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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 [2−7]. 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) [8−19] 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 [20−23]. The nonfactorizable terms may appear for several reasons, including soft gluon exchange and FSIrescattering effects [20−23]. The rescattering effects on the outgoing mesons have been studied in detail for bottom meson decays [24−25]. 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 [29−30]. 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 [31−33]. Initially, data on branching fractions of
D→ˉKπ decays seemed to requirea1≈c1=1.26,a2≈c2=−0.51, leading to destructive interference between color-favored (CF) and color-suppressed (CS) processes forD+→ˉK0π+, thereby implying theNc→∞ limit [34]. However, later measurement ofˉB→Dπ meson decays did not favor this result empirically, as these decays requirea1≈1.03, anda2≈0.23, i.e., a positive value ofa2 , in sharp contrast to the expectations based on the largeNc 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 forB−→π−D0, seem to favorNc=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 (
ˉB→PP/PV ) emitting decays [1−24]. There exist four L=1 states: scalar(JPC=0++), axial-vectors(JPC=1++ and1+−) , 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:ˉB→P(0−+)+S(0++),ˉB→P(0−+)+A(1++),ˉB→P(0−+)+A′(1+−),ˉB→P(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
D→ˉKπ/ˉKρ/ˉK∗π 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:D→ˉKa1/πˉK1/πˉK−1/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-favoredˉB→πD/ρD/πD∗ decay modes bear a universal ratio equal toα within the experimental errors. Extension of this universality toˉB→a1D/πD1/πD′1/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.
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To study the two-body hadronic
B− decays, we consider the effective weak Hamiltonian [37]Hw=GF√2VcbV∗ud[c1(¯du)(¯cb)+c2(¯cu)(¯db)],
(1) where
Vud andVcb are the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements [1],Vud=0.975,Vcb=0.041,
ˉq1q2=ˉq1γμ(1−γ5)q2 denotes color singlet V−A 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
ˉq1λaq2≡ˉq1γμ(1−γ5)λaq2 represents the color octet current, and performing similar treatment on the other operator(ˉcu)(ˉdb), the weak Hamiltonian finally becomesHCFw=GF√2VcbV∗ud[a1(ˉdu)H(ˉcb)H+c2H8w],
(4) HCSw=GF√2VcbV∗ud[a2(ˉcu)H(ˉdb)H+c1˜H8w],
(5) for the CF and CS processes, respectively, where
a1,2=c1,2+c2,1Nc,
(6) H8w=128∑a=1(¯cλau)(ˉdλab),˜H8w=128∑a=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.
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In this section, we describe our approach by analysing
ˉB→PP decay mode. The branching fraction for B-meson decay into two pseudoscalar mesons is related to its decay amplitude as follows:B(ˉB→P1P2)=τB|GF√2VcbV∗ud|2p8πm2B|A(ˉB→P1P2)|2,
(8) where
τB denotes the lifetime of B-mesons measured to be [1]τˉB0=(1.519±0.004)×10−12 s,τB−=(1.638±0.004)×10−12 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−(m1−m2)2]}1/122.
Using the isospin framework,
ˉB→πD 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(ˉB0→π−D+)=1√3[AπD3/322eiδπD3/2+√2AπD1/122eiδπD1/2],A(ˉB0→π0D0)=1√3[√2AπD3/322eiδπD3/2−AπD1/122eiδπD1/2],A(B−→π−D0)=√3AπD3/322eiδπD3/2.
(9) These equations lead to the following relations:
AπD1/2=[|A(ˉB0→π−D+)|2+|A(ˉB0→π0D0)|2−13|A(B−→π−D0)|2]1/2,AπD3/322=√13|A(B−→π−D0)|.
