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 $\mathrm{\Gamma }[X(3872)\to {\chi }_{cJ}{\pi }^0]$ 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 $I=0$ $D{\overline{D}}^{*}+\mathrm{c}.\mathrm{c}$ 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 the 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 the effective theory.

• Received 21 May 2015

DOI:https://doi.org/10.1103/PhysRevD.92.034019