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