The role of triangle singularity in isospin breaking process $J/\psi\to\Lambda\bar{\Lambda}\pi$ and the possible evidence of $\Sigma^{*}(\frac{1}{2}^{-})$ states

With the preparations given in Sec. III, we proceed to present the numerical calculations and corresponding discussions. In Fig. 1, the charge configuration with primary decay process where $J/\psi$ decaying into two neutral $\Sigma^{(\ast)0}$ particles is not included. It is because under this situation the isospin factor of $V_{3}$ vertex, i.e., the interaction between $\Sigma^{\ast 0}$ and $\Sigma^{0}\pi^{0}$, is zero, due to the Clebsch-Gorden coefficient of isospin factor $\langle 1,0;1,0|1,0\rangle=0$. Thus, the neutral channel will have no contribution to the process. For a single loop diagram in Fig. 1, after multiplying the isospin factors of all the vertices $V_{i}(i=1,...,4)$, the signs of the amplitudes of the two kinds of charge configurations in the triangle loop are opposite. Therefore, if the $\Sigma^{(\ast)}$ particles have no mass differences in charges, the contributions of these two charge configurations will cancel each other and make the total amplitude of the process still be zero. However, if we take into account the practical masses differences of charge states ParticleDataGroup:2022pth , we will find some changes on the lineshapes in $\Lambda\pi~{}(\bar{\Lambda}\pi)$ invariant mass spectrum, as shown in Fig. 2.

In Fig. 2, for instance, we choose Fig. 1 $(b)$ to illustrate the lineshapes of the $m_{\Lambda\pi}$ invariant mass spectrum. Here, $\bar{\Sigma}^{\ast\pm}$ and $\Sigma^{\ast 0}$ are assigned as $\bar{\Sigma}^{\ast}(1670)^{\pm}$ and $\Sigma^{\ast}(1381)^{0}$, respectively, with all coupling constants set to 1. In RPP ParticleDataGroup:2022pth , the masses of $\Sigma^{-}$ and $\Sigma^{+}$ are 1197.449 MeV and 1189.37 MeV, respectively, and the mass of $\Sigma^{\ast}(1670)$ is just around 1662 MeV ParticleDataGroup:2022pth . After applying Coleman-Norton theorem , the positions of triangle singularities are given as 1.40909 GeV and 1.40073 GeV for the different kinds of charge intermediate states, respectively. From Fig. 2, it can be observed that the lineshapes of the black solid line and the red dashed line exhibit slight differences in the following two aspects. One difference is that the central value of the peak in the black solid line is slightly larger than that in the red dashed line due to variations in the positions of the triangle singularities. Another distinction is that the height of the black solid line is slightly lower than that of the red dashed line, attributed to differences in the phase spaces. Crucially, the non-zero total contribution of the green dot-dashed line arises from the mass distinction of $\Sigma^{\mp}$ in these two charge configurations. Consequently, a resonance-like structure emerges around 1.4 GeV in the $\Lambda\pi$ invariant mass spectrum, with a width of approximately 20 MeV.

In Fig. 2, we only take into account the contribution of Fig. 1 $(b)$. Given that Fig. 1 $(a)$ is simply the conjugate of Fig. 1 $(b)$, it is conceivable that the lineshapes of the $\bar{\Lambda}\pi$ invariant mass spectrum will be identical if we solely consider the contribution from Fig. 1 $(a)$. To account into all the contributions, we should sum the amplitudes of these two diagrams coherently, and the interference effects between them are necessary. In Fig. 3, we give the Dalitz plots in addition with their projections on the invariant masses, where both the two terms in Fig. 1 are included. In Fig. 1, the $\Sigma^{\ast\pm}~{}(\bar{\Sigma}^{\ast\pm})$ is set as $\Sigma^{\ast}(1670)^{\pm}~{}(\bar{\Sigma}^{\ast}(1670)^{\pm})$, and the $\Sigma^{\ast 0}~{}(\bar{\Sigma}^{\ast 0})$ is set as $\Sigma^{\ast}(1381)^{0}~{}(\bar{\Sigma}^{\ast}(1381)^{0})$.