(10) The experimental values [1]
B(ˉB0→π−D+)=(2.52±0.13)×10−3,B(ˉB0→π0D0)=(2.63±0.14)×10−4,B(B−→π−D0)=(4.68±0.13)×10−3,
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(ˉB→πD)=Af(ˉB→πD)+Anf(ˉB→πD),
(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
ˉB→πD decays:Af(ˉB0→π−D+)=a1fπ(m2B−m2D)FˉBD0(m2π)=(2.180±0.099)GeV3,
(13) Af(ˉB0→π0D0)=−1√2a2fD(m2B−m2π)FˉBπ0(m2D)=−(0.111±0.021)GeV3,
(14) Af(B−→π−D0)=a1fπ(m2B−m2D)FˉBD0(m2π)+a2fD(m2B−m2π)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)(1−q2/m2s),
(17) where the pole masses ms are given by the lowest lying meson with the appropriate quantum numbers, i.e.,
JP=0+ forF0(0) and1− forF1(0). For numerical estimation, we take scalar mesons carrying the quantum number of the corresponding weak currents, which arems(0+)=5.78 GeV andms(0+)=6.80 GeV [2−21, 41, 42]. Form-factorsF0(0) atq2=0 are taken from [43] as followsFˉBπ0(0)=(0.27±0.05),FˉBD0(0)=(0.66±0.03).
(18) Exploiting the isospin relations
Af1/2(ˉB→πD)=1√3{√2Af(ˉB0→π−D+)−Af(ˉB0→π0D0)},Af3/2(ˉB→πD)=1√3{Af(ˉB0→π−D+)+√2Af(ˉB0→π0D0)},
(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 +
ˉB→πD process:Anf(ˉB0→π−D+)=13c2(⟨πD‖H8w‖ˉB⟩3/322+2⟨πD‖H8w‖ˉB⟩1/122),Anf(ˉB0→π0D0)=√23c1(⟨πD‖˜H8w‖ˉB⟩3/322−⟨πD‖˜H8w‖ˉB⟩1/122),Anf(B−→π−D0)=c2⟨πD‖H8w‖ˉB⟩3/322+c1⟨πD‖˜H8w‖ˉB⟩3/322,
(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 andAπD3/322exp , respectively, from Eq. (11). These bear the following ratio:α=(Anf1/2Anf3/2)ˉB→πD=0.229±0.042.
(23) Such isospin formalism can easily be extended to
ˉB→ρD andˉB→πD∗ decays, as the isospin structure of these decay modes is exactly the same as that ofˉB→πD. Following the procedure discussed for theˉB→πD mode, we calculate the ratio of non-factorizable isospin parts forˉB→ρD andˉB→πD∗ decay modes, given as follows for the sake of comparison:Anf1/2(ˉB→ρD)Anf3/2(ˉB→ρD)≃Anf1/2(ˉB→πD∗)Anf3/2(ˉB→πD∗)≃Anf1/2(ˉB→πD)Anf3/2(ˉB→πD).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 modesˉB→πD/ρD/πD∗ are in consistent agreement with each other.In fact, these relations can be expressed in a generic form as
B−++B00=τˉB03τB−B0−{1+[α+(√2−α)Af−+−(1+√2α)Af00A0−]2},
(25) with
α=Anf1/2/Anf3/2, where the subscripts of branching fractionsB−+,B00,B0− denote the charge states of the non-charm and charm mesons emitted in each case.Af−+ andAf00 factorizable amplitudes denote the charge state of the mesons forˉB0− decays.A0− , the total decay amplitude, is obtained from theB−− decay asA0−=√B0−τB−×(kinematicfactor),
(26) where the kinematic factors for
ˉB→PP andˉB→PV are as follows:kinematic factor for ˉB→PP=|GF√2VcbV∗ud|2p8πm2B,
(27) kinematic factor for ˉB→PV=|GF√2VcbV∗ud|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 asB(ˉB0→π−D+)+B(ˉB0→π0D0)=(0.28±0.02)0/0Theo,=(0.28±0.01)0/0Expt;
(29) B(ˉB0→ρ−D+)+B(ˉB0→ρ0D0)=(0.76±0.13)0/0Theo,=(0.79±0.12)0/0Expt;
(30) B(ˉB0→π−D∗+)+B(ˉB0→π0D∗0)=(0.29±0.04)0/0Theo,=(0.30±0.01)0/0Expt.
