Study of B c → J / ψ π , η c π decays with perturbative QCD approach

Abstract

The B

c
→J/ψ
π, η

c

π decays are studied with the perturbative QCD approach. It is found that the form factors
A0,1,2Bc® J/yA_{0,1,2}^{B_{c}\to J/{\psi}}
and
F0Bc®hcF_{0}^{B_{c}\to{\eta}_{c}}
for the B

c
→J/ψ, η

c
transitions and the branching ratios are sensitive to the parameters ω, v, f

J/ψ
and
fhcf_{{\eta}_{c}}
, where ω and v are the parameters of the charmonium wave functions for a Coulomb potential and the harmonic-oscillator potential, respectively,
and f

J/ψ
and
fhcf_{{\eta}_{c}}
are the decay constants of the J/ψ and η

c
mesons, respectively. The large branching ratios and the clear signals of the final states make the B

c
→J/ψ
π, η

c

π decays the prospective channels for measurements at the hadron colliders.

Full text

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arXiv:0808.3619v3 [hep-ph] 28 Feb 2009
Study of Bc→ J/ψπ, ηcπ decays with perturbative QCD approach
Junfeng Sun,1,2Dongsheng Du,3and Yueling Yang1
1College of Physics and Information Engineering,
Henan Normal University, Xinxiang 453007, China∗
2Theoretical Physics Center for Science Facilities,
Chinese Academy of Sciences, P.O.Box 918(4), Beijing 100049, China
3Institute of High Energy Physics, Chinese Academy of Sciences,
P.O.Box 918(4), Beijing 100049, China
Abstract
The Bc→ J/ψπ, ηcπ decays are studied with the perturbative QCD approach. It is found that
the form factors ABc→J/ψ
0,1,2
and FBc→ηc
0
for the Bc→ J/ψ, ηctransitions and the branching ratios are
sensitive to the parameters ω, v, fJ/ψand fηc, where ω and v are the parameters of the charmonium
wave functions for Coulomb potential and harmonic oscillator potential, respectively, fJ/ψand fηc
are the decay constants of the J/ψ and ηcmesons, respectively. The large branching ratios and the
clear signals of the final states make the Bc→ J/ψπ, ηcπ decays to be the prospective channels
for measurements at the hadron colliders.
PACS numbers: 12.39.St 13.25.Hw
∗Mailing address
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I.INTRODUCTION
The Large Hadron Collider (LHC) is schedule to run in this year. At the era of the LHC,
there is still a room for B physics. The study of the decays of B mesons is important and
interesting for the determination of the flavor parameters of the Standard Model (SM), the
exploration of CP violation, the search of new physics beyond SM, and so on. The decays of
Bu,dmesons have been investigated widely by the detectors at the e+e−colliders, such as the
CLEO, Babar, Belle. The Bcmeson could be produced abundantly and studied detailedly
at the hadron colliders, such as the Tevatron and LHC. The study of the Bcmesons will
highlight the advantages of B physics.
Compared with the Bu,dmesons, the Bcmesons have some special properties: (1) The
Bcmesons are the “double heavy-flavored” binding systems. We can study the two heavy
flavors of both b and c quarks simultaneously with the Bc mesons. (2) The Bc mesons
have much rich decay modes, because they have sufficiently large mass and that either b
or c quarks can decay individually. The potential decays of the Bc mesons permit us to
over-constrain quantities determined by the Bu,dmeson decays.
It is estimated that one could expect around 5 × 1010Bcevents per year at LHC [1]. The
nonleptonic decays of the Bcmesons have been studied in previous literature [1, 2]. The
theoretical status of the Bcmeson was reviewed in [1]. In this paper, we will concentrate
on the Bc→ J/ψπ, ηcπ decays using the perturbative QCD approach. There are several
reasons :
(i) From the experimental point of view, the decay modes containing the signal of J/ψ
meson are among the most easily reconstructible Bcdecay modes, due to the narrow-
peak of J/ψ and the high purity J/ψ → ℓ+ℓ−. For example, the Bcmesons are firstly
discovered via Bc→ J/ψℓν by the CDF Collaboration in 1998 [3]. Recently the CDF
and D0 Collaborations announced their accurate measurements on the Bcmesons via
Bc→ J/ψπ mode [4, 5]. Especially, compared with the semi-leptonic decays where
the neutrino momentum is not detected directly, all final-state particles are detectable
for the Bc→ J/ψπ, ηcπ decays. It is estimated that the ATLAS detector would be
able to record about 5600 events of Bc→ J/ψπ per year [1]. So Bc→ J/ψπ, ηcπ
decays may be two of the most prospective channels for measurements.
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(ii) From the phenomenological point of view: In recent years, several attractive methods
have been proposed to study the nonleptonic B decays, such as the QCD factorization
[6], perturbative QCD method (pQCD) [7, 8, 9], soft and collinear effective theory
[10, 11], and so on. The study of Bcdecays provides opportunities to test the kT and
collinear factorizations, to check the various treatments for the entanglement of differ-
ent energy modes, to deepen our understanding on perturbative and nonperturbative
contributions. These methods developed recently are widely applied to the nonleptonic
two-body Bu,d,sdecays in literature, but with very few application of these methods
on the Bcmeson decays. The appealing feature of the pQCD factorization [7, 8, 9]
is that form factors can be computed in terms of wave functions (nonperturbative
contributions) and hard kernels (perturbative contributions arising from hard gluon
exchange) assuming that additional soft contributions are suppressed by the Sudakov
