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The ˉB0sJ/ψπ0η decay and the a0(980)f0(980) mixing

  • We study the ˉB0sJ/ψf0(980) and ˉB0sJ/ψa0(980) reactions, and pay attention to the different sources of isospin violation and mixing of f0(980) and a0(980) resonances where these resonances are dynamically generated from meson–meson interactions. We find that the main cause of isospin violation is isospin breaking in the meson–meson transition T matrices, and the other source is that the loops involving kaons in the production mechanism do not cancel due to the different masses of charged and neutral kaons. We obtain a branching ratio for a0(980) production of the order of 5×106. Future experiments can address this problem, and the production rate and shape of the π0η mass distribution will definitely help to better understand the nature of scalar resonances.
  • The Bs decay into J/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. The cˉc quarks give rise to the J/ψ and the extra sˉs, which appear in the Cabibbo favored decay mode, have isospin I=0. It is a rather clean process and indeed, in the LHCb experiment [2] the f0(980) resonance was seen as a strong peak in the invariant mass distribution of π+π. The way π+π are produced is studied in detail in Ref. [3]. The sˉs pair of quarks is hadronized, introducing a ˉqq pair with vacuum quantum numbers, and KˉK in I=0 plus ηη are produced, which are allowed to interact within the chiral unitary approach [47] to produce the f0(980) resonance, which is dynamically generated from the interaction of pseudoscalar pairs and couples mostly to KˉK. With such a clean process producing I=0, one finds a very interesting place to produce the a0(980), via isospin violation, and add extra information to the subject of the f0(980)a0(980) mixing that has stimulated much work. Indeed, there are many works devoted to this subject [836] 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 the a0(980) resonance can be produced in the BsJ/ψπ0η decay is tied to the nature of f0(980) and a0(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, particularly KˉK in the case of f0 and a0, that give rise to the resonances. The K+K and K0ˉK0 loops cancel for I=1 starting from the I=0 combination of the hadronized sˉs quarks, but only if the masses of K+ and K0 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 the a0f0 mixing to this mass difference is shared by most theoretical studies, starting from Ref. [8]. However, as shown in Ref. [38] in the study of Dse+νea0(980), isospin breaking takes place in the loop for KˉK 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 the a0(980) and f0(980) resonances, which has originated much debate in the literature.

    Figure 1

    Figure 1.  Diagram for ˉB0s decay into J/ψ and a primary sˉs pair.

    We study the process of a0 and f0 production, following the lines of Refs. [3] and [38], and by taking experimental information on the BsJ/ψπ+π reaction, we make predictions for the rate of BsJ/ψπ0η production and the shape of the π0η mass distribution. The branching fraction obtained for this latter decay is of the order of 5×106, 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 ˉB0sJ/ψπ+π(π0η) reaction is depicted in Fig. 1, having an sˉs pair with isospin I=0 at the end. Note that the light scalars f0(980) and a0(980) have I=0,1, respectively. The production of f0(980) is isospin conserved, while the production of a0(980) is isospin forbidden and involves isospin violation.

    To obtain π+π or π0η in the final state in Fig. 1, we need to hadronize the sˉs pair by introducing an extra ˉqq pair with vacuum quantum numbers. We start with the qˉq matrix M in SU(3),

    M=(uˉuuˉduˉsdˉudˉddˉssˉusˉdsˉs).

    (1)

    Next, we write the matrix M in terms of pseudoscalar mesons, assuming that the η is η8 of SU(3),

    MP=(12π0+16η+13ηπ+K+π12π0+16η+13ηK0KˉK023η+13η),

    (2)

    which is often used in chiral perturbation theory [4]. On the other hand, when we consider the Bramon ηη mixing [41], the matrix M can be written as

    MP(m)=(12π0+13η+16ηπ+K+π12π0+13η+16ηK0KˉK013η+23η).

    (3)

    Since the η is inessential in the dynamical generation of the f0(980) and a0(980) resonances [4], we will ignore the η in the present work.

    After hadronization of the sˉs component, we obtain

    sˉsH=isˉqiqiˉs=iP3iPi3=(P2)33.

    (4)

    In the case without ηη mixing, the matrix P of Eq. (2) is used, and then the hadron component H in Eq. (4) is given by

    H=KK++ˉK0K0+23ηη.