(31) All theoretical values match well within experimental errors.
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In this section, we study the Cabibbo-favored p-wave meson emitting decays in the channels
ˉB→PA/PT/PS involving b + u→ c + 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,ˉB→a1D 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 toˉB→a1D/πD1/πD′1/πD2/πD0 decay modes, as the isospin structure of these decay modes is exactly the same as that ofˉB→πD mode.Channels Branching fraction of decays Experimental branching fractions [1] ˉB→PA B(ˉB0→a1−D+)B(B−→a1−D0)B(B−→π−D1(2.420)0)B(B−→π−D′1(2.427)0)×B(D′1(2.427)0→π−D∗+) (6.0±3.3)×10−3(4±4)×10−3(1.5±0.6)×10−3(5.0±1.2)×10−4 ˉB→PT B(B−→π−D∗2(2.462)0)×B(D∗2(2.462)0→π−D+)B(B−→π−D∗2(2.462)0)×B(D∗2(2.462)0→π−D∗+) (3.56±0.24)×10−4(2.2±1.1)×10−4 ˉB→PS B(B−→π−D∗0(2.400)0)×B(D∗0(2.400)0→π−D+) (6.4±1.4)×10−4 Table 1. Experimental data for p-wave meson emitting decays [1].
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1. Axial-Vector meson spectroscopy
Experimentally [1], two types of axial-vector mesons exist with different charge conjugation properties, i.e.,
3P1(JPC=1++) and1P1(JPC=1+−), which behave well with respect to the quark modelqˉq assignments observed; strange and charmed states are given by mixture of3P1 and1P1 states. In contrast, hidden-flavor diagonal3P1 and1P1 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 contentuˉd,uˉu−dˉd/√2 , anddˉus :a+1,a01anda−1.
(32) ii. Four isoscalars
f1(1.285),f1(1.420),f1′(1.512), andχc1(3.511) have been observed, of whichf1(1.420) is a multiquark state in the form of aKˉKπ bound state [44] orKˉK∗ deuteron-state [42].For
1P1 multiplet:i. Isovector
b1(1.229) with flavor content the same as given in (32):b+1,b01andb−1.
(33) ii. Three isoscalars
h1(1.170),h′1(1.380) , andhc1(3.526). The C-parity ofh′1(1.380) and spin and parity ofhc1(3.526) remain to be confirmed.The proximity of
a1(1.230) andf1(1.285) , and to a lesser extent that ofb1(1.229) andh1(1.170) , indicate the ideal mixing for both1++ and1+− diagonal states.The states involving a strange quark of
A(JPC=1++) andA′(JPC=1+−) multiplets mix to generate the physical states in the following manner [45−47]:K1(1.270)=K1Asinθ1+K1A′cosθ1,K_1(1.400)=K1Acosθ1−K1A′sinθ1,
(34) where
K1A andK1,A′ denote the strange partners ofa1(1.230) andb1(1.229) , respectively. The Particle Data Group [1] assumes that the mixing is maximal, i.e.,θ1=45∘, whereasτ1→K1(1.270)/K1(1.400)+vτ data yieldθ1=±37∘ andθ1=±58∘ [48]. However, the study ofD→K1(1.270)π;K1(1.400)π decays rules out positive mixing-angle solutions. AsD→K−1(1.400)π+ is largely suppressed forθ1=−37∘ , the solutionθ1=−58∘ [49] is experimentally favored.The mixing of charmed
(cˉu andcˉd) and strange charmed(cˉs) state mesons is given in a similar manner:D1(2.420)=D1AsinθD1+D1A′cosθD1,D′1(2.427)=D1AcosθD1−D1A′sinθD1,
(35) and
Ds1(2.460)=Ds1AsinθDs1+Ds1A′cosθDs1,D_s1(2.535)=Ds1AcosθDs1−Ds1A′sinθDs1.
