A Theoretical Prediction for Exclusive Decays $B \rightarrow K(K^*) η_c$

6 Pages·0.07 MB·English

by  Mohammad R. Ahmady

A Theoretical Prediction for Exclusive Decays
B K(K )η
∗ c
→
Mohammad R. Ahmady and Roberto R. Mendel
Department of Applied Mathematics
4
9 The University of Western Ontario
9 London, Ontario, Canada
1

n
a
J

5
2

1 ABSTRACT
v
7
2
3
1
0 ∗
The decay rates for the exclusive B decays B Kη and B K η are
4 c c
→ →
9 calculated in the context of the heavy quark effective theory. We obtain
/ ∗ ∗
h Γ(B Kη )/Γ(B Kψ) = 1.6 0.2 and Γ(B K η )/Γ(B K ψ) =
c c
p → → ± → →
0.39 0.04. These results lead to estimates BR(B Kη ) = (0.11
p- 0.02)±% and BR(B K∗η ) = (0.05 0.01)% if we use→the cecntral curren±t
c
e → ∗ ±
h experimental values for B (K,K )ψ branching ratios.
→
:
v
i
X
r
a
Withtherecentprogressinthemeasurement ofsomenonleptonicBdecaychannels[1],
∗
the exclusive decays B K(K )η could be within experimental reach very soon. In
c
→
this paper, we calculate the decay rate of the above processes in the context of the heavy
quark effective theory. The branching ratio for these channels are obtained in terms of
∗
decay rates Γ(B K(K )ψ), which have been measured [1]
→
The effective Hamiltonian relevant to the process b scc¯can be written as [2]:
→
H = Cs¯γ (1 γ )bc¯γ (1 γ )c , (1)
eff µ 5 µ 5
− −
where
G
F ∗
C = (c +c /3)V V .
√2 1 2 cs cb
c and c are QCD improved coefficients, V andV are elements of CKM mixing ma-
1 2 cs cb
trix. Assuming factorization and using the definition for f , the decay constant for the
ηc
psuedoscalar charmonium state:
f qµ =< 0 c¯γµγ c η > ,
ηc | 5 | c
leads to H for B X η , where X is hadronic final state containing a strange quark
eff s c s
→
H = Cf s¯γ (1 γ )bqµ .
eff ηc µ − 5
∗
To calculate the exclusive decay rates Γ(B Kη ) and Γ(B K η ) we need to
c c
→ →
evaluate the following matrix elements:
′
< K(p) s¯γ (1 γ )b B(p) > , (2)
µ 5
| − |
∗ ′
< K (p,ǫ) s¯γ (1 γ )b B(p) > . (3)
µ 5
| − |
Using heavy quark effective theory, i.e. assuming that bottom and strange quark are
both heavy compare to Λ , we can write (2) and (3) in terms of one universal Isgur-
QCD
Wise function [3]:
′ ′ ′
< K(v ) s¯γ (1 γ )b B(v) >= √m m ξ(v.v )(v +v ) , (4)
| µ − 5 | K B µ µ
∗ ′
< K (v ,ǫ) s¯γ (1 γ )b B(v) >=
µ 5
| − | (5)
√mK∗mBξ(v.v′) (1+v.v′)ǫµ −vµ′(ǫ.v)+iǫµαβγvαv′βǫγ .
′ h i
where v and v are velocities of initial and final state mesons respectively. Consequently,
the decay rates are as follows:
C2f2 m 2
Γ(B Kη ) = ηcg(m ,m ,m )m m2 1 K (1+v.v′)2 ξ(v.v′) 2 , (6)
→ c 16π B K ηc K B − m | |
(cid:18) B(cid:19)
1
Γ(B → K∗ηc) = C126fπη2cg(mB,mK∗,mηc)mK∗m2B 1+ mmK∗ 2 v.v′2 −1 |ξ(v.v′)|2 , (7)
(cid:18) B (cid:19)
(cid:16) (cid:17)
where
c2 b2 2 4b2c2 1/2
g(a,b,c) = 1 ,
 − a2 − a2! − a4 
 
and
m2 +m2 m2
v.v′ = B K(∗) − ηc .
2m m
B K(∗)
′ ∗
The value of v.v for B Kη and B K η is 3.69 and 2.04 respectively.
c c
→ →
The decay rates for similar processes involving ψ has been calculated in reference [4]
C2f2
ψ
Γ(B Kψ) = g(m ,m ,m )m
B K ψ K
→ 16π ×
2 (8)
(m m )
′ 2 B K ′ ′ 2
(1+v.v ) − 2(1+v.v ) ξ(v.v) ,
" m2 − #| |
ψ
C2f2
Γ(B K∗ψ) = ψg(mB,mK∗,mψ)mK∗
→ 16π ×
2 (9)
′2 (mB +mK∗) ′ ′ ′ 2
(v.v 1) +2(1+2v.v )(1+v.v ) ξ(v.v ) ,
" − m2 #| |
ψ
where
m2 +m2 m2
v.v′ = B K(∗) − ψ .
2m m
B K(∗)
′ ∗
v.v 3.55 for B Kψ and 2.02 for B K ψ. Using (6), (7), (8) and (9) we
≈ → ≈ →
obtain:
Γ(B Kη ) ξ(v.v′ = 3.69) 2f2
→ c = (13.0GeV2)| | ηc , (10)
Γ(B Kψ) ξ(v.v′ = 3.55) 2 f2
→ | | ψ
Γ(B K∗η ) ξ(v.v′ = 2.10) 2f2
→ c = (3.2GeV2)| | ηc . (11)
Γ(B K∗ψ) ξ(v.v′ = 2.02) 2 f2
→ | | ψ
We would like to point out that although the strange quark is not particularly heavy
and significant corrections to eqns. (4) and (5) are expected, these corrections are likely
to cancel out in the ratios (10) and (11).
The ratio of the Isgur-Wise functions in (10) and (11) are expected to be close to one
as their arguments are nearly equal. For example, taking
9
′ 2 ′ 2
ξ(v.v ) = exp m (1 (v.v ) ) , (12)
"256β2 K(∗) − #
2
which is obtained using the model of Isgur, Wise, Grinstein and Scora [5] and the
parameter β = 0.295 GeV is fixed by the best fit to the measured decay rates B
∗ ′ →
(K,K )+(ψ,ψ ) [6] , this ratio is evaluated to be:
′ 2
ξ(v.v = 3.69)
| | 0.82 ,
ξ(v.v′ = 3.55) 2 ≈
| ′ |2 (13)
ξ(v.v = 2.10)
| | 0.81 ,
ξ(v.v′ = 2.02) 2 ≈
| |
leading to
Γ(B Kη ) f2
→ c = (10.7GeV2) ηc , (14)
Γ(B Kψ) f2
→ ψ
Γ(B K∗η ) f2
→ c = (2.6GeV2) ηc . (15)
Γ(B K∗ψ) f2
→ ψ
The numerical coefficient in (14) and (15) differs from those obtained from pole approx-
imation and BSW model [7]. To evaluate the ratio of the decay constants, we use the
potantialmodelrelations relatingthese constants tothe valueofthemeson wavefunction
at the origin:
12
f = Ψ (0) ,
ηc smηc ηc (16)
f = 12m Ψ (0) .
ψ ψ ψ
q
The wavefunctions of pseudoscalar meson η and vector meson ψ are not expected to be
c
very different. Using a simple perturbation theory argument, we make an estimate [8]
2
Ψ (0)
| ηc | = 1.4 0.1 . (17)
2
Ψ (0) ±
ψ
| |
Consequently, the ratio of the decay constants turns out to be:
f2 1 Ψ (0) 2
ηc = | ηc | 0.15 0.01GeV−2 . (18)
f2 m m Ψ (0) 2 ≈ ±
ψ ηc ψ ψ
| |
Inserting (13) and (15) in (14) and (15) we obtain:
Γ(B Kη )
c
→ = 1.6 0.2 ,
Γ(B Kψ) ±
→ ∗ (19)
Γ(B K η )
c
→ = 0.39 0.04 .
Γ(B K∗ψ) ±
→
3
Using the central values of the measurements [9, 10]
BR(B Kψ) = 0.071% ,
→
and
∗
BR(B K ψ) = 0.135% ,
→
result in our predictions:
BR(B Kη ) = (0.11 0.02)% ,
→ ∗c ± (20)
BR(B K η ) = (0.05 0.01)% .
c
→ ±
In conclusion, we estimated the ratios of the two body nonleptonic decay rates of the
∗
B meson to η and ψ for K and K final states using heavy quark effective theory. For K
c
final state, the decay rate to η is larger than the decay to ψ by a factor 1.6. However,
c
∗ ≈
for K final state, it is ψ that is produced more than twice as often as η . Finally, due
c
to likely cancellations in the estimated ratios, we do not expect significant corrections
to our results because of our neglect of subleading terms in the heavy quark expansion
or our use of a non-relativistic formalism to relate f to f .
ηc ψ
Acknowledgements
This work was supported in part by the Natural Sciences and Engineering Research
Council of Canada. M. A. would like to thank the Department of Applied Mathematics
at UWO for warm hospitality during his visit.
4
References
[1] T. E. Browder, et al., CLEO Collaboration, CLNS-93-1226.
[2] N.G. Deshpande, J. Trampetic and K. Panose, Phys. Lett. B214(1988)467.
[3] N. Isgur and M. Wise, Phys. Lett. B232(1989)113; Phys. Lett. B237(1990)527.
[4] M. R. Ahmady and D. Liu, Phys. Lett. B302 (1993) 491.
[5] B. Grinstein, N. Isgur, D. Scora and M. Wise, Phys. Rev. D39 (1989) 799.
[6] M. R. Ahmady and D. Liu, hep-ph 9311335, Submitted to Phys. Lett. B.
[7] N. G. Deshpande and J. Trampetic, Preprint OITS 509.
[8] M. R. Ahmady and R. R. Mendel, hep-ph 9401315, Submitted to Phys. Lett. B.
[9] Particle Data Group, Phys. Rev. D45 (1992) number 11, Part II. If we would use
∗
instead the new measurements for B K(K )ψ [See Ref. [1]], the results in eqn.
→
(20) would be changed accordingly.
[10] We have used the charge averages Γ(B K(K∗)ψ) = 1/2[Γ(B+ K+(K∗+)ψ)+
◦ ◦ ∗◦ → →
Γ(B K (K )ψ)] from Ref. [9].
→
5