Triangle singularity in the $J/\psi \rightarrow K^+ K^- f_0(980)(a_0(980))$ decays

We study the $J/\psi \rightarrow K^+ K^- f_0(980)(a_0(980))$ reaction and find that the mechanism to produce this decay develops a triangle singularity around $M_{\rm inv}(K^- f_0/K^- a_0) \approx 1515$~MeV. The differential width $d\Gamma / dM_{\rm inv}(K^- f_0/K^- a_0)$ shows a rapid growth around the invariant mass being 1515~MeV as a consequence of the triangle singularity of this mechanism, which is directly tied to the nature of the $f_0(980)$ and $a_0(980)$ as dynamically generated resonances from the interaction of pseudoscalar mesons. The branching ratios obtained for the $J/\psi \rightarrow K^+ K^- f_0(980)(a_0(980))$ decays are of the order of $10^{-5}$, accessible in present facilities, and we argue that their observation should provide relevant information concerning the nature of the low-lying scalar mesons.


I. INTRODUCTION
Discussed already in Ref. [1] and systematized by Landau in Ref. [2], the triangle singularities (TS) were fashionable in the sixties [3][4][5][6][7][8] and efforts were done to understand some reactions through TS mechanisms [9,10]. A triangle mechanism stems from the decay of a particle A into 1 + 2, followed by the decay of 1 into 3 + B, and posterior fusion of 2 + 3 to give a new particle C (see Fig. 1(a)) or 2 + 3 (see Fig. 1(b), rescattering), or a different pair of particles. It was shown in Ref. [2] that when all these particles, 1, 2, 3, can be placed on shell in the corresponding Feynman diagram, a singularity can develop in the corresponding amplitude. The conditions for the singularity are made more specific by Coleman-Norton [7] showing that particles 1 and B have to be parallel in the A rest frame and the process has to be possible at the classical level. Analytical expressions of these conditions can be see in Ref. [11] and in a simpler form in Ref. [12]. The formalism of Ref. [12] allows one to see the explicit effect of the width of particle 1 in the shape of the singularity, and this is explicitly shown in Ref. [13] where some considerations are made concerning the Schmid theorem [8], which states that in the case of the mechanism with 2 + 3 → 2 + 3 (rescattering) the triangle singularity can be absorbed by the tree level diagram A → 1 + 2 (1 → 3 + B). In Ref. [13] it is shown that this only occurs in the limit of zero width for particle 1, where the triangle mechanism is negligible with respect to the tree level one.
where q on is the on-shell momentum of the φ and q a− the on-shell K − momentum in the loop, antiparallel to the φ. It should be noted that the TS condition of Eq. (1) is now fulfilled only in a very narrow window of K + K − energies between 987 MeV and 993 MeV, where the f 0 (980) and a 0 (980) peak. Away but close to the point where the TS appears, the amplitudes are no longer singular in the Γ φ → 0 limit, but a peak structure still remains by inertia. Yet, this feature confers the amplitude a special signature that makes the shapes different to ordinary cases of f 0 (980) or a 0 (980) production, which is tied again to the dynamical nature of these resonances formed from the pseudoscalarpseudoscalar interaction in coupled channels. The reaction, hence, contains relevant information concerning the nature of the f 0 (980) and a 0 (980) resonances.

II. FORMALISM
Our mechanism for the J/ψ → K + K − f 0 (980)(a 0 (980)) reaction is depicted in Fig. 2. There is a primary decay of J/ψ → K + φK − , a second decay of φ → K − K + , and the posterior fusion of K − K + to produce the f 0 (980) or a 0 (980) resonance. Given the complicated dynamics of the whole process, our strategy to provide absolute numbers for the decay width and mass distributions consists of taking the information for the primary step J/ψ → K + φK − from experiment and the rest can be calculated reliably. In view of this, we study in a first step the reaction J/ψ → K + φK − .