From Fig. 3, it is evident that the triangle singularity effects occurring in this process will result in two intersecting bands in both the $\bar{\Lambda}\pi-\Lambda\pi$ and $\Lambda\bar{\Lambda}-\Lambda\pi$ Dalitz plots. Therefore, as a result of this intersection, the peak in the $\bar{\Lambda}\pi$ invariant mass spectrum will be significantly enhanced, increasing the likelihood of observing this peak in experiments. In the bottom-left subfigure, a minor cusp emerges around 1.33 GeV, coinciding with the location of the $\Sigma\pi$ threshold. A sudden decrease around 1.82 GeV is also observed, attributed to the limitation of phase space.

In the bottom-right panel of Fig. 3, when only Fig. 1(b) is taken into account, a flat plateau will be observed in the $\Lambda\bar{\Lambda}$ invariant mass spectrum from 2.6 GeV to 2.9 GeV. Conversely, as a result of the intersection of the two bands, a small peak will emerge on the plateau, located around 2.85 GeV.

Although Fig. 3 has shown the basic phenomenon of our predicted triangle singularity, the currently missing status of $\Sigma^{\ast}(1381)$ indicates that choosing different $\Sigma^{\ast 0}~{}(\bar{\Sigma}^{\ast 0})$ to study the phenomena must be necessary. Thus, in Fig. 4, $\Sigma^{\ast}(1381)$ is changed to the well established $\Sigma^{\ast}(1385)$ and similar contents as Fig. 3 are presented.

As shown in Fig. 4, when $\Sigma^{\ast 0}~{}(\bar{\Sigma}^{\ast 0})$ changes from $\Sigma^{\ast}(1381)^{0}~{}(\bar{\Sigma}^{\ast}(1381)^{0})$ to $\Sigma^{\ast}(1385)^{0}~{}(\bar{\Sigma}^{\ast}(1385)^{0})$, the lineshapes change a lot. From the plots in the first row, the bands in the Dalitz plots are no longer uniform, indicating a typical $P-$wave contribution. From the bottom row, it can also be observed that both the $\bar{\Lambda}\pi$ and $\Lambda\bar{\Lambda}$ invariant mass spectra are significantly suppressed. The changes can be attributed to the $J^{P}$ quantum number of the exchanged $\Sigma^{\ast 0}~{}(\bar{\Sigma}^{\ast 0})$. If the quantum number of $\Sigma^{\ast}(1381)$ is $J^{P}=(\frac{1}{2})^{-}$, the interactions between $\Sigma^{\ast}(1381)$ and $\Sigma\pi$ as well as $\Lambda\pi$ are both in $S-$wave. For the $\Sigma^{\ast}(1385)$, the quantum number is $J^{P}=(\frac{3}{2})^{+}$, indicating that both its interactions with $\Sigma\pi$ and $\Lambda\pi$ transition to $P-$wave. The interactions in higher partial waves may diminish the impact of triangle singularities and threshold cusps, a conclusion that has been previously explored and presented in Refs. Swanson:2014tra ; Shen:2020gpw .

To validate this hypothesis, it would be intriguing to examine an alternative configuration, where $\Sigma^{\ast 0}~{}(\bar{\Sigma}^{\ast 0})$ corresponds to $\Sigma^{\ast}(1381)^{0}~{}(\bar{\Sigma}^{\ast}(1381)^{0})$, and $\Sigma^{\ast\pm}~{}(\bar{\Sigma}^{\ast\pm})$ corresponds to $\Sigma^{\ast}(1620)^{\pm}~{}(\bar{\Sigma}^{\ast}(1620)^{\pm})$. At this point, given that the $J^{P}$ values of $\Sigma^{\ast}(1670)$ and $\Sigma^{\ast}(1620)$ are $(\frac{3}{2})^{-}$ and $(\frac{1}{2})^{-}$ respectively, the interactions between $\Sigma^{\ast\pm}~{}(\bar{\Sigma}^{\ast\pm})$ and $\Lambda\pi^{\pm}(\bar{\Lambda}\pi^{\pm})$ transition from $D-$wave to $S-$wave. These modifications are reflected in the results depicted in Fig. 5.

