Tentative estimates of $\mathcal{B}(X(3872)\to {\pi }^0{\pi }^0{\chi }_{c1})$ and $\mathcal{B}(X(3872)\to {\pi }^{+}{\pi }^{-}{\chi }_{c1})$

Abstract

The rates of the $X(3872)\to {\pi }^0{\pi }^0{\chi }_{c1}$ and $X(3872)\to {\pi }^{+}{\pi }^{-}{\chi }_{c1}$ decays are estimated in the model of the triangle loop diagrams with charmed $D^{\ast }\bar{D}D$ and ${\bar{D}}^{\ast }D\bar{D}$ mesons in the loops. There are the triangle logarithmic singularities in the physical region of the $X(3872)\to {\pi }^0{\pi }^0{\chi }_{c1}$ decay which manifest themselves as narrow peaks in the ${\pi }^0{\chi }_{c1}$ mass spectrum near the $D^0{\bar{D}}^0$ threshold. The model predicts approximately the same branching fractions of the $X(3872)\to {\pi }^0{\pi }^0{\chi }_{c1}$ and $X(3872)\to {\pi }^{+}{\pi }^{-}{\chi }_{c1}$ decays at the level of about (0.8-1.7)$\times 10^{-4}$. A distinct prediction of the model is the value of the ratio $\mathcal{R}=\mathcal{B}(X(3872)\to {\pi }^{+}{\pi }^{-}{\chi }_{c1})/\mathcal{B}(X(3872)\to {\pi }^0{\pi }^0{\chi }_{c1})\approx 1.1$. It weakly depends on the $X(3872)$ resonance parameters and indicates a significant violation of the isotopic symmetry according to which one would expect $\mathcal{R}=2$.

Comments: 11 pages, 6 figures; v2, clarifications added, published in Phys. Rev. D