# Isospin violation effect and three-body decays of the $T_{cc}^{+}$ state

Zhi-Feng Sun1,2,3,4<EMAIL_ADDRESS>Ning Li5<EMAIL_ADDRESS>Xiang
Liu1,2,3,4<EMAIL_ADDRESS>1School of Physical Science and Technology,
Lanzhou University, Lanzhou 730000, China
2Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
3MoE Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou,
Gansu 730000, China
3Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China
5School of Physics, Sun Yat-Sen University, Guangzhou 510275, China

###### Abstract

In this work, we make a study of $T_{cc}^{+}$ state observed by the LHCb
collaboration in 2021. In obtaining the effective potentials using the One-
Boson-Exchange Potential Model we use an exponential form factor, and find
that in the short and medium range, the contributions of the $\pi$, $\rho$ and
$\omega$ exchanges are comparable while in the long range the pion-exchange
contribution is dominant. Based on the assumption that $T_{cc}^{+}$ is a
loosely bound state of $D^{*}D$, we focus on its three-body decay using the
meson-exchange method. Considering that the difference between the thresholds
of $D^{*+}D^{0}$ and $D^{*0}D^{+}$ is even larger than the binding energy of
$T_{cc}^{+}$, the isospin-breaking effect is amplified by the small binding
energy of $T_{cc}^{+}$. Explicitly including such an isospin-breaking effect
we obtain, by solving the Schrödinger equation, that the probability of the
isoscalar component is about $91\%$ while that of the isovector component is
around $9\%$ for $T_{cc}^{+}$. Using the experimental value of the mass of
$T_{cc}^{+}$ as an input, we obtain the wave function of $T_{cc}^{+}$ and
further obtain its width via the three-body hadronic as well as the radiative
decays. The total width we obtain is in agreement with the experimental value
of the LHCb measurement with a unitarised Breit-Wigner profile. Conversely,
the current results support the conclusion that $T_{cc}^{+}$ is a hadronic
molecule of $D^{*}D$.

##  I Introduction

The search for the exotic hadrons and the exploration of their nature have
been important goals of the particle physics Brambilla:2019esw ; Chen:2016qju
; Guo:2017jvc ; Liu:2019zoy ; Meng:2022ozq ; Chen:2022asf ; Liu:2024uxn . The
doubly heavy multiquark is particularly important, as its study can deepen our
understanding of the nonperturbative strong interaction in the low-energy
region and enrich our identification of the hadronic spectrum. In 2021, the
LHCb Collaboration announced the observation of the doubly charmed tetraquark
$T_{cc}^{+}$ in the $D^{0}D^{0}\pi^{+}$ mass distribution LHCb:2021vvq ;
LHCb:2021auc . This discovery is a turning point in hadron physics, providing
the first evidence for a clearly exotic meson state with two charm quarks.
Using a relativistic P-wave two-body Breit-Wigner function with a Blatt-
Weisskopf form factor as the natural resonance profile, the location of the
peak relative to the $D^{*+}D^{0}$ mass threshold, $\delta m_{BW}$, and the
width, $\Gamma_{BW}$, are determined to be LHCb:2021vvq

$\displaystyle\delta m_{BW}$ $\displaystyle=$ $\displaystyle-273\pm 61\pm
5^{+11}_{-14}\ \text{keV},$ $\displaystyle\Gamma_{BW}$ $\displaystyle=$
$\displaystyle 410\pm 165\pm 43^{+18}_{-38}\ \text{keV}.$

In order to assess the fundamental properties of near-threshold resonances,
the LHCb Collaboration uses advanced parametrisations, i.e., a unitarised
Breit-Wigner profile. The extracted mass relative to the $D^{*+}D^{0}$
threshold and the width are given respectively by LHCb:2021auc

$\displaystyle\delta m_{U}$ $\displaystyle=$ $\displaystyle-360\pm
40^{+4}_{-0}\ \text{keV},$ $\displaystyle\Gamma_{U}$ $\displaystyle=$
$\displaystyle 48\pm 2^{+0}_{-14}\ \text{keV}.$

Before this discovery, there were many works searching for the structures with
two heavy and two light quarks Ballot:1983iv ; Zouzou:1986qh ; Heller:1986bt ;
Carlson:1987hh ; Silvestre-Brac:1992kaa ; Manohar:1992nd ; Brink:1998as ;
Cook:2002am ; Gelman:2002wf ; Janc:2004qn ; Vijande:2006jf ; Ebert:2007rn ;
Vijande:2009kj ; Yang:2009zzp ; Dias:2011mi ; Ohkoda:2012hv ; Du:2012wp ;
Li:2012ss ; Ikeda:2013vwa ; Luo:2017eub ; Mehen:2017nrh ; Fontoura:2019opw ;
Xu:2017tsr ; Francis:2016hui ; Francis:2018jyb ; Agaev:2019qqn ; Tan:2020ldi ;
Yang:2019itm ; Cheng:2020wxa ; Lipkin:1986dw ; Navarra:2007yw ; Semay:1994ht ;
Vijande:2007rf ; Lee:2009rt ; Karliner:2017qjm ; Eichten:2017ffp ;
Wang:2017uld ; Junnarkar:2018twb ; Liu:2019stu ; Ding:2020dio ; Molina:2010tx
; Pepin:1996id ; Vijande:2003ki ; Feng:2013kea ; Deng:2018kly ; Liu:2020nil ;
Park:2018wjk ; Lu:2020rog ; Braaten:2020nwp . The striking feature that
$T_{cc}^{+}$ is extremely close to the threshold. $T_{cc}^{+}$ has motivated
much work investigating this state within the picture of the hadronic
molecular state Chen:2021vhg ; Feijoo:2021ppq ; Fleming:2021wmk ; Meng:2021jnw
; Albaladejo:2021vln ; Chen:2021tnn ; Dai:2021vgf ; Deng:2021gnb ; Du:2021zzh
; Ke:2021rxd ; Ling:2021bir ; Liu:2019yye ; Chen:2021cfl ; Dong:2021bvy ;
Xin:2021wcr ; Huang:2021urd ; Ren:2021dsi ; Zhao:2021cvg ; Agaev:2022ast ;
He:2022rta ; Padmanath:2022cvl ; Albaladejo:2022sux ; Cheng:2022qcm ;
Abreu:2022sra ; Chen:2022vpo ; Jia:2022qwr ; Wang:2022jop ; Dai:2023mxm ;
Li:2023hpk ; Hu:2021gdg ; Azizi:2021aib ; Abreu:2022lfy ; Vidana:2023olz ;
Dai:2023cyo ; He:2023ucd ; Jia:2023hvc ; Lei:2023ttd ; Sakai:2023syt ;
Montesinos:2023qbx , tetraquark state Azizi:2021aib ; Abreu:2022lfy ;
Chen:2021tnn ; Agaev:2021vur ; Jin:2021cxj ; Wu:2022gie ; Weng:2021hje ;
Abreu:2021jwm ; Kim:2022mpa ; Noh:2023zoq ; Liu:2023vrk ; Wang:2024vjc , and
many other explanations Shi:2022slq ; Kinugawa:2023fbf ; Dai:2023kwv ;
Ma:2023int ; Chen:2021tnn ; Yan:2021wdl ; Deng:2022cld ; Wang:2023iaz .

