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p-wave mesons emitting weak decays of bottom mesons

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Maninder Kaur, Supreet Pal Singh and R. C. Verma. p-wave mesons emitting weak decays of bottom mesons[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad9893
Maninder Kaur, Supreet Pal Singh and R. C. Verma. p-wave mesons emitting weak decays of bottom mesons[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ad9893 shu
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p-wave mesons emitting weak decays of bottom mesons

  • Department of Physics, Punjabi University, Patiala – 147002, India

Abstract: This paper is the extension of our previous work entitled ''Searching a systematics for nonfactorizable contributions to B and ˉB0 hadronic decays''. Obtaining the factorizable contributions from the spectator-quark model for Nc=3, a systematics was identified among the isospin reduced amplitudes for the nonfactorizable terms among ˉBDπ/Dπ/Dρ decay modes. This systematics enables us to derive a generic formula to help predict the branching fractions for ˉB0 decays. Inspired by this observation, we extend our analysis to p-wave meson emitting decays of Bmeson ˉBPA/PT/PS, particularly ˉBa1D/πD1/πD1/πD2/πD0, which have similar isospin structures and make predictions for ˉB0 decays, for which experimental measurements are not yet available.

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    I. INTRODUCTION
    • At present, large amounts of information is available on the decays of the heavy flavor hadrons, and more measurements are expected in future experiments. Worldwide, several groups at Fermi lab, Cornell, LHC-CERN, KEK, DESY, and Beijing Electron Collider, among others, have been working to provide wide knowledge of heavy flavor physics. One of the goals of heavy flavor hadron physics is also to elucidate the relationship among the particles of different generations [1].

      Heavy, charm, and bottom mesons have revealed many channels for leptonic, semi-leptonic and hadronic decays. The b quark is especially interesting in this respect, as it has W-mediated transitions to both first-generation (u) and second-generation (c) quarks. The Standard Model (SM) is reasonably successful in explaining the leptonic and semileptonic decays, but the issue of weak hadronic decays is yet to be settled, and these decays have posed serious problems due to the strong interaction interference with the weak interactions responsible for these decays [27]. Initially, the weak hadronic decays of charm and bottom mesons were expected to have less interference due to the strong interactions, as their decay products carry large momenta. However, their measurements have revealed the contrary. The prominent reason being that experiments are producing data at the hadronic level, whereas the theory (SM) deals with quarks and leptons. Presently, the problem of Hadronization (formation of hadrons from quarks), being a low-energy phenomenon, cannot yet be resolved from first principles. In fact, understanding of the decays becomes more complicated as the produced hadrons in the weak hadronic decays can participate in the Final State Interactions (FSI) [819] caused by the strong interactions at the hadronic level. Therefore, analysis of weak hadronic decays requires phenomenological treatment, for which symmetry principles and quark models are often employed to explore the dynamics involved.

      Even the weak interaction vertex itself is also affected through gluon-exchange among the quarks involved. At W-mass scale, hard gluons exchange effects are calculable using the perturbative QCD. Usually, factorization of weak matrix elements is performed in terms of certain form-factors and decay constants. Besides these high-energy gluon exchanges, there exist possible soft gluon exchanges around the W- vertex, which generate nonfactorizable contributions in the weak matrix elements [2023]. The nonfactorizable terms may appear for several reasons, including soft gluon exchange and FSIrescattering effects [2023]. The rescattering effects on the outgoing mesons have been studied in detail for bottom meson decays [2425]. Besides that, flavor SU(3) symmetry and the Factorization Assisted Topological (FAT) approach have been employed for the study of such nonfactorizable contributions, as they have the advantage of absorbing various kinds of lump-sum contributions in terms of a few parameters, which can be fixed empirically [26, 27]. Extensive work has also been conducted to treat nonfactorizable contributions, such as the QCD factorization approach based on collinear factorization theorem [28] and the perturbative QCD factorization approach [2930]. Unfortunately, it is not straightforward to calculate such terms, which are nonperturbative in nature and require empirical data to investigate their behaviour.

      In the naïve factorization scheme, the nonfactorizable contribution for the decay amplitudes is completely ignored, and the two QCD coefficients a1 and a2 are fixed from the experimental data [3133]. Initially, data on branching fractions of DˉKπ decays seemed to require a1c1=1.26,a2c2=0.51, leading to destructive interference between color-favored (CF) and color-suppressed (CS) processes for D+ˉK0π+, thereby implying the Nc limit [34]. However, later measurement of ˉBDπ meson decays did not favor this result empirically, as these decays require a11.03, and a20.23, i.e., a positive value of a2, in sharp contrast to the expectations based on the large Nc limit, because the final state particles leave the interaction region very quickly, allowing little time for final state interactions, and soft-gluon exchange becomes less important [35]. Thus, B-meson decays, revealing constructive interference between CF and CS diagrams for BπD0, seem to favor Nc=3(real value).

      It has been found experimentally that two-body decays dominate the spectrum. Bottom meson decays to two s-wave mesons (pseudoscalar and vector mesons) have been studied reasonably well. Theoretical focus has also, so far, been on the s-wave meson (ˉBPP/PV) emitting decays [124]. There exist four L=1 states: scalar (JPC=0++), axial-vectors (JPC=1++and 1+), and tensor (JPC=2++) mesons. All these p-wave states and vector mesons decay to s-wave mesons through strong interactions, so these are called meson resonances. These states are generally produced either in scattering experiments or as decay products of heavy flavour mesons. Thus, we investigated the following decay modes involving one p-wave meson in the final state:

      ˉBP(0+)+S(0++),ˉBP(0+)+A(1++),ˉBP(0+)+A(1+),ˉBP(0+)+T(2++).

