ZJOU-PHY-TH-07-02
NJNU-TH-07-11
A Study of B0d → J/Ψη(′) Decays in the pQCD Approach
Xin Liua∗, Zhen-Jun Xiaob†, Hui-Sheng Wangc
a. Department of Physics, Zhejiang Ocean University,
Zhoushan, Zhejiang 316000, P.R. China
b. Department of Physics and Institute of Theoretical Physics,
Nanjing Normal University, Nanjing, Jiangsu 210097, P.R. China and
c. Department of Applied Mathematics and Physics,
Anhui University of Technology and Science,
Wuhu, Anhui 241000, P.R. China
(Dated: November 4, 2018)
Abstract
Motivated by the very recent measurement of the branching ratio of B0d → J/ψη decay, we
calculate the branching ratios of Bd
0 → J/ψη and Bd0 → J/Ψη′ decays in the perturbative
QCD (pQCD) approach. The pQCD predictions for the branching ratios of considered decays
are: BR(B0d → J/Ψη) = (1.96
+9.68
−0.65) × 10−6, which is consistent with the first experimental
measurement within errors; while BR(B0d → J/Ψη′) = (1.09
+3.76
−0.25) × 10−6, very similar with
B0d → J/Ψη decay and can be tested by the forthcoming LHC experiments. The measurements
of these decay channels may help us to understand the QCD dynamics in the corresponding
energy scale, especially the reliability of pQCD approach to these kinds of B meson decays.
PACS numbers: 13.25.Hw, 12.38.Bx, 14.40.Nd
∗ liuxin@zjou.edu.cn
† xiaozhenjun@njnu.edu.cn
http://arxiv.org/abs/0704.0395v1
Very recently, the first observation of B0d → J/Ψη decay was reported by Belle Collab-
oration [1], and the branching ratio measured is
BR(B0d → J/Ψη) = (9.5± 1.7(stat)± 0.8(syst))× 10−6, (1)
which is consistent with the currently available theoretical predictions [1, 2, 3].
Up to now, the theoretical calculations for the branching ratios of Bd → J/Ψη(′) decays
were obtained by using the heavy quark factorization approximation in Ref. [2], or from
the measured J/Ψπ0 and J/ΨK0 branching ratios[3, 4, 5] based on the assumption of
the SU(3) flavor symmetry of strong interaction. In this paper, we will calculate the
branching ratios of B0d → J/Ψη and B0d → J/Ψη(′) decays directly by employing the
low energy effective Hamiltonian [6] and the perturbative QCD (pQCD) factorization
approach [7, 8, 9].
The paper is organized as follows: we present the formalism used in the calculation of
B0d → J/ψη(′) decays in Sec. I. In Sec. II, we show the numerical results and compare
them with the measured values. A short summery and some conclusions are also included
in this section.
I. FORMALISM AND PERTURBATIVE CALCULATIONS
The pQCD approach has been developed earlier from the QCD hard-scattering ap-
proach [7], and has been used frequently to calculate various B meson decay channels
[7, 8, 9, 10]. For two body charmless hadronic Bd,s → Mη(′) (here M stands for the
pseudo-scalar or vector light mesons composed of the light quarks u, d, s) decays, the
pQCD predictions generally agree well with the measured values [9, 10, 11].
In Refs. [12, 13], the authors calculated B → D∗sK,D
s and Bs → D(∗)+D(∗)−
decays and found that the pQCD approach works well for such decays. Here we try to
apply the pQCD approach to calculate the B meson decays involving the heavier J/Ψ
meson as one of the two final state mesons.
A. Formulism
In pQCD approach, the decay amplitude of B → J/ΨP (P = η, η(′) here) decay can
bo written conceptually as the convolution,
A(B →M1M2) ∼
d4k1d
4k3 Tr
C(t)ΦB(k1)ΦJ/Ψ(k2)ΦP (k3)H(k1, k2, k3, t)
, (2)
where the term “Tr” denotes the trace over Dirac and color indices. C(t) is the Wilson
coefficient which results from the radiative corrections at short distance. In the above
convolution, C(t) includes the harder dynamics at larger scale thanMB scale and describes
the evolution of local 4-Fermi operators from mW (the W boson mass) down to t ∼
Λ̄MB) scale, where Λ̄ ≡ MB −mb. The function H(k1, k2, k3, t) is the hard part and
can be calculated perturbatively. The function ΦM is the wave function which describes
hadronization of the quark and anti-quark to the mesonM . While the functionH depends
on the process considered, the wave function ΦM is independent of the specific process.
Using the wave functions determined from other well measured processes, one can make
quantitative predictions here.
Using the light-cone coordinates the B meson and the two final state meson momenta
can be written as
(1, 1, 0T ), P2 =
(1, r2, 0T ), P3 =
(0, 1− r2, 0T ), (3)
respectively, where r = MJ/Ψ/MB, and the light meson masses m
η have been ne-
glected. The longitudinal polarization vector of the J/Ψ meson, ǫL, is given by ǫL =
2MJ/Ψ
(1,−r2, 0T ). Putting the light (anti-) quark momenta in B, J/Ψ and η(
′) mesons
as k1, k2, and k3, respectively, we can choose
k1 = (x1P
1 , 0,k1T ), k2 = (x2P
2 , 0,k2T ), k3 = (0, x3P
3 ,k3T ). (4)
Then, for B → J/Ψη decay for example, the integration over k−1 , k−2 , and k+3 in eq.(2)
will lead to
A(B → J/Ψη′) ∼
dx1dx2dx3b1db1b2db2b3db3
C(t)ΦB(x1, b1)ΦJ/Ψ(x2, b2)Φη(x3, b3)H(xi, bi, t)St(xi) e
−S(t)] ,(5)
where bi is the conjugate space coordinate of kiT , and t is the largest energy scale in
functionH(xi, bi, t). The large logarithms ln(mW/t) are included in the Wilson coefficients
C(t). The large double logarithms (ln2 xi) on the longitudinal direction are summed by
the threshold resummation [14], and they lead to St(xi) which smears the end-point
singularities on xi. The last term, e
−S(t), is the Sudakov form factor which suppresses the
soft dynamics effectively [15]. Thus it makes the perturbative calculation of the hard
part H applicable at intermediate scale, i.e., MB scale. We will calculate analytically the
function H(xi, bi, t) for the considered decays in the first order in αs expansion and give
the convoluted amplitudes in next section.
