Heavy-meson molecules with dynamical pions

Abstract

We investigate heavy-meson molecules by treating the pion as a dynamical degree of freedom. We construct an energy-independent Hamiltonian that couples the two- and three-body Fock spaces. The two-body sector contains a pseudoscalar meson and a vector meson $H_1{H}_2^{\ast }$, while the three-body sector consists of two pseudoscalar mesons and a pion $H_1{H}_2\pi$. The $H^{\ast }$ self-energy, the $H_1{H}_2^{\ast }$ pion-exchange interaction, and the $H_1{H}_2\pi$ three-body propagation are consistently generated through the coupling between the two Fock sectors. We use the Gaussian expansion and complex scaling methods to solve the coupled Schrödinger equation. Previous approaches usually reduce three-body effects to energy-dependent two-body kernels and solve the equations iteratively. Our approach obtains the poles by solving a single linear generalized eigenvalue problem. The eigenvectors also provide explicit wave functions in both Fock sectors, allowing a qualitative analysis of channel weights and spatial structure. We benchmark the framework in $T_{cc}^{+}$ and $X(3872)$. The benchmark results are consistent with the current experimental pole determinations. We then apply this formalism to the $BB\pi$, $B\bar{B}\pi$, $DB\pi$ and $D\bar{B}\pi$ systems. We predict one isoscalar bound state in each of these systems. The $DB\pi$ and $D\bar{B}\pi$ systems also contain resonances lying below and above their three-body threshold, respectively. We propose complementary searches for these resonances through $D{B}^{\ast }\to DB\gamma$ and $D\bar{B}\pi$. This coupled Fock-space framework can be readily extended to other few-body systems in which unstable constituents can decay and change the particle number.

arXiv comment18 pages, 10 figures