The decays of $$\bar{B}^0$$B¯0, $$\bar{B}^0_s$$B¯s0 and $$B^-$$B- into $$\eta _c$$ηc plus a scalar or vector meson

We investigate the decays of $$\bar{B}^0_s$$B¯s0, $$\bar{B}^0$$B¯0 and $$B^-$$B- into $$\eta _c$$ηc plus a scalar or vector meson in a theoretical framework by taking into account the dominant process for the weak decay of $$\bar{B}$$B¯ meson into $$\eta _c$$ηc and a $$q\bar{q}$$qq¯ pair. After hadronization of this $$q\bar{q}$$qq¯ component into pairs of pseudoscalar mesons we obtain certain weights for the pseudoscalar meson-pseudoscalar meson components. In addition, the $$\bar{B}^0$$B¯0 and $$\bar{B}^0_s$$B¯s0 decays into $$\eta _c$$ηc and $$\rho ^0$$ρ0, $$K^*$$K∗ are evaluated and compared to the $$\eta _c$$ηc and $$\phi $$ϕ production. The calculation is based on the postulation that the scalar mesons $$f_0(500)$$f0(500), $$f_0(980)$$f0(980) and $$a_0(980)$$a0(980) are dynamically generated states from the pseudoscalar meson-pseudoscalar meson interactions in S-wave. Up to a global normalization factor, the $$\pi \pi $$ππ, $$K \bar{K}$$KK¯ and $$\pi \eta $$πη invariant mass distributions for the decays of $$\bar{B}^0_s \rightarrow \eta _c \pi ^+ \pi ^-$$B¯s0→ηcπ+π-, $$\bar{B}^0_s \rightarrow \eta _c K^+ K^-$$B¯s0→ηcK+K-, $$\bar{B}^0 \rightarrow \eta _c \pi ^+ \pi ^-$$B¯0→ηcπ+π-, $$\bar{B}^0 \rightarrow \eta _c K^+ K^-$$B¯0→ηcK+K-, $$\bar{B}^0 \rightarrow \eta _c \pi ^0 \eta $$B¯0→ηcπ0η, $$B^- \rightarrow \eta _c K^0 K^-$$B-→ηcK0K- and $$B^- \rightarrow \eta _c \pi ^- \eta $$B-→ηcπ-η are predicted. Comparison is made with the limited experimental information available and other theoretical calcualtions. Further comparison of these results with coming LHCb measurements will be very valuable to make progress in our understanding of the nature of the low lying scalar mesons, $$f_0(500), f_0(980)$$f0(500),f0(980) and $$a_0(980)$$a0(980).

Estimations of the branch ratios for some of these decays have been done by employing the perturbative QCD factorization approach [5,6]. Also, in Ref. [7] the decay widths of B 0 s → η c f 0 (980) and B 0 s → η c φ were evaluated in the light-front quark model. The conclusions of Ref. [7] are that the mostly dominant contribution for the B 0 s → η c π + π − decay is from the f 0 (980) and the f 0 (980) should be a KK molecule or a tetraquark state, at least the pure quarkantiquark component of f 0 (980) is small.
For the B 0 s → J/ψπ + π − decay, a simple theoretical method based on the final state interaction of mesons provided by the chiral unitary approach has been applied in Ref. [8], where the theoretical results are in agreement with the data. The work of Ref. [8] isolates the dominant weak decay mechanism into J/ψ and a qq pair. Then, the qq pair is hadronized and meson-meson pairs are formed with a certain weight. The final state interaction of the meson-meson components, described in terms of chiral unitary approach, gives rise to the f 0 (980) and f 0 (500) resonances. The approach of Ref. [8] was successfully extended to study other weak B and D decays in Refs. [9][10][11][12][13][14][15][16] (see also Ref. [17] for an extensive review). Other theoretical work has also been done within the perturbative QCD approach in Ref. [18]. Recently, another approach has been used in Ref. [19] using effective Hamiltonians, transversity form factors and implementing the meson-meson final sate interaction. In addition to the π + π − production, theB 0 s decay into J/ψ and K + K − is also studied in Ref. [19] and compared to experimental measurements. In Ref. [20], the work of Ref. [19] on theB 0 s decay was extended to theB 0 decay, where theB 0 → J/ψπ η and J/ψ KK were investigated. It was found that the crucial information on πη scattering can be deduced from thē B 0 → J/ψπ η and J/ψ KK decays [20].
Following this line of research, the purpose of this paper is to investigate the reactions ofB 0 ,B 0 s and B − decaying into η c plus a scalar or vector meson. We evaluate the π + π − and K + K − invariant mass distributions in theB 0 s decays into η c π + π − and η c K + K − and the π + π − , K + K − and πη production in theB 0 decay into η c and this pair of mesons through scalar of vector meson. At the same time, we investigate also the B − → η c K 0 K − and B − → η c π − η decays. Up to a global factor, one can compare the strength of those invariant mass distributions.
