ratios for the CKM-favored and CKM-suppressed decays are .... Employing the factorization scheme, we calculate the decay amplitudes for CKM-favored.
DECAYS OF BOTTOM MESONS EMITTING PSEUDOSCALAR AND TENSOR MESONS IN ISGW II MODEL
Neelesh Sharma and R.C. Verma Department of Physics, Punjabi University, Patiala-147 002, INDIA.
Abstract In this paper, we investigate phenomenologically two-body weak decays of the bottom mesons emitting pseudoscalar and tensor mesons. Decay amplitudes are obtained using the factorization scheme in the improved ISGW II model. Branching ratios for the CKM-favored and CKM-suppressed decays are calculated.
PACS no (s): 13.25.Hw, 12.39.Jh, 12.39.St
1
I. INTRODUCTION Experimental results are available for the branching ratios of several B-meson decay modes. Many theoretical works have been done to understand exclusive hadronic B decays in the framework of the generalized factorization, QCD factorization or flavor SU(3) symmetry. Weak hadronic decays of the B-mesons are expected to provide a rich phenomenology yielding a wealth of information for testing the standard model and for probing strong interaction dynamics. However, these decays involve nonperturbative strong processes which cannot be calculated from the first principles. Thus phenomenological approaches [1-5] have generally been applied to study them using factorization hypothesis. It involves the expansion of the transition amplitudes in terms of a few invariant form factors which provide essential information on the structure of the mesons and the interplay of the strong and weak interactions. This scheme has earlier been employed to study the weak hadronic decays of B-meson to s-wave mesons [5-12]. B-mesons, being heavy, can also emit heavier mesons such as p-wave mesons, which have attracted theoretical attention recently. However, there exist a few works on the hadronic B decays [13-17] that involve a tensor meson in the final state using the frameworks of flavor SU(3) symmetry and the generalized factorization. In the next few years new experimental data on rare decays of B mesons would become available from the B factories such as Belle, Babar, BTeV, LHC. It is expected that improved measurements or new bounds will be obtained on the branching ratios for various decay modes and many decay modes with small branching ratios may also be observed for the first time. In this paper, we analyze two-body hadronic decays of B − , B 0 and B s0 mesons to pseudoscalar (P (0-)) meson and tensor (T (2+)) meson, for whom the experiments have provided the following branching ratios [18,19]: B ( B − → π − D20 ) = (7.8 ± 1.4) × 10−4 , B ( B − → π − f 2 ) = (8.2 ± 2.5) × 10 −6 , 0.4 B ( B − → K − f 2 ) = (1.3+−0.5 ) × 10 −6 ,
B ( B − → ηK 2− ) = (9.1 ± 3.0) × 10−6 , B ( B 0 → ηK 20 ) = (9.6 ± 2.1) × 10−6 , B ( B 0 → D 0 f 2 ) = (1.2 ± 0.4) ×10 −4 , B ( B 0 → π ∓ a2± ) =< 3.0 × 10 −4 , −
−
0 2 − + s 2
−6
B ( B → π K ) = < 6.9 × 10 , B ( B 0 → D a ) = < 1.9 × 10 −4 , B ( B 0 → π + K 2− ) = < 1.8 × 10 −5 , B ( B 0 → π − D2+ ) =< 2.2 × 10 −3 .
2
(1)
Employing the factorization scheme, we calculate the decay amplitudes for CKM-favored and CKM-suppressed modes involving b → c and b → u transitions in the Isgur-ScoraGrinstein-Wise (ISGW II) model [2, 3]. In general, W-annihilation and W-exchange diagrams may also contribute to these decays under consideration. Normally, such contributions are expected to be suppressed due to the helicity and color arguments and are neglected in this work. The paper is organized as follows: In Sec. II, we present meson spectroscopy. Methodology for calculating B → PT is provided in Sec. III. Sec. IV deals with numerical results and discussions. Summary and conclusions are given in the last section.
II. MESON SPECTROSCOPY Experimentally [18], the tensor meson sixteen-plet comprises of an isovector a 2 (1.318) , strange isospinor K 2* (1.429) , charm SU(3) triplet D2* (2.457) , Ds*2 (2.573) and three isoscalars f 2 (1.275) , f 2′(1.525) and χ c 2 (3.555) . These states behave well with respect to the quark model assignments, though the spin and parity of the charm isosinglet Ds*2 (2.573) remain to be confirmed. The numbers given within parentheses indicate mass (in GeV units) of the respective mesons. χ c 2 (3.555) is assumed to be pure (cc ) state, and mixing of the isoscalar states is defined as:
f 2 (1.275) = f 2' (1.525)
1 (uu + dd ) cos φT + ( ss ) sin φT , 2
1 (uu + dd ) sin φT − ( ss ) cos φT , 2
(2)
where φT = θ (ideal ) − θ T ( physical ) and θT ( physical ) = 27 [18]. Similarly, for η and η ′ states of well established pseudoscalar sixteen-plet, we use 1 (uu + dd ) sin φP − ( ss ) cos φ P , 2 1 η ′(0.958) = (uu + dd ) cos φ P + ( ss ) sin φ P , 2
η (0.547) =
(3)
where φ p = θ (ideal ) − θ p ( physical ) and we take θ P ( physical ) = − 15.4 [18]. ηc is taken as
ηc (2.979) = (cc ) .