Upon comparing the panels in Fig. 3 and Fig. 5, it becomes evident that when the interactions between $\Sigma^{\ast\pm}~{}(\bar{\Sigma}^{\ast\pm})$ and $\Lambda\pi^{\pm}(\bar{\Lambda}\pi^{\pm})$ transition from $D-$wave to $S-$wave, the peak height is significantly enhanced. Furthermore, two distinct bands also emerge, precisely positioned at the thresholds of $\Sigma\pi$ and $\bar{\Sigma}\pi$ respectively. This means that the cusp effect at this time is enhanced a lot. Furthermore, from the invariant mass spectra in the bottom row, it can be seen that when the partial wave of interaction is reduced, the triangle singularity effects are enhanced a lot, with a factor around 2.0. Thus, we may also get the conclusion that higher partial wave will suppress the effect of triangle singularity in addition with the cusp effect.

If the triangle singularity effects mentioned in the previous subsection can be detected in experiments, discussions on the background are absolutely essential. Similar to our previous works Huang:2020kxf ; Huang:2021olv , the background contributions are depicted as shown in Fig. 6.

We illustrate the two possible mechanisms for this isospin process at the tree level in Fig. 6. Clearly, in the background, either the interaction between $J/\psi$ and $\Lambda\bar{\Sigma}^{\ast 0}$ or the interaction between $\Lambda^{\ast}$ and $\Lambda\pi$ violates the isospin conservation law of strong interactions. Therefore, these strength of interactions must be similar as that of the electroweak type. According to Ref. Donald:PA , typically, the relative strengths of the strong, electromagnetic, and weak forces are approximately $1$, $10^{-2}$, and $10^{-7}$ respectively. Hence, it is reasonable to assume that either the interaction between $J/\psi$ and $\Lambda\bar{\Sigma}^{\ast 0}$ or the interaction between $\Lambda^{\ast}$ and $\Lambda\pi$ occurs through electromagnetic process at the quark level. Upon consulting RPP ParticleDataGroup:2022pth , we observe that the branching ratios for the $J/\psi\to\Sigma^{\ast}(1385)^{-}\bar{\Sigma}^{+}$ and $J/\psi\to\Sigma^{\ast}(1385)^{0}\bar{\Lambda}$ processes are $(3.1\pm 0.5)\times 10^{-4}$ and $<4.1\times 10^{-6}$, respectively. In other words, the ratio between them is at most 100. Therefore, the inference regarding the type of interactions at the $J/\psi\Lambda\bar{\Sigma}^{\ast 0}$ and $\Lambda^{\ast}\Lambda\pi$ vertices is justified.

The aim of this study is to investigate whether the triangle singularity effect remains observable after accounting for the background. The key focus area is the invariant mass of $\Lambda\pi~{}(\bar{\Lambda}\pi)$ around 1.4 GeV. Near this region, there are two well established resonances $\Lambda(1405)$ and $\Sigma^{\ast}(1385)$ for the diagrams shown in Fig. 6. However, apart from the $J/\psi\to\Sigma^{\ast}(1385)^{0}\bar{\Lambda}$ process, the $J/\psi\to\Lambda(1405)\bar{\Lambda}$ and $\Lambda(1405)\to\Lambda\pi$ processes have not been observed ParticleDataGroup:2022pth . This implies that the background originating from $\Lambda(1405)$ cannot be calculated with high precision. Fortunately, the branching ratio of $J/\psi\to\Lambda\bar{\Sigma}^{-}\pi^{+}$ have been measured by previous experiments as $(8.3\pm 0.7)\times 10^{-4}$ ParticleDataGroup:2022pth . If we assume that this process is dominated by $\Lambda(1405)$, the coupling constant $g_{J/\psi\Lambda(1405)\Lambda}$ then can be extracted. Taking into account the isospin-breaking effect of approximately $1\%$, we can calculate the contribution of $\Lambda(1405)$ to $J/\psi\to\Lambda\bar{\Lambda}\pi^{0}$.

Since the $J/\psi\to\Lambda\bar{\Sigma}^{-}\pi^{+}$ process primarily occurs through $\Lambda(1405)$, the triangle singularity effect remains prominent when factoring in the background contribution around 1.4 GeV in the $\Lambda\pi~{}(\bar{\Lambda}\pi)$ invariant mass spectrum. This suggests that detecting it in future experiments holds promise.