To end this introduction, we would like to mention that, up to an arbitrary normalization, one can obtain the invariant mass distributions and relate the different mass distributions with no parameters fitted to the data. This is due to the unified picture that the chiral unitary approach provides for the final state interaction of mesons. In this sense, predictions on the coming measurements should be most welcome, and if supported by experiment, it can give us more information about the nature of these low lying scalar mesons, f 0 (500), f 0 (980) and a 0 (980), which are dynamically generated states from the interaction of pseudoscalar mesons using a meson-meson interaction derived from the chiral Lagrangians [21,22].
This article is organized as follows. In Sect. 2, we present the theoretical formalism of the decays ofB 0 ,B 0 s and B − decays into η c plus a scalar meson, explaining in detail the hadronization and final state interactions of the meson-meson pairs. Numerical results and discussions are presented in Sect. 3, followed by a summary in the last section.

Scalar meson production
The leading contributions to the decays ofB 0 s ,B 0 and B − into η c plus a scalar meson is the Cabibbo favored b → ccs process and the Cabibbo suppressed b → ccd process. In the following we will discuss the production mechanisms for these decays.
In addition to the charm quark decay process described above, in principle one should consider also the diagrams in which the qq pair in the final state annihilate into gluons which can then hadronize into pseudoscalar meson pairs. However, these contributions are smaller than the charm quark decay process, because these disconnected diagrams should be suppressed according to the Okubo-Zeig-Iizuka (OZI) rule. On the other hand, including such contributions, the decay amplitudes would become more complex due to additional parameters and we cannot determine or constrain these parameters because of the scarce information about them. Hence, we will leave these contributions to a future work when more experimental data become available.
Following Refs. [8,23], in Fig. 1 we show the diagrams at the quark level that are responsible for theB 0 s ,B 0 , and B − decays into η c and another pair of quarks: ss in the case of theB 0 s decay (Fig. 1a), dd in the case ofB 0 decay (Fig. 1b),  and dū for the B − decay (Fig. 1c). TheB 0 s decay involves the V cs , Cabibbo favored Cabibbo-Kobayashi-Maskawa matrix element, and theB 0 and B − decays involves the V cd Cabibbo suppressed one, which makes the widths large in theB 0 s case compared to theB 0 and B − decays. 1 In order to produce two mesons, the qq pair has to hadronize, which one can implement adding an extraqq pair with the quantum numbers of the vacuum,ūu +dd +ss, as shown also in Fig. 1. Next step corresponds to writing the qq(ūu +dd +ss) combination in terms of pairs of mesons. Following the work of Refs. [8,17] we obtain 2 dd(ūu +dd +ss) ≡ π − π + + 1 2 where the η terms have been neglected because the η has large mass and has very small effect here. It is worth to 1 The use of charge-conjugate modes is implied throughout this paper. 2 There are small changes here with respect to Ref. [8], where the singlet of SU(3) was ignored (see also in Ref. [13], where more details can be found). In this work, we take the standard mixing of the η and η in terms of a singlet η 1 and an octet η 8 of SU (3): Diagrammatic representations of the production of π + π − , π 0 η, K + K − , π − η, and K − K 0 via direct plus re-scattering mechanisms inB 0 (a),B 0 s (b) and B − (c) decays mention that these results shown in Eqs. (1)-(3) can be also obtained with the scalar form factors [24,25].
After the production of a meson-meson (MM) pair, the final state interaction between the mesons takes place, which can be parameterized by the re-scattering shown in Fig. 2 at the hadronic level. Since we consider only the S-wave interaction between the pseudo-scalar meson and pseudoscalar meson, we will have the contributions from only the scalar mesons. In Fig. 2, we also show the tree level diagrams for the ππ, KK and πη production.
The decay amplitudes for a final production of the different meson pairs are given by [26] forB 0 s decays, and forB 0 decays, and forB − decays. In Eqs. (4)-(10), V P is the production vertex which contains all dynamical factors common to all the above seven decays. We shall assume V P as constant and fit it to the experimental date. The G MM are the loop functions of two meson propagators. The t MM→MM are the scattering matrices and they are calculated in Ref. [8] following Ref. [27]. Note that we can easily obtain t π − η→π − η , t K 0 K − →K 0 K − and t π − η→K 0 K − using isospin symmetry, With all the ingredients obtained above, one can write down the invariant mass distributions for those decays as follows [29] where MB j is the mass ofB 0 ,B 0 s , or B − , while M inv is the invariant mass of the final MM pair. The p η c is the η c momentum in the rest frame ofB j andp M is the momentum of one pseudo-scalar meson in the rest frame of MM pair, which are, where λ is the Källen function with λ(x, y, z) = (x − y − z) 2 − 4yz, and m 1 and m 2 are the masses of the two mesons of the M M pair, respectively.

