# Predicting another doubly charmed molecular resonance
$T_{cc}^{\prime+}(3876)$

Rui Chen1,2<EMAIL_ADDRESS>Qi Huang3,6<EMAIL_ADDRESS>Xiang
Liu4,5,6<EMAIL_ADDRESS>Shi-Lin Zhu2<EMAIL_ADDRESS>1Key Laboratory of
Low-Dimensional Quantum Structures and Quantum Control of Ministry of
Education, Department of Physics and Synergetic Innovation Center for Quantum
Effects and Applications, Hunan Normal University, Changsha 410081, China
2Center of High Energy Physics, Peking University, Beijing 100871, China
3School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China
4School of Physical Science and Technology, Lanzhou University, Lanzhou
730000, China
5Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China
6Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, and Frontiers Science Center for Rare Isotopes, Lanzhou
University, Lanzhou 730000, China

###### Abstract

The isospin breaking effect plays an essential role in generating hadronic
molecular states with a very tiny binding energy. Very recently, the LHCb
Collaboration observed a very narrow doubly charmed tetraquark $T_{cc}^{+}$ in
the $D^{0}D^{0}\pi$ mass spectrum, which lies just below the $D^{0}D^{*+}$
threshold around 273 keV. In this work, we study the $D^{0}D^{*+}/D^{+}D^{*0}$
interactions with the one-boson-exchange effective potentials and consider the
isospin breaking effect carefully. We not only reproduce the mass of the newly
observed $T_{cc}^{+}$ very well in the doubly charmed molecular tetraquark
scenario, but also predict the other doubly charmed partner resonance
$T_{cc}^{\prime+}$ with $m=3876~{}\text{MeV}$, and $\Gamma=412~{}\text{keV}$.
The prime decay modes of the $T_{cc}^{\prime+}$ are $D^{0}D^{+}\gamma$ and
$D^{+}D^{0}\pi^{0}$. The peculiar characteristic mass spectrum of the
$D^{0}D^{*+}/D^{+}D^{*0}$ molecular systems can be applied to identify the
doubly charmed molecular states.

Introduction—As an important and effective approach to shed light on the non-
perturbative behavior of the quantum chromodynamics (QCD), the study of the
hadron spectroscopy has become an active research field. Among abundant
research issues around the hadron spectroscopy, searching for the exotic
hadronic matter is full of challenges and opportunities at the birth of quark
model GellMann:1964nj ; Zweig:1981pd ; Zweig:1964jf . There exist different
exotic hadronic matters like glueball, hybrid, multiquark states.

Very recently, the LHCb Collaboration reported a very narrow state
$T_{cc}^{+}$ in the $D^{0}D^{0}\pi^{+}$ mass spectrum with the significance
over 10 $\sigma$ LHCb:2021vvq ; LHCb:2021auc . It is the firstly observed
doubly charmed tetraquark as its simplest valence quark component is
$cc\bar{u}\bar{d}$. Its spin-parity is assumed as $J^{P}=1^{+}$. Its mass with
respect to the $D^{0}D^{*+}$ threshold and width are

$\displaystyle\delta m$ $\displaystyle=$ $\displaystyle
m_{T_{cc}^{+}}-(m_{D^{0}}+m_{D^{*+}})=-273\pm 61\pm
5^{+11}_{-14}~{}\text{keV}/c^{2},$ $\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle 410\pm 165\pm 43^{+18}_{-38}~{}\text{keV},$

respectively. This important observation shall push the exploration of exotic
hadronic matter into a new era.

Since this doubly charmed tetraquark $T_{cc}^{+}$ locates just below the
$DD^{*}$ threshold, the doubly charmed meson-meson molecule is the very
attractive and probable explanation of the newly $T_{cc}^{+}$ state. The
predicted mass of the $DD^{*}$ molecule with $I(J^{P})=0(1^{+})$ Li:2012ss ;
Xu:2017tsr  is very consistent with the mass of the newly $T_{cc}^{+}$. The
binding energy is very small, which is similar to the case of the $X(3872)$ as
an isoscalar $D\bar{D}^{*}$ molecule. The $T_{cc}^{+}$ and $X(3872)$ shares
the same one-pion-exchange force and nearly the same isospin breaking pattern.
The very tiny binding energy can amplify the isospin breaking effect Li:2012cs
, which are also important for single heavy tetraquarks Gutsche:2016cml . In
fact, the interactions between the $D^{0}D^{*+}$ and $D^{+}D^{*0}$ are almost
the same. Therefore, if the $T_{cc}^{+}$ is the $D^{0}D^{*+}$ molecule, there
should exist the other $D^{+}D^{*0}$ molecule just below the $D^{+}D^{*0}$
threshold. When we recall the experimental information in the
$D^{0}D^{0}\pi^{+}$ mass spectrum, one can find a minor structure existing
between the $D^{0}D^{*+}$ and $D^{+}D^{*0}$ threshold, which may correspond to
the $D^{+}D^{*0}$ bound state.

