11institutetext: Departemen Fisika, FMIPA, Universitas Indonesia, Depok 16424,
Indonesia 22institutetext: Few-Body System in Physics Laboratory, RIKEN
Nishina Center, Wako 351-0198, Japan 33institutetext: Research Center for
Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan

# The Properties of Radially Excited Charmonia
in the Light Front Quark Model

Muhammad Ridwan 11 Ahmad Jafar Arifi Corresponding author:
<EMAIL_ADDRESS>Terry Mart 11

###### Abstract

Investigating the properties of excited charmonia is important to clarify its
internal structure. In this paper, we present the mass spectra (MS) and decay
constants (DC) for charmonia up to 3$S$ states calculated by means of the
light-front quark model based on a variational approach. In particular, we
consider the QCD-motivated effective Hamiltonian, which includes both
confinement (linear and screened) and Coulomb-like potentials. Furthermore,
since the existence of the nature of heavy quark symmetry, we treat hyperfine
interactions perturbatively. We developed the harmonic oscillator expansion
method to approximate the wave function (WF) for excited states. We found that
the results of our theoretical calculations, using screened potentials rather
than linear ones, are in good agreement with experimental data. By looking at
the mass and decay constant result, we found that our result on the $\psi$(3S)
state matched the properties of the $\psi$(4040) resonance.

Keywords: Charmonia, LFQM, Harmonic oscillator expansion method, Mass spectra,
Decay cosntant

##  1 Introduction

Since 50 years ago, Quantum Chromodynamics (QCD) Gross:2022hyw  has been
understood as a fundamental theory of strong interaction between quarks and
gluon with a non-perturbative nature in low-energy region and asymptotic
freedom in high-energy limit. With the emergence of a connection between QCD
and constituent quark model (CQM), QCD becomes the most useful tool for
describing hadrons, including baryons ($qqq$) and mesons ($q\bar{q}$).
However, quark models cannot provide a detailed explanation of exotic hadrons,
such as the $X$, $Y$, and $Z$ states observed in experiments Hosaka:2016pey .
Additionally, since the finding of the heavy charmonium state $J/\psi$ in 1974
Khare:1999zz , experimentalists have also identified excited states of
charmonium, particularly within the $\psi$ family ParticleDataGroup:2022pth ,
including $\psi(4040)$, $\psi(4160)$, and $\psi(4415)$. While numerous unknown
hadrons have been confirmed in experiments, theorists are still working to
develop suitable models that can explain the properties of hadrons, including
some basic properties, namely mass spectra (MS) and decay constants (DC).

Since Dirac proposed the three relativistic dynamics in 1949 Dirac:1949cp ,
light-front dynamics (LFD) has emerged as an effective tool for describing
hadron structure. In addition to employing a relativistic framework, LFD also
offers distinctive features, including a rational energy-momentum relation and
seven time-independent kinematic quantities. Hence, calculating the properties
of all hadrons becomes easier. In particular, the light-front quark model
(LFQM) offers a distinctive way of constructing hadron models by treating
hadrons as relativistic bound states, making the model provide reliable
predictions for hadron properties.

There have been several efforts to determine the properties of mesons,
including the calculation of MS and DC of the ground state
Choi:2015variational , as well as investigation of the basis expansion effects
of $\Phi(k)=\sum_{n=1}^{2}\phi^{HO}_{nS}(k)$ in the calculation up to the $2S$
state in LFQM Arifi:2022pal . The present work is mainly intended to calculate
the properties of the $c\bar{c}$ charmonia, including MS and DC up to the $3S$
states. In particular, we investigate the effect of the inclusion of
additional 3$S$ harmonic oscillator (HO) basis
$\Phi(k)=\sum_{n=1}^{3}\phi^{\rm HO}_{nS}(k)$ in the radial part of the WF.

##  2 Model Description

In LFQM, our formalism utilizes the hadron state approximation through the
leading-order Fock state expansion. The meson structure is considered at rest
as an interacting bound system of effectively dressed quarks $q\bar{q}$, which
satisfies the eigenvalue equation of the effective Hamiltonian

$H_{q\bar{q}}\ket{\Psi_{q\bar{q}}}=M_{q\bar{q}}\ket{\Psi_{q\bar{q}}}$ (1)

where $M_{q\bar{q}}$ and $\Psi_{q\bar{q}}$ are the mass eigenvalue and
eigenfunction, respectively. The Hamiltonian is given by

