One-photon decay of the tetraquark state $X(3872)\to \gamma +J/\psi$ in a relativistic constituent quark model with infrared confinement

We further explore the consequences of treating the $X(3872)$ meson as a tetraquark bound state by analyzing its one-photon decay $X\to \gamma +J/\psi$ in the framework of our approach developed in previous papers which incorporates quark confinement in an effective way. To introduce electromagnetism we gauge a nonlocal effective Lagrangian describing the interaction of the $X(3872)$ meson with its four constituent quarks by using the $P$-exponential path-independent formalism. We calculate the matrix element of the transition $X\to \gamma +J/\psi$ and prove its gauge invariance. We evaluate the $X\to \gamma +J/\psi$ decay width and the longitudinal/transverse composition of the $J/\psi$ in this decay. For a reasonable value of the size parameter of the $X(3872)$ meson we find consistency with the available experimental data. We also calculate the helicity and multipole amplitudes of the process, and describe how they can be obtained from the covariant transition amplitude by covariant projection.

Figure 1

Feynman diagrams describing the decay $X\to \gamma +J/\psi$.

Figure 2

The dependence of the decay widths $\Gamma (X_l\to \gamma +J/\psi )$ and $\Gamma (X_l\to J/\psi 2\pi )$ on the size parameter ${\Lambda }_X$.