      Branching fractions for some of the decay modes have been measured experimentally [1]. Kinematically, these decays are expected to be suppressed; however, the measured branching fractions of these modes are rather large. Therefore, it is desirable to study B- meson decays emitting p-wave mesons, which requires theoretical understanding.

      Our group performed a thorough study of nonfactorizable contributions by using isospin analysis for DˉKπ/ˉKρ/ˉKπ decay modes and recognized a systematics for the ratio of nonfactorized reduced amplitudes. It is worth noting that this systematics was also found to be consistent with p-wave meson emitting decays of charm mesons: DˉKa1/πˉK1/πˉK1/πˉK0/ˉKa2 [22]. In our previous work [36], it was found that the nonfactorizable contributions in the respective 1/2 and 3/2 isospin reduced amplitudes for Cabibo-favored ˉBπD/ρD/πD decay modes bear a universal ratio equal to α within the experimental errors. Extension of this universality to ˉBa1D/πD1/πD1/πD2/πD0 is hoped to yield useful predictions for their branching fractions. Therefore, in this paper, we extend our analysis to investigate nonfactorizable terms in the p-wave mesons emitting decays.

      The remainder of this paper is organised as follow. In Section II, the weak Hamiltonian is expressed as a sum of two particle-generating factorizable and nonfactorizable contributions to the hadronic decays of B-mesons. In Section III, we introduce the methodology of our approach by analysing s-wave mesons emitting decays of bottom mesons. In Section IV, detailed analysis of p-wave meson emitting decays is presented. Summary and conclusions are given in the final section.

    II. WEAK HAMILTONIAN
    • To study the two-body hadronic Bdecays, we consider the effective weak Hamiltonian [37]

      Hw=GF2VcbVud[c1(¯du)(¯cb)+c2(¯cu)(¯db)],

      (1)

      where Vud and Vcb are the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements [1],

      Vud=0.975,Vcb=0.041,

      ˉq1q2=ˉq1γμ(1γ5)q2 denotes color singlet VA Dirac current, and the QCD coefficients at the bottom mass scale are taken as [27, 38]

      c1=1.132,c2=0.287.

      (2)

      In the standard factorization scheme, the current operators in the weak Hamiltonian are expressed in terms of the fundamental quark fields. It is appropriate to have the Hamiltonian in a form such that one of these currents carries the same quantum numbers as one of the mesons emitted in the final state of bottom meson decays. Consequently, the hadronic matrix elements of the Hamiltonian operator Hw receive contribution from the operator itself and from its Fierz transformation. For instance, separating the factorizable and nonfactorizable parts of (ˉdu)(ˉcb)using the Fierz identity [39] as

      (ˉdu)(ˉcb)=1Nc(ˉcu)(ˉdb)+12(ˉcλau)(ˉdλab),

      (3)

      where ˉq1λaq2ˉq1γμ(1γ5)λaq2 represents the color octet current, and performing similar treatment on the other operator (ˉcu)(ˉdb), the weak Hamiltonian finally becomes

      HCFw=GF2VcbVud[a1(ˉdu)H(ˉcb)H+c2H8w],

      (4)

      HCSw=GF2VcbVud[a2(ˉcu)H(ˉdb)H+c1˜H8w],

      (5)

      for the CF and CS processes, respectively, where

      a1,2=c1,2+c2,1Nc,

      (6)

      H8w=128a=1(¯cλau)(ˉdλab),˜H8w=128a=1(ˉdλau)(ˉcλab).

      (7)

      The subscript H in (4) and (5) indicates the change from quark current to hadron field operator. Matrix elements of the first terms in (4) and (5) lead to the factorizable contributions, and the second terms, involving the color octet currents, generate nonfactorizable contributions.

    III. ANALYSIS OF s-WAVE MESON EMITTING DECAYS OF ˉB-MESONS
    • In this section, we describe our approach by analysing ˉBPP decay mode. The branching fraction for B-meson decay into two pseudoscalar mesons is related to its decay amplitude as follows:

      B(ˉBP1P2)=τB|GF2VcbVud|2p8πm2B|A(ˉBP1P2)|2,

      (8)

      where τB denotes the lifetime of B-mesons measured to be [1]

      τˉB0=(1.519±0.004)×1012 s,τB=(1.638±0.004)×1012 s,

      and p is the magnitude of the 3-momentum of the final state particles in the rest frame of the parent B-meson:

      p=|p1|=|p2|=12mB{[m2B(m1+m2)2]×[m2B(m1m2)2]}1/122.

      Using the isospin framework, ˉBπD decay amplitudes are represented in terms of isospin reduced amplitudes (AπD1/2, AπD3/2) and the strong interaction phases (δπD1/2,δπD3/2) in respective Isospin –1/2 and 3/2 final states as follows:

      A(ˉB0πD+)=13[AπD3/322eiδπD3/2+2AπD1/122eiδπD1/2],A(ˉB0π0D0)=13[2AπD3/322eiδπD3/2AπD1/122eiδπD1/2],A(BπD0)=3AπD3/322eiδπD3/2.

