# Nature of $X(3872)$ from its radiative decay

Shuo-Ying Yu 1 Xian-Wei Kang 1,2<EMAIL_ADDRESS>1 Key Laboratory of Beam
Technology of the Ministry of Education, College of Nuclear Science and
Technology, Beijing Normal University, Beijing 100875, China
2 Institute of Radiation Technology, Beijing Academy of Science and
Technology, Beijing 100875, China

###### Abstract

We study the radiative decay of $X(3872)$ based on the assumption that
$X(3872)$ is regarded as a $c\overline{c}$ charmonium with quantum number
$J^{PC}=1^{++}$ ($J,\,P,\,C$ represent the spin, parity and charge
conjugation, respectively). The form factors of $X(3872)$ transitions to
$J/\psi\gamma$ and $\psi^{\prime}\gamma$ ($\psi^{\prime}$ denotes $\psi(2S)$
throughout the paper) are calculated in the framework of the covariant light-
front quark model. The phenomenological wave function of a meson depends on
the parameter $\beta$, whose inverse essentially describes the confinement
scale. In the present work, the parameters $\beta$ for the vector $J/\psi$ and
$\psi^{\prime}$ mesons will be determined through their decay constants, which
are obtained from the experimental values of their partial decay widths to the
electron-positron pair. For $X(3872)$, we determined the value of $\beta$ by
the decay width of $X(3872)\rightarrow\psi^{\prime}\gamma$. Then, we examined
the width of $X(3872)\to J/\psi\gamma$ in a manner of parameter-free
prediction and compared it with the experimental value. As a result, an
inconsistency or contradiction occurs between the widths of $X(3872)\to
J/\psi\gamma$ and $X(3872)\to\psi^{\prime}\gamma$. We thus conclude that
$X(3872)$ cannot be a pure $c\overline{c}$ resonance and that other components
are necessary in its wave function.

##  I introduction

Traditionally there exists two types of hadrons in nature: mesons composed of
quarks and anti-quarks and baryons composed of three quarks. As a key topic in
hadron physics, researchers have put much effort into both theoretical studies
and experimental searches for exotic states beyond the aforementioned
configurations, such as tetraquark, pentaquark, glueball and hybrid states. In
addition, some peaks are only a manifestation of the threshold effects
Haidenbauer:2015yka ; Guo:2019twa . In 2003, the Belle Collaboration observed
a narrow resonance state $X(3872)$ in the mass spectrum of
$\pi^{+}\pi^{-}J/\psi$ Belle:2003nnu , which is considered to be the first
candidate of an exotic state. Later, many measurements were made on its mass
and decay properties. The LHCb Collaboration determined the quantum number of
$X(3872)$ as $J^{PC}=1^{++}$ (with $J,\,P,\,C$ denoting the spin, parity and
charge conjugation value, respectively) by performing a full five-dimensional
amplitude analysis of the angle correlation LHCb:2013kgk ; LHCb:2015jfc . The
signals of $X(3872)\rightarrow D^{0}\bar{D}^{*0}$ BESIII:2020nbj ,
$\pi^{+}\pi^{-}J/\psi$ LHCb:2020fvo , $\omega J/\psi$ BESIII:2019qvy ,
$\pi^{0}\chi_{c1}$ BESIII:2019esk , $J/\psi\gamma$ Belle:2011wdj and
$\psi^{\prime}\gamma$LHCb:2014jvf  were also observed. For more details, the
Particle Data Group (PDG) can be referenced ParticleDataGroup:2022pth . We
will focus on the radiative decays $X(3872)\to J/\psi\gamma$ and
$\psi^{\prime}\gamma$. The most recent measurement of their ratio
$\frac{\text{Br}(X(3872)\rightarrow\psi(2S)\gamma)}{\text{Br}(X(3872)\rightarrow
J/\psi\gamma)}$ (Br denotes the branching ratio) is from the BESIII
collaboration, with a value $<0.59$ at the 90% confidence level (C.L.)
BESIII:2020nbj . It agrees with the upper limit $<2.1$ at 90% C.L. set by the
Belle collaboration while marginally agreeing with the BaBar value $3.4\pm
1.4$ BaBar:2008flx  and challenging the LHCb value $2.46\pm 0.70$ LHCb:2014jvf
within two standard deviations. Taking into account the model predictions,
BESIII disfavors the interpretation of a pure charmonium state compared to
other interpretations BESIII:2020nbj . Until 2020, significant progress was
made in measuring the width of $X(3872)$ by LHCb LHCb:2020xds ; LHCb:2020fvo .
Now, the PDG average is $\Gamma_{X(3872)}=(1.1\pm 0.21)$ MeV.

As shown in Ref. Kang:2016jxw , the data on the line shape of $X(3872)$ are
not sufficient to determine its pole structure from the perspective of
compositeness Li:2021cue ; Baru:2003qq ; Kinugawa:2023fbf : either a bound
state or a virtual state of $\bar{D}D^{*}$ is possible for $X(3872)$. See also
the review Guo:2017jvc  for more discussions. Consequently, it is necessary to
exploit the $X(3872)$ decay information to investigate its inner structure
from another point of view. Among the various decay channels, radiative decay
can be measured experimentally and treated in theory in an easier way. In this
work, we assume that $X(3872)$ is a pure charmonium state. The plausibility of
such a hypothesis is tested by comparing the calculated width
$\Gamma(X(3872)\rightarrow J/\psi(\psi^{\prime})\gamma)$ to the experimental
widths. The form factors describing the radiative transition of $X(3872)\to
J/\psi,\psi^{\prime}$ are calculated in the covariant light-front quark model
(CLFQM). This quark model involves a free parameter $\beta$ that appears in
the wave function of a hadron. The values of $\beta$ for $J/\psi$ and
$\psi^{\prime}$ will be fixed by their decay constants, or more precisely
speaking, by the values of $\Gamma(J/\psi(\psi^{\prime})\to e^{+}e^{-})$.
Taking the decay width $\Gamma(X(3872)\rightarrow\psi^{\prime}\gamma)$
Workman:2022ynf  as input, we can fix the parameter $\beta$ for $X(3872)$.
Then, the calculation of the width $\Gamma(X(3872)\rightarrow J/\psi\gamma)$
will be a parameter-free prediction. The comparison between it and the
experimentally measured width will definitely provide a criterion for how the
interpretation of a conventional charmonium works for $X(3872)$.

This paper is organized as follows. In Sec. II, we derive the expressions of
the form factors and decay width in CLFQM. In Sec. III, we show our numerical
results and provide a discussion. The conclusion is given in Sec. IV.

##  II The decay of $X(3872)\to J/\psi\gamma,\,\psi^{\prime}\gamma$

###  II.1 Notation

We take the covariant light-front quark model used in Jaus:1991cy ;
Jaus:1989au ; Jaus:1996np ; Jaus:1999zv ; Cheng:2003sm ; Choi:2007se ;
Shi:2016cef ; Chang:2019obq  to calculate the width of radiative decay. In
Ref. Ke:2011jf , the radiative decay of $X(3872)$ is investigated for the case
of $2^{-+}$. In the light-front framework, a momentum $p$ is expressed as

$\displaystyle p^{\mu}$ $\displaystyle=$
$\displaystyle(p^{-},p^{+},p_{\bot}),$ $\displaystyle p_{\bot}$
$\displaystyle=$
$\displaystyle(p^{1},p^{2}),\,\,p^{-}=p^{0}-p^{3},\,\,p^{+}=p^{0}+p^{3},$ (1)

and thus

$p^{2}=p^{+}p^{-}-p_{\bot}^{2},\,\,\,p\cdot
q=\frac{1}{2}(p^{-}q^{+}+p^{+}q^{-})-p_{\bot}q_{\bot}.$ (2)

The incoming (outgoing) meson has momentum
$P^{\prime}(^{\prime\prime})=p_{1}^{\prime}(^{\prime\prime})+p_{2}$, where
$p_{1}^{\prime}(^{\prime\prime})$ and $p_{2}$ are the momenta of the off-shell
active quark and spectator quark, respectively, with masses
$m^{\prime}_{1}(^{\prime\prime})$ and $m_{2}$. We defined
$P=P^{\prime}+P^{\prime\prime}$ and $q=P^{\prime}-P^{\prime\prime}$. An
illustration is shown in Fig. 1 111In principle, there are diagrams of the
photon that couples directly to the quark-quark-vector meson vertex. However,
as shown in Ref. Ganbold:2021nvj  those contributions from the bubble diagrams
are small and not more than 10% of the result. Thus they are omitted without
loss of our accuracy for a conclusion. These momenta can be expressed in terms
of the internal variables ($x_{i},p^{\prime}_{\bot}$),

$\displaystyle
p_{1}^{\prime}(^{\prime\prime})^{+}=x_{1}P^{{}^{\prime}(^{\prime\prime})+},\,\,\,p_{2}^{\prime}(^{\prime\prime})^{+}=x_{2}P^{\prime}(^{\prime\prime})^{+}$
$\displaystyle
p^{\prime}(^{\prime\prime})_{1\bot}=x_{1}P^{\prime}(^{\prime\prime})_{\bot}+p^{\prime}(^{\prime\prime})_{\bot},$
$\displaystyle
p^{\prime}(^{\prime\prime})_{2\bot}=x_{2}P^{{}^{\prime}(^{\prime\prime})}_{\bot}-p^{\prime}(^{\prime\prime})_{\bot},$
$\displaystyle p^{\prime\prime}_{\bot}=p^{\prime}_{\bot}-x_{2}q_{\bot},$ (3)

with $x_{1}+x_{2}=1$. The following physical quantities are defined for
subsequent derivations:

$\displaystyle M_{0}^{\prime
2}=(e_{1}^{\prime}+e_{2})^{2}=\frac{p_{\bot}^{\prime 2}+m_{1}^{\prime
2}}{x_{1}}+\frac{p_{\bot}^{\prime 2}+m_{2}^{2}}{1-x_{1}},$ $\displaystyle
e_{1}^{\prime}=\sqrt{p_{\bot}^{\prime 2}+p_{z}^{\prime 2}+m_{1}^{\prime
2}},\,\,e_{2}=\sqrt{p_{\bot}^{\prime 2}+p_{z}^{\prime 2}+m_{2}^{2}},$
$\displaystyle
p_{z}^{\prime}=\frac{(1-x_{1})M_{0}^{\prime}}{2}-\frac{p_{\bot}^{\prime
2}+m_{2}^{2}}{2(1-x_{1})M_{0}^{\prime}},$ $\displaystyle M_{0}^{\prime\prime
2}=\frac{p_{\bot}^{\prime\prime 2}+m_{1}^{\prime\prime
2}}{x_{1}}+\frac{p_{\bot}^{\prime\prime 2}+m_{2}^{2}}{1-x_{1}},$
$\displaystyle e_{1}^{\prime\prime}=\sqrt{p_{\bot}^{\prime\prime
2}+p_{z}^{\prime\prime 2}+m_{1}^{\prime\prime 2}},$ $\displaystyle
p_{z}^{\prime\prime}=\frac{(1-x_{1})M_{0}^{\prime\prime}}{2}-\frac{p_{\bot}^{\prime\prime
2}+m_{2}^{2}}{2(1-x_{1})M_{0}^{\prime\prime}}.$ (4)

Here, $M_{0}^{\prime 2}$ ($M_{0}^{\prime\prime 2}$) can be interpreted as the
kinetic invariant mass squared of the incoming (outgoing) $q\overline{q}$
system, and $e_{i}$ is the energy of Quark $i$.

###  II.2 Form factors

Figure 1: Feynman diagrams for radiative transitions. The uppercase
$P^{\prime}$ and $P^{\prime\prime}$ denote the four-momentum of the initial
and final meson, respectively. The lower case $p_{1,2}$ is the momentum of the
spectator quark, and other momenta with prime or double prime in the
superscript correspond to the active quarks involving a photon emission shown
by a wavy line.

The transition amplitude for $X(3872)\to J/\psi\,\gamma,\psi^{\prime}\gamma$
can be written as

$\displaystyle
A(X(3872)\rightarrow\psi\gamma)=\epsilon^{*\alpha}(q)\epsilon^{\prime\mu}(P^{\prime})\epsilon^{\prime\prime*\nu}(P^{\prime\prime})\mathcal{A}_{\alpha\mu\nu}$
$\displaystyle\mathcal{A}_{\alpha\mu\nu}=\varepsilon_{\alpha\nu\beta\eta}P^{\beta}q^{\eta}[P_{\mu}f_{1}(q^{2})+q_{\mu}f_{2}(q^{2})]$
$\displaystyle\qquad\quad+\varepsilon_{\alpha\mu\beta\eta}P^{\beta}q^{\eta}[P_{\nu}f_{3}(q^{2})+q_{\nu}f_{4}(q^{2})]$
$\displaystyle\qquad\quad+\varepsilon_{\alpha\mu\nu\rho}[P^{\rho}f_{5}(q^{2})+q^{\rho}f_{6}(q^{2})]$
$\displaystyle\qquad\quad+\varepsilon_{\mu\nu\beta\eta}P^{\beta}q^{\eta}[P_{\alpha}f_{7}(q^{2})+q_{\alpha}f_{8}(q^{2})],$
(5)

where $\epsilon^{\prime}$ denotes the polarization of $X(3872)$,
$\epsilon^{\prime\prime}$ denotes the polarization of $J/\psi$ or
$\psi^{\prime}$, and $\epsilon$ denotes the polarization of the photon.
$f_{i}(q^{2})$ with $i=1\cdots 8$ is the $q^{2}$-dependent form factor that
encodes the underlying dynamics. By considering the properties of the
polarization vectors, the following can be obtained:

$\displaystyle\epsilon^{*\alpha}(q)\cdot q_{\alpha}=0$
$\displaystyle\epsilon^{\prime\mu}(P^{\prime})\cdot
P^{\prime}_{\mu}=\epsilon^{\prime\mu}(P^{\prime})\cdot\frac{1}{2}(P+q)_{\mu}=0$
$\displaystyle\epsilon^{\prime\prime*\nu}(P^{\prime\prime})\cdot
P^{\prime\prime}_{\nu}=\epsilon^{\prime\prime*\nu}(P^{\prime\prime})\cdot\frac{1}{2}(P-q)_{\nu}=0,$
(6)

$\mathcal{A}_{\alpha\mu\nu}$ will be reduced to, and only the nonvanishing
terms are maintained as follows:

$\displaystyle\mathcal{A}_{\alpha\mu\nu}$ $\displaystyle=$
$\displaystyle\varepsilon_{\alpha\nu\beta\eta}P^{\beta}q^{\eta}P_{\mu}f_{m}(q^{2})+\varepsilon_{\alpha\mu\beta\eta}P^{\beta}q^{\eta}P_{\nu}f_{p}(q^{2})$
(7)
$\displaystyle+\varepsilon_{\alpha\mu\nu\rho}[P^{\rho}f_{5}(q^{2})+q^{\rho}f_{6}(q^{2})]$
$\displaystyle+\varepsilon_{\mu\nu\beta\eta}P^{\beta}q^{\eta}P_{\alpha}f_{7}(q^{2}),$

where $f_{m}(q^{2})=f_{1}(q^{2})-f_{2}(q^{2})$,
$f_{p}(q^{2})=f_{3}(q^{2})+f_{4}(q^{2})$. Imposing the gauge invariance
$q^{\alpha}A_{\alpha\mu\nu}=0$, we further obtained

$\displaystyle\mathcal{A}_{\alpha\mu\nu}$ $\displaystyle=$
$\displaystyle\varepsilon_{\alpha\nu\beta\eta}P^{\beta}q^{\eta}P_{\mu}f_{m}(q^{2})+\varepsilon_{\alpha\mu\beta\eta}P^{\beta}q^{\eta}P_{\nu}f_{p}(q^{2})$
(8) $\displaystyle+\varepsilon_{\alpha\mu\nu\rho}q^{\rho}f_{6}(q^{2}).$

For the calculation of radiative decay, we employed the covariant light-front
quark model based on the assumption that $X(3872)$ is regarded as an axial
vector ($A$) meson of $c\overline{c}$, while $J/\psi$ ($\psi^{\prime}$) is a
vector ($V$) meson. The vertex functions for the axial vector meson and vector
meson states are written as Cheng:2003sm

$\displaystyle-
iH_{A}^{\prime}\left(\gamma_{\mu}+\frac{(p_{1}^{\prime}-p_{2})_{\mu}}{W_{A}}\right)\gamma_{5},$
$\displaystyle
iH_{V}^{\prime\prime}\left(\gamma_{\mu}-\frac{(p_{1}^{\prime\prime}-p_{2})_{\mu}}{W_{V}}\right),$
(9)

where $H_{A}^{\prime}$ and $H_{V}^{\prime\prime}$ are the wave functions of
the axial vector meson and vector meson that will be given below. The Feynman
amplitude corresponding to Fig. 1 can be written as

$\displaystyle A_{\alpha\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{(2\pi)^{4}}\int
d^{4}p_{1}^{\prime}H_{A}^{\prime}H_{V}^{\prime\prime}$ (10)
$\displaystyle\times\left(\frac{2e}{3}\frac{S_{\alpha\mu\nu}^{a}}{N_{1}^{\prime}N_{1}^{\prime\prime}N_{2}}+\frac{2e}{3}\frac{S_{\alpha\mu\nu}^{b}}{{N_{2}^{\prime}N_{2}^{\prime\prime}N_{1}}}\right),$

where $N_{c}=3$, $2e/3$ is the electric charge of the charm quark,
$N_{i}=p_{i}^{2}-m_{i}^{2}+i0^{+}$, $N^{\prime}_{i}=p^{\prime 2}_{i}-m^{\prime
2}_{i}+i0^{+}$, and $N^{\prime\prime}_{i}=p^{\prime\prime
2}_{i}-m^{\prime\prime 2}_{i}+i0^{+}$. The terms $S_{\alpha\mu\nu}^{a}$ and
$S_{\alpha\mu\nu}^{b}$ are the traces corresponding to Fig. 1(a) and Fig.
1(b), respectively. They are shown as follows:

$\displaystyle S_{\alpha\mu\nu}^{a}$ $\displaystyle=$
$\displaystyle\text{Tr}\Big{[}(\gamma_{\mu}+\frac{(p_{1}^{\prime}-p_{2})_{\mu}}{W_{A}^{\prime}})\gamma_{5}(-p_{2}\mkern-10.5mu/+m_{2})(\gamma_{\nu}-\frac{(p_{1}^{\prime\prime}-p_{2})_{\nu}}{W_{V}^{\prime\prime}})(p_{1}^{\prime\prime}\mkern-10.5mu/+m_{1}^{\prime\prime})\gamma_{\alpha}(p_{1}^{\prime}\mkern-10.5mu/+m_{1}^{\prime})\Big{]},$
$\displaystyle S_{\alpha\mu\nu}^{b}$ $\displaystyle=$
$\displaystyle\text{Tr}\Big{[}(\gamma_{\mu}+\frac{(p_{1}-p_{2}^{\prime})_{\mu}}{W_{A}^{\prime}})\gamma_{5}(-p_{2}^{\prime}\mkern-10.5mu/+m_{2}^{\prime})\gamma_{\alpha}(-p_{2}^{\prime\prime}\mkern-10.5mu/+m_{2}^{\prime\prime})(\gamma_{\nu}-\frac{(p_{1}-p_{2}^{\prime\prime})_{\nu}}{W_{V}^{\prime\prime}})(p_{1}\mkern-10.5mu/+m_{1})\Big{]}.$
(11)

Explicitly, the following can be obtained:

$\displaystyle S_{\alpha\mu\nu}^{a}$ $\displaystyle=$
$\displaystyle\varepsilon_{\alpha\mu\nu\rho}\Big{[}-2p_{1}^{\prime\rho}(N_{1}^{\prime\prime}-2m_{2}m_{1}^{\prime\prime}+m_{1}^{\prime\prime
2}-2m_{1}^{\prime\prime}m_{1}^{\prime}+N_{1}^{\prime}-q^{2})+q^{\rho}(N_{1}^{\prime\prime}+4m_{2}m_{1}^{\prime}$
(12) $\displaystyle+m_{1}^{\prime\prime
2}-2m_{1}^{\prime\prime}m_{1}^{\prime}+m_{1}^{\prime
2}+N_{1}^{\prime}-q^{2})+P^{\rho}(N_{1}^{\prime\prime}+m_{1}^{\prime\prime
2}-2m_{1}^{\prime\prime}m_{1}^{\prime}+m_{1}^{\prime
2}+N_{1}^{\prime}-q^{2})\Big{]}$
$\displaystyle+2\varepsilon_{\alpha\mu\beta\eta}q^{\beta}p_{1}^{\prime\eta}\Big{(}P_{\nu}+q_{\nu}-2p_{1\nu}^{\prime}\Big{)}+2\varepsilon_{\alpha\nu\beta\eta}q^{\beta}p_{1}^{\prime\eta}\Big{(}P_{\mu}+q_{\mu}-2p_{1\mu}^{\prime}\Big{)}$
$\displaystyle+\varepsilon_{\mu\nu\beta\eta}\Big{[}2P^{\beta}p_{1}^{\prime\eta}(2p_{1\alpha}^{\prime}-q^{\alpha})-2P^{\beta}q^{\eta}p_{1\alpha}^{\prime}-2q^{\beta}p_{1}^{\prime\eta}q_{\alpha}\Big{]}$
$\displaystyle+\frac{\varepsilon_{\alpha\nu\beta\eta}}{W_{A}^{\prime}}(P_{\mu}+q_{\mu}-4p_{1\mu}^{\prime})\Big{[}p_{1}^{\prime\eta}q^{\beta}(m_{1}^{\prime}+m_{1}^{\prime\prime}-2m_{2})+P^{\beta}p_{1}^{\prime\eta}(m_{1}^{\prime\prime}+m_{1}^{\prime})+P^{\beta}q^{\eta}m_{1}^{\prime}\Big{]}$
$\displaystyle+\frac{\varepsilon_{\alpha\mu\beta\eta}}{W_{V}^{\prime\prime}}(P_{\nu}+3q_{\nu}-4p_{1\nu}^{\prime})\Big{[}-p_{1}^{\prime\eta}q^{\beta}(m_{1}^{\prime}+m_{1}^{\prime\prime}+2m_{2})+P^{\beta}p_{1}^{\prime\eta}(+m_{1}^{\prime}-m_{1}^{\prime\prime})-P^{\beta}q^{\eta}m_{1}^{\prime}\Big{]}$
$\displaystyle+\frac{\varepsilon_{\alpha\beta\eta\rho}P^{\beta}q^{\eta}p_{1}^{\prime\rho}(P_{\nu}+3q_{\nu}-4p_{1\nu}^{\prime})(P_{\mu}+q_{\mu}-4p_{1\mu}^{\prime})}{2W_{A}^{\prime}W_{V}^{\prime\prime}}.$

In the light-front formalism, the integration measure is

$d^{4}p^{\prime}_{1}=\frac{1}{2}P^{\prime+}dp_{1}^{\prime-}dx_{1}d^{2}p^{\prime}_{\bot}.$
(13)

As discussed in Ref. Jaus:1999zv , if it is assumed that the vertex function
($H_{A}^{\prime}$, $H_{V}^{\prime\prime}$) has no pole in the upper complex
$p_{1}^{\prime-}$ plane, the CLFQM used here and the standard light-front
formulism where the covariance property is not satisfied will lead to
identical results at the one-loop level. We assume such a situation holds
here. Then, the integration over $dp_{1}^{\prime-}$ picks up a residue at
$p_{2}=\hat{p}_{2}$, where $\hat{p}_{2}^{2}=m_{2}^{2}$. Thus, the following
replacements will be obtained:

$\displaystyle N_{1}^{\prime}\rightarrow\hat{N}_{1}^{\prime}=x_{1}(M^{\prime
2}-M_{0}^{\prime 2}),$ $\displaystyle
N_{1}^{\prime\prime}\rightarrow\hat{N}_{1}^{\prime\prime}=x_{1}(M^{\prime\prime
2}-M_{0}^{\prime\prime 2}),$ $\displaystyle
H_{A}^{\prime}\rightarrow\hat{H}_{A}^{\prime}\equiv h_{A}^{\prime},\quad
H_{V}^{\prime\prime}\rightarrow\hat{H}_{V}^{\prime\prime}\equiv
h_{V}^{\prime\prime},$ $\displaystyle
W_{A}^{\prime}\rightarrow\hat{W}_{A}^{\prime}\equiv w_{A}^{\prime},\quad
W_{V}^{\prime\prime}\rightarrow\hat{W}_{V}^{\prime\prime}\equiv
w_{V}^{\prime\prime},$ $\displaystyle\int
d^{4}p_{1}^{\prime}\frac{H_{A}^{\prime}H_{V}^{\prime\prime}}{N_{1}^{\prime}N_{1}^{\prime\prime}N_{2}}S_{\alpha\mu\nu}^{a}$
$\displaystyle\qquad\rightarrow-i\pi\int
dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{A}^{\prime}h_{V}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}\hat{S}_{\alpha\mu\nu}^{a},$
(14)

with Cheng:2003sm

$\displaystyle h_{A}^{\prime}=\sqrt{\frac{3}{2}}(M^{\prime 2}-M_{0}^{\prime
2})\sqrt{\frac{x_{1}x_{2}}{N_{c}}}\frac{1}{\sqrt{2}\widetilde{M}_{0}^{\prime}}\frac{\widetilde{M}_{0}^{\prime
2}}{2\sqrt{3}M_{0}^{\prime}}\varphi^{\prime}(nP),$ $\displaystyle
h_{V}^{\prime\prime}=(M^{\prime\prime 2}-M_{0}^{\prime\prime
2})\sqrt{\frac{x_{1}x_{2}}{N_{c}}}\frac{1}{\sqrt{2}\widetilde{M}_{0}^{\prime\prime}}\varphi^{\prime\prime}(nS),$
$\displaystyle w_{A}^{\prime}=\frac{\widetilde{M}_{0}^{\prime
2}}{m_{1}^{\prime}-m_{2}},\quad
w_{V}^{\prime\prime}=M_{0}^{\prime\prime}+m_{1}^{\prime\prime}+m_{2},$
$\displaystyle\widetilde{M}_{0}^{\prime}(^{\prime\prime})=\sqrt{M_{0}^{\prime}(^{\prime\prime})^{2}-(m_{1}^{\prime}(^{\prime\prime})-m_{2})^{2}}.$
(15)

Considering $X(3872)$ as a $2P$ charmonium, we need the following wave
functions ($1S$ for $J/\psi$ and $2S$ for $\psi^{\prime}$) Hwang:2008qi

$\displaystyle\varphi(1S)$ $\displaystyle=$ $\displaystyle
4\left(\frac{\pi}{\beta^{2}}\right)^{\frac{3}{4}}\sqrt{\frac{dp_{z}}{dx_{2}}}\exp\left(-\frac{p_{z}^{2}+p_{\bot}^{2}}{2\beta^{2}}\right),$
$\displaystyle\varphi(2S)$ $\displaystyle=$ $\displaystyle
4\sqrt{\frac{8}{3}}\left(\frac{\pi}{\beta^{2}}\right)^{\frac{3}{4}}\sqrt{\frac{dp_{z}}{dx_{2}}}\left(\frac{p_{z}^{2}+p_{\bot}^{2}}{2\beta^{2}}-\frac{3}{4}\right)$
$\displaystyle\times\exp\left(-\frac{p_{z}^{2}+p_{\bot}^{2}}{2\beta^{2}}\right),$
$\displaystyle\varphi(2P)$ $\displaystyle=$ $\displaystyle
4\left(\frac{\pi}{\beta^{2}}\right)^{\frac{3}{4}}\sqrt{\frac{dp_{z}}{dx_{2}}}\sqrt{\frac{5}{\beta^{2}}}\left(1-\frac{2}{5}\frac{p_{z}^{2}+p_{\bot}^{2}}{2\beta^{2}}\right)$
(16)
$\displaystyle\times\exp\left(-\frac{p_{z}^{2}+p_{\bot}^{2}}{2\beta^{2}}\right).$

with

$\frac{dp_{z}^{\prime}(^{\prime\prime})}{dx_{2}}=\frac{e_{1}^{\prime}(^{\prime\prime})e_{2}}{x_{1}x_{2}M_{0}^{\prime}(^{\prime\prime})}.$
(17)

For the practical calculation, the symbols $p_{z}$ and $p_{\bot}$ in Eq.
(II.2) should be associated with the superscript ′ for the initial meson and
′′ for the final meson.

