A pion-driven near-threshold enhancement in the $π\,T_{cc}$ system

Hadronic molecules are commonly viewed as products of near-threshold two-body dynamics. We investigate whether an experimentally established molecule can also serve as a building block for a more complex exotic hadron by studying pion scattering off $T^+_{cc}(3875)$. Treating $T_{cc}$ as a correlated isoscalar $D D^*$ molecular cluster, we describe the resulting $\pi D D^*$ dynamics in the $I(J^P)=1(1^-)$ channel within the Fixed Center Approximation to the Faddeev equations. The full three-body amplitude develops a pronounced enhancement slightly above the $\pi T_{cc}$ threshold, centered at $M_{\rm peak}\simeq 4055$~MeV, with an apparent width of approximately $60$~MeV. The associated behavior of the real and imaginary parts of the amplitude is compatible with a resonant-like interpretation. These results show that the lightest hadron can play an active dynamical role in promoting an observed hadronic molecule into a building block of a heavier near-threshold structure generated by the three-body dynamics. We propose searching for this enhancement in the $T_{cc}\pi$ invariant-mass distribution, particularly through the $D^0D^0\pi^+\pi^-$ final state.