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The
Bs decay intoJ/ψ and two mesons is an excellent source of information on meson dynamics. At the quark level the decay proceeds via internal emission [1], as shown in Fig. 1. Thecˉc quarks give rise to theJ/ψ and the extrasˉs , which appear in the Cabibbo favored decay mode, have isospinI=0 . It is a rather clean process and indeed, in the LHCb experiment [2] thef0(980) resonance was seen as a strong peak in the invariant mass distribution ofπ+π− . The wayπ+π− are produced is studied in detail in Ref. [3]. Thesˉs pair of quarks is hadronized, introducing aˉqq pair with vacuum quantum numbers, andKˉK inI=0 plusηη are produced, which are allowed to interact within the chiral unitary approach [4–7] to produce thef0(980) resonance, which is dynamically generated from the interaction of pseudoscalar pairs and couples mostly toKˉK . With such a clean process producingI=0 , one finds a very interesting place to produce thea0(980) , via isospin violation, and add extra information to the subject of thef0(980)−a0(980) mixing that has stimulated much work. Indeed, there are many works devoted to this subject [8–36] and some cases where, due to a triangle singularity, the amount of isospin breaking (we prefer this language rather than mixing, since there is not a universal mechanism for the mixing and it depends upon the particular reaction) is abnormally large [26, 27, 37]. The way thea0(980) resonance can be produced in theBs→J/ψπ0η decay is tied to the nature off0(980) anda0(980) , since the resonances are dynamically generated by the pseudoscalar–pseudoscalar (PP ) interaction [4]. It is the meson–meson loops in the Bethe–Salpeter equation, particularlyKˉK in the case off0 anda0 , that give rise to the resonances. TheK+K− andK0ˉK0 loops cancel forI=1 starting from theI=0 combination of the hadronizedsˉs quarks, but only if the masses ofK+ andK0 are taken as equal. When the mass difference is considered, then the isospin is automatically broken and some peaks appear for the isospin-violating decay modes which are rather narrow and are tied to the kaon mass differences. The relation of thea0−f0 mixing to this mass difference is shared by most theoretical studies, starting from Ref. [8]. However, as shown in Ref. [38] in the study ofDs→e+νea0(980) , isospin breaking takes place in the loop forKˉK propagation in the decay but also in the same meson–meson scattering matrix, which enters the evaluation of the process, something already noticed in Ref. [39]. Yet, the two sources of isospin violation are different depending on the reaction studied, hence the importance of studying the isospin violation in different processes to gain information on the way the violation is produced and its dependence on the nature of thea0(980) andf0(980) resonances, which has originated much debate in the literature.We study the process of
a0 andf0 production, following the lines of Refs. [3] and [38], and by taking experimental information on theBs→J/ψπ+π− reaction, we make predictions for the rate ofBs→J/ψπ0η production and the shape of theπ0η mass distribution. The branching fraction obtained for this latter decay is of the order of5×10−6 , well within the range of rates already measured and reported by the PDG [40], which should stimulate its measurement in the future. -
The mechanism at the quark level for the
ˉB0s→J/ψπ+π−(π0η) reaction is depicted in Fig. 1, having ansˉs pair with isospinI=0 at the end. Note that the light scalarsf0(980) anda0(980) haveI=0,1 , respectively. The production off0(980) is isospin conserved, while the production ofa0(980) is isospin forbidden and involves isospin violation.To obtain
π+π− orπ0η in the final state in Fig. 1, we need to hadronize thesˉs pair by introducing an extraˉqq pair with vacuum quantum numbers. We start with theqˉq matrix M in SU(3),M=(uˉuuˉduˉsdˉudˉddˉssˉusˉdsˉs).
(1) Next, we write the matrix M in terms of pseudoscalar mesons, assuming that the
η isη8 of SU(3),M→P=(1√2π0+1√6η+1√3η′π+K+π−−1√2π0+1√6η+1√3η′K0K−ˉK0−√23η+√13η′), (2) which is often used in chiral perturbation theory [4]. On the other hand, when we consider the Bramon
η−η′ mixing [41], the matrix M can be written asM→P(m)=(1√2π0+1√3η+1√6η′π+K+π−−1√2π0+1√3η+1√6η′K0K−ˉK0−1√3η+√23η′).
