Triangle singularity enhancing isospin violation in $\bar B_s^0 \to J/\psi \pi^0 f_0(980)$

We perform calculations for the $\bar B_s^0 \to J/\psi \pi^0 f_0(980)$ and $\bar B_s^0 \to J/\psi \pi^0 a_0(980)$ reactions, showing that the first one is isospin-suppressed while the second one is isospin-allowed. The reaction proceeds via a triangle mechanism, with $\bar B_s^0 \to J/\psi K^* \bar K +c.c.$, followed by the decay $K^* \to K\pi$ and a further fusion of $K\bar K$ into the $f_0(980)$ or $a_0(980)$. We show that the mechanism develops a singularity around the $\pi^0 f_0(980)$ or $\pi^0 a_0(980)$ invariant mass of 1420 MeV where the $\pi^0 f_0$ and $\pi^0 a_0$ decay modes are magnified and also the ratio of $\pi^0 f_0$ to $\pi^0 a_0$ production. Using experimental information for the $\bar B_s^0 \to J/\psi K^* \bar K +c.c.$ decay, we are able to obtain absolute values for the reactions studied which fall into the experimentally accessible range. The reactions proposed and the observables evaluated, when contrasted with actual experiments should be very valuable to obtain information on the nature of the low lying scalar mesons.


I. INTRODUCTION
Triangle singularities (TS) are capturing the attention of hadron physics (see talk in the latest hadron conference [1]). Introduced by Landau in 1959 [2], the TS stems from a mechanism that can be represented by a Feynman diagram with a loop with three propagators.
An external particle A decays into two particles 1 and 2. Particle 2 decays into particle 3 and an external particle B, and then particles 1 and 3 merge into an external particle C. The loop contains the particles 1, 2, 3 as internal particles. Under certain circumstances which correspond to having the possibility of the process occurring at the classical level, a singularity in the amplitude develops [3]. This occurs when all the intermediate particles are placed on-shell and are colinear. The amplitude becomes infinite if the internal particles have zero width. However, the fact that particle 2 can decay into 3 + B implies that it has a width, and the infinite amplitude turns into a finite peak, which can be identified experimentally. A reformulation of the problem, in the light of present computing facilities (at the level of a simple PC), offers a more intuitive and practical approach to this issue [4]. The existence of a singularity for a given mechanism is established by means of a single equation, q on = q a− (see Eq. (18) of Ref. [4]), where q on is the on-shell momentum of particle 1 in the decay of A → 1 + 2, and q a− is the smallest momentum for particle 2, when 2 + 3 merge on shell to give the moving particle C, with particles 2 and B having momenta in opposite directions (this situation allows the Coleman Norton theorem [3] to be fulfilled). Clear as the problem is, no experimental examples were found for long time, but the situation has reversed recently. Suggestions to find TS in different reactions were done in Ref. [5]. In particular, it was suggested that the peak seen by the COMPASS collaboration that was initially associated to a new resonance, the a 1 (1420) [6], was a consequence of a triangle singularity that reinforced the a 1 (1260) decay into πf 0 (980). Detailed calculations clearly reaching this conclusion were done in Refs. [7,8]. Similarly, arguments have been given in Ref. [9] that the f 1 (1420) resonance, catalogued as such in the PDG [10], does not correspond to a resonance, but it is a manifestation of the f 1 (1285) decay into KK * , with the "πa 0 (980) decay mode" claimed in Ref. [11] corresponding to a TS enhanced decay mode of the f 1 (1285). Another example is given by the f 2 (1810) "resonance", which as shown in Ref. [12], comes naturally from a TS involving K * K * production, followed by K * → πK andK * K fusing into a 1 (1260). Some awakening to the TS was spurred by the suggestion that the P c (4450) peak seen by the LHCb collaboration [13,14] might correspond to a TS [15,16], but the follow-up work in Ref. [4] showed that for the preferred quantum numbers J P = 3/2 − , 5/2 + of the experimental analysis this could not be the explanation.
The TS has also helped to explain some peculiar experimental features of different reactions, like the peak around √ s = 2110 MeV of the γp → K + Λ(1405) reaction [17], explained in Ref. [18] through a TS, or the πN * (1535) contribution to the γp → π 0 ηp reaction [19], also explained through such a mechanism in Ref. [20]. A possible φp resonance, the hiddenstrange analogue of the P c state, was investigated in the Λ + c → π 0 φp decay by considering a triangle singularity mechanism [21], where the obtained φp invariant mass distribution agrees with the existing Belle data [22]. Other examples can be found in a more detailed description in Ref. [23].