    (5)

    In the case with ηη mixing, one uses matrix P(m) of Eq. (3), and obtains

    H=KK++ˉK0K0+13ηη,

    (6)

    differing only in the ηη component, which affects the production of f0 but not the production of a0. We define the weight of the PP components in H as

    hK+K=1,hK0ˉK0=1,hηη=23,h(m)ηη=13.

    (7)

    One can see that neither Eq. (5) nor Eq. (6) contains π+π or π0η, but they can be produced by the final state interaction of the KˉK and ηη components, as depicted in Fig. 2. The transition matrix from the PP state to π+π or π0η is represented by the circle behind the meson–meson loop in Fig. 2, which contains the information of f0(980) and a0(980) respectively. According to the method in Ref. [4] (the chiral unitary approach), these resonances are the result of the PP interaction in the coupled channels KˉK,ππ,πη,ηη.

    Figure 2

    Figure 2.  Final state interaction of the hadron components leading to π+π or π0η in the final state.

    By using the unitary normalization [4, 38], the amplitude for the ˉB0sJ/ψπ+π decay, as a function of the π+π invariant mass Minv(π+π), is given by [38]

    tπ+π=C[hK+KGK+K(Minv(π+π))TK+K,π+π(Minv(π+π))+hK0ˉK0GK0ˉK0(Minv(π+π))TK0ˉK0,π+π(Minv(π+π))+hηη×2×12Gηη(Minv(π+π))Tηη,π+π(Minv(π+π))],

    (8)

    and the amplitude for the ˉB0sJ/ψπ0η decay, as a function of the π0η invariant mass Minv(π0η), is given by [38]

    tπ0η=C[hK+KGK+K(Minv(π0η))TK+K,π0η(Minv(π0η))+hK0ˉK0GK0ˉK0(Minv(π0η))TK0ˉK0,π0η(Minv(π0η))+hηη×2×12Gηη(Minv(π0η))Tηη,π0η(Minv(π0η))],

    (9)

    with C an arbitrary normalization constant which is canceled in the ratio of the f0 and a0 production rates. For the case with ηη mixing, the corresponding amplitudes can be obtained by replacing hηη with h(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-off qmax [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 the ij transition and can be obtained by solving the Bethe–Salpeter (BS) equation with six PP coupled channels π+π, π0π0, K+K, K0ˉK0, ηη and π0η, in a matrix form,

    T=[1VG]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 with ηη mixing, respectively.

    The differential decay width for ˉB0sJ/ψπ0η or ˉB0sJ/ψπ+π decay is given by

    dΓdMinv(ij)=1(2π)314M2ˉB0s13p2J/ψpJ/ψ˜pπ|tij|2,

    (12)

    where ij=π+π or π0η, Minv(ij) is the invariant mass of the final π+π or π0η, tπ+π and tπ0η are the amplitudes from Eq. (8) and Eq. (9) respectively, pJ/ψ is the J/ψ momentum in the ˉB0s rest frame, and ˜pπ is the pion momentum in the rest frame of the π+π or π0η system,

    pJ/ψ=λ1/2(M2ˉB0s,M2J/ψ,M2inv)2MˉB0s,

    (13)

    ˜pπ={λ1/2(M2inv,m2π,m2π)2Minv,forπ+πproduction,λ1/2(M2inv,m2π,m2η)2Minv,forπ0ηproduction,

    (14)

    with λ(x2,y2,z2)=x2+y2+z22xy2yz2zx the Källen function. In Eq. (12), the factor 13p2J/ψ stems from the fact that we need a p-wave to match angular momentum in the 010+ transition and we take a vertex of type pJ/ψcosθ.

    We follow Ref. [38] and take the cut-off qmax=600 MeV and 650 MeV for the cases without ηη mixing and with ηη mixing respectively, with which the f0(980) and a0(980) resonances can be dynamically produced well from the PP interaction. The π+π and π0η mass distributions dΓdMinv(ij) are shown in Fig. 3 for the case without ηη mixing and in Fig. 4 for the case with ηη 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

    Figure 3.  (color online) Minv(π+π) mass distribution for ˉB0sJ/ψf0(980),f0(980)π+π decay, and Minv(π0η) mass distribution for ˉB0sJ/ψa0(980),a0(980)π0η decay. Inset: Magnified π0η. (Without ηη mixing).