$H_{q\bar{q}}=H_{0}+V_{q\bar{q}}=\sqrt{m_{q}^{2}+\boldsymbol{k}^{2}}+\sqrt{m_{\bar{q}}^{2}+\boldsymbol{k}^{2}}+V_{q\bar{q}},$
(2)

where $H_{0}$ is the kinetic energy part of $q\bar{q}$ with three-momentum
$\boldsymbol{k}$. The effective potential $V_{q\bar{q}}$ is given by

$V_{q\bar{q}}=V_{\rm Conf}+V_{\rm Coul}+V_{\rm Hyp},$ (3)

where we use the linear confinement

$V_{\rm Conf}=a+br,$ (4)

where $a$ and $b$ are confinement parameters. The screened effect is given by

$V_{\rm Conf}=a+\frac{b(1-{\rm e}^{-\mu r})}{\mu}.$ (5)

where $\mu$ is the screened parameter. The effective one-gluon exchanges for
the $S$-wave mesons are assumed for the Coulomb and hyperfine interaction
potentials, i.e.,

$V_{\rm Coul}=-\frac{4\alpha_{s}}{3r},\hskip 14.22636pt{\rm and}\hskip
14.22636ptV_{\rm
Hyp}=\frac{2\expectationvalue{\boldsymbol{S}_{q}\cdot\boldsymbol{S}_{\bar{q}}}}{3m_{q}m_{\bar{q}}}\nabla^{2}V_{\text{Coul}}.$
(6)

The coupling constant $\alpha_{s}$ in this work is assumed to be a free
parameter, whereas the values of
$\expectationvalue{\boldsymbol{S}_{q}\cdot\boldsymbol{S}_{\bar{q}}}$ are $1/4$
($-3/4$) for the vector (pseudoscalar) mesons, respectively. In addition, we
consider the $\nabla^{2}V_{\text{Coul.}}$ as the Dirac delta function
$\delta^{3}(r)$.

The light-front wave function can be expressed as

$\Psi^{M}_{q\bar{q}}=\Psi^{JJ_{z}}_{\lambda_{q},\lambda_{\bar{q}}}(x,\boldsymbol{k}_{\perp})=\Phi(x,\boldsymbol{k}_{\perp})\mathcal{R}^{JJ_{z}}_{\lambda_{q},\lambda_{\bar{q}}}(x,\boldsymbol{k}_{\perp}),$
(7)

where $\Phi(x,\boldsymbol{k}_{\perp})$ represents the radial wave function,
and
$\mathcal{R}^{JJ_{z}}_{\lambda_{q},\lambda_{\bar{q}}}(x,\boldsymbol{k}_{\perp})$
is the spin-orbit wave function derived from the Melosh transformation. The
covariant forms of $\mathcal{R}^{JJ_{z}}_{\lambda_{q}\lambda_{\bar{q}}}$ are
given by

$\mathcal{R}^{JJ_{z}}_{\lambda_{q}\lambda_{\bar{q}}}=\frac{1}{\sqrt{2}\tilde{M}_{0}}\bar{u}_{\lambda_{q}}(p_{q})\Gamma_{P(V)}v_{\lambda_{\bar{q}}}(p_{\bar{q}}),$
(8)

where $\Gamma_{P(V)}$ is the vertex for pseudoscalar(vector) meson,

$\Gamma_{P}=\gamma_{5},\hskip 14.22636pt{\rm and}\hskip
14.22636pt\Gamma_{V}=-\not{\epsilon}(J_{z})+\frac{\epsilon\cdot(p_{q}-p_{\bar{q}})}{M_{0}+m_{q}+m_{\bar{q}}},$
(9)

with $\tilde{M}_{0}\equiv\sqrt{M_{0}^{2}-(m_{q}-m_{\bar{q}})^{2}}$. The
polarization vectors
$\epsilon^{\mu}(J_{z})=(\epsilon^{+},\epsilon^{-},\boldsymbol{\epsilon}_{\perp})$
are given by

$\epsilon^{\mu}(\pm
1)=\left(0,\frac{2}{P^{+}}\bm{\epsilon}_{\perp}(\pm)\cdot\boldsymbol{P}_{\perp},\boldsymbol{\epsilon}_{\perp}(\pm)\right),\hskip
14.22636pt{\rm and}\hskip
14.22636pt\epsilon^{\mu}(0)=\frac{1}{M_{0}}\left(P^{+},\frac{-M^{2}_{0}+\boldsymbol{P}^{2}_{\perp}}{P^{+}},\boldsymbol{P}_{\perp}\right),$
(10)