On the other hand, the issue of isospin violation in production of the f 0 (980) or a 0 (980) resonances, and their mixing, has been a recurrent topic [24][25][26][27][28]. While trying to establish a "f 0 − a 0 mixing parameter" from different reactions, the concept had to be abandoned because it was shown that the amount of isospin violation was very much reaction dependent.
Particularly, it was shown in Refs. [29,30] that the large isospin violation in the η(1405) → π 0 f 0 (980) decay [31] was due to a TS. Since then, a search for TS enhanced isospin-violating reactions producing the f 0 (980) or a 0 (980) resonances has been initiated. In Ref. [32] the f 1 (1285) decays into the isospin-allowed π 0 a 0 (980) mode and the isospin-forbidden π 0 f 0 (980) mode were studied and the latter was confirmed a few months later in a BESIII experiment [33]. More recently the D + s → π + π 0 a 0 (980)(f 0 (980)) reaction has been suggested as an example of isospin violation (D + s → π + π 0 f 0 (980)) enhanced by a TS [23]. In this reaction, the D + s decays into π + and a quark pair ss which hadronizes in two mesons in isospin I = 0. The TS emerges from the decay mode D + s → π + (K + K * − + K 0K * 0 ) followed byK * → π 0K and KK merging into the a 0 (980) (the isospin-allowed mode). The mechanism produces a TS at around 1420 MeV of the invariant mass of π 0 a 0 (980), M inv (π 0 a 0 (980)) . The isospinforbidden D + s → π + π 0 f 0 (980) mode emerges from the lack of cancellation between the K 0K 0 and K + K − intermediate states in the loops, and it is shown that the mode is enhanced with respect to the isospin-allowed mode around the TS peak.
Following this line of research, in this work we present a different reaction,B 0 s → J/ψπ 0 f 0 (980)(a 0 (980)), in which the f 0 (980) production mode is also isospin-forbidden.
The reaction has different dynamics than the D + s → π + π 0 f 0 (980)(a 0 (980)) but shares some features concerning the TS. We also observe an enhancement of dΓ/dM inv (π 0 f 0 ) and dΓ/dM inv (π 0 a 0 ) around M inv = 1420 MeV, and the ratio of these two distributions also peaks around this value of the invariant mass. These features are tied to the picture of the f 0 (980) and a 0 (980) as dynamically generated states from the interaction of pseudoscalar mesons, and their experimental confirmation will be relevant to gain further insight into the nature of the low lying scalar mesons.
We describe theB 0 s → J/ψπ 0 f 0 (980)(a 0 (980)) reaction. In a first step we show in Fig.  1(a) the basic decay ofB 0 s into J/ψ(cc) and a pair of quarks ss. This mechanism proceeds via internal emission [34,35], and leaving apart the bcW vertex, needed for the decay, the second vertex, W cs, is Cabibbo favored. The next step consists of the hadronization of ss to give a pair of mesons, which is shown in Fig. 1(b). Following the step of Refs. [35,36], we can write where i runs over the quarks u, d, s, and M is the qq matrix in SU(3). We can write the M matrix in terms of pseudoscalar mesons, Φ, or vector mesons, V , as For reasons that will become clear later, we choose for one M the matrix Φ and for the other the matrix V and we get the possible combinations for s(ūu +dd +ss)s, or In the triangle diagram that we shall discuss briefly, theKK * orK * K will convert into π 0 f 0 or π 0 a 0 , which have C-parity positive. This means that in order to get this final state we must take the C-parity positive combination ofKK * andK * K, which under the implicit prescription CK * = −K * that we use is given by and the process that we are interested in is The strength of this process is obtained by using the experimental branching ratio for which has a branching fraction [10,37] Br In the experiment of Ref. [37], the K 0 π + or K − π + are both producing the K * + ,K * 0 , from where one concludes that the rate for B 0 s → J/ψK 0K * 0 is one fourth of the rate of Eq. (8), since the complex conjugate part of Eq. (8) equals the rate of B 0 s → J/ψK 0 K − π + . Since we are interested in the strength of the amplitude for the process of Eq. (7) with KK * ,KK * having C-parity positive, we assume that both C-parity positive and negative would give the same contribution (we shall come back to this point) and then conclude that But, since K * 0 → K + π − , K 0 π 0 with strengths 2 3 , 1 3 respectively, we have We also take the structure for the amplitude of this decay, suited to the production of two vectors, as in Refs. [38,39] As usual, we take the lowest possible angular momentum, but we shall check the consistency later. The coefficient C is obtained by comparing the strength of Eq. (10) with the integral over the invariant masses of J/ψK * 0 and K * 0K 0 . We have [10] The sum over polarizations of tB0 s →J/ψK * 0K 0 2 is given by Thus, If we want to obtain dΓB 0 s →J/ψK * 0K0 dM inv (J/ψK * 0 ) we integrate the double differential width over dM inv (K * 0K 0 ) and conversely, if we wish to get dΓB 0 s →J/ψK * 0K0 dM inv (K * 0K 0 ) we integrate the double differential width with respect to dM inv (J/ψK * 0 ). The limits of the integration are given by the PDG [10].