B. The B0d → J/Ψη(
′) Decays
The low energy effective Hamiltonian for decay modes B0d → J/ψη(
′) can be written as
Heff =
[VcbV
cd (C1(µ)O
1(µ) + C2(µ)O
2(µ))] , (6)
with the four-fermion operators
Oc1 = d̄αγ
µ(1− γ5)cβ · c̄βγµ(1− γ5)bα , Oc2 = d̄αγµ(1− γ5)cα · c̄βγµ(1− γ5)bβ (7)
where the Wilson coefficients Ci(µ) (i = 1, 2), we will use the leading order (LO) expres-
sions, although the next-to-leading order (NLO) results already exist in the literature [6].
This is the consistent way to cancel the explicit µ dependence in the theoretical formulae.
For the renormalization group evolution of the Wilson coefficients from higher scale to
lower scale, we use the formulae as given in Ref.[16] directly.
FIG. 1: Typical Feynman diagrams contributing to the Cabibbo- and color- suppressed B0d →
J/Ψη(
′) decays.
As for B meson wavefunction, we make use of the same parameterizations as used
in the studies of different processes [16]. For vector J/ψ meson, in terms of the nota-
tion in Ref. [17], we decompose the nonlocal matrix elements for the longitudinally and
transversely polarized J/ψ mesons into
ΦJ/Ψ(x) =
mJ/ψǫ/ LΨ
L(x) + ǫ/ LP/Ψ
, (8)
Here, ΨL denote for the twist-2 distribution amplitudes, and Ψt for the twist-3 distri-
bution amplitudes. x represents the momentum fraction of the charm quark inside the
charmonium.
The J/ψ meson asymptotic distribution amplitudes read as [18]
ΨL(x) = 9.58
x(1− x)
x(1 − x)
1− 2.8x(1− x)
Ψt(x) = 10.94
(1− 2x)2
x(1− x)
1− 2.8x(1− x)
. (9)
It is easy to see that both the twist-2 and twist-3 DAs vanish at the end points due to
the factor [x(1− x)]0.7.
From the effective Hamiltonian (6), the Feynman diagrams corresponding to the con-
sidered decay are shown in Fig.1. With the meson wave functions and Sudakov factors,
the hard amplitude is given as
Feη = 8πCFm
dx1dx3
b1db1b3db3 φB(x1, b1)
(1− r2)
(1 + x3(1− r2))φAη (x3, b3) + r0(1− 2x3)
·φPη (x3, b3)
(1− 2x3) + r2(1 + 2x3)
φTη (x3, b3)
·αs(t1e) he(x1, x3, b1, b3) exp[−Sab(t1e)]
1− (1− x1)r2φPη (x3, b3)− x1r2φAη (x3, b3)
·αs(t2e)he(x3, x1, b3, b1) exp[−Sab(t2e)]
. (10)
where r0 = m
0/mB; CF = 4/3 is a color factor. The function he, the scales t
e and the
Sudakov factors Sab are displayed in Appendix A.
For the non-factorizable diagrams 1(c) and 1(d), all three meson wave functions are
involved. The integration of b3 can be performed using δ function δ(b3 − b1), leaving only
integration of b1 and b2. For the concerned operators, the corresponding decay amplitude
Meη =
dx1dx2 dx3
b1db1b2db2 φBs(x1, b1)
2rrcφ
J/Ψ(x2, b2)φ
η (x3, b2)− 4rr0rcφtJ/Ψ(x2, b2)φTη (x3, b2)
2 + x3(1− 2r2)
φLJ/Ψ(x2, b2)φ
η (x3, b2)
x3r0 + (x2 − x3)r0r2
φLJ/Ψ(x2, b2)φ
η (x3, b2)
·αs(tf )hf(x1, x2, x3, b1, b2) exp[−Scd(tf )]} . (11)
where rc = mc/mB,mc is the mass for c quark.
For the B0d → J/Ψη′ decay, the Feynman diagrams are obtained by replacing the η
meson in Fig. 1 with the meson η′. The corresponding expressions of decay amplitudes will
be similar with those as given in Eqs.(10-11), since the η and η′ are all light pseudoscalar
mesons and have the similar wave functions. The expressions of B0d → J/Ψη′ decay can
be obtained simply by the following replacements
φAη −→ φAη′ , φPη −→ φPη′ , φTη −→ φTη′ , r0 −→ r′0. (12)
For the η−η′ system, there exist two popular mixing basis: the octet-singlet basis and
the quark-flavor basis [19, 20]. Here we use the quark-flavor basis [19] and define
ηq = (uū+ dd̄)/
2, ηs = ss̄. (13)
The physical states η and η′ are related to ηq and ηs through a single mixing angle φ,
= U(φ)
cosφ − sinφ
sin φ cos φ
. (14)
The three input parameters fq, fs and φ in the quark-flavor basis have been extracted
from various related experiments [19, 20]
fq = (1.07± 0.02)fπ, fs = (1.34± 0.06)fπ, φ = 39.3◦ ± 1.0◦, (15)
where fπ = 130 MeV. In the numerical calculations, we will use these mixing parameters
as inputs. It worth of mentioning that the effects of possible gluonic component of η′
meson will not considered here since it is small in size [10, 21, 22].
