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

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 δ between the excitation energy of the D∗ and the pion mass is smaller than either by a factor ∼20. We therefore assign δ an order Q2 in our counting. This provides a consistent framework for a double expansion in both δ/mπ and the ratio of mπ to the high-energy scales in this system. It ensures that amplitudes have the correct behavior 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.