Experimentally we have these two mass distributions in Fig. 10 of Ref. [37], and one finds a peak around 1500 MeV in the K * K mass distribution, which cannot be obtained from the structure of Eq. (11). The structure indicates that there is a term like the one in Eq. (11) and another one that would come from the interaction of K * K to give a resonance around 1500 MeV. Consistent with the implicit s-wave for K * K given by the structure of Eq. (11), we take the f 1 (1510) resonance and a structure of the type where 4000 4200 4400 4600 4800 5000
In Fig. 2, we show both K * 0K 0 and J/ψK * 0 mass distributions compared with experiment. We take M f 1 = 1518 MeV and Γ f 1 = 98 MeV compatible with the data of the PDG [10] and the parameter a = −1.2 to agree with the data in Ref. [37]. We see that we account for the bulk of the K * 0K 0 data, and the mass distribution of J/ψK * 0 , which is not fitted, agrees well with the data. It is clear that the K * 0K 0 mass distribution in Fig. 2 also has some resonance-like structures around 1750 MeV and 2100 MeV, but their strength is much smaller than at the peak of 1518 MeV and there is also some extra strength around 1600 MeV. We neglect these higher resonance contributions, but it is clear that we account for most of the strength of the distribution.
Since the structure proposed provides a reasonable description of the data, we can see that the reaction 0 − → 1 − 1 − 0 − (s-wave) respects parity. Inasmuch as CP is a very good symmetry in weak reactions, if parity is conserved, so is C parity. SinceB 0 s is an equal mixture of CP positive and negative, we must also expect an equal mixture of CP positive and negative for KK * andKK * , and with P also conserved, an equal mixture of C parity states.
B. Triangle diagram mechanism forB 0 s → J/ψπ 0 f 0 (a 0 ) In the former subsection we studied on theB 0 s → J/ψK * 0K 0 decay in order to estimate the strength of the transition of Eq. (7). Next we show how the J/ψπ 0 f 0 (a 0 ) is produced using this input. We look now into the related, and unavoidable, mechanism depicted in Fig. 3. In this mechanism, theB 0 s decays into the J/ψKK * (orK * K), the K * (orK * ) decays into πK (or πK), and then the K andK merge to give the a 0 (980) or f 0 (980) in the final state.
The evaluation of the diagrams requires the use of the K * → Kπ amplitude, which comes from the vector(V)-pseudoscalar(P)-pseudoscalar(P) Lagrangian with the trace in SU (3), g = M V 2fπ , m V ∼ 800 MeV the vector mass, f π = 93 MeV the decay constant of pion, and Φ and V given by Eqs. (2), (3). The K * 0 → π 0 K 0 and K * 0 → π 0K 0 amplitudes, stemming from Eq. (17), have opposite signs, and the same happens with K * + → π 0 K + and K * − → π 0 K − . Hence, diagrams Fig. 3(a) and 3(c) with the minus sign give the same contribution and so do Fig. 3(b) and 3(d) with the minus sign. Should we have the C-parity negative K * K combination the (−) sign would be replaced by a (+) sign and the diagrams would cancel, as it should be since π 0 f 0 , π 0 a 0 are C-parity positive.
For the amplitude of the diagram of Fig. 3(a) for π 0 a 0 production, we obtain where g a 0 , K 0K 0 is the coupling of the a 0 resonance to K 0K 0 , and P 0 = M inv (π 0 a 0 ) in the π 0 a 0 rest frame.
