Semileptonic decays $B_c\to(η_c,J/Ψ)lν$ in the perturbative QCD approach

0.28 MB·English

by  Wen-Fei Wang

Semileptonic decays B (η ,J/Ψ)lν in the perturbative QCD
c c
→
approach
Wen-Fei Wang, Ying-Ying Fan and Zhen-Jun Xiaoa
Department of Physics and Institute of Theoretical Physics,
Nanjing Normal University, Nanjing,
Jiangsu 210023, People’s Republic of China
(Dated: January 31, 2013)
3
1 Abstract
0 − −
In this paper we study the semileptonic decays of B (η ,J/ψ)l ν¯. We firstly evaluate
2 c → c l
the B (η ,J/Ψ) transition form factors F (q2), F (q2), V(q2) and A (q2) by employing
n c → c 0 + 0,1,2
a the pQCD factorization approach, and then we calculate the branching ratios for all considered
J
semileptonic decays. Based on the numerical results and the phenomenological analysis, we find

0
that: (a) the pQCD predictions for the values of the B η and B J/Ψ transition form
3 c c c
→ →
factors agree well with those obtained by using other methods; (b) the pQCD predictions for

h] the branching ratios of the considered decays are Br(Bc− → ηce−ν¯e(µ−ν¯µ)) = (4.41+−11..2029)×10−3,
p Br(B− η τ−ν¯ ) = (1.37+0.37) 10−3, Br(B− J/Ψe−ν¯ (µ−ν¯ )) = (10.03+1.33) 10−3, and
- c → c τ −0.34 × c → e µ −1.18 ×
p Br(B− J/Ψτ−ν¯ ) = (2.92+0.40) 10−3; and (c) we also define and calculate two ratios of
e c → τ −0.34 ×
the branching ratios R and R , which will be tested by LHCb and the forthcoming Super-B
h ηc J/Ψ
[ experiments.

2
v PACS numbers: 13.25.Hw, 12.38.Bx,14.40.Nd
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v
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[email protected]
1
I. INTRODUCTION
The charmed B meson was found by the CDF Collaboration at Tevatron [1] fifteen
c
years ago. It is a pseudoscalar ground state of two heavy quarks b and c, which can decay
individually. Being below the B D threshold, B meson can only decay through weak
c
−
interactions, it is an ideal system to study weak decays of heavy quarks. Due to the different
decay rate of the two heavy quarks, the B meson decays are rather different from those
c
of B or B meson. Although the phase space in c s transition is smaller than that in
s
→
b c decays, but the Cabibbo-Kobayashi-Maskawa (CKM) matrix element V 1 is
cs
→ | | ∼
much larger than V 0.04. Thus the c-quark decays provide the dominant contribution
cb
| | ∼
(about 70%) to the decay width of B meson [2]. At LHC experiments, around 5 1010 B
c c
×
events per year are expected [2, 3], which provide a very good platform to study various B
c
meson decay modes.
Infact, theB decayshavebeenstudiedintensively bymanyauthors[4–13]. InRef.[4], for
c
example, Dhir and Verma presented a detailed analysis of the B decays in the Bauer-Stech-
c
Wirbel(BSW)framework, whiletheauthorsoftheRefs.[5–7]studiedtheB mesondecaysin
c
the non-relativistic or relativistic quark model. In the perturbative QCD (pQCD) approach,
furthermore, various B decay modes have also been studied for example in Refs.[11–13].
c
In this paper, we will study the semileptonic decays of B (η ,J/Ψ)lν¯ (here l stands
c c l
→
for leptons e,µ, and τ.) in the pQCD factorization approach [14, 15]. The lowest order
diagrams for B (η ,J/Ψ)lν¯ are displayed in Fig.1, where B stands for B meson and M
c c l c
→
for η or J/Ψ meson, and the leptonic pairs come from the b-quark’s weak decay. In this
c
work, we firstly calculate the q2-dependent form factors for B (η ,J/Ψ) transitions, and
c c
→
then we will give the branching ratios of the considered semileptonic decay modes.
The structure of this paper is as below: after this introduction, we collect the distribution
amplitudes of the B meson and the η ,J/Ψ mesons in Sec.II. In Sec.III, based on the k
c c T
factorization theorem, we calculate and present the expressions for the B (η ,J/Ψ) tran-
c c
→
sition form factors in the large recoil regions. The numerical results and relevant discussions
are given in Sec. IV, a short summary will also be included in this section.