In Ref. Li:2012ss  we investigated the possible tetraquarks with double charm.
We found that the interaction between $D^{*}$ and $D$ is attractive for the
system $D^{*}D$ with quantum number $I(J^{P})=0(1^{+})$, and it can form a
loosely bound state. In the present work, we still consider $T_{cc}^{+}$ as
the $D^{*+}D^{0}/D^{*0}D^{+}$ molecular state. Since the mass difference of
$D^{*+}D^{0}$ and $D^{*0}D^{+}$ is so large compared to the small binding
energy of $T_{cc}^{+}$, the isospin violation effect should not be neglected,
and in the present work we explicitly take this effect into account.

In the compact tetraquark picture, the heavy diquark-antiquark symmetry
contributes a strong attractive force for the doubly heavy systems, so that
this scheme can generate a deep bound state relative to the corresponding two-
meson threshold. In order to clarify whether the hadronic molecular picture is
correct, in this work we develop a method to calculate the three-body decay of
$T_{cc}^{+}$ based on $T_{cc}^{+}$ being a hadronic molecule of $D^{*}D$. Note
that in Refs. Meng:2021jnw ; Ling:2021bir ; Jia:2023hvc , the authors have
already studied the three-body decay of $T_{cc}^{+}$. Unlike these works, we
study the three-body decay of $T_{cc}^{+}$ in the framework of the
$D^{*+}D^{0}/D^{*0}D^{+}$ molecular state based on the one-boson-exchange
model, where its molecular structure is represented by the wave functions
obtained by solving the Schrödinger equation.

In this work, we also make an improvement to the one-boson-exchange model. In
the past, to calculate the effective potentials a monopole form factor was
introduced at each vertex to suppress the high-momentum (or short-range)
contribution. However, when the mass of the exchanged particle is large, the
monopole form factor does not work very well due to the suppression by its
numerator. We therefore choose an exponentially parameterized form factor in
the derivation of the effective potential to regulate the short-range
interaction.

We also extend the effective Lagrangians from
$[\mathrm{SU(3)_{L}}\otimes\mathrm{SU(3)_{R}}]_{\mathrm{global}}\otimes[\mathrm{SU(3)_{V}}]_{\mathrm{local}}$
to
$[\mathrm{U(3)_{L}}\otimes\mathrm{U(3)_{R}}]_{\mathrm{global}}\otimes[\mathrm{U(3)_{V}}]_{\mathrm{local}}$,
in which $\pi$, $K$, $\eta^{(\prime)}$ are identified as the pseudoscalar
Goldstone bosons, while $\rho$, $\omega$, $K^{*}$ and $\phi$ are taken as the
hidden local symmetry gauge bosons.

The paper is organized as follows. In Sec. II, we will present the effective
potentials with an exponentially parameterized form factor. The formalism for
the three-body decay of $T_{cc}^{+}$ will be discussed explicitly in Sec. III.
The numerical results are then presented and analyzed in Sec. IV. In the final
section V we will summarize our results and draw conclusions.

##  II Effective Potential

In this work, we consider the Lagrangians by extending the hidden local
symmetry
$[\mathrm{SU(3)_{L}}\otimes\mathrm{SU(3)_{R}}]_{\mathrm{global}}\otimes[\mathrm{SU(3)_{V}}]_{\mathrm{local}}$
to
$[\mathrm{U(3)_{L}}\otimes\mathrm{U(3)_{R}}]_{\mathrm{global}}\otimes[\mathrm{U(3)_{V}}]_{\mathrm{local}}$.
In such case, the exchanged light pseudoscalar and vector mesons can be
grouped into compact matrices, respectively, which are as follows

$\displaystyle M$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}+\frac{\eta^{\prime}}{\sqrt{3}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}+\frac{\eta^{\prime}}{\sqrt{3}}&K^{0}\\\
K^{-}&\bar{K}^{0}&-\frac{2}{\sqrt{6}}\eta+\frac{\eta^{\prime}}{\sqrt{3}}\end{array}\right),$
(4) $\displaystyle\hat{\rho}^{\mu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{array}\right)^{\mu}.$ (8)

Taking into account also the heavy quark spin symmetry, the Lagrangian of the
interaction between the light meson and the heavy meson containing a charm or
bottom quark is constructed as Casalbuoni:1996pg ; Casalbuoni:1992gi

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
ig\text{Tr}\left[H^{(Q)}_{b}\gamma_{\mu}\gamma_{5}A^{\mu}_{ba}\bar{H}^{(Q)}_{a}\right]$
(9)
$\displaystyle+i\beta\text{Tr}\left[H^{(Q)}_{b}v_{\mu}(V^{\mu}_{ba}-\rho^{\mu}_{ba})\bar{H}^{(Q)}_{a}\right]$
$\displaystyle+i\lambda\text{Tr}\left[H^{(Q)}_{b}\sigma_{\mu\nu}F^{\mu\nu}_{ba}\bar{H}^{(Q)}_{a}\right],$

where

$\displaystyle H^{(Q)}_{a}$ $\displaystyle=$
$\displaystyle\frac{1+\not{v}}{2}\left[P^{*\mu}_{a}\gamma_{\mu}-P_{a}\gamma_{5}\right],$
(10) $\displaystyle\bar{H}^{(Q)}_{a}$ $\displaystyle=$
$\displaystyle\gamma_{0}H^{(Q){\dagger}}_{a}\gamma_{0}=\left[P^{*^{\dagger}\mu}_{a}\gamma_{\mu}+P^{\dagger}_{a}\gamma_{5}\right]\frac{1+\not{v}}{2},$
(11) $\displaystyle A^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\xi^{\dagger}\partial^{\mu}\xi-\xi\partial^{\mu}\xi^{\dagger}\right),$
(12) $\displaystyle V^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\xi^{\dagger}\partial^{\mu}\xi+\xi\partial^{\mu}\xi^{\dagger}\right),$
(13) $\displaystyle F_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\rho_{\nu}-\partial_{\nu}\rho_{\mu}-\left[\rho_{\mu},\rho_{\nu}\right],$
(14)

$P=(D^{0},D^{+},D_{s}^{+})$ and $P^{*}=(D^{*0},D^{*+},D_{s}^{*+})$ are the
heavy pseudoscalar and vector meson fields, respectively.
$\xi=e^{iM/f_{\pi}}$, and
$\rho_{\mu}=\frac{ig_{V}}{\sqrt{2}}\hat{\rho}_{\mu}$. $A^{\mu}$ is the axial
current, while $V^{\mu}$ is the vector current. In the heavy quark limit, we
apply the static limit, i.e., the four-velocity of the heave meson is taken as
$v^{\mu}=(1,0,0,0)$.