      (9)

      These equations lead to the following relations:

      AπD1/2=[|A(ˉB0πD+)|2+|A(ˉB0π0D0)|213|A(BπD0)|2]1/2,AπD3/322=13|A(BπD0)|.

      (10)

      The experimental values [1]

      B(ˉB0πD+)=(2.52±0.13)×103,B(ˉB0π0D0)=(2.63±0.14)×104,B(BπD0)=(4.68±0.13)×103,

      yield

      AπD1/2=±(1.273±0.065) GeV3,AπD3/2=±(1.323±0.018) GeV3.

      (11)

      We express decay amplitude as the sum of the factorizable and nonfactorizable parts,

      A(ˉBπD)=Af(ˉBπD)+Anf(ˉBπD),

      (12)

      arising from the respective terms of the weak Hamiltonian given in (4) and (5).

      Using the factorization scheme, spectator-quark parts of the decay amplitudes arising from W-emission diagrams are derived for the following classes of ˉBπD decays:

      Af(ˉB0πD+)=a1fπ(m2Bm2D)FˉBD0(m2π)=(2.180±0.099)GeV3,

      (13)

      Af(ˉB0π0D0)=12a2fD(m2Bm2π)FˉBπ0(m2D)=(0.111±0.021)GeV3,

      (14)

      Af(BπD0)=a1fπ(m2Bm2D)FˉBD0(m2π)+a2fD(m2Bm2π)FˉBπ0(m2D)=(2.339±0.103)GeV3.

      (15)

      Numerical inputs for the decay constant

      fD=(0.207±0.009) GeV,fπ=(0.131±0.002)GeV,

      (16)

      are taken from the leptonic decays of D and π mesons, respectively [40].

      Assuming the nearest pole dominance, the momentum dependence of the form-factors appearing in the decay amplitudes given in Eqs. (13)−(15) is taken as

      F0(q2)=F0(0)(1q2/m2s),

      (17)

      where the pole masses ms are given by the lowest lying meson with the appropriate quantum numbers, i.e., JP=0+ for F0(0) and 1 for F1(0). For numerical estimation, we take scalar mesons carrying the quantum number of the corresponding weak currents, which are ms(0+)=5.78 GeV and ms(0+)=6.80 GeV [221, 41, 42]. Form-factors F0(0) at q2=0 are taken from [43] as follows

      FˉBπ0(0)=(0.27±0.05),FˉBD0(0)=(0.66±0.03).

      (18)

      Exploiting the isospin relations

      Af1/2(ˉBπD)=13{2Af(ˉB0πD+)Af(ˉB0π0D0)},Af3/2(ˉBπD)=13{Af(ˉB0πD+)+2Af(ˉB0π0D0)},

      (19)

      we obtain

      Af1/122=(1.845±0.082) GeV3,Af3/322=(1.168±0.060) GeV3.

      (20)

      We write the non-factorizable part of the decay amplitudes in terms of isospin CG coefficients [21, 22] as scattering amplitudes for the spurion +ˉBπD process:

      Anf(ˉB0πD+)=13c2(πDH8wˉB3/322+2πDH8wˉB1/122),Anf(ˉB0π0D0)=23c1(πD˜H8wˉB3/322πD˜H8wˉB1/122),Anf(BπD0)=c2πDH8wˉB3/322+c1πD˜H8wˉB3/322,

      (21)

      where the spurion is a fictitious particle carrying the quantum numbers of the weak Hamiltonian. At present, there are no available techniques to calculate these quantities exactly from the theory of strong interactions. Therefore, subtracting the factorizable part (20) from the experimental decay amplitudes (11), we determine the nonfactorizable isospin reduced amplitudes as

      Anf1/2=(0.572±0.105) GeV3,Anf3/2=(2.491±0.062) GeV3.

      (22)

      By choosing positive and negative values for AπD1/122exp and AπD3/322exp, respectively, from Eq. (11). These bear the following ratio:

      α=(Anf1/2Anf3/2)ˉBπD=0.229±0.042.

      (23)

      Such isospin formalism can easily be extended to ˉBρD and ˉBπDdecays, as the isospin structure of these decay modes is exactly the same as that of ˉBπD. Following the procedure discussed for the ˉBπD mode, we calculate the ratio of non-factorizable isospin parts for ˉBρD and ˉBπD decay modes, given as follows for the sake of comparison:

      Anf1/2(ˉBρD)Anf3/2(ˉBρD)Anf1/2(ˉBπD)Anf3/2(ˉBπD)Anf1/2(ˉBπD)Anf3/2(ˉBπD).0.200±0.0960.211±0.1090.229±0.042

      (24)

      Note that the ratios of α=Anf1/2/Anf3/2 for all three decay modes ˉBπD/ρD/πD are in consistent agreement with each other.

