Heavy hadron chiral perturbation theory ($\mathrm{HH}\chi \mathrm{PT}$) and XEFT are applied to the decays $X(3872)\to \psi (2S)\gamma$ and $\psi (4040)\to X(3872)\gamma$ under the assumption that the $X(3872)$ is a molecular bound state of neutral charm mesons. In these decays the emitted photon energies are 181 MeV and 165 MeV, respectively, so $\mathrm{HH}\chi \mathrm{PT}$ can be used to calculate the underlying $D^0{\overline{D}}^{0*}+{\overline{D}}^0{D}^{0*}\to \psi (2S)\gamma$ or $\psi (4040)\to (D^0{\overline{D}}^{0*}+{\overline{D}}^0{D}^{0*})\gamma$ transition. These amplitudes are matched onto XEFT to obtain decay rates. The decays receive contributions from both long-distance and short-distance processes. We study the polarization of the $\psi (2S)$ in the decay $X(3872)\to \psi (2S)\gamma$ and the angular distribution of $X(3872)$ in the decay $\psi (4040)\to X(3872)\gamma$ and find they can be used to differentiate between different decay mechanisms as well as discriminate between $2^{-+}$ and $1^{++}$ quantum number assignments of the $X(3872)$.

• Received 10 February 2011

DOI:https://doi.org/10.1103/PhysRevD.83.094009