For B0d → J/Ψη decay, by combining the contributions from different diagrams, the
total decay amplitude can be written as
M(B0d → J/Ψη) = VcbV ∗cdF1(φ)
FeηfJ/Ψ
+MeηC2
where the relevant mixing parameter is F1(φ) = cos φ/
It should be mentioned that the Wilson coefficients Ci = Ci(t) in Eq. (16) should
be calculated at the appropriate scale t using equations as given in the Appendices of
Ref. [16]. Here the scale t in the Wilson coefficients should be taken as the same scale
appeared in the expressions of decay amplitudes in Eqs. (10) and (11). This is the way
in pQCD approach to eliminate the scale dependence. In order to estimate the effect of
higher order corrections, however, we introduce a scale factor at = 1.0± 0.2 and vary the
scale tmax as described in Appendix A.
Similarly, the decay amplitudes for B0d → J/Ψη′ decay can be obtained easily from
Eq.(16) by the following replacements of F1(φ) → F ′1(φ) = sinφ/
II. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we will calculate the branching ratios for those considered decay modes.
The input parameters and the wave functions to be used are given in Appendix B. In
numerical calculations, central values of input parameters will be used implicitly unless
otherwise stated.
With the complete decay amplitudes, we can obtain the decay width for the considered
decays,
Γ(B0d → J/ψη(
′)) =
(1− r2)
M(B0d → J/ψη(
. (17)
By employing the quark-flavor scheme of η−η′ system and using the mixing parameters
as given in Eq. (15), one finds the branching ratios for the considered two decays with
error bars as follows:
Br( B0d → J/Ψη) =
1.96+0.71−0.50(ωb)
+9.65
−0.39(at)
+0.32
+0.13(a2)
+0.14
−0.13(fJ/Ψ)
× 10−6, (18)
Br( B0d → J/Ψη′) =
1.09+0.32−0.24(ωb)
+3.73
+0.01(at)
+0.28
+0.01(a2)
+0.08
−0.07(fJ/Ψ)
× 10−6, (19)
where the main errors are induced by the uncertainties of ωb = 0.40 ± 0.05 GeV, at =
1.0 ± 0.2, a2 = 0.115 ± 0.115 and fJ/Ψ = 0.405 ± 0.014 GeV , respectively. One can see
that the pQCD predictions are sensitive to the variations of ωb and at.
For B0d → J/Ψη decay, the central value of the pQCD prediction for Br(B0d → J/Ψη)
is a factor of 4 smaller than the measured value as given in Eq. (1) [1]. But the pQCD
prediction is in fact still consistent with Belle’s first measurement if we take the large
theoretical and experimental errors into account. By varying the scale factor at in the
range of at = [0.8, 1.0], for example, the central value of Br(B → J/Ψη) will change in the
range of [0.2, 1.1]×10−5 accordingly. It is not difficult to understand such at dependence.
Since the J/Ψ meson is much heavier than light mesons, and therefore moving not as fast
as those light meson when B meson is decaying. So a small decrease of the scale ti will
lead to a larger Wilson coefficients C1,2(t) and αs(ti), and consequently results in a larger
decay rate.
For B0d → J/Ψη′ decay, only experimental upper limit (at 90% C.L) is available now:
BR(B0 → J/Ψη′) < 6.3 × 10−5 [4, 5]. The pQCD prediction for the branching ratio of
B0d → J/Ψη′ decay is very similar in magnitude with that of B0d → J/Ψη, consistent with
the upper limit and will be tested in the forthcoming LHC experiments.
At the leading order, only the tree Feynman diagrams as shown in Fig. 1 contribute to
B0d → J/Ψη(′) decays. There exists no CP violation in these decays within the standard
model, since there is only one kind of Cabibbo-Kabayashi-Muskawa (CKM) phase involved
in the corresponding decay amplitudes, as can be seen from eq. (16).
In short, we calculated the branching ratios of B0d → J/Ψη and B0d → J/Ψη′ decays at
the leading order by using the pQCD factorization approach. Besides the usual factoriz-
able diagrams, the non-factorizable spectator diagrams are also calculated analytically in
the pQCD approach. By keeping the transverse momentum kT , the end-point singularity
disappears in our calculation.
From our calculations and phenomenological analysis, we found the following results:
• Using the quark-flavor scheme, the pQCD predictions for the branching ratios are
Br(B0d → J/Ψη) =
1.96+9.68−0.65
× 10−6, (20)
Br(B0d → J/Ψη′) =
1.09+3.76−0.25
× 10−6, (21)
where the various errors as specified previously have been added in quadrature.
• The major theoretical errors of the pQCD predictions are induced by the uncertain-
ties of the hard energy scale ti’s and the parameters ωb.
Acknowledgments
X. Liu would like to acknowledge the financial support of The Scientific Research
Start-up Fund of Zhejiang Ocean University under Grant No.21065010706. This work was
partially supported by the National Natural Science Foundation of China under Grant
No.10575052, and by the Specialized Research Fund for the Doctoral Program of Higher
Education (SRFDP) under Grant No. 20050319008.
APPENDIX A: RELATED FUNCTIONS
We show here the function hi’s, coming from the Fourier transformations of the function
H(0),
he(x1, x3, b1, b3) = K0
x1x3(1− r2)mBb1
θ(b1 − b3)K0
x3(1− r2)mBb1
x3(1− r2)mBb3
+ θ(b3 − b1)K0
x3(1− r2)mBb3
x3(1− r2)mBb1
St(x3), (A1)
hf(x1, x2, x3, b1, b2) =
θ(b2 − b1)I0(MB
x1x3(1− r2)b1)K0(MB
x1x3(1− r2)b2)
+ (b1 ↔ b2)
K0(MBF(1)b2), for F
(1) > 0
0 (MB
| b2), for F 2(1) < 0
, (A2)
where J0 is the Bessel function, K0 and I0 are the modified Bessel functions with
K0(−ix) = −(π/2)Y0(x) + i(π/2)J0(x), and F(j)’s are defined by
F 2(1) = (x1 − x2)x3(1− r2) + r2c , (A3)
F 2(2) = (x1 − x2)x3(1− r2) + r2c . (A4)
The threshold resummation form factor St(xi) is adopted from Ref. [17]
St(x) =
21+2cΓ(3/2 + c)√
πΓ(1 + c)
[x(1 − x)]c, (A5)
where the parameter c = 0.3. This function is normalized to unity.