(3) Since the
η′ is inessential in the dynamical generation of thef0(980) anda0(980) resonances [4], we will ignore theη′ in the present work.After hadronization of the
sˉs component, we obtainsˉs→H=∑isˉqiqiˉs=∑iP3iPi3=(P2)33.
(4) In the case without
η−η′ mixing, the matrixP of Eq. (2) is used, and then the hadron component H in Eq. (4) is given byH=K−K++ˉK0K0+23ηη.
(5) In the case with
η−η′ mixing, one uses matrixP(m) of Eq. (3), and obtainsH=K−K++ˉK0K0+13ηη,
(6) differing only in the
ηη component, which affects the production off0 but not the production ofa0 . We define the weight of thePP components in H ashK+K−=1,hK0ˉK0=1,hηη=23,h(m)ηη=13.
(7) One can see that neither Eq. (5) nor Eq. (6) contains
π+π− orπ0η , but they can be produced by the final state interaction of theKˉK andηη components, as depicted in Fig. 2. The transition matrix from thePP state toπ+π− orπ0η is represented by the circle behind the meson–meson loop in Fig. 2, which contains the information off0(980) anda0(980) respectively. According to the method in Ref. [4] (the chiral unitary approach), these resonances are the result of thePP interaction in the coupled channelsKˉK,ππ,πη,ηη .Figure 2. Final state interaction of the hadron components leading to
π+π− orπ0η in the final state.By using the unitary normalization [4, 38], the amplitude for the
ˉB0s→J/ψπ+π− decay, as a function of theπ+π− invariant massMinv(π+π−) , is given by [38]tπ+π−=C[hK+K−⋅GK+K−(Minv(π+π−))⋅TK+K−,π+π−(Minv(π+π−))+hK0ˉK0⋅GK0ˉK0(Minv(π+π−))⋅TK0ˉK0,π+π−(Minv(π+π−))+hηη×2×12⋅Gηη(Minv(π+π−))⋅Tηη,π+π−(Minv(π+π−))],
(8) and the amplitude for the
ˉB0s→J/ψπ0η decay, as a function of theπ0η invariant massMinv(π0η) , is given by [38]tπ0η=C[hK+K−⋅GK+K−(Minv(π0η))⋅TK+K−,π0η(Minv(π0η))+hK0ˉK0⋅GK0ˉK0(Minv(π0η))⋅TK0ˉK0,π0η(Minv(π0η))+hηη×2×12⋅Gηη(Minv(π0η))⋅Tηη,π0η(Minv(π0η))],
(9) with
C an arbitrary normalization constant which is canceled in the ratio of thef0 anda0 production rates. For the case withη−η′ mixing, the corresponding amplitudes can be obtained by replacinghηη withh(m)ηη in Eqs. (8) and (9).In Eqs. (8) and (9),
Gi is the loop function of the two intermediate pseudoscalar mesons, which is regularized with a three momentum cut-offqmax [4],Gi(√s)=∫qmax0q2dq(2π)2w1+w2w1w2[s−(w1+w2)2+iϵ],
(10) with
wj=√m2j+→q2 and√s the centre-of-mass energy of the two mesons in the loop.Ti,j is the total amplitude for thei→j transition and can be obtained by solving the Bethe–Salpeter (BS) equation with sixPP coupled channelsπ+π− ,π0π0 ,K+K− ,K0ˉK0 ,ηη andπ0η , in a matrix form,T=[1−VG]−1V,
(11) where the matrix V is the kernel of the BS equation. Its elements
Vij are the s-wave transition potentials which can be taken from Eq. (A3) and Eq. (A4) of Ref. [38], corresponding to the cases without and withη−η′ mixing, respectively.The differential decay width for
ˉB0s→J/ψπ0η orˉB0s→J/ψπ+π− decay is given bydΓdMinv(ij)=1(2π)314M2ˉB0s13p2J/ψpJ/ψ˜pπ|tij|2,
(12) where
ij=π+π− orπ0η ,Minv(ij) is the invariant mass of the finalπ+π− orπ0η ,tπ+π− andtπ0η are the amplitudes from Eq. (8) and Eq. (9) respectively,pJ/ψ is theJ/ψ momentum in theˉB0s rest frame, and˜pπ is the pion momentum in the rest frame of theπ+π− orπ0η system,pJ/ψ=λ1/2(M2ˉB0s,M2J/ψ,M2inv)2MˉB0s,
(13) ˜pπ={λ1/2(M2inv,m2π,m2π)2Minv,forπ+π−production,λ1/2(M2inv,m2π,m2η)2Minv,forπ0ηproduction,
(14) with
λ(x2,y2,z2)=x2+y2+z2−2xy−2yz−2zx the Källen function. In Eq. (12), the factor13p2J/ψ stems from the fact that we need a p-wave to match angular momentum in the0−→1−0+ transition and we take a vertex of typepJ/ψcosθ . -