After expanding the Lagrangians in Eq. (9), we have the terms we need for our
calculation

$\displaystyle\mathcal{L}_{P^{(*)}P^{(*)}M}$ $\displaystyle=$
$\displaystyle-i\frac{2g}{f_{\pi}}\epsilon_{\alpha\mu\nu\lambda}v^{\alpha}P^{*\mu}\partial^{\nu}MP^{*\lambda{\dagger}}$
(15)
$\displaystyle-\frac{2g}{f_{\pi}}(P\partial^{\lambda}MP_{\lambda}^{*{\dagger}}+P_{\lambda}^{*}\partial^{\lambda}MP^{\dagger}),$
$\displaystyle\mathcal{L}_{P^{(*)}P^{(*)}V}$ $\displaystyle=$
$\displaystyle-\sqrt{2}\beta g_{V}P(v\cdot\hat{\rho})P^{\dagger}$ (16)
$\displaystyle-2\sqrt{2}\lambda
g_{V}\epsilon_{\lambda\mu\alpha\beta}v^{\lambda}(P\partial^{\alpha}\hat{\rho}^{\beta}P^{*\mu{\dagger}}+P^{*\mu}\partial^{\alpha}\hat{\rho}^{\beta}P^{\dagger})$
$\displaystyle+\sqrt{2}\beta
g_{V}P^{\mu*}(v\cdot\hat{\rho})P_{\mu}^{*{\dagger}}$
$\displaystyle-i2\sqrt{2}\lambda
g_{V}P^{*\mu}(\partial_{\mu}\hat{\rho}_{\nu}-\partial_{\nu}\hat{\rho}_{\mu})P^{*\nu{\dagger}}.$

With the Lagrangians above, we derive the effective potentials, which are
related to the scattering amplitudes by

$\displaystyle V(\bm{q})$ $\displaystyle=$
$\displaystyle-\frac{\mathcal{M}(\bm{q})}{4\sqrt{m_{1}m_{2}m_{3}m_{4}}},$ (17)

where $m_{i}\ (i=1,2,3,4)$ is the mass of the heavy meson in the initial or
final state. By performing the Fourier transformation, we get the effective
potentials in coordinate space

$\displaystyle V(r)$ $\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{3}}\int
d^{3}qe^{i\bm{q}\cdot\bm{r}}V(\bm{q})F^{2}(\bm{q}).$ (18)

Here, $F(\bm{q})$ is the form factor, which suppresses the contribution of
high momenta, i.e., small distance. And the presence of such a form factor is
dictated by the extended (quark) structure of the hadrons Machleidt:1987hj .
In this work, we adopt the exponentially parameterized form factor

$\displaystyle F(\bm{q})$ $\displaystyle=$ $\displaystyle
e^{(q_{0}^{2}-\bm{q}^{2})/\Lambda^{2}},$ (19)

where $q_{0}$ is the zero-th component of the four momentum of exchanged
meson, and $\Lambda$ is the cutoff. In the past, a monople form factor is
usually used,

$\displaystyle F_{M}(\bm{q})$ $\displaystyle=$
$\displaystyle\frac{\Lambda^{2}-m_{ex}^{2}}{\Lambda^{2}-q_{0}^{2}+\bm{q}^{2}}.$
(20)

However, in the case of the heavier $\rho$, $\omega$ or $\phi$ exchanges,
$F_{M}(\bm{q})$ is suppressed by the numerator $\Lambda^{2}-m_{ex}^{2}$, which
may affect the final results. In this work, we will compare the effective
potentials and the probability distributions obtained with these two form
factors.

Table 1: The effective potentials. | $D^{*0}D^{+}$ | $D^{*+}D^{0}$
---|---|---
$D^{*0}D^{+}$ | $-V^{D}_{\rho^{0}}+V^{D}_{\omega}+V^{C}_{\rho^{-}}+V^{C}_{\pi^{-}}$ | $2V^{D}_{\rho^{-}}-\frac{1}{2}V^{C}_{\rho^{0}}+\frac{1}{2}V^{C}_{\omega}-\frac{1}{2}\tilde{V}^{C}_{\pi^{0}}+\frac{1}{6}V^{C}_{\eta}+\frac{1}{3}V^{C}_{\eta^{\prime}}$
$D^{*+}D^{0}$ | $2V^{D}_{\rho^{-}}-\frac{1}{2}V^{C}_{\rho^{0}}+\frac{1}{2}V^{C}_{\omega}-\frac{1}{2}\tilde{V}^{C}_{\pi^{0}}+\frac{1}{6}V^{C}_{\eta}+\frac{1}{3}V^{C}_{\eta^{\prime}}$ | $-V^{D}_{\rho^{0}}+V^{D}_{\omega}+V^{C}_{\rho^{+}}+V^{C}_{\pi^{+}}$

The expressions of the effective potentials are listed in Table 1 with the
definition of the following functions

$\displaystyle V^{D}_{V}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\beta^{2}g_{V}^{2}(\bm{\epsilon}_{1}\cdot\bm{\epsilon}_{3}^{\dagger})Y(\Lambda,m_{V},q_{0},r),$
(21) $\displaystyle V^{C}_{V}$ $\displaystyle=$ $\displaystyle
2\lambda^{2}g_{V}^{2}\left[\frac{2}{3}\bm{\epsilon}_{1}\cdot\bm{\epsilon}^{\dagger}_{4}\bigtriangledown^{2}Y(\Lambda,\tilde{m}_{V},q_{0},r)\right.$
(22)
$\displaystyle\left.-\frac{1}{3}S(\hat{\bm{r}},\bm{\epsilon}_{1},\bm{\epsilon}_{4}^{\dagger})r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial
r}Y(\Lambda,\tilde{m}_{V},q_{0},r)\right],$ $\displaystyle V^{C}_{p}$
$\displaystyle=$
$\displaystyle\frac{g^{2}}{f_{\pi}^{2}}\left[\frac{1}{3}\bm{\epsilon}_{1}\cdot\bm{\epsilon}^{\dagger}_{4}\bigtriangledown^{2}Y(\Lambda,\tilde{m}_{p},q_{0},r)\right.$
(23)
$\displaystyle\left.+\frac{1}{3}S(\hat{\bm{r}},\bm{\epsilon}_{1},\bm{\epsilon}_{4}^{\dagger})r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial
r}Y(\Lambda,\tilde{m}_{p},q_{0},r)\right],$ $\displaystyle\tilde{V}^{C}_{p}$
$\displaystyle=$
$\displaystyle\frac{g^{2}}{f_{\pi}^{2}}\left[\frac{1}{3}\bm{\epsilon}_{1}\cdot\bm{\epsilon}^{\dagger}_{4}\bigtriangledown^{2}U(\Lambda,\tilde{m}_{p}^{\prime},q_{0},r)\right.$
(24)
$\displaystyle\left.+\frac{1}{3}S(\hat{\bm{r}},\bm{\epsilon}_{1},\bm{\epsilon}_{4}^{\dagger})r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial
r}U(\Lambda,\tilde{m}_{p}^{\prime},q_{0},r)\right].$