In a concrete calculation in the light front quark model, one usually works in
the $q^{+}=0$ frame, where one only needs to consider the valence quark part,
and all other non-valence contributions e.g., the pair creations from the
vacuum (the so-called Z diagram) vanish. By making the contour integral over
$p_{1}^{\prime-}$ for the four-dimensional integration, one gets the above hat
quantities. These matrix elements will acquire a spurious $\omega$ dependence,
with $\omega=(2,0,0_{\perp})$ as a constant. Thus the covariance is lost. The
reason is due to the missing contribution in the
$p_{1}^{\prime-}=p_{1}^{\prime\prime-}=0$ region. Once such contribution is
correctly included, the covariance will be restored. W. Jaus finds an
effective way for inclusion of such zero mode contribution as the necessary
condition for the covariance, and the net results are Jaus:1999zv :

$\displaystyle\hat{p}^{\prime}_{1\mu}\rightarrow$ $\displaystyle
P_{\mu}A_{1}^{(1)}+q_{\mu}A_{2}^{(1)}$
$\displaystyle\hat{p}^{\prime}_{1\mu}\hat{p}^{\prime}_{1\nu}\rightarrow$
$\displaystyle
g_{\mu\nu}A_{1}^{(2)}+P_{\mu}P_{\nu}A_{2}^{(2)}+(P_{\mu}q_{\nu}+q_{\mu}P_{\nu})A_{3}^{(2)}$
$\displaystyle+q_{\mu}q_{\nu}A_{4}^{(2)}$
$\displaystyle\hat{p}^{\prime}_{1\mu}\hat{p}^{\prime}_{1\nu}\hat{p}^{\prime}_{1\alpha}\rightarrow$
$\displaystyle(g_{\mu\nu}P_{\alpha}+g_{\mu\alpha}P_{\nu}+g_{\nu\alpha}P_{\mu})A_{1}^{(3)}$
(18)
$\displaystyle+(g_{\mu\nu}q_{\alpha}+g_{\mu\alpha}q_{\nu}+g_{\nu\alpha}q_{\mu})A_{2}^{(3)}$
$\displaystyle+P_{\mu}P_{\nu}P_{\alpha}A_{3}^{(3)}$
$\displaystyle+(P_{\mu}P_{\nu}q_{\alpha}+P_{\mu}q_{\nu}P_{\alpha}+q_{\mu}P_{\nu}P_{\alpha})A_{4}^{(3)}$
$\displaystyle+(q_{\mu}q_{\nu}P_{\alpha}+q_{\mu}P_{\nu}q_{\alpha}+P_{\mu}q_{\nu}q_{\alpha})A_{5}^{(3)}$
$\displaystyle+q_{\mu}q_{\nu}q_{\alpha}A_{6}^{(3)},$

with those coefficients $A_{i}^{(j)}$ given by

$\begin{split}&A_{1}^{(1)}=\frac{x_{1}}{2},\,\,A_{2}^{(1)}=\frac{x_{1}}{2}-\frac{p^{\prime}_{\bot}\cdot
q_{\bot}}{q^{2}}\\\ &A_{1}^{(2)}=-p^{\prime
2}_{\bot}-\frac{(p^{\prime}_{\bot}\cdot
q_{\bot})^{2}}{q^{2}},\,\,A_{2}^{(2)}=(A_{1}^{(1)})^{2}\\\
&A_{3}^{(2)}=A_{1}^{(1)}A_{2}^{(1)},\,\,A_{4}^{(2)}=(A_{2}^{(1)})^{2}-\frac{A_{1}^{(2)}}{q^{2}}\\\
&A_{1}^{(3)}=A_{1}^{(1)}A_{1}^{(2)},\,\,A_{2}^{(3)}=A_{1}^{(2)}A_{2}^{(1)}\\\
&A_{3}^{(3)}=A_{1}^{(1)}A_{2}^{(2)},\,\,A_{4}^{(3)}=A_{2}^{(1)}A_{2}^{(2)}\\\
&A_{5}^{(3)}=A_{1}^{(1)}A_{4}^{(2)},\,\,A_{6}^{(3)}=A_{2}^{(1)}A_{4}^{(2)}-\frac{2}{q^{2}}A_{2}^{(1)}A_{1}^{(2)}\\\
\end{split}$ (19)

By matching with Eq. (8), the expressions of the form factors can be obtained
as follows:

$\displaystyle f_{m}^{a}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{2e}{3}\frac{N_{c}}{16\pi^{3}}\int
dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{A}^{\prime}h_{V}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}(-4)\left\\{\frac{1}{w_{A}^{\prime}}\Big{[}(m_{1}^{\prime}-m_{1}^{\prime\prime})(A_{3}^{(2)}-A_{4}^{(2)})\right.$
$\displaystyle+(m_{1}^{\prime}+m_{1}^{\prime\prime}-2m_{2})\times(A_{2}^{(2)}-A_{3}^{(2)})+m_{1}^{\prime}(A_{2}^{(1)}-A_{1}^{(1)})\Big{]}+A_{2}^{(2)}-A_{3}^{(2)}$
$\displaystyle\left.-\frac{1}{w_{A}^{\prime}w_{V}^{\prime\prime}}(2A_{2}^{(3)}-2A_{1}^{(3)})\right\\},$
$\displaystyle f_{p}^{a}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{2e}{3}\frac{N_{c}}{16\pi^{3}}\int
dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{A}^{\prime}h_{V}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}(-4)\left\\{\frac{1}{w_{V}^{\prime\prime}}\Big{[}(m_{1}^{\prime}-m_{1}^{\prime\prime})(A_{3}^{(2)}+A_{4}^{(2)}\right.$
$\displaystyle-
A_{2}^{(1)})+(m_{1}^{\prime}+m_{1}^{\prime\prime}+2m_{2})\times(A_{2}^{(2)}+A_{3}^{(2)}-A_{1}^{(1)})-m_{1}^{\prime}(A_{1}^{(1)}+A_{2}^{(1)}$
$\displaystyle\left.-1)\Big{]}+A_{1}^{(1)}-A_{2}^{(2)}-A_{3}^{(2)}-\frac{1}{w_{A}^{\prime}w_{V}^{\prime\prime}}(2A_{1}^{(3)}+2A_{2}^{(3)}-2A_{1}^{(2)})\right\\},$
$\displaystyle f_{6}^{a}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{2e}{3}\frac{N_{c}}{16\pi^{3}}\int
dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{A}^{\prime}h_{V}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}(-4)\left\\{\frac{1}{w_{A}^{\prime}}(m_{1}^{\prime}+m_{1}^{\prime\prime}-2m_{2})A_{1}^{(2)}\right.$
(20)
$\displaystyle+\frac{1}{w_{V}^{\prime\prime}}(m_{1}^{\prime}+m_{1}^{\prime\prime}+2m_{2})A_{1}^{(2)}-\frac{1}{4}(1-2A_{2}^{(1)})\Big{[}-q^{2}+\hat{N}_{1}^{\prime}+\hat{N}_{1}^{\prime\prime}$
$\displaystyle\left.+(m_{1}^{\prime}-m_{1}^{\prime\prime})^{2}\Big{]}-A_{2}^{(1)}(m_{1}^{\prime\prime}m_{2}-m_{1}^{\prime}m_{2})-m_{1}^{\prime}m_{2}\phantom{\frac{1}{w_{A}^{\prime}}}\right\\}.$

The form factors corresponding to Fig. 1(b), i.e., the expressions of
$f_{m}^{b}(q^{2}),f_{p}^{b}(q^{2}),f_{6}^{b}(q^{2})$, can be obtained from Eq.
(II.2) by the interchanges

$m_{1}^{\prime}\rightarrow m_{2}^{\prime},\,m_{1}^{\prime\prime}\rightarrow
m_{2}^{\prime\prime},\,m_{2}\rightarrow m_{1}.$ (21)

In practice, all these quark masses are the mass of the charm quark, and the
contribution of Fig. 1(b) is the same as that of Fig. 1(a). The form factors
will finally be

$\displaystyle
f_{i}(q^{2})=f_{i}^{a}(q^{2})+f_{i}^{b}(q^{2})=2f_{i}^{a}(q^{2}),$ (22)

with $i$ denoting the indices $m,\,p$, and 6. For the radiative decay
considered here, only the form factor values at $q^{2}=0$ are relevant.

