Decay X(3872) → π0π+π- and S-wave D0<(D)over bar>0 → π+π- scattering length

The isospin-breaking decay X(3872) -> (D*(D) over bar+(D) over bar *D)-> pi D-0 (D) over bar -> 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)->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(D0) approximate to 3.73 GeV. The analysis shows that BR(X(3872)->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 vertical bar alpha(D0 (D) over bar0 ->pi+pi-)''vertical bar approximate to 1/(2m(D*+)) approximate to 0.25 GeV-1.