In this work, we perform a coupled channel analysis of the
$D^{0}D^{*+}/D^{+}D^{*0}$ interactions by adopting the OBE effective potential
and considering both the isospin breaking effect and $S-D$ wave mixing effect.
We not only find that the newly $T_{cc}^{+}$ state perfectly matches the very
shallow bound $D^{0}D^{*+}/D^{+}D^{*0}$ molecular explanation with
$J^{P}=1^{+}$, but also obtain a coupled $T_{cc}^{\prime+}$ resonance with
$m=3876~{}\text{MeV}$, $\Gamma=412~{}\text{keV}$. The $T_{cc}^{+}$ molecular
state is mainly composed of the $S-$wave $D^{0}D^{*+}$ component with over 70%
probability. The dominant channel for the $T_{cc}^{\prime+}$ resonance is the
$S-$wave $D^{+}D^{*0}$ component. The quantitative analysis on the strong
decay behavior for the $T_{cc}^{\prime+}$ resonance indicates the
$D^{+}D^{0}\pi^{0}$ channel is more important than the $D^{0}D^{0}\pi^{+}$,
which explains why the experimental events for the $T_{cc}^{\prime+}$ in the
$D^{0}D^{0}\pi^{+}$ mass spectrum are not so significant. Nevertheless, it is
likely to search for the $T_{cc}^{\prime+}$ state in the $D^{0}D^{+}\gamma$
and $D^{+}D^{0}\pi^{0}$ final states.

$D^{0}D^{*+}/D^{+}D^{*0}$ interactions and the isospin breaking effect— In
this work, a crucial input is the interaction between the charmed mesons. So
far, the understanding of the realistic interactions is not enough due to the
poor experimental data. Several phenomenological models have been put forward,
such as the multiquark model, the QCD sum rule, the OBE model, the effective
field theory (see reviews Chen:2016qju ; Liu:2019zoy ; Guo:2017jvc  for
details. Especially the $T_{cc}$ has been reviewed extensively in Ref.
Liu:2019zoy ). We adopt the OBE model to study the mass spectrum of the
$D^{0}D^{*+}/D^{+}D^{*0}$ system with $J^{P}=1^{+}$ and consider the
$\pi/\sigma/\eta/\rho/\omega$ meson exchange interactions.

The general procedure for deducing the OBE effective potentials is organized
as follows. After constructing the effective Lagrangians, one can easily write
down the OBE scattering amplitude $\mathcal{M}[M_{1}{M}_{2}\to M_{3}{M}_{4}]$
for the $M_{1}{M}_{2}\to M_{3}{M}_{4}$ process. The OBE effective potentials
in the momentum space can be related to the corresponding scattering
amplitudes by a Breit approximation,
$\mathcal{V}(\bm{q})=-\mathcal{M}[M_{1}{M}_{2}\to
M_{3}{M}_{4}]/\sqrt{16m_{M_{1}}m_{{M}_{2}}m_{M_{3}}m_{{M}_{4}}}$. In order to
obtain the OBE effective potentials in the coordinate space, we further
perform a Fourier transformation,
$\mathcal{V}(\bm{r})=\int\frac{d^{3}\bm{q}e^{i\bm{q}\cdot\bm{r}}}{(2\pi)^{3}}\mathcal{V}(\bm{q})\mathcal{F}^{2}(q^{2},m_{E}^{2})$.
Here, we introduce a monopole form factor at every interaction verteces to
compensate the off-shell effect of the exchanged meson, which has the form of
$\mathcal{F}^{2}(q^{2},m_{E}^{2})=(\Lambda^{2}-m_{E}^{2})/(q^{2}-m_{E}^{2})$.
$\Lambda$, $m_{E}$, and $q$ stand for the cutoff, mass and four-momentum of
the exchanged particle, respectively.