###  II.3 Decay width

The decay width will be conveniently expressed in the helicity basis. We then
define the helicity amplitude as
$A_{\lambda^{\prime}\lambda^{\prime\prime}\lambda_{\gamma}}$, with
$\lambda^{\prime},\,\lambda^{\prime\prime},\,\lambda_{\gamma}$ denoting the
helicity of $X(3872)$, $J/\psi$ (or $\psi^{\prime}$), and the photon,
respectively. In the rest frame of the initial meson $X(3872)$, we can obtain
the explicit representations of the momenta and polarization vectors as
follows Dubnicka:2011mm :

$\displaystyle P^{\prime\mu}=(M^{\prime},0,0,0),\quad
P^{\prime\prime\mu}=(E_{1},0,0,|\bm{q}|),$ $\displaystyle
q^{\mu}=(|\bm{q}|,0,0,-|\bm{q}|),$
$\displaystyle\epsilon^{\prime\mu}_{\pm}=\frac{1}{\sqrt{2}}(0,\mp
1,-i,0),\,\,\epsilon^{\prime\mu}_{0}=(0,0,0,1),$
$\displaystyle\epsilon^{\prime\prime\mu}_{\pm}=\frac{1}{\sqrt{2}}(0,\mp
1,-i,0),\,\,\epsilon^{\prime\prime\mu}_{0}=\frac{1}{M^{\prime\prime}}(|\bm{q}|,0,0,E_{1}),$
$\displaystyle\epsilon^{\mu}_{\pm}(\gamma)=\frac{1}{\sqrt{2}}(0,\pm 1,-i,0),$
(23)

where $|{\bm{q}}|=\frac{M^{\prime 2}-M^{\prime\prime 2}}{2M^{\prime}}$ and
$E_{1}=\frac{M^{\prime 2}+M^{\prime\prime 2}}{2M^{\prime}}$ is the energy of
the vector meson. Due to the conservation of angular momentum,
$\lambda^{\prime}=\lambda^{\prime\prime}-\lambda_{\gamma}$, and the
nonvanishing amplitudes for $X(3872)\to J/\psi\gamma,\,\psi^{\prime}\gamma$
decay will be

$\displaystyle A_{+0-}$ $\displaystyle=$
$\displaystyle-i\frac{M^{\prime}|\bm{q}|}{M^{\prime\prime}}(f_{6}+2M^{\prime}|\bm{q}|f_{p}),$
$\displaystyle A_{0++}$ $\displaystyle=$ $\displaystyle
i|\bm{q}|(f_{6}+2M^{\prime}|\bm{q}|f_{m}),$ $\displaystyle A_{-0+}$
$\displaystyle=$ $\displaystyle-A_{+0-},\quad A_{0--}=-A_{0++}.$ (24)

The last two equations follow from the parity relation and have also been
verified by calculation. Obviously, we can also adopt the convention of the
polarization vectors used in Refs. Zhang:2020dla ; Ivanov:2019nqd . The
difference lies only in the definition of the momentum direction and the
resulting change in the polarization vector forms. We verified that they give
identical physical results. Stated differently, this transition includes the
$E1$ and $M2$ types, which are characterized by the $\bm{q}$ and $\bm{q}^{2}$
for the near-threshold behavior for the amplitude. Of course, both $S$ and $D$
waves are included, and combined together in Eq. (II.3). The helicity relation
$\lambda^{\prime}=\lambda^{\prime\prime}-\lambda_{\gamma}$ does not imply $S$
wave solely, since the projection of any orbital angular momentum onto the
linear momentum vanishes.

The decay width can be calculated by

$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle\frac{|\bm{q}|}{8\pi
M^{\prime
2}}\left(|A_{+0-}|^{2}+|A_{-0+}|^{2}+|A_{0--}|^{2}+|A_{0++}|^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{|\bm{q}|^{3}}{\pi}\left(\frac{f_{6}+2M^{\prime}|\bm{q}|f_{p}}{4M^{\prime\prime
2}}+\frac{f_{6}^{2}}{4M^{\prime
2}}+\frac{f_{6}f_{m}|\bm{q}|}{M^{\prime}}+f_{m}^{2}|\bm{q}|^{2}\right).$

The dimension of the amplitude
$A_{\lambda^{\prime}\lambda^{\prime\prime}\lambda_{\gamma}}$ is [mass]1, while
the form factors $f_{6},f_{p}$, and $f_{m}$ have dimensions of [mass]0,
[mass]-2 and [mass]-2, respectively.

##  III numerical results and discussions

By using Eqs. (II.2) and (II.3) in Sec. II, we will calculate the width of
radiative decay. We take the constituent quark mass $m_{c}=1.4$ GeV
Cheng:2003sm ; Jaus:1989au ; Jaus:1996np  for calculation. In the wave
function of a meson, there is a parameter $\beta$ that needs to be determined.
For vector mesons (V) $J/\psi$ and $\psi^{\prime}$, we fix the $\beta$ value
by their decay constants, which are extracted from the decay width to the
$e^{+}e^{-}$ pair through the following equation:

$\Gamma(V\rightarrow
e^{+}e^{-})=\frac{4\pi}{3}\frac{4}{9}\alpha^{2}\frac{f_{V}^{2}}{M^{\prime}},$
(26)

where $\alpha$ is the fine structure constant, $f_{V}$ is the decay constant,
and $M^{\prime}$ is the mass of the vector meson. By taking
$\text{Br}(J/\psi\rightarrow e^{+}e^{-})=(5.971\pm 0.032)\%$,
$\text{Br}(\psi^{\prime}\rightarrow e^{+}e^{-})=(7.93\pm 0.17)\times 10^{-3}$,
$\Gamma(J/\psi)=(92.6\pm 1.7)$ keV and $\Gamma(\psi(2S))=(294\pm 8)$ keV
Workman:2022ynf , we can obtain the decay constants $f_{J/\psi}=415.49$ MeV
and $f_{\psi^{\prime}}=294.35$ MeV as our central values. The uncertainties
are very small. The formula for the decay constant of vector mesons in the
LFQM is given by Cheng:2003sm

$\displaystyle f_{V}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{4\pi^{3}M^{\prime}}\int
dxd^{2}p^{\prime}_{\bot}\frac{h_{V}^{\prime}}{x(1-x)(M^{\prime
2}-M_{0}^{\prime 2})}$ (27) $\displaystyle\times\Big{[}xM_{0}^{\prime
2}-m_{1}^{\prime}(m_{1}^{\prime}-m_{2})-p_{\bot}^{\prime 2}$
$\displaystyle\quad+\frac{m_{1}^{\prime}+m_{2}}{M_{0}^{\prime}+m_{1}^{\prime}+m_{2}}p_{\bot}^{\prime
2}\Big{]},$

from which we fix the parameters $\beta_{J/\psi}=0.631$ GeV and
$\beta_{\psi^{\prime}}=0.487$ GeV.

Nowadays the PDG reported the $X(3872)$ width as $\Gamma(X(3872)=(1.19\pm
0.21)$ MeV Workman:2022ynf  based on the LHCb measurements LHCb:2020fvo ;
LHCb:2020xds . But this corresponds to the Breit-Weigner (BW) width. The BW
parametrization is not an appropriate parametrization for a very near-
threshold state. LHCb collaboration has discussed their pole searches
LHCb:2020xds  and very recently BESIII did such activity too BESIII:2023hml .
We will take the central value of 0.4 MeV for the X(3872) pole width
BESIII:2023hml  for illustration despite the large uncertainties in the pole
parameters, and this width value also agrees with the reported full width at
the half maximum for the line shape BESIII:2023hml . Taking the branching
ratio values $\text{Br}(X(3872)\rightarrow\psi^{\prime}\gamma)=(4.5\pm 2.0)\%$
and $\text{Br}(X(3872)\rightarrow J/\psi\gamma)=(8\pm 4)\times 10^{-3}$
Workman:2022ynf , we obtain
$\Gamma_{\text{exp}}(X(3872)\rightarrow\psi^{\prime}\gamma)=(1.8\pm 0.8)\times
10^{-2}$ MeV and $\Gamma_{\text{exp}}(X(3872)\to J/\psi\gamma)=(3.2\pm
1.6)\times 10^{-3}$ MeV as the “experimental” values for the partial decay
widths. Those uncertainties purely come from the ones for the branching
ratios. From the partial decay width to $\psi^{\prime}\gamma$, we are able to
fix the $\beta$ value for $X(3872)$, $\beta_{X(3872)}$, as
$0.56^{+0.04}_{-0.03}$ GeV. Then the theoretical value for $\Gamma(X(3872)\to
J/\psi\gamma)$ will be predicted to $(9.1^{+1.7}_{-1.5})\times 10^{-1}$ MeV,
which deviates the aforementioned “experimental” number too much.
Consequently, the scenario of a pure $c\overline{c}$ assignment for $X(3872)$
will encounter difficulty in reconciling the widths to $J/\psi\gamma$ and
$\psi^{\prime}\gamma$.