The Sudakov factors used in the text are defined as
Sab(t) = s
x1mB/
2, b1
x3mB/
2, b3
(1− x3)mB/
2, b3
ln(t/Λ)
− ln(b1Λ)
ln(t/Λ)
− ln(b3Λ)
, (A6)
Scd(t) = s
x1mB/
2, b1
x2mB/
2, b2
(1− x2)mB/
2, b2
x3mB/
2, b1
(1− x3)mB/
2, b1
ln(t/Λ)
− ln(b1Λ)
ln(t/Λ)
− ln(b2Λ)
, (A7)
where the function s(q, b) are defined in the Appendix A of Ref. [16]. The scale ti’s in the
above equations are chosen as
t1e = at ·max(
x3(1− r2)MB, 1/b1, 1/b3),
t2e = at ·max(
x1(1− r2)MB, 1/b1, 1/b3),
tf = at ·max(
x1x3(1− r2)MB,
(x1 − x2)x3(1− r2) + r2cMB, 1/b1, 1/b2), (A8)
where at = 1.0±0.2 and r =MJ/Ψ/MB. The scale ti’s are chosen as the maximum energy
scale appearing in each diagram to kill the large logarithmic radiative corrections.
APPENDIX B: INPUT PARAMETERS AND WAVE FUNCTIONS
The masses, decay constants, QCD scale and B0d meson lifetime are
(f=4)
= 250MeV, fπ = 130MeV, fJ/Ψ = 405MeV,
0 = 1.08GeV, MB0d = 5.28MeV, MJ/Ψ = 3.097GeV,
MW = 80.41GeV, τB0
= 1.54× 10−12s. (B1)
For the CKM matrix elements, here we adopt the Wolfenstein parametrization for the
CKM matrix, and take λ = 0.2272, A = 0.818, ρ = 0.221 and η = 0.340 [4].
For the B meson wave function, we adopt the model
φB(x, b) = NBx
2(1− x)2exp
(ωbb)
, (B2)
where ωb is a free parameter and we take ωb = 0.40± 0.05 GeV in numerical calculations,
and NB = 91.745 is the normalization factor for ωb = 0.40 for the B meson.
The wave function for dd̄ components of η(′) meson is given by
Φηdd̄(p, x, ζ) ≡
P/φAηdd̄(x) +m
(x) + ζm
0 (v/n/− v · n)φTηdd̄(x)
, (B3)
where p and x are the momentum and the momentum fraction of ηdd̄ respectively, while
φAηdd̄, φ
and φTηdd̄ represent the axial vector, pseudoscalar and tensor components of the
wave function respectively. We here assume that the wave function of ηdd̄ is same as the
π wave function based on SU(3) flavor symmetry. The parameter ζ is either +1 or −1
depending on the assignment of the momentum fraction x.
The explicit expression of chiral enhancement scale m
0 = m
0 is given by [21]
[m2η cos
2 φ+m2η′ sin
(m2η′ −m2η) cosφsinφ], (B4)
and numerically m
0 = 1.07MeV for mη = 547.5 MeV, mη′ = 957.8 MeV, fq = 1.07fπ,
fs = 1.34fπ and φ = 39.3
For the distribution amplitude φAηq , φ
and φTηq , we utilize the results for π meson
obtained from the light-cone sum rule [23] including twist-3 contributions:
φAηq(x) =
fqx(1 − x)
1 + a
5(1− 2x)2 − 1
21(1− 2x)4 − 14(1− 2x)2 + 1
, (B5)
φPηq(x) =
30η3 −
3(1− 2x)2 − 1
−3η3ω3 −
ρ2ηq −
ρ2ηq(s)a
35(1− 2x)4 − 30(1− 2x)2 + 3
,(B6)
φTηq(x) =
fq(1− 2x)
+ (5η3 −
η3ω3 −
ρ2ηq −
ρ2ηqa
2 )(10x
2 − 10x+ 1)
, (B7)
with the updated Gegenbauer moments [24]
2 = 0.115, a
4 = −0.015, ρηq = 2mq/mqq, η3 = 0.015, ω3 = −3.0. (B8)
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(1999).
[24] P. Ball and R. Zwicky, Phys. Rev. D 71, 014015 (2005); P. Ball, V.M. Braun, and A. Lenz,
J. High Energy Phys. 0605 (2006) 004.
http://arxiv.org/abs/hep-ex/0603003
http://www.slac.stanford.edu/xorg/hfag
http://arxiv.org/abs/hep-ph/0701146
http://arxiv.org/abs/hep-ph/0703162
http://arxiv.org/abs/hep-ph/0703187
	Formalism and Perturbative Calculations
	Formulism
	 The Bd0 J/(') Decays
	Numerical results and Discussions
	Acknowledgments
	Related Functions 
	Input parameters and wave functions
	References

  Motivated by the very recent measurement of the branching ratio of ${B_d^0}
\to J/\psi \eta$ decay, we calculate the branching ratios of ${B_d}^0 \to
J/\psi \eta$ and ${B_d}^0 \to J/\Psi \eta'$ decays in the perturbative QCD
(pQCD) approach. The pQCD predictions for the branching ratios of considered
decays are: $BR(B_d^0 \to J/\Psi \eta) = (1.96 ^{+9.68}_{-0.65}) \times
10^{-6}$, which is consistent with the first experimental measurement within
errors; while $BR(B_d^0 \to J/\Psi \eta') = (1.09 ^{+3.76}_{-0.25}) \times
10^{-6}$, very similar with $B_d^0 \to \jpsi \eta$ decay and can be tested by
the forthcoming LHC experiments. The measurements of these decay channels may
help us to understand the QCD dynamics in the corresponding energy scale,
especially the reliability of pQCD approach to these kinds of B meson decays.

