Proposal to detect a moving triangle singularity in $\psi(2S)\to\pi^{+}\pi^{-}K^{+}K^{-}$ process

Before presenting our numerical results, we need to determine the relevant coupling constants needed first. For the coupling constants $g_{\psi(2S)J/\psi\eta}$, $g_{J/\psi\rho^{0}\pi^{0}}$, $g_{\rho^{0}\pi^{+}\pi^{-}}$, $g_{a_{2}(1320)^{0}K^{+}K^{-}}$ and $g_{\psi(2S)a_{2}(1320)^{0}\rho^{0}}$, since experiments have measured the branching ratios of the corresponding processes Zyla:2020zbs , these coupling constants can be extracted from the data given in RPP Zyla:2020zbs , both of which are collected in Table 1.

For the $g_{\psi(2S)a_{0}(980)\rho^{0}}$, since the branching ratio of $\psi(2S)\to a_{0}(980)\rho$ process is still absent in RPP Zyla:2020zbs , by assuming that $J/\psi$ and $\psi(2S)$ have very similar properties, we naively estimate the coupling constants between $\psi(2S)$, $a_{0}(980)$, and $\rho$ by the following way. From RPP, we can get that the branching ratios of $J/\psi\to f_{0}(980)\omega$, $J/\psi\to f_{0}(980)\phi$, $J/\psi\to f_{0}(980)\phi\to\phi\pi^{+}\pi^{-}$, and $\psi(2S)\to f_{0}(980)\phi\to\phi\pi^{+}\pi^{-}$ processes are $(1.4\pm 0.5)\times 10^{-4}$, $(3.2\pm 0.9)\times 10^{-4}$, $(2.59\pm 0.34)\times 10^{-4}$, and $(7.5\pm 3.3)\times 10^{-5}$, respectively Zyla:2020zbs . Thus the branching ratio of $\psi(2S)\to a_{0}(980)\rho$ process might be estimated as

Then the coupling constant for the vertex of $\psi(2S)\to a_{0}(980)^{0}\rho^{0}$ is calculated as $g_{\psi(2S)a_{0}(980)^{0}\rho^{0}}=(9.6\pm 3.1)\times 10^{-4}$ GeV.

With all the preparations above, now we can present our numerical results. First of all, we want to show the behavior of the loop diagram of $\psi(2S)\to\pi^{+}\pi^{-}K^{+}K^{-}$ process. In Fig. 3, the Dalitz plot of the loop diagram only is presented, where a thin band totally caused by the triangle singularity can be clearly seen. In this figure, $m_{\pi^{+}\pi^{-}}$ and $m_{K^{+}K^{-}}$ are limited in [0.59 GeV, 0.96 GeV] and [1.04 GeV, 1.31 GeV] respectively. Here, the interval of $m_{\pi^{+}\pi^{-}}$ just covers the $\rho$ meson as $[m_{\rho}-1.3\Gamma_{\rho},m_{\rho}+1.3\Gamma_{\rho}]$, and the range of $m_{K^{+}K^{-}}$ is just around the range where the triangle singularity is happening. From Fig. 3, it is clear that when $m_{\pi^{+}\pi^{-}}$ is closer to the mass of $\rho$ meson, the brightness of the thin band is higher. In addition, when $m_{\pi^{+}\pi^{-}}$ is smaller than $m_{\rho}-\Gamma_{\rho}$ (0.62 GeV) or larger than $m_{\rho}+\Gamma_{\rho}$ (0.92 GeV), the color of the thin band is too gloomy to be distinguished from the region where the triangle singularity doesn’t happen, i.e., the strength of the triangle singularity is suppressed tremendously by the propagator of $\rho$ meson when the difference between $m_{\pi^{+}\pi^{-}}$ and $m_{\rho}$ is larger than $\Gamma_{\rho}$. Thus, in the loop diagram of $\psi(2S)\to\pi^{+}\pi^{-}K^{+}K^{-}$ process, when $m_{\pi^{+}\pi^{-}}$ changes in the interval [0.59 GeV, 0.96 GeV], it already contains most of the contribution of the triangle singularity. In the current calculation, we will just focus on the physics in this phase space range, and such cut will be perfect by reducing the interference from the background.