(36) However, in the heavy quark limit, the physical mass eigenstates with (JP = 1+) are
P3/21 andP1/21 rather than3P1 and1P1 states, as the heavy quark spinSQ decouples from the other degrees of freedom, such thatSQ 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|1P1⟩−√23|3P1⟩.
(37) However, beyond the heavy quark limit, there is still a small mixing between
P3/21 andP1/21 states, denoted byD1(2.420)=D1/21cosθ2+D3/21sinθ2,D′1(2.427)=−D1/21sinθ2+D3/21cosθ2,
(38) Likewise, for strange axial-vector charmed mesons,
Ds1(2.460)=D1/2s1cosθ3−D3/2s1sinθ3,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 detailedˉB→D∗ππ analysis [50, 51], whileθ3≈7∘ was obtained from the quark potential model [49]. We now considerˉB→a1D andˉB→πD1/πD′1 decays in the following subsections.2.
ˉB→a1D decay modeIn this section, we illustrate methodology of our approach by analysing
ˉB→PA decay mode. The branching fraction for B-mesons decay into pseudoscalar and axial vector mesons is related to its decay amplitude as follows:B(ˉB→PA)=τB|GF√2VcbV∗ud|2p38πm2A|A(ˉB→PA)|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−(mP−mA)2}]1/2.
Using the isospin framework,
ˉB→a1D 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 areA(¯B0→a−1D+)=1√3[Aa1D3/2eiδa1D3/2+√2Aa1D1/2eiδa1D1/2],A(¯B0→a01D0)=1√3[√2Aa1D3/2eiδa1D3/2−Aa1D1/2eiδa1D1/2],A(B−→a−1D0)=√3Aa1D3/2eiδa1D3/2.
(41) These equations lead to the following relations:
Aa1D1/2=[|A(ˉB0→a−1D+)|2+|A(ˉB0→a01D0)|2−13|A(B−→a−1D0)|2]1/2,Aa1D3/2=√13|A(B−→a−1D0)|,
(42) and using the experimental value
B(B−→a1−D0)=(4±4)×10−3 , we getA(B−→a−1D0)=(0.25±0.25) GeV2.
(43) The isospin formalism assists us in deriving a generic relation among the branching fractions of
ˉB→a1D decays as follows:B(ˉB0→a−1D+)+B(ˉB0→a01D0)=τˉB03τB−B(B−→a−1D0){1+[α+(√2−α)Af(ˉB0→a−1D+)−(1+√2α)Af(ˉB→a01D0)A(B−→a−1D0)]2},
(44) where
α≡Anf1/2/Anf3/2=0.22, from the analysis of s-wave meson emitting decays ofB -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 ofD→πˉK decay mode.Now, we obtain factorizable amplitudes for
ˉB0 decays asAf(ˉB0→a1−D+)=2a1ma1fa1F1ˉBD(m2a1)=(0.369±0.016)GeV2,
(45) Af(ˉB0→a10D0)=−1√2a2ma1fDV0ˉBa1(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 followingq2 dependence:VˉBa10(q2)=VˉBa10(0)[1−a(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(B−→a1−D0)=(4±4)×10−3 , we predictB(ˉB0→a−1D+)+B(ˉB→a01D0)={(4.7±0.7)×10−3forB(B−→a−1D0)=4×10−3,(5.6±0.3)×10−3forB(B−→a−1D0)=8×10−3,
(50) which are barely touching the experimental value of
B(ˉB0→a−1D+)=(6.4±3.3)×10−3. There are several existing model calculations for the
ˉB→A 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 theˉB→a1 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.Table 2. Form-factor of the
ˉB→a1 transitions 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(ˉB0→a1D) withB(B−→a1−D0) for different values of form-factor.∑B(ˉB0→decays)≡B(ˉB0→a−1D+)+B(ˉB→a01D0)
with respect to
B(B−→a1−D0) for different values of form factor,V0ˉBa1(0)=0.14 and1.20 (the dashed line corresponds toV0ˉBa1(0)=0.14, and the solid line corresponds toV0ˉBa1(0)=1.20 ), which enhances our prediction by a factor of1.19, i.e.,B(ˉB0→a−1D+)+B(ˉB→a01D0)=(5.6±0.3)×10−3.