3
(4)
III. METHODOLOGY A. Weak Hamiltonian For bottom changing ∆b = 1 decays, the weak Hamiltonian involves the bottom changing current, J µ = (c b)Vcb + (u b)Vub , where (qi q j ) ≡ qi γ µ (1 − γ 5 )q j denotes the weak V-A current. Hamiltonian is then given below: i)
(5) QCD modified weak
for decays involving b → c transition,
HW =
GF 2
{VcbVud* [a 1 (cb)(du ) + a2 (db)(cu )] + VcbVcs* [a 1 (cb)( sc) + a2 ( sb)(cc)] +
(6a)
* cb us
V V [a 1 (cb)( su ) + a2 ( sb)(cu )] + VcbVcd* [a 1 (cb)(dc) + a2 (db)(cc)]}, ii)
for decays involving b → u transition, HW =
GF 2
{VubVcs* [a 1 (ub)( sc) + a2 ( sb)(uc)] + VubVud* [a 1 (ub)(du ) + a2 (db)(uu )] +
(6b)
* ub us
V V [a 1 (ub)( su ) + a2 ( sb)(uu )] + VubVcd* [a 1 (ub)(dc) + a2 (db)(uc)]}, where qi q j ≡ qiγ µ (1 − γ 5 )q j denotes the weak V-A current and Vij are the well-known CKM matrix elements, a1 and a2 are the QCD coefficients. By factorizing matrix elements of the four-quark operator contained in the effective Hamiltonian (6), one can distinguish three classes of decays [20]: •
class I transition caused by color favored diagram: the corresponding decay 1 amplitudes are proportional to a1 , where a1 ( µ ) = c1 ( µ ) + c 2 ( µ ) , and N c is Nc the number of colors.
4
•
•
class II transition caused by color suppressed diagram: the corresponding decay amplitudes in this class are proportional to a2 i.e. for the color suppressed modes 1 a 2 (µ ) = c2 (µ ) + c1 ( µ ). Nc class III transition caused by both color favored and color suppressed diagrams: these decays experience the interference of color favored and color suppressed diagrams.
We follow the general convention of large N c limit to fix the QCD coefficients a1 ≈ c1 and a 2 ≈ c 2 , where [20,21]:
c1 ( µ ) = 1.12 , c2 ( µ ) = −0.26 at µ ≈ mb2 .
(7)
B. Decay Amplitudes and Rates The decay rate formula for B → PT decays is given by 2
mB pC5 2 Γ ( B → P T ) = A( B → P T ) , 2 mT 12 π mT
(8)
where pC 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. The factorization scheme in general expresses the weak decay amplitude as the G product of matrix elements of weak currents (up to the weak scale factor of F × CKM 2 elements × QCD factor),
PT HW B ≈ P J µ 0 T J µ B + T J µ 0 P J µ B . However, the matrix elements
T (qµ ) J µ 0
(9)
vanish due to the tracelessness of the
polarization tensor ∈µυ of spin 2 meson and the auxiliary condition Remaining matrix elements are expressed as:
q µ ∈µυ = 0 [19].
P (k µ ) J µ 0 = −if P k µ , T ( PT ) J µ B ( PB ) = ih ∈µυλρ ∈*υα PBα ( PB + PT )λ ( PB − PT ) ρ + k ∈*µυ PBυ * + b+ (∈αβ PBα PBβ )[( PB + PT ) µ + b− ( PB − PT ) µ ],
5
(10)
in the ISGW model [3] which yields
PT H W B = −if P (∈*µυ PBµ PBυ ) F B →T (mP2 ),
(11)
where
F B →T (mP2 ) = k (mP2 ) + (mB2 − mT2 )b+ (mP2 ) + m 2p b− (mP2 ).
(12)
Thus A( B → P T ) =
GF × (CKM factors × QCD factors × CG factors ) × f P F B →T (mP2 ) . (13) 2
C. Form Factors in the ISGW II Model
The form factors have the following expressions in the ISGW II quark model, for B → T transitions [3]: k=
b+ + b− =
md 2β B
) F5( k ) , (1 + ω
md md βT2 ( b+ +b− ) βT2 1 F , − 2 2 5 4 2md mb m Bβ B β BT 2m B β BT
b+ − b− = −
md 2mb m T β B
(14)
md mb βT2 βT2 md βT2 ( b+ −b− ) 1 F , + − 1 − 2 2 2 5 2µ + m B β BT 4β BT 2m B β BT
where m −1 m 1 F5( k ) = F5 ( ~ B ) 2 ( ~T ) 2 , mB mT m −5 m 1 F5( b+ + b− ) = F5 ( ~ B ) 2 ( ~T ) 2 , mB mT
(15)
m −3 m − 1 F5( b+ −b− ) = F5 ( ~ B ) 2 ( ~T ) 2 , mB mT
t (≡ q 2 ) dependence is given by
− 1= ω
tm − t , 2mB mT 6
(16)
and the common scale factor 1/ 2
m F5 = T m B
βT β B 2 βBT
5/ 2
−3
1 2 , 1 + 18 h (tm − t )
(17)
16 33 − 2n f
(18)
where h2 =
3md2 3 1 + + 2 4mb mq 2mB mT β BT mB mT
α S (µQM ) ], ln[ α m ( ) S q
and
β2BT =
1 2 βB + βT2 . 2
(
)
~ is the sum of the mesons constituent quarks masses, m is the hyperfine averaged m physical masses, nf is the number of active flavors, which is taken to be five in the present case, tm = (mB − mT ) 2 is the maximum momentum transfer and −1
1 1 µ+ = + , m m b d
(19)
Here, md is the spectator quark mass in the decaying particle. For Bs → T transitions, md is replaced with ms . We take the following constituent quark masses (in GeV): mu = md = 0.33, ms = 0.55, mc = 1.82, mb = 5.20,
(20)
which are taken from the ISGW II model [3] in which treats mesons as composed of the constituent quarks. Values of the parameter β for different s-wave and p-wave mesons are given in the Table I. We obtain the form factors describing B → T transitions which are given in Table II at q2 = tm. For the sake of comparison, the form factors obtained in the ISGW I quark model [2], are given in parentheses in Table II.
IV. NUMERICAL RESULTS AND DISCUSSIONS Sandwiching the weak Hamiltonian (6) between the initial and final states, we obtain decay amplitudes of B − , B 0 and B s0 mesons for various decay modes as given in the Tables III, IV, V(a) and V(b). For numerical calculations, we use the following values of the decay constants (given in GeV) of the pseudoscalar mesons [13, 18, 21]: f π = 0.131 , f K = 0.160 , f D = 0.223 , f Ds = 0.294 , f η = 0.133 , f η ′ = 0.126 and f ηc = 0.400 .
