Investigating the two-pion exchange of the double charm $DD^*$ chiral interactions and $T_{cc}$
Under chiral effective field theory, we study the $S$-wave $DD^*$ interactions up to second chiral order at one-loop level, which contain full contact, one-pion-exchange (OPE) and two-pion-exchange (TPE) contributions. Here, we adopt a new subtraction scheme of the two-particle-reducible contributions, and attempt different regularization schemes uniquely for the TPE contributions since they are highly divergent on the momentum transfer. Under these schemes we conclude that, in the $I=0$ channel, the TPE contribution is repulsive, then the competition between this powerful repulsing TPE and the other two (contact and OPE) results in a quite weak attraction. This explains why $T_{cc}$ has an extremely small binding energy if treated as the $I=0$ $DD^*$ bound state. This feature resembles that of hidden charm $D\bar{D}^*$ as we investigated in previous work [Phys. Rev. D 105, 034013 (2022)], which also interpreted the extremely near-threshold phenomenon of $X(3872)$. In addition, we also solve the Bethe-Salpeter equation with the chiral interactions for a consistency check.