where $\boldsymbol{\epsilon}_{\perp}(\pm 1)=\mp\left(1,\pm i\right)/\sqrt{2}$.
Since we only consider the radial part, the spin-orbit part satisfies the
unitary condition, i.e.,
$\Braket{\mathcal{R}^{JJ_{z}}_{\lambda_{q}\lambda_{\bar{q}}}}{\mathcal{R}^{JJ_{z}}_{\lambda_{q}\lambda_{\bar{q}}}}=1$.
The light-front wave function is written in terms of the invariant Lorentz
variable $x_{i}=p^{+}_{i}/P^{+}$, $\boldsymbol{k}_{\perp
i}=\boldsymbol{p}_{\perp i}-x_{i}\boldsymbol{p}_{\perp}$, helicity
$\lambda_{i}$, $P^{\mu}=(P^{+},P^{-},\boldsymbol{p}_{\perp})$ is the meson’s
momentum and $p^{\mu}$ is the $i$-th ($i$=1,2) constituent quark’s momentum,
and define $x\equiv x_{1}$ with
$\boldsymbol{k}_{\perp}\equiv\boldsymbol{k}_{\perp 1}$. Furthermore, the
three-momentum $\boldsymbol{k}=(k_{z},\boldsymbol{k}_{\perp})$ can be
described as $\boldsymbol{k}=(x,\boldsymbol{k}_{\perp})$ through the
relationship,

$k_{z}=\left(x-\frac{1}{2}\right)M_{0}+\frac{m^{2}_{\bar{q}}-m^{2}_{q}}{2M_{0}},$
(11)

where

$M_{0}^{2}=\frac{\boldsymbol{k}_{\bot}^{2}+m_{q}^{2}}{x}+\frac{\boldsymbol{k}_{\bot}^{2}+m_{\bar{q}}^{2}}{1-x}$
(12)

the boost-invariant meson mass squared is introduced in Eq. (12). Note that
the normalization is taken into account on our radial part following the
variable transformation
$\\{k_{z},\boldsymbol{k}_{\bot}\\}\to\\{x,\boldsymbol{k}_{\bot}\\}$ emerges
and influence the Jacobian factor $\frac{\partial k_{z}}{\partial x}$

$\frac{\partial k_{z}}{\partial
x}=\frac{M_{0}}{4x(1-x)}\left[1-\frac{(m_{q}^{2}-m_{\bar{q}}^{2})^{2}}{M_{0}^{4}}\right],$
(13)

For the ground state, we use the harmonic oscillator expansion method (HOEM)
in our radial wave function as

$\displaystyle\phi_{1S}^{H0}(\boldsymbol{k})$ $\displaystyle=$
$\displaystyle\frac{1}{\pi^{3/4}\beta^{3/2}}e^{-\boldsymbol{k}^{2}/2\beta^{2}},$
(14) $\displaystyle\phi_{2S}^{H0}(\boldsymbol{k})$ $\displaystyle=$
$\displaystyle\frac{(2k^{2}-3\beta^{2})}{\sqrt{6}\pi^{3/4}\beta^{7/2}}e^{-\boldsymbol{k}^{2}/2\beta^{2}},$
(15) $\displaystyle\phi_{3S}^{H0}(\boldsymbol{k})$ $\displaystyle=$
$\displaystyle\frac{(15\beta^{4}-20\beta^{2}k^{2}+4k^{4})}{2\sqrt{30}\pi^{3/4}\beta^{11/2}}e^{-\boldsymbol{k}^{2}/2\beta^{2}},$
(16)

with $k=|\boldsymbol{k}|$ and

$\phi_{nS}^{H0}(x,\boldsymbol{k}_{\bot})=\sqrt{2(2\pi)^{3}}\sqrt{\frac{\partial
k_{z}}{\partial x}}\phi_{nS}^{H0}(\boldsymbol{k}),$ (17)

whereas $\beta$ is the parameter that is inversely proportional to the width
of the WF. In this calculation, $\beta$ is considered as the variational
parameter. The WF $\phi_{nS}$ include with the Jacobian factor $\partial
k_{z}/\partial x$ satisfy the normalization:

$\int\frac{\differential
x\differential^{2}\boldsymbol{k}_{\bot}}{2(2\pi)^{3}}|\phi_{nS}^{H0}(x,\bm{k}_{\bot})|^{2}=1.$
(18)

Furthermore, we expand the basis as a linear combination of the $1S$, $2S$,
and $3S$ HO bases. It can be written in a matrix form