      In fact, these relations can be expressed in a generic form as

      B++B00=τˉB03τBB0{1+[α+(2α)Af+(1+2α)Af00A0]2},

      (25)

      with α=Anf1/2/Anf3/2, where the subscripts of branching fractions B+,B00,B0 denote the charge states of the non-charm and charm mesons emitted in each case. Af+ and Af00 factorizable amplitudes denote the charge state of the mesons for ˉB0decays. A0, the total decay amplitude, is obtained from the B decay as

      A0=B0τB×(kinematicfactor),

      (26)

      where the kinematic factors for ˉBPP and ˉBPV are as follows:

      kinematic factor for ˉBPP=|GF2VcbVud|2p8πm2B,

      (27)

      kinematic factor for ˉBPV=|GF2VcbVud|2p38πm2V,

      (28)

      where p is the magnitude of the three-momentum of the final-state particle in the rest frame of the B-meson, and mB and mV denote the masses of the B-meson and vector meson, respectively.

      Taking the average value of α = 0.22, the predicted sum of the branching fractions of ˉB0 decays [37] is given as

      B(ˉB0πD+)+B(ˉB0π0D0)=(0.28±0.02)0/0Theo,=(0.28±0.01)0/0Expt;

      (29)

      B(ˉB0ρD+)+B(ˉB0ρ0D0)=(0.76±0.13)0/0Theo,=(0.79±0.12)0/0Expt;

      (30)

      B(ˉB0πD+)+B(ˉB0π0D0)=(0.29±0.04)0/0Theo,=(0.30±0.01)0/0Expt.

      (31)

      All theoretical values match well within experimental errors.

    IV. ANALYSIS OF p-WAVE MESON EMITTING DECAYS OF ˉB-MESONS
    • In this section, we study the Cabibbo-favored p-wave meson emitting decays in the channels ˉBPA/PT/PS involving b + uc + d/s transitions. Naively, one may expect these decays to be kinematically suppressed due to the large masses of the p-wave resonance. However, it has been found that their measured branching fractions compete well with those of the s-wave meson emitting decays of bottom mesons. On the experimental side, branching fractions of a few such decays have been measured, as shown in the Table 1. Among them, ˉBa1D decays have clean values for their branching fractions, whereas other branching fractions are measured in the composite form. From the results of s-wave analysis, we can extend this isospin formalism to ˉBa1D/πD1/πD1/πD2/πD0 decay modes, as the isospin structure of these decay modes is exactly the same as that of ˉBπD mode.

      ChannelsBranching fraction of decaysExperimental branching fractions [1]
      ˉBPAB(ˉB0a1D+)B(Ba1D0)B(BπD1(2.420)0)B(BπD1(2.427)0)×B(D1(2.427)0πD+)(6.0±3.3)×103(4±4)×103(1.5±0.6)×103(5.0±1.2)×104
      ˉBPTB(BπD2(2.462)0)×B(D2(2.462)0πD+)B(BπD2(2.462)0)×B(D2(2.462)0πD+)(3.56±0.24)×104(2.2±1.1)×104
      ˉBPSB(BπD0(2.400)0)×B(D0(2.400)0πD+)(6.4±1.4)×104

      Table 1.  Experimental data for p-wave meson emitting decays [1].

    • A. ˉBPA MODE

    • 1. Axial-Vector meson spectroscopy

      Experimentally [1], two types of axial-vector mesons exist with different charge conjugation properties, i.e., 3P1(JPC=1++) and 1P1(JPC=1+), which behave well with respect to the quark model qˉq assignments observed; strange and charmed states are given by mixture of 3P1 and 1P1states. In contrast, hidden-flavor diagonal 3P1 and 1P1 states have opposite C-parity and therefore cannot mix. The following non-strange and uncharmed mesons have been observed (mass in GeV):

      For 3P1 multiplet:

      i. Isovector a1(1.230)with the quark content uˉd,uˉudˉd/2, and dˉus:

      a+1,a01anda1.

      (32)

      ii. Four isoscalars f1(1.285),f1(1.420),f1(1.512), and χc1(3.511) have been observed, of which f1(1.420) is a multiquark state in the form of a KˉKπ bound state [44] or KˉK deuteron-state [42].

      For 1P1 multiplet:

      i. Isovector b1(1.229) with flavor content the same as given in (32):

      b+1,b01andb1.

      (33)

      ii. Three isoscalars h1(1.170),h1(1.380), and hc1(3.526).The C-parity of h1(1.380) and spin and parity of hc1(3.526) remain to be confirmed.

      The proximity of a1(1.230) and f1(1.285), and to a lesser extent that of b1(1.229) and h1(1.170), indicate the ideal mixing for both 1++ and 1+diagonal states.

      The states involving a strange quark of A(JPC=1++) and A(JPC=1+)multiplets mix to generate the physical states in the following manner [4547]:

      K1(1.270)=K1Asinθ1+K1Acosθ1,K_1(1.400)=K1Acosθ1K1Asinθ1,

      (34)

      where K1A and K1,A denote the strange partners of a1(1.230) and b1(1.229), respectively. The Particle Data Group [1] assumes that the mixing is maximal, i.e., θ1=45, whereas τ1K1(1.270)/K1(1.400)+vτ data yield θ1=±37 and θ1=±58 [48]. However, the study of DK1(1.270)π;K1(1.400)π decays rules out positive mixing-angle solutions. As DK1(1.400)π+ is largely suppressed for θ1=37, the solution θ1=58 [49] is experimentally favored.

