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We compare two existing approaches to calculating the decay of molecularquarkonium 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 themolecular quarkonium proceeds through a triangle diagram with charmed mesons inthe loop. We argue this approach predicts excessively large rates for$\Gamma[X(3872) \to \chi_{cJ}\pi^0]$ unless both charged and neutral mesons areincluded and a cancellation between these contributions is arranged to suppressthe decay rates. This cancellation occurs naturally if the $X(3872)$ isprimarily in the $I=0$ $D \bar{D}^{*} +c.c.$ scattering channel. Thefactorization approach to molecular decays calculates the rates in terms oftree-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 equivalentto hadronic loops approach if the cutoff on the loop integrations is taken tobe a few hundred MeV or smaller, as is appropriate when the charged $D$ mesonshave been integrated out of XEFT.

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