7
(21)
Finally, we calculate branching ratios of B-meson decays in CKM-favored and CKMsuppressed modes involving b → c and b → u transitions. The results are given in column III of the Tables VI, VII, VIII(a) and VIII(b) for various possible modes. We make the following observations:
1. B → PT decays involving b → c transition a)
∆b = 1, ∆C = 1, ∆S = 0 mode : i. Calculated branching ratio B ( B − → π − D20 ) = 6.7×10-4 agrees well with
the
experiment
value
[19]
(7.8 ± 1.4) × 10−4 ,
and
-4
B ( B 0 → π − D2+ ) = 6.1×10 , is well below the experimental upper
limit < 2.2 × 10 −3 . ii. Branching ratios of other dominant modes are B ( B − → D 0 a2− ) = 1.8×10-4, B ( Bs0 → π − Ds+2 ) = 7.1×10-4, and B ( Bs0 → D 0 K 20 ) = 1.1×10-4. We hope that these values are within the reach of the furure experiments. iii. Decays B 0 → D 0 a 20 and B 0 → D 0 f 2 have branching ratios of the order of 10-5, since these involve color-suppressed spectator process. The branching ratio of B 0 → D 0 f 2′ decay is further suppressed due to the f 2 − f 2′ mixing being close to the ideal mixing. B 0 → π 0 D20 / ηD20 / η ′D20 / D + a2− / Ds+ K 2− / K − Ds+2
iv. Decays 0 s
0
0 2
+ s
and
− 2
B → K D / D a are forbidden in the present analysis due to the vanishing matrix element between the vacuum and tensor meson. However, these may occur through an annihilation mechanism. The decays B 0 → π 0 D20 / D + a2− may also occur through elastic final state interactions (FSIs).
b) ∆b = 1, ∆C = 0, ∆S = −1 mode : i. Dominant modes are found to have branching ratios: B ( B − → Ds− D20 ) = 6.8×10-4, B ( B − → ηc K 2− ) = 1.4×10-4,
8
= 6.4×10-4, B ( B 0 → ηc K 20 )
B ( B 0 → Ds− D2+ )
1.3×10-4,
=
B ( Bs0 → Ds− Ds−2 ) = 7.7×10-4 and B ( Bs0 → ηc f 2′) = 1.3×10-4.
ii.
Decays
B − → D 0 Ds−2 / Ds− D20 / K − χ c 2 (1P ) ,
/ Ds− D2+ / K − χ c 2 (1P )
B 0 → D + Ds−2
Bs0 → π 0 χ c 2 / ηχ c 2 / η ′χ c 2 / D + D2−
and
/ D 0 D20 / Ds+ Ds−2 / D − D2+ / D 0 D20 / ηc a20 are forbidden in our work. Penguin
diagrams
may
cause
B − → D 0 Ds−2 / Ds− D20 and
B 0 → D + Ds−2 / Ds− D2+ decays, however these are likely to remain suppressed as these decays require cc pair to be created.
c)
∆b = 1, ∆C = 0, ∆S = 0 mode : i. For dominant decays, we predict B ( B − → D − D20 ) = 2.5×10-5, B ( B 0 → D − D2+ ) = 2.4×10-5 and B ( Bs0 → D − Ds+2 ) = 2.9×10-5.
ii.
B − → D 0 D2− / π − χ c 2 (1P ) ,
Decays
B 0 → D 0 D20 / Ds− Ds+2
/ D + D2− / D 0 D20 / Ds+ Ds−2 / π 0 χ c 2 (1P ) / ηχ c 2 (1P ) /η ′χ c 2 (1P ) 0 s
0
+ s
and
− 2
B → K χ c 2 / D D are forbidden in our analysis. Annihilation diagrams, elastic FSI and penguin diagrams may generate these decays to the naked charm mesons. However, decays emitting charmonium χ c 2 (1P ) remains forbidden in the ideal mixing limit.
d)
∆b = 1, ∆C = 1, ∆S = −1 mode : i. Branching ratios of the dominant decays are B ( B − → K − D20 ) = 4.8×10-5, B ( B 0 → K − D2+ ) 5.2×10-5. ii. Decays
= 4.5×10-5 and B ( Bs0 → K − Ds+2 ) =
B 0 → K 0 D20 / D + K 2−
and
Bs0 → π− D2+ / π0 D20 / ηD20
/ η ′D20 / D + a 2− / D 0 a 20 / Ds+ K 2− are forbidden in our analysis. Annihilation diagrams do not contribute to these decays. However, these may acquire nonzero branching ratios through elastic FSI.
9
2. B → PT decays involving b → u transition a)
∆b = 1, ∆C = 0, ∆S = 0 mode : i.
B ( B − → π − f 2 ) = 7.1×10-6 is in good agreement with the
experimental value (8.2±2.5)×10-6 and B ( B 0 → π − a 2+ ) = 1.3×10-5 is well below the experimental upper limit < 3.0 ×10 −4 . ii.
B − → K 0 K 2− / K − K 20 ,
B 0 → K + K 2− / K 0 K 20 / K 0 K 20 / K 0 K 20 /
K − K 2+ / π + a2− and Bs0 → K + a 2− / K 0 a 20 / K 0 f 2 / K 0 f 2′ are forbidden in the present analysis. Annihilation process and FSIs may generate these decays.
iii.