II. KINEMATICS AND THE WAVE FUNCTIONS
In the B meson rest frame, with the m (m) stands for the mass of the B (η or J/Ψ)
c Bc c c
meson, and r = m/m , the momenta of B and η (J/Ψ) mesons are defined in the same
Bc c c
way as in Ref. [16]
m m
p1 = Bc(1,1,0⊥), p2 = Bc(rη+,rη−,0⊥), (1)
√2 √2
where η+ = η + η2 1 and η− = η η2 1 with the definition of the η as of the form
− − −
p mp m2 q2
η = Bc 1+ − , (2)
2m m2
(cid:20) Bc (cid:21)
where q2 = (p p )2 is the invariant mass ofthe lepton pairs. The momenta ofthe spectator
1 2
−
quarks in B and η (J/Ψ) mesons are parameterized as
c c
m m m
k1 = (0,x1 Bc,k1⊥), k2 = (x2 Bcrη+,x2 Bcrη−,k2⊥). (3)
√2 √2 √2
2
FIG. 1. The typical Feynman diagrams for the semileptonic decays B (η ,J/Ψ)lν¯, where B
c c
→
stands for B meson and M for η or J/Ψ meson.
c c
For the J/Ψ meson, we define its polarization vector ǫ as
1
ǫL = (η+, η−,0⊥), ǫT = (0,0,1), (4)
√2 −
where ǫ and ǫ denotes the longitudinal and transverse polarization of the J/Ψ meson.
L T
In the calculations, one can ignore the k contributions of B meson. Furthermore, one
T c
can assume that the b and c quark in B meson are on the mass shell approximately and
c
treat its wave function as a δ function. In this work, we use the same distribution amplitude
for B meson as those used in Refs.[11, 12]
c
i
Φ (x) = [(p/+m )γ φ (x)] (5)
Bc √2N Bc 5 Bc αβ
c
with
f
φ (x) = Bc δ(x m /m ) (6)
Bc 2√2N − c Bc
c
where m is the mass of c-quark.
c
For pseudoscalar meson η , the wave function is the form of
c
i
Φ (x) = γ [p/φv(x)+m φs(x)]. (7)
ηc √2N 5 ηc
c
The twist-2 and twist-3 asymptotic distribution amplitudes, φv(x) and φs(x), can be written
as [17, 18]
0.7
f x(1 x)
φv(x) = 9.58 ηc x(1 x) − ,
2√2N − 1 2.8x(1 x)
c (cid:20) − − (cid:21)
0.7
f x(1 x)
φs(x) = 1.97 ηc − . (8)
2√2N 1 2.8x(1 x)
c (cid:20) − − (cid:21)
As for the vector J/Ψ meson, we take the wave function as follows,
1
ΦL (x) = m /ǫ φL(x)+/ǫ p/φt(x) ;
J/Ψ √2N J/Ψ L L
c(cid:26) (cid:27)
1
ΦT (x) = m /ǫ φV(x)+/ǫ p/φT(x) . (9)
J/Ψ √2N J/Ψ T T
c(cid:26) (cid:27)
3
And the asymptotic distribution amplitudes of J/Ψ meson read as [17]
f x(1 x) 0.7
φL(x) = φT(x) = 9.58 J/Ψ x(1 x) − ,
2√2N − 1 2.8x(1 x)
c (cid:20) − − (cid:21)
f x(1 x) 0.7
φt(x) = 10.94 J/Ψ (1 2x)2 − ,
2√2N − 1 2.8x(1 x)
c (cid:20) − − (cid:21)
f x(1 x) 0.7
φV(x) = 1.67 J/Ψ [1+(2x 1)2] − . (10)
2√2N − 1 2.8x(1 x)
c (cid:20) − − (cid:21)
Here, φL andφT denotefor thetwist-2 distribution amplitudes, andφt andφV for thetwist-3
distribution amplitudes.
III. FORM FACTORS AND SEMILEPTONIC DECAYS
As is well-known, the form factors of B P,V,S1 transitions have been calculated
(s)
→
by many authors in the pQCD factorization approach and other methods[19, 20], and the
pQCD predictions are generally consistent with those from other methods.