By taking the a 0 (980) mass slightly above the KK threshold to apply Eq. (18) of Ref. [4], we find that there is a triangle singularity for this diagram at M inv (π 0 a 0 ) ∼ 1424 MeV. The singularity turns into a finite peak around that mass where most of the strength of the mechanism is concentrated. We take advantage of this fact because then, recalling that the TS places the internal particles on-shell, the on-shell K * 0 momentum in the loop in the frame of π 0 a 0 at rest is 163 MeV/c. This allows us to ignore the ǫ 0 component of K * 0 , which only introduces corrections of order (p K * /m K * ) 2 with a coefficient that renders this correction smaller than 1% (see appendix of Ref. [39]). Then t of Eq. (18) becomes Next, as done in Refs. [4,32], we perform the q 0 integration analytically, leaving a d 3 q integral to be performed numerically. In addition, since k is the only momentum not integrated in Eq. (19) (we evaluate t in the rest frame of π 0 a 0 where P = 0 ), we can replace d 3 q q · · · by k d 3 q q· k k 2 · · · and then t of Eq. (19) can be rewritten as with where

C. Invariant mass distribution
The invariant mass distribution for π 0 a 0 is given by and Next we consider that the a 0 will be seen in the π 0 η mass distribution for the decay of the a 0 and look at the double differential mass distribution in M inv (π 0 a 0 ) and M inv (π 0 η).
This is done in detail in Ref. [38] and we write the final result given by where now C 2 Γ B 0 s is taken from Eq. (14), and t K 0K 0 , π 0 η is the scattering amplitude for K 0K 0 → π 0 η which is calculated using the chiral unitary approach [40], but keeping the masses of the K 0 , K + different, which introduces some isospin breaking in the PP → PP scattering amplitudes. In Eq. (31) the momenta are given by Eqs. (26), (29), replacing m 2 a 0 with M 2 inv (π 0 η), andq So far we have only considered the contribution of the diagram of Fig. 3(a). We must consider explicitly the contribution of diagram Fig. 3(b) and multiply by two to account for Fig. 3(c) and 3(d). This is done replacing t T t K 0K 0 , π 0 η by The production of the f 0 (980), which is related to the π + π − channel, proceeds in the same way. If we look into the π + π − decay channel, all we must do is to replace π 0 η in Eqs. (31) and (33) by π + π − , substituting M inv (π 0 a 0 ) → M inv (π 0 f 0 ) and

III. RESULTS
As we have mentioned, we expect the TS to appear at M inv (π 0 a 0 ) or M inv (π 0 f 0 ) ≈ 1424 MeV. In Fig. 4 we show the double mass distribution as a function of M inv (R) (i.e. M inv (π 0 η) or M inv (π + π − )) for fixed values of M inv (π 0 a 0 ) or M inv (π 0 f 0 ). We take three values around the peak of the TS, 1320 MeV, 1420 MeV and 1500 MeV.
As we can see for M inv (π 0 R) (R = a 0 , f 0 ) at 1420 MeV, we get a large strength for a 0 production as well as f 0 , compared to the other two M inv (π 0 R) masses, which are away from the TS invariant mass. The effect of the TS can be more clearly seen in Fig. 5, where we have integrated the double mass distribution over M inv (R) (i.e. M inv (π 0 η) or M inv (π + π − )).  Fig. 5 and we observe that both the π 0 a 0 and π 0 f 0 mass distributions have a clear peak around M inv (π 0 R) = 1420 MeV.
In π 0 a 0 production there is a bump around 1420 MeV, clearly attributable to the TS, while we also observe a neater peak around 1500 MeV, whose origin is obviously the resonance shape of the original K * K production shown in Fig. 2. Curiously, in the π 0 f 0 production the situation is reversed and the peak appears at 1420 MeV, while at 1500 MeV there is just . The inset magnifies the π 0 f 0 distribution. Fig. 6, and we see that this ratio also peaks around the mass of the TS, although shifted a bit to lower invariant masses. The resonant shape of the K * K production has no role in this ratio, because the factor |F (M inv (π 0 R))| 2 is the same in the two distributions and cancels in the ratio. In other words, the TS enhances the isospin-violating mode π 0 f 0 in absolute terms, but also relative to the isospin-allowed π 0 a 0 mode.
