Production of the weakly bound $T_{cc}^{+}$ state combining the coalescence model with its mass spectrum

We investigate the production of the doubly charmed molecular state $T_{cc}^{+}$ within the coalescence model. The formation probability is evaluated with the Wigner distribution function. In contrast to previous studies, which usually adopt a Gaussian wave function, we obtain the molecular wave function by numerically solving the coupled-channel Schrödinger equation with the one-boson-exchange potential fitted to the mass spectrum of $T_{cc}^{+}$. The resulting wave function, well described by a Lorentzian in momentum space (a Yukawa-type exponential in coordinate space), is the universal form of a weakly bound $S$-wave molecule and thus more suitable for such a state. As the input of the coalescence model, we use the transverse-momentum distributions of the constituent $D$ and $D^{\ast }$ mesons fitted to the ALICE and LHCb data, and also generated by PYTHIA for comparison. We compute the transverse-momentum distributions of $T_{cc}^{+}$ in $pp$ and Pb-Pb collisions. The resulting spectrum, together with its insensitivity to the binding energy, highlights the universal character of the coalescence production of a weakly bound molecule, which is governed by the long-range tail of the wave function rather than by short-range details. These results provide a theoretical reference for future experimental measurements of $T_{cc}^{+}$.