Exclusive
Bc -meson decays
at LHC
A.V. Luchinsky
Institute for High Energy Physics, Protvino, Russia
1. Introduction
Bc -mesons, that is the states, that in valence approximation are build from b- and
c-quarks, takes an intermediate place between usual heavy quarkoina with hidden
flavor. Thus, they give an opportunity for an independent check of different models, that are used in describing charmonium and bottomonium mesons. In addition,
due to different flavor of constituent quarks, main decays modes of the ground
state are caused by weak interaction, so the investigation of these states can give
also additional information about electroweak sector of the Standard Model.
Experimentally very little is known currently about Bc -mesons. Only mass
and lifetime of the ground state was measured and two decay modes with vector
J /  -meson production were observed [1,2]. It is expected, that at LHC collider
large number of Bc(*) -mesons will be produced, so reliable theoretical consideration
of other decay modes is required. In the current paper, following the works [3,4],
exclusive decays into vector or pseudoscalar heavy quarkonia ( J / - or Bs(*) mesons) and a set of up to four  -mesons are considered.
On the quark level these decays correspond to weak decay of one of the constituent quarks, while the other quark is a spectator. So in the framework of factorization theorem the squared matrix elements of these decays can be splited into two
parts. The first factor describes the weak decay of Bc -meson into heavy quarkonium and virtual W-boson, while the other corresponds to hadronization of this boson into final system of light mesons. The first part can be described in terms of Bc meson form-factors, while for the latter the introduction of spectral functions is
convenient. With the help of this method the branching fractions of the considered
decays and distributions over squared transferred momentum are calculated.
The rest of the paper is organized as follows. In the next section the expressions for branching fractions distributions, Bc -meson form-factors and spectral
functions are given. In section 3 numerical values for branching fractions of the
considered decays are presented. Last section is devoted to conclusion.
2. Branching fractions of Bc  V ( P)  n decays and q 2 -distributions
Figure 1. Typical diagram for Bc  V ( P)  R decays
In the framework of the factorization theorem the decays Bc  V ( P)  n can be
considered as weak decays of one of the constituent quarks (b-quark in the case of
Bc  J /  n decay and c-quark for Bc  Bc(*)  n reaction), see Fig.1. As a result, the amplitude can factorized into two parts [5]:
M Bc  V ( P)  n  
GFVQQ'Vud
2
a1Q H   R ,
where H is the Bc  V ( P)  W vertex, while   is the effective polarization vector
of the light meson system. The parameter a1Q describes the effect of higher order
corrections.
The vertex Bc  V ( P)  W can be written in the following form [6]:

R
H  f  q 2 p   f  q 2 q 

for pseudoscalar meson in the final state and
H   F0A q q    FA q 2 p  FA q 2 q  iFV q 2 e  p q 
for decay into vector meson. In these expressions p  p1  p2 , q  p1  p2 , p1 and p2
are momenta of initial and final quarkonia, and   is the polarization vector of final vector meson. The form-factors f  q 2  , FA,0 q 2 , and FV q 2  can be determined,
for example, from analysis of Bc -meson semileptonic decays or different theoretical methods (QCD sum rules, etc).
The transition of virtual W-boson into a system of light mesons can be expressed through contribution of various resonances with suitable quantum numbers
[7] (  - or a 0 -mesons, for example). If only integrated branching fractions or q 2 distributions are considered, however, there is no need in such detailed description.
One can integrate over the phase space of light mesons system and introduce transverse and longitudinal spectral functions according to following relation [8]:
1
2
 d W  R   
R
R*
 q q  q 2 g   TR q 2   q q  LR q 2 
These spectral functions can be determined from theoretical and experimental
analysis of other processes, such as      R decays or electron-positron annihilation.
Using presented above relations it is easy to obtain the following expressions
2
for q -distributions in Bc  V ( P)  n decays:
 

2 2
2
2
2
dBc  P  n  GFVQQ ' a1

 f 2 M 14  2  TR  M 12  M 22   LR  f 2 q 2  LR  2 f  f  q 2 M 12  M 22   TR
2
dq
32M 1
2 2
2
2

dBv  V  R  GFV QQ ' a1 M 12 R  A 2  q 2 M 22

 T  F0 12
  2    4 M 14 FA  8 2 q 2 M 22 FV
2
2
2
dq
128 M 2
M1



2
2
2
2
A Q
 2M 1  M 2  q  Re F0 F .
2

In these expressions M 1, 2 are initial and final quarkonia masses, and  is the velocity of of V(P)-meson.
In our article the exponential parameterization of Bc -meson form-factors is
used:


Fi q 2   Fi 0 exp c1q 2  c2 q 4 ,
where the parameters, obtained from QCD sum rules [9 ] presented in table 1.
Fi 0
c1 , GeV
c2 , GeV
2
4
f
f
F0A
FA
FA
FV
1.3
 5.8
8.1
0.15
1.8
1.08
0.30
0.30
0.30
0.30
0.30
0.30
0.069
0.069
0.069
0.069
0.069
Table 1. Parameters of Bc  V ( P)  W form-factors
0.069
If two-particle decays ( Bc  V P    or Bc  V P    ) are considered, the
spectral functions are proportional to corresponding  -functions. In the latter case,
however, the width of  -meson is not taken into account. If there are more particles in the final state, the spectral functions can be determined from analysis of q 2 distributions in  -lepton decays and electron-positron annihilation. It turns out,
that they can be approximated with the following Breit-Wigner-like expressions:
 s  4m2
 s   1.35  10 
2

2
T
3
 s  9m2
s

 T3 s   5.86  10 5 
 s  16m2
s

 T4 s   1.8  10 4 
2



2



4
1  0.64 s
,
s  0.572  0.013
1  190
s
s  1.06
2

 0.048
2
,
 1  5.07 s  8.36s 2

,
2
2
 s  1.83  0.61


Here s is measured in GeV and the spectral functions itself are dimensionless.

4. Branching fractions and q 2 -distributions
Using presented in the previous section relations it is easy to obtain the branching
fractions of the considered in our article decays (see Table 2). One can easily see
that for small number of particles in the final state the branching fractions of
Bc  Bs(*)  n decays are larger than the branching fractions of Bc  J /  n decays, while for higher number of  -mesons the situation is opposite. The reason
for this difference is simple. In the case of one or two  -mesons production the
difference between CKM mixing angles is important. The reaction b  c  W is
Cabbibo-suppressed, so the branching fractions of Bc -meson decays with vector
charmonium in the final state are small in comparison with corresponding branching fractions with Bs(*) -meson production. If there are more final particles, the
phase-space suppression plays a role. Since the difference between Bc - and Bs(*) meson masses is small, this suppression is stronger, than CKM-enhancement.


2
3
4
Bc  J /  R
0.17
0.48
0.48
0.77
0.40
Bc  Bs  R
18
7.6
6.1
0.096
0.0064
Bc  Bs*  R
7
21
13
0.23
0.015
Table 2. Branching fractions of Bc  V ( P)  n decays (in %)
Comparing third and fourth columns of this table one can see the role of  -meson
width in Bc  V ( P)  2 decays. In the case of J / production this width can be
safely neglected and the branching fractions are almost equal. In the case of
Bc  Bs(*)  2 decays the inclusion of  -meson width changes the branching fractions significantly. The reason is clear: the width of the  -meson is tiny in comparison with the difference of Bc - and J / -meson masses, and comparable with
M B  M B . So, in the former case this width can be neglected, and for the latter reaction it should be taken into account.
c
s
5. Conclusion
In this paper exclusive Bc -meson decays into pseudoscalar or vector heavy quarkonium and a set of light mesons are considered. The presented branching fractions
show, that it is possible to observe some of these decays at LHC. Moreover, it
turns out, that it is necessary to take into account the widths of resonances (for example,  -meson in the case of 2 -meson production) to get the reliable results.
The method used in this paper is based on spectral functions technique,
when the phase-space of the light mesons is integrated out. These functions are
usually determined from analysis of  -lepton decays or e  e  -annihilation. In considered here decays, however, a larger region of the transferred momentum is
available, so they can be used also in studying the physics of light mesons.
It is clear, that only branching fraction of the considered here decays and distributions over transferred momentum can be obtained with the used in this paper
method. For experimental purposes a more detailed description of the W  n
transition is required. This is the task of our future work.
This work was financially supported by Russian Foundation for Basic Research (grants #09-02-00132-a, 07-02-00417-a), noncommercial foundation “Dynasty” and by President Grants #MK-110.2008.2, MK-406.2010.2
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