The relevant effective Lagrangians to describe the interactions between the
$S-$wave charmed mesons and the light mesons are constructed in terms of heavy
quark symmetry and chiral symmetry Yan:1992gz ; Wise:1992hn ; Burdman:1992gh ;
Casalbuoni:1996pg ; Falk:1992cx ; Ding:2008gr , i.e.,

$\displaystyle\mathcal{L}_{{\mathcal{P}}^{(*)}{\mathcal{P}}^{(*)}\sigma}$
$\displaystyle=$
$\displaystyle-2g_{s}{\mathcal{P}}_{b}^{{\dagger}}{\mathcal{P}}_{b}\sigma-2g_{s}{\mathcal{P}}_{b}^{*}\cdot{\mathcal{P}}_{b}^{*{\dagger}}\sigma,$
(1)
$\displaystyle\mathcal{L}_{{\mathcal{P}}^{(*)}{\mathcal{P}}^{(*)}\mathbb{P}}$
$\displaystyle=$
$\displaystyle-\frac{2g}{f_{\pi}}\left({\mathcal{P}_{b}}{\mathcal{P}}_{a\lambda}^{*{\dagger}}+{\mathcal{P}}^{*}_{b\lambda}{\mathcal{P}}_{a}^{{\dagger}}\right)\partial^{\lambda}\mathbb{P}_{ba}$
(2)
$\displaystyle-i\frac{2g}{f_{\pi}}v^{\alpha}\varepsilon_{\alpha\mu\nu\lambda}{\mathcal{P}}_{b}^{*\mu}{\mathcal{P}}_{a}^{*\lambda{\dagger}}\partial^{\nu}\mathbb{P}_{ba},$
$\displaystyle\mathcal{L}_{{\mathcal{P}}^{(*)}{\mathcal{P}}^{(*)}\mathbb{V}}$
$\displaystyle=$ $\displaystyle-\sqrt{2}\beta
g_{V}{\mathcal{P}}_{b}\cdot{\mathcal{P}}_{a}^{{\dagger}}v\cdot\mathbb{V}_{ba}$
(3) $\displaystyle-2\sqrt{2}\lambda
g_{V}v^{\lambda}\varepsilon_{\lambda\mu\alpha\beta}\left({\mathcal{P}}_{b}{\mathcal{P}}^{*\mu{\dagger}}_{a}+{\mathcal{P}}_{b}^{*\mu}{\mathcal{P}}_{a}^{{\dagger}}\right)\partial^{\alpha}\mathbb{V}^{\beta}_{ba}$
$\displaystyle-\sqrt{2}\beta
g_{V}{\mathcal{P}}_{b}^{*}\cdot{\mathcal{P}}_{a}^{*{\dagger}}v\cdot\mathbb{V}_{ba}$
$\displaystyle-i2\sqrt{2}\lambda
g_{V}{\mathcal{P}}_{b}^{*\mu}{\mathcal{P}}_{a}^{*\nu{\dagger}}\left(\partial_{\mu}\mathbb{V}_{\nu}-\partial_{\nu}\mathbb{V}_{\mu}\right)_{ba}$

with the pseudoscalar mesons fields $\mathcal{P}^{T}=(D^{+},~{}D^{0})$ and
vector mesons fields $\mathcal{P}^{*T}=(D^{*+},~{}D^{*0})$.
$v=(1,\textbf{0})$. The light pseudoscalar meson matrix $\mathbb{P}$ and the
light vector meson matrix $\mathbb{V}_{\mu}$ have the conventional forms of

$\displaystyle\left.\begin{array}[]{c}{\mathbb{P}}={\left(\begin{array}[]{ccc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&K^{0}\\\
K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}}\eta\end{array}\right)},\\\
{\mathbb{V}}_{\mu}={\left(\begin{array}[]{ccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{array}\right)}_{\mu},\end{array}\right.$ (12)

respectively. We employ the coupling constants $g_{s}=0.76$ Machleidt:1987hj ;
Wang:2019nwt , $g=0.59$ Isola:2003fh ; Zyla:2020zbs , $f_{\pi}=0.132$ GeV,
$\beta=0.9$ Isola:2003fh , $\lambda=0.56~{}\text{GeV}^{-1}$ Isola:2003fh , and
$g_{V}=5.8$.

