy-light form factors,
couplings and decay constants require the correct description of the light
quark’s propagator as solutions of the DSE or gap equation, while the meson’s
wave functions are obtained from Poincaré-invariant solutions of the
corresponding BSE. The couplings we discussed in here were mostly obtained
with simplified models of these wave function. Their calculation with the
heavy-light BSAs presented in Section 4 is in progress and results will be
available shortly.

## Acknowledgments

While the ongoing pandemic unfortunately didn’t allow for an on-site meeting
in Mexico City, we thank the organizers for the opportunity of a “virtual
presentation” and a well-organized on-line conference. This work is supported
by the Brazilian agencies FAPESP, grant no. 2018/20218-4, and CNPq, grant no.
428003/2018-4, and is part of the project “INCT-Física Nuclear e Aplicações”,
no. 464898/2014-5. F.E.S. is a CAPES-PNPD postdoctoral fellow, contract no.
88882.314890/2013-01.

## References

  * [1] R. Aaij et al. [LHCb], Phys. Rev. Lett. 110 (2013) no.22, 221601 doi:10.1103/PhysRevLett.110.221601
  * [2] R. Aaij et al. [LHCb], Phys. Rev. Lett. 122 (2019) no.21, 211803 doi:10.1103/PhysRevLett.122.211803
  * [3] B. El-Bennich, M. A. Ivanov and C. D. Roberts, Nucl. Phys. B Proc. Suppl. 199, 184-190 (2010) doi:10.1016/j.nuclphysbps.2010.02.026
  * [4] B. El-Bennich, A. Furman, R. Kamiński, L. Leśniak and B. Loiseau, Phys. Rev. D 74 (2006), 114009 doi:10.1103/PhysRevD.74.114009
  * [5] B. El-Bennich, A. Furman, R. Kamiński, L. Leśniak, B. Loiseau and B. Moussallam, Phys. Rev. D 79 (2009), 094005 [erratum: Phys. Rev. D 83 (2011), 039903] doi:10.1103/PhysRevD.83.039903
  * [6] T. E. O. Ericson, B. Loiseau and A. W. Thomas, Phys. Rev. C 66 (2002), 014005 doi:10.1103/PhysRevC.66.014005
  * [7] S. K. Choi et al. [Belle], Phys. Rev. Lett. 91 (2003), 262001 doi:10.1103/PhysRevLett.91.262001
  * [8] B. Aubert et al. [BaBar], Phys. Rev. D 71 (2005), 071103 doi:10.1103/PhysRevD.71.071103
  * [9] D. Acosta et al. [CDF], Phys. Rev. Lett. 93 (2004), 072001 doi:10.1103/PhysRevLett.93.072001
  * [10] V. M. Abazov et al. [D0], Phys. Rev. Lett. 93 (2004), 162002 doi:10.1103/PhysRevLett.93.162002
  * [11] X. Liu, B. Zhang and S. L. Zhu, Phys. Lett. B 645 (2007), 185-188 doi:10.1016/j.physletb.2006.12.031
  * [12] A. Martínez Torres, K. P. Khemchandani, F. S. Navarra, M. Nielsen and L. M. Abreu, Phys. Rev. D 90, no.11, 114023 (2014) [erratum: Phys. Rev. D 93, no.5, 059902 (2016)] doi:10.1103/PhysRevD.90.114023
  * [13] J. Haidenbauer and G. Krein, Phys. Rev. D 89 (2014) no.11, 114003 doi:10.1103/PhysRevD.89.114003
  * [14] R. Aaij et al. [LHCb], Phys. Rev. Lett. 115 (2015), 072001 doi:10.1103/PhysRevLett.115.072001
  * [15] D. Kharzeev, Proc. Int. Sch. Phys. Fermi 130 (1996), 105-131 doi:10.3254/978-1-61499-215-8-105
  * [16] Z. E. Meziani and S. Joosten, doi:10.1142/9789811214950_0048
  * [17] G. Krein and T. C. Peixoto, Few Body Syst. 61 (2020) no.4, 49 doi:10.1007/s00601-020-01581-1
  * [18] Y. Z. Xu, S. Chen, Z. Q. Yao, D. Binosi, Z. F. Cui and C. D. Roberts, [arXiv:2107.03488 [hep-ph]].