Here we clarify more on the uncertainty for our result. The central value of
$\beta_{X(3872)}$ is required to reproduce the central value of the
$\Gamma(X(3872)\rightarrow\psi^{\prime}\gamma)$, while its asymmetric errors
are obtained by sampling $\Gamma(X(3872)\rightarrow\psi^{\prime}\gamma)$
within one standard deviation range. From the produced set of
$\beta_{X(3872)}$ numbers, the maximum and minimum values are picked out.
Obviously, the value of $\Gamma(X(3872)\to J\psi\gamma)$ takes into account
the uncertainty of $\beta_{X(3872)}$ through the set of the numbers for
$\beta_{X(3872)}$. Moreover, we also associate $\Gamma(X(3872)\to
J\psi\gamma)$ with another uncertainty by a roughly 10% of its central value
due to the neglected contribution from the $\gamma$-quark-antiquark-meson
vertex (see the footnote).

The above results show that assigning $X(3872)$ by a $c\bar{c}$ state is much
disfavored from the viewpoint of its radiative decays. However, other
configurations are possible. For example, there is indeed a calculation of
radiative decay based on the tetraquark assumption in Ref. Dubnicka:2011mm .
There they find a consistency of their theoretical prediction of
$\Gamma(X(3872)\to J/\psi\gamma)/\Gamma(X(3872)\to J/\psi 2\pi)$ with the
available experimental measurements, by choosing a reasonable value of the
size parameter of the $X(3872)$ meson (as a parameter of their model). But
note that the calculated quantity therein is the decay width ratio of
$\Gamma(X(3872)\to J/\psi\gamma)/\Gamma(X(3872)\to J/\psi 2\pi)$, and here we
concern the ratio of $\Gamma(X(3872)\to
J/\psi\gamma)/(X(3872)\to\psi^{\prime}\gamma)$. The calculation from the
$D\bar{D}^{*}$ molecular assumption was done in Refs. Aceti:2012cb ;
Guo:2014taa . The calculation from the coupled channels
$D\bar{D}^{*}+c\bar{c}$ was done in Ref. Badalian:2012jz . Future works along
the line of investigating the nature of $X(3872)$ through its radiative decays
are still meaningful and encouraged.

##  IV Conclusion

The experimental measurements on the $X(3872)$ decay width and branching
ratios have made great progress. Assuming that $X(3872)$ is regarded as a
conventional charmonium with the quantum number $J^{PC}=1^{++}$, we calculated
the transition form factors for $X(3872)$ decaying to a photon and $J/\psi$
($\psi^{\prime}$) in the framework of the covariant light-front quark model.
In this approach, we need to determine the values of $\beta$ appearing in the
wave functions of mesons, where $\beta_{J/\psi}$ and $\beta_{\psi^{\prime}}$
are fixed by their decay constants, extracted from their partial decay widths
to the $e^{+}e^{-}$ pair. $\beta_{X(3872)}$ will be fixed by the width of
$\Gamma(X(3872)\rightarrow\psi^{\prime}\gamma)$ reported by the PDG. In this
manner, we can cleanly predict the decay width $\Gamma(X(3872)\rightarrow
J/\psi\gamma)$. It turns out that the predicted partial decay width of
$\Gamma(X(3872)\rightarrow J/\psi\gamma)$ is too large to assign $X(3872)$ as
a traditional charmonium state. Or stated differently, the probability that
$X(3872)$ is a pure $c\overline{c}$ resonance is rather small.

## Acknowledgment

We thank Prof. Yu-Bing Dong, Prof. Xiang Liu for discussions, and Prof. V. O.
Galkin for his careful reading and suggestions. Prof. Hong-Wei Ke should be
especially acknowledged for his patient discussions in the early stage of our
work. This work is supported by the National Natural Science Foundation of
China (NSFC) under Project No. 12275023.

## References

  * [1] Johann Haidenbauer, Christoph Hanhart, Xian-Wei Kang, and Ulf-G. Meißner.  Origin of the structures observed in $e^{+}e^{-}$ annihilation into multipion states around the $\bar{p}p$ threshold.  Phys. Rev. D, 92(5):054032, 2015.
  * [2] Feng-Kun Guo, Xiao-Hai Liu, and Shuntaro Sakai.  Threshold cusps and triangle singularities in hadronic reactions.  Prog. Part. Nucl. Phys., 112:103757, 2020.
  * [3] S. K. Choi et al.  Observation of a narrow charmonium-like state in exclusive $B^{\pm}\to K^{\pm}\pi^{+}\pi^{-}J/\psi$ decays.  Phys. Rev. Lett., 91:262001, 2003.
  * [4] R Aaij et al.  Determination of the X(3872) meson quantum numbers.  Phys. Rev. Lett., 110:222001, 2013.
  * [5] Roel Aaij et al.  Quantum numbers of the $X(3872)$ state and orbital angular momentum in its $\rho^{0}J\psi$ decay.  Phys. Rev. D, 92(1):011102, 2015.
  * [6] Medina Ablikim et al.  Study of Open-Charm Decays and Radiative Transitions of the $X(3872)$.  Phys. Rev. Lett., 124(24):242001, 2020.
  * [7] Roel Aaij et al.  Study of the $\psi_{2}(3823)$ and $\chi_{c1}(3872)$ states in $B^{+}\rightarrow\left(J\psi\pi^{+}\pi^{-}\right)K^{+}$ decays.  JHEP, 08:123, 2020.
  * [8] M. Ablikim et al.  Study of $e^{+}e^{-}\to\gamma\omega J/\psi$ and Observation of $X(3872)\to\omega J/\psi$.  Phys. Rev. Lett., 122(23):232002, 2019.
  * [9] M. Ablikim et al.  Observation of the decay $X(3872)\to\pi^{0}\chi_{c1}(1P)$.  Phys. Rev. Lett., 122(20):202001, 2019.
  * [10] V. Bhardwaj et al.  Observation of $X(3872)\to J/\psi\gamma$ and search for $X(3872)\to\psi^{\prime}\gamma$ in B decays.  Phys. Rev. Lett., 107:091803, 2011.
  * [11] Roel Aaij et al.  Evidence for the decay $X(3872)\rightarrow\psi(2S)\gamma$.  Nucl. Phys. B, 886:665–680, 2014.
  * [12] R. L. Workman et al.  Review of Particle Physics.  PTEP, 2022:083C01, 2022.
  * [13] Bernard Aubert et al.  Evidence for $X(3872)\to\psi_{2S}\gamma$ in $B^{\pm}\to X_{3872}K^{\pm}$ decays, and a study of $B\to c\bar{c}\gamma K$.  Phys. Rev. Lett., 102:132001, 2009.
  * [14] R. Aaij et al.  Study of the lineshape of the $\chi_{c1}(3872)$ state.  Phys. Rev. D, 102(9):092005, 2020.
  * [15] Xian-Wei Kang and J. A. Oller.  Different pole structures in line shapes of the $X(3872)$.  Eur. Phys. J. C, 77(6):399, 2017.
  * [16] Yan Li, Feng-Kun Guo, Jin-Yi Pang, and Jia-Jun Wu.  Generalization of Weinberg’s compositeness relations.  Phys. Rev. D, 105(7):L071502, 2022.
  * [17] V. Baru, J. Haidenbauer, C. Hanhart, Yu. Kalashnikova, and Alexander Evgenyevich Kudryavtsev.  Evidence that the a(0)(980) and f(0)(980) are not elementary particles.  Phys. Lett. B, 586:53–61, 2004.
  * [18] Tomona Kinugawa and Tetsuo Hyodo.  Compositeness of $T_{cc}$ and $X(3872)$ with decay and coupled-channel effects.  3 2023.
  * [19] Feng-Kun Guo, Christoph Hanhart, Ulf-G. Meißner, Qian Wang, Qiang Zhao, and Bing-Song Zou.  Hadronic molecules.  Rev. Mod. Phys., 90(1):015004, 2018.  [Erratum: Rev.Mod.Phys. 94, 029901 (2022)].
  * [20] R. L. Workman et al.  Review of Particle Physics.  PTEP, 2022:083C01, 2022.
  * [21] Wolfgang Jaus.  Relativistic constituent quark model of electroweak properties of light mesons.  Phys. Rev. D, 44:2851–2859, 1991.
  * [22] W. Jaus.  Semileptonic Decays of B and d Mesons in the Light Front Formalism.  Phys. Rev. D, 41:3394, 1990.
  * [23] Wolfgang Jaus.  Semileptonic, radiative, and pionic decays of B, B* and D, D* mesons.  Phys. Rev. D, 53:1349, 1996.  [Erratum: Phys.Rev.D 54, 5904 (1996)].
  * [24] W. Jaus.  Covariant analysis of the light front quark model.  Phys. Rev. D, 60:054026, 1999.
  * [25] Hai-Yang Cheng, Chun-Khiang Chua, and Chien-Wen Hwang.  Covariant light front approach for s wave and p wave mesons: Its application to decay constants and form-factors.  Phys. Rev. D, 69:074025, 2004.
  * [26] Ho-Meoyng Choi.  Decay constants and radiative decays of heavy mesons in light-front quark model.  Phys. Rev. D, 75:073016, 2007.
  * [27] Yan-Liang Shi.  Revisiting radiative decays of $1^{+-}$ heavy quarkonia in the covariant light-front approach.  Eur. Phys. J. C, 77(4):253, 2017.
  * [28] Qin Chang, Li-Ting Wang, and Xiao-Nan Li.  Form factors of $V^{\prime}\to V^{\prime\prime}$ transition within the light-front quark models.  JHEP, 12:102, 2019.
  * [29] Hong-Wei Ke and Xue-Qian Li.  What do the radiative decays of X(3872) tell us.  Phys. Rev. D, 84:114026, 2011.
  * [30] Gurjav Ganbold, Thomas Gutsche, Mikhail A. Ivanov, and Valery E. Lyubovitskij.  Radiative transitions of charmonium states in the covariant confined quark model.  Phys. Rev. D, 104(9):094048, 2021.
  * [31] Chien-Wen Hwang.  Study of quark distribution amplitudes of 1S and 2S heavy quarkonium states.  Eur. Phys. J. C, 62:499–509, 2009.
  * [32] Stanislav Dubnicka, Anna Z. Dubnickova, Mikhail A. Ivanov, Juergen G. Koerner, Pietro Santorelli, and Gozyal G. Saidullaeva.  One-photon decay of the tetraquark state $X(3872)\to\gamma+J/\psi$ in a relativistic constituent quark model with infrared confinement.  Phys. Rev. D, 84:014006, 2011.
  * [33] Lu Zhang, Xian-Wei Kang, Xin-Heng Guo, Ling-Yun Dai, Tao Luo, and Chao Wang.  A comprehensive study on the semileptonic decay of heavy flavor mesons.  JHEP, 02:179, 2021.
  * [34] Mikhail A. Ivanov, Jürgen G. Körner, Jignesh N. Pandya, Pietro Santorelli, Nakul R. Soni, and Chien-Thang Tran.  Exclusive semileptonic decays of D and Ds mesons in the covariant confining quark model.  Front. Phys. (Beijing), 14(6):64401, 2019.
  * [35] Medina Ablikim et al.  A coupled-channel analysis of the $X(3872)$ lineshape with BESIII data.  9 2023.
  * [36] F. Aceti, R. Molina, and E. Oset.  The $X(3872)\to J/\psi\gamma$ decay in the $D\bar{D}^{*}$ molecular picture.  Phys. Rev. D, 86:113007, 2012.
  * [37] Feng-Kun Guo, C. Hanhart, Yu. S. Kalashnikova, Ulf-G. Meißner, and A. V. Nefediev.  What can radiative decays of the X(3872) teach us about its nature?  Phys. Lett. B, 742:394–398, 2015.
  * [38] A. M. Badalian, V. D. Orlovsky, Yu. A. Simonov, and B. L. G. Bakker.  The ratio of decay widths of X(3872) to $\psi^{\prime}\gamma$ and $J/\psi\gamma$ as a test of the X(3872) dynamical structure.  Phys. Rev. D, 85:114002, 2012.


