$η^{(\prime)}$ productions in semileptonic B decays

0.15 MB·

by  Chuan-Hung Chen

η( ) productions in semileptonic B decays
′
Chuan-Hung Chen1,2 and Chao-Qiang Geng3,4
∗ †
1Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan
2National Center for Theoretical Sciences, Taiwan
3Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan
4Theory Group, TRIUMF, 4004 Wesbrook Mall,
7
Vancouver, B.C V6T 2A3, Canada
0
0
2 (Dated: February 2, 2008)

n
a Abstract
J

1 Inspired by the new measurements on B η()ℓν¯ from the BaBar Collaboration, we examine
− ′ ℓ
→

3 the constraint on the flavor-singlet mechanism, proposed to understand the large branching ratios
v
6 for B η K decays. Based on the mechanism, we study the decays of B¯ η()ℓ+ℓ and find
4 → ′ d,s → ′ −
2
that they are sensitive to the flavor-singlet effects. In particular, we show that the decay branching
8
0
6 ratios of B¯d,s η′ℓ+ℓ− can be as large as O(10−8) and O(10−6), respectively.
→
0
/
h
p
-
p
e
h
:
v
i
X
r
a
∗ Email: [email protected]
† Email: [email protected]
1
Until now, the unexpected large branching ratios (BRs) for the decays B η K are still
′
→
mysterious phenomena among the enormous measured exclusive B decays at B factories
[1, 2]. One of the most promising mechanisms to understand the anomaly is to introduce a
flavor-singlet state, produced by the two-gluon emitted from the light quarks in η() [3, 4].
′
In this mechanism, the form factors in the B η() transitions receive leading power correc-
′
→
tions. Consequently, the authors in Ref. [5] have studied the implication on the semileptonic
decays of B¯ Pℓν with P = η() and ℓ = e, µ. In particular, they find that the decay
ℓ ′
→
BRs of the η modes can be enhanced by one order of magnitude. Recently, the BaBar
′
Collaboration [6] has measured the semileptonic decays with the data as follows:
BR(B+ ηℓ+ν ) = (0.84 0.27 0.21) 10 4 < 1.4 10 4(90% C.L.),
ℓ − −
→ ± ± × ×
BR(B+ η ℓ+ν ) = (0.33 0.60 0.30) 10 4 < 1.3 10 4(90% C.L.). (1)
′ ℓ − −
→ ± ± × ×
Although the measurements in Eq. (1) are only 2.55σ and 0.95σ significances, respectively,
it is important to examine if they give some constraints on the form factors due to the
flavor-singlet state in the decays of B¯ η()ℓν¯ . It should be also interesting to investi-
′ ℓ
→
gate the implication of the measurements in Eq. (1) by concentrating on the flavor-singlet
contributions on the flavor changing neutral current (FCNC) decays of B¯ η()ℓ+ℓ .