Vector meson production
The B meson decays weakly into η c meson in addition to a vector meson. At the quark level, this process is illustrated in Fig. 1 without the qq hadronzition to two mesons. According to this figure, the cc quark pair converts into an η c , while the remaining qq can be related to the vector mesons, ρ 0 , φ or K * 0 , which are largely qq states [28,29]. Therefore, in terms of quarks the vector mesons are given by (uū−dd), φ = ss, and K * 0 = ds. Then the decay amplitudes for the vector mesons production are where p V (V = φ, ρ 0 , K * 0 orK * 0 ) is the vector momentum in the rest frame ofB j , As in Refs. [8,9], in a 0 − → 0 − 1 − transition, we shall need an L = 1 orbital angular momentum for the vector meson to match angular momentum conservation. Hence, we assume the vertex as V P p V cosθ and we also assume V P to be constant. Then we can determine ratios of vector mesons production with respect to the φ production which was measured by the LHCb collaboration [2], The  [29].
Then we obtain the branching ratios by taking Br(B 0 s → η c φ) = (5.01 ± 0.87) × 10 −4 [2] as input, where the errors are from the experimental branching ratio of Br(B 0 s → η c φ). The experimental result for Br(B 0 → η cK * 0 ) is (6.3 ± 0.9) × 10 −4 [29], from where we can see that the agreement between our theoretical calculation and the experiment is good within error. There are no experimental results for the branching ratios of Br(B 0 s → η c K * 0 ) and Br(B 0 → η c ρ 0 ). Our predictions here will be tested by the future LHCb and Belle-II experiment.
On the other hand, if we integrated Eq. (14), up to one free parameter V P , we can extract the contribution from f 0 (980) for the decay ofB 0 s → η c π + π − , since, in our production mechanism, the main contribution for this decay is f 0 (980). Then, one can determine V P as In Fig. 3, we show the π + π − and K + K − invariant mass distributions for theB 0 s → η c π + π − andB 0 s → η c K + K − , respectively. As one can see, the f 0 (980) production is clearly dominant while there is no evident signal for the f 0 (500). For theB 0 s → η c K + K − decay, the K + K − distribution gets maximum strength just above the K + K − threshold and then falls down gradually. This is due to the effect of the f 0 (980) Fig. 3 π + π − and K + K − invariant mass distributions forB 0 s → η c π + π − and η c K + K − resonance below the KK threshold. 4 Starting from the dominant weak decay process shown in Fig. 1 we have η c and ss production in theB 0 s decay. Because the ss pair has isospin zero, and the strong interaction hadronization conserves it, even if the K + K − system can have I = 0 or 1, the process of formation guarantees that this is an I = 0 state and the shape of the K + K − distribution is due to the f 0 (980) with I = 0.
The strength for the K + K − distribution is small compared to the one of the f 0 (980) at its peak for the π + π − distribution, but the integrated strength over the invariant mass of K + K − is of the same order of magnitude as that for the strength below the peak of the f 0 (980) going to π + π − . It is interesting to compare this with experiment. First by integrating the strength of the K + K − invariant mass distribution over its invariant mass, up to M inv (K + K − ) = 1200 MeV, 5 we find a ratio Here, we should mention that we are calculating only the S-wave contribution of the K + K − invariant mass distribution, hence, contributions from higher waves, such as φ (Pwave), f 2 (1525) (D-wave) etc, are not included.
Secondly, if we stick to a band of energies around the φ meson peak, 990 < M inv (K + K − ) < 1050 MeV, as done in Ref. [34] for theB 0 s → J/ψ K + K − , we get the S-wave fraction 4 The pole position for f 0 (980) is obtained as: √ s R = 981.5 − i5.5 (MeV). 5 We should mention that the chiral unitary approach that we use only makes reliable predictions up to 1200 MeV [27]. One should not use the model for higher invariant masses. With this perspective we will have to admit uncertainties in the mass distributions, particularly at invariant masses higher than 1200 MeV [31][32][33]. Fig. 4 π + π − , π 0 η, and K + K − invariant mass distributions forB 0 → η c π + π − , η c K + K − , and η c π 0 η, respectively where Br[B 0 s → η c φ] = (5.01 ± 0.87) × 10 −4 [2] and the branching fraction of 0.489 for φ decay into K + K − has been taken [29]. This value, one of our model predictions, could be tested by future experiment.
