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

Understand

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.