The B η form factors induced by vector currents are defined in Refs. [16, 21, 22]
c c
→
m2 m2 m2 m2
η (p ) c¯(0)γ b(0) B (p ) = (p +p ) Bc − q F (q2)+ Bc − q F (q2)(,11)
h c 2 | µ | c 1 i 1 2 µ − q2 µ + q2 µ 0
(cid:20) (cid:21)
where q = p p is the momentum transfer to the lepton pairs, and m is the mass of η
1 2 c
−
meson. In order to cancel the poles at q2 = 0, F (0) should be equal to F (0). For the sake
+ 0
of the calculation, it is convenient to define the auxiliary form factors f (q2) and f (q2)
1 2
η (p ) c¯(0)γ b(0) B (p ) = f (q2)p +f (q2)p . (12)
c 2 µ c 1 1 1µ 2 2µ
h | | i
In terms of f (q2) and f (q2) the form factor F (q2) and F (q2) read
1 2 + 0
1
F (q2) = f (q2)+f (q2) ,
+ 1 2
2
1 (cid:2) q(cid:3)2 1 q2
F (q2) = f (q2) 1+ + f (q2) 1 . (13)
0 2 1 m2 m2 2 2 − m2 m2
(cid:20) Bc − (cid:21) (cid:20) Bc − (cid:21)
The form factors F and F (or f and f ) of the B η transition are dominated by a
+ 0 1 2 c c
→
single gluon exchange in the leading-order and in the large recoil regions. And in the hard-
scattering kernel, the transverse momentum is retained to regulate the endpoint singularity.
The factorization formula for the B η form factors in pQCD approach is written as [19]
c
→
dz+d2z dy−d2y
η (p ) c¯(0)γ b(0) B (p ) = g2C N dx dx d2k d2k T T
h c 2 | µ | c 1 i s F c 1 2 1T 2T (2π)3 (2π)3
Z
e−ik2·y η (p ) c¯′(y)c (0) 0 eik1·z 0 c¯ (0)b′(z) B (p ) Tγβ;αδ. (14)
× h c 2 | γ β | i h | α δ | c 1 i Hµ
1
Here P,V,S denote the pseudoscalar,vector and scalar meson respectively.
4
In the transverse configuration b-space and by including the Sudakov form factors and
the threshold resummation effects, we obtain the B η form factors f (q2) and f (q2) as
c 1 2
→
following,
f (q2) = 8πm2 rC dx dx b db b db φ (x )
1 Bc F 1 2 1 1 2 2 Bc 1
Z Z
2[φs(x ) rx φv(x )] h (x ,x ,b ,b ) α (t )exp[ S (t )]
2 2 2 1 1 2 1 2 s 1 ab 1
× − · · −
n
x η+(η+φv(x ) 2φs(x ))
+ 4r φs(x ) 2rφv(x )+ 1 2 − 2
c 2 2
" − η2 1 #
−
h (x ,x ,b ,b ) α (t )exp[ S (tp)] , (15)
2 1 2 1 2 s 2 ab 2
× · −
o
f (q2) = 8πm2 C dx dx b db b db φ (x )
2 Bc F 1 2 1 1 2 2 Bc 1
Z Z
[2φv(x ) 4rx (φs(x ) ηφv(x ))]
2 2 2 2
× − −
hn (x ,x ,b ,b ) α (t )exp[ S (t )]
1 1 2 1 2 s 1 ab 1
× · −
x (η+φv(x ) 2φs(x ))
+ 4rφs(x ) 2r φv(x ) 1 2 − 2
2 c 2
" − − η2 1 #
−
h (x ,x ,b ,b ) α (t )exp[ S p(t )] , (16)
2 1 2 1 2 s 2 ab 2
× · −
o
where C = 4/3 is a color factor, r is the same as in Eqs.(1,3), while r = m /m , and m
F c c Bc c
is the mass of c-quark. The functions h and h , the scales t , t and the Sudakov factors
1 2 1 2
S are given in the Appendix A of this paper.