On the other hand, for the case of J/ψ production, the experimental result for the S-wave fraction is (1.1 ± 0.1 +0.2 −0.1 )% [34], while the theoretical calculation for this value in Ref. [19] is about (1.1 ∼ 1.95)% and the value of (1.7 ± 0.3)% was obtained in Ref. [12]. The decays of B 0 s into η c plus a scalar meson is in S-wave and thus we should expect the S-wave fraction for theB 0 s → η c f 0 (980) reasonably bigger than forB 0 s → J/ψ f 0 (980). We come back now to the decays of theB 0 . In Fig. 4 we show the theoretical results for the π + π − , K + K − and π 0 η invariant mass distributions forB 0 → η c π + π − , η c K + K − , and η c π 0 η. In theB 0 decays into η c plus a scalar meson, we had the hadronization of a dd pair, which contains I = 0 and 1. But, the π + π − in S-wave can only be in I = 0, hence the peaks for the π + π − distribution due to the f 0 (500) and f 0 (980) excitation. While for the ρ 0 vector meson production, we see that the ρ 0 contribution peaks around 770 MeV, and has larger strength than the f 0 (500) contribution, but at invariant masses around 500 MeV and bellow, the strength of the f 0 (500) dominates over the one of the ρ 0 meson. The strength of the peak of ρ 0 distribution is about a factor 2 larger than that of the f 0 (500) distribution. For the K + K − production in theB 0 decay, we have considered both the I = 0 [ f 0 (980)] and I = 1 [a 0 (980)] contribution. This makes the line shape of K + K − invariant mass distributions different with the one shown in Fig. 3 for theB 0 s decay. One can see that, from Fig. 4, the strength of the f 0 (980) excitation is very small compared to that of the f 0 (500) (the broad peak to the left). Note that because of the experimental resolution the f 0 (980) peak would not appear so narrow in  [8,10], we can extract the f 0 (500) contribution to the branching ratio by assuming a smooth background below the f 0 (980) peak, we find with error from the uncertainty of V P shown in Eq. (32). Then we find ratios, R 1 and R 2 which is consistent with the ones obtained in Refs. [5,6]:  [5] with the perturbative QCD factorization approach. We hope that the future experimental measurements can clarify this issue.
In Fig. 4, the π 0 η invariant mass distribution has a sizeable strength, bigger than that for the S-wave π + π − and K + K − distributions. As one can see, we get the typical cusp structure of the a 0 (980). This prediction is tied exclusively to the weights of the starting meson meson channels in Eq. (1) and the final state interaction in Eqs. (6), (7), and (8). Hence, this is a prediction of this approach, not tied to any experimental input.
Next, we show the results for B − decay in Fig. 5, where the strength for the π − η invariant mass distribution is two times as big as the one ofB 0 → η c π 0 η shown in Fig. 4. For the K 0 K − mass distribution we see that the position of the peak has moved to higher invariant masses compared to the K + K − invariant mass spectrum of theB 0 → η c K + K − or B 0 s → η c K + K − decays. In fact, the K 0 K − invariant mass distribution in the B − decay due to the a 0 (980), which is seen in the figures, is much wider than that of the f 0 (980). It would be most instructive to see all these features in future experiments, such as LHCb and Belle-II.

Summary
We have performed a study of the ππ, πη and KK invariant mass distributions forB 0 s → η c π + π − ,B 0 s → η c K + K − , B 0 → η c π + π − ,B 0 → η c K + K − ,B 0 → η c π 0 η, B − → η c π − η, and B − → η c K 0 K − . We take the dominant mechanism for the weak decay of the B meson, going to η c and a qq pair that, upon hadronization, leads to ππ, πη, and KK in the final state, and this interaction is basically mediated by the scalar mesons, f 0 (500), f 0 (980), and a 0 (980). In addition, we have also studied the decay of these B mesons into η c plus a vector meson, ρ 0 , φ,K * 0 , and K * 0 .
Up to a global factor, 7 which is determined to the experimental measurement, we can compare the strength of the ππ, πη and KK invariant mass distributions. For theB 0 s → η c K + K − , only the f 0 (980) resonance contributes to the K + K − mass distribution, but in the case of theB 0 → η c K + K − , both the f 0 (980) and a 0 (980) resonances contribute to its strength. The strength of the KK invariant mass distribution in theB 0 s decay is much larger than the one inB 0 decay, which is because theB 0 s decay is Cabibbo favored process, while theB 0 decay is the Cabibbo suppressed process. In the case of theB 0 → η c π 0 η, one finds a cusp structure for the a 0 (980) and its strength is much larger than the one for theB 0 → η c π + π − decay around the f 0 (980) peak. We also have made predictions for η c ρ 0 , η c K * 0 , and η cK * 0 productions that can be tested in a future partial wave analysis of these decays. In particular, we find that the ρ 0 production is bigger then the f 0 (500) production inB 0 decay by a factor 2.
Our theoretical results shown here are predictions for ongoing experiments at LHCb and Belle-II, and comparison of the observed results with our predictions will be most useful to make progress in our understanding of the mesonmeson interaction and the nature of the low lying scalar mesons. We would suggest the experimentalists carry out