We follow Ref. [38] and take the cut-off
qmax=600 MeV and650 MeV for the cases withoutη−η′ mixing and withη−η′ mixing respectively, with which thef0(980) anda0(980) resonances can be dynamically produced well from the PP interaction. Theπ+π− andπ0η mass distributionsdΓdMinv(ij) are shown in Fig. 3 for the case withoutη−η′ mixing and in Fig. 4 for the case withη−η′ mixing, respectively. By comparing Fig. 3 and Fig. 4, one finds that the results of the two figures are very similar, and the difference between them can serve as an estimate of the uncertainties of our formalism.Figure 3. (color online)
Minv(π+π−) mass distribution forˉB0s→ J/ψf0(980),f0(980)→π+π− decay, andMinv(π0η) mass distribution forˉB0s→J/ψa0(980),a0(980)→π0η decay. Inset: Magnifiedπ0η . (Withoutη−η′ mixing).Figure 4. (color online)
Minv(π+π−) mass distribution forˉB0s→ J/ψf0(980),f0(980)→π+π− decay, andMinv(π0η) mass distribution forˉB0s→J/ψa0(980),a0(980)→π0η decay. Inset: Magnifiedπ0η . (Withη−η′ mixing).Now, let us look at the
π+π− andπ0η mass distributions in Fig. 4 withη−η′ mixing. One can see a strong peak forf0(980) production in theπ+π− mass distribution and a small peak fora0(980) production in theπ0η mass distribution. Here the shape ofa0(980) resonance is quite narrow, considerably different to the standard cusp-like shape (with a width of about 120 MeV) of the ordinary production ofa0(980) in an isospin allowed reaction [42]. If isospin were conserved, one would find thea0(980) production with zero strength. The small peak ofa0(980) in Fig. 4 indicates that isospin violation takes places in theˉB0s→J/ψπ0η reaction. According to Eq. (A4) of Ref. [38], we haveVK+K−,π0η=−VK0ˉK0,π0η for the transition potentials. Hence, if we use average masses for kaons, there will be a precise cancellation of the first two terms of the amplitudetπ0η in Eq. (9), resulting on zero strength fora0(980) production. On the contrary, using the physical masses for the neutralK0 and the chargedK+ in the formalism results in the production of thea0(980) resonance with a narrow shape related to the difference of mass between the charged and neutral kaons. In our picture, there are two sources of isospin violation: one is theK+ ,K0 mass difference for the explicitK+K− andK0ˉK0 loops in Fig. 2, and the other is from the T matrix involving rescattering in Fig. 2.It is interesting to investigate the effects of these two sources of isospin violation. For that, we follow Ref. [38] and define the ratio R, which reflects the amount of the isospin violation, as
R=Γ(ˉB0s→J/ψa0(980),a0(980)→π0η)Γ(ˉB0s→J/ψf0(980),f0(980)→π+π−),
(15) with decay widths
Γ[ˉB0s→J/ψa0(980) ,a0(980)→π0η] andΓ[ˉB0s→J/ψf0(980),f0(980)→π+π−] obtained by integrating Eq. (12) over the invariant massMinv(ij) .Under several different assumptions related to the two sources of isospin violation, we evaluate the ratio R. The results are shown in Table 1.
no η−η′ mixingI.V. both in T matrix and in explicit KˉK loops (Case 1)3.1×10−2 I.V. only in T matrix (Case 2) 3.5×10−2 I.V. only in explicit KˉK loops (Case 3)7.0×10−4 with η−η′ mixingI.V. both in T matrix and in explicit KˉK loops (Case 4)3.7×10−2 I.V. only in T matrix (Case 5) 4.1×10−2 I.V. only in explicit KˉK loops (Case 6)9.7×10−4 Table 1. Values of R with different assumptions. (In the table, I.V. denotes isospin violation.)