$\displaystyle\begin{pmatrix}\Phi_{1S}\\\ \Phi_{2S}\\\ \Phi_{3S}\end{pmatrix}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}c_{12}c_{13}&s_{12}s_{13}&s_{13}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}&c_{12}c_{13}-s_{12}s_{23}s_{13}&s_{23}c_{13}\\\
s_{12}c_{23}-c_{12}s_{23}s_{13}&-c_{12}c_{23}-s_{12}c_{23}s_{13}&c_{23}c_{13}\end{pmatrix}\begin{pmatrix}\phi_{1S}^{\rm
HO}\\\ \phi_{2S}^{\rm HO}\\\ \phi_{3S}^{\rm HO}\end{pmatrix}$ (19)
$\displaystyle=$
$\displaystyle\begin{pmatrix}c_{1}^{1S}&c_{2}^{1S}&c_{3}^{1S}\\\
c_{1}^{2S}&c_{2}^{2S}&c_{3}^{2S}\\\
c_{1}^{3S}&c_{2}^{3S}&c_{3}^{3S}\end{pmatrix}\begin{pmatrix}\phi_{1S}^{\rm
HO}\\\ \phi_{2S}^{\rm HO}\\\ \phi_{3S}^{\rm HO}\end{pmatrix}\,,$

with $c_{ij}(s_{ij})=\cos\theta_{ij}(\sin\theta_{ij})$. Note that the matrix
form in $R_{M}$ we used is quite similar to the CKM matrix Chau:1984fp , but
we ignore the CP violation terms. In our LFQM analysis, the heavy mesons (in
this case, charmonia) have a number of parameters, i.e., the constituent quark
masses ($m_{c}$), the potential parameters $(a,b,\alpha_{s},\mu)$, the HO
parameter $\beta_{cc}$ for both pseudoscalar and vector charmonia. We begin
with determining the parameters’ values by fitting the MS based on the
variational calculation

$\frac{\partial\bra{\Psi_{q\bar{q}}}H_{0}\ket{\Psi_{q\bar{q}}}}{\partial\beta}=0,$
(20)

and then computing the other meson properties such as decay constant. We note
that the hyperfine interaction is treated perturbatively.

###  2.1 Mass Spectra

The mass formula of 1S, 2S, and 3S state charmonia with three HO bases is
expressed as

$M_{q\bar{q}}=\expectationvalue{H_{0}}+\expectationvalue{V_{\rm
Conf}}+\expectationvalue{V_{\rm Coul}}+\expectationvalue{V_{\rm Hyp}},$ (21)

where the formula is derived by taking the expectation values of the effective
Hamiltonian

$\bra{\Psi_{q\bar{q}}}(H_{0}+V_{q\bar{q}})\ket{\Psi_{q\bar{q}}}.$ (22)

The contribution from the kinetic term is given by

$\displaystyle\expectationvalue{H_{0}}$ $\displaystyle=$
$\displaystyle\frac{\beta}{120\sqrt{\pi}}\sum_{j=1,2}\biggl{[}(120c_{1}^{2}-120\sqrt{6}c_{1}c_{2}+180c_{2}^{2}+60\sqrt{30}c_{1}c_{3}-180\sqrt{5}c_{2}c_{3}$
$\displaystyle+225c_{3}^{2}+40c_{2}^{2}+8\sqrt{30}c_{1}c_{3}-104\sqrt{5}c_{2}c_{3}+260c_{3}^{2}z_{i}^{2})z_{i}{\rm
e}^{z_{i}/2}K_{1}\left[\frac{z_{i}}{2}\right]$
$\displaystyle-4(10c_{2}^{2}-26\sqrt{5}c_{2}c_{3}+2\sqrt{30}c_{1}c_{3}+65c_{3}^{2})z_{i}^{2}(z_{i}^{2}-3){\rm
e}^{z_{i}/2}K_{2}\left[\frac{z_{i}}{2}\right]$
$\displaystyle+15\sqrt{\pi}\biggl{(}4(-6c_{2}^{2}+9\sqrt{5}c_{2}c_{3}-15c_{3}^{2}+2\sqrt{6}(c_{1}c_{2}-\sqrt{5}c_{1}c_{3}))$
$\displaystyle\times
U(-\tfrac{1}{2},-2,z_{i})+28(\sqrt{5}c_{2}c_{3}-5c_{3}^{2})U(-\tfrac{1}{2},-4,z_{i})+63c_{3}^{2}U(-\tfrac{1}{2},-5,z_{i})\biggr{)}\biggl{]},$

where $z_{i}=m_{i}^{2}/\beta^{2}$, $K_{n}(x)$ is the modified Bessel function
of second kind of order n, and $U(a,b,z)$ is the Tricomi’s (confluent
hypergeometric) function. The contribution from the linear confinement
$(a+br)$ is given by