Regarding the loop diagram, experimental data on the branching ratios of the $J/\psi\to\Sigma^{\ast}(1670)\bar{\Sigma}$ and $J/\psi\to\Sigma^{\ast}(1620)\bar{\Sigma}$ processes are currently unavailable, and as a result, the corresponding coupling constants cannot be accurately estimated. When comparing the $J/\psi\to\Sigma^{\ast}(1385)\bar{\Sigma}$, $J/\psi\to\Sigma^{\ast}(1670)\bar{\Sigma}$, and $J/\psi\to\Sigma^{\ast}(1620)\bar{\Sigma}$ processes, it is observed that the $J/\psi\to\Sigma^{\ast}(1385)\bar{\Sigma}$ process takes place in the $S-$wave, whereas the others occur in the $P-$wave. The branching ratio of the $J/\psi\to\Sigma^{\ast}(1620/1670)\bar{\Sigma}$ process may be slightly lower than that of $J/\psi\to\Sigma^{\ast}(1385)\bar{\Sigma}$. Taking into account that the phase space of the $J/\psi\to\Sigma^{\ast}(1385)\bar{\Sigma}$ process is slightly larger, this will also result in a slightly lower branching ratio for the $J/\psi\to\Sigma^{\ast}(1620/1670)\bar{\Sigma}$ processes. Considering these two points, we assume that the branching ratio of the $J/\psi\to\Sigma^{\ast}(1385)\bar{\Sigma}$ process is approximately 5 times greater than that of the $J/\psi\to\Sigma^{\ast}(1620/1670)\bar{\Sigma}$ processes, i.e., $\frac{Br(J/\psi\to\Sigma^{\ast}(1620/1670)\bar{\Sigma})}{Br(J/\psi\to\Sigma^{\ast}(1385)\bar{\Sigma})}=\frac{1}{5}$. As for the interactions between $\Sigma^{\ast}(1381)$ and $\Lambda\pi/\Sigma\pi$, we assume that they are similar to those of $\Sigma^{\ast}(1385)$ in order to extract the coupling constants.

All the coupling constants mentioned in this study can be compiled in Table 1 and Table 2. Table 1 consists of the coupling constants extracted from existing experimental data, while Table 2 comprises the coupling constants derived from estimated branching ratios.

With the these coupling constants, the significance of triangle singularity related to the background can be estimated. In Fig. 7, as an example, we display the $\bar{\Lambda}\pi$ invariant mass spectrum, which includes the contributions of both background and the loop diagram with the $\Sigma^{\ast}(1670)\Sigma^{\ast}(1381)~{}(\bar{\Sigma}^{\ast}(1670)\bar{\Sigma}^{\ast}(1381))$ configuration. In the background, we include both the contributions from $\Lambda(1405)$ and $\Sigma^{\ast}(1385)$ as intermediate states.

As shown in Fig. 7, around 1.4 GeV, the peak position of the triangle singularity is nearly identical, though slightly higher, than the background attributed to $\Lambda(1405)$. It is clear that the contribution from the background is suppressed, likely due to the background originating from an isospin-breaking process. However, the contribution from the loop diagram is almost the same as that of tree diagram. After integrating the $\bar{\Lambda}\pi$ invariant mass spectrum, we have determined the branching ratios for the loop, the $\Sigma(1385)$ and the $\Lambda(1405)$ background components to be approximately $6.76\times 10^{-6}$, $1.18\times 10^{-5}$, and $9.27\times 10^{-6}$, respectively. This indicates that the contributions of the triangle singularity and background are comparable. Although the position of the triangle singularity coincides with that of the $\Lambda(1405)$, it is evident from Fig. 7 that their lineshapes differ. This results in the triangle singularity effect being much sharper than the resonance structure. This results in the triangle singularity effect being much sharper than the resonance structure.

After estimating the backgrounds, the significance of the triangle singularity is determined using the variable

which represents the relative magnitude between the triangle singularity and background in the $\bar{\Lambda}\pi$ invariant mass spectrum. This ratio excludes the interference contamination from the loop diagrams and background, and it circumvents the phase angle issue as discussed in Refs. Huang:2020kxf ; Huang:2021olv , as mentioned in the introduction. If the triangle singularity effect can still be identified from this ratio, its detection in experiments is assured.