Additionally, a thin and bright curve in Fig. 3 covers almost $30$ MeV for the invariant mass of $K^{+}K^{-}$, which indeed proves our previous argument in Sec. II that there exists a moving triangle singularity in Fig. 2 $(a)$. When getting the coordinate of the point on the curve, we can easily find that when the value of the $m_{\pi^{+}\pi^{-}}$ is smaller, the $m_{K^{+}K^{-}}$ will be larger. Actually, this phenomena is easy to be understood from the physics of the triangle singularity. Since the mass of $\psi(2S)$ is fixed, for the $\psi(2S)\to J/\psi\eta$ process, the velocity of the emitted $\eta$ meson is also fixed. However, for the $J/\psi\to\rho^{0}\pi^{0}$ process, when the $m_{\pi^{+}\pi^{-}}$ is smaller, it is equivalent to having the mass of the emitted $\rho^{0}$ smaller, then it will cause the velocity of $\pi^{0}$ become larger. Obviously, under this situation $\pi^{0}$ can catch up with $\eta$ easier, which lets the position of triangle singularity in the invariant mass spectrum of $K^{+}K^{-}$ become larger.

To check the behavior of the triangle singularity in the $K^{+}K^{-}$ invariant mass spectrum, an integration on $m_{\pi^{+}\pi^{-}}$ is carried out for the intervals as $[m_{\rho}-\Gamma_{\rho},m_{\rho}+\Gamma_{\rho}]$, $[m_{\rho}-\frac{\Gamma_{\rho}}{2},m_{\rho}+\frac{\Gamma_{\rho}}{2}]$ and $[m_{\rho}-\frac{\Gamma_{\rho}}{4},m_{\rho}+\frac{\Gamma_{\rho}}{4}]$. The results are shown in Fig. 4. For each range of integration, the loop diagram (Fig. 2 $(a)$) produces a broad bump in the $K^{+}K^{-}$ invariant mass spectrum, and we find that the width of bumps will become narrower when $\Delta\Gamma$ is reduced. It indicates that this broad width depends on the integration range of $m_{\pi^{+}\pi^{-}}$, rather than on the triangle singularity only. Actually, the peak due to triangle singularity here should be very sharp, because the intermediate states, $J/\psi$, $\eta$ and $\pi^{0}$ are all very narrow. It was proven in our previous work on $\psi(2S)\to p\bar{p}\eta/p\bar{p}\pi^{0}$ processes Huang:2020kxf , where the triangle singularity caused by the pure $J/\psi\eta p$ loop generates a very sharp structure with width around $1$ MeV.

Now let us explain why the integration range of $m_{\pi^{+}\pi^{-}}$ affects the width of the bump in the invariant mass spectrum of $K^{+}K^{-}$. As we have mentioned in Sec. II, the change of $m_{\pi^{+}\pi^{-}}$ will cause the movement of the peak position due to the triangle singularity in the $K^{+}K^{-}$ invariant mass spectrum. In addition, the integration range of $m_{\pi^{+}\pi^{-}}$ is $[m_{\rho}-\Gamma_{\rho},m_{\rho}+\Gamma_{\rho}]$, where each value of $m_{\pi^{+}\pi^{-}}$ can have a triangle singularity. As a result, the broad structure actually is constructed by a series of narrow bumps purely due to the triangle singularities. Obviously, if we reduce $\Delta\Gamma$, due to the fact that the triangle singularity with $m_{\pi^{+}\pi^{-}}$ out of $[m_{\rho}-\Delta\Gamma,m_{\rho}+\Delta\Gamma]$ is suppressed tremendously, the triangle singularity located around $m_{K^{+}K^{-}}=1.17$ GeV, which is the $m_{EF}$ solved from Eq.(1) with the $m_{CD}=m_{\pi^{+}\pi^{-}}=m_{\rho}=0.77$ MeV, will give the largest contribution. At that time, the signal of the triangle singularity will be very sharp, which will give the same behavior as that in $\psi(2S)\to p\bar{p}\eta/p\bar{p}\pi^{0}$ processes Huang:2020kxf .