We also notice that the present data favour
B(B−→a−1D0) being on the higher side. A new measurement of branching fractions of these decays would clarify the situation.3.
ˉB→πD1(2.420) decay modeWe start by writing the generic formula explicitly for
B−→π−D1(2.420)0 B(ˉB0→π−D+1)+B(ˉB0→π0D01)=τˉB03τB−B(B−→π−D01)[1+{α+(√2−α)Af(ˉB0→π−D+1)−(1+√2α)Af(ˉB0→π0D01)A(B−→π−D01)}2].
(51) We take
α≡Anf1/2Anf3/2=0.22 from the analysis of s-wave meson emitting decays ofB -mesons (23).From the experimental branching
B(B−→π−D01)=(1.5±0.6)×10−3 , we getA(B−→π−D01)=(0.213±0.040) GeV2,
(52) Now, we obtain factorizable amplitudes for
ˉB0− decays as follows:Af(ˉB0→π−D1+)=2a1mD1fπV0ˉBD1(m2π)=0.332 GeV2,Af(ˉB0→π0D10)=−1√22a2mD1fD1F1ˉBπ(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
ˉB→D1 form factor is given byVˉBD10(m2π)=VˉBD1/2l0(m2π)cosθ1+VˉBD3/210(m2π)sinθ1.
(56) The form-factors
VˉBD1/210(m2π) andVˉBD3/210(m2π) are taken from CLFQM [40] results with the followingq2 dependence:VˉBD1/210(q2)=VˉBD1/210(0)(1−a(q2m2B)−b(q2m2B)2),
(57) V0ˉBD3/21(q2)=V0ˉBD3/21(0)(1−a(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
ˉB→π form 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(ˉB0→πD1)≡B(ˉB0→π−D+1)+B(ˉB0→π0D01)={(4.7±1.7)×10−4forθ1=−8.1∘(4.9±1.7)×10−4forθ1=−3.3∘
(61) Here, we also plot variation of the sum of
B(ˉB0→π−D+1) andB(ˉB0→π0D01) with respect toB(B−→π−D10) in Fig. 2, in the light of experimental error inB(B−→π−D10). 4.
ˉB→πD′1(2.427) decay modeThe generic formula for
B−→π−D′1(2.427)0 takes the following form:B(ˉB0→π−D′+1)+B(ˉB0→π0D′01)=τˉB03τB−B(B−→π−D′01)[1+{α+(√2−α)Af(ˉB0→π−D′+1)−(1+√2α)Af(ˉB0→π0D′01)A(B−→π−D′01)}2].
(62) Here, we also take
α≡Anf1/2Anf3/2=0.22 .Using experimental branching
B(B−→π−D′1(2.427)0)×B(D′1(2.427)0→π−D∗−) given in Table 1, assuming that theD′01 width is saturated byπD∗ [50] and then using isospin sum rule,B(B−→π−D′1(2.427)0)×B(D′1(2.427)0→π−D∗+)=2/3,
(63) we obtain
B(B−→π−D′01)=(7.5±1.7)×10−4 , which yieldsA(B−→π−D′01)=0.213 GeV2.
(64) Now, we obtain factorizable amplitudes for
ˉB0− decays, given byAf(ˉB0→π−D′+1)=2a1mD′1fπV0ˉBD′3/21(m2π)=0.106 GeV2.
Af(ˉB0→π0D′01)=−1√22a2mD′1fD′3/21F1ˉBπ(m2D′3/21)=−0.029 GeV2.
