Decay X(3872)→π0π+π− and S-wave D0D0→π+π− scattering length

The isospin-breaking decay X(3872)→(D*D + D*D)→π0DD→π0π+π− is discussed. In its amplitude there is a triangle logarithmic singularity, due to which the dominant contribution to BR(X(3872)→π0π+π−) comes from the production of the π+π− system in a narrow interval of the invariant mass mπ+π− near the value of 2mD0 ≈ 3.73 GeV. The analysis shows that BR(X(3872)→π0π+π−) 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 |αD0D0→π+π−′′|≈1/(2mD*+) ≈ 0.25 GeV−1.