d,s ′ −
→
We start by writing the effective Hamiltonians for B η()ℓν and B¯ η()ℓ+ℓ at
− ′ ℓ ′ −
→ →
quark level in the SM as
G V
= F ubu¯γ (1 γ )bℓ¯γµ(1 γ )ν , (2)
I µ 5 5 ℓ
H √2 − −
G α λq′
= F em t H Lµ +H L5µ , (3)
II 1µ 2µ
H √2π
(cid:2) (cid:3)
respectively, with
2m
H = Ceff(µ)q¯γ P b bC (µ)q¯iσ qνP b,
1µ 9 ′ µ L − q2 7 ′ µν R
H = C q¯γ P b,
2µ 10 ′ µ L
Lµ = ℓ¯γµℓ, L5µ = ℓ¯γµγ ℓ, (4)
5
where α is the fine structure constant, λq′ = V V , m is the current b-quark mass, q
em t tb t∗q′ b
is the momentum transfer, P = (1 γ )/2 and C are the Wilson coefficients (WCs)
L(R) 5 i
∓
with their explicit expressions given in Ref. [7]. In particular, Ceff, which contains the
9
2
contribution from the on-shell charm-loop, is given by [7]
Ceff(µ) = C (µ)+(3C (µ)+C (µ))h(z,s) ,
9 9 1 2 ′
8 m 8 8 4 2
h(z,s) = ln b lnz + + x (2+x) 1 x 1/2
′
−9 µ − 9 27 9 − 9 | − |
ln √1 x+1 iπ, for x 4z2/s < 1,
 (cid:12)√1−−x−1(cid:12)− ≡ ′ (5)
×  2(cid:12)arctan(cid:12) 1 , for x 4z2/s > 1,
(cid:12) √(cid:12)x 1 ≡ ′
−

where h(z,s) describes the one-loop matrix elements of operators O = s¯ γµP b c¯ γ P c
′ 1 α L β β µ L α
and O = s¯γµP b c¯γ P c [7] with z = m /m and s = q2/m2. Here, we have ignored the
2 L µ L c b ′ b
resonant contributions [8, 9] as they are irrelevant to our analysis. In Table I, we show the
values of dominant WCs at µ = 4.4 GeV in the next-to-leading-logarithmic (NLL). We note
that since the value of h(z,s) is less than 2, the influence of the charm-loop is much less
′
| |
than C which are dominated by the top-quark contributions.
9,10
TABLE I: WCs at µ = 4.4 GeV in the NLL order.
C C C C C
1 2 7 9 10
0.226 1.096 0.305 4.344 4.599
− − −
To study the exclusive semileptonic decays, the hadronic QCD effects for the B¯ P
→
transitions are parametrized by
P q P q
P(p ) q¯γµb B¯(p ) = fP(q2) Pµ · qµ +fP(q2) · q ,
h P | ′ | B i + (cid:18) − q2 (cid:19) 0 q2 µ
fP(q2)
P(p ) q¯iσ qνb B¯(p ) = T P qq q2P , (6)
P ′ µν B µ µ
h | | i m +m · −
B P (cid:2) (cid:3)
with P = (p + p ) and q = (p p ) . Consequently, the transition amplitudes
µ B P µ µ B P µ
−
associated with the interactions in Eqs. (2) and (3) can be written as
√2G V
= F ubfP(q2)ℓ¯ p ℓ, (7)
MI π + 6 P
G α λq′
= F em t m˜ ℓ¯ p ℓ+m˜ ℓ¯ p γ ℓ , (8)
II 97 P 10 P 5
M √2π 6 6
(cid:2) (cid:3)
for B¯ Pℓν¯ and B¯ Pℓ+ℓ , respectively, with
ℓ −
→ →
2m
m˜ = CefffP(q2)+ b C fP(q2), m˜ = C fP(q2). (9)
97 9 + m +m 7 T 10 10 +
B P
3
Since we concentrate on the productions of the light leptons, we have neglected the terms
explicitly related to m . By choosing the coordinates for various particles:
ℓ
q = ( q2, 0, 0, 0), p = (E , 0, 0, p~ ),
B B P
| |
p
p = (E , 0, 0, p~ ), p = E (1, sinθ , 0, cosθ ), (10)
P P P ℓ ℓ ℓ ℓ
| |
where E = (m2 q2 m2)/(2 q2), p~ = E2 m2 and θ is the polar angle, the
P B − − P | P| P − P ℓ
differential decay rates for B pPℓν¯ and B¯ pPℓ+ℓ as functions of q2 are given by
− ℓ d −
→ →
dΓ G2 V 2m3 2
I = F| ub| B (1 s+mˆ2)2 4mˆ2 fP(q2)Pˆ , (11)
dq2 3 26π3 q − P − P (cid:16) + P(cid:17)
·
dΓII = G2Fαe2mm3B λq′ 2 (1 s+mˆ2)2 4mˆ2Pˆ2 m˜ 2 + m˜ 2 , (12)
dq2 3 29π5 | t | q − P − P P | 97| | 10|
· (cid:0) (cid:1)
respectively, with Pˆ = 2√s p~ /m = (1 s mˆ2)2 4smˆ2, mˆ = m /m and s =
P | P| B − − P − P P P B
p
q2/m2 .
