Weakly-bound hadronic molecule near a 3-body threshold

The X(3872) seems to be a loosely-bound hadronic molecule whose constituents are two charm mesons. A novel feature of this molecule is that the mass difference of the constituents is close to the mass of a lighter meson that can be exchanged between them, namely, the pion. We analyze this feature in a simple model with spin-0 mesons only. Various observables are calculated to next-to-leading order in the interaction strength of the exchanged meson. Renormalization requires summing a geometric series of next-to-leading order corrections. The dependence of observables on the ultraviolet cutoff can be removed by renormalizations of the mass of the heaviest meson, the coupling constant for the contact interaction between the heavy mesons, and short-distance coefficients in the operator product expansion. The next-to-leading order correction has an unphysical infrared divergence at the threshold of the two heavier mesons that can be eliminated by a further resummation that takes into account the nonzero width of the heaviest meson.