    Figure 4

    Figure 4.  (color online) Minv(π+π) mass distribution for ˉB0sJ/ψf0(980),f0(980)π+π decay, and Minv(π0η) mass distribution for ˉB0sJ/ψa0(980),a0(980)π0η decay. Inset: Magnified π0η. (With ηη mixing).

    Now, let us look at the π+π and π0η mass distributions in Fig. 4 with ηη mixing. One can see a strong peak for f0(980) production in the π+π mass distribution and a small peak for a0(980) production in the π0η mass distribution. Here the shape of a0(980) resonance is quite narrow, considerably different to the standard cusp-like shape (with a width of about 120 MeV) of the ordinary production of a0(980) in an isospin allowed reaction [42]. If isospin were conserved, one would find the a0(980) production with zero strength. The small peak of a0(980) in Fig. 4 indicates that isospin violation takes places in the ˉB0sJ/ψπ0η reaction. According to Eq. (A4) of Ref. [38], we have VK+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 amplitude tπ0η in Eq. (9), resulting on zero strength for a0(980) production. On the contrary, using the physical masses for the neutral K0 and the charged K+ in the formalism results in the production of the a0(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 the K+, K0 mass difference for the explicit K+K and K0ˉK0 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(ˉB0sJ/ψa0(980),a0(980)π0η)Γ(ˉB0sJ/ψf0(980),f0(980)π+π),

    (15)

    with decay widths Γ[ˉB0sJ/ψa0(980), a0(980)π0η] and Γ[ˉB0sJ/ψf0(980),f0(980)π+π] obtained by integrating Eq. (12) over the invariant mass Minv(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.

    Table 1

    Table 1.  Values of R with different assumptions. (In the table, I.V. denotes isospin violation.)
    no ηη mixingI.V. both in T matrix and in explicit KˉK loops (Case 1)3.1×102
    I.V. only in T matrix (Case 2)3.5×102
    I.V. only in explicit KˉK loops (Case 3)7.0×104
    with ηη mixingI.V. both in T matrix and in explicit KˉK loops (Case 4)3.7×102
    I.V. only in T matrix (Case 5)4.1×102
    I.V. only in explicit KˉK loops (Case 6)9.7×104
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    From Table 1, we observe that the ratio R with ηη mixing (Case 4) is about 20% bigger than that without ηη 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 explicit KˉK loops, being at least one order of magnitude larger. This fact is interesting, since in our picture the f0(980) and a0(980) resonances are dynamically generated from the PP interaction with the information on their nature contained in the T matrix. For the ˉB0sJ/ψπ+π(π0η) decay, neither the π+π nor the π0η can be directly produced from sˉs 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, with f0(980) and a0(980) resonances as dynamically generated states from the PP interaction. The production rate of the f0(980) (a0(980)) resonance in the ˉB0sJ/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 of f0(980) and a0(980) resonances and to investigate the isospin violation.

    From the PDG [40], the experimental branching ratio of the ˉB0sJ/ψf0(980),f0(980)π+π decay reads

    Br[ˉB0sJ/ψf0(980),f0(980)π+π]=(1.28±0.18)×104.

    (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[ˉB0sJ/ψa0(980),a0(980)π0η]={(3.95±0.56)×106,forCase1;(4.74±0.67)×106,forCase4.

    (17)

    This branching ratio is of the order of 5×106, not too small considering that several rates of the order of 107 are tabulated in the PDG [40]. The branching ratio and the shape of the π0η mass distribution of the ˉB0sJ/ψπ0η decay provide relevant information on the nature of the a0(980) resonance. Experimental measurements will be very valuable.

    In the present work, we study the isospin allowed decay process ˉB0sJ/ψπ+π and the isospin forbidden decay process ˉB0sJ/ψπ0η, paying attention to the different sources of isospin violation.

