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 $|\alpha''_{ D^0\bar D^0 \to\pi^+\pi^-}|\approx1/(2m_{ D^{ *+}})\approx0.25\ \mbox{ GeV}^{ -1}$.
Categories: hep-ph hep-ex nucl-th