The J/ψ → K + K − φ decay can proceed via S-wave. We take its amplitude, suited to the production of two vectors, as in Refs. [26,35]: where ǫ(J/ψ) and ǫ(φ) are the polarization vectors of the J/ψ and φ, respectively. Then we write the K − φ invariant mass distribution of this decay as where p K + is the momentum of the K + in the J/ψ rest frame, and p K − is the momentum of the K − in the K − φ rest frame: Here λ(x, y, z) is the Källen function λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2yz − 2xz. After summing over polarizations of Eq. (2), we can simplify Eq. (3) to be Thus, the branching ratio of the J/ψ → K + K − φ decay can be calculated through where the integration is performed from The branching ratio of the J/ψ → K + K − φ decay has been experimentally measured to be [30] Br(J/ψ → K + K − φ) = (8.3 ± 1.2) × 10 −4 .
Using this as the input, we can determine the value of the constant A, satisfying: where the error is taken from Eq. (9).
B. Triangle diagram mechanism for the J/ψ → K + K − f0(980)(a0(980)) decays In the former subsection we studied the J/ψ → K + K − φ decay. In this subsection we show how the J/ψ → K + K − f 0 (980)(a 0 (980)) decays can be produced using this input. To do this we look into the triangle diagrams depicted in Fig. 2. The two mechanisms of Fig. 2(a) and Fig. 2(b) are clearly distinguishable. The kinematics of the reaction around the TS point provide for the mechanism of Fig. 2(a) a momentum p K + ≈ 1114 MeV/c and p K − ≈ 170 MeV/c, and opposite for the mechanism of Fig. 2(b). Because of this, there is also no interference between these mechanisms. The two mechanisms give the same width and we can study just one of them.
We take the first diagram Fig. 2(a) and the J/ψ → K + K − f 0 (980) decay as an example, and write down its amplitude as: where in the last equation we have summed over the φ polarization. The φ → K + K − vertex is obtained from the P -wave Lagrangian [43,44], where P and V are the ordinary pseudoscalar and vector meson SU (3) matrices, m V is the vector mass (m V ∼ 800 MeV), and f π is the pion decay constant (f π = 93 MeV). This Lagrangian produces a vertex and the ǫ 0 (φ) component can be neglected as shown in Appendix A of Ref. [35], since the three-momentum of the φ is very small compared to its mass for φ on-shell in that diagram. The f 0 (980) → K + K − and a 0 (980) → K + K − vertices are obtained from the chiral unitary approach of Ref. [19] with g f0,K + K − = 2567 MeV and g a0,K + K − = 3875 MeV. Note that these two vertices will be more carefully examined in the next subsection. Then we follow Refs. [12,40] and perform analytically the q 0 integration in Eq. (11) in the K − f 0 (980) rest frame, with the result where In addition, since k is the only momentum not integrated in Eq. (14), we can replace Then the amplitude t of Eq. (14) can be rewritten as where As in Ref. [12], the above integration is regularized with the factor θ(q max − | q * |), where q * is the momentum of the K − in the rest frame of f 0 (980), with q max = 600 MeV as it is needed in the chiral unitary approach that reproduces the f 0 (980) [41,42]. After summing over polarizations in Eq. (21), we obtain Now we can write the K − f 0 (980) invariant mass distribution of the J/ψ → K + K − f 0 (980) decay as where p ′ K + is the momentum of the K + in the J/ψ rest frame, and p ′ K − = k is the momentum of the K − in the K − f 0 (980) rest frame: Recalling Eq. (10) and Eq. (23), we obtain the differential branching ratio of the J/ψ → K + K − f 0 (980) decay to be The case for a 0 (980) production is identical replacing g f0,K + K − by g a0,K + K − .
In the former subsection we studied the triangle diagram mechanism for the J/ψ → K + K − f 0 (980)(a 0 (980)) decays. In this subsection we further consider that the f 0 (980)/a 0 (980) will be seen in the π + π − /π 0 η mass distribution, as depicted in Fig. 3. Take the first diagram Fig. 3(a) as an example, the J/ψ first decays into K + K − φ, next the φ decays into K + K − , then the K − and K + merge to give the f 0 (980), and finally the f 0 (980) decays into π + π − . We can write down its amplitude as: This amplitude t ′ is very similar to t given in Eq. (11) in the former subsection, just with g f0,K + K − replaced by the transition amplitude t K + K − ,π + π − . We can follow the same procedure to simplify it to be where t ′ T is also very similar to t T given in Eq. (22), just with the following replacements: The K + K − → π + π − and K + K − → π 0 η scattering has been studied in detail in Refs. [41,42] within the chiral unitary approach, where altogether six channels were taken into account, including π + π − , π 0 π 0 , K + K − , K 0K 0 , ηη, and π 0 η. In the present study we use this as input, and we shall see simultaneously both the f 0 (980) (with I = 0) and a 0 (980) (with I = 1) productions. Now we can write down the double differential mass distribution for the J/ψ → K + K − f 0 (980) → K + K − π + π − reaction, as a function of M inv (K − f 0 ) and M inv (π + π − ) [26]: where p ′′ K + is the momentum of the K + in the J/ψ rest frame, p ′′ K − = k ′ is the momentum of the K − in the K − f 0 (980) rest frame, and p ′′ π + is the momentum of the π + in the π + π − rest frame: Recalling Eq. (10) and Eq. (29), we obtain the double differential branching ratio of the J/ψ With trivial changes, replacing π + π − by π 0 η, we get the corresponding expressions for π 0 η production.