In the above, $\tilde{m}_{V}^{2}=m_{V}^{2}-q_{0}^{2}$,
$\tilde{m}_{p}^{2}=m_{p}^{2}-q_{0}^{2}$, $\tilde{m}_{p}^{\prime
2}=q_{0}^{2}-m_{p}^{2}$ and
$q_{0}=\frac{m^{2}_{2}-m^{2}_{1}+m^{2}_{3}-m^{2}_{4}}{2(m_{3}+m_{4})}$. The
functions $Y$ and $U$ are defined as follows

$\displaystyle Y(\Lambda,\mu,q_{0},r)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}e^{i\bm{q}\cdot\bm{r}}\frac{1}{\bm{q}^{2}+\mu^{2}-i\epsilon}e^{2(q_{0}^{2}-\bm{q}^{2})/\Lambda^{2}}$
(25) $\displaystyle=$
$\displaystyle-\frac{e^{2q_{0}^{2}/\Lambda^{2}}}{(2\pi)^{2}r}\frac{\partial}{\partial
r}\left\\{\frac{\pi}{2\mu}\left[e^{-\mu r}+e^{\mu r}+e^{-\mu r}\right.\right.$
$\displaystyle\times\text{erf}\left(\frac{r\Lambda}{2\sqrt{2}}-\frac{\sqrt{2}\mu}{\Lambda}\right)-e^{\mu
r}\text{erf}\left(\frac{r\Lambda}{2\sqrt{2}}\right.$
$\displaystyle\left.\left.\left.+\frac{\sqrt{2}\mu}{\Lambda}\right)\right]e^{2\mu^{2}/\Lambda^{2}}\right\\},$
$\displaystyle U(\Lambda,\mu,q_{0},r)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}e^{i\bm{q}\cdot\bm{r}}\frac{1}{\bm{q}^{2}-\mu^{2}-i\epsilon}e^{2(q_{0}^{2}-\bm{q}^{2})/\Lambda^{2}}$
(26) $\displaystyle=$
$\displaystyle\frac{e^{2q_{0}^{2}/\Lambda^{2}}}{(2\pi)^{2}r}\frac{\partial}{\partial
r}\left\\{\pi\left[-\frac{1}{2i\mu}\left(e^{-i\mu
r}\text{erf}\left(\frac{r\Lambda}{2\sqrt{2}}\right.\right.\right.\right.$
$\displaystyle\left.\left.-\frac{\sqrt{2}i\mu}{\Lambda}\right)-e^{i\mu
r}\text{erf}\left(\frac{r\Lambda}{2\sqrt{2}}+\frac{\sqrt{2}i\mu}{\Lambda}\right)\right)$
$\displaystyle\left.\left.-\frac{i}{\mu}\cos(\mu
r)\right]e^{-2\mu^{2}/\Lambda^{2}}\right\\}.$

The derivation of Eqs. (25)-(26) can be found in the Appendix of Ref.
He:2024aej .

Using the Gaussian expansion method (GEM) Hiyama:2003cu , we solve the
coupled-channel Schrödinger equation to find the bound state solutions,

$\displaystyle\left(\hat{K}+\hat{M}+\hat{V}\right)\Psi=E\Psi.$ (27)

Here,
$\hat{K}=\text{diag}(-\frac{\tilde{\Delta}}{2\mu_{1}},-\frac{\tilde{\Delta}}{2\mu_{2}},\cdots)$,
$\hat{M}=\text{diag}(0,M_{2}-M_{1},M_{3}-M_{1},\cdots)$. For the central-force
field problem, the full wave function can be decoupled into the radial part
and the angular components. After integrating out the angular part, one
obtains a one-dimensional radial Schödinger equation where the double
derivative operator $\tilde{\Delta}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial}{\partial r}\right)-\frac{l(l+1)}{r^{2}}$. For the
function $U(\Lambda,\mu,q_{0},r)$, which is complex, we use the real part in
order to solve the radial part of the stationary Schrödinger equation, which
is real.

##  III Three Body Decay of $T_{cc}^{+}$

The main task of the current work is to derive a practical general formula for
the calculation of the strong and radiative three-body decay of $T_{cc}^{+}$.
Here, the $T_{cc}^{+}$ state is considered as the $D^{*+}D^{0}/D^{*0}D^{+}$
molecular state. The decays occur first with the t-channel exchange and then
sequentially the vector charmed meson $D^{*}$ decays into $\pi D$ or $\gamma
D$. The widths of the strong and radiative decay are

$\displaystyle\Gamma_{T_{cc}^{+}\to\pi^{+,0}D^{0,+}D^{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{32m_{T_{cc}^{+}}^{3}}\int
dm_{34}^{2}dm_{45}^{2}$ (28)
$\displaystyle\times\frac{1}{3}\sum\limits_{S_{z}^{T}}|\mathcal{M}_{T_{cc}^{+}\to\pi^{+,0}D^{0,+}D^{0}}|^{2}\frac{1}{S},$

and

$\displaystyle\Gamma_{T_{cc}^{+}\to\gamma D^{+}D^{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{32m_{T_{cc}^{+}}^{3}}\int
dm_{34}^{2}dm_{45}^{2}$ (29)
$\displaystyle\times\frac{1}{3}\sum\limits_{S_{z}^{T},S_{z}^{\gamma}}|\mathcal{M}_{T_{cc}^{+}\to\gamma
D^{+}D^{0}}|^{2},$

respectively, where $S_{z}^{T}$ and $S_{z}^{\gamma}$ are the spin magnetic
quantum numbers of $T_{cc}^{+}$ and the photon respectively. ${1}/{S}$ is the
symmetry factor, i.e., ${1}/{2!}$ for $T_{cc}^{+}\to\pi^{+}D^{0}D^{0}$, $1$
for $T_{cc}^{+}\to\pi^{0}D^{+}D^{0}$ and $1$ for $T_{cc}^{+}\to\gamma
D^{+}D^{0}$. $\mathcal{M}_{T_{cc}^{+}\to\pi^{+}D^{0}D^{0}}$,
$\mathcal{M}_{T_{cc}^{+}\to\pi^{0}D^{+}D^{0}}$ and
$\mathcal{M}_{T_{cc}^{+}\to\gamma D^{+}D^{0}}$ are the amplitudes
corresponding to the Feynman diagrams in Fig. 1. In fact, there should be the
same number of diagrams with pseudoscalar(vector) charmed meson associated
with the vector(pseudoscalar) charmed meson in the t-channel process. However,
after specifically calculating such diagrams, we found that their
contributions are close to zero. Therefore, we do not show such diagrams in
Fig. 1. The Feynmann diagrams Fig. 1 (a)-(d) are calculated using the
Lagrangians in Eqs. (15)-(16) to obtain the strong decay amplitudes. To
calculate the radiative decay amplitude, however, we need two more vertices
describing the interactions of $T_{cc}^{+}$ with $\gamma D^{*0}D^{0}$ and
$\gamma D^{*+}D^{+}$, i.e.,