    First, we have J/ψsˉs production in the ˉB0s decay, via internal emission as shown in Fig. 1. After the hadronization of sˉs into meson–meson components, we obtain KˉK pairs and ηη, while π+π and π0η are not produced at this step. Therefore, to see π+π or π0η in the final state, rescattering of the KˉK or ηη components is needed to produce π+π and π0η 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 for f0(980) production. We also observe that the shape of the π0η mass distribution is very different from the shape of the common a0(980) production in isospin-allowed reactions, and it is related to the difference in mass between the charged and neutral kaons. In the production of a0(980) we find two sources of isospin violation: one is that the loops containing K+K or K0ˉK0 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] and ˉB0s mesonic decay both produce an sˉs pair at the end, and the two resonances of f0(980) and a0(980) are produced dynamically by the interaction of pseudoscalar mesons through the chiral unitary approach. The results of D+s semileptonic decay are consistent with the experimental upper bound. We also calculate the branching ratio of ˉB0sJ/ψa0(980) for a0(980) production, and the values are not too small, of the order of 5×106. Our results provide a reference basis for experiments, which we expect to be carried out in the near future.

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  • [1] L. L. Chau, Phys. Rept. 95, 1-94 (1983) doi: 10.1016/0370-1573(83)90043-1
    [2] R. Aaij et al. [LHCb], Phys. Lett. B 698, 115-122 (2011) doi: 10.1016/j.physletb.2011.03.006
    [3] W. H. Liang and E. Oset, Phys. Lett. B 737, 70-74 (2014) doi: 10.1016/j.physletb.2014.08.030
    [4] J. A. Oller and E. Oset, Nucl. Phys. A 620, 438-456 (1997); [Erratum: Nucl. Phys. A 652, 407-409 (1999)]
    [5] N. Kaiser, Eur. Phys. J. A 3, 307-309 (1998) doi: 10.1007/s100500050183
    [6] M. P. Locher, V. E. Markushin, and H. Q. Zheng, Eur. Phys. J. C 4, 317-326 (1998) doi: 10.1007/s100529800766
    [7] J. Nieves and E. Ruiz Arriola, Nucl. Phys. A 679, 57-117 (2000) doi: 10.1016/S0375-9474(00)00321-3
    [8] N. N. Achasov, S. A. Devyanin, and G. N. Shestakov, Phys. Lett. B 88, 367-371 (1979) doi: 10.1016/0370-2693(79)90488-X
    [9] N. N. Achasov, S. A. Devyanin, and G. N. Shestakov, Yad. Fiz. 33, 1337-1348 (1981); Sov. J. Nucl. Phys. 33, 715 (1981)
    [10] N. N. Achasov and G. N. Shestakov, Phys. Rev. D 56, 212-220 (1997) doi: 10.1103/PhysRevD.56.212
    [11] O. Krehl, R. Rapp, and J. Speth, Phys. Lett. B 390, 23-28 (1997) doi: 10.1016/S0370-2693(96)01425-6
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Jia-Ting Li, Jia-Xin Lin, Gong-Jie Zhang, Wei-Hong Liang and E. Oset. The ˉB0sJ/ψπ0η decay and the a0(980)f0(980) mixing[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac6cd5
Jia-Ting Li, Jia-Xin Lin, Gong-Jie Zhang, Wei-Hong Liang and E. Oset. The ˉB0sJ/ψπ0η decay and the a0(980)f0(980) mixing[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac6cd5 shu
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The ˉB0sJ/ψπ0η decay and the a0(980)f0(980) mixing

    Corresponding author: Wei-Hong Liang, liangwh@gxnu.edu.cn, Corresponding author
    Corresponding author: E. Oset, oset@ific.uv.es
  • 1. Department of Physics, Guangxi Normal University, Guilin 541004, China
  • 2. Guangxi Key Laboratory of Nuclear Physics and Technology, Guangxi Normal University, Guilin 541004, China
  • 3. Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC Institutos de Investigación de Paterna, Aptdo.22085, 46071 Valencia, Spain

Abstract: We study the ˉB0sJ/ψf0(980) and ˉB0sJ/ψa0(980) reactions, and pay attention to the different sources of isospin violation and mixing of f0(980) and a0(980) resonances where these resonances are dynamically generated from meson–meson interactions. We find that the main cause of isospin violation is isospin breaking in the meson–meson transition T matrices, and the other source is that the loops involving kaons in the production mechanism do not cancel due to the different masses of charged and neutral kaons. We obtain a branching ratio for a0(980) production of the order of 5×106. Future experiments can address this problem, and the production rate and shape of the π0η mass distribution will definitely help to better understand the nature of scalar resonances.