ZJOU-PHY-TH-07-02
NJNU-TH-07-11
A Study of B0d → J/Ψη(′) Decays in the pQCD Approach
Xin Liua∗, Zhen-Jun Xiaob†, Hui-Sheng Wangc
a. Department of Physics, Zhejiang Ocean University,
Zhoushan, Zhejiang 316000, P.R. China
b. Department of Physics and Institute of Theoretical Physics,
Nanjing Normal University, Nanjing, Jiangsu 210097, P.R. China and
c. Department of Applied Mathematics and Physics,
Anhui University of Technology and Science,
Wuhu, Anhui 241000, P.R. China
(Dated: November 4, 2018)
Abstract
Motivated by the very recent measurement of the branching ratio of B0d → J/ψη decay, we
calculate the branching ratios of Bd
0 → J/ψη and Bd0 → J/Ψη′ decays in the perturbative
QCD (pQCD) approach. The pQCD predictions for the branching ratios of considered decays
are: BR(B0d → J/Ψη) = (1.96
+9.68
−0.65) × 10−6, which is consistent with the first experimental
measurement within errors; while BR(B0d → J/Ψη′) = (1.09
+3.76
−0.25) × 10−6, very similar with
B0d → J/Ψη decay and can be tested by the forthcoming LHC experiments. The measurements
of these decay channels may help us to understand the QCD dynamics in the corresponding
energy scale, especially the reliability of pQCD approach to these kinds of B meson decays.
PACS numbers: 13.25.Hw, 12.38.Bx, 14.40.Nd
∗ liuxin@zjou.edu.cn
† xiaozhenjun@njnu.edu.cn
http://arxiv.org/abs/0704.0395v1
Very recently, the first observation of B0d → J/Ψη decay was reported by Belle Collab-
oration [1], and the branching ratio measured is
BR(B0d → J/Ψη) = (9.5± 1.7(stat)± 0.8(syst))× 10−6, (1)
which is consistent with the currently available theoretical predictions [1, 2, 3].
Up to now, the theoretical calculations for the branching ratios of Bd → J/Ψη(′) decays
were obtained by using the heavy quark factorization approximation in Ref. [2], or from
the measured J/Ψπ0 and J/ΨK0 branching ratios[3, 4, 5] based on the assumption of
the SU(3) flavor symmetry of strong interaction. In this paper, we will calculate the
branching ratios of B0d → J/Ψη and B0d → J/Ψη(′) decays directly by employing the
low energy effective Hamiltonian [6] and the perturbative QCD (pQCD) factorization
approach [7, 8, 9].
The paper is organized as follows: we present the formalism used in the calculation of
B0d → J/ψη(′) decays in Sec. I. In Sec. II, we show the numerical results and compare
them with the measured values. A short summery and some conclusions are also included
in this section.
I. FORMALISM AND PERTURBATIVE CALCULATIONS
The pQCD approach has been developed earlier from the QCD hard-scattering ap-
proach [7], and has been used frequently to calculate various B meson decay channels
[7, 8, 9, 10]. For two body charmless hadronic Bd,s → Mη(′) (here M stands for the
pseudo-scalar or vector light mesons composed of the light quarks u, d, s) decays, the
pQCD predictions generally agree well with the measured values [9, 10, 11].
In Refs. [12, 13], the authors calculated B → D∗sK,D
s and Bs → D(∗)+D(∗)−
decays and found that the pQCD approach works well for such decays. Here we try to
apply the pQCD approach to calculate the B meson decays involving the heavier J/Ψ
meson as one of the two final state mesons.
A. Formulism
In pQCD approach, the decay amplitude of B → J/ΨP (P = η, η(′) here) decay can
bo written conceptually as the convolution,
A(B →M1M2) ∼
d4k1d
4k3 Tr
C(t)ΦB(k1)ΦJ/Ψ(k2)ΦP (k3)H(k1, k2, k3, t)
, (2)
where the term “Tr” denotes the trace over Dirac and color indices. C(t) is the Wilson
coefficient which results from the radiative corrections at short distance. In the above
convolution, C(t) includes the harder dynamics at larger scale thanMB scale and describes
the evolution of local 4-Fermi operators from mW (the W boson mass) down to t ∼
Λ̄MB) scale, where Λ̄ ≡ MB −mb. The function H(k1, k2, k3, t) is the hard part and
can be calculated perturbatively. The function ΦM is the wave function which describes
hadronization of the quark and anti-quark to the mesonM . While the functionH depends
on the process considered, the wave function ΦM is independent of the specific process.
Using the wave functions determined from other well measured processes, one can make
quantitative predictions here.