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
eg_{D^{0}D^{*0}\gamma}\epsilon^{\mu\nu\alpha\beta}\partial_{\mu}A_{\nu}v_{\alpha}D^{*0}_{\beta}D^{0{\dagger}},$
(30) $\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
eg_{D^{+}D^{*+}\gamma}\epsilon^{\mu\nu\alpha\beta}\partial_{\mu}A_{\nu}v_{\alpha}D^{*+}_{\beta}D^{+{\dagger}}.$
(31)

The elementary charge $e=0.303$. To determine the value of the couplings
$g_{D^{0}D^{*0}\gamma}$ and $g_{D^{+}D^{*+}\gamma}$, we use the decay width of
$D^{*}$ in Ref. Ling:2021bir . And we get $g_{D^{0}D^{*0}\gamma}=1.911$ GeV-1
and $g_{D^{+}D^{*+}\gamma}=0.478$ GeV-1.

Figure 1: The diagrams of the strong and radiative decays of $T_{cc}^{+}$. The
exchanged mesons are $\pi$, $\eta$, $\eta^{\prime}$, $\rho$ and $\omega$. The
same number of diagrams with charmed pseudoscalar (vector) meson connected to
vector (pseudoscalar) meson are not shown because their contributions are
close to zero, see the main text.

Before calculating the amplitudes corresponding to the diagrams in Fig. 1, we
apply a power counting to estimate the contribution of each diagram to
simplify the calculation. The order of a quantity $X$ is defined as
$\mathcal{O}(X)=-\log_{10}X$. Neglecting the higher-order contributions, we
obtain the following decay amplitudes

$\displaystyle\mathcal{M}_{T_{cc}^{+}\to\pi^{+}D^{0}D^{0}}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{2\pi
m_{T_{cc}^{+}}}{E_{D^{*0}}E_{D^{+}}}}\int_{0}^{\infty}drrj_{0}(k_{5}r)u_{S}^{D^{*0}D^{+}}(r)\mathcal{A}_{\rho^{-}}^{(a)}$
$\displaystyle-\sqrt{\frac{2\pi
m_{T_{cc}^{+}}}{E_{D^{*0}}E_{D^{+}}}}\int_{0}^{\infty}drrj_{0}(k_{4}r)u_{S}^{D^{*0}D^{+}}(r)\mathcal{A}_{\rho^{-}}^{(b)},$
$\displaystyle\mathcal{M}_{T_{cc}^{+}\to\pi^{0}D^{+}D^{0}}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{2\pi
m_{T_{cc}^{+}}}{E_{D^{*0}}E_{D^{+}}}}\int_{0}^{\infty}drrj_{0}(k_{5}r)u_{S}^{D^{*0}D^{+}}(r)\mathcal{A}_{\rho^{-}}^{(c)}$
$\displaystyle-\sqrt{\frac{2\pi
m_{T_{cc}^{+}}}{E_{D^{*+}}E_{D^{0}}}}\int_{0}^{\infty}drrj_{0}(k_{4}r)u_{S}^{D^{*+}D^{0}}(r)\mathcal{A}_{\rho^{+}}^{(d)},$
$\displaystyle\mathcal{M}_{T_{cc}^{+}\to\gamma D^{+}D^{0}}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{2\pi
m_{T_{cc}^{+}}}{E_{D^{*0}}E_{D^{+}}}}\int_{0}^{\infty}drrj_{0}(k_{5}r)u_{S}^{D^{*0}D^{+}}(r)\mathcal{A}_{\rho^{-}}^{(e)}$
$\displaystyle-\sqrt{\frac{2\pi
m_{T_{cc}^{+}}}{E_{D^{*+}}E_{D^{0}}}}\int_{0}^{\infty}drrj_{0}(k_{4}r)u_{S}^{D^{*+}D^{0}}(r)\mathcal{A}_{\rho^{+}}^{(f)}.$

Note here that we neglect the D-wave contributions which are much smaller than
those of the S-wave. In Eqs. (LABEL:eq21)-(LABEL:eq23), the specific
expressions of $\mathcal{A}^{(a)}_{\rho^{-}},\ldots$ are

