The $X(3872)\rightarrow J/ψπγ$ and $X(3872)\rightarrow J/ψππγ$ decays

We study the $ρ$ and $ω$ meson contribution to the radiative decays $X(3872)\rightarrow J/ψπγ$ and $X(3872)\rightarrow J/ψππγ$. The $X(3872)\rightarrow J/ψπγ$ is dominated by the $ω$ meson. As for the $X(3872)\rightarrow J/ψππγ$, the contributions of the cascade decays through the $ρ$ and $ω$ mesons are strongly suppressed with respect to the diagrams which proceed either through the $ψ(2S)$ or the three body decay of $ρ$. The branching ratios of $X(3872)\rightarrow J/ψπγ$ and $X(3872)\rightarrow J/ψππγ$ are $(8.10^{+3.50}_{-2.88})\times10^{-3}$ and $(2.38\pm1.06)\%$, which may be accessible by the BESIII and LHCb Collaborations. Especailly, the $X(3872)\rightarrow J/ψπγ$ and $X(3872)\rightarrow J/ψπ^+π^- γ$ decays can be employed to extract the couplings $g_{Xψω}$ and $g_{Xψρ}$, which probe the isoscalar and isovector components of the X(3872) wave function respectively.