Isospin violation effect and three-body decays of the Tcc⁺ state

In this work, we make a study of Tcc⁺ 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 π, ρ and ω exchanges are comparable while in the long range the pion-exchange contribution is dominant. Based on the assumption that Tcc⁺ 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*+D⁰ and D*0D⁺ is even larger than the binding energy of Tcc⁺, the isospin-breaking effect is amplified by the small binding energy of Tcc⁺. 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 Tcc⁺. Using the experimental value of the mass of Tcc⁺ as an input, we obtain the wave function of Tcc⁺ 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 Tcc⁺ is a hadronic molecule of D^*D.