ab
For the charged current B η lν¯, the quark level transitions are the b clν¯ transition
c c l l
→ →
with the effective Hamiltonian [23]
G
(b clν¯) = FV c¯γ (1 γ )b ¯lγµ(1 γ )ν , (17)
eff l cb µ 5 5 l
H → √2 − · −
where G = 1.16637 10−5GeV−2 is the Fermi-coupling constant. With the two form
F
×
factors F (q2) and F (q2), we can write down the differential decay width of the decay mode
+ 0
B η lν¯ as [9, 24]
c c l
→
dΓ(B η lν¯) G2 V 2 q2 m2 (q2 m2)2 m2 m2 q2 2
c → c l = F| cb| − l − l Bc − − m2
dq2 192π3m3 (q2)2 s q2 s 4q2 −
Bc (cid:0) (cid:1)
m2 +2q2 q2 (m m)2 q2 (m +m)2 F2(q2)
× l − Bc − − Bc +
+3nm(cid:0)2 m2 (cid:1)m(cid:2)2 2F2(q2) , (cid:3)(cid:2) (cid:3) (18)
l Bc − 0
o
(cid:0) (cid:1)
±
where m and m is the mass of the lepton and η respectively. If the produced lepton is e
l c
±
or µ , the corresponding mass terms of the lepton could be neglected.
5
The form factors V(q2) and A (q2) for B J/Ψ transition are defined in the same
0,1,2 c
→
way as in Refs. [16, 21, 22] and are written explicitly as,
V(q2) = 4πm2 C dx dx b db b db φ (x ) (1+r)
Bc F 1 2 1 1 2 2 Bc 1 ·
Z Z
2 φT(x ) rx φV(x ) h (x ,x ,b ,b ) α (t )exp[ S (t )]
2 2 2 1 1 2 1 2 s 1 ab 1
× − · · −
n
(cid:2) x (cid:3)
+ 2r+ 1 φV(x ) h (x ,x ,b ,b ) α (t )exp[ S (t )] , (19)
2 2 1 2 1 2 s 2 ab 2
" η2 1! #· · −
− o
p
A (q2) = 8πm2 C dx dx b db b db φ (x )
0 Bc F 1 2 1 1 2 2 Bc 1
Z Z
1 r2x +2rx η φL(x )+r(1 2x )φt(x )
2 2 2 2 2
× − −
nh1(cid:2)((cid:0)x1,x2,b1,b2) αs(t(cid:1)1)exp[ Sab(t1)] (cid:3)
× · −
x x (η +r(1 2η2))
+ r2 +r + 1 rx η + 1 − φL(x )
c 1 2
" 2 − 2 η2 1 ! #
−
h (x ,x ,b ,b ) α (t )exp[ Sp(t )] , (20)
2 1 2 1 2 s 2 ab 2
× · −
o
r
A (q2) = 8πm2 C dx dx b db b db φ (x )
1 Bc F 1 2 1 1 2 2 Bc 1 · 1+r
Z Z
2(1+rx η)φV(x ) 2(2rx η)φT(x )
2 2 2 2
× − −
nh1(cid:2)(x1,x2,b1,b2) αs(t1)exp[ Sab(t1)] (cid:3)
× · −
+ (2r x +2rη)φV(x ) h (x ,x ,b ,b ) α (t )exp[ S (t )] , (21)
c 1 2 2 1 2 1 2 s 2 ab 2
− · · −
o
(cid:2) (cid:3)
(1+r)2(η r) 1+r
A (q2) = − A (q2) 8πm2 C dx dx b db b db φ (x )
2 2r(η2 1) · 1 − Bc F 1 2 1 1 2 2 Bc 1 · η2 1
− Z Z −
(η(1 r2x ) r(1+x 2x η2))φL(x )+ 1+2r2x rη(1+2x ) φt(x )
2 2 2 2 2 2 2
× − − − −
nh1(cid:2)(x1,x2,b1,b2) αs(t1)exp[ Sab(t1)] (cid:0) (cid:1) (cid:3)
× · −
1 x x
+ x rη η2 1+ r +r2 1 η+r 1 r 1 +x η2 φL(x )
1 c c 1 2
− 2 − − 2 − − 2
(cid:20) (cid:18) (cid:19) (cid:21)
p (cid:16) (cid:17) (cid:16) (cid:17)
h (x ,x ,b ,b ) α (t )exp[ S (t )] , (22)
2 1 2 1 2 s 2 ab 2
× · −
o
wherte r = m /m , C and r are the same as in Eqs.(15,16). The expressions of the
J/Ψ Bc F c
hard function h and h , hard scales t and t , and Sudakov function S are all given in
1 2 1 2 ab
the Appendix A. One should note that the pQCD predictions for the form factors f (q2),
1,2
V(q2) and A (q2) as given in Eqs.(15,16,19-22) are all leading order results. The NLO
0,1,2
contributions to the form factors of B (π,K) transitions as given in Refs. [25, 26] do not
→
apply here because of the large mass of c-quark and (η ,J/Ψ) mesons.