B − → π − K 20 and B 0 → π + K 2− are also forbidden in the present analysis which may be generated through annihilation diagram or elastic FSI.
b) ∆b = 1, ∆C = −1, ∆S = −1 mode : i. Branching ratios B ( B − → Ds− a20 ) = 2.0×10-5, B ( B − → Ds− f 2′) = 2.2×10-5, B ( B 0 → Ds− a2+ ) = 3.8×10-5 and B ( Bs0 → Ds− K 2+ ) = 2.6×10-5 have relatively large branching ratios. ii. Decays
B − → π 0 Ds−2 / ηDs−2 / η ′Ds−2 / K 0 D2− / K − D20 / D − K 20 ,
B 0 → K 0 D20 / π + Ds−2 and Bs0 → K + Ds−2 / π + D2− / π 0 D20 / ηD20 / η ′D20
/ D 0 a20 / D − a2+ are forbidden in the present analysis. Annihilation and FSIs may generate these decays.
c)
∆b = 1, ∆C = −1, ∆S = 0 mode : i.
Branching ratios of B ( B 0 → D 0 f 2 ) = 3.6×10-8 is smaller than the experimental value (1.2 ± 0.4) ×10 −4 . It may be noted that Wannihilation and W-exchange diagrams may also contribute to the B decays under consideration. Normally, such contributions are expected to be suppressed due to the helicity and color arguments. Including the factorizable contribution of such diagrams, the decay
10
amplitude of B 0 → D 0 f 2 get modified to (leaving aside the scale G factor F VubVcd* ) 2 A( B 0 → D 0 f 2 ) =
1 a2 f D cos φT F B → f 2 (mD2 ) + 2 1 a2 f B cos φT F f2 → D (mB2 ) . (22) 2
Using f B = 0.176 GeV, we find that the experimental branching ratio B ( B 0 → D 0 f 2 ) requires F f 2 → D (mB2 ) = -9.99 GeV. This in turn enhances the branching ratio for B − → D − f 2 to 1.2×10-4. ii.
Dominant decay is B( B 0 → D − a 2+ ) = 1.2×10-6 and next order dominant decays are B( B − → D − f 2 ) = 6.9×10-7 B( B − → D − a 20 ) = 6.5×10-7 and B( B s0 → D − K 2+ ) = 8.3×10-7.
iii.
B − → K 0 Ds−2 / π0 D2− / π− D20 / ηD2− / η′D2− / Ds− K 20 / ηc D2− ,
Decays
B 0 → K + Ds−2 / π + D2− / π 0 D20 / ηD20 / η ′D20 / Ds− K 2+ / ηc D2− 0 s
0
and
0 2
B → K D are forbidden in the present analysis. Annihilation diagrams may generate these decays.
d)
∆b = 1, ∆C = 0, ∆S = −1 mode : i. B ( B − → K − f 2 ) = 0.54×10-6 is smaller than the experimental value −6 (1.3+−0.4 0.5 ) × 10 . This decay mode is also likely to have contribution from the W-annihilation and W-exchange processes. Including the factorizable contribution of such diagrams, the decay amplitudes of B − → K − f 2 get modified to (putting aside the scale factor GF VubVus* ) 2
A( B − → K − f 2 ) =
11
1 a1 f K cos φT F B → f 2 (mK2 ) + 2 1 a1 f B cos φT F f2 → K (mB2 ) . 2
(23)
As it is not possible to evaluate the form factor F f2 → K at mB2 even in the phenomenological models, it is treated as a free parameter. Taking f B = 0.176 GeV, we find that the experimental branching −6 requires F f2 → K (mB2 ) = ratio B ( B − → K − f 2 ) = (1.3+−0.4 0.5 ) × 10 0.083 GeV. This value in turn enhances the branching ratio for B − → K − f 2 through the W-annihilation contibution to 1.3×10-6.
ii.
Branching ratios of B ( B − → ηK 2− ) = 1.2×10-8 is small than the experimental value (9.1 ± 3.0) × 10−6 . Similar to B − → K − f 2 decay, this decay mode is also likely to have contribution from the Wannihilation and W-exchange processes. Including the factorizable contribution of such diagrams, the decay amplitudes of B → ηK 2 G get modified to (leaving aside the scale factor F VubVus* ) 2 1 a2 fη sin φ P F B → K 2 (mη2 ) + 2 1 a2 f B sin φ P F K 2 →η (mB2 ) 2 1 A( B 0 → ηK 20 ) = a2 fη sin φ P F B → K 2 (mη2 ) + 2 1 a2 f B sin φ P F K 2 →η (mB2 ) . 2
A( B − → ηK 2− ) =
(24)
For f B = 0.176 GeV, we find that the experimental branching ratio B ( B − → ηK 2− ) = (9.1 ± 3.0) × 10−6 requires F K 2 →η (mB2 ) =
- 3.03
0
GeV. This in turn enhances the branching ratio for B → ηK 20 to 8.1×10-6, which is consistent with the experimental value (9.6 ± 2.1) × 10−6 . iii.
Decays B − → π − K 20 / K 0 a2− , B 0 → π + K 2− / K 0 a20
/ K 0 f 2 / K 0 f 2′ and Bs0 → K + K 2− / K 0 K 20 / π + a2− / π 0 a20 / π − a2+ / ηa20 / K 0 K 20 / η ′a20 are forbidden in the present analysis. Annihilation and FSIs may generate these decays.
