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

## TL;DR

This paper analyzes the isospin-breaking decay of the X(3872) particle into three pions, highlighting the role of a triangle singularity and estimating the decay's branching ratio near 10^{-3} to 10^{-4}.

## Contribution

It introduces a detailed analysis of the decay process involving a triangle logarithmic singularity and estimates the branching ratio based on the scattering length.

## Key findings

- Decay BR estimated at 10^{-3} to 10^{-4}
- Dominant contribution from narrow invariant mass interval
- Triangle singularity significantly influences decay amplitude

## Abstract

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