III. RESULTS
Firstly, we show our results for the J/ψ → K + K − f 0 (980)(a 0 (980)) decays, which were previously studied in Sec. II B. Let us begin by showing in Fig. 4 the contribution of the triangle loop defined in Eq. (22). The TS condition of Eq. (1) requires all K + K − φ intermediate particles to be on shell. This forces m f0(a0) > m K + + m K − . On the other hand, if we go to energies a bit bigger than that, Eq. (1) is no longer fulfilled. There is hence a very narrow window of f 0 (a 0 ) masses where the TS condition is exactly fulfilled, i.e., from 987 MeV to 993 MeV. In view of this we plot in Fig. 4 the real, imaginary parts and modulus of t T of Eq. (22) for different masses of f 0 (a 0 ). The magnitude depends on the f 0 (a 0 ) mass, independent on whether we have f 0 or a 0 , since the different couplings to K + K − have been factorized out of the integral of t T . We show the results for six different masses. The first two are inside the window of energies where the TS appears, the other four are outside. We observe a neat peak in the first two cases, which gets broader gradually as we depart from the TS window. Note that the peak of the imaginary part is related to the triangle singularity, while the one of the real part is related to the K − φ threshold, as discussed in Refs. [35,36].
dMinv(K − f0/K − a0) , the differential branching ratio of the J/ψ → K + K − f 0 (980)(a 0 (980)) decays defined in Eq. (27), as a function of M inv (K − f 0 /K − a 0 ) in Fig. 5. We plot the results for three selected masses of f 0 (a 0 ), 989 MeV, 987 MeV, and 981 MeV. The results for f 0 or a 0 production differ only in a factor because of the different couplings g f0,K + K − or g a0,K + K − . We observe a peak in The peak is clear for the first mass of 989 MeV, but gradually the upper part of the spectrum falls down more slowly. This is due to the factor p ′3 K − in Eq. (27), which raises fast as M inv (K − f 0 /K − a 0 ) increases. If we remove this factor the peaks are sharper. Next we integrate We can see that the results for the integrated branching ratios depend on the mass assumed for the f 0 (a 0 ) resonance. For the same mass, the f 0 or a 0 production rates differ by the ratio of the square of their couplings to K + K − . In view of the changing shape and strength of the results on the mass assumed for the f 0 (a 0 ) resonance, we apply next the method discussed in Sec. II C, taking into account the mass distribution of the f 0 (980) and a 0 (980) reflected by the t K + K − ,π + π − and t K + K − ,π 0 η amplitudes. In Fig. 6 we plot |t ′ T |, the triangle loop defined in Eq. (29) for the J/ψ → K + K − f 0 (980) → K + K − π + π − reaction, as a function of M inv (π + π − ) by fixing M inv (K − f 0 ) = 1496 MeV, 1516 MeV, and 1536 MeV. The distribution gets its largest strength when M inv (K − f 0 ) is near 1516 MeV. The triangle loop |t ′ T | for the J/ψ → K + K − a 0 (980) → K + K − π 0 η reaction is the same as this one. We also show 1 in the left panel of Fig. 7, that is the double differential branching ratio of the J/ψ → K + K − f 0 (980) → K + K − π + π − reaction defined in Eq. (36). We show this as a function of M inv (π + π − ) by fixing M inv (K − f 0 ) = 1496 MeV, 1516 MeV, and 1536 MeV. A strong peak can be found when M inv (π + π − ) is around 980 MeV, corresponding to the f 0 (980). Consequently, most of the contribution comes from M inv (π + π − ) ∈ [900, 1050] MeV, and we can restrict the integral in M inv (π + π − ) to this region when calculating the . Unlike in Fig. 6, the strength for M inv (K − f 0 ) = 1536 MeV is a bit bigger than that for 1516 MeV. This is because of the factor p ′′ K − 3 in Eq. (36). Similarly, we show in the right panel of Fig. 7. Again, we can restrict M inv (π 0 η) to the region  27) is for f0 production, and for a0 production one multiplies it by (g a 0 , T | for the J/ψ → K + K − f0(980) → K + K − π + π − reaction, defined in Eq. (29), as a function of Minv(π + π − ). The red, black, and blue curves are obtained by setting Minv(K − f0) = 1496 MeV, 1516 MeV, and 1536 MeV, respectively. The triangle loop |t ′ T | for the J/ψ → K + K − a0(980) → K + K − π 0 η reaction is the same as the this one.
