Hadronic Loops versus Factorization in EFT calculations of $X(3872) \to \chi_{cJ} \pi^0$

We compare two existing approaches to calculating the decay of molecular quarkonium states to conventional quarkonia in effective field theory, using $X(3872) \to \chi_{cJ} \pi^0$ as an example. In one approach the decay of the molecular quarkonium proceeds through a triangle diagram with charmed mesons in the loop. We argue this approach predicts excessively large rates for $\Gamma[X(3872) \to \chi_{cJ}\pi^0]$ unless both charged and neutral mesons are included and a cancellation between these contributions is arranged to suppress the decay rates. This cancellation occurs naturally if the $X(3872)$ is primarily in the $I=0$ $D \bar{D}^{*} +c.c.$ scattering channel. The factorization approach to molecular decays calculates the rates in terms of tree-level transitions for the $D$ mesons in the $X(3872)$ to the final state, multiplied by unknown matrix elements. We show that this approach is equivalent to hadronic loops approach if the cutoff on the loop integrations is taken to be a few hundred MeV or smaller, as is appropriate when the charged $D$ mesons have been integrated out of XEFT.