c
6
The effective Hamiltonian for the decay modes B J/Ψlν¯ is the same as B η lν¯,
c l c c l
→ →
butcorrespondingdifferentialdecaywidthsaredifferent. ForB J/Ψlν¯,wehave[9,27,28]
c l
→
dΓ (B J/Ψlν¯) G2 V 2 q2 m2 (q2 m2)2 m2 m2 q2 2
L c → l = F| cb| − l − l Bc − − m2
dq2 192π3m3 (q2)2 s q2 s 4q2 −
Bc (cid:0) (cid:1)
m2 +2q2
3m2λ(q2)A2(q2)+ l
× l 0 4m2
(
2
λ(q2)
(m2 m2 q2)(m +m)A (q2) A (q2) , (23)
× Bc − − Bc 1 − m +m 2
" Bc # )
dΓ±(Bc → J/Ψlν¯l) = G2F|Vcb|2 q2 −m2l (q2 −m2l)2 m2Bc −m2 −q2 2 m2
dq2 192π3m3 q2 s q2 s 4q2 −
Bc (cid:0) (cid:1)
2
V(q2) (m +m)A (q2)
(m2 +2q2)λ(q2) Bc 1 , (24)
× ( l "mBc +m ∓ λ(q2) # )
where m = m , and λ(q2) = (m2 +m2 q2)2 4m2 m2 is thpe phase-space factor. The
J/Ψ Bc − − Bc
combined transverse and total differential decay widths are defined as
dΓT dΓ+ dΓ− dΓ dΓL dΓT
= + , = + . (25)
dq2 dq2 dq2 dq2 dq2 dq2
IV. NUMERICAL RESULTS AND DISCUSSIONS
In the numerical calculations we use the following input parameters (here masses and
decay constants in unit GeV ) [11, 29]
(f=4)
Λ = 0.287, m = 2.981, m = 3.097, m = 6.277,
M¯S ηc J/Ψ Bc
m = 1.275 0.025, m = 1.777, f = 0.489, τ = (0.45 0.04) ps,
c ± τ Bc Bc ±
V = (41.2+1.1) 10−3, f = (0.420 0.050), f = (0.405 0.014). (26)
| cb| −0.5 × ηc ± J/Ψ ±
By using the expressions in Eqs.(15,16,19-22) and the definitions in Eq. (13) we calculate
the values of the form factors F (q2), F (q2), V(q2), A (q2), A (q2) and A (q2) for given
0 + 0 1 2
value of q2 in the region of 0 q2 (m m)2. But one should note that the pQCD
≤ ≤ Bc −
predictions for the considered form factors are reliable only for small values of q2. For the
form factors in the larger q2 region, one has to make an extrapolation for them from the
lower q2 region to larger q2 region. In this work we make the extrapolation by using the
formula in Refs. [9, 31]
F(q2) = F(0) exp a q2 +b (q2)2 . (27)
· · ·
where F stands for the form factors F ,V,A(cid:2) , and a,b are(cid:3)the parameters to be deter-
0,+ 0,1,2
mined by the fitting procedure.
The numerical values of the form factors F ,V and A at q2 = 0 and their fitted
0,+ 0,1,2
parameters are listed in Table I. The first error of the pQCD predictions for the form factors
7
1.2 1.2 1.2
2 F(q)00.8 2 F(q)0+.8 2 V(q)0.8
0.4 0.4 0.4
0.0 0.0 0.0
0 2 2 2 4 6 0 2 2 2 4 6 0 2 2 2 4 6
q (G eV) q (G eV) q (G eV)

1.2 1.2 1.2
2q)0.8 2q)0.8 2q)0.8
A(0 A(1 A(2
0.4 0.4 0.4
0.0 0.0 0.0
0 2 4 6 0 2 4 6 0 2 4 6
2 2 2 2 2 2
q (G eV) q (G eV) q (G eV)
FIG. 2. The pQCD predictions for the q2-dependence of the form factors F ,F ,V,A ,A and A .