It is interesting to see the sources of isospin violation. They are tied to the differences of the K 0 , K + masses, but they influence both t T in the triangle singularity as well as the two-body scattering matrices t ij for KK → π 0 η and KK → π + π − . To show the effects independently, we take the middle mass M inv (π 0 R) in Fig. 4 and show the π 0 f 0 production in two cases: One assuming equal K masses in the evaluation of the KK → π 0 η, π + π − amplitudes (isospin symmetry in the meson scattering amplitudes) and keeping different K masses in the triangle loop evaluation, t T , and another case in which we take equal K masses in t T but different masses in the meson amplitudes. The results can be seen in Fig. 7, where the "Total" line contains isospin violation both in t T and t ij , same as in Fig. 4. We can see that both effects are important and they add to the total amplitude producing π 0 f 0 . These results are similar to those found in the study of the χ c1 → π 0 f 0 (980)(π + π − ) and χ c1 → π 0 a 0 (980)(π 0 η) in Ref. [41]. In the figure one can observe two structures to the right of the invariant mass distribution corresponding to the K + K − and K 0K 0 thresholds.
It is interesting to compare the behavior of Fig. 6 with what we should expect if there is no triangle singularity. For this purpose we use the same formalism but artificially change the mass of the K * to 300 MeV and its width to zero. This guarantees that when K andK are close to on-shell to produce the f 0 or a 0 , the K * is far off-shell and acts as a point-like interaction. Then we would have a mechanism as depicted in Fig. 8. The result for the new ratio can be seen in Fig. 9.
The results are interesting. We can see that the ratio is practically constant between 1300 MeV and 1500 MeV. It ranges from 9.3 × 10 −3 to 10.2 × 10 −3 in that range, while in Fig. 6 it changes in a factor two in that range. Note also that in Fig. 6 the results are about a factor six bigger than in Fig. 9, indicating the importance of the TS inducing the isospin-violating mode of π 0 f 0 .
Finally, in order to estimate the total rate forB 0 s → J/ψπ 0 f 0 andB 0 s → J/ψπ 0 a 0 , we integrate dΓ dM inv (π 0 R) in Fig. 5 over the π 0 R invariant mass in the range [ 1200 MeV, 1600 MeV ] of invariant masses of the figure and we find dM inv (π 0 f 0 ) dM inv (π + π − ) for fixed M inv (π 0 f 0 ) = 1420 MeV, for two cases, isospin violation only in t T and isospin violation only in KK → π + π − . Note: in this figure, the label of the longitudinal axis is If we take into account the π 0 π 0 decay channel of the f 0 (980), which is one half of the π + π − , These rates are within present observation capability at LHCb.

IV. CONCLUSIONS
We have made a study of theB 0 s → J/ψπ 0 f 0 (980) (a 0 (980)) decay which proceeds via a triangle mechanism in which there is first the decayB 0 s → J/ψK * K orB 0 s → J/ψK * K and posterior fusion of KK to give the f 0 (980) or a 0 (980) resonance. The primary process at quark level isB 0 s → J/ψ ss, with the ss hadronizing into K * K −K * K, which guarantees isospin I = 0 for this combination. This means that the isospin-allowed π 0 R (R = f 0 , a 0 ) final state is π 0 a 0 , while the π 0 f 0 mode is isospin-suppressed. Yet, the explicit consideration of the K + , K 0 different masses gives a contribution for J/ψπ 0 f 0 (980) at the end, with a shape for the f 0 (980) in the π + π − mass distribution tied to the difference of masses of K + , K 0 and, hence, much narrower than the standard f 0 (980) shape seen in the isospin-allowed modes.
This shape and strength are tied to the dynamically generated nature of the f 0 (980) and a 0 (980) as coming from the interactions of pseudoscalar mesons.
The shape obtained for this isospin-suppressed mode is in agreement with other experiments where the f 0 is also obtained with isospin-violating mechanisms. The novelty in the reaction proposed is that the triangle mechanism develops a triangle singularity at an invariant mass M inv (π 0 f 0 ) of about 1420 MeV. Around this invariant mass the production of both the J/ψπ 0 f 0 and J/ψπ 0 a 0 modes are enhanced, and more notably the ratio of the J/ψπ 0 f 0 to J/ψπ 0 a 0 production also shows a peak around the triangle singularity point.
This evidences the role of this triangle singularity in reinforcing isospin violation in the reaction. We also showed that the isospin-violating amplitude has two sources, one from the consideration of the different K masses in the triangle loop, and the other one from the isospin violation in the meson-meson amplitudes, coming again from the consideration of different meson masses in the coupled channels unitary approach used to generate these amplitudes.
Using experimental input from the B 0 s → J/ψK * K + c.c. decay, we can make absolute predictions for the branching fractions ofB 0 s → J/ψπ 0 f 0 (980) (a 0 (980)) and find them within measurable range.
The predictions made, and their accessibility within present experimental facilities, should give a strong motivation to perform these experiments, which will provide valuable information on the nature of the low lying scalar mesons.