Decay $X(3872)\to {\pi }^0{\pi }^{+}{\pi }^{-}$ and $S$-wave $D^0{\bar{D}}^0\to {\pi }^{+}{\pi }^{-}$ scattering length

Abstract

The isospin-breaking decay $X(3872)\to (D^{\ast }\bar{D}+{\bar{D}}^{\ast }D)\to {\pi }^{0}D\bar{D}\to {\pi }^0{\pi }^{+}{\pi }^{-}$ is discussed. In its amplitude there is a triangle logarithmic singularity, due to which the dominant contribution to $BR(X(3872)\to {\pi }^0{\pi }^{+}{\pi }^{-})$ comes from the production of the ${\pi }^{+}{\pi }^{-}$ system in a narrow interval of the invariant mass $m_{{\pi }^{+}{\pi }^{-}}$ near the value of $2m_{D^0}\approx 3.73$ GeV. The analysis shows that $BR(X(3872)\to {\pi }^0{\pi }^{+}{\pi }^{-})$ can be expected at the level of $10^{-3}$--$10^{-4}$. This estimate includes, in particular, the assumption that the $S$-wave inelastic scattering length $|{\alpha }_{D^0{\bar{D}}^0\to {\pi }^{+}{\pi }^{-}}^{\prime \prime }|\approx 1/(2m_{D^{\ast +}})\approx 0.25\text{mbox}{GeV}^{-1}$.

Comments: 7 pages, 6 figures; v2: journal version; clarifications and references added