Power counting for three-body decays of a near-threshold state

TL;DR

This paper introduces a novel power counting scheme for effective field theories of near-threshold states with unstable constituents, enabling systematic expansions and accurate threshold behavior modeling.

Contribution

It develops a new power counting method that treats small scales systematically, improving the theoretical description of near-threshold states like the X(3872).

Findings

Provides a consistent double expansion framework.

Ensures correct three-body threshold behavior.

Derives results consistent with previous approximations.

Abstract

We propose a new power counting for the effective field theory describing a near-threshold state with unstable constituents, such as the X(3872) meson. In this counting, the momenta of the heavy particles, the pion mass and the excitation energy of the unstable constituent -- the D* in the case of the X -- are treated as small scales, of order Q. The difference, delta, between the excitation energy of the D* and the pion mass is smaller than either by a factor ~20. We therefore assign delta an order Q^2 in our counting. This provides a consistent framework for a double expansion in both delta/m_pi and the ratio of m_pi to the high-energy scales in this system. It ensures that amplitudes have the correct behaviour at the three-body threshold. It allows us to derive, within an effective theory, various results which have previously been obtained using physically-motivated approximations.