Using the light-cone coordinates the B meson and the two final state meson momenta
can be written as
(1, 1, 0T ), P2 =
(1, r2, 0T ), P3 =
(0, 1− r2, 0T ), (3)
respectively, where r = MJ/Ψ/MB, and the light meson masses m
η have been ne-
glected. The longitudinal polarization vector of the J/Ψ meson, ǫL, is given by ǫL =
2MJ/Ψ
(1,−r2, 0T ). Putting the light (anti-) quark momenta in B, J/Ψ and η(
′) mesons
as k1, k2, and k3, respectively, we can choose
k1 = (x1P
1 , 0,k1T ), k2 = (x2P
2 , 0,k2T ), k3 = (0, x3P
3 ,k3T ). (4)
Then, for B → J/Ψη decay for example, the integration over k−1 , k−2 , and k+3 in eq.(2)
will lead to
A(B → J/Ψη′) ∼
dx1dx2dx3b1db1b2db2b3db3
C(t)ΦB(x1, b1)ΦJ/Ψ(x2, b2)Φη(x3, b3)H(xi, bi, t)St(xi) e
−S(t)] ,(5)
where bi is the conjugate space coordinate of kiT , and t is the largest energy scale in
functionH(xi, bi, t). The large logarithms ln(mW/t) are included in the Wilson coefficients
C(t). The large double logarithms (ln2 xi) on the longitudinal direction are summed by
the threshold resummation [14], and they lead to St(xi) which smears the end-point
singularities on xi. The last term, e
−S(t), is the Sudakov form factor which suppresses the
soft dynamics effectively [15]. Thus it makes the perturbative calculation of the hard
part H applicable at intermediate scale, i.e., MB scale. We will calculate analytically the
function H(xi, bi, t) for the considered decays in the first order in αs expansion and give
the convoluted amplitudes in next section.
B. The B0d → J/Ψη(
′) Decays
The low energy effective Hamiltonian for decay modes B0d → J/ψη(
′) can be written as
Heff =
[VcbV
cd (C1(µ)O
1(µ) + C2(µ)O
2(µ))] , (6)
with the four-fermion operators
Oc1 = d̄αγ
µ(1− γ5)cβ · c̄βγµ(1− γ5)bα , Oc2 = d̄αγµ(1− γ5)cα · c̄βγµ(1− γ5)bβ (7)
where the Wilson coefficients Ci(µ) (i = 1, 2), we will use the leading order (LO) expres-
sions, although the next-to-leading order (NLO) results already exist in the literature [6].
This is the consistent way to cancel the explicit µ dependence in the theoretical formulae.
For the renormalization group evolution of the Wilson coefficients from higher scale to
lower scale, we use the formulae as given in Ref.[16] directly.
FIG. 1: Typical Feynman diagrams contributing to the Cabibbo- and color- suppressed B0d →
J/Ψη(
′) decays.
As for B meson wavefunction, we make use of the same parameterizations as used
in the studies of different processes [16]. For vector J/ψ meson, in terms of the nota-
tion in Ref. [17], we decompose the nonlocal matrix elements for the longitudinally and
transversely polarized J/ψ mesons into
ΦJ/Ψ(x) =
mJ/ψǫ/ LΨ
L(x) + ǫ/ LP/Ψ
, (8)
Here, ΨL denote for the twist-2 distribution amplitudes, and Ψt for the twist-3 distri-
bution amplitudes. x represents the momentum fraction of the charm quark inside the
charmonium.
The J/ψ meson asymptotic distribution amplitudes read as [18]
ΨL(x) = 9.58
x(1− x)
x(1 − x)
1− 2.8x(1− x)
Ψt(x) = 10.94
(1− 2x)2
x(1− x)
1− 2.8x(1− x)
. (9)
It is easy to see that both the twist-2 and twist-3 DAs vanish at the end points due to
the factor [x(1− x)]0.7.
From the effective Hamiltonian (6), the Feynman diagrams corresponding to the con-
sidered decay are shown in Fig.1. With the meson wave functions and Sudakov factors,
the hard amplitude is given as
Feη = 8πCFm
dx1dx3
b1db1b3db3 φB(x1, b1)
(1− r2)
(1 + x3(1− r2))φAη (x3, b3) + r0(1− 2x3)
·φPη (x3, b3)
(1− 2x3) + r2(1 + 2x3)
φTη (x3, b3)
·αs(t1e) he(x1, x3, b1, b3) exp[−Sab(t1e)]
1− (1− x1)r2φPη (x3, b3)− x1r2φAη (x3, b3)
·αs(t2e)he(x3, x1, b3, b1) exp[−Sab(t2e)]
. (10)
where r0 = m
0/mB; CF = 4/3 is a color factor. The function he, the scales t
e and the
Sudakov factors Sab are displayed in Appendix A.
For the non-factorizable diagrams 1(c) and 1(d), all three meson wave functions are
involved. The integration of b3 can be performed using δ function δ(b3 − b1), leaving only
integration of b1 and b2. For the concerned operators, the corresponding decay amplitude
Meη =
dx1dx2 dx3
b1db1b2db2 φBs(x1, b1)
2rrcφ
J/Ψ(x2, b2)φ
η (x3, b2)− 4rr0rcφtJ/Ψ(x2, b2)φTη (x3, b2)
2 + x3(1− 2r2)
φLJ/Ψ(x2, b2)φ
η (x3, b2)
x3r0 + (x2 − x3)r0r2
φLJ/Ψ(x2, b2)φ
η (x3, b2)
·αs(tf )hf(x1, x2, x3, b1, b2) exp[−Scd(tf )]} . (11)
where rc = mc/mB,mc is the mass for c quark.
For the B0d → J/Ψη′ decay, the Feynman diagrams are obtained by replacing the η
meson in Fig. 1 with the meson η′. The corresponding expressions of decay amplitudes will
be similar with those as given in Eqs.(10-11), since the η and η′ are all light pseudoscalar
mesons and have the similar wave functions. The expressions of B0d → J/Ψη′ decay can
be obtained simply by the following replacements
φAη −→ φAη′ , φPη −→ φPη′ , φTη −→ φTη′ , r0 −→ r′0. (12)
For the η−η′ system, there exist two popular mixing basis: the octet-singlet basis and
the quark-flavor basis [19, 20]. Here we use the quark-flavor basis [19] and define
ηq = (uū+ dd̄)/
2, ηs = ss̄. (13)
The physical states η and η′ are related to ηq and ηs through a single mixing angle φ,
= U(φ)
cosφ − sinφ
sin φ cos φ
. (14)
The three input parameters fq, fs and φ in the quark-flavor basis have been extracted
from various related experiments [19, 20]
fq = (1.07± 0.02)fπ, fs = (1.34± 0.06)fπ, φ = 39.3◦ ± 1.0◦, (15)
where fπ = 130 MeV. In the numerical calculations, we will use these mixing parameters
as inputs. It worth of mentioning that the effects of possible gluonic component of η′
meson will not considered here since it is small in size [10, 21, 22].
