Melting down a tetraquark: $D^{\ast}D^{(\ast)}$ interactions and $T_{cc}(3875)^+$ in a hot environment

We discuss the modification of the properties of the tetraquark-like $T_{cc}(3875)^+$ and its heavy quark spin partner, $T_{cc}(4016)^{*+}$ immersed in a hot bath of pions. We consider these exotic states as purely isoscalar $DD^\ast$ and $D^\ast D^\ast$ $S$-wave bound states, respectively. Finite temperature effects are incorporated through the $D$ and $D^\ast$ state-of-the-art thermal spectral functions calculated in [G. Montana et al., Phys. Rev. D, 102 (2020) 096020], using the imaginary-time formalism. We find important modifications of the $DD^\ast$ and $D^\ast D^\ast$ scattering amplitudes already for $T=80$ MeV, and show that the hot-bath lineshapes of these tetraquark-like states strongly depend on their Weinberg molecular content. We find that the thermal $T_{cc}(3875)^+$ and $T_{cc}(4016)^{*+}$ spectral functions change more rapidly with temperature for high molecular probabilities $P_0$. For large values of $P_0$, the widths significantly increase with temperature, leading to the melting of these exotic states for temperatures larger than 80 MeV. For small molecular components, the changes in the spectral functions of these states due to temperature become significantly less important. All these results show that any future experimental determination of the $D^{(\ast)}D^*$ scattering amplitudes at finite temperature will provide valuable insights into the molecular content of the $T_{cc}(3875)^+$ and $T_{cc}(4016)^{*+}$ exotics.