$\displaystyle\mathcal{A}_{\rho^{-}}^{(a)}$ $\displaystyle=$ $\displaystyle
i\frac{4g\beta^{2}g_{V}^{2}}{f_{\pi}}m_{D^{0}}m_{D^{*+}}\sqrt{m_{D^{*0}}m_{D^{+}}}$
(35)
$\displaystyle\times\frac{Y(\Lambda,\tilde{M}_{1},\tilde{q}_{1},r)}{k^{2}-m_{D^{*+}}^{2}+im_{D^{*+}}\Gamma_{D^{*+}}}\bm{\epsilon}_{1}\cdot\bm{k}_{3},$
$\displaystyle\mathcal{A}_{\rho^{-}}^{(b)}$ $\displaystyle=$ $\displaystyle
i\frac{4g\beta^{2}g_{V}^{2}}{f_{\pi}}m_{D^{0}}m_{D^{*+}}\sqrt{m_{D^{*0}}m_{D^{+}}}$
(36)
$\displaystyle\times\frac{Y(\Lambda,\tilde{M}_{1},\tilde{q}_{1},r)}{k^{\prime
2}-m_{D^{*+}}^{2}+im_{D^{*+}}\Gamma_{D^{*+}}}\bm{\epsilon}_{1}\cdot\bm{k}_{3},$
$\displaystyle\mathcal{A}_{\rho^{-}}^{(c)}$ $\displaystyle=$
$\displaystyle-i\frac{2\sqrt{2}g\beta^{2}g_{V}^{2}}{f_{\pi}}m_{D^{+}}m_{D^{*+}}\sqrt{m_{D^{*0}}m_{D^{0}}}$
(37)
$\displaystyle\times\frac{Y(\Lambda,\tilde{M}_{1},\tilde{q}_{1},r)}{k^{2}-m_{D^{*+}}^{2}+im_{D^{*+}}\Gamma_{D^{*+}}}\bm{\epsilon}_{1}\cdot\bm{k}_{3},$
$\displaystyle\mathcal{A}_{\rho^{+}}^{(d)}$ $\displaystyle=$ $\displaystyle
i\frac{2\sqrt{2}g\beta^{2}g_{V}^{2}}{f_{\pi}}m_{D^{0}}m_{D^{*0}}\sqrt{m_{D^{*+}}m_{D^{+}}}$
(38)
$\displaystyle\times\frac{Y(\Lambda,\tilde{M}_{2},\tilde{q}_{2},r)}{k^{\prime
2}-m_{D^{*0}}^{2}+im_{D^{*0}}\Gamma_{D^{*0}}}\bm{\epsilon}_{1}\cdot\bm{k}_{3},$
$\displaystyle\mathcal{A}_{\rho^{-}}^{(e)}$ $\displaystyle=$
$\displaystyle-i2eg_{D^{+}D^{*+}\gamma}\beta^{2}g_{V}^{2}m_{D^{+}}m_{D^{*+}}\sqrt{m_{D^{*0}}m_{D^{0}}}$
(39)
$\displaystyle\times\frac{Y(\Lambda,\tilde{M}_{1},\tilde{q}_{1},r)[\bm{\epsilon}_{1}\cdot(\bm{k}_{3}\times\bm{\epsilon}_{3}^{\dagger})]}{k^{2}-m_{D^{*+}}^{2}+im_{D^{*+}}\Gamma_{D^{*+}}},$
$\displaystyle\mathcal{A}_{\rho^{+}}^{(f)}$ $\displaystyle=$
$\displaystyle-i2eg_{D^{0}D^{*0}\gamma}\beta^{2}g_{V}^{2}m_{D^{0}}m_{D^{*0}}\sqrt{m_{D^{*+}}m_{D^{+}}}$
(40)
$\displaystyle\times\frac{Y(\Lambda,\tilde{M}_{2},\tilde{q}_{2},r)[\bm{\epsilon}_{1}\cdot(\bm{k}_{3}\times\bm{\epsilon}_{3}^{\dagger})]}{k^{\prime
2}-m_{D^{*0}}^{2}+im_{D^{*0}}\Gamma_{D^{*0}}}.$

Here, $j_{0}(x)={\sin x}/{x}$ is the spherical Bessel function of order 0.
$\tilde{q}_{1}=\frac{m^{2}_{D^{+}}-m^{2}_{D^{*0}}+m^{2}_{D^{*+}}-m^{2}_{D^{0}}}{2(m_{D^{*+}}+m_{D^{0}})}$,
$\tilde{q}_{2}=\frac{m^{2}_{D^{0}}-m^{2}_{D^{*+}}+m^{2}_{D^{*0}}-m^{2}_{D^{+}}}{2(m_{D^{*0}}+m_{D^{+}})}$,
$\tilde{M}_{1}=\sqrt{m_{\rho^{-}}^{2}-\tilde{q}_{1}^{2}}$, and
$\tilde{M}_{2}=\sqrt{m_{\rho^{+}}^{2}-\tilde{q}_{2}^{2}}$.
$u_{0}^{D^{*0}D^{+}}/r$ and $u_{0}^{D^{*+}D^{0}}/r$ are the S-wave radial wave
functions of $D^{*0}D^{+}$ and $D^{*+}D^{0}$ channels, respectively.
$k^{\mu}=k_{3}^{\mu}+k_{4}^{\mu}$ is the four-momentum of the intermediate
$D^{*}$ in Fig. 1 (a), (c) and (e), while
$k^{\prime\mu}=k_{3}^{\mu}+k_{5}^{\mu}$ is the four-momentum of the
intermediate $D^{*}$ meson in Fig. 1 (b), (d) and (f).

Note that the amplitudes in Eqs. (LABEL:eq21)-(LABEL:eq23) have the same
tensor structure as those in Refs. Meng:2021jnw ; Ling:2021bir , i.e.,

$\displaystyle\tilde{\mathcal{M}}_{T_{cc}^{+}\to\pi^{+}D^{0}D^{0}}$
$\displaystyle\sim$ $\displaystyle
k_{\pi^{+}\mu}\frac{-g^{\mu\nu}+k^{\mu}k^{\nu}/m_{D^{*}}^{2}}{k^{2}-m_{D^{*}}^{2}+im_{D^{*}}\Gamma_{D^{*}}}\epsilon_{\nu}$
(41) $\displaystyle\sim$ $\displaystyle\bm{k}_{\pi^{+}}\cdot{\bm{\epsilon}},$
$\displaystyle\tilde{\mathcal{M}}_{T_{cc}^{+}\to\pi^{0}D^{+}D^{0}}$
$\displaystyle\sim$ $\displaystyle
k_{\pi^{0}\mu}\frac{-g^{\mu\nu}+k^{\mu}k^{\nu}/m_{D^{*}}^{2}}{k^{2}-m_{D^{*}}^{2}+im_{D^{*}}\Gamma_{D^{*}}}\epsilon_{\nu}$
(42) $\displaystyle\sim$ $\displaystyle\bm{k}_{\pi^{0}}\cdot\bm{\epsilon},$
$\displaystyle\tilde{\mathcal{M}}_{T_{cc}^{+}\to\gamma D^{+}D^{0}}$
$\displaystyle\sim$
$\displaystyle\frac{-g^{\mu\nu}+k^{\mu}k^{\nu}/m_{D^{*}}^{2}}{k^{2}-m_{D^{*}}^{2}+im_{D^{*}}\Gamma_{D^{*}}}\epsilon^{\mu\nu\alpha\beta}k_{\gamma\mu}\epsilon_{\gamma\nu}k_{\alpha}\epsilon_{\nu}$
(43) $\displaystyle\sim$
$\displaystyle\bm{\epsilon}\cdot(\bm{k}_{\gamma}\times\bm{\epsilon}_{\gamma}).$

##  IV Numerical results

With the analytic results given before, we solve the Schrödinger equation to
obtain the radial wave function, and then we calculate the decay width based
on the radial wave function. In this section we will present the numerical
results, perform the discussion and draw the conclusion.

###  IV.1 Comparison of the effective potentials with different form factors

Figure 2: The effective potentials for the $D^{*+}D^{0}\to D^{*+}D^{0}$
process. (a) and (b) correspond to the exponential form factor with different
interaction range, and (c) and (d) the monopole form factor with different
interaction range. In all cases, the cutoff $\Lambda$ is chosen to be $1000$
MeV.

