The $X(3872)\to J\mathrm{/}\psi \pi \gamma$ and $X(3872)\to J\mathrm{/}\psi \pi \pi \gamma$ decays

We study the $\rho$ and $\omega$ meson contribution to the radiative decays $X(3872)\to J\mathrm{/}\psi \pi \gamma$ and $X(3872)\to J\mathrm{/}\psi \pi \pi \gamma$. The $X(3872)\to J\mathrm{/}\psi \pi \gamma$ is dominated by the $\omega$ meson. As for the $X(3872)\to J\mathrm{/}\psi \pi \pi \gamma$, the contributions of the cascade decays through the $\rho$ and $\omega$ mesons are strongly suppressed with respect to the diagrams which proceed either through the $\psi (2S)$ or the three body decay of $\rho$. The branching ratios of $X(3872)\to J\mathrm{/}\psi \pi \gamma$ and $X(3872)\to J\mathrm{/}\psi \pi \pi \gamma$ are $(8.10_{-2.88}^{+3.50})\times {10}^{-3}$ and $(2.38\pm 1.06)\mathrm{\%}$, which may be accessible by the BESIII and LHCb Collaborations. Especailly, the $X(3872)\to J\mathrm{/}\psi \pi \gamma$ and $X(3872)\to J\mathrm{/}\psi {\pi }^{+}{\pi }^{-}\gamma$ decays can be employed to extract the couplings $g_{X\psi \omega }$ and $g_{X\psi \rho }$, which probe the isoscalar and isovector components of the X(3872) wave function respectively.