Binding energy of the $X(3872)$ at unphysical pion masses

TL;DR

This paper investigates how the binding energy of the $X(3872)$ particle depends on pion mass variations using a renormalisable chiral effective field theory, revealing the disappearance of the bound state at large pion masses and the importance of pion dynamics.

Contribution

It introduces a parameter-free leading-order calculation within a modified Weinberg formulation to study the pion mass dependence of the $X(3872)$ binding energy, highlighting the role of higher-order interactions and pion dynamics.

Findings

$X$-pole disappears at large unphysical pion masses.

Higher-order contact interactions can strengthen the bound state at physical pion masses.

Pion dynamics and $Dar{D}\pi$ effects are crucial for chiral extrapolations.

Abstract

Chiral extrapolation of the $X(3872)$ binding energy is investigated using the modified Weinberg formulation of chiral effective field theory for the $D \bar{D}^*$ scattering. Given its explicit renormalisability, this approach is particularly useful to explore the interplay of the long- and short-range $D \bar{D}^*$ forces in the $X(3872)$ from studying the light-quark (pion) mass dependence of its binding energy. In particular, the parameter-free leading-order calculation shows that the $X$-pole disappears for unphysical large pion masses. On the other hand, without contradicting the naive dimensional analysis, the higher-order pion-mass-dependent contact interaction can change the slope of the binding energy at the physical point yielding the opposite scenario of a stronger bound $X$ at pion masses larger than its physical value. An important role of the pion dynamics and of the 3-body $D\bar{D}\pi$ effects for chiral extrapolations of the $X$-pole is emphasised. The results of the present study should be of practical value for the lattice simulations since they provide a non-trivial connection between lattice points at unphysical pion masses and the physical world.