12
V. SUMMARY AND CONCLUSIONS In this paper, we have studied hadronic weak decays of bottom mesons emitting pseudoscalar and tensor mesons. The matrix elements T (qµ ) J µ 0 vanish due to the tracelessness of the polarization tensor ∈µυ of spin 2 meson and the auxiliary condition
q µ ∈µυ = 0 . Therefore, either color-favored or color-suppressed diagrams contribute. Therefore, the analysis of these decays is free of constructive or destructive interference for color-favored and color-suppressed diagrams. We employ ISGW II model [3] to determine the B → T form factors appearing in the decay matrix element of weak currents involving b → c and b → u transitions. Consequently, we have obtained the decay amplitudes and calculated the branching ratios of B → PT decays in CKMfavored and CKM-suppressed modes. We make the following conclusions: Decays involving b → c transition have larger branching ratios of the order of 10-4 to 10-8 and decays involving b → u transition have branching ratios of the order of 10-5 to 10-11. Dominant decay modes involving b → c transition are B( B − → Ds− D20 ) = 6.8×10-4, B( B − → π − D20 ) = 6.7×10-4, B( B 0 → Ds− D2+ ) = 6.4×10-4, B( B 0 → π − D2+ ) = 6.1×10-4, B( B − → D 0 a 2− ) = 1.8×10-4,
B( B − → ηc K 2− ) = 1.4×10-4, B( B 0 → η c K 20 ) =
1.3×10-4, B( Bs0 → Ds− Ds−2 ) = 7.7×10-4, B( B s0 → π − D s+2 ) = 7.1×10-4, B( B s0 → η c f 2′ ) = 1.3×10-4 and B( B s0 → D 0 K 20 ) = 1.1×10-4. Experimentally, the branching ratios of only five decay modes are measured and upper limits are available for six other decays. We find that the calculated branching ratio B ( B − → π − f 2 ) = 7.1×10-6 is in good agreement with the experimental value (8.2±2.5)×10-6 whereas B( B − → K − f 2 ) = 5.4×10-7 is smaller −6 than the experimental value (1.3+−0.4 0.5 ) × 10 . B -decay requires contribution from Wannihilation diagram to bridge the gap between theoretical and experimental value. In contrast to the charm meson decays, the experimental data show constructive interference for B meson decays involving both the color-favored and color-suppressed diagrams, giving a1 = 1.10 ± 0.08 and a 2 = 0.20 ± 0.02 . In the present analysis, the decay amplitude is proportional to only one QCD coefficient either a1 (for color favored
diagram) or a2 (for color suppressed diagram), therefore our results remains unaffected from the interference pattern. We also compare our results with branching ratios calculated in the other models [17,23,24]. The predicted branching ratios in KLO [17] shown in 3rd column of tables VI, VII, VIII(a) and VIII(b) are generally smaller as compared to the present branching ratios because of the difference in the form factors since different quark masses have been used in the two works. Branching ratios have also been calculated by Cheng [23]. His predictions B( B − → π − D20 ) = 6.7×10-4 and B( B 0 → π − D2+ ) = 6.1×10-4 match well with the numerical branching ratios obtained in the present work. However, the other branching ratios B( B − → Ds− D20 ) = 4.2×10-4, B( B 0 → Ds− D2+ ) = 3.8×10-4 and
13
B( B s0 → π − D s+2 ) = 3.8×10-4 are different from our results owing to the different values
used for the decay constant f Ds . MQ [24] have recently studied few charmless decays of B → PT mode. Some of the branching ratios are smaller than our numerical value of branching ratios while the others are large as compare to the present predictions, particularly, for η, η′ emitting decays. The disagreement with their predictions may be attributed due to the difference in the form factors obtained in covariant light-front approach (CLF) and inclusion of the non-factorizable contributions in their results. It may be noted that the form factors at small q 2 obtained in the CLF and ISGW II quark model
agrees within 40% [3]. However, when q 2 increases h(q 2 ) , b+ (q 2 ) and b− (q 2 ) increases more rapidly in the light front model than in the ISGW II model. Another important fact is that the behavior of the form factor k in both models is different. The Belle collaboration is currently searching for some B → PT modes and their preliminary results indicate that the branching ratios for these may not be very small compared to B → PP modes. We hope our predictions would be within the reach of the current experiments. Observation of these decays in the B experiments such as Belle, Babar, BTeV, LHC and so on will be crucial in testing the ISGW II and other quark models as well as validity of the factorization scheme.
Acknowledgement One of the authors (N. S.) is thankful to the University Grant Commission, New Delhi, for the financial assistance.
References
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5. T. Mannel, W. Roberts and Z. Razak, Phys. Lett. B 248 (1990) 392; M.J. Dugan and B. Grinstein, ibid. 255 (1991) 583, and references therein; A Ali and T. Mannel, ibid. 264 (1991) 447; G. Kramer, T. Mannel and W.F. Palmer, Z. Phys. C 55 (1992) 497; A.F. Falk, M.B. Wise and I. Dunietz, Phys. Rev. D 51 (1995) 1183. 6. M. Neubert, V. Rieckert, B. Stech, and Q. P. Xu, in Heavy Flavours, edited by A.J. Buras and H. Lindner, World Scientific, Singapore, 1992 and references therein; S. Stone, talk presented at 5th Int. Symposium on Heavy Flavor Physics, Montrêal, July 6-10, 1993. 7. T.E. Browder and K. Honscheid, Prog. Part. Nucl. Phys. 35 (1995) 81, and references therein. 8. M. Gourdin, A.N. Kamal and X.Y. Pham, Phys. Rev. Lett. 73 (1994) 3197; A.N. Kamal and T.N. Pham, Phys. Rev. D 50 (1994) 395; M. Gourdin et al., Phys. Lett. B 333 (1994) 507. 9. M.S. Alam et al., Phys. Rev. D 50 (1994) 43. 10. S.M. Sheikholeslami and R.C. Verma, Int. J Mod. Phys. A 7 (1992). 11. G. Kramer and W.F. Palmer, Phys. Rev. D 46 (1992) 3197. 12. C.E. Carlson and J. Milana, Phys. Rev. D 49 (1994) 5908. 13. A.C. Katoch and R.C. Verma, Phys. Rev. D 52 (1995) 1717; 55 (1997) 7316 (E). 14. G. Lopez Castro and J.H. Munoz, Phys. Rev. D 55 (1997) 5581. 15. J.H. Munoz, A.A. Rojas and G. Lopez Castro, Phys. Rev. D 59 (1999) 077504. 16. C.S. Kim, B.H. Lim and Sechul Oh, Eur. Phys. J. C 22 (2002) 683; Eur. Phys. J. C 22 (2002) 683; C.S. Kim, J.P. Lee and Sechul Oh, hep-ph/0205262; Phys. Rev. D 67 (2003) 014002. 17. C.S. Kim, J.P. Lee and Sechul Oh, Phys. Rev. D 67 (2003) 014011. 18. C. Amsler et al. (Particle Data Group), Phys. Lett. B 667 (2008). 19. Belle Collaboration, K. Abe et al., Phys. Rev. D 69 (2004) 112002. 20. Sechul Oh, Phys. Rev. D 60 (1999) 034006; M. Grounau and J.L. Rosner, ibdi. 61, (2000) 073008.