When we consider the isospin breaking effect and the $S-D$ wave mixing effect,
the discussed channels for the coupled $D^{0}D^{*+}/D^{+}D^{*0}$ systems
include $D^{0}D^{*+}({}^{3}S_{1})$, $D^{0}D^{*+}({}^{3}D_{1})$,
$D^{+}D^{*0}({}^{3}S_{1})$, and $D^{+}D^{*0}({}^{3}D_{1})$. The OBE effective
potentials for the coupled $D^{0}{D}^{*+}/D^{+}D^{*0}$ system with
$J^{P}=1^{+}$ read as

$\displaystyle V=\left(\begin{array}[]{cc}\mathcal{V}^{D^{0}{D}^{*+}\to
D^{0}D^{*+}}&\mathcal{V}^{D^{+}{D}^{*0}\to D^{0}D^{*+}}\\\
\mathcal{V}^{D^{0}{D}^{*+}\to D^{+}D^{*0}}&\mathcal{V}^{D^{+}{D}^{*0}\to
D^{+}D^{*0}}\end{array}\right),$ (15)

with

$\displaystyle\mathcal{V}^{D^{0}{D}^{*+}\to D^{0}D^{*+}}$ $\displaystyle=$
$\displaystyle-g^{2}_{s}\mathcal{Y}_{\sigma}+\frac{g^{2}}{3f^{2}_{\pi}}\mathcal{Z}_{\pi
0}^{\prime}-\frac{1}{4}\beta^{2}g_{V}^{2}\left(\mathcal{Y}_{\rho}-\mathcal{Y}_{\omega}\right)$
(16) $\displaystyle+\frac{2}{3}\lambda^{2}g^{2}_{V}\mathcal{X}_{\rho 0},$
$\displaystyle\mathcal{V}^{D^{+}{D}^{*0}\to D^{0}D^{*+}}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{6f^{2}_{\pi}}\left(\mathcal{Z}_{\pi
2}-\frac{1}{3}\mathcal{Z}_{\eta
2}\right)+\frac{1}{2}\beta^{2}g_{V}^{2}\mathcal{Y}_{\rho}$ (17)
$\displaystyle-\frac{1}{3}\lambda^{2}g^{2}_{V}\left(\mathcal{X}_{\rho
2}-\mathcal{X}_{\omega 2}\right),$ $\displaystyle\mathcal{V}^{D^{+}{D}^{*0}\to
D^{+}D^{*0}}$ $\displaystyle=$
$\displaystyle-g^{2}_{s}\mathcal{Y}_{\sigma}+\frac{g^{2}}{3f^{2}_{\pi}}\mathcal{Z}_{\pi
1}-\frac{1}{4}\beta^{2}g_{V}^{2}\left(\mathcal{Y}_{\rho}-\mathcal{Y}_{\omega}\right)$
(18) $\displaystyle+\frac{2}{3}\lambda^{2}g^{2}_{V}\mathcal{X}_{\rho 1}.$

The variables in the above equations are

$\displaystyle\left.\begin{array}[]{ll}q_{0}=m_{D^{*+}}-m_{D^{0}},&q_{1}=m_{D^{*0}}-m_{D^{+}},\\\
q_{2}=\frac{m_{D^{*+}}^{2}+m_{D^{+}}^{2}-m_{D^{0}}^{2}-m_{D^{*0}}^{2}}{2(m_{D^{0}}+m_{D^{*+}})},\\\
\Lambda_{i}^{2}=\Lambda^{2}-q_{i}^{2},&m_{\rho/\omega
i}^{2}=m_{\rho/\omega}^{2}-q_{i}^{2},\\\ m_{\pi
0}^{2}=q_{0}^{2}-m_{\pi^{+}}^{2},&m_{\pi 1}=m_{\pi^{+}}^{2}-q_{1}^{2},\\\
m_{\pi 2}=m_{\pi^{0}}^{2}-q_{2}^{2},&m_{\eta
2}=m_{\eta}^{2}-q_{2}^{2}.\end{array}\right.$