  * [19] F. E. Serna, R. C. da Silveira, J. J. Cobos-Martínez, B. El-Bennich and E. Rojas, Eur. Phys. J. C 80 (2020) no.10, 955 doi:10.1140/epjc/s10052-020-08517-3
  * [20] F. E. Serna and B. El-Bennich, PoS CHARM2020 (2021), 047 doi:10.22323/1.385.0047
  * [21] M. Neubert, Phys. Rept. 245, 259-396 (1994) doi:10.1016/0370-1573(94)90091-4
  * [22] A. V. Manohar and M. B. Wise, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 10, 1-191 (2000)
  * [23] B. El-Bennich, C. D. Roberts and M. A. Ivanov, PoS QCD-TNT-II, 018 (2012) doi:10.22323/1.136.0018
  * [24] G. Burdman and J. F. Donoghue, Phys. Lett. B 280 (1992), 287-291 doi:10.1016/0370-2693(92)90068-F
  * [25] M. B. Wise, Phys. Rev. D 45 (1992) no.7, R2188 doi:10.1103/PhysRevD.45.R2188
  * [26] R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Rept. 281, 145-238 (1997) doi:10.1016/S0370-1573(96)00027-0
  * [27] A. Anastassov et al. [CLEO], Phys. Rev. D 65, 032003 (2002) doi:10.1103/PhysRevD.65.032003
  * [28] J. P. Lees et al. [BaBar], Phys. Rev. D 88 (2013) no.5, 052003 [erratum: Phys. Rev. D 88 (2013) no.7, 079902 ] doi:10.1103/PhysRevD.88.052003
  * [29] S. G. Matinyan and B. Müller, Phys. Rev. C 58, 2994-2997 (1998) doi:10.1103/PhysRevC.58.2994
  * [30] Z. w. Lin and C. M. Ko, Phys. Rev. C 62, 034903 (2000) doi:10.1103/PhysRevC.62.034903
  * [31] Y. s. Oh, T. Song and S. H. Lee, Phys. Rev. C 63, 034901 (2001) doi:10.1103/PhysRevC.63.034901
  * [32] M. E. Bracco, M. Chiapparini, F. S. Navarra and M. Nielsen, Prog. Part. Nucl. Phys. 67, 1019-1052 (2012) doi:10.1016/j.ppnp.2012.03.002
  * [33] F. L. Braghin, [arXiv:2109.02203 [hep-ph]].
  * [34] G. N. Zeminiani, J. J. Cobos-Martínez and K. Tsushima, Eur. Phys. J. A 57, no.8, 259 (2021) doi:10.1140/epja/s10050-021-00569-7
  * [35] A. Sibirtsev, K. Tsushima and A. W. Thomas, Eur. Phys. J. A 6, 351-359 (1999) doi:10.1007/s100500050353
  * [36] T. Mizutani and A. Ramos, Phys. Rev. C 74, 065201 (2006) doi:10.1103/PhysRevC.74.065201
  * [37] J. Haidenbauer, G. Krein, U. G. Meissner and A. Sibirtsev, Eur. Phys. J. A 33, 107-117 (2007) doi:10.1140/epja/i2007-10417-3
  * [38] R. Molina, D. Gamermann, E. Oset and L. Tolos, Eur. Phys. J. A 42, 31-42 (2009) doi:10.1140/epja/i2009-10853-y
  * [39] J. Haidenbauer, G. Krein, U. G. Meissner and L. Tolos, Eur. Phys. J. A 47, 18 (2011) doi:10.1140/epja/i2011-11018-3
  * [40] Y. Yamaguchi, S. Ohkoda, S. Yasui and A. Hosaka, Phys. Rev. D 84, 014032 (2011) doi:10.1103/PhysRevD.84.014032
  * [41] B. B. Malabarba, K. P. Khemchandani and A. M. Torres, [arXiv:2107.03496 [hep-ph]].