(65) Here,
fD′1=−fD1/21sinθ1+fD3/21cosθ1,
(66) VˉBD′10(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(ˉB0→πD′1)≡B(ˉB0→π−D′+1)+B(ˉB0→π0D′01)={(8.8±0.4)×10−4forθ1=−8.1∘(8.4±0.4)×10−4forθ1=−3.3∘
(68) Considering the uncertainty of the experimental branching
B(B−→π−D′1(2.427)0) , here, we also plot the variation of∑B(ˉB0→πD′1) with respect toB(B−→π−D′01) in Fig. 3. -
Experimentally [1], the tensor meson sixteen-plet comprises of isovector
a2(1.320), strange iso-spinorK∗2(1.430), charm tripletD∗2(2.460),D∗s2(2.573), and three isoscalarsf2(1.270),f′2(1.525), andχc2(1P). These states behave well with respect to quark model assignments. ForˉB→PT decays, only one mode has been observed [1],B(B−→π−D∗02(2.460)), and more data are expected to come in near future.The generic formula for
ˉB→πD∗2 decays is given byB(ˉB0→π−D∗+2)+B(ˉB0→π0D∗02)=τˉB03τB−B(B−→π−D∗02)[1+{α+(√2−α)Af(ˉB0→π−D∗+2)−(1+√2α)Af(ˉB0→π0D∗02)A(B−→π−D∗02)}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(B−→π−D∗2(2.462)0)×B(D∗2(2.462)0→π−D+)=(3.56±0.24)×10−4,
(70) B(B−→π−D∗2(2.462)0)×B(D∗2(2.462)0→π−D∗−)=(2.2±1.0)×10−4,
(71) to arrive at
B(B−→π−D∗2(2.462)0)×B(D∗2(2.462)0→π−D+,π−D∗+)=(5.7±1.1)×10−4.
(72) Using
B(D∗2(2.462)0→π−D+,π−D∗+) = 2/3 following from the isospin symmetry and assuming that theD∗02 width is saturated byπD andπD∗ [50−59], we getB(B−→π−D∗02(2.462))=(8.6±1.7)×10−4.
(73) We use the branching fraction formula
B(ˉB→PT)=τB|GF√2VcbV∗ud|2m2Bp512πm4T|A(ˉB→PT)|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(B−→π−D∗02)=(6.5±0.6)×10−2 GeV.
(75) The factorization parts of the weak decay amplitudes for
ˉB→PT decays are expressed as the product of matrix elements of weak currents (up to the weak scale factor ofGF√2×CKM elements ×QCD factors ):⟨PT|Hw|B⟩=⟨P|Jμ|0⟩⟨T|Jμ|B⟩+⟨T|Jμ|0⟩⟨P|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 tensorεμν of spin 2 meson and the auxiliary conditionqμεμν=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)⟩=ihε∗μνPBα(PB+PT)λ(PB−PT)ρ+kεμν∗PνB+b+(εαβ∗PαBPBβ)[(PB+PT)μ+b−(PB−PT)μ],
(78) in the ISGW2 model [5]. The matrix elements simplify to
A(ˉB→PT)=−ifPFˉBT(m2P),
(79) where
FBT(m2P)=k(m2P)+(m2B−m2T)b+(m2P)+m2Pb−(m2P).