This is a preprint of the following chapter: Nicolás-Carlock J.R. & Luna-Pla
I., Corruptomics, published in Corruption Networks, edited by Granados O.M.,
Nicolás-Carlock J.R., 2021, Springer, reproduced with permission of Springer.
The final authenticated version is available online at
https://doi.org/10.1007/978-3-030-81484-7_9

# Corruptomics

José R. Nicolás-Carlock1,∗ & Issa Luna-Pla1 1Institute of Legal Research,
National Autonomous University of Mexico, Mexico<EMAIL_ADDRESS>
###### Abstract.

Corruption studies must evolve to match the complexity of the modern world.
Here, we present three main problems in corruption analysis that need to be
address: the complexity of the corruption phenomenon itself and its context,
the complexity of the analytical description, and the complexity of the
perspectives that different disciplines bring to the table. In this regard, we
argue that the interdisciplinary framework of complex systems and network
science represent a promising analytical approach to move forward in this
endeavor. Furthermore, current research efforts in this direction indicate the
dawn of a new interdisciplinary discipline for corruption studies in the 21st
century.

###### Key words and phrases:

corruption analysis, network science, complexity science

> “Injustice anywhere is a threat to justice everywhere. We are caught in an
> inescapable network of mutuality, tied in a single garment of destiny.
> Whatever affects one directly, affects all indirectly.” - Martin Luther King
> Jr.

Corruption is one of the most prominent global policy challenges of the 21st
century. During the last few decades, corruption not only has been extensively
addressed in the public policy arena but also, it has become a very active
academic research field [1, 2]. As an academic subject, corruption is mostly
regarded as a fundamental societal problem that researchers from diverse
disciplinary traditions aim to address along four main interdependent axes:
conceptualizations and definitions, measuring methods and techniques,
modelling of causes and consequences, and control or tackling strategies.
However, although relevant advances have been made, the challenge to design
theoretical and technical frameworks that are able to handle the complexity of
this phenomenon remains highly contested [3]. This is due to three main
problems in corruption analysis: (I) the complexity of the nature of the
phenomenon itself and its context, (II) the complexity of the analytical
description, and (III) the complexity of the different perspectives that each
discipline brings to the table.

I. The complexity of the nature of the phenomenon itself and its context.
Human beings are complex and corruption is inherently hard to tackle due to
the complex nature of human behavior. As any other human activity, corruption
occurs within the intricate structure and dynamics of the social, economic and
political systems of society. As such, corrupt behavior manifests as a non-
separable activity that tends to be hidden and interwoven within the multiple
activities that could be deem as non-corrupt for a given setting, from the
micro group dynamics that take place within the structure of government
institutions, small or big corporations, and civil life, to the macro
interactions that take place among them. In addition, the systems that
characterize our complex societies are not independent of each other but
constitute a system of systems whose structure and dynamics are always
evolving in response to changes in the corresponding social and regulation
context. Notably, global dynamics have radically changed in the last few
years. The flow of people, materials, and information have remarkably
increased the levels of interactions at different spatial and administrative
scales and consequently, the interdependencies among social, economic and
political systems have become stronger. Our heterogeneous world is more
connected than ever. This not only represents great cooperation opportunities
for development but also highly systemic threats at regional and global
scales, what happens in one place or sector can have effects on other
seemingly unrelated ones, putting the analysis of corruption on a whole
another level [4, 5].

II. The complexity of the analytical description. The analysis of corruption
is complex and has evolved over a long period of time. Nowadays, this analysis
is done under an international consensus that has put individual behavior and
indiscretions at the center of modern corruption thinking [3]. Values, norms
and ideas, that still play an important role in our understanding of what is
acceptable or not in society, vary from place to place, and from time to time,
making an objective understanding of moral and ethical issues a great
challenge [6]. In an effort to be more objective, the analysis of corruption
has opted for a more pragmatic and empirical approach.

On the conceptual dimension, contemporary thinking is dominated by four main
approaches: legal definitions, the “abuse of entrusted power” criterion,
economic or business-oriented, and the so-called “legal” corruption. Among
these, the first one has become the most popular since its adoption from
Transparency International, the Organization for Economic Cooperation and
Development, the World Bank and the International Monetary Fund. However, the
subjective elements embedded in this definition, such as what constitutes
“abuse” and its bias towards the public sector, make it object of continuous
critic and debate. On the measuring front, aggregated indices rise as the most
popular as they have provided an overall and broad picture of global
corruption. These still face criticism due to changes in their methodology or
the perception/experience-based analysis since that makes the interpretation
or comparison of results difficult. This has lead to the creation of other
promising indicators that address more specific matters, as well as more
sophisticated approaches based on proxies to corruption that have the capacity
to describe macro features from micro data. On the modelling challenge, the
consensus is low on what are the general causes of corruption. The different
models based on structural forces, rational agents, principals or
discretionary criteria provide insight for some settings but not all,
therefore, these too are not free from debate due to the emergence of
interesting collective dilemmas and the clash of social realities across the
globe. On the control arena, the strategies to tackle corruption are not
universal and, in direct relation with the causes of the problem at hand,
strategies depend on the specific context. National regulations and
international instruments aim to solve this problem but their results are not
clear and there’s no catalogue of solutions that work for all places at all
times [3].

III. The complexity of different disciplinary perspectives. As multifaceted as
the phenomenon is, corruption analysis has been greatly enriched by the
insights of researchers from different disciplines and schools of thought. To
this day, these are mostly from political science, economy, sociology,
anthropology, or law. In the effort to come up with a general
interdisciplinary corruption theory, these researchers have to deal not only
with the complex nature of phenomenon itself, the inherent heterogeneity and
complexity of the systems where it takes place, or the subtleties of the four
dimensions that comprise modern analysis, but also, with the clash of
different ideas and perspectives about social reality that each discipline
brings to the table [7]. This is a complex scenario where consensus on
fundamental aspects is hard to achieve and, therefore, a general
interdisciplinary theory of corruption has remained elusive [3].