In the above expressions, we have defined several useful functions,

$\displaystyle\mathcal{Y}_{E}$ $\displaystyle=$
$\displaystyle\mathcal{O}_{1}[J^{P}]Y(\Lambda,m_{E},r)$ $\displaystyle=$
$\displaystyle\mathcal{O}_{1}[J^{P}]\left(\frac{e^{-m_{E}r}-e^{-\Lambda^{2}r}}{4\pi
r}-\frac{\Lambda^{2}-m^{2}_{E}}{8\pi\Lambda}e^{-\Lambda r}\right),$
$\displaystyle\mathcal{Z}_{Ei}$ $\displaystyle=$
$\displaystyle\left(\mathcal{O}_{1}[J^{P}]\nabla^{2}+\mathcal{O}_{2}[J^{P}]r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial r}\right)Y(\Lambda_{i},m_{Ei},r),$
$\displaystyle\mathcal{Z}_{Ei}^{\prime}$ $\displaystyle=$
$\displaystyle\left(\mathcal{O}_{1}[J^{P}]\nabla^{2}+\mathcal{O}_{2}[J^{P}]r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial r}\right)$
$\displaystyle\times\left(\frac{\cos(m_{Ei}r)-e^{-\Lambda_{i}r}}{4\pi
r}-\frac{\left(\Lambda_{i}^{2}+m_{Ei}^{2}\right)\exp(-\Lambda_{i}r)}{8\pi\Lambda_{i}}\right),$
$\displaystyle\mathcal{X}_{Ei}$ $\displaystyle=$
$\displaystyle\left(2\mathcal{O}_{1}[J^{P}]\nabla^{2}-\mathcal{O}_{2}[J^{P}]r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial r}\right)Y(\Lambda_{i},m_{Ei},r).$

Here, $\mathcal{O}_{1}[J^{P}]$ and $\mathcal{O}_{2}[J^{P}]$ are the spin-spin
interactions $\bm{\epsilon}_{1}\cdot\bm{\epsilon}_{3}^{{\dagger}}$,
$\bm{\epsilon}_{1}\cdot\bm{\epsilon}_{4}^{{\dagger}}$ and the tensor force
operators
$S\left(\hat{r},\bm{\epsilon}_{1},\bm{\epsilon}_{3}^{{\dagger}}\right)$,
$S\left(\hat{r},\bm{\epsilon}_{1},\bm{\epsilon}_{4}^{{\dagger}}\right)$ with
$S(\hat{r},\bm{x},\bm{y})=3(\hat{r}\cdot\bm{x})(\hat{r}\cdot\bm{y})-(\bm{x}\cdot\bm{y})$,
respectively. In our calculations, these operators are replaced by numerical
matrices $\langle f|\mathcal{O}|i\rangle$, $|i\rangle$ and $\langle f|$ stand
for the spin-orbit wave functions for the initial and final states,
respectively, i.e.,

$\displaystyle\left.\begin{array}[]{c}\bm{\epsilon}_{1}\cdot\bm{\epsilon}_{3}^{{\dagger}}\\\
\bm{\epsilon}_{1}\cdot\bm{\epsilon}_{4}^{{\dagger}}\end{array}\right.\mapsto\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right),\quad\left.\begin{array}[]{c}S\left(\hat{r},\bm{\epsilon}_{1},\bm{\epsilon}_{3}^{{\dagger}}\right)\\\
S\left(\hat{r},\bm{\epsilon}_{1},\bm{\epsilon}_{4}^{{\dagger}}\right)\end{array}\right.\mapsto\left(\begin{array}[]{cc}0&-\sqrt{2}\\\
-\sqrt{2}&1\end{array}\right).$ (28)

Here, we can also find an approximate relation between the OPE effective
potentials, like $V_{\pi}^{D^{+}D^{*0}\to
D^{0}D^{*+}}\sim-2V_{\pi}^{D^{0}D^{*+}\to D^{0}D^{*+}}$, where the slight
deviation from the exact relation is due to the different exchanged momentum
of the exchanged mesons.

We first solve the coupled channel Shr$\ddot{\text{o}}$dinger equation and
obtain the bound state properties for the coupled $D^{0}D^{*+}/D^{+}D^{*0}$
system with $J^{P}=1^{+}$. When the cutoff takes a reasonable value
$\Lambda=1.16$ GeV, we find a very shallow bound state with a binding energy
$E=-259.90$ keV. Thus, its mass $M=m_{D^{0}}+m_{D^{*+}}+E$ overlaps with the
mass of the $T_{cc}^{+}$. In Figure 1, we present the radial wave functions
for the doubly charmed $D^{0}D^{*+}/D^{+}D^{*0}$ molecular state with
$J^{P}=1^{+}$. It is a typical loosely bound molecular state. Its root-mean-
square radius is 6.28 fm, which is much larger than the size of its
components. The $S-$wave $D^{0}D^{*+}$ and $D^{+}D^{*0}$ components are
dominant. Their probabilities $\int d^{3}r|\psi_{i}(r)|^{2}/\sum_{i}\int
d^{3}r|\psi_{i}(r)|^{2}$ are 72.52% and 25.85%, respectively.