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    I.   INTRODUCTION
    • The Bs decay into J/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. The cˉc quarks give rise to the J/ψ and the extra sˉs, which appear in the Cabibbo favored decay mode, have isospin I=0. It is a rather clean process and indeed, in the LHCb experiment [2] the f0(980) resonance was seen as a strong peak in the invariant mass distribution of π+π. The way π+π are produced is studied in detail in Ref. [3]. The sˉs pair of quarks is hadronized, introducing a ˉqq pair with vacuum quantum numbers, and KˉK in I=0 plus ηη are produced, which are allowed to interact within the chiral unitary approach [47] to produce the f0(980) resonance, which is dynamically generated from the interaction of pseudoscalar pairs and couples mostly to KˉK. With such a clean process producing I=0, one finds a very interesting place to produce the a0(980), via isospin violation, and add extra information to the subject of the f0(980)a0(980) mixing that has stimulated much work. Indeed, there are many works devoted to this subject [836] 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 the a0(980) resonance can be produced in the BsJ/ψπ0η decay is tied to the nature of f0(980) and a0(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, particularly KˉK in the case of f0 and a0, that give rise to the resonances. The K+K and K0ˉK0 loops cancel for I=1 starting from the I=0 combination of the hadronized sˉs quarks, but only if the masses of K+ and K0 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 the a0f0 mixing to this mass difference is shared by most theoretical studies, starting from Ref. [8]. However, as shown in Ref. [38] in the study of Dse+νea0(980), isospin breaking takes place in the loop for KˉK 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 the a0(980) and f0(980) resonances, which has originated much debate in the literature.

      Figure 1.  Diagram for ˉB0s decay into J/ψ and a primary sˉs pair.

      We study the process of a0 and f0 production, following the lines of Refs. [3] and [38], and by taking experimental information on the BsJ/ψπ+π reaction, we make predictions for the rate of BsJ/ψπ0η production and the shape of the π0η mass distribution. The branching fraction obtained for this latter decay is of the order of 5×106, well within the range of rates already measured and reported by the PDG [40], which should stimulate its measurement in the future.

    II.   FORMALISM
    • The mechanism at the quark level for the ˉB0sJ/ψπ+π(π0η) reaction is depicted in Fig. 1, having an sˉs pair with isospin I=0 at the end. Note that the light scalars f0(980) and a0(980) have I=0,1, respectively. The production of f0(980) is isospin conserved, while the production of a0(980) is isospin forbidden and involves isospin violation.

      To obtain π+π or π0η in the final state in Fig. 1, we need to hadronize the sˉs pair by introducing an extra ˉqq pair with vacuum quantum numbers. We start with the qˉq matrix M in SU(3),

      M=(uˉuuˉduˉsdˉudˉddˉssˉusˉdsˉs).

      (1)

      Next, we write the matrix M in terms of pseudoscalar mesons, assuming that the η is η8 of SU(3),

      MP=(12π0+16η+13ηπ+K+π12π0+16η+13ηK0KˉK023η+13η),

      (2)

      which is often used in chiral perturbation theory [4]. On the other hand, when we consider the Bramon ηη mixing [41], the matrix M can be written as

      MP(m)=(12π0+13η+16ηπ+K+π12π0+13η+16ηK0KˉK013η+23η).

      (3)

      Since the η is inessential in the dynamical generation of the f0(980) and a0(980) resonances [4], we will ignore the η in the present work.

      After hadronization of the sˉs component, we obtain

      sˉsH=isˉqiqiˉs=iP3iPi3=(P2)33.

      (4)

      In the case without ηη mixing, the matrix P of Eq. (2) is used, and then the hadron component H in Eq. (4) is given by

      H=KK++ˉK0K0+23ηη.

      (5)

      In the case with ηη mixing, one uses matrix P(m) of Eq. (3), and obtains

      H=KK++ˉK0K0+13ηη,

      (6)

      differing only in the ηη component, which affects the production of f0 but not the production of a0. We define the weight of the PP components in H as

      hK+K=1,hK0ˉK0=1,hηη=23,h(m)ηη=13.