0 + 0 1 2
Withthesolidlinesstandforthecentralvalues,andthebandsshowtheerrorsofthecorresponding
form factors.
in Table I comes from the uncertainty of the decay constants of η and/or J/Ψ mesons, and
c
the second error comes from the uncertainty of m = 1.275 0.025 [29]. For the parameters
c
±
(a,b), theirerrorsfromthedecayconstantsf ,f orfromm = 1.275 0.025arenegligibly
ηc J/Ψ c ±
small and not shown here explicitly. As a comparison, we also present some results obtained
by other authors based on different methods in Table II. One should note that the definition
of the B J/Ψ transition form factors in this paper are different from those in Ref. [6]
c
→
(ISK) and [5] (HNV). From Table II, we find that our results agree well with the results in
other literatures. In Fig.2, we show the pQCD predictions for the q2-dependence of those
form factors, where the solid lines stand for the central values, and the bands show the
theoretical errors of the corresponding form factors.
TABLE I. The pQCD predictions for form factors F ,V,A at q2 = 0 and the parametrization
0,+ 0,1,2
constants a and b for B η and B J/Ψ transitions.
c c c
→ →
F(0) a b
FBc→ηc 0.48 0.06 0.01 0.037 0.0007
0 ± ±
FBc→ηc 0.48 0.06 0.01 0.055 0.0014
+ ± ±
VBc→J/Ψ 0.42 0.01 0.01 0.065 0.0015
± ±
ABc→J/Ψ 0.59 0.02 0.01 0.047 0.0017
0 ± ±
ABc→J/Ψ 0.46 0.02 0.01 0.038 0.0015
1 ± ±
ABc→J/Ψ 0.64 0.02 0.01 0.064 0.0041
2 ± ±
By using the relevant formulas and the input parameters as defined or given in previous
sections, it is straightforward to calculate the branching ratios for all the considered decays.
8
TABLE II. B η ,J/Ψ transition form factors at q2 = 0 evaluated in this paper and in other
c c
→
literatures.
pQCD WSL[9] EFG[7] ISK[6] HNV[5] DV[4]
FBc→ηc = FBc→ηc 0.48 0.61 0.47 0.61 0.49 0.58
0 +
VBc→J/Ψ 0.42 0.74 0.49 0.83 0.61 0.91
ABc→J/Ψ 0.59 0.53 0.40 0.57 0.45 0.58
0
ABc→J/Ψ 0.46 0.50 0.50 0.56 0.49 0.63
1
ABc→J/Ψ 0.64 0.44 0.73 0.54 0.56 0.74
2
By making the numerical integration over the physical range of q2, we find the pQCD
predictions for the branching ratios of considered decay modes:
Br B− η e−ν¯ (µ−ν¯ ) = 4.41+1.11(f ) 0.39(τ )+0.24(V )+0.22(m ) 10−3,
c → c e µ −0.99 ηc ± Bc −0.11 cb −0.21 c ×
Br B− η τ−ν¯ = 1.37+0.34(f ) 0.12(τ )+0.07(V )+0.07(m ) 10−3,
(cid:0) c → c τ(cid:1) (cid:0) −0.31 ηc ± Bc −0.03 cb −0.06 c (cid:1)×
Br B− J/Ψe−ν¯ (µ−ν¯ ) = 10.03+0.71(f ) 0.89(τ )+0.54(V )+0.41(m ) 10−3,
c → (cid:0) e µ (cid:1) (cid:0) −0.68 J/Ψ ± Bc −0.24 cb −0.27 (cid:1)c ×
Br B− J/Ψτ−ν¯ = 2.92+0.21(f ) 0.26(τ )+0.16(V )+0.12(m ) 10−3.(28)
(cid:0) c → τ(cid:1) (cid:0) −0.20 J/Ψ ± Bc −0.07 cb −0.08 c (cid:1)×
where the m(cid:0)ajor theoretical e(cid:1)rror(cid:0)s come from the uncertainties of the input par(cid:1)ameters f ,
ηc
f , V , τ and m as given explicitly in Eq. (26).