      The mixing of charmed (cˉu and cˉd) and strange charmed (cˉs) state mesons is given in a similar manner:

      D1(2.420)=D1AsinθD1+D1AcosθD1,D1(2.427)=D1AcosθD1D1AsinθD1,

      (35)

      and

      Ds1(2.460)=Ds1AsinθDs1+Ds1AcosθDs1,D_s1(2.535)=Ds1AcosθDs1Ds1AsinθDs1.

      (36)

      However, in the heavy quark limit, the physical mass eigenstates with (JP = 1+) are P3/21 and P1/21 rather than 3P1 and 1P1 states, as the heavy quark spin SQ decouples from the other degrees of freedom, such that SQ and the total angular momentum of the light antiquark are each good quantum numbers. Therefore, heavy quark symmetry leads to

      |P3/21=23|1P1+13|3P1,|P2/11=13|1P123|3P1.

      (37)

      However, beyond the heavy quark limit, there is still a small mixing between P3/21 and P1/21 states, denoted by

      D1(2.420)=D1/21cosθ2+D3/21sinθ2,D1(2.427)=D1/21sinθ2+D3/21cosθ2,

      (38)

      Likewise, for strange axial-vector charmed mesons,

      Ds1(2.460)=D1/2s1cosθ3D3/2s1sinθ3,D_s1(2.535)=D1/2s1sinθ3+D3/2s1cosθ3,

      (39)

      where the mixing angle θ2=(5.7±2.4) was obtained by the Belle Collaboration through a detailed ˉBDππ analysis [50, 51], while θ37 was obtained from the quark potential model [49]. We now consider ˉBa1D and ˉBπD1/πD1 decays in the following subsections.

      2. ˉBa1D decay mode

      In this section, we illustrate methodology of our approach by analysing ˉBPA decay mode. The branching fraction for B-mesons decay into pseudoscalar and axial vector mesons is related to its decay amplitude as follows:

      B(ˉBPA)=τB|GF2VcbVud|2p38πm2A|A(ˉBPA)|2,

      (40)

      where p is the magnitude of the 3-momentum of the final state particles in the rest frame of the parent B- meson, and mA denotes the mass of axial-vector meson.

      p=|p1|=|p2|=12mB[{m2B(mP+mA)2}×{m2B(mPmA)2}]1/2.

      Using the isospin framework, ˉBa1D decay amplitudes are represented in terms of isospin reduced amplitudes (Aa1D1/2, Aa1D3/2), and the strong interaction phases (δa1D1/2,δa1D3/2) in respective Isospin –1/2 and 3/2 final states are

      A(¯B0a1D+)=13[Aa1D3/2eiδa1D3/2+2Aa1D1/2eiδa1D1/2],A(¯B0a01D0)=13[2Aa1D3/2eiδa1D3/2Aa1D1/2eiδa1D1/2],A(Ba1D0)=3Aa1D3/2eiδa1D3/2.

      (41)

      These equations lead to the following relations:

      Aa1D1/2=[|A(ˉB0a1D+)|2+|A(ˉB0a01D0)|213|A(Ba1D0)|2]1/2,Aa1D3/2=13|A(Ba1D0)|,

      (42)

      and using the experimental value B(Ba1D0)=(4±4)×103, we get

      A(Ba1D0)=(0.25±0.25) GeV2.

      (43)

      The isospin formalism assists us in deriving a generic relation among the branching fractions of ˉBa1D decays as follows:

      B(ˉB0a1D+)+B(ˉB0a01D0)=τˉB03τBB(Ba1D0){1+[α+(2α)Af(ˉB0a1D+)(1+2α)Af(ˉBa01D0)A(Ba1D0)]2},

      (44)

      where αAnf1/2/Anf3/2=0.22, from the analysis of s-wave meson emitting decays of B-mesons, inspired by the analysis of charm meson decays [22], where it has been observed that the p-wave mesons bear the same ratio as that of DπˉK decay mode.

      Now, we obtain factorizable amplitudes for ˉB0 decays as

      Af(ˉB0a1D+)=2a1ma1fa1F1ˉBD(m2a1)=(0.369±0.016)GeV2,

      (45)

      Af(ˉB0a10D0)=12a2ma1fDV0ˉBa1(m2D)=(0.0033±0.0001)GeV2,

      (46)

      where the decay constants are taken from [42]:

      fa1=(0.203±0.018)GeV,fD=(0.207±0.009)GeV.

      (47)

      The form-factor VBa10(m2D) is obtained from CLFQM [40] results with the following q2 dependence:

      VˉBa10(q2)=VˉBa10(0)[1a(q2m2B)b(q2m2B)2],

      (48)

      where

      VˉBa10(0)=0.14±0.01,a=1.66±0.04,b=1.11±0.08,

      (49)

      and

      FˉBD1(0)=FˉBD0(0)=(0.67±0.01),

      which has already been used in (18). Finally, taking B(Ba1D0)=(4±4)×103, we predict

      B(ˉB0a1D+)+B(ˉBa01D0)={(4.7±0.7)×103forB(Ba1D0)=4×103,(5.6±0.3)×103forB(Ba1D0)=8×103,

      (50)

      which are barely touching the experimental value of B(ˉB0a1D+)=(6.4±3.3)×103.