As mentioned above, we use the exponential form factor in our calculation. For
comparison, we take the $D^{*+}D^{0}\to D^{*+}D^{0}$ process as an example,
and plot the effective potentials with the exponential form factor (upper
panel) and the monopole form factor (lower panel) in Fig. 2. It can clearly be
seen that for the exponential form factor the pion-exchange contribution and
the vector-meson-exchange contribution are comparable in the short and medium
interaction range whereas in the long range the pion-exchange contribution is
dominant (see Fig. 2 (a) and (b)). In the case of the monopole form factor,
the vector exchange contribution is relatively small compared to the pion
exchange for the whole interaction range. This is because the masses of the
vector mesons are larger than that of pion, and their contributions are
suppressed by the numerator of the monopole form factor
$F_{M}(\bm{q})=\frac{\Lambda^{2}-m_{ex}^{2}}{\Lambda^{2}-q_{0}^{2}+\bm{q}^{2}}$,
where the cutoff $\Lambda$ is around 1000 MeV.

###  IV.2 Isospin Symmetry Violation

The threshold difference betwen $D^{*0}D^{+}$ and $D^{*+}D^{0}$ is about
$1.34$ MeV, which is much larger than the the small binding energy of
$T_{cc}^{+}$. Therefore, the isospin symmetry does not fully hold. Here, we
explicitly consider the isospin symmetry violation effect for $T_{cc}^{+}$.

In the following, we apply two types of bases, one is for the different
channels (labelled by 1) and the other is for the different isospin states
(labelled by 2). The wave functions of $T_{cc}^{+}$ in these two bases are

$\displaystyle\psi_{T_{cc}^{+}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\frac{u^{D^{*0}D^{+}}_{S}}{r}|^{3}S_{1}\rangle\\\
\frac{u^{D^{*0}D^{+}}_{D}}{r}|^{3}D_{1}\rangle\\\
\frac{u^{D^{*+}D^{0}}_{S}}{r}|^{3}S_{1}\rangle\\\
\frac{u^{D^{*+}D^{0}}_{D}}{r}|^{3}D_{1}\rangle\end{array}\right),\
\psi_{T_{cc}^{+}}^{\prime}=\left(\begin{array}[]{c}-\frac{u^{D^{*0}D^{+}}_{S}+u^{D^{*+}D^{0}}_{S}}{\sqrt{2}r}|^{3}S_{1}\rangle\\\
-\frac{u^{D^{*0}D^{+}}_{D}+u^{D^{*+}D^{0}}_{D}}{\sqrt{2}r}|^{3}D_{1}\rangle\\\
\frac{u^{D^{*0}D^{+}}_{S}-u^{D^{*+}D^{0}}_{S}}{\sqrt{2}r}|^{3}S_{1}\rangle\\\
\frac{u^{D^{*0}D^{+}}_{D}-u^{D^{*+}D^{0}}_{D}}{\sqrt{2}r}|^{3}D_{1}\rangle\end{array}\right),$
(52)

respectively. These two wave functions are related by each other via the
relation $\psi_{T_{cc}^{+}}^{\prime}=K\psi_{T_{cc}^{+}}$ with the
transformation matrix

$\displaystyle K$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}-\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}&0\\\
0&-\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}&0\\\
0&\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}\end{array}\right).$ (57)

With the effective potentials shown in Sec. II, we solve the Schrödinger
equation using the basis $1$. The numerical results are given in Table 2. When
the cutoff $\Lambda$ is fixed at 782-798 MeV, we obtain a loosely bound state
of $D^{*}D$ with a binding energy $274-359$ keV, which is in agreement with
the experimental value from LHCb. The radial probability distributions of the
different channels are shown in Fig. 3, in which the cutoffs are fixed at
$798$ MeV and $1187$ MeV for the respective exponential and monopole form
factors, so that the binding energies of these two cases are both around $360$
keV. It can clearly be seen that the probability distributions of the S-wave
are almost the same. Since the S-wave contributions are dominant, replacing
the monopole form factor with the exponential form factor does not change the
results too much.

Table 2: The numerical results of the binding energy ($E$), root-mean-square radius ($R_{rms}$) and possibilities ($P$) of different channels. $\Lambda$, $E$, $R_{rms}$ and the probabilities are in units of MeV, keV, fm and $\%$, respectively. $\Lambda$ | $E$ | $R_{rms}$ | $P_{D^{*0}D^{+}(^{3}S_{1})}$ | $P_{D^{*0}D^{+}(^{3}D_{1})}$ | $P_{D^{*+}D^{0}(^{3}S_{1})}$ | $P_{D^{*+}D^{0}(^{3}D_{1})}$
---|---|---|---|---|---|---
782 | 200.3 | 6.7 | 23.9 | 0.1 | 75.7 | 0.2
786 | 236.2 | 6.2 | 25.3 | 0.1 | 74.3 | 0.2
790 | 274.5 | 5.7 | 26.6 | 0.1 | 73.0 | 0.2
794 | 315.3 | 5.4 | 27.8 | 0.1 | 71.9 | 0.2
798 | 358.6 | 5.1 | 28.9 | 0.1 | 70.7 | 0.2

Figure 3: The radial probability distributions for different channels. (a) and
(b) are the results using the exponential form factor with $\Lambda=798$ MeV,
while (c) and (d) are the results using the monopole form factor with
$\Lambda=1187$ MeV.

Considering the basis $2$, the probability of the isovector component within
$T_{cc}^{+}$ is

$\displaystyle\rho_{10}$ $\displaystyle=$ $\displaystyle\int
dr\frac{\left[u^{D^{*0}D^{+}}_{S}+u^{D^{*+}D^{0}}_{S}\right]^{2}+\left[u^{D^{*0}D^{+}}_{D}+u^{D^{*+}D^{0}}_{D}\right]^{2}}{2},$
(58)

while that of the isoscalar component is

$\displaystyle\rho_{00}$ $\displaystyle=$ $\displaystyle\int
dr\frac{\left[u^{D^{*0}D^{+}}_{S}-u^{D^{*+}D^{0}}_{S}\right]^{2}+\left[u^{D^{*0}D^{+}}_{D}-u^{D^{*+}D^{0}}_{D}\right]^{2}}{2}.$
(59)

Numerically, if the cutoff is chosen as $790-798$ MeV, the probabilities of
$D^{*+}D^{0}$ and $D^{*0}D^{+}$ are 73.3%-71.0% and 26.7%-29.0% respectively.
The probability of the isoscalar component is 90.2%-92.2% while that of the
isovector component is 9.8%-7.8%, revealing a large isospin symmetry violation
effect for the loosely bound state of $T_{cc}^{+}$.