J/Ψ | cb| Bc c
From the pQCD predictions for the form factors F ,V,A as given in Table I and the
0,+ 0,1,2
pQCD predictions for the branching ratios of B (η ,J/Ψ)lν as given in Eq.(28), we find
c c
→
the following points:
(i) The form factor F (0) equals to F (0) by definition, but they have different q2-
0 +
dependence. The error bands of F (q2) and F (q2) in Fig.2 are larger than that
0 +
of V(q2) and A (q2). The reason is that the uncertainty of the decay constant f
0,1,2 ηc
in B η transition is larger than the one of f in B J/Ψ transition.
c c J/Ψ c
→ →
(ii) The pQCD predictions for the form factors as listed in Table II agree well with those
obtained by using other methods.
(iii) The pQCD predictions for the branching ratios of the four decay modes B
c
(η ,J/Ψ)lν are at the order of 10−3. Because of its large mass of τ lepton, t→he
c
decays involving a τ in the final state has a smaller decay rate than those with light
− −
e or µ . Since the ratio of the branching ratios has smaller theoretical error than the
decay rates themselves, we here define two ratios R and R , the pQCD predictions
ηc J/Ψ
for them are the following
− −
Br(B η l ν¯)
R = c → c l 3.2, for l = (e,µ), (29)
ηc Br(B− η τ−ν¯ ) ≈
c → c τ
− −
Br(B J/Ψl ν¯)
R = c → l 3.4, for l = (e,µ). (30)
J/Ψ Br(B− J/Ψτ−ν¯ ) ≈
c → τ
These relations will be tested by LHCb and the forthcoming Super-B experiments.
9
− −
Inshort we calculated thebranching ratiosofthesemileptonic decays B (η ,J/ψ)l ν¯
c → c l
in the pQCD factorization approach. We first calculated the relevant form factors by em-
ploying the pQCD factorization approach, and then evaluated the branching ratios for all
consideredsemileptonicB decays. Basedonthenumerical resultsandthephenomenological
c
analysis, we find that
(i) For B (η ,J/Ψ) transitions, the LO pQCD predictions for the form factors
c c
→
F (0),V(0) and A (0) agree with those derived by using other different meth-
0,+ 0,1,2
ods.
(ii) The pQCD predictions for the branching ratios of the considered decay modes are:
Br B− η e−ν¯ (µ−ν¯ ) = (4.41+1.22) 10−3,
c → c e µ −1.09 ×
Br B− η τ−ν¯ = (1.37+0.37) 10−3,
(cid:0) c → c τ(cid:1) −0.34 ×
Br(B− J/Ψe−ν¯ (µ−ν¯ )) = (10.03+1.33) 10−3,
c → (cid:0) e µ (cid:1) −1.18 ×
Br B− J/Ψτ−ν¯ = (2.92+0.40) 10−3, (31)
c → τ −0.34 ×
(cid:0) (cid:1)
where the individual theoretical errors in Eq.(28) have been added in quadrature.
(iii) We also defined two ratios of the branching ratios R and R and presented the
ηc J/Ψ
corresponding pQCD predictions, which will be tested by LHCb and the forthcoming
Super-B experiments.
ACKNOWLEDGMENTS
This work is supported in part by the National Natural Science Foundation of China
under Grant No. 10975074 and 11235005.
Appendix A: Relevant functions
In this appendix, we present the functions appeared in the previous sections. The thresh-
old resummation factors S (x) is adopted from Ref. [19]:
t
21+2cΓ(3/2+c)
S = [x(1 x)]c, (A1)
t
√πΓ(1+c) −
and we here set the parameter c = 0.3. The hard functions h and h come form the Fourier
1 2
transform and can be written as
h (x ,x ,b ,b ) = K (β b )[θ(b b )I (α b )K (α b )
1 1 2 1 2 0 1 1 1 2 0 1 2 0 1 1
− (A2)
+θ(b b )I (α b )K (α b )]S (x ),
2 1 0 1 1 0 1 2 t 2
−
h (x ,x ,b ,b ) = K (β b )[θ(b b )I (α b )K (α b )
2 1 2 1 2 0 2 2 1 2 0 2 2 0 2 1
− (A3)
+θ(b b )I (α b )K (α b )]S (x ),
2 1 0 2 1 0 2 2 t 1
−
with α = m √x rη, β = m √x x rη+, α = m √x rη+ and β = β . The functions
1 Bc 2 1 Bc 1 2 2 Bc 1 2 1
K and I are modified Bessel functions.
0 0
10