15
21. D. Spehler and S.F. Novaes, Phys. Rev. D 44 (1991) 3990. 22. Rohit Dhir, Neelesh Sharma and R.C. Verma, J. Phys. G: Nucl. Part. Phys. 35 (2008) 085002. 23. H.Y Cheng, Phys. Rev. D 68 (2003) 094005. 24. J. Muñoz and N. Quintero, J. Phys. G: Nucl. Part. Phys. 36 (2009) 095004.
Table I. The parameter β for s-wave and p-wave mesons in the ISGW II model
Quark content β S (GeV)
ud
us
ss
cu
cs
ub
sb
0.41
0.44
0.53
0.45
0.56
0.43
0.54
β P (GeV)
0.28
0.30
0.33
0.33
0.38
0.35
0.41
Table II. Form factors of B → T transition at q2 = tm in the ISGW II quark model
Transition B → a2
k
b+
b-
0.432
-0.013
0.015
B → f2 B → f 2′
0.425
-0.014
0.014
0.533
-0.013
0.015
B → K2
0.480
-0.015
0.015
B → D2
0.677
-0.013
0.013
Bs → f 2
0.423
-0.015
0.016
Bs → f 2′
0.572
-0.016
0.017
Bs → K 2
0.492
-0.013
0.015
Bs → Ds 2
0.854
-0.015
0.016
16
Table III. Decay amplitudes of B → PT decays in CKM-favored mode involving b → c transition
Decay
Amplitude G × F VcbVud* 2
∆b = 1, ∆C = 1, ∆S = 0 B − → π − D20
a1 f π F B→ D2 (mπ2 )
B − → D 0 a 2−
a2 f D F B → a2 (mD2 )
B 0 → π − D2+
a1 f π F B→ D2 (mπ2 )
B 0 → D 0 a 20 B 0 → D0 f2 B 0 → D 0 f 2′
1 a2 f D F B → a2 (mD2 ) 2
−
1 a2 f D cos φT F B → f 2 (mD2 ) 2 1 a2 f D sin φT F B → f 2′ (mD2 ) 2
B s0 → π − D s+2
a1 f π F Bs → Ds 2 (mπ2 )
B s0 → D 0 K 20
a2 f D F Bs → K 2 (mD2 )
∆b = 1, ∆C = 0, ∆S = −1 −
− s
0 2
B →D D
×
GF
2
VcbVcs*
a1 f Ds F B → D2 (mD2 s )
B − → ηc K 2−
a2 fηc F B → K 2 (mη2c )
B 0 → Ds− D2+
a1 f Ds F B → D2 (mD2 s )
B 0 → η c K 20
a2 fηc F B → K 2 (mη2c )
Bs0 → Ds− Ds+2
a1 f Ds F Bs → Ds 2 (mD2 s )
Bs0 → η c f 2
a2 fηc sin φT F Bs → f 2 (mη2c )
B s0 → η c f 2′
- a2 fηc cos φT F Bs → f 2 (mη2c )
′
17
Table IV. Decay amplitudes of B → PT decays in CKM-suppressed mode involving b → c transition
Decay
Amplitude G × F VcbVus* 2
∆b = 1, ∆C = 1, ∆S = −1 B − → K − D20
a1 f K F B → D2 (mK2 )
B − → D 0 K 2−
a2 f D F B → K 2 (mD2 )
B 0 → K − D2+
a1 f K F B → D2 (mK2 )
B 0 → D 0 K 20
a2 f D F B → K 2 (mD2 )
Bs0 → K − Ds+2
a1 f K F Bs → Ds 2 (mK2 )
Bs0 → D 0 f 2
a2 f D sin φT F Bs → f 2 (mD2 )
B s0 → D 0 f 2′
− a2 f D cos φT F Bs → f 2′ (mD2 )
∆b = 1, ∆C = 0, ∆S = 0 −
−
×
0 2
GF
2
VcbVcd*
B →D D
a1 f D F B → D2 (mD2 )
B − → η c a 2−
a2 fηc F B →a2 (mη2c )
B 0 → D − D2+
a1 f D F B → D2 (mD2 )
B 0 → η c a 20 B 0 → ηc f 2 B 0 → η c f 2′
-
1 a2 f ηc F B →a2 (mη2c ) 2
1 a2 fηc cos φT F B → f 2 (mη2c ) 2 1 a2 fηc sin φT F B → f 2′ (mη2c ) 2
Bs0 → D − Ds+2
a1 f D F Bs → Ds 2 (mD2 )
Bs0 → η c K 20
a2 f ηc F Bs → K 2 (mη2c )
18
Table V (a). Decay amplitudes of B → PT decays involving b → u transition
Decay
Amplitude G × F VubVcs* 2
∆b = 1, ∆C = −1, ∆S = −1 B − → D 0 K 2−
a2 f D F B → K 2 (mD2 )
B − → Ds− a 20
1 a1 f Ds F B → a2 (mD2 s ) 2
B − → Ds− f 2
1 a1 f Ds cos φT F B → f 2 (mD2 s ) 2 1 a1 f Ds sin φT F B → f 2′ (mD2 s ) 2
B − → D s− f 2′ B 0 → D 0 K 20
a2 f D F B → K 2 (mD2 )
B 0 → Ds− a 2+
a1 f Ds F B → a2 (mD2 s )
Bs0 → D 0 f 2
a2 f D sin φT F Bs → f 2 (mD2 )
Bs0 → D 0 f 2′
− a2 f D cos φT F Bs → f 2 (mD2 )