B
To discuss the P = η() modes, we employ the quark-flavor scheme to describe the states
′
η and η , expressed by [10, 11]
′
η cosφ sinφ η
q
  =  −   (13)
η sinφ cosφ η
′ s
    
with η = (uu¯+dd¯)/√2, η = ss¯ and φ = 39.3 1.0 . Based on this scheme, it is found
q s ◦ ◦
±
that the form factors in Eq. (6) at q2 = 0 with the flavor-singlet contributions are given by
[4]
cosφ f 1 f f
fη(0) = qfπ(0)+ √2cosφ q sinφ s fsing(0),
i √2 f i √3 (cid:18) f − f (cid:19) i
π π π
fη′(0) = sinφfqfπ(0)+ 1 √2sinφfq +cosφfs fsing(0), (14)
i √2 f i √3 (cid:18) f f (cid:19) i
π π π
where i = +,T, f = (1.07 0.02)f , f = (1.34 0.06)f [11] and fsing(0) denote the
q ± π s ± π i
unknown transition form factors in the flavor-singlet mechanism. We note that the flavor-
singlet contributions to B¯ η() have also been considered in the soft collinear effective
′
→
theory [12]. For the q2-dependence form factors fπ (q2), we quote the results calculated
+(T)
by the light-cone sum rules (LCSR) [13], given by
fπ (0)
fπ (q2) = +(T) (15)
+(T) (1 q2/m2 )(1 α q2/m2 )
− B∗ − +(T) B∗
4
with f+π(T)(0) = 0.27, α+(T) = 0.52(0.84) and mB∗ = 5.32 GeV. Since f+si,nTg(q2) are unknown,
as usual, we parametrize them to be [5]
fsing (0)
fsing (q2) = +(T) (16)
+(T) (1 q2/m2 )(1 β q2/m2 )
− B∗ − +(T) B∗
with β being the free parameters. We will demonstrate that the BRsfor the semileptonic
+(T)
decays are not sensitive to the values of β , but those of fsing (0). Moreover, based on the
+(T) +(T)
result of fπ(0) fπ(0) in the large energy effective theory (LEET) [14], we may relate the
+ ∼ T
singlet form factors of fsing(0) and fsing(0). Explicitly, we assume that fsing(0) fsing(0).
+ T T ∼ +
Note that this assumption will not make a large deviation from the real case since the effects
of fP(q2) on the dilepton decays are small due to C >> C in Eq. (9). Hence, the value of
T 9 7
fsing(0) could be constrained by the decays B¯ η()ℓν¯ .
+ → ′ ℓ
Before studying the effects of fsing(q2) on the BRs of semileptonic decays, we examine
+
the β dependence on the BRs. By taking V = 3.67 10 3 and Eqs. (11) and (16), in
+ ub −
| | ×
Table II, we present BR(B η()ℓν¯ ) and BR(B η()ℓ+ℓ ) with fsing(0) = 0.2 and
− → ′ ℓ d → ′ − +
various values of β . From the table, we see clearly that the errors of BRs induced by the
+
errors of 60% in β are less than 7% and 14% for the η and η modes, respectively. Hence,
+ ′
it is a good approximation to take the q2 dependence for fsing(q2) to be the same as fπ(q2).
+ +
Consequently, the essential effect on the BRs for semileptonic decays is the value of fsing(0).