In Fig. 5, we continue to reduce $\Delta\Gamma$ to present the $K^{+}K^{-}$ invariant mass spectrum of the loop diagram (Fig. 2 $(a)$). The bump caused by triangle singularity becomes much sharper, however, its strength also becomes much smaller. In summary, a very important information that we can get from the above explanation is that the broad bump in Fig. 4 is actually a superposition of a series of sharp triangle singularities with different positions. Then, through changing the integration range of $m_{\pi^{+}\pi^{-}}$, i.e., the value of $\Delta\Gamma$, the width of the peak in the invariant mass spectrum of $K^{+}K^{-}$ can be adjusted to a suitable value. As Ref.Huang:2020kxf points out, the width is too narrow to be detected because of the limitation of the resolution of the detector. Thus, by comparing the structure at all $\Delta\Gamma$ as shown in Figs. 4 and 5, we find that $\Delta\Gamma=16$ MeV looks the best choice.

Finally, Fig. 6 presents the movement of the triangle singularity. Here, we carry out the integration on $m_{\pi^{+}\pi^{-}}$ within the interval $[m_{\rho}+\Delta m-\Delta\Gamma,m_{\rho}+\Delta m+\Delta\Gamma]$, where $\Delta m$ is the shift from the center mass of $\rho$ and $\Delta\Gamma$ is fixed as 16 MeV. As shown in this figure, when $\Delta m$ changes, the positions of the peaks caused by the triangle singularities change explicitly. Furthermore, when $|\Delta m|$ become larger, the strength of the peak is reduced because of the suppression from the propagator of $\rho$ meson. For each peak, the width is around $3$ MeV because of the same value of $\Delta\Gamma=16$ MeV used.

Apparently, to study if our triangle singularity in the $\psi(2S)\to\pi^{+}\pi^{-}K^{+}K^{-}$ process can be observed in experiments, discussions on the loop diagram only is far from enough. In principle, we should consider the contributions of the tree diagrams given by Fig. 2 (b), i.e., the background, to see the possibility.

Similarly as done in Sec. III.2, we also present the Dalitz plot first, and the result is given by Fig. 7. In this Dalitz plot, two bright spots located around 1 and 1.35 GeV are for the contributions of $M=a_{0}(980)$ and $a_{2}(1320)$ from the tree diagrams as shown in Fig. 2 (b), respectively. These two bright spots indicate that the contribution of tree diagram dominate in our interested phase space range. However, fortunately, in Fig. 7, we can still slightly see a subtle thin band at the same location of Fig. 3, which tells us that the effect caused by our triangle singularity may still be observable. Thus, next, our task is to find a proper way to make the signal of the triangle singularity still be visible after including the background.

In the first step, we draw the similar figure as Fig. 4 but including the background contribution as shown in Fig. 8. Unfortunately, the broad peaks due to the triangle singularity almost disappear and just leave some twists there. It is natural to continue reducing the value of $\Delta\Gamma$, because the peak structure of the loop becomes much sharper as shown in Fig.5. Then we set $\Delta\Gamma$ as 4 MeV, 8 MeV and 16 MeV to see the change of the $K^{+}K^{-}$ invariant mass spectrum, and the corresponding numerical results are given in Fig. 9. Now the peaks due to the triangle singularity are clearly seen even taking into account the contribution of the tree diagram, which means that it may be possible for experiments to observe our triangle singularity by adding cuts on the invariant mass of $\pi^{+}\pi^{-}$. This phenomenum is understandable. For each fixed value of $m_{\pi^{+}\pi^{-}}$, in the invariant mass spectrum of $K^{+}K^{-}$, the triangle singularity only dominates in a very small range. However, the background contribution is everywhere and almost flat. Then, through the integration, these background contributions are summed together for every point in the invariant mass spectrum of $K^{+}K^{-}$. Once we reduce the value of $\Delta\Gamma$ roughly with a factor 10, i.e., from $\Delta\Gamma=\Gamma_{\rho}\sim 140$ MeV to $16$ MeV, as shown by the dashed blue lines in Fig.4 and Fig.5, the background contribution is also suppressed roughly with a factor 10, while from Fig.4 and Fig.5 the peak strength of $d\Gamma/dm_{K^{+}K^{-}}$ only reduces from $3.8\times 10^{-10}$ to $1.2\times 10^{-10}$. In other words, the contribution of background will be suppressed much faster than that of signal when $\Delta\Gamma$ is reduced. Thus, if the integral interval of $m_{\pi^{+}\pi^{-}}$ is too large, the discrepancy between the contributions of tree and loop diagrams will be too large. Furthermore, as discussed before, when $\Delta\Gamma$ is larger, the peak structure of the triangle singularity becomes broader. Thus, the signal of the triangle singularity is buried, which will make our triangle singularity invisible. Thus, to observe the peak due to the triangle singularity, we need a small interval range of $m_{\pi^{+}\pi^{-}}$. However, from Fig. 9 we can see that we can not cut $m_{\pi^{+}\pi^{-}}$ as small as possible because if the integral interval $m_{\pi^{+}\pi^{-}}$ is too small, not only the overall magnitude of the $K^{+}K^{-}$ invariant mass spectrum will be too small, but also the width of the signal will be too narrow, which will make it too difficult for experiments to observe this signal of the triangle singularity. Thus, in our view, to detect the triangle singularity in future experiments, we should find a balance between the cut on $m_{\pi^{+}\pi^{-}}$, the overall magnitude of $K^{+}K^{-}$ invariant mass spectrum and the significance of the signal of the triangle singularity, which we will discuss in detail in Sec. III.4.

