Detecting the pure triangle singularity effect through the $\psi (2S)\to p\overline{p}\eta /p\overline{p}{\pi }^0$ process

In this work, the triangle singularity mechanism is investigated in the $\psi (2S)\to p\overline{p}\eta /p\overline{p}{\pi }^0$ process. The triangle loop composed by $J/\psi$, $\eta$, and $p$ has a singularity in the physical kinematic range for the $\psi (2S)\to p\overline{p}\eta /p\overline{p}{\pi }^0$ process, and it would generate a very narrow peak in the invariant mass spectrum of $p\eta (\pi )$ around 1.56387 GeV, which is far away from both the threshold and relative resonances. In these processes, all the involved vertices are constrained by the experimental data. Thus, we can make a precise model independent prediction here. It turns out that the peak in the $p\eta$ invariant mass spectrum is visible, while it is very small in the $p{\pi }^0$ invariant mass spectrum. We expect this effect shown in $p\overline{p}\eta$ final state can be observed by the Beijing Spectrometer and Super Tau-Charm Facility in the future.

Figure 1

Kinematical mechanism of the production of a triangle singularity.

Figure 2

The Feynman diagrams describing the process $\psi (2S)\to p\overline{p}\eta /p\overline{p}\pi$. (a) Loop diagram where triangle singularity happens; (b) tree diagram that called “background."

Figure 3

The fitted results of the cross section of ${\pi }^{-}p\to \eta n$. The blue points are the experimental data given in Refs. [68, 69, 70, 71, 72, 73], the red solid line is our fitted result, and the black, orange, and purple dashed lines are the contributions of the background, $N(1535)$ and $N(1650)$, respectively.

Figure 4

The $p\eta$ invariant mass spectrum of ${\mathcal{M}}^{\text{Loop}}$. The green solid line represents that $\alpha$ in the form factor is 1 and that both $N(1535)$ and $N(1650)$ are included. The blue dashed line represents that $\alpha$ is 2 and that both $N(1535)$ and $N(1650)$ are included. The red dash-dot line represents that $\alpha$ is 1 and that only $N(1535)$ is considered.

Figure 5

The $p\eta$ invariant mass distributions of the $\psi (2S)\to p\overline{p}\eta$ process. For black solid line, we consider the interface between ${\mathcal{M}}^{\text{Loop}}$ and ${\mathcal{M}}^{\text{Tree}}$ with the relative phase angle being 0, and here both $N(1535)$ and $N(1650)$ exchange are included and $\alpha$ is set to 1. For the red dash-dot line, there includes one more tree diagram $\psi (2S)\to \eta (J/\psi \to \overline{p}p)$ with the relative phase angle being 0 and having a $m_{p\overline{p}}<3.067\mathrm{GeV}$ cut following Ref. [75].

Figure 6

The Dalitz plot of the $\psi (2S)\to p\overline{p}\eta$ process after considering the contribution of Fig. 2 (the vertical band), the tree diagrams $\psi (2S)\to \eta (J/\psi \to \overline{p}p)$ (the very thin band lies on the lower left corner) and $\psi (2S)\to p({\overline{N}}^{*}\to \overline{p}\eta )$ (the horizontal band). For the triangle diagram, only $N(1535)$ exchange is included and $\alpha$ is set to 1. Here, the red dashed line denotes the $m_{p\overline{p}}<3.067\mathrm{GeV}$ cut as given in Ref. [75]. To make all the bands visible, we increase the strength of triangle diagram by a factor ${10}^4$ and the $\overline{N}(1535)$ tree diagram by a factor ${10}^2$.

Figure 7

The tree diagram of $\psi (2S)\to p\overline{p}{\pi }^0$, where the contributions of $N(1440)$, $N(1520)$, $N(1535)$, and $N(1650)$ are included, and all the involved phase angles are 0.

Figure 8

The $\alpha$ dependence of the triangle singularity caused by the $J/\psi \eta p$ loop.

Figure 9

The $p{\pi }^0$ invariant mass distribution of the $\psi (2S)\to p\overline{p}{\pi }^0$ process after considering the interface between ${\mathcal{M}}^{\text{Loop}}$ and ${\mathcal{M}}^{\text{Tree}}$ (the phase angle is 0), where $\alpha$ is set to 1.

Figure 10

The peaks caused by the triangle singularity with different assumptions of the width of $J/\psi$.

Figure 11

The dependence behavior of $I$ on $\cos \theta$ for different values of $g$.