  * [42] L. M. Abreu, K. P. Khemchandani, A. Martínez Torres, F. S. Navarra and M. Nielsen, Phys. Rev. C 97, no.4, 044902 (2018) doi:10.1103/PhysRevC.97.044902
  * [43] J. Haidenbauer and G. Krein, Phys. Rev. D 91, no.11, 114022 (2015) doi:10.1103/PhysRevD.91.114022
  * [44] G. Montaña, À. Ramos, L. Tolos and J. M. Torres-Rincon, Phys. Rev. D 102, no.9, 096020 (2020) doi:10.1103/PhysRevD.102.096020
  * [45] V. M. Belyaev, V. M. Braun, A. Khodjamirian and R. Ruckl, Phys. Rev. D 51, 6177-6195 (1995) doi:10.1103/PhysRevD.51.6177
  * [46] B. El-Bennich, M. A. Ivanov and C. D. Roberts, Phys. Rev. C 83, 025205 (2011) doi:10.1103/PhysRevC.83.025205
  * [47] B. El-Bennich, G. Krein, L. Chang, C. D. Roberts and D. J. Wilson, Phys. Rev. D 85, 031502 (2012) doi:10.1103/PhysRevD.85.031502
  * [48] B. El-Bennich, M. A. Paracha, C. D. Roberts and E. Rojas, Phys. Rev. D 95, no.3, 034037 (2017) doi:10.1103/PhysRevD.95.034037
  * [49] M. E. Bracco, M. Chiapparini, F. S. Navarra and M. Nielsen, Phys. Lett. B 659, 559-564 (2008) doi:10.1016/j.physletb.2007.11.066
  * [50] M. E. Bracco, M. Chiapparini, A. Lozea, F. S. Navarra and M. Nielsen, Phys. Lett. B 521, 1-6 (2001) doi:10.1016/S0370-2693(01)01192-3
  * [51] B. Osório Rodrigues, M. E. Bracco, M. Nielsen and F. S. Navarra, Nucl. Phys. A 852, 127-140 (2011) doi:10.1016/j.nuclphysa.2011.01.001
  * [52] B. Osório Rodrigues, M. E. Bracco and M. Chiapparini, Nucl. Phys. A 929, 143-156 (2014) doi:10.1016/j.nuclphysa.2014.06.008
  * [53] A. Cerqueira, B. Osório Rodrigues, M. E. Bracco and M. Nielsen, Nucl. Phys. A 936, 45-58 (2015) doi:10.1016/j.nuclphysa.2015.01.007
  * [54] Z. G. Wang, Nucl. Phys. A 796 (2007), 61-82 doi:10.1016/j.nuclphysa.2007.09.004
  * [55] Z. G. Wang, Eur. Phys. J. C 52 (2007), 553-560 doi:10.1140/epjc/s10052-007-0404-6
  * [56] C. Y. Cui, Y. L. Liu and M. Q. Huang, Phys. Lett. B 711, 317-326 (2012) doi:10.1016/j.physletb.2012.04.015
  * [57] K. U. Can, G. Erkol, M. Oka, A. Ozpineci and T. T. Takahashi, Phys. Lett. B 719, 103-109 (2013) doi:10.1016/j.physletb.2012.12.050
  * [58] R. Khosravi and M. Janbazi, Phys. Rev. D 87, no.1, 016003 (2013) doi:10.1103/PhysRevD.87.016003
  * [59] R. Khosravi and M. Janbazi, Phys. Rev. D 89, no.1, 016001 (2014) doi:10.1103/PhysRevD.89.016001
  * [60] M. Janbazi and R. Khosravi, Eur. Phys. J. C 78, no.7, 606 (2018) doi:10.1140/epjc/s10052-018-6069-5
  * [61] M. R. Seyedhabashi, E. Kazemi, M. Janbazi and N. Ghahramany, Nucl. Phys. A 1000, 121846 (2020) doi:10.1016/j.nuclphysa.2020.121846
  * [62] C. E. Fontoura, J. Haidenbauer and G. Krein, Eur. Phys. J. A 53, no.5, 92 (2017) doi:10.1140/epja/i2017-12289-2
  * [63] A. Ballon-Bayona, G. Krein and C. Miller, Phys. Rev. D 96, no.1, 014017 (2017) doi:10.1103/PhysRevD.96.014017
  * [64] T. M. Aliev and K. Şimşek, [arXiv:2107.02735 [hep-ph]].
  * [65] G. M. de Divitiis et al. [UKQCD], JHEP 10, 010 (1998)
  * [66] H. Ohki, H. Matsufuru and T. Onogi, Phys. Rev. D 77, 094509 (2008) doi:10.1103/PhysRevD.77.094509
  * [67] D. Becirevic, B. Blossier, E. Chang and B. Haas, Phys. Lett. B 679, 231-236 (2009) doi:10.1016/j.physletb.2009.07.031
  * [68] D. Becirevic and B. Haas, Eur. Phys. J. C 71, 1734 (2011) doi:10.1140/epjc/s10052-011-1734-y
  * [69] D. Becirevic and F. Sanfilippo, Phys. Lett. B 721, 94-100 (2013) doi:10.1016/j.physletb.2013.03.004
  * [70] J. M. Flynn et al. [RBC and UKQCD], Phys. Rev. D 93, no.1, 014510 (2016) doi:10.1103/PhysRevD.93.014510
  * [71] F. Bernardoni et al. [ALPHA], Phys. Lett. B 740 (2015), 278-284 doi:10.1016/j.physletb.2014.11.051
  * [72] W. Detmold, C. J. D. Lin and S. Meinel, Phys. Rev. D 85 (2012), 114508 doi:10.1103/PhysRevD.85.114508
  * [73] F. L. Braghin and W. F. De Sousa, J. Phys. G 47 (2020) no.4, 045110 doi:10.1088/1361-6471/ab694a
  * [74] F. L. Braghin, Phys. Rev. D 103 (2021) no.9, 094028 doi:10.1103/PhysRevD.103.094028
  * [75] F. L. Braghin, [arXiv:2108.02748 [hep-ph]].