From Table 1, we observe that the ratio R with
η−η′ mixing (Case 4) is about 20% bigger than that withoutη−η′ mixing (Case 1). By comparing the values of R for Case 2 and Case 3 (or, for Case 5 and Case 6), we find that the isospin violation in the T matrix has a more important effect than that in the explicitKˉK loops, being at least one order of magnitude larger. This fact is interesting, since in our picture thef0(980) anda0(980) resonances are dynamically generated from thePP interaction with the information on their nature contained in the T matrix. For theˉB0s→J/ψπ+π−(π0η) decay, neither theπ+π− nor theπ0η can be directly produced fromsˉs hadronization [see Eqs. (5) and (6)], hence there is no contribution from the tree level. Instead, they are produced through the rescattering mechanism of Fig. 2, withf0(980) anda0(980) resonances as dynamically generated states from thePP interaction. The production rate of thef0(980) (a0(980) ) resonance in theˉB0s→J/ψπ+π−(π0η) decay is sensitive to the resonance information contained in the T matrix. Therefore, this mode is particularly suitable to test the nature off0(980) anda0(980) resonances and to investigate the isospin violation.From the PDG [40], the experimental branching ratio of the
ˉB0s→J/ψf0(980),f0(980)→π+π− decay readsBr[ˉB0s→J/ψf0(980),f0(980)→π+π−]=(1.28±0.18)×10−4.
(16) By using the ratio R in Table 1 and the branching ratio of Eq. (16), the branching ratio for
a0(980) production can be obtained,Br[ˉB0s→J/ψa0(980),a0(980)→π0η]={(3.95±0.56)×10−6,forCase1;(4.74±0.67)×10−6,forCase4.
(17) This branching ratio is of the order of
5×10−6 , not too small considering that several rates of the order of10−7 are tabulated in the PDG [40]. The branching ratio and the shape of theπ0η mass distribution of theˉB0s→J/ψπ0η decay provide relevant information on the nature of thea0(980) resonance. Experimental measurements will be very valuable. -
In the present work, we study the isospin allowed decay process
ˉB0s→J/ψπ+π− and the isospin forbidden decay processˉB0s→J/ψπ0η , paying attention to the different sources of isospin violation.First, we have
J/ψsˉs production in theˉB0s decay, via internal emission as shown in Fig. 1. After the hadronization ofsˉs into meson–meson components, we obtainKˉK pairs andηη , whileπ+π− andπ0η are not produced at this step. Therefore, to seeπ+π− orπ0η in the final state, rescattering of theKˉK orηη components is needed to produceπ+π− andπ0η at the end. The picture shows that the weak decay amplitudes are proportional to the T matrix of the meson–meson transitions. We can obtain information about the violation of isospin from these magnitudes. In Figs. 3 and 4, we observe a clear signal forf0(980) production. We also observe that the shape of theπ0η mass distribution is very different from the shape of the commona0(980) production in isospin-allowed reactions, and it is related to the difference in mass between the charged and neutral kaons. In the production ofa0(980) we find two sources of isospin violation: one is that the loops containingK+K− orK0ˉK0 do not cancel due to the different mass between the charged and neutral kaons, and the other is that the transition T matrix of the meson–meson interaction already contains some isospin violation. In fact, we find that the contribution from isospin violation in the T matrix is far more important than the contribution of the explicit loops in the weak decay, being at least one order of magnitude larger. The study here shows that this reaction is very sensitive to the way the resonances are generated.The
D+s semileptonic decay [38] andˉB0s mesonic decay both produce ansˉs pair at the end, and the two resonances off0(980) anda0(980) are produced dynamically by the interaction of pseudoscalar mesons through the chiral unitary approach. The results ofD+s semileptonic decay are consistent with the experimental upper bound. We also calculate the branching ratio ofˉB0s→J/ψa0(980) fora0(980) production, and the values are not too small, of the order of5×10−6 . Our results provide a reference basis for experiments, which we expect to be carried out in the near future.