(80) Now, we obtain factorizable amplitude values for
ˉB0− decays,Af(ˉB0→π−D∗+2)=a1fπFˉBD∗2(m2π)={0.070forFˉBD∗2(m2π)=0.52,0.051forFˉBD∗2(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 factorFˉBD2(m2n)=0.52,0.38, taken from the CLFQM [40] and ISGW models [3]:Af(ˉB0→π0D∗02)=−1√2a2fD2∗FˉBπ(m2D∗2)=0,
(82) Af(ˉB0→π0D∗02) becomes zero due to vanishing of the decay constant of theD∗2 meson. Finally, using (69) forα=0.22 , we predictB(ˉB0→π−D∗+2)+B(ˉB0→π0D∗02)={(5.7±0.4)×10−4forFˉBD∗2(m2π)=0.52;(4.1±0.4)×10−4forFˉBD∗2(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 experimentalB(B−→π−D∗2(2.460)0), we show the increasing behavior of∑B(ˉB0→πD2) with respect toB(B−→π−D∗02) in Fig. 4 for both choices, shown as dashed and thick lines, respectively. -
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 spinorK∗0(1.429), one isoscalarχc0(1P)(3.145), and charm tripletD∗0(2.400),D∗s0(2.480), behave well with respect to quark model assignments. ForˉB→PS decays, only one mode has been observed [1],B(B−→π−D∗00(2.400)), and more data are expected to come in the near future.Writing the generic formula explicitly for
ˉB→πD∗0 decays,B(ˉB0→π−D∗+0)+B(ˉB0→π0D∗00)=τˉB03τB−B(B−→π−D∗00)[1+{α+(√2−α)Af(ˉB0→π−D∗+0)−(1+√2α)Af(ˉB0→π0D∗00)A(B−→π−D∗00)}2].
(84) To obtain the branching fraction
B(B−→π−D∗00) from the experimental valueB(B−→π−D∗0(2.400)0)×B(D∗0(2.400)0)→π−D+)=(6.4±1.4)×10−4,
(85) given in Table 1, we employ isospin symmetry, which gives
Γ(D∗00→π−D+)Γ(D∗00→π0D0)+Γ(D∗00→π−D+)=23,
(86) and realizing the saturation of strong
D∗00 decays withD∗00→πD modes [52], we estimateB(B−→π−D∗0(2.400)0)=(9.6±2.1)×10−4,
(87) for our analysis. Using this estimate and decay rate formula, similar to that of
ˉB→PP, B(ˉB→PS)=τB|GF√2VcbV∗ud|2p8πm2B|A(ˉB→PS)|2,
(88) and we get
A(B−→π−D∗00)=(1.06±0.32)×10−4 GeV3.
(89) We then obtain factorizable amplitudes for
ˉB0− decays, which are given asAf(ˉB0→π−D0∗+)=a1fπ(m2B−m2D∗0)FˉBD∗0(m2π)=0.824 GeV3,
Af(ˉB0→π0D0∗0)=−1√2a2fD∗0(m2B−m2π)FˉBπ(m2D∗0)=−0.0522 GeV3
(90) Numerical values are calculated using the decay constants [42],
fπ=(0.131±0.002)GeV,fD∗0=(0.107±0.013)GeV.
(91) and
FˉBD∗0(m2π) from the CLFQM [40] results, i.e.,FˉBD∗0(q2)=FˉBD∗0(0)(1−a(q2m2B)−b(q2m2B)2),
(92) where
FˉBD∗0(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(ˉB0→πD∗0)≡B(ˉB0→π−D∗+0)+B(ˉB→π0D∗00)=(4.8±0.6)×10−4,
(94) for
α=0.22. Here, we also plot the variation of∑B(ˉB0→πD0∗) with respect toB(B−→π−D∗00) in Fig. 5, which also shows increasing behaviour. -
In our previous work, we conducted isospin analysis of CKM-favored two-body weak decays of bottom mesons
ˉB→PP/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 amplitudeAnf1/2 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 inˉB→PA/PT/PS channels, particularly for theˉB→a1D/πD1/πD′1/πD2/πD0 decays, which have the same isospin structure as that ofˉB→πD/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 inB(ˉB→a1D)=(4±4)×10−3 and the form-factorFˉBa1(0) is not uniquely known, looking at these uncertainties, we plot the variation of∑B(ˉB0→deccys) with respect toB(B−→a1−D0) for extreme values ofV0ˉBa1(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
ˉB→πD1/πD′1/πD2/πD0 decay modes, which have a similar isospin structure, and make predictions forˉB0− decays. 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.
p-wave mesons emitting weak decays of bottom mesons
- Received Date: 2024-04-05
- Available Online: 2025-02-15
Abstract: This paper is the extension of our previous work entitled ''Searching a systematics for nonfactorizable contributions to