      There are several existing model calculations for the ˉBA form factors: the ISGW2 model [4], constituent quark–meson model (CQM) [52], QSR [53], LCSR [54], and more recently, the pQCD approach [55]. For the sake of comparison, results for the ˉBa1 transition form factors are given in Table 2 for these approaches, which show relatively significant differences because these approaches differ in their treatment of dynamics of the form-factors. Specfically, VˉBa10=1.20 obtained in the quark–meson model and 1.01 in the ISGW2 model are larger than the values obtained by other approaches.

      ˉBa1CLFQM [40]ISGW2 [3]CQM [52]QSR [53]LCSR [54]pQCD [55]
      V00.14±0.011.011.200.23±0.050.30±0.050.34±0.07

      Table 2.  Form-factor of the ˉBa1transitions at maximum recoil (q2 =0). The results of CQM and QSR have been rescaled according to the form-factor definition.

      Considering these uncertainties, in Fig. 1, we present variation of the sum of

      Figure 1.  (color online) Variation of B(ˉB0a1D) with B(Ba1D0) for different values of form-factor.

      B(ˉB0decays)B(ˉB0a1D+)+B(ˉBa01D0)

      with respect to B(Ba1D0)for different values of form factor, V0ˉBa1(0)=0.14 and 1.20 (the dashed line corresponds to V0ˉBa1(0)=0.14, and the solid line corresponds to V0ˉBa1(0)=1.20), which enhances our prediction by a factor of 1.19, i.e.,

      B(ˉB0a1D+)+B(ˉBa01D0)=(5.6±0.3)×103.

      We also notice that the present data favour B(Ba1D0) being on the higher side. A new measurement of branching fractions of these decays would clarify the situation.

      3. ˉBπD1(2.420) decay mode

      We start by writing the generic formula explicitly for BπD1(2.420)0

      B(ˉB0πD+1)+B(ˉB0π0D01)=τˉB03τBB(BπD01)[1+{α+(2α)Af(ˉB0πD+1)(1+2α)Af(ˉB0π0D01)A(BπD01)}2].

      (51)

      We take αAnf1/2Anf3/2=0.22 from the analysis of s-wave meson emitting decays of B-mesons (23).

      From the experimental branching B(BπD01)=(1.5±0.6)×103 , we get

      A(BπD01)=(0.213±0.040) GeV2,

      (52)

      Now, we obtain factorizable amplitudes for ˉB0 decays as follows:

      Af(ˉB0πD1+)=2a1mD1fπV0ˉBD1(m2π)=0.332 GeV2,Af(ˉB0π0D10)=122a2mD1fD1F1ˉBπ(m2D1)=0.012 GeV2,

      (53)

      where the decay constants are given by

      fD1=fD1/21cosθ1+fD3/21sinθ1,

      (54)

      and

      fD1/21=(0.179±0.035)GeV,fD3/21=(0.054±0.013)GeV,fπ=(0.131±0.002)GeV,

      (55)

      are taken from [42]. Required ˉBD1 form factor is given by

      VˉBD10(m2π)=VˉBD1/2l0(m2π)cosθ1+VˉBD3/210(m2π)sinθ1.

      (56)

      The form-factors VˉBD1/210(m2π) and VˉBD3/210(m2π) are taken from CLFQM [40] results with the following q2 dependence:

      VˉBD1/210(q2)=VˉBD1/210(0)(1a(q2m2B)b(q2m2B)2),

      (57)

      V0ˉBD3/21(q2)=V0ˉBD3/21(0)(1a(q2m2B)b(q2m2B)2),

      (58)

      where

      V0ˉBD1/21(0)=0.11±0.01,a=1.08±0.02,b=0.08±0.03;

      (59)

      V0ˉBD3/21(0)=0.52±0.01,a=1.14±0.04,b=0.34±0.02.

      (60)

      The ˉBπform factor,

      FˉBπ1(0)=FˉBπ0(0)=(0.27±0.05),

      has already been used in (18). For the charm meson mixing angle θ1=(5.7±2.4), we predict,

      B(ˉB0πD1)B(ˉB0πD+1)+B(ˉB0π0D01)={(4.7±1.7)×104forθ1=8.1(4.9±1.7)×104forθ1=3.3

      (61)

      Here, we also plot variation of the sum of B(ˉB0πD+1) and B(ˉB0π0D01) with respect to B(BπD10) in Fig. 2, in the light of experimental error in B(BπD10).

      Figure 2.  (color online) Variation of B(ˉB0πD1) with B(BπD01).

      4. ˉBπD1(2.427) decay mode

      The generic formula for BπD1(2.427)0 takes the following form:

      B(ˉB0πD+1)+B(ˉB0π0D01)=τˉB03τBB(BπD01)[1+{α+(2α)Af(ˉB0πD+1)(1+2α)Af(ˉB0π0D01)A(BπD01)}2].

      (62)

      Here, we also take αAnf1/2Anf3/2=0.22.

      Using experimental branching B(BπD1(2.427)0)×B(D1(2.427)0πD) given in Table 1, assuming that the D01 width is saturated by πD [50] and then using isospin sum rule,

      B(BπD1(2.427)0)×B(D1(2.427)0πD+)=2/3,

      (63)

      we obtain B(BπD01)=(7.5±1.7)×104, which yields

      A(BπD01)=0.213 GeV2.

      (64)

      Now, we obtain factorizable amplitudes for ˉB0decays, given by

      Af(ˉB0πD+1)=2a1mD1fπV0ˉBD3/21(m2π)=0.106 GeV2.