Figure 1: The radial wave functions for the doubly charmed
$D^{0}D^{*+}/D^{+}D^{*0}$ molecular state with $J^{P}=1^{+}$.

We further study the phase shifts for the coupled $D^{0}D^{*+}/D^{+}D^{*0}$
systems with $J^{P}=1^{+}$ to search for the doubly charmed resonant
tetraquark state. Here, we adopt the same OBE effective potentials for the
coupled $D^{0}D^{*+}/D^{+}D^{*0}$ systems with $J^{P}=1^{+}$ and the same
cutoff value $\Lambda=1.16$ GeV. In general, a typical Breit-Wigner resonance
appears in the position with $\delta=(n+1/2)\pi$, where the cross section
$\sigma(E)$ reaches the maximum $\sigma_{\text{Max}}(E_{0})$, and $E_{0}$
corresponds to the mass of the resonance. The width of the resonance reads as
$\Gamma=2/\left(\frac{d\delta(E)}{{d}E}\right)_{E_{0}}$. As shown in the
Figure 2, there exists a doubly charmed resonance $T_{cc}^{\prime+}$ in the
phase shift for the $S-$ wave $D^{0}D^{*+}$ channel, its mass and width are

$\displaystyle m=3876~{}\text{MeV},\quad\Gamma=412~{}\text{keV},$ (29)

respectively. In fact, the $T_{cc}^{\prime+}$ state is not a shape-type
resonance but a Feshbach-type resonance. Once we turn off the contribution
from the $D^{+}D^{*0}$ channel, it disappears. Thus, the isospin breaking
effect plays a very important role in forming the $T_{cc}^{\prime+}$ state.

Figure 2: Phase shifts for the coupled $D^{0}D^{*+}/D^{+}D^{*0}$ system with
$J^{P}=1^{+}$. Here, we adopt the same cutoff value $\Lambda=1.16$ GeV, the
dotted line shows the mass position of the obtained doubly charmed resonance.

To summarize, we have obtained a loosely bound doubly charmed molecular
tetraquark and a doubly charmed resonance from the $D^{0}D^{*+}/D^{+}D^{*0}$
interactions with $J^{P}=1^{+}$. In Figure 3, we fit the $D^{0}D^{0}\pi^{+}$
mass spectrum with our obtained mass for the $T_{cc}^{+}$ molecule state and
our obtained mass and width for the $T_{cc}^{\prime+}$ resonance. We use the
experimental value $\Gamma=410$ keV for the decay width of the $T_{cc}^{+}$ in
the fitting LHCb:2021vvq . Our fit is consistent with the experimental data.
Therefore, after considering the isospin breaking effect, the $T_{cc}^{+}$
state matches the loosely bound doubly charmed molecular tetraquark
explanation very well. Since the probabilities ratio for the $S-$wave
$D^{0}D^{*+}$ and $D^{+}D^{*0}$ components is $2.8:1$, the isospin breaking
effect does play an important role in generating this doubly charmed molecular
tetraquark. Simultaneously, there exists a doubly charmed resonance
$T_{cc}^{\prime+}$ between the $D^{0}D^{*+}$ and $D^{+}D^{*0}$ mass
thresholds. These two doubly charmed states are the mixture of the
$D^{0}D^{*+}$ and $D^{+}D^{*0}$ components after considering the isospin
breaking effect, which satisfies

$\displaystyle\left(\begin{array}[]{c}|T_{cc}^{+}\rangle\\\
|T_{cc}^{\prime+}\rangle\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\text{cos}\theta&\text{sin}\theta\\\
-\text{sin}\theta&\text{cos}\theta\end{array}\right)\left(\begin{array}[]{c}|D^{0}D^{*+}\rangle\\\
|D^{+}D^{*0}\rangle\end{array}\right).$ (36)

By using the probability ratio for the $D^{0}D^{*+}$ and $D^{+}D^{*0}$
components in the $T_{cc}^{+}$ state, the mixing angle is $\theta=\pm
30.8^{\circ}$. In this scenario, the $S-$wave $D^{+}D^{*0}$ channel is the
dominant channel for the doubly charmed $T_{cc}^{\prime+}$ resonant
tetraquark.