+
With V = 8.1 10 3 [15] and Eqs. (11) and (12), the decay BRs of B η()ℓν¯ and
td − − ′ ℓ
| | × →
TABLE II: BRs of B η()ℓν¯ ( in units of 10 4) and B¯ η()ℓ+ℓ ( in units of 10 7) with
− ′ ℓ − d ′ − −
→ →
fsing(0) = 0.2 and β = 0.2, 0.4, 0.5, 0.6 and 0.8, respectively.
+ +
β B ηℓν¯ B η ℓν¯ B¯ ηℓ+ℓ B¯ η ℓ+ℓ
+ − ℓ − ′ ℓ d − d ′ −
→ → → →
0.2 0.61 1.37 0.080 0.190
0.4 0.62 1.46 0.082 0.204
0.5 0.63 1.52 0.083 0.211
0.6 0.64 1.58 0.085 0.220
0.8 0.67 1.73 0.088 0.242
B¯ η()ℓ+ℓ as functions of fsing(0) are shown in Fig. 1. In Table III, we also explicitly
d → ′ − +
display the BRs with fsing(0) = 0, 0.1 and 0.2. From Table III, we find that without the
+
flavor-singlet effects, the result for BR(B ηℓν¯ ) is a factor of 2 smaller than the central
− ℓ
→
5
3
−40 1.2 (a) −40 2.5 (b)
1 1
ν)l 0.9 ν)l 2
ηl η’l 1.5
→ 0.6 →
−R(B 0.3 −R(B 0.51
B B
0 0
0 0.1 0.2 0 0.1 0.2
sing sing
f (0) f (0)
+ +
0.1 0.4
−70 (c) −70 (d)
− +η)1ll0.08 −+η)1’ll 00..23
→ →
Bd Bd 0.1
R(0.06 R(
B B
0
0 0.1 0.2 0 0.1 0.2
sing sing
f (0) f (0)
+ +
FIG. 1: BRs of (a)[(b)] B η[]ℓν¯ (in units of 10 4) and (c)[(d)] B¯ η[]ℓ+ℓ (in units of
− ′ ℓ − d ′ −
→ →
10 7) as functions of fsing(0), where the horizontal solid and dashed lines in (a) denote the central
− +
and upper and lower values of the current data at 90% C.L., while the dashed line in (b) is the
upper limit of the data.
TABLE III: BRs of B η()ℓν¯ ( in units of 10 4) and B¯ η()ℓ+ℓ ( in units of 10 7) with
− ′ ℓ − d ′ − −
→ →
φ= 39.3 and fsing(0) = 0.0, 0.1 and 0.2.
◦ +
fsing(0) B ηℓν¯ B η ℓν¯ B¯ ηℓ+ℓ B¯ η ℓ+ℓ
+ − → ℓ − → ′ ℓ d → − d → ′ −
0.0 0.41 0.20 0.06 0.03
0.1 0.52 0.71 0.07 0.10
0.2 0.63 1.53 0.08 0.21
value of the BaBar data in Eq. (1). Clearly, if the data shows a correct tendency, it indicates
that there exist some mechanisms, such as the one with the flavor-singlet state, to enhance
the decay of B¯ η as illustrated in Table III. Moreover, as shown in Fig. 1b and Table III,
→
the decays of B η ℓν are very sensitive to fsing(0). In particular, the current data has
− → ′ ℓ +
constrained that
fsing(0) 0.2. (17)
+ ≤
It is interesting to note that for fsing(0) = 0.2, BR(B¯ η ℓ+ℓ ) = 0.21 10 7, which is as
+ d → ′ − × −
6
large as BR(B π ℓ+ℓ ), while that of B¯ ηℓ+ℓ is slightly enhanced. In addition, it
− − − d −
→ →
is easy to see that the flavor-singlet contributions could result in the BRs of the η modes
′
to be over than those of the η ones.