$\displaystyle\expectationvalue{V^{\rm Lin}_{\rm Conf}}$ $\displaystyle=$
$\displaystyle
a+\frac{b}{\beta\sqrt{\pi}}\biggl{(}2c_{1}^{2}+3c_{2}^{2}+\frac{15}{4}c_{3}^{2}-2\sqrt{\frac{2}{3}}c_{1}c_{2}-\sqrt{\frac{2}{15}}c_{1}c_{3}-\sqrt{5}c_{2}c_{3}\biggr{)},\quad\quad$
(24)

and those of screened confinement $(a+{b(1-{\rm e}^{-\mu r})}/{\mu})$ is given
by

$\displaystyle\expectationvalue{V^{\rm Scr}_{\rm Conf}}$ $\displaystyle=$
$\displaystyle
a+\frac{b}{\mu}+\frac{b}{3840\sqrt{\pi}\beta^{10}\mu}\biggl{[}2\beta\mu\bigl{(}32(60c_{1}^{2}-4\sqrt{6}(10c_{1}c_{2}+\sqrt{5}c_{1}c_{3})$
(25)
$\displaystyle+15(8c_{2}^{2}-4\sqrt{5}c_{2}c_{3}+11c_{3}^{2}))\beta^{8}+32(40c_{2}^{2}-48\sqrt{5}c_{2}c_{3}+115c_{3}^{2}$
$\displaystyle+2\sqrt{6}(-5c_{1}c_{2}+2\sqrt{5}c_{1}c_{3})\beta^{6}\mu^{2}+8(10c_{2}^{2}-28\sqrt{5}c_{2}c_{3}$
$\displaystyle+2\sqrt{30}c_{1}c_{3}+89c_{3}^{2})\beta^{4}\mu^{4}+-8(\sqrt{5}c_{2}c_{3}-6c_{3}^{2})\beta^{2}\mu^{6}+c_{3}^{2}\mu^{8}\bigr{)}$
$\displaystyle-\sqrt{\pi}{\rm
e}^{\mu^{2}/4\beta^{2}}(3840\beta^{10}+1920(c_{1}^{2}+3c_{2}^{2}+5c_{3}^{2}-\sqrt{6}c_{1}c_{2}$
$\displaystyle-2\sqrt{5}c_{2}c_{3})\beta^{8}\mu^{2}+160(9c_{2}^{2}+30c_{3}^{2}+\sqrt{30}c_{1}c_{3}-2\sqrt{6}c_{1}c_{2}$
$\displaystyle-12\sqrt{5}c_{2}c_{3})\beta^{6}\mu^{4}+16(5c_{2}^{2}+50c_{3}^{2}-15\sqrt{5}c_{2}c_{3}+\sqrt{30}c_{1}c_{3})\beta^{4}\mu^{6}$
$\displaystyle+50c_{3}^{2}-8\sqrt{5}c_{2}c_{3}\beta^{2}\mu^{8}+c_{3}^{2}\mu^{10}){\rm
erfc}\left[\frac{\mu}{2\beta}\right]\biggl{]},$

where the error function ${\rm erfc}(z)=1-{\rm erf}(z)$. The Coulomb-like and
Hyperfine potentials are given by

$\displaystyle\expectationvalue{V_{\rm Coul}}$ $\displaystyle=$
$\displaystyle-\frac{\beta\alpha_{s}}{45\sqrt{\pi}}\biggl{(}120c_{1}^{2}+100c_{2}^{2}+89c_{3}^{2}+40\sqrt{6}c_{1}c_{2}+12\sqrt{30}c_{1}c_{3}$
(26) $\displaystyle+44\sqrt{5}c_{2}c_{3}\biggr{)},$
$\displaystyle\expectationvalue{V_{\rm Hyp}}$ $\displaystyle=$
$\displaystyle\frac{\expectationvalue{\boldsymbol{S}_{q}\cdot\boldsymbol{S}_{\bar{q}}}\beta^{3}\alpha_{s}}{3m_{c}m_{\bar{c}}\sqrt{\pi}}\biggl{(}\frac{32}{3}c_{1}^{2}+16c_{2}^{2}+20c_{3}^{2}+32\sqrt{\frac{2}{3}}c_{1}c_{2}+16\sqrt{\frac{10}{3}}c_{1}c_{3}$
(27) $\displaystyle+16\sqrt{5}c_{2}c_{3}\biggr{)},$

respectively. Note that the meson’s mass is able to be calculated by replacing
$c_{i}$ in the mass formula with $c_{i}^{1S},c_{i}^{2S},$ or $c_{i}^{3S}$ for
$1S$, $2S$, and $3S$ state, respectively.