      (7)

      One can see that neither Eq. (5) nor Eq. (6) contains π+π or π0η, but they can be produced by the final state interaction of the KˉK and ηη components, as depicted in Fig. 2. The transition matrix from the PP state to π+π or π0η is represented by the circle behind the meson–meson loop in Fig. 2, which contains the information of f0(980) and a0(980) respectively. According to the method in Ref. [4] (the chiral unitary approach), these resonances are the result of the PP interaction in the coupled channels KˉK,ππ,πη,ηη.

      Figure 2.  Final state interaction of the hadron components leading to π+π or π0η in the final state.

      By using the unitary normalization [4, 38], the amplitude for the ˉB0sJ/ψπ+π decay, as a function of the π+π invariant mass Minv(π+π), is given by [38]

      tπ+π=C[hK+KGK+K(Minv(π+π))TK+K,π+π(Minv(π+π))+hK0ˉK0GK0ˉK0(Minv(π+π))TK0ˉK0,π+π(Minv(π+π))+hηη×2×12Gηη(Minv(π+π))Tηη,π+π(Minv(π+π))],

      (8)

      and the amplitude for the ˉB0sJ/ψπ0η decay, as a function of the π0η invariant mass Minv(π0η), is given by [38]

      tπ0η=C[hK+KGK+K(Minv(π0η))TK+K,π0η(Minv(π0η))+hK0ˉK0GK0ˉK0(Minv(π0η))TK0ˉK0,π0η(Minv(π0η))+hηη×2×12Gηη(Minv(π0η))Tηη,π0η(Minv(π0η))],

      (9)

      with C an arbitrary normalization constant which is canceled in the ratio of the f0 and a0 production rates. For the case with ηη mixing, the corresponding amplitudes can be obtained by replacing hηη with h(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-off qmax [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 the i\to j transition and can be obtained by solving the Bethe–Salpeter (BS) equation with six PP coupled channels \pi^+\pi^- , \pi^0\pi^0 , K^+K^- , K^0\bar{K}^0 , \eta\eta and \pi^0\eta , in a matrix form,

      T = [1-V\,G]^{-1}\, V,

      (11)

      where the matrix V is the kernel of the BS equation. Its elements V_{ij} 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 with \eta-\eta' mixing, respectively.

      The differential decay width for \bar B_s^0\to J/\psi\pi^0\eta or \bar B_s^0\to J/\psi\pi^+\pi^- decay is given by

      \frac{{\rm{d}} \Gamma}{{\rm{d}} M_{\rm{inv}}(ij)} = \frac{1}{(2\pi)^3} \; \frac{1}{4M_{\bar B_s^0}^2}\; \frac{1}{3}\; p_{J/\psi}^2 \; p_{J/\psi}\; \tilde{p}_{\pi}\; |t_{ij}|^2,

      (12)

      where ij = \pi^+ \pi^- or \pi^0 \eta , M_{\rm{inv}}(ij) is the invariant mass of the final \pi^+ \pi^- or \pi^0 \eta , t_{\pi^+ \pi^-} and t_{\pi^0 \eta} are the amplitudes from Eq. (8) and Eq. (9) respectively, p_{J/\psi} is the J/\psi momentum in the \bar B_s^0 rest frame, and \tilde{p}_{\pi} is the pion momentum in the rest frame of the \pi^+ \pi^- or \pi^0 \eta system,

      p_{J/\psi} = \frac{\lambda^{1/2}(M^2_{\bar B^0_s},M^2_{J/\psi},M_{\rm{inv}}^2)}{2M_{\bar B^0_s}},

      (13)

      {\tilde p_\pi } = \left\{ {\begin{array}{*{20}{l}} {\dfrac{{{\lambda ^{1/2}}(M_{{\rm{inv}}}^2,m_\pi ^2,m_\pi ^2)}}{{2{M_{{\rm{inv}}}}}},}&{{\rm{for}}\;{\pi ^ + }{\pi ^ - }\;{\rm{production}},}\\ {\dfrac{{{\lambda ^{1/2}}(M_{{\rm{inv}}}^2,m_\pi ^2,m_\eta ^2)}}{{2{M_{{\rm{inv}}}}}},}&{{\rm{for}}\;{\pi ^0}\eta \;{\rm{production}},} \end{array}} \right.