The challenge is great and in order to move forward, it’s imperative to adopt
new ideas and perspectives that enable us to handle and embrace the complexity
of our world instead of avoid it. Here, complexity or complex systems science
represent a promising approach in this endeavour:

  * •

First, we must understand that the complexity of society is nothing but the
result of our collective doing as we become and create the systems that shape
our social, economic and political environments. In other words, we not only
are but also create the complex networks in which corruption takes place.
Therefore, our connections and interactions at the different scales, sectors
and regions keep valuable information that can be tapped, modelled and studied
in order to understand the principles that govern such systems and give us the
opportunity to develop strategies and interventions tailored to the structure
and dynamics of the problem at hand. In that regard, complexity science is an
interdisciplinary approach to the study of collective phenomena in natural,
social and technical systems that has been successful in the analysis of the
structure and dynamics of such systems in terms of the relationships among
their parts and their environment [8, 9]. The main idea in complex systems is
that a collection of interacting components behaves in way not predicted by
the components in isolation or disconnected. Interactions and dependencies
matter more than the nature of the parts. Therein, the collection of
interacting parts is best understood as a whole, rather than disconnected. As
such, when corruption behavior manifest as a non-separable activity that tends
to be hidden and intertwined among the multiple activities that occur within
the structure and dynamics of social, economic, political and technical
systems, then, activities that could be deemed as corrupt are best understood
systemically, this is, from the collective behavior and features of connected
individuals or organizations acting as a whole, rather than from their
particular characteristics in isolation.

  * •

Second, given the multifaceted nature of corruption, the importance of a
comprehensive and general analytical framework – not necessarily universal but
that unifies the different disciplinary perspectives – and that encompasses
proper conceptualizations and definitions of corrupt practice cannot be
overstated, given that definitions determine what gets modelled and what
researchers look for in data, in such a way that unsuited definitions can
cause misleading measurements, mistaken interpretations of causes and
consequences, and ultimately inappropriate policy suggestions. Achieving a
unified framework seems daunting, but here again, complexity science presents
itself as an interesting and relevant example on this matter, as a discipline
that on its own is dealing with a similar challenge [10]. For that, let us
recall that complexity science is an interdisciplinary approach to the study
of natural, social and technical systems that has created an still-evolving
analytical framework by drawing concepts and tools from disciplines such as
physics, chemistry, biology, ecology, sociology, mathematics and computer
science [11]. In this way, complex systems are often studied in terms of
networks, self-organization, evolution, non-linearity, scaling, and emergence.
Remarkably, this framework has been applied to different systems and settings
leading to relevant insights into crime, terrorism, war, disease spreading,
financial markets, democracy and other social subjects [12, 13, 14, 15]. In
the case of corruption, the concepts and tools of complexity and networks have
been applied to tackle specific matters, such as the conceptualization of
corruption as a networked phenomenon [16], measuring and modelling of
political corruption [17, 18, 19], corruption in public procurement and
corruption scandals [20, 21, 22, 23], and ways to identify, counteract and
control phenomena such as cartel formation, money laundering and tax evasion
by means of data and artificial intelligence [24, 25, 26].

  * •

Lastly, anti-corruption is not an endeavor of isolated and disconnected
individuals. The efforts made in the public policy arena have shown that
groups of people, private institutions, and governments must learn to
cooperate and work purposely in order for any strategy to be effective and
successful. In this world, we are not only connected at multiple scales but
also we are interdependent members of this highly complex system known as
society: whatever affects one directly, affects all indirectly. Complex
corruption networks are tackled with complex anti-corruption networks, and
corruption studies and anti-corruption strategies must evolve to match the
complexity of our reality.

Modern technological advances by themselves are not sufficient to suggest that
we have the upper-hand in the fight against corruption since, in the same way
those emerging technologies allow for greater transparency, cooperation and
development, they could also allow for more sophisticated corruption
mechanisms and challenges. It comes down to us to make the best out of the
modern tools and achievements obtained so far in the long history of
corruption analysis and move forward. The research done so far to address
corruption, such as the one presented all along the chapters of this book,
constitute an effort in this direction, as well as example of the scope,
possibilities, and potential of a new approach to corruption analysis that is
open to perspectives, methods and disciplines that go beyond the traditional
schools of thought in order to give place to something different and hopefully
useful.

We might not be any closer to a corruption free era, but we are positive that
we are at the dawn of a new paradigm in corruption and anti-corruption studies
that could take us closer to that goal. Let us consider this an opportunity to
create a new discipline that embraces the complexities of the phenomenon, that
takes into account the structure and dynamics of the networks that shape our
societies, that takes advantage of technology, data and empirical evidence,
that pulls insight from the full spectrum of the prism of interdisciplinary
science, and finally, that dares to take corruption studies closer to an exact
science. Let us consider and know henceforth this new discipline as
corruptomics: corruption analysis for the 21st century.

## References

  * [1] S. Rose-Ackerman and B. J. Palifka, Corruption and government: Causes, consequences, and reform.  Cambridge university press, 2016.
  * [2] A. Mungiu-Pippidi and P. M. Heywood, A research agenda for studies of corruption.  Edward Elgar Publishing, 2020.
  * [3] D. Hough, Analysing corruption.  Agenda Publishing Limited, 2017.
  * [4] D. Helbing, “Globally networked risks and how to respond,” Nature, vol. 497, no. 7447, pp. 51–59, 2013.
  * [5] J. Balsa-Barreiro, A. Vié, A. J. Morales, and M. Cebrián, “Deglobalization in a hyper-connected world,” Palgrave Communications, vol. 6, no. 1, pp. 1–4, 2020.
  * [6] V. Capraro and M. Perc, “Grand challenges in social physics: in pursuit of moral behavior,” Frontiers in Physics, vol. 6, p. 107, 2018.
  * [7] G. De Graaf, P. Von Maravic, and P. Waagenar, The good cause: Theoretical perspectives on corruption.  Verlag Barbara Budrich, 2010.
  * [8] Y. Bar-Yam, Making things work: solving complex problems in a complex world.  Knowledge Industry, 2004.
  * [9] A. L. Barabási, Network science.  Cambridge University Press, 2016.
  * [10] J. Ladyman and K. Wiesner, What Is a Complex System? Yale University Press, 2020.
  * [11] M. Mitchell, Complexity: A guided tour.  Oxford University Press, 2009.
  * [12] D. Helbing, D. Brockmann, T. Chadefaux, K. Donnay, U. Blanke, O. Woolley-Meza, M. Moussaid, A. Johansson, J. Krause, S. Schutte, et al., “Saving human lives: What complexity science and information systems can contribute,” Journal of statistical physics, vol. 158, no. 3, pp. 735–781, 2015.
  * [13] G. Caldarelli, “A perspective on complexity and networks science,” Journal of Physics: Complexity, vol. 1, no. 2, p. 021001, 2020.
  * [14] T. Eliassi-Rad, H. Farrell, D. Garcia, S. Lewandowsky, P. Palacios, D. Ross, D. Sornette, K. Thébault, and K. Wiesner, “What science can do for democracy: a complexity science approach,” Humanities and Social Sciences Communications, vol. 7, no. 1, pp. 1–4, 2020.
  * [15] J. Kertész and J. Wachs, “Complexity science approach to economic crime,” Nature Reviews Physics, 2020.
  * [16] W. Slingerland, Network corruption: when social capital becomes corrupted.  Eleven International Publishing, 2018.
  * [17] H. V. Ribeiro, L. G. A. Alves, A. F. Martins, E. K. Lenzi, and M. Perc, “The dynamical structure of political corruption networks,” Journal of Complex Networks, vol. 6, pp. 989–1003, 01 2018.
  * [18] T. Colliri and L. Zhao, “Analyzing the bills-voting dynamics and predicting corruption-convictions among brazilian congressmen through temporal networks,” Scientific Reports, vol. 9, no. 1, p. 16754, 2019.
  * [19] P. C. Solimine, “Political corruption and the congestion of controllability in social networks,” Applied Network Science, vol. 5, no. 1, pp. 1–20, 2020\.
  * [20] S. I. Fierăscu, The networked phenomenon of state capture.  PhD thesis, Central European University, 2017.
  * [21] I. Luna-Pla and J. Nicolás-Carlock, “Corruption and complexity: a scientific framework for the analysis of corruption networks,” Applied Network Science, vol. 5, no. 1, p. 13, 2020.
  * [22] J. Wachs, M. Fazekas, and J. Kertész, “Corruption risk in contracting markets: a network science perspective,” International Journal of Data Science and Analytics, 2020.
  * [23] M. Fazekas and J. Wachs, “Corruption and the network structure of public contracting markets across government change,” Politics and Governance, vol. 8, no. 2, pp. 153–166, 2020.
  * [24] J. Wachs and J. Kertész, “A network approach to cartel detection in public auction markets,” Scientific reports, vol. 9, no. 1, pp. 1–10, 2019.
  * [25] O. Garcia-Bedoya, O. Granados, and J. C. Burgos, “Ai against money laundering networks: the colombian case,” Journal of Money Laundering Control, 2020\.
  * [26] M. Zumaya, R. Guerrero, E. Islas, O. Pineda, C. Gershenson, G. Iñiguez, and C. Pineda, “Identifying tax evasion in mexico with tools from network science and machine learning,” arXiv preprint arXiv:2104.13353, 2021.

This is a preprint of the following chapter: Nicolás-Carlock J.R. & Luna-Pla
I., Corruptomics, published in Corruption Networks, edited by Granados O.M.,
Nicolás-Carlock J.R., 2021, Springer, reproduced with permission of Springer.
The final authenticated version is available online at
https://doi.org/10.1007/978-3-030-81484-7_9