The role of the pion in the lineshape of the $X(3872)$

TL;DR

This paper investigates how long-range pion interactions influence the properties of the $X(3872)$, a loosely bound meson molecule, revealing they contribute to decay width but do not significantly alter the effective range.

Contribution

It provides a quantitative analysis of pion effects on the $X(3872)$ using non-relativistic quantum mechanics, highlighting their limited impact on the effective range and their role in decay processes.

Findings

Pion interactions do not produce a large negative effective range.

Pion interactions contribute to the decay width of $X(3872)$.

The approach applies to other exotic states like $T_{cc}^+$.

Abstract

We determine the contribution of long-range pion interactions to the $X(3872)$ dynamics, assuming it is a loosely bound $D^0 \bar{D}^{*0}$ molecule. Our result is based on the distorted wave Born approximation in non-relativistic quantum mechanics. Despite their long-range nature, we find that pion interactions cannot produce a large and negative effective range. Nonetheless, they introduce imaginary parts. In particular, they contribute to the total decay width of the $X(3872)$ with a term associated with, but not precisely corresponding to, the $D^*$ width. Our approach can also be applied to the recently discovered $T_{cc}^+$ states.