      (14)

      with \lambda(x^2, y^2, z^2) = x^2+y^2+z^2-2xy-2yz-2zx the Källen function. In Eq. (12), the factor \dfrac{1}{3}\, p_{J/\psi}^2 stems from the fact that we need a p-wave to match angular momentum in the 0^- \to 1^-\, 0^+ transition and we take a vertex of type p_{J/\psi}\, \cos \theta .

    III.   RESULTS
    • We follow Ref. [38] and take the cut-off q_{\rm{max}} = 600 MeV and 650 MeV for the cases without \eta-\eta' mixing and with \eta-\eta' mixing respectively, with which the f_0(980) and a_0(980) resonances can be dynamically produced well from the PP interaction. The \pi^+ \pi^- and \pi^0 \eta mass distributions \frac{{\rm{d}} \Gamma}{{\rm{d}} M_{\rm{inv}}(ij)} are shown in Fig. 3 for the case without \eta-\eta' mixing and in Fig. 4 for the case with \eta-\eta' 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) M_{\rm inv}(\pi^+ \pi^-) mass distribution for \bar B_s^0 \to J/\psi f_0(980), f_0(980) \to \pi^+ \pi^- decay, and M_{\rm inv}(\pi^0 \eta) mass distribution for \bar B_s^0 \to J/\psi a_0(980), a_0(980) \to \pi^0 \eta decay. Inset: Magnified \pi^0 \eta. (Without \eta-\eta' mixing).

      Figure 4.  (color online) M_{\rm inv}(\pi^+ \pi^-) mass distribution for \bar B_s^0 \to J/\psi f_0(980), f_0(980) \to \pi^+ \pi^- decay, and M_{\rm inv}(\pi^0 \eta) mass distribution for \bar B_s^0 \to J/\psi a_0(980), a_0(980) \to \pi^0 \eta decay. Inset: Magnified \pi^0 \eta. (With \eta-\eta' mixing).

      Now, let us look at the \pi^+ \pi^- and \pi^0 \eta mass distributions in Fig. 4 with \eta-\eta' mixing. One can see a strong peak for f_0(980) production in the \pi^+ \pi^- mass distribution and a small peak for a_0(980) production in the \pi^0 \eta mass distribution. Here the shape of a_0(980) resonance is quite narrow, considerably different to the standard cusp-like shape (with a width of about 120 MeV) of the ordinary production of a_0(980) in an isospin allowed reaction [42]. If isospin were conserved, one would find the a_0(980) production with zero strength. The small peak of a_0(980) in Fig. 4 indicates that isospin violation takes places in the \bar{B}_s^0 \to J/\psi \pi^0 \eta reaction. According to Eq. (A4) of Ref. [38], we have V_{K^+K^-,\pi^0\eta} = -V_{K^0 \bar K^0,\pi^0\eta} 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 amplitude t_{\pi^0\eta} in Eq. (9), resulting on zero strength for a_0(980) production. On the contrary, using the physical masses for the neutral K^0 and the charged K^+ in the formalism results in the production of the a_0(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 the K^+ , K^0 mass difference for the explicit K^+ K^- and K^0 \bar K^0 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 = \frac{\Gamma(\bar B_s^0 \to J/\psi a_0(980), a_0(980) \to \pi^0 \eta )} {\Gamma(\bar B_s^0 \to J/\psi f_0(980), f_0(980) \to \pi^+ \pi^-)},

      (15)