Furthermore, by changing the integral interval of $m_{\pi^{+}\pi^{-}}$, Fig. 10 presents the ”moving triangle singularity”. In Fig. 10, the integration on $m_{\pi^{+}\pi^{-}}$ is carried out within the interval $[m_{\rho}+\Delta m-\Delta\Gamma,m_{\rho}+\Delta m+\Delta\Gamma]$, where $\Delta m$ is the divergence from the center mass of $\rho$ meson and $\Delta\Gamma$ is fixed as 16 MeV. From Fig. 10 we can see that when $|\Delta m|=$35 MeV, the center value of the signal caused by our triangle singularity can move about 3 MeV, in addition, when $\Delta m\neq 0$, both the significance of the signal and the overall magnitude of $K^{+}K^{-}$ invariant mass spectrum are decreased, which is easy to be understood since when $\Delta m\neq 0$, both of them will be suppressed by the propagator of $\rho$ meson. Thus, it requires to change the cuts on $m_{\pi^{+}\pi^{-}}$ in experiments to observe the movement of the triangle singularity, which, in our view, must be a very interesting topic and help us understand more about the triangle singularity itself.

At last, it is worthy to make a discussion to justify that it is almost enough to use these tree diagrams instead of all possible background contribution. We have do other two calculation to estimate the background. From RPP, we have the branching ratio of the $\psi(2S)\to\pi^{+}\pi^{-}K^{+}K^{-}$ process is $7.2\times 10^{-4}$ Zyla:2020zbs . However, in our calculation the branching ratios of $\psi(2S)\to\rho^{0}a^{0}_{0}(980)\to\pi^{+}\pi^{-}K^{+}K^{-}$ and $\psi(2S)\to\rho^{0}a^{0}_{2}(1320)\to\pi^{+}\pi^{-}K^{+}K^{-}$ are around $5\times 10^{-6}$. It indicates that both of them are not the dominant processes for the $\psi(2S)\to\pi^{+}\pi^{-}K^{+}K^{-}$ reaction. From Refs. BaBar:2007ptr ; BaBar:2011btv , the main contribution is from $\psi(2S)\to K^{*}(892)\bar{K}^{*}(892)$. Thus, we assume the branching ratio of $\psi(2S)\to K^{*}(892)\bar{K}^{*}(892)\to\pi^{+}\pi^{-}K^{+}K^{-}$ is $7.2\times 10^{-4}$, i.e., all $\pi^{+}\pi^{-}K^{+}K^{-}$ final states are from $K^{*}(892)\bar{K}^{*}(892)$. But we find that $d\Gamma/dm_{K^{+}K^{-}}$ with the integration range of $m_{\pi^{+}\pi^{-}}$ from $m_{\rho}-\Gamma_{\rho}$ to $m_{\rho}+\Gamma_{\rho}$ is around $10^{-11}$, which is an order of magnitude smaller than that of the tree diagrams of $a^{0}_{0}(980)$ and $a^{0}_{2}(1320)$. Furthermore, if we just consider a phase space distribution, the $d\Gamma/dm_{K^{+}K^{-}}$ by integrating $m_{\pi^{+}\pi^{-}}$ from $m_{\rho}-\Gamma_{\rho}$ to $m_{\rho}+\Gamma_{\rho}$ is around $10^{-10}$ which is still smaller than the contribution of the $a^{0}_{0}(980)$ and $a^{0}_{2}(1320)$ resonances. By these two comparisons, we believe that in the range which is sensitive for the signal of the triangle singularity, the main background is roughly from the tree diagram with $a^{0}_{0}(980)$ and $a^{0}_{2}(1320)$ resonances as calculated above.