Using the ratio determined from Eq. (12), one can plot the distributions in $[\Sigma^{\ast\pm},\Sigma^{\ast 0}]=[\Sigma^{\ast}(1670)^{\pm},\Sigma^{\ast}(1381)^{0}]$, $[\Sigma^{\ast}(1670)^{\pm},\Sigma^{\ast}(1385)^{0}]$, $[\Sigma^{\ast}(1620)^{\pm},\Sigma^{\ast}(1381)^{0}]$ cases. The results are presented in Fig. 8, Fig. 9, and Fig. 10, respectively.

Figs. 8-10 display the relative contribution from the triangle singularity for all $[\Sigma^{\ast\pm},\Sigma^{\ast 0}]$ cases. It is evident that a peak around 1.4 GeV emerges in the $\bar{\Lambda}\pi$ invariant mass spectrum, despite the significant differences in lineshapes. The narrow peaks are a result of the widths of the triangle singularity and the peak in the background being 20 MeV and 60 MeV, respectively. This indicate that the triangle singularity effect is significant compared to the background, and the signal peak of triangle singularity is much sharper than the resonance structure. Especially, for the $\Sigma^{\ast}(1620)^{\pm}~{}(\bar{\Sigma}^{\ast}(1620)^{\pm})$ and $\Sigma^{\ast}(1381)^{0}~{}(\bar{\Sigma}^{\ast}(1381)^{0})$ cases, the contribution of the triangle singularity at the peak is approximately four times larger than that of the resonance, as illustrated in Fig. 10.

In Fig. 11, we illustrate the peak structure from the loop and background, with the triangle singularity peak dominating in the region around 1.4 GeV. The ratio of each background is much smaller than that of the loop diagrams, and the branching ratios of the loop diagrams and all the backgrounds to $J/\psi$ are estimated to be approximately $1.56\times 10^{-5}$ and $2.4\times 10^{-5}$, respectively. It has been observed that there is a significant contribution from the long tails to the tree diagrams. This phenomenon suggests that if both $\Sigma^{\ast}(1620)$ and $\Sigma^{\ast}(1381)$ indeed exist, the peak around 1.4 GeV in the $\bar{\Lambda}\pi$ invariant mass spectrum may be predominantly influenced by the triangle singularity effect.

In above calculations, we have included two $\Sigma^{\ast}(\frac{1}{2}^{-})$ states in the singularity calculation. One is the not-well-established $\Sigma^{\ast}(1620)$, the other is the still missing $\Sigma^{\ast}(1381)$. From Figs. 8-10, it is evident that both of their effects on the final results are substantial. For $\Sigma^{\ast}(1381)$, the triangle singularity effect attributed to it is more significant than that caused by $\Sigma^{\ast}(1385)$. This can be observed by comparing the lineshapes in Fig. 8 and Fig. 9. This is the outcome of the interactions of $\Sigma^{\ast}(1381)$ and $\Sigma^{\ast}(1385)$ with $\Sigma\pi/\Lambda\pi$ transitioning from $S-$wave to $P-$wave, respectively. Consequently, the triangle singularity effect will be greatly suppressed for the $P-$wave case, indicating that the influence of the triangle singularity generated via $\Sigma^{\ast}(1385)$ has almost no impact on the width of the entire peak.

Regarding $\Sigma^{\ast}(1620)$, its presence directly results in the triangle singularity when $\Sigma^{\ast 0}$ is considered to be the $\Sigma^{\ast}(1381)$. The $\Sigma^{\ast}(1670)\Lambda\pi$ interaction occurs in $D-$wave, while that of $\Sigma^{\ast}(1620)\Lambda\pi$ takes place in $S$-wave. If one replaces $\Sigma^{\ast}(1670)$ with $\Sigma^{\ast}(1620)$, it will significantly enhance the triangle singularity. Therefore, the loop diagram will have the most significant impact on the $J/\psi\to\Lambda\bar{\Lambda}\pi^{0}$ process around 1.4 GeV in the $\Lambda\pi~{}(\bar{\Lambda}\pi)$ invariant mass spectrum.

In conclusion, if future experiments can confirm the predictions of this study, it will not only validate the concept of triangle singularity but also provide a secondary investigation into the existence of the two $\Sigma^{\ast}(\frac{1}{2}^{-})$ states around 1.4 GeV and 1.6 GeV.