M inv (π 0 η) ∈ [900, 1050] MeV, and perform the integration The (single) differential branching ratios obtained are shown in Fig. 8. We see a clear structure around 1515 MeV coming from the peak of the triangle loop t ′ T , but we also observe strong contribution from the larger K − f 0 /K − a 0 invariant masses produced by the p ′′ Br(J/ψ → K + K − a 0 (980) → K + K − π 0 η) = 5.2 × 10 −6 . (46) These rates should be multiplied by two to account for the mechanisms of Fig 3(b) and Fig 3(d), if one looks for J/ψ → K + K − f 0 (980)(a 0 (980)) independently of which of the K's is the fast one.
As we can see, the explicit consideration of the t K + K − ,π + π − and t K + K − ,π 0 η changes the final shape of the differential width and integrated branching ratio. We should note that in the case of the f 0 production the method of Sec. II B accounts for all the decay modes of the f 0 , π + π − and π 0 π 0 , the latter with a strength 1/2 compared to the one of π + π − . To compare the result of Eq. (45) with those of Eq. (39) we must multiply the result of Eq. (45) by 3/2 and then the results are more similar. The discrepancies are bigger in the case of the a 0 production. This should not be a surprise since the a 0 (980) is a border line state between a bound KK state and a threshold cusp, as a consequence of which the coupling of a 0 to K + K − has large uncertainties, in which case, the method used in Sec. II C is more reliable. (b) the differential branching ratio 1 , defined in Eq. (44), as a function of Minv(K − a0).

IV. CONCLUSION
We have studied the J/ψ → K + K − f 0 (980)(a 0 (980)) decays and have seen that they are driven by a triangle singularity, peaking at M inv (K − f 0 /K − a 0 ) ≈ 1515 MeV. The process proceeds as follows: In a first step the J/ψ decays to K + K − φ. The K + and K − momenta are very distinct in the process and we select for our study the mode with K + with large momentum and K − with small momentum. The opposite case provides the same contribution. In a second step the φ decays into K + K − and the primary K − together with the K + from the φ decay merge to give the f 0 (980) or a 0 (980) resonance. The mechanism implicitly assumes that the f 0 (980) and a 0 (980) resonances are not produced directly but are a consequence of the final state interaction of the K + K − . This is the basic finding of the chiral unitary approach where these two resonances are the consequence of the pseudoscalar-pseudoscalar interaction in coupled channels and not qq objects.
Using as input empirical information from the J/ψ → K + K − φ decay, we are able to determine the double differential decay width in terms of the K − f 0 /K − a 0 invariant mass and the π + π − /π 0 η from the decay of the f 0 (980)/a 0 (980), respectively. We find very distinct shapes of the double differential distributions, and the single one, dΓ/dM inv (K − f 0 /K − a 0 ), with a sharp raise of this magnitude around M inv (K − f 0 /K − a 0 ) ≈ 1515 MeV, where the triangle singularity appears. All these features are tied to the nature of the f 0 (980) and a 0 (980) resonances as dynamically generated from the interaction of pseudoscalar mesons, and its experimental observation should bring valuable information on the important issue of the nature of the low-lying scalar mesons. We find the branching ratios of the order of 10 −5 , which are accessible in present facilities, where many J/ψ branching ratios of the order of 10 −6 ∼ 10 −7 have already been measured.