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

The isospin-breaking decay $X(3872)\to(D^*\bar D+ \bar D^D) \to\pi^0D\bar D\to\pi^0\pi^+\pi^-$ 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 $|\alpha’'{D^0\bar D^0 \to\pi^+\pi^-}|\approx1/(2m{D^{+}})\approx0.25\ \mbox{GeV}^{-1}$.