For B0d → J/Ψη decay, by combining the contributions from different diagrams, the
total decay amplitude can be written as
M(B0d → J/Ψη) = VcbV ∗cdF1(φ)
FeηfJ/Ψ
+MeηC2
where the relevant mixing parameter is F1(φ) = cos φ/
It should be mentioned that the Wilson coefficients Ci = Ci(t) in Eq. (16) should
be calculated at the appropriate scale t using equations as given in the Appendices of
Ref. [16]. Here the scale t in the Wilson coefficients should be taken as the same scale
appeared in the expressions of decay amplitudes in Eqs. (10) and (11). This is the way
in pQCD approach to eliminate the scale dependence. In order to estimate the effect of
higher order corrections, however, we introduce a scale factor at = 1.0± 0.2 and vary the
scale tmax as described in Appendix A.
Similarly, the decay amplitudes for B0d → J/Ψη′ decay can be obtained easily from
Eq.(16) by the following replacements of F1(φ) → F ′1(φ) = sinφ/
II. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we will calculate the branching ratios for those considered decay modes.
The input parameters and the wave functions to be used are given in Appendix B. In
numerical calculations, central values of input parameters will be used implicitly unless
otherwise stated.
With the complete decay amplitudes, we can obtain the decay width for the considered
decays,
Γ(B0d → J/ψη(
′)) =
(1− r2)
M(B0d → J/ψη(
. (17)
By employing the quark-flavor scheme of η−η′ system and using the mixing parameters
as given in Eq. (15), one finds the branching ratios for the considered two decays with
error bars as follows:
Br( B0d → J/Ψη) =
1.96+0.71−0.50(ωb)
+9.65
−0.39(at)
+0.32
+0.13(a2)
+0.14
−0.13(fJ/Ψ)
× 10−6, (18)
Br( B0d → J/Ψη′) =
1.09+0.32−0.24(ωb)
+3.73
+0.01(at)
+0.28
+0.01(a2)
+0.08
−0.07(fJ/Ψ)
× 10−6, (19)
where the main errors are induced by the uncertainties of ωb = 0.40 ± 0.05 GeV, at =
1.0 ± 0.2, a2 = 0.115 ± 0.115 and fJ/Ψ = 0.405 ± 0.014 GeV , respectively. One can see
that the pQCD predictions are sensitive to the variations of ωb and at.
For B0d → J/Ψη decay, the central value of the pQCD prediction for Br(B0d → J/Ψη)
is a factor of 4 smaller than the measured value as given in Eq. (1) [1]. But the pQCD
prediction is in fact still consistent with Belle’s first measurement if we take the large
theoretical and experimental errors into account. By varying the scale factor at in the
range of at = [0.8, 1.0], for example, the central value of Br(B → J/Ψη) will change in the
range of [0.2, 1.1]×10−5 accordingly. It is not difficult to understand such at dependence.
Since the J/Ψ meson is much heavier than light mesons, and therefore moving not as fast
as those light meson when B meson is decaying. So a small decrease of the scale ti will
lead to a larger Wilson coefficients C1,2(t) and αs(ti), and consequently results in a larger
decay rate.
For B0d → J/Ψη′ decay, only experimental upper limit (at 90% C.L) is available now:
BR(B0 → J/Ψη′) < 6.3 × 10−5 [4, 5]. The pQCD prediction for the branching ratio of
B0d → J/Ψη′ decay is very similar in magnitude with that of B0d → J/Ψη, consistent with
the upper limit and will be tested in the forthcoming LHC experiments.
At the leading order, only the tree Feynman diagrams as shown in Fig. 1 contribute to
B0d → J/Ψη(′) decays. There exists no CP violation in these decays within the standard
model, since there is only one kind of Cabibbo-Kabayashi-Muskawa (CKM) phase involved
in the corresponding decay amplitudes, as can be seen from eq. (16).
In short, we calculated the branching ratios of B0d → J/Ψη and B0d → J/Ψη′ decays at
the leading order by using the pQCD factorization approach. Besides the usual factoriz-
able diagrams, the non-factorizable spectator diagrams are also calculated analytically in
the pQCD approach. By keeping the transverse momentum kT , the end-point singularity
disappears in our calculation.
From our calculations and phenomenological analysis, we found the following results:
• Using the quark-flavor scheme, the pQCD predictions for the branching ratios are
Br(B0d → J/Ψη) =
1.96+9.68−0.65
× 10−6, (20)
Br(B0d → J/Ψη′) =
1.09+3.76−0.25
× 10−6, (21)
where the various errors as specified previously have been added in quadrature.
• The major theoretical errors of the pQCD predictions are induced by the uncertain-
ties of the hard energy scale ti’s and the parameters ωb.
Acknowledgments
X. Liu would like to acknowledge the financial support of The Scientific Research
Start-up Fund of Zhejiang Ocean University under Grant No.21065010706. This work was
partially supported by the National Natural Science Foundation of China under Grant
No.10575052, and by the Specialized Research Fund for the Doctoral Program of Higher
Education (SRFDP) under Grant No. 20050319008.