With the same model, the authors of Ref. Li:2012ss  performed a coupled-
channel analysis on the $D^{(*)}D^{(*)}$ system without considering the
isospin breaking effects. They found that the OBE interactions from the
$DD^{*}$ state with $I(J^{P})=0(1^{+})$ is stronger attractive than those in
the isovector state. When we consider the isospin breaking effects, there
still exist two bound states, which corresponds to the $T_{cc}$ molecular
state and $T_{cc}^{\prime}$ resonance, as shown in the Eq. (36), the
probabilities for the $D^{0}D^{*+}$ and $D^{+}D^{*0}$ components are not the
same any more as in the without isospin breaking case.

Figure 3: The fit to the experimental data of the $D^{0}D^{0}\pi^{+}$ mass
spectrum through the obtained masses and width for the doubly molecule and the
doubly resonance. Here, $\Gamma_{1}^{*}=410$ keV is the experimental width of
the newly $T_{cc}^{+}$ LHCb:2021vvq . The dash lines label the $D^{0}D^{*+}$
and $D^{+}D^{*0}$ mass thresholds.

The decay behavior for the exotic state is very helpful to understand their
inner structures. Very recently, several groups discussed the strong and
electromagnetic decay behavior for the newly $T_{cc}^{+}$ state Meng:2021jnw ;
Ling:2021bir . The strong and electromagnetic decay amplitudes for the
$T_{cc}^{+}$ and $T_{cc}^{\prime+}$ can be expressed as

$\displaystyle\mathcal{A}_{T_{cc}^{+}\to D^{0}D^{0}\pi^{+}}$ $\displaystyle=$
$\displaystyle\text{cos}\theta\mathcal{A}_{D^{0}D^{*+}\to
D^{0}D^{0}\pi^{+}}+\text{sin}\theta\mathcal{A}_{D^{+}D^{*0}\to
D^{0}D^{0}\pi^{+}},$ $\displaystyle\mathcal{A}_{T_{cc}^{+}\to
D^{+}D^{0}\pi^{0}}$ $\displaystyle=$
$\displaystyle\text{cos}\theta\mathcal{A}_{D^{0}D^{*+}\to
D^{+}D^{0}\pi^{0}}+\text{sin}\theta\mathcal{A}_{D^{+}D^{*0}\to
D^{+}D^{0}\pi^{0}},$ $\displaystyle\mathcal{A}_{T_{cc}^{\prime+}\to
D^{0}D^{0}\pi^{+}}$ $\displaystyle=$
$\displaystyle-\text{sin}\theta\mathcal{A}_{D^{0}D^{*+}\to
D^{0}D^{0}\pi^{+}}+\text{cos}\theta\mathcal{A}_{D^{+}D^{*0}\to
D^{0}D^{0}\pi^{+}},$ $\displaystyle\mathcal{A}_{T_{cc}^{\prime+}\to
D^{+}D^{0}\pi^{0}}$ $\displaystyle=$
$\displaystyle-\text{sin}\theta\mathcal{A}_{D^{0}D^{*+}\to
D^{+}D^{0}\pi^{0}}+\text{cos}\theta\mathcal{A}_{D^{+}D^{*0}\to
D^{+}D^{0}\pi^{0}},$ $\displaystyle\mathcal{A}_{T_{cc}^{+}\to
D^{0}D^{+}\gamma}$ $\displaystyle=$
$\displaystyle\text{cos}\theta\mathcal{A}_{D^{0}D^{*+}\to
D^{0}D^{+}\gamma}+\text{sin}\theta\mathcal{A}_{D^{+}D^{*0}\to
D^{0}D^{+}\gamma},$ $\displaystyle\mathcal{A}_{T_{cc}^{\prime+}\to
D^{0}D^{+}\gamma}$ $\displaystyle=$
$\displaystyle-\text{sin}\theta\mathcal{A}_{D^{0}D^{*+}\to
D^{0}D^{+}\gamma}+\text{cos}\theta\mathcal{A}_{D^{+}D^{*0}\to
D^{0}D^{+}\gamma}.$