      Af(ˉB0π0D01)=122a2mD1fD3/21F1ˉBπ(m2D3/21)=0.029 GeV2.

      (65)

      Here,

      fD1=fD1/21sinθ1+fD3/21cosθ1,

      (66)

      VˉBD10(m2π)=VˉBD1/210(m2π)sinθ1+VˉBD3/210(m2π)cosθ1.

      (67)

      We use numerical values for the decay constants and form-factors, as presented in the previous case. Finally, we predict

      B(ˉB0πD1)B(ˉB0πD+1)+B(ˉB0π0D01)={(8.8±0.4)×104forθ1=8.1(8.4±0.4)×104forθ1=3.3

      (68)

      Considering the uncertainty of the experimental branching B(BπD1(2.427)0), here, we also plot the variation of B(ˉB0πD1) with respect to B(BπD01) in Fig. 3.

      Figure 3.  Variation of B(ˉB0πD1) with B(BπD10).

    • B. ˉBπD2 decay mode

    • Experimentally [1], the tensor meson sixteen-plet comprises of isovector a2(1.320), strange iso-spinor K2(1.430), charm triplet D2(2.460),Ds2(2.573), and three isoscalars f2(1.270),f2(1.525), and χc2(1P). These states behave well with respect to quark model assignments. For ˉBPT decays, only one mode has been observed [1], B(BπD02(2.460)), and more data are expected to come in near future.

      The generic formula for ˉBπD2 decays is given by

      B(ˉB0πD+2)+B(ˉB0π0D02)=τˉB03τBB(BπD02)[1+{α+(2α)Af(ˉB0πD+2)(1+2α)Af(ˉB0π0D02)A(BπD02)}2].

      (69)

      We proceed to calculate various quantities on the right-hand side. We combine both the results given in Table 1, i.e.,

      B(BπD2(2.462)0)×B(D2(2.462)0πD+)=(3.56±0.24)×104,

      (70)

      B(BπD2(2.462)0)×B(D2(2.462)0πD)=(2.2±1.0)×104,

      (71)

      to arrive at

      B(BπD2(2.462)0)×B(D2(2.462)0πD+,πD+)=(5.7±1.1)×104.

      (72)

      Using B(D2(2.462)0πD+,πD+) = 2/3 following from the isospin symmetry and assuming that the D02 width is saturated by πD and πD [5059], we get

      B(BπD02(2.462))=(8.6±1.7)×104.

      (73)

      We use the branching fraction formula

      B(ˉBPT)=τB|GF2VcbVud|2m2Bp512πm4T|A(ˉBPT)|2,

      (74)

      where p is the magnitude of the three-momentum of the final-state particle in the rest frame of B-meson and mB and mT denote masses of the B-meson and tensor meson, respectively.

      By using the experimental value (72), we get

      A(BπD02)=(6.5±0.6)×102 GeV.

      (75)

      The factorization parts of the weak decay amplitudes for ˉBPTdecays are expressed as the product of matrix elements of weak currents (up to the weak scale factor of GF2×CKM elements ×QCD factors ):

      PT|Hw|B=P|Jμ|0T|Jμ|B+T|Jμ|0P|Jμ|B.

      (76)

      The matrix elements P|PJμ|0Jμ|0 and P|PJμ|BJμ|B are given below. The hadronic current creating meson from the vacuum is given by

      P|PJμ|0Jμ|0=ifBPB,

      (77)

      where PB is the four-momentum of the pseudoscalar meson. However, the matrix elements T|Jμ|0 vanish due to the tracelessness of the polarization tensor εμν of spin 2 meson and the auxiliary condition qμεμν=0 [60]. Thus, the tensor meson cannot be produced from the V-A current. Relevant B → T matrix elements are expressed as follow:

      T(PT)|Jμ|B(PB)=ihεμνPBα(PB+PT)λ(PBPT)ρ+kεμνPνB+b+(εαβPαBPBβ)[(PB+PT)μ+b(PBPT)μ],

      (78)

      in the ISGW2 model [5]. The matrix elements simplify to

      A(ˉBPT)=ifPFˉBT(m2P),

      (79)

      where

      FBT(m2P)=k(m2P)+(m2Bm2T)b+(m2P)+m2Pb(m2P).

      (80)

      Now, we obtain factorizable amplitude values for ˉB0decays,

      Af(ˉB0πD+2)=a1fπFˉBD2(m2π)={0.070forFˉBD2(m2π)=0.52,0.051forFˉBD2(m2π)=0.38,

      (81)

      using the decay constant values fπ=(0.131±0.002)GeV, as already used in the previous sections [40], and the form factor FˉBD2(m2n)=0.52,0.38, taken from the CLFQM [40] and ISGW models [3]:

      Af(ˉB0π0D02)=12a2fD2FˉBπ(m2D2)=0,

      (82)

      Af(ˉB0π0D02)becomes zero due to vanishing of the decay constant of the D2 meson. Finally, using (69) for α=0.22, we predict

      B(ˉB0πD+2)+B(ˉB0π0D02)={(5.7±0.4)×104forFˉBD2(m2π)=0.52;(4.1±0.4)×104forFˉBD2(m2π)=0.38;

      (83)

      for the two choices of FˉBD2(m2n)=0.52,0.38, respectively, which may be tested in future experiments. Considering the ambiguity of the experimental B(BπD2(2.460)0), we show the increasing behavior of B(ˉB0πD2) with respect toB(BπD02) in Fig. 4 for both choices, shown as dashed and thick lines, respectively.

