An X(3872)-Like Peak in $$B\rightarrow (J/\psi \pi ^+\pi ^-) K\pi $$ Due to Triangle Singularity

Triangle mechanisms for $B^0\to (J/\psi\pi^+\pi^-) K^+\pi^-$ are studied. Experimentally, an $X(3872)$ peak has been observed in this process. When the final $(J/\psi\pi^+\pi^-)\pi$ invariant mass is around $D^*\bar D^*$ threshold, one of triangle causes a singularity and generates sharp $X(3872)$-like $J/\psi\pi^+\pi^-$ distribution. The Breit-Wigner width fitted to spectrum 3871.68 MeV (a few keV above $D^{*0}\bar{D}^0$ threshold) $\sim$0.4 MeV, respectively.These parameters hardly depends on choice model parameters. Comparing with precisely measured width, $3871.69\pm 0.17$ $< 1.2$ agreement remarkable. studying signal from process, non-resonant contribution be understood advance. We also study charge analogous process (J/\psi\pi^0\pi^-) K^+\pi^0$. A similar exists $X^-(3876)$-like peak.