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

TL;DR

This paper compares hadronic loop and factorization methods for calculating the decay of the $X(3872)$ into $ ext{chi}_{cJ} ext{pi}^0$, highlighting their equivalence under certain conditions and implications for decay rate predictions.

Contribution

It demonstrates the equivalence of hadronic loop and factorization approaches in EFT calculations of $X(3872)$ decays when using a small cutoff, clarifying the conditions for accurate decay rate predictions.

Findings

Hadronic loop approach predicts large decay rates unless charged and neutral mesons are included.

Cancellation between charged and neutral meson contributions occurs naturally for $I=0$ $D ar{D}^*$ scattering.

Factorization approach is equivalent to hadronic loops with a small cutoff in loop integrations.

Abstract

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.