Revisit the isospin violating decays of $X(3872)$

In this work, we revisit the isospin violating decays of $X(3872)$ in a coupled-channel effective field theory. In the molecular scheme, the $X(3872)$ is interpreted as the bound state of $\bar{D}^{*0}D^0/\bar{D}^0D^{*0}$ and $D^{*-}D^+/D^-D^{*+}$ channels. In a cutoff-independent formalism, we relate the coupling constants of $X(3872)$ with the two channels to the molecular wave function. The isospin violating decays of $X(3872)$ are obtained by two equivalent approaches, which amend some deficiencies about this issue in literature. In the quantum field theory approach, the isospin violating decays arise from the coupling constants of $X(3872)$ to two di-meson channels. In the quantum mechanics approach, the isospin violating is attributed to wave functions at the origin. We illustrate that how to cure the insufficient results in literature. Within the comprehensive analysis, we bridge the isospin violating decays of $X(3872)$ to its inner structure. Our results show that the proportion of the neutral channel in $X(3872)$ is over $80\%$. As a by-product, we calculate the strong decay width of $X(3872)\to \bar{D}^0 D^0\pi^0$ and radiative one $X(3872)\to \bar{D}^0 D^0\gamma$. The strong decay width and radiative decay width are about 30 keV and 10 keV, respectively, for the binding energy from $-300$ keV to $-50$ keV.