  * [76] A. Bashir, L. Chang, I. C. Cloët, B. El-Bennich, Y. X. Liu, C. D. Roberts and P. C. Tandy, Commun. Theor. Phys. 58, 79-134 (2012) doi:10.1088/0253-6102/58/1/16
  * [77] L. Chang, I. C. Cloët, B. El-Bennich, T. Klähn and C. D. Roberts, Chin. Phys. C 33, 1189-1196 (2009) doi:10.1088/1674-1137/33/12/022
  * [78] B. El-Bennich, J. P. B. C. de Melo, B. Loiseau, J. P. Dedonder and T. Frederico, Braz. J. Phys. 38 (2008), 465-471 doi:10.1590/S0103-97332008000400016
  * [79] E. Rojas, B. El-Bennich and J. P. B. C. de Melo, Phys. Rev. D 90 (2014), 074025 doi:10.1103/PhysRevD.90.074025
  * [80] E. Rojas, B. El-Bennich, J. P. B. C. De Melo and M. A. Paracha, Few Body Syst. 56 (2015) no.6-9, 639-644 doi:10.1007/s00601-015-1020-x
  * [81] B. El-Bennich, G. Krein, E. Rojas and F. E. Serna, Few Body Syst. 57 (2016) no.10, 955-963 doi:10.1007/s00601-016-1133-x
  * [82] F. F. Mojica, C. E. Vera, E. Rojas and B. El-Bennich, Phys. Rev. D 96 (2017) no.1, 014012 doi:10.1103/PhysRevD.96.014012
  * [83] F. E. Serna, B. El-Bennich and G. Krein, Phys. Rev. D 96 (2017) no.1, 014013 doi:10.1103/PhysRevD.96.014013
  * [84] F. E. Serna, C. Chen and B. El-Bennich, Phys. Rev. D 99 (2019) no.9, 094027 doi:10.1103/PhysRevD.99.094027
  * [85] L. Albino, A. Bashir, L. X. G. Guerrero, B. E. Bennich and E. Rojas, Phys. Rev. D 100 (2019) no.5, 054028 doi:10.1103/PhysRevD.100.054028
  * [86] L. Albino, A. Bashir, B. El-Bennich, E. Rojas, F. E. Serna and R. C. da Silveira, [arXiv:2108.06204 [nucl-th]].
  * [87] R. C. da Silveira, F. E. Serna and B. El-Bennich, in preparation.
  * [88] F. T. Hawes and M. A. Pichowsky, Phys. Rev. C 59 (1999), 1743-1750 doi:10.1103/PhysRevC.59.1743
  * [89] M. S. Bhagwat and P. Maris, Phys. Rev. C 77 (2008), 025203 doi:10.1103/PhysRevC.77.025203
  * [90] A. Höll, C. D. Roberts and S. V. Wright, [arXiv:nucl-th/0601071 [nucl-th]].
  * [91] P. Qin, S. x. Qin and Y. x. Liu, Phys. Rev. D 101 (2020) no.11, 114014 doi:10.1103/PhysRevD.101.114014
  * [92] N. Nakanishi, Phys. Rev. 138 (1965), B1182-B1192 doi:10.1103/PhysRev.138.B1182
  * [93] P. A. Zyla et al. [Particle Data Group], PTEP 2020 (2020) no.8, 083C01 doi:10.1093/ptep/ptaa104
  * [94] S. Aoki et al. [Flavour Lattice Averaging Group], Eur. Phys. J. C 80 (2020) no.2, 113 doi:10.1140/epjc/s10052-019-7354-7
  * [95] C. McNeile, C. T. H. Davies, E. Follana, K. Hornbostel and G. P. Lepage, Phys. Rev. D 86 (2012), 074503 doi:10.1103/PhysRevD.86.074503
  * [96] V. Lubicz et al. [ETM], Phys. Rev. D 96 (2017) no.3, 034524 doi:10.1103/PhysRevD.96.034524