###  2.2 Decay Constant

All pseudoscalar(vector) mesons $P$($V$) with four-momentum $P^{\mu}$ and mass
$M_{P}$ ($M_{V}$) have decay constants, namely $f_{P}$ and $f_{V}$, which are
defined in the matrix transition form as

$\displaystyle\bra{0}\bar{q}\gamma^{\mu}\gamma_{5}q\ket{P}$ $\displaystyle=$
$\displaystyle if_{P}P^{\mu},$
$\displaystyle\bra{0}\bar{q}\gamma^{\mu}q\ket{V(P,\lambda)}$ $\displaystyle=$
$\displaystyle f_{V}M_{V}\epsilon^{\mu}(\lambda),$ (28)

where $\epsilon^{\mu}(\lambda)$ is the polarization vector of a vector meson.
Solving the left and right-hand side of Eq. (2.2) simultaneously, we obtain
the explicit form of $f_{P(V)}$

$\displaystyle f_{P}$ $\displaystyle=$ $\displaystyle
2\sqrt{6}\int\frac{\differential
x\differential^{2}\boldsymbol{k}_{\bot}}{2(2\pi)^{3}}\frac{{\Phi}(x,\boldsymbol{k}_{\bot})}{\sqrt{\mathcal{A}^{2}+\mathbf{k}_{\bot}^{2}}}\leavevmode\nobreak\
\mathcal{A},$ (29) $\displaystyle f_{V}$ $\displaystyle=$ $\displaystyle
2\sqrt{6}\int\frac{\differential
x\differential^{2}\boldsymbol{k}_{\bot}}{2(2\pi)^{3}}\frac{{\Phi}(x,\boldsymbol{k}_{\bot})}{\sqrt{\mathcal{A}^{2}+\boldsymbol{k}_{\bot}^{2}}}\left[\mathcal{A}+\frac{2\boldsymbol{k}_{\bot}^{2}}{D_{0}}\right],$
(30)

respectively, where $\mathcal{A}=(1-x)m_{q}+xm_{\bar{q}}$ and and
$D_{0}=M_{0}+m_{q}+m_{\bar{q}}$.

##  3 Result and Discussion

Before calculating the mass spectra and decay constants, it is necessary to
determine seven parameters through the variational principle applied to
$\Phi_{1S}$ in the mass formula. The values of these parameters are listed in
Table 1. Here, we only fix the widely-known string tension $b=0.18$ GeV2.
Additionally, we assume $\theta_{13}=\theta_{23}$ to simplify the
parameterization that reasonably describes the experimental data. Initially,
we considered $\theta_{12}=\theta_{13}=\theta_{23}$ and found that the
predicted result deviates considerably from the data. Because of that, we
assume two mixing angles for the basis expansion coefficients in this work. We
obtain the $\theta_{12}=12.12^{\circ}$ and $\theta_{13}=8.44^{\circ}$ where
$\theta_{12}>\theta_{13}$. As for comparison, the angle value $\theta_{12}$ is
quite similar to Ref. Arifi:2022pal , but they consider only two HO bases such
that ($\theta_{13}=\theta_{23}=0$) and use it to calculate observables for the
$1S$ and $2S$ states.

Table 1: Model parameters that include constituent quark mass $m$, potential parameters $(a$ and $b)$ as well as screening parameter $\mu$ and variational parameters $\beta$ (GeV). $\theta_{12}$ | $\theta_{13}=\theta_{23}$ | $m_{c}$ | $a$ | $b$ | $\mu$ | $\alpha_{s}$ | $\beta_{c\bar{c}}$
---|---|---|---|---|---|---|---
$12.12$ | $8.44$ | $1.61$ | $-0.41$ | 0.18 | $0.027$ | $0.402$ | $0.5417$

In the case of the quark mass, our obtained value of 1.61 GeV is quite similar
to the value in the Godfrey-Isgur (GI) model Godfrey:1985xj . However, our
results for the potential parameters, particularly the strong coupling
constant $\alpha_{s}$, differ somewhat from Arifi’s work. Moreover, our value
for the screening parameter $\mu$ is a bit different from that in Gao’s work
Gao:2022Screened , where they obtained $\mu=0.045$ GeV2. As for the
variational parameter $\beta$ on the HO expansion, we obtain
$\beta_{c\bar{c}}=0.5417$ GeV. We note that other parameters such as the
mixing angle and potential parameter will influence the value of $\beta$. For
example, Arifi et al., obtained $\beta_{c\bar{c}}=0.592$ GeV where they only
apply linear potential and $\theta=12^{\circ}$. This is quite essential since
the $\beta$ parameter determines the the wave function characteristics and is
always inversely proportional to the width of the wave function Wang:2021hho .
As a result, it indirectly affects the determination of MS and DC.