      with decay widths \Gamma[\bar B_s^0 \to J/\psi a_0(980) , a_0(980) \to \pi^0 \eta ] and \Gamma[\bar B_s^0 \to J/\psi f_0(980), f_0(980) \to \pi^+ \pi^-] obtained by integrating Eq. (12) over the invariant mass M_{\rm{inv}}(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 \eta-\eta' mixingI.V. both in T matrix and in explicit K\bar K loops (Case 1)3.1\times 10^{-2}
      I.V. only in T matrix (Case 2)3.5\times 10^{-2}
      I.V. only in explicit K\bar K loops (Case 3)7.0\times 10^{-4}
      with \eta-\eta' mixingI.V. both in T matrix and in explicit K\bar K loops (Case 4)3.7\times 10^{-2}
      I.V. only in T matrix (Case 5)4.1\times 10^{-2}
      I.V. only in explicit K\bar K loops (Case 6)9.7\times 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 \eta-\eta' mixing (Case 4) is about 20% bigger than that without \eta-\eta' 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 explicit K\bar K loops, being at least one order of magnitude larger. This fact is interesting, since in our picture the f_0(980) and a_0(980) resonances are dynamically generated from the PP interaction with the information on their nature contained in the T matrix. For the \bar B_s^0 \to J/\psi \pi^+ \pi^- (\pi^0 \eta) decay, neither the \pi^+ \pi^- nor the \pi^0 \eta can be directly produced from s\bar s 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, with f_0(980) and a_0(980) resonances as dynamically generated states from the PP interaction. The production rate of the f_0(980) ( a_0(980) ) resonance in the \bar B_s^0 \to J/\psi \pi^+ \pi^- (\pi^0 \eta) decay is sensitive to the resonance information contained in the T matrix. Therefore, this mode is particularly suitable to test the nature of f_0(980) and a_0(980) resonances and to investigate the isospin violation.

      From the PDG [40], the experimental branching ratio of the \bar B_s^0 \to J/\psi f_0(980), f_0(980) \to \pi^+ \pi^- decay reads

      {\mathrm{Br}}[\bar B_s^0 \to J/\psi f_0(980), f_0(980) \to \pi^+ \pi^-] = (1.28 \pm 0.18)\times 10^{-4}.

      (16)

      By using the ratio R in Table 1 and the branching ratio of Eq. (16), the branching ratio for a_0(980) production can be obtained,

      \begin{aligned}[b]& {\rm{Br}}[\bar B_s^0 \to J/\psi {a_0}(980),{a_0}(980) \to {\pi ^0}\eta ] \\=& \left\{ {\begin{array}{*{20}{l}} {(3.95 \pm 0.56) \times {{10}^{ - 6}},}&{{\rm{for}}\;{\rm{Case}}\;1;}\\ {(4.74 \pm 0.67) \times {{10}^{ - 6}},}&{{\rm{for}}\;{\rm{Case}}\;4.} \end{array}} \right. \end{aligned}

      (17)

      This branching ratio is of the order of 5\times 10^{-6} , not too small considering that several rates of the order of 10^{-7} are tabulated in the PDG [40]. The branching ratio and the shape of the \pi^0 \eta mass distribution of the \bar B_s^0 \to J/\psi \pi^0 \eta decay provide relevant information on the nature of the a_0(980) resonance. Experimental measurements will be very valuable.

    IV.   CONCLUSIONS
    • In the present work, we study the isospin allowed decay process \bar B_s^0 \to J/\psi \pi^+ \pi^- and the isospin forbidden decay process \bar B_s^0 \to J/\psi \pi^0 \eta , paying attention to the different sources of isospin violation.

      First, we have J/\psi\; s\bar s production in the \bar B_s^0 decay, via internal emission as shown in Fig. 1. After the hadronization of s \bar s into meson–meson components, we obtain K\bar K pairs and \eta \eta , while \pi^+ \pi^- and \pi^0 \eta are not produced at this step. Therefore, to see \pi^+\pi^- or \pi^0\eta in the final state, rescattering of the K \bar K or \eta \eta components is needed to produce \pi^+ \pi^- and \pi^0 \eta 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 for f_0(980) production. We also observe that the shape of the \pi^0 \eta mass distribution is very different from the shape of the common a_0(980) production in isospin-allowed reactions, and it is related to the difference in mass between the charged and neutral kaons. In the production of a_0(980) we find two sources of isospin violation: one is that the loops containing K^+ K^- or K^0 \bar K^0 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] and \bar B_s^0 mesonic decay both produce an s \bar s pair at the end, and the two resonances of f_0(980) and a_0(980) are produced dynamically by the interaction of pseudoscalar mesons through the chiral unitary approach. The results of D_s^+ semileptonic decay are consistent with the experimental upper bound. We also calculate the branching ratio of \bar B_s^0 \to J/\psi a_0(980) for a_0(980) production, and the values are not too small, of the order of 5\times 10^{-6} . Our results provide a reference basis for experiments, which we expect to be carried out in the near future.

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