APPENDIX A: RELATED FUNCTIONS
We show here the function hi’s, coming from the Fourier transformations of the function
H(0),
he(x1, x3, b1, b3) = K0
x1x3(1− r2)mBb1
θ(b1 − b3)K0
x3(1− r2)mBb1
x3(1− r2)mBb3
+ θ(b3 − b1)K0
x3(1− r2)mBb3
x3(1− r2)mBb1
St(x3), (A1)
hf(x1, x2, x3, b1, b2) =
θ(b2 − b1)I0(MB
x1x3(1− r2)b1)K0(MB
x1x3(1− r2)b2)
+ (b1 ↔ b2)
K0(MBF(1)b2), for F
(1) > 0
0 (MB
| b2), for F 2(1) < 0
, (A2)
where J0 is the Bessel function, K0 and I0 are the modified Bessel functions with
K0(−ix) = −(π/2)Y0(x) + i(π/2)J0(x), and F(j)’s are defined by
F 2(1) = (x1 − x2)x3(1− r2) + r2c , (A3)
F 2(2) = (x1 − x2)x3(1− r2) + r2c . (A4)
The threshold resummation form factor St(xi) is adopted from Ref. [17]
St(x) =
21+2cΓ(3/2 + c)√
πΓ(1 + c)
[x(1 − x)]c, (A5)
where the parameter c = 0.3. This function is normalized to unity.
The Sudakov factors used in the text are defined as
Sab(t) = s
x1mB/
2, b1
x3mB/
2, b3
(1− x3)mB/
2, b3
ln(t/Λ)
− ln(b1Λ)
ln(t/Λ)
− ln(b3Λ)
, (A6)
Scd(t) = s
x1mB/
2, b1
x2mB/
2, b2
(1− x2)mB/
2, b2
x3mB/
2, b1
(1− x3)mB/
2, b1
ln(t/Λ)
− ln(b1Λ)
ln(t/Λ)
− ln(b2Λ)
, (A7)
where the function s(q, b) are defined in the Appendix A of Ref. [16]. The scale ti’s in the
above equations are chosen as
t1e = at ·max(
x3(1− r2)MB, 1/b1, 1/b3),
t2e = at ·max(
x1(1− r2)MB, 1/b1, 1/b3),
tf = at ·max(
x1x3(1− r2)MB,
(x1 − x2)x3(1− r2) + r2cMB, 1/b1, 1/b2), (A8)
where at = 1.0±0.2 and r =MJ/Ψ/MB. The scale ti’s are chosen as the maximum energy
scale appearing in each diagram to kill the large logarithmic radiative corrections.
APPENDIX B: INPUT PARAMETERS AND WAVE FUNCTIONS
The masses, decay constants, QCD scale and B0d meson lifetime are
(f=4)
= 250MeV, fπ = 130MeV, fJ/Ψ = 405MeV,
0 = 1.08GeV, MB0d = 5.28MeV, MJ/Ψ = 3.097GeV,
MW = 80.41GeV, τB0
= 1.54× 10−12s. (B1)
For the CKM matrix elements, here we adopt the Wolfenstein parametrization for the
CKM matrix, and take λ = 0.2272, A = 0.818, ρ = 0.221 and η = 0.340 [4].
For the B meson wave function, we adopt the model
φB(x, b) = NBx
2(1− x)2exp
(ωbb)
, (B2)
where ωb is a free parameter and we take ωb = 0.40± 0.05 GeV in numerical calculations,
and NB = 91.745 is the normalization factor for ωb = 0.40 for the B meson.
The wave function for dd̄ components of η(′) meson is given by
Φηdd̄(p, x, ζ) ≡
P/φAηdd̄(x) +m
(x) + ζm
0 (v/n/− v · n)φTηdd̄(x)
, (B3)
where p and x are the momentum and the momentum fraction of ηdd̄ respectively, while
φAηdd̄, φ
and φTηdd̄ represent the axial vector, pseudoscalar and tensor components of the
wave function respectively. We here assume that the wave function of ηdd̄ is same as the
π wave function based on SU(3) flavor symmetry. The parameter ζ is either +1 or −1
depending on the assignment of the momentum fraction x.
The explicit expression of chiral enhancement scale m
0 = m
0 is given by [21]
[m2η cos
2 φ+m2η′ sin
(m2η′ −m2η) cosφsinφ], (B4)
and numerically m
0 = 1.07MeV for mη = 547.5 MeV, mη′ = 957.8 MeV, fq = 1.07fπ,
fs = 1.34fπ and φ = 39.3
For the distribution amplitude φAηq , φ
and φTηq , we utilize the results for π meson
obtained from the light-cone sum rule [23] including twist-3 contributions:
φAηq(x) =
fqx(1 − x)
1 + a
5(1− 2x)2 − 1
21(1− 2x)4 − 14(1− 2x)2 + 1
, (B5)
φPηq(x) =
30η3 −
3(1− 2x)2 − 1
−3η3ω3 −
ρ2ηq −
ρ2ηq(s)a
35(1− 2x)4 − 30(1− 2x)2 + 3
,(B6)
φTηq(x) =
fq(1− 2x)
+ (5η3 −
η3ω3 −
ρ2ηq −
ρ2ηqa
2 )(10x
2 − 10x+ 1)
, (B7)
with the updated Gegenbauer moments [24]
2 = 0.115, a
4 = −0.015, ρηq = 2mq/mqq, η3 = 0.015, ω3 = −3.0. (B8)
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http://arxiv.org/abs/hep-ex/0603003
http://www.slac.stanford.edu/xorg/hfag
http://arxiv.org/abs/hep-ph/0701146
http://arxiv.org/abs/hep-ph/0703162
http://arxiv.org/abs/hep-ph/0703187
	Formalism and Perturbative Calculations
	Formulism
	 The Bd0 J/(') Decays
	Numerical results and Discussions
	Acknowledgments
	Related Functions 
	Input parameters and wave functions
	References