      Figure 4.  (color online) Variation of B(ˉB0πD2) with B(BπD02) for different values of form-factor.

    • C. ˉBπD0 decay mode

    • The scalar mesons mostly appear as the hadronic resonances and have large decay widths. There will exist several resonances and decay channels within a short mass interval. The overlaps between resonances and background make it considerably difficult to resolve the scalar mesons. The scalar-meson family has been the most difficult one to identify as a standard sixteen-plet. Experimentally [1], the following states of scalar meson sixteen-plet, isovector a0(0.980), strange spinor K0(1.429), one isoscalar χc0(1P)(3.145),and charm triplet D0(2.400),Ds0(2.480), behave well with respect to quark model assignments. For ˉBPS decays, only one mode has been observed [1], B(BπD00(2.400)), and more data are expected to come in the near future.

      Writing the generic formula explicitly for ˉBπD0 decays,

      B(ˉB0πD+0)+B(ˉB0π0D00)=τˉB03τBB(BπD00)[1+{α+(2α)Af(ˉB0πD+0)(1+2α)Af(ˉB0π0D00)A(BπD00)}2].

      (84)

      To obtain the branching fraction B(BπD00) from the experimental value

      B(BπD0(2.400)0)×B(D0(2.400)0)πD+)=(6.4±1.4)×104,

      (85)

      given in Table 1, we employ isospin symmetry, which gives

      Γ(D00πD+)Γ(D00π0D0)+Γ(D00πD+)=23,

      (86)

      and realizing the saturation of strong D00 decays with D00πD modes [52], we estimate

      B(BπD0(2.400)0)=(9.6±2.1)×104,

      (87)

      for our analysis. Using this estimate and decay rate formula, similar to that of ˉBPP,

      B(ˉBPS)=τB|GF2VcbVud|2p8πm2B|A(ˉBPS)|2,

      (88)

      and we get

      A(BπD00)=(1.06±0.32)×104 GeV3.

      (89)

      We then obtain factorizable amplitudes for ˉB0decays, which are given as

      Af(ˉB0πD0+)=a1fπ(m2Bm2D0)FˉBD0(m2π)=0.824 GeV3,

      Af(ˉB0π0D00)=12a2fD0(m2Bm2π)FˉBπ(m2D0)=0.0522 GeV3

      (90)

      Numerical values are calculated using the decay constants [42],

      fπ=(0.131±0.002)GeV,fD0=(0.107±0.013)GeV.

      (91)

      and FˉBD0(m2π)from the CLFQM [40] results, i.e.,

      FˉBD0(q2)=FˉBD0(0)(1a(q2m2B)b(q2m2B)2),

      (92)

      where

      FˉBD0(0)=(0.27±0.01),a=1.08±0.04,b=0.23±0.02,

      (93)

      and the form-factor FˉBπ(0)=0.27±0.05 was already given in previous sections.

      Finally, we predict

      B(ˉB0πD0)B(ˉB0πD+0)+B(ˉBπ0D00)=(4.8±0.6)×104,

      (94)

      for α=0.22. Here, we also plot the variation of B(ˉB0πD0) with respect to B(BπD00) in Fig. 5, which also shows increasing behaviour.

      Figure 5.  (color online) Variation of B(ˉB0πD0) with B(BπD00).

    V. SUMMARY AND CONCLUSIONS
    • In our previous work, we conducted isospin analysis of CKM-favored two-body weak decays of bottom mesons ˉBPP/PV,occurring through W-emission quark diagrams. Obtaining the factorizable contributions from the spectator-quark model for Nc = 3 (real value), we have determined nonfactorizable reduced isospin amplitudes from the experimental data for these modes. We have observed that in all the decay modes, the nonfactorizable isospin reduced amplitude Anf1/2 bears the same ratio asAnf3/2 within the experimental errors. In the charm sector, a systematics observed for the charm mesons decaying to s-wave mesons has been found to be consistent with their p-wave meson emitting decays [22]. Encouraged by the success for the s-wave emitting decays in the bottom meson sector [36], we have extended isospin analysis to the p-wave meson emitting decays in ˉBPA/PT/PS channels, particularly for the ˉBa1D/πD1/πD1/πD2/πD0 decays, which have the same isospin structure as that of ˉBπD/ρD/πD cases.

      To include the effects of nonfactorizable contributions, for these cases, we exploit the generic formula to predict the sum of the branching fractions of ˉB0 decays in these channels. As there are large errors involved in B(ˉBa1D)=(4±4)×103 and the form-factor FˉBa1(0) is not uniquely known, looking at these uncertainties, we plot the variation of B(ˉB0deccys) with respect to B(Ba1D0) for extreme values of V0ˉBa1(0)=0.14 and 1.20, which enhances our prediction by a factor of 1.19. Our predictions will be tested in future experiments.

      We extend our analysis to ˉBπD1/πD1/πD2/πD0 decay modes, which have a similar isospin structure, and make predictions for ˉB0decays. It is hoped that the predictions made in this paper will help experimentalists to identify the p-wave meson emitting decays of the heaviest bottom mesons.

Reference (60)

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