# RareB

Authors: Berin Belma Şirvanlı
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

The rareB
s
→
γ
ν
ν
- decay with polarized
photon is studied in the framework of a family nonuniversal Z
′ model. The branching ratio and photon polarization asymmetry to the model
parameters are calculated and compared with the Standard Model. Deviations from the Standard
Model will indicate the presence of new physics.


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## Body

## 1. Introduction

One of the main aims of the LHC is investigatingB meson decays. Two distinct aspects may indicate why we should investigate B-mesons.From theoretical aspects, the rare decays occur at loop level in the SM. Also the flavor changing neutral current (FCNC) transitions are forbidden at the tree level in Standard Model (SM). Besides, these processes only occur at the loop level in the weak interactions. Any finding about the FCNC transitions could be quite beneficial. These are plenty of areas in which they could be useful for calculation such as Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and leptonic decay constant. Moreover, these rare decays are able to prove new physics beyond the SM. Moreover, when the photon is emitted from external charged leptons, it is proportional to lepton mass which gives small contribution. They are helicity suppressed by a factor of m
l
2
/
m
B
2. Despite this suppression factor, this decay, that is, B
→
l
+
l
-, has been observed at LHC [1]. If we inspect τ channel, there is no such suppression.From experimental point of view, their observation can be difficult because of the low efficiency. In case of massless neutrino, theB
→
ν
ν
- decay is forbidden in the SM because of the helicity conservation. In the rare B
→
γ
ν
ν
- decay, since the helicity suppression is removed, we expect the larger branching ratio (BR). For the rare BR
(
B
s
→
γ
ν
ν
-
) decay, branching ratio is of the order of ~1
0
-
8, so that these decays might have very clear experimental signature.If the rareB
s decay has the same order branching ratio as that of B
→
l
+
l
-, it can be measurable in the near future. In [2], this process was studied also within the framework of a family nonuniversal Z
′ model.The purpose of this study was to examine the rareB
s
→
γ
ν
ν
- decay in family nonuniversal Z
′ model by taking into account the polarization of the photon. B
s
→
γ
ν
ν
- decay has been investigated in the SM by using the constituent quark model and pole models [3] for the determination of the leptonic decay constants f
B.In case of massless neutrino, this decay was studied within the framework of the light-cone QCD sum rules method in [4]. In [4], the Hamiltonian formed of a single term representing the four vector interactions of the left handed neutrinos. But, according to the results obtained from the Super Kamiokande experiment [5, 6], the neutrinos have mass which could have right handed components. Similar FCNC decays in detail are studied in [7, 8]. We note that the rare B
s
→
γ
ν
ν
- decay has been previously studied in [9, 10] in a model independent way. The final state photon can be revealed by examining the polarization in a radiative decay mode like B
s
→
γ
ν
ν
-. Investigating the