Finally, in this subsection we will discuss in detail how to detect the predicted triangle singularity in the $\psi(2S)\to\pi^{+}\pi^{-}K^{+}K^{-}$ process in future experiments. As discussed in the above subsection, if the integral interval of $m_{\pi^{+}\pi^{-}}$ is too large, the signal of the triangle singularity would be invisible in the invariant mass spectrum of $K^{+}K^{-}$ since the ratio between the contributions of loop and tree diagrams is negligible. Thus, when analyzing the experimental data of $\psi(2S)\to\pi^{+}\pi^{-}K^{+}K^{-}$ process, experimentalists can not get the signal of the triangle singularity directly from the full $K^{+}K^{-}$ invariant mass spectrum, which just integrate the completed range of $m_{\pi^{+}\pi^{-}}$. Therefore, to extract the triangle singularity, the experimentalists should make cuts on the $m_{\pi^{+}\pi^{-}}$ around the $m_{\rho}$ region, and then divide them in several bins of $m_{\pi^{+}\pi^{-}}$.

Now let us discuss how to make the cut of $m_{\pi^{+}\pi^{-}}$ to show the peak due to triangle singularity on the $m_{K^{+}K^{-}}$ spectrum. Here, we need to balance two facts, the visible peak structure and enough statistic events, and there exists one variable $\Delta\Gamma$ to control them. Furthermore, because of the limitation of the resolution of the detector, for example, currently the resolution of BESIII is about 4 MeV Ablikim:2019hff , the width of peaks to be observed should not be too small. Thus, by comparing various lines in Figs.8 and 9, we think that $\Delta\Gamma$=16 MeV maybe a good choice. In this situation, the width of the signal is enlarged to about 3 MeV, and the signal almost takes 5%, which requires almost 500 events at least to control the statistical fluctuations.

Now let us derive the expression to estimate the events number through $d\Gamma/dm_{K^{+}K^{-}}$ as follows,

where $N_{\psi(2S)}$ is the number of $\psi(2S)$ events, $\Gamma_{\psi(2S)}$ is the width of $\psi(2S)$, $\Delta m$ is the bin width, $N_{\mathrm{\Delta m}}$ is the events that the bin contains, and the integration $\int_{\Delta m}\frac{d\Gamma}{dm_{K^{+}K^{-}}}dm_{K^{+}K^{-}}$ is the total width under each bin. Then with Eq. (III.4), as an example, we now analyse if the current BESIII experiment can observe the triangle singularity. According to Ref. Ablikim:2019hff , nowadays BESIII has 800 million $\psi(2S)$, while in the future, the number of $\psi(2S)$ will increase to 4 billion Ablikim:2019hff . In addition, currently the resolution of BESIII is about 4 MeV Ablikim:2019hff . By taking $\Delta\Gamma$=16 MeV, if we have 4 billion $\psi(2S)$, then we will have about 10 events per MeV, also, the width of the signal is enlarged to about 3 MeV. The enhancement of the signal is about 5%, which tells it will be very hard for BESIII to observe the triangle singularity.

Recently, we notice that there is a talk on the updated simulation progress of STCF STCF:talk . In this talk, we find that in the future STCF, we can get 640 billion $\psi(2S)$ even per year STCF:talk , which means even using the $\Delta\Gamma$=16 MeV cut on $m_{\pi^{+}\pi^{-}}$, for the $K^{+}K^{-}$ invariant mass spectrum, we will have about 1600 events per MeV per year, also, since the enhancement of the signal is about 5%, we can get 200 events of the enhancement per year. In addition, STCF has excellent resolution STCF:talk , which makes us believe STCF will be a very nice platform for searching this triangle singularity.

If the future, experiments can really observe our triangle singularity after doing some cuts on $m_{\pi^{+}\pi^{-}}$. Experimentalists are encouraged to change the cuts to verify the movement of our triangle singularity, i.e., experiments can continue to check the results given in Fig. 10. In our view, both the detection of our triangle singularity and the observation of its movement are very important and interesting topics. They will not only help us verify the concept of triangle singularity, but also help us understand more about its properties.