If we only consider the contributions from the tree diagram as shown in Ref.
Meng:2021jnw ; Li:2021zbw , we can obtain
$\mathcal{R}_{1}={\Gamma[T_{cc}^{+}\to
D^{0}D^{0}\pi^{+}]}/{\Gamma[T_{cc}^{\prime+}\to
D^{0}D^{0}\pi^{+}]}={\text{cos}^{2}\theta}:{\text{sin}^{2}\theta}=2.80$, which
explains why the significance for the $T_{cc}^{\prime+}$ state is a little
small compared to the $T_{cc}^{+}$. After neglecting the very small
contribution from the $D^{*+}\to D^{+}\gamma$ process,
$\mathcal{R}_{2}={\Gamma[T_{cc}^{+}\to
D^{0}D^{+}\gamma]}:{\Gamma[T_{cc}^{\prime+}\to
D^{0}D^{+}\gamma]}={\text{sin}^{2}\theta}:{\text{cos}^{2}\theta}=0.35$. For
the $D^{+}D^{0}\pi^{0}$ final states, the decay width ratio between these two
doubly charmed tetraquark is a little complicated due to undetermined partial
decay width $\Gamma[D^{*0}\to D^{0}\pi^{0}]$. Assuming that $\Gamma[D^{*0}\to
D^{0}\pi^{0}]$ is of the same order or even larger than $\Gamma[D^{*0}\to
D^{0}\pi^{0}]$, $\mathcal{R}_{3}={\Gamma[T_{cc}^{+}\to
D^{+}D^{0}\pi^{0}]}/{\Gamma[T_{cc}^{\prime+}\to D^{+}D^{0}\pi^{0}]}$ is less
than 0.7. Therefore, the $D^{0}D^{+}\gamma$ and $D^{+}D^{0}\pi^{0}$ decay mode
shall be the prime channels to search for the $T_{cc}^{\prime+}$ state.

Summary—At the European Physical Society conference on high energy physics
2021, the LHCb Collaboration reported the observation of the doubly charmed
tetraquark $T_{cc}^{+}$ in the $D^{0}D^{0}\pi^{+}$ mass spectrum. Stimulated
by its very near threshold property, we perform an isospin breaking effect
analysis on the $D^{0}D^{*+}/D^{+}D^{*0}$ interactions. We adopt the OBE model
and consider the $S-D$ wave mixing effect. Our results indicate the newly
observed $T_{cc}^{+}$ is consistent with the $D^{0}D^{*+}/D^{+}D^{*0}$ doubly
charmed molecular tetraquark with $J^{P}=1^{+}$, while the probabilities for
the $S-$wave $D^{0}D^{*+}$ and $D^{+}D^{*0}$ components are 72.51% and 25.85%,
respectively.

Using the same OBE effective potentials, we obtain a coupled
$D^{0}D^{*+}/D^{+}D^{*0}$ doubly charmed resonance $T_{cc}^{\prime+}$, whose
mass and decay width are $m=3876~{}\text{MeV}$, $\Gamma=412~{}\text{keV}$,
respectively. Based on the $T_{cc}^{+}$ and $T_{cc}^{\prime+}$ being the
mixture of the $D^{0}D^{*+}$ and $D^{+}D^{*0}$ components, we further discuss
their strong and electromagnetic decay properties. Our quantitative analysis
indicates that it is a little difficult to identify the predicted
$T_{cc}^{\prime+}$ doubly charmed tetraquark in the $D^{0}D^{0}\pi^{+}$.
However, it is promising to search for the $T_{cc}^{\prime+}$ state in the
$D^{0}D^{+}\gamma$ and $D^{+}D^{0}\pi^{0}$ decay modes.

We strongly encourage our experimental colleague to pay more attention to the
structure between the $D^{0}D^{*+}$ and $D^{+}D^{*0}$ thresholds with more
precise data. If this substructure can be confirmed in the near future, it
shall provide very strong evidence of the existence of the doubly charmed
molecules.

Acknowledgments—This work is supported by the National Natural Science
Foundation of China under Grants 11975033 and 12070131001, the China National
Funds for Distinguished Young Scientists under Grant No. 11825503, National
Key Research and Development Program of China under Contract No.
2020YFA0406400, and the 111 Project under Grant No. B20063, the Fundamental
Research Funds for the Central Universities under Grants No.
lzujbky-2021-sp24. R. C. is supported by the National Postdoctoral Program for
Innovative Talent.

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