Table 2: Mass spectra [MeV] with linear and screened potentials. We also show other model predictions for comparison. State | $M_{\rm lin.}$ | $M_{\rm scr.}$ | Exp. ParticleDataGroup:2022pth  | GI Godfrey:1985xj  | RQM Ebert:2003RQM  | NRQCD Soni:2017wvy
---|---|---|---|---|---|---
$\eta_{c}(1S)$ | 3010.8 | 3002.3 | 2983.9 $\pm$ 0.4 | 2970 | 2979 | 2989
$\eta_{c}(2S)$ | 3641.3 | 3614.5 | 3637.5 $\pm$ 1.1 | 3620 | 3588 | 3602
$\eta_{c}(3S)$ | 4076.3 | 4028.1 | … | 4060 | 3991 | 4058
$J/\psi(1S)$ | 3110.8 | 3102.4 | 3098.9 $\pm$ 0.01 | 3100 | 3096 | 3094
$\psi(2S)$ | 3706.6 | 3679.8 | 3686.1$\pm$ 0.06 | 3680 | 3686 | 3681
$\psi(3S)$ | 4127.3 | 4079.1 | 4039 $\pm$ 1 | 4100 | 4088 | 4129

First, let us discuss the obtained masses of the charmonia. Here, we use two
confinement potentials separately to study its effect. While $M_{\rm lin.}$
denotes the mass spectra using linear potential, $M_{\rm scr.}$ denotes the
mass spectra using screened potential. We present our prediction along with
the experimental value from PDG as well as the results from other theoretical
models in Table 2. We can clearly see that $M_{\rm scr.}$ for all charmonium
states are lower than those in $M_{\rm lin.}$ and the discrepancy increases as
the radial excitation level increases. It happens because of the exponential
term in the screened potential, which leads to a decrease in mass as the
distance increases. We note that the excited state wave function is more
extended. It is worth noting that the inclusion of the mixing angle in our
calculation, whether for $M_{\rm lin.}$ or $M_{\rm scr.}$, effectively reduces
the predicted mass. A similar observation can be found in Arifi’s work
Arifi:2022pal . Besides our predictions and experiments, results from the GI
model Godfrey:1985xj  are quite similar to the experiment, although there is a
considerable difference on $\psi(3S)$. For instance, some of the states on
charmonia that employ the relativistic effect on the Relativized Quark Model
(RQM) Ebert:2003RQM  model will reduce the mass result compared to the Non-
Relativistic Quantum Chromodynamics (NRQCD) Soni:2017wvy  model although this
model also uses the linear potential. Interestingly, our prediction and other
theoretical models on the $\psi(3S)$ state lead to the $\psi(4040)$ instead of
$\psi(4160)$ and $\psi(4415)$ as well. A review from Ref. Mo:2010bw  suggests
that $\psi(4040)$ corresponds to the $\psi(3S)$ and $\psi(4415)=\psi(5S)$,
while the assignment of $\psi(4160)$ to a specific state has not yet been
determined.

Table 3: Predicted decay constant [MeV] for charmonia. State | $f_{\rm Theo.}$ | Exp. ParticleDataGroup:2022pth  | Lattice Davies:2010ip ; Davies:2012ogo  | Sum Rules Becirevic:2016rfu  | LFQM (CJ) Choi:2009Semileptonic
---|---|---|---|---|---
$\eta_{c}$ ($1S$) | 349.1 | 335(75) | 395(2.4) | 387(7) | 326
$J/\psi$ ($1S$) | 396.7 | 407(5) | 405(6) | 418(9) | 360
|  |  | BLFQ Li:2017mlw  | RQM Azhothkaran:2020ipl  | RQM2 Lakhina:2006vg
$\eta_{c}$ ($2S$) | 224 | … | 299(68) | 173 | 240
$\psi$ ($2S$) | 285.6 | 294(5) | 312(73) | 261 | 293
|  |  | NRQM Soni:2017wvy  | RQM Azhothkaran:2020ipl  | RQM2 Lakhina:2006vg
$\eta_{c}$ ($3S$) | 163.9 | … | 249 | 127 | 193
$\psi$ ($3S$) | 230.2 | 238(5) | 230 | 206 | 258

Here, we have also calculated the decay constants for the charmonium states.
Our predictions, along with a comparison to the experimental values and the
results from other theoretical models, are presented in Table 3. To determine
the experimental value of the decay constant for each charmonium state, we
extract it using the leptonic partial width $\Gamma_{e^{-}e^{+}}$ for vector
charmonium and the diphoton partial width $\Gamma_{\gamma\gamma}$ for
pseudoscalar charmonium. It can be observed that the decay constant decreases
as the radial excitation level increases, establishing the hierarchy
$f(1S)>f(2S)>f(3S)$, where the ground state has a larger decay constant than
the excited states. Unfortunately, experimental results for $\eta_{c}(2S)$ and
$\eta_{c}(3S)$ cannot be found because there is no experimental data available
for determining the partial width required to calculate the decay constant in
the PDG.