factor in the heavy quark limit. Although there is still some controversy about the
pQCD method, for example, the problem of gauge invariant [12], the pQCD method
has been extensively used in the past to study nonleptonic B decays with fairly good
phenomenological results [13]. In this paper, we will take the Bc→ J/ψπ, ηcπ decays
as examples to discuss the Bcdecays in the perturbative QCD method.
(iii) From the theoretical point of view: The Bc→ J/ψπ, ηcπ decays are similar to the
Bq → D(∗)
Bq→ D(∗)
the Bq → D(∗)
that the Bc meson and the J/ψ (or ηc) meson are heavy quarkonia and could be
qπ (where q = u, d, s) decays with the “spectator quark” ansatz. The
qπ decays have been studied with the pQCD method [15]. Compared with
qπ decays, the Bc→ J/ψπ, ηcπ decays are easy to deal with because
described approximatively by nonrelativistic dynamics. Given mBc≃ mb+ mc, the
wave function of the Bcmesons would be close to δ(x−mc/mBc) in the nonrelativistic
limit (where mBc, mb, and mcare the masses of the Bcmesons, b quark, and c quark,
respectively; x denotes the momentum fraction of the c quark in the Bcmeson). The
wave functions for pion are well-defined. The only parameter is the wave function of
the J/ψ (or ηc) meson. So the Bc→ J/ψπ, ηcπ decays provide good platform to test
quark potential models derived from QCD.
This paper is organized as follows: In Section II, we discuss the theoretical framework and
compute the decay amplitudes for Bc→ J/ψπ, ηcπ with the perturbative QCD approach.
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The section III is devoted to the numerical results. Finally, we summarize in Section IV.
II. THEORETICAL FRAMEWORK AND THE DECAY AMPLITUDES
A.The effective Hamiltonian
Using the operator product expansion and renormalization group (RG) equation, the low
energy effective Hamiltonian for Bc→ Xc¯ cπ decay can be written as (where Xc¯ c= J/ψ, ηc):
Heff =GF
√2VcbV∗
ud
?
C1(µ)Q1+ C2(µ)Q2
?
+ H.c.,(1)
where VcbV∗
udis the CKM factor accounting for the strengths of the concerned nonleptonic
decay processes. The parameters Ci(µ) are Wilson coefficients which have been evaluated
to the next-to-leading order with the perturbation theory. The expressions of the local
operators are
Q1= [¯ cαγµ(1 − γ5)bα][¯dβγµ(1 − γ5)uβ],Q2= [¯ cαγµ(1 − γ5)bβ][¯dβγµ(1 − γ5)uα], (2)
where α, β are color indices. The essential problem obstructing the calculation of decay
amplitude is how to evaluate the hadronic matrix elements of the local operators.
B. Hadronic matrix elements
The calculation of the hadronic matrix elements is difficult due to the nonperturbative
effects arising from the strong interactions. Phenomenologically, the simplest approach
to hadronic matrix elements is the Bauer-Stech-Wirbel (BSW) model [14] based on color
transparency and naive factorization hypothesis, where the hadronic matrix elements are
parameterized into the product of the decay constants and the transition form factors.
One defect of the rough BSW method is that the hadronic matrix elements cannot cancel
the renormalization scheme- and scale- dependence of the Wilson coefficients. To remedy
this problem, the “nonfactorizable” contributions must be taken into account. Using the
Brodsky-Lepage approach [16], the hadronic matrix elements can be written as the convo-
lution of a hard-scattering amplitude, including some perturbative QCD contributions, and
meson wave functions.
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Recently, a modified perturbative QCD formalism has been proposed under the kT fac-
torization framework [7, 8, 9]. The Sudakov effects are introduced to modify the endpoint
behavior. The decay amplitudes are factorized into three convolution factors: the “harder”
functions, the heavy quark decay subamplitudes, and the nonperturbative meson wave func-
tions, which are characterized by the W±boson mass mW, the typical scale t of the decay
processes, and the hadronic scale ΛQCD, respectively. Using the resummation technique and
the RG treatment, the final decay amplitudes can be expressed as
A(Bc→Xc¯ cπ)∝C(t)⊗H(t)⊗ΦBc(x1,b1)⊗ΦXc¯ c(x2,b2)⊗Φπ(x3,b3),(3)
where the Wilson coefficient C(t) is calculated in perturbative theory at scale of mW and
evolved down to the typical scale t using the RG equations, ⊗ denotes the convolution over
parton kinematic variables, H(t) is the hard-scattering subamplitude, the wave functions
Φ(x,b) absorb nonperturbative long-distance dynamics, x is the longitudinal momentum
fraction of the valence quark of the meson, b is the conjugate variable of the transverse
momentum of the valence quark of the meson. According the arguments in [7, 8, 9], the
amplitude of Eq.(3) is free from the renormalization scale dependence.
C.Kinematic variables
For convenience, the kinematics variables are described in the terms of the light cone
coordinate. The momenta of the valence quarks and hadrons in the rest frame of the Bc
meson are defined by
p1=
mBc
√2(1,1,?0⊥),
mBc
√2(1,r2
mBc
√2(0,1 − r2
k1= x1p1,
k2= x2p2+ (0,0,?k2⊥),
k3= x3p3+ (0,0,?k3⊥),
n2= (1,0,0),
p2=
Xc¯ c,?0⊥),ǫ?=
n3= (0,1,0),
1
√2rψ(1,−r2
ψ,?0),
p3=
Xc¯ c,?0⊥),
where the notation of momenta of piand kiare displayed in FIG.1. The null vectors n2and
n3are the plus and minus directions, respectively. The mass of the π meson is neglected.
The momentum of the π meson is chosen to be parallel to the null vector n3. The mass
ratios are rXc¯ c= mXc¯ c/mBc, rb= mb/mBc, rc= mc/mBc.
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