Triangle mechanisms for \(B^0\rightarrow (J/\psi \pi ^+\pi ^-) K^+\pi ^-\) 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 depend 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 \(B^0\rightarrow (J/\psi \pi ^0\pi ^-) K^+\pi ^0\). A similar triangle singularity exists and generates an \(X^-(3876)\)-like peak.

研究了\(B^0\rightarrow (J/\psi \pi ^+\pi ^-) K^+\pi ^-\) 的三角机制。实验中，在此过程中观察到X (3872) 峰。当最终的\((J/\psi \pi ^+\pi ^-)\pi \)不变质量在\(D^*\bar{D}^*\)阈值附近时，三角机制之一导致三角形奇点，并在\(J/\psi \pi ^+\pi ^-\)不变质量分布中产生类似X (3872) 的尖峰。Breit-Wigner 质量和光谱拟合宽度为 3871.68 MeV（比\(D^{*0}\bar{D}^0\)阈值高几 keV ）和\(\sim \) 分别为 0.4 MeV。这些 Breit-Wigner 参数几乎不依赖于模型参数的选择。与精确测量的X (3872) 质量和宽度\(3871.69\pm 0.17\)  MeV 和\(< 1.2\)  MeV 相比，一致性是显着的。在研究来自该过程的X (3872) 信号时，必须提前了解这种非共振贡献。我们还研究了类似电荷的过程\(B^0\rightarrow (J/\psi \pi ^0\pi ^-) K^+\pi ^0\)。一个相似的三角形奇点存在并产生一个\(X^-(3876)\) -like 峰值。