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

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 Schr\"{o}dinger 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 $D^*D$.