Our investigation of the flavor-singlet effects can be extended to the dileptonic decays of
B¯ η()ℓ+ℓ [16]. In the following, we use the notation with a tilde at the top to represent
s ′ −
→
the form factors associated with B . Hence, similar to Eq. (14), we express the form factors
s
for B¯ η() with the flavor-singlet effects at q2 = 0 to be
s ′
→
1 f f
f˜η(0) = sinφf˜ηs[mη](0)+ √2cosφ q sinφ s f˜sing(0),
+ − + √3 (cid:18) f − f (cid:19) +
K K
f˜η′(0) = cosφf˜ηs[mη′](0)+ 1 √2sinφ fq +cosφ fs f˜sing(0). (18)
+ + √3 (cid:18) f f (cid:19) +
K K
For the q2-dependence form factors of f˜ηs[mη(′)], we adopt the results calculated by the
+,T
constituent quark model (CQM) [17], given by
f˜ηs[mη(′)](0)
f˜ηs[mη(′)](q2) = +,T (19)
+,T 1 aη(′) q2/m2 +bη(′) (q2/m2 )2
− +,T Bs +,T Bs
with f˜ηs[mη](0) = f˜ηs[mη′](0) = f˜ηs[mη](0) = 0.36, f˜ηs[mη′](0) = 0.39, aη = aη′ = 0.60,
+ + T T + +
bη = bη′ = 0.20, aη = aη′ = 0.58 and bη = bη′ = 0.18. By using m = 5.37 GeV and
+ + T T T T Bs
V = 0.04 instead of m and V in Eq. (12), we present the BRs of B¯ η()ℓ+ℓ in
ts B td s ′ −
− →
Table IV. We also display the BRs as functions of f˜sing(0) in Fig. 2. As seen from Table IV
+
TABLE IV: BRs of B¯ η()ℓ+ℓ ( in units of 10 7) with φ = 39.3 and f˜sing(0) = 0.0, 0.1 and
s → ′ − − ◦ +
0.2.
f˜sing(0) 0.0 0.1 0.2
+
B¯ ηℓ+ℓ 3.71 3.27 2.84
s −
→
B¯ η ℓ+ℓ 3.35 5.97 9.35
s ′ −
→
and Fig. 2, due to the flavor-singlet effects, the BRs of B¯ η ℓ+ℓ are enhanced and could
s ′ −
→
be as large as O(10 6) with around a factor of 3 enhancement, whereas those of B¯ ηℓ+ℓ
− s −
→
decrease as increasing f˜sing(0), which can be tested in future hadron colliders.
+
Insummary,wehavestudiedtheeffectsoftheflavor-singletstateontheη() productionsin
′
thesemileptonic Bdecays. Interms of theconstraints fromthecurrent data ofB η()ℓν ,
− ′ ℓ
→
7
4 20
−70 3.5 −70 15
1 1
−+η)ll 3 −+η)’ll 10
→ →
Bs Bs
R(2.5 R( 5
B B
2 0
0 0.1 0.2 0 0.1 0.2
~sing ~sing
f (0) f (0)
(a) (b)
+ +
FIG. 2: (a)[(b)] BRs (in units of 10 7) of B¯ η[]ℓ+ℓ as functions of f˜sing(0).
− s → ′ − +
we have found that the BRs of B¯ η ℓ+ℓ could be enhanced to be O(10 8) and O(10 6),
d,s ′ − − −
→
respectively. Finally, we remark that the flavor-singlet effects could result in the BRs of the
B¯ η modes to be larger than those of B¯ η and the statement is reversed if the effects
d,s ′
→ →
are neglected.
Note added: After we presented the paper, Charng, Kurimoto and Li [18] calculated the
flavor singlet contribution to the B η(′) transition form factors from the gluonic content
→
of η(′) in the large recoil region by using the perturbative QCD (PQCD) approach. Here,
we make some comparisons as follows:
1. While Ref. [18] gives a theoretical calculation on the flavor singlet contribution to the
form factors in the PQCD, we consider the direct constraint from the experimental data.