###  IV.3 The Three-Body Decay of $T_{cc}^{+}$

In Ref. LHCb:2021vvq , a relativistic P-wave two-body Breit-Wigner function
with a Blatt-Weisskopf form factor is used as the natural resonance profile,
while in Ref. LHCb:2021auc , a unitarised Breit-Wigner profile is used in the
analysis. The experimental widths are $410\pm 165\pm 43^{+18}_{-38}$ keV and
$48\pm 2^{+0}_{-14}$ keV, respectively. In this subsection, we will calculate
the three-body decay width to investigate which result is reasonable. Using
the analytic formulae given in Sec. III and the radial wave function of
$T_{cc}^{+}$, we calculate the strong and radiative three-body decay of
$T_{cc}^{+}$ and list the numerical results in Table 3.

For strong decays of $T_{cc}^{+}$, there are two possible channels, i.e.,
$T_{cc}^{+}\to D^{0}D^{0}\pi^{+}$ and $T_{cc}^{+}\to D^{0}D^{+}\pi^{0}$. Note
that the decay $T_{cc}^{+}\to D^{+}D^{+}\pi^{0}$ is kinetically forbidden. If
we fix the cutoff $\Lambda$ at $782-798$ MeV, the total width of these two
channels is $17.3-22.4$ keV. The partial decay widths of $T_{cc}^{+}\to
D^{0}D^{0}\pi^{+}$ and $T_{cc}^{+}\to D^{0}D^{+}\pi^{0}$ are $14.8$ keV and
$7.6$ keV respectively for the cutoff $\Lambda=798$ MeV. Note that if the
isospin symmetry is exact, in which case the masses of $D^{(*)+}$ and
$D^{(*)0}$ are equal, then these two partial widths should be exactly the
same. This means that the difference between the partial widths is caused by
the isospin symmetry violation effect.

For radiative decays, the only possible channel is $T_{cc}^{+}\to
D^{0}D^{+}\gamma$. If the cutoff is fixed as $782-798$ MeV, the width of the
radiative decay is $0.7-1.0$ keV.

Consequently, the total decay width of $T_{cc}^{+}$ is $23.4$ keV with
$\Lambda=798$ MeV, which is close to the lower limit of the experimental value
$48\pm 2^{+0}_{-14}$ keV using a unitarised Breit-Wigner profile. This result
therefore supports the conclusion that $T_{cc}^{+}$ is a hadronic molecule
composed of $D^{*+}D^{0}/D^{*0}D^{+}$.

Table 3: The numerical results of the three-body decay widths as a function of the cutoff. $\Gamma_{R}$, $\Gamma_{S}$ and $\Gamma_{\text{tot}}$ are the radiative, strong and total decay widths. $\Lambda$ (MeV) | 782 | 786 | 790 | 794 | 798
---|---|---|---|---|---
$\Gamma_{R}$ (keV) | 0.7 | 0.8 | 0.9 | 0.9 | 1.0
$\Gamma_{S}$ (keV) | 17.3 | 18.6 | 19.9 | 21.2 | 22.4
$\Gamma_{\text{tot}}$ (keV) | 18.0 | 19.4 | 20.7 | 22.1 | 23.4

##  V Summary

In the present work, we revisit the tetraquark $T_{cc}^{+}$ observed by the
LHCb collaboration in 2021. We consider $T_{cc}^{+}$ as a
$D^{*+}D^{0}/D^{*0}D^{+}$ molecular state with the isospin symmetry violation
effect explicitly included using the one-boson-exchange potential model. Based
on this result, we focus on the three-body strong and radiative decays of
$T_{cc}^{+}$ which can give us much more information about the structure of
$T_{cc}^{+}$.

In order to calculate the effective potentials, we use the exponential form
factor $F(\bm{q})=e^{(q_{0}^{2}-\bm{q}^{2})/\Lambda^{2}}$ instead of the
monopole form factor
$F_{M}=\frac{\Lambda^{2}-m_{ex}^{2}}{\Lambda^{2}-q_{0}^{2}+\bm{q}^{2}}$ which
is frequently used in the one-boson-exchange potential model. We find that for
the monopole form factor the contribution of $\rho$ and $\omega$ exchanges is
much smaller than that of the pion-exchange over the whole interaction range.
This is due to the suppression of the numerator of the monopole form factor.
However, if we use the exponential form factor, we find a different behavior
of the potentials, i.e., in the short and medium interaction range (around
$0-0.3$ fm and $0.3-2$ fm, respectively), the vector-meson-exchange
contributions are comparable to those of the pion-exchange, while in the long
range the pion-exchange potential is dominant.

In this work, we use the exponential form factor in our calculation. After
solving the Schrödinger equation, we obtain a loosely bound state of
$D^{*+}D^{0}/D^{*0}D^{+}$ with binding energy of $358.6$ keV if the cutoff is
fixed at $798$ MeV. On the other hand, if we choose a cutoff $790$ MeV, the
binding energy is $274.5$ keV. LHCb shows that the binding energy of the
$T_{cc}^{+}$ is $-273\pm 61\pm 5^{+11}_{-14}$ keV and $-360\pm 40^{+4}_{-0}$
keV, when using a relativistic P-wave two-body Breit-Wigner function with a
Blatt-Weisskopf form factor as the natural resonance profile and a unitarised
Breit-Wigner profile, respectively. Thus, from a spectrum perspective, our
results support that $T_{cc}^{+}$ can be explained as a
$D^{*+}D^{0}/D^{*0}D^{+}$ molecular state.

Since the threshold difference between $D^{*+}D^{0}$ and $D^{*0}D^{+}$ is
large compared to the small binding energy of $T_{cc}^{+}$, the isospin
symmetry is violated for the $D^{*+}D^{0}/D^{*0}D^{+}$ system. After including
an isospin breaking effect, we find that the probability of the isoscalar
component is about 91% and that of the isovector component is about 9%.

Since LHCb adopted two methods to analyze the experimental data of the
$T_{cc}^{+}$, there are two different results of its mass and width. In order
to check which one is more reasonable, we study the three-body decay of
$T_{cc}^{+}$ and develop for the first time a general method to calculate the
three-body decay width. We find that if we take the experimental value of the
binding energy, around $360$ keV, as an input the total width is $23.4$ keV,
which is close to the experimental value using a unitarised Breit-Wigner
profile. Therefore, our results for the three-body decay width of $T_{cc}^{+}$
also support that the $T_{cc}^{+}$ is a hadronic molecule of
$D^{*+}D^{0}/D^{*0}D^{+}$.

## Acknowledgments

We would like to thank Prof. Shi-Lin Zhu for suggestive discussion. This
project is supported by the Fundamental Research Funds for the Central
Universities under Grant No. lzujbky-2022-sp02, the National Natural Science
Foundation of China (NSFC) under Grant Nos. 11705069, 12335001, 11965016 and
12247101, the project for top-notch innovative talents of Gansu province, the
National Key Research and Development Program of China under Contract No.
2020YFA0406400, and the Guangdong Basic and Applied Basic Research Foundation
under Grant No. 2023A1515011704.

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