Bs0 → Ds− K 2+
a1 f Ds F Bs → K 2 (mD2 s )
∆b = 1, ∆C = 0, ∆S = 0 −
0
B →π a
− 2
B − → ηa 2− B − → η ′a 2− B − → π − a 20 B− → π − f2 B − → π − f 2′
×
GF
2
VubVud*
1 a2 f π F B→ a2 (mπ2 ) 2 1 a2 fη sin φ P F B →a2 (mη2 ) 2 1 a2 fη ′ cos φ P F B →a2 (mη2′ ) 2 1 a1 fπ F B→a2 (mπ2 ) 2 1 a1 fπ cos φT F B → f 2 (mπ2 ) 2 1 a1 f π sin φT F B → f 2′ (mπ2 ) 2
B 0 → π − a 2+
a1 f π F B→a2 (mπ2 )
B 0 → π 0 a 20
1 − a2 f π F B→ a2 (mπ2 ) 2
19
B 0 → π 0 f2 B 0 → π 0 f 2′ B 0 → ηa 20 B 0 → ηf 2 B 0 → ηf 2′ B 0 → η ′a 20 B 0 → η ′f 2 B 0 → η ′f 2′ B s0 → π 0 K 20
1 a2 f π cos φT F B → f 2 (mπ2 ) 2 1 a2 fπ sin φT F B → f 2′ (mπ2 ) 2 1 − a2 fη sin φ P F B →a2 (mη2 ) 2 1 a2 fη sin φ P cos φT F B → f 2 (mη2 ) 2 1 a2 fη sin φP sin φT F B → f 2′ (mη2 ) 2 1 − a2 fη ′ cos φ P F B →a2 (mη2′ ) 2 1 a2 fη ′ cos φ P cos φT F B → f 2 (mη2′ ) 2 1 a2 fη ′ cos φ P sin φT F B → f 2′ (mη2′ ) 2 1 a2 fπ F Bs → K 2 (mπ2 ) 2
Bs0 → π − K 2+
a1 f π F Bs → K 2 (mπ2 )
B s0 → ηK 20
1 a2 fη sin φ P F Bs → K 2 (mη2 ) 2 1 a2 fη ′ cos φ P F Bs → K 2 (mη2′ ) 2
Bs0 → η ′K 20
20
Table V (b). Decay amplitudes of B → PT decays involving b → u transition
Decay
∆b = 1, ∆C = 0, ∆S = −1 B − → K − a 20 B− → K − f2
Amplitude G × F VubVus* 2 1 a1 f K F B → a2 (mK2 ) 2
1 a1 f K cos φT F B → f 2 (mK2 ) 2 1 a1 f K sin φT F B → f 2′ (mK2 ) 2 1 a2 fπ F B→ K 2 (mπ2 ) 2 1 a2 fη sin φ P F B → K 2 (mη2 ) 2 1 a2 fη ′ cos φ P F B → K 2 (mη2′ ) 2
B − → K − f 2′ B − → π 0 K 2− B − → ηK 2− B − → η ′K 2− B 0 → K − a 2+
a1 f K F B →a2 (mK2 )
B 0 → π 0 K 20
1 a2 fπ F B→ K 2 (mπ2 ) 2
B 0 → ηK 20
1 a2 fη sin φ P F B → K 2 (mη2 ) 2 1 a2 fη ′ cos φ P F B → K 2 (mη2′ ) 2 1 a2 fπ sin φT F Bs → f 2 (mπ2 ) 2 1 − a2 f π cos φT F Bs → f 2′ (mπ2 ) 2 1 a2 fη sin φ P sin φT F Bs → f 2 (mη2 ) 2
B 0 → η ′K 20 B s0 → π 0 f 2 Bs0 → π 0 f 2′ Bs0 → ηf 2 Bs0 → K − K 2+
a1 f K F Bs → K 2 (mK2 )
Bs0 → ηf 2′
1 a2 f η sin ϕP cos ϕT F Bs → f2′ (mη2 ) 2 1 a2 fη ′ cos φ P sin φT F Bs → f 2 (mη2′ ) 2 1 − a2 fη ′ cos φ P cos φT F Bs → f 2′ (mη2′ ) 2
B s0 → η ′f 2 B s0 → η ′f 2′
-
21
∆b = 1, ∆C = −1, ∆S = 0 −
−
B →D a
×
0 2
B− → D− f2 B − → D − f 2′
B 0 → D 0 f2 B 0 → D 0 f 2′
2
VubVcd*
1 a1 f D F B →a2 (mD2 ) 2 1 a1 f D cos φT F B → f 2 (mD2 ) 2 1 a1 f D sin φT F B → f 2′ (mD2 ) 2
B − → D 0 a 2− B 0 → D 0 a 20
GF
a2 f D F B → a2 (mD2 ) -
1 a2 f D F B → a2 (mD2 ) 2
1 a2 f D cos φT F B → f 2 (mD2 ) 2 1 a2 f D sin φT F B → f 2′ (mD2 ) 2
B 0 → D − a 2+
a1 f D F B →a2 (mD2 )
B s0 → D − K 2+
a1 f D F Bs → K 2 (mD2 )
Bs0 → D 0 K 20
a2 f D F Bs → K2 (mD2 )
22
Table VI. Branching ratios of B → PT decays in CKM-favored mode involving b → c transition
Decay
Branching ratios This Work KLO ∆b = 1, ∆C = 1, ∆S = 0 6.7×10-4 3.5×10-4 B − → π − D20
B − → D 0 a 2−
1.8×10-4
1.0×10-4
B 0 → π − D2+
6.1×10-4
3.3×10-4
B 0 → D 0 a 20
8.2×10-5
4.8×10-4
B 0 → D0 f2
8.8×10-5
5.3×10-5
B 0 → D 0 f 2′
1.7×10-6
0.62×10-6
B s0 → π − D s+2
7.1×10-4
-
-4
-
1.1×10 B s0 → D 0 K 20 ∆b = 1, ∆C = 0, ∆S = −1 6.8×10-4 B − → Ds− D20
4.9×10-4
B − → ηc K 2−