In general, our results for the decay constant are in good agreement with the
experimental data, especially for $\psi$ mesons. However, for the $1S$ state,
we obtained different results compared to other LFQM by the Choi-Ji (CJ)
scheme Choi:2009Semileptonic . This difference may be attributed to their
focus being primarily on the ground state mesons with various quark flavors
and variations in parameter determination during the variational analysis. We
found that the results for $\eta_{c}(1S)$ are consistent with the experimental
data but deviate from the lattice data Davies:2010ip ; Davies:2012ogo  and sum
rules Becirevic:2016rfu . For the $J/\psi(1S)$, our prediction is comparable
to the experimental data and other calculations. In a similar framework of
light-front dynamics, basis light-front quantization (BLFQ) Li:2017mlw
produces a higher result than ours for the $2S$ states, particularly showing a
moderate difference between BLFQ and experimental result for $\psi(2S)$. For
the $3S$ state, our predictions for $\psi(3S)$ are generally consistent with
the experimental data and other calculations. However, our result on
$\eta_{c}$ has a large discrepancy compared to either RQM Azhothkaran:2020ipl
or NRQM Soni:2017wvy , but not quite differ to RQM2 Lakhina:2006vg . This
discrepancy arises from differences in the treatment of certain parameters.
Specifically, Lakhina’s work Lakhina:2006vg  defines the running constant
$\alpha_{s}$ with a logarithmic scale applied to their denominator, whereas
Azhothkaran Azhothkaran:2020ipl  and Soni’s work Soni:2017wvy  takes into
account QCD corrections, including $\delta_{P}$ (for pseudoscalar meson) and
$\delta_{V}$ (for vector meson), in their decay constant calculation.
Consequently, our result appears to be in better agreement with Lakhina’s
findings than with the others.

##  4 Conclusion and Outlook

We have analyzed the properties of charmonia ($\eta_{c},J/\psi$) from the
ground state ($1S$) to the second radial excitation ($3S$) using the light-
front quark model. In this study, we calculated their mass spectra along with
decay constants by employing the harmonic oscillator expansion method.
Furthermore, we expand the basis as a linear combination of
$\phi_{1S},\phi_{2S}$, and $\phi_{3S}$. To obtain the parameters
$\beta_{cc},a,b,\alpha_{s},\mu$, and $\theta_{ij}$ we carried out the
variational analysis using the mass formula we have derived.

For our mass spectra calculation, we consider the QCD-motivated effective
Hamiltonian with several potentials such as confinement (linear and screened),
Coulomb-like, and hyperfine term. In this work, we consider the hyperfine
potential perturbatively since its contribution is rather small. Our
predictions for mass spectra up to $3S$ states with the screened potential are
consistent with the experimental data and other theoretical models. Also, our
predictions for decay constants are in good agreement with experimental data
and other theoretical models. We note that $f_{P(V)}^{exp}$ are extracted from
$\Gamma_{e^{-}e^{+}}$ ($\Gamma_{\gamma\gamma}$) for vector (pseudoscalar)
mesons. Interestingly, we find our model can reproduce the hierarchy of decay
constant $f_{1S}>f_{2S}>f_{3S}$ which is mainly due to the increasing number
of the basis functions. If we use a single HO basis function, the hierarchy of
the decay constant will be opposite Arifi:2022pal .

For future work, we would like to consider other mesons that contain bottom
quarks such as $B_{c}^{(*)}$ mesons and bottomonia ($\eta_{b},\Upsilon$).
Computing other properties such as distribution amplitudes Arifi:2023uqc  and
radiative decaysChoi:2007se  with the obtained light-front wave function is
also certainly of interest. We would like also to apply the smeared function
Hong:2022sht  instead of the Dirac delta function in the spin-spin
interaction. It is important to note that the Harmonic Oscillator (HO) basis
function is assumed for the wave function in the present work, but one should
study different basis functions further to solve the Hamiltonian that can lead
to more accurate predictions.

## Acknowledgements

M.R. thanks the ISCPMS 2023 organizer for providing the oral presentation for
his work. The work of A.J.A. is supported by the RIKEN special postdoctoral
researcher (SPDR) program. The work of T.M. is supported by the PUTI Q2 grant
from Universitas Indonesia (UI), under contract No.
NKB-663/UN2.RST/HKP.05.00/2022. The hospitality of the master program in
physics at UI is also gratefully acknowledged.

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