The conclusion that the singlet contribution is negligible (large) in the B η(′) form factors
→
in Ref. [18] is the same as ours. However, the overall ratios in Ref. [18] for the singlet
contributions are about a factor 4 smaller than ours. On the other hand, as stressed in
Ref. [18], they have used a small Gegenbauer coefficient, which corresponds smaller gluonic
contributions. For a larger allowed value of the the Gegenbauer coefficient in Ref. [4], the
overall ratios will be a few factors larger. In other words, the real gluonic contributions rely
on future experimental measurements.
2. Although our assumption of fsing(0) fsing(0) seems to be somewhat different from the
T ∼ +
PQCD calculation as pointed out in Ref. [18] due to an additional term, the numerical
values of fsing(0) = 0.042 and fsing(0) = 0.035 by the PQCD [18] do not change our
+ T
assumption very much. After all, as stated in point 1 that there exist large uncertainties
8
for the wave functions in the PQCD. In addition, the difference actually is not important
for our results as explained in the text.
Acknowledgments
This work is supported in part by the National Science Council of R.O.C. under Grant
#s:NSC-95-2112-M-006-013-MY2, NSC-94-2112-M-007-(004,005) and NSC-95-2112-M-007-
059-MY3.
[1] BELLE Collaboration, K. Abe et al., arXiv:hep-ex/0603001.
[2] BABAR Collaboration, B. Aubert et al., Phys. Rev. Lett. 94, 191802 (2005);
arXiv:hep-ex/0608005.
[3] P. Kroll and K. Passek-Kumericki, Phys. Rev. D 67, 054017 (2003) [arXiv:hep-ph/0210045].
[4] M. Beneke and M. Neubert, Nucl. Phys. B651, 225 (2003) [arXiv:hep-ph/0210085].
[5] C.S. Kim, S. Oh and C. Yu, Phys. Lett. B590, 223 (2004) [arXiv:hep-ph/0305032].
[6] Babar Collaboration, B. Aubert et al., arXiv:hep-ex/0607066.
[7] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys 68, 1230 (1996)
[arXiv:hep-ph/9512380].
[8] C. S. Lim, T. Morozumi and A. I. Sanda, Phys. Lett. B 218, 343 (1989); N. G. Deshpande,
J.TrampeticandK.Panose,Phys.Rev.D39,1461(1989); P.J.O’DonnellandH.K.K.Tung,
Phys. Rev. D 43, 2067 (1991).
[9] C.H. Chen and C.Q. Geng, Phys. Rev. D66, 034006 (2002) [arXiv:hep-ph/0207038].
[10] J. Schechter, A. Subbaraman and H. Weigel, Phys. Rev. D 48, 339 (1993).
[11] T. Feldmann, P. Kroll and B. Stech, Phys. Rev. D58, 114006(1998) [arXive:hep-ph/9802409].
[12] A. R. Williamson and J. Zupan, Phys. Rev. D 74, 014003 (2006) [Erratum-ibid. D 74, 03901
(2006)] [arXiv:hep-ph/0601214].
[13] P. Ball and R. Zwicky, Phys. Rev. D71, 014015 (2005) [arXiv:hep-ph/0406232]; Phys. Lett.
B625, 225 (2005) [arXiv:hep-ph/0507076].
[14] M.J. Dugan and B. Grinstein, Phys. Lett. B255, 583 (1991); J. Charles et. al., Phys. Rev.
9
D60, 014001 (1999).
[15] Particle Data Group, W.M. Yao et al., J. Phys. G 33, 1 (2006).
[16] C.Q. Geng and C.C. Liu, J. Phys. G 29, 1103 (2003) [arXiv:hep-ph/0303246].
[17] D. Melikhov and B. Stech, Phys. Rev. D62, 014006 (2000) [arXiv:hep-ph/0001113].
[18] Y. Y. Charng, T. Kurimoto and H. n. Li, Phys. Rev. D 74, 074024 (2006)
[arXiv:hep-ph/0609165].
10