1.4×10-4
1.1×10-4
B 0 → Ds− D2+
6.4×10-4
4.6×10-4
B 0 → η c K 20
1.3×10-4
9.6×10-5
Bs0 → Ds− Ds−2
7.7×10-4
-
Bs0 → η c f 2
2.7×10-6
-
B s0 → η c f 2′
1.3×10-4
-
23
Table VII. Branching ratios of B → PT decays in CKM-suppressed mode involving b → c transition
Decay
Branching ratios This Work KLO ∆b = 1, ∆C = 1, ∆S = −1 4.8×10-5 2.5×10-5 B − → K − D20
B − → D 0 K 2−
8.7×10-6
7.3×10-6
B 0 → K − D2+
4.5×10-5
2.4×10-5
B 0 → D 0 K 20
8.1×10-6
6.8×10-6
Bs0 → K − Ds+2
5.2×10-5
-
Bs0 → D 0 f 2
9.9×10-8
-
6.7×10-6 B s0 → D 0 f 2′ ∆b = 1, ∆C = 0, ∆S = 0 2.5×10-5 B − → D − D20
2.2×10-5
B − → η c a 2−
9.2×10-6
4.9×10-6
B 0 → D − D2+
2.4×10-5
2.1×10-5
B 0 → η c a 20
4.3×10-6
2.3×10-6
B 0 → ηc f 2
4.8×10-6
2.7×10-6
B 0 → η c f 2′
6.7×10-8
0.02×10-6
Bs0 → D − Ds+2
2.9×10-5
-
Bs0 → η c K 20
6.9×10-6
-
24
Table VIII (a). Branching ratios of B → PT decays involving b → u transition Decays
Branching ratios This Work KLO
MQ
∆b = 1, ∆C = −1, ∆S = −1 1.3×10-6 B − → D 0 K 2−
1.2×10-6
-
B − → Ds− a 20
2.0×10-5
9.4×10-6
-
B − → Ds− f 2
2.2×10-5
1.×10-5
-
B − → D s− f 2′
4.3×10
-7
1.2×10
-6
3.8×10
-5
Bs0 → Ds− K 2+
2.6×10
-5
-
-
Bs0 → D 0 f 2
1.5×10-8
-
-
1.0×10-6 Bs0 → D 0 f 2′ ∆b = 1, ∆C = 0, ∆S = 0 6.7×10-6 B − → π − a 20
-
-
2.6×10-6
4.38×10-6 -
B 0 → D 0 K 20 B 0 → Ds− a 2+
0.12×10 1.1×10
-
1.8×10
-5
-
7.1×10-6
-
B − → π − f 2′
-7
-
B − → π 0 a 2−
0.38×10
-6
-
-6
B− → π − f2
1.5×10
-6
-
0.001×10
-6
0.015×10-6
B − → ηa 2−
0.23×10-6
0.29×10-6
45.8×10-6
B − → η ′a 2−
0.13×10-6
1.31×10-6
71.3×10-6
B 0 → π − a 2+
13.0×10-6
4.88×10-6
8.19×10-6
B 0 → π 0 a 20
0.18×10-6
0.0003×10-6
0.007×10-6
B 0 → π 0 f2
1.9×10-7
-
-
B 0 → π 0 f 2′
-9
-
3.9×10
-6
B 0 → ηa 20
0.11×10
B 0 → ηf 2
1.1×10-7
-
B 0 → ηf 2′
-9
-
2.4×10
0.14×10
-6
0.62×10
-6
25.2×10-6 -
-6
43.3×10-6
B 0 → η ′a 20
0.06×10
B 0 → η ′f 2
6.3×10-8
-
-
B 0 → η ′f 2′
1.3×10
-9
-
-
Bs0 → π − K 2+
7.8×10
-6
-
-
B s0 → π 0 K 20
2.2×10-7
-
-
B s0 → ηK 20
1.3×10-7
-
-
Bs0 → η ′K 20
7.5×10-8
-
-
25
Table VIII (b). Branching ratios of B → PT decays involving b → u transition
Decays
Branching ratios This Work
∆b = 1, ∆C = 0, ∆S = −1 0.51×10-6 B − → K − a 20
KLO
MQ
0.31×10-6
0.39×10-6 -
B− → K − f2
5.4×10-7
-
B − → K − f 2′
-8
-
1.5×10
-6
-6
0.15×10-6
B − → π 0 K 2−
0.02×10
B − → ηK 2−
0.01×10-6
0.03×10-6
1.19×10-6
B − → η ′K 2−
0.007×10-6
1.40×10-6
2.70×10-6
B 0 → K − a 2+
0.95×10-6
0.58×10-6
0.73×10-6
B 0 → π 0 K 20
0.02×10-6
0.08×10-6
0.13×10-6
B 0 → ηK 20
0.01×10-6
0.03×10-6
1.09×10-6
B 0 → η ′K 20
0.006×10-6
1.3×10-6
2.46×10-6
Bs0 → K − K 2+
5.9×10-7
-
B s0 → π 0 f 2
1.9×10-10
-
Bs0 → π 0 f 2′
1.4×10-8
-
Bs0 → ηf 2
1.1×10-10
-
Bs0 → ηf 2′
8.3×10-9
-
Bs0 → η ′f 2
6.5×10-11
-
4.7×10-9 B s0 → η ′f 2′ ∆b = 1, ∆C = −1, ∆S = 0 6.5×10-7 B − → D − a 20
-
6.9×10
-7
-
B − → D − f 2′
1.4×10
-7
-
B − → D 0 a 2−
7.3×10-8
-
B 0 → D − a 2+
1.2×10
-6
-
B 0 → D 0 a 20
3.4×10
-8
-
B 0 → D 0 f2
3.6×10-8
-
B 0 → D 0 f 2′
-10
-
8.3×10
-7
-
4.6×10
-8
-
B− → D− f2
B s0 → D − K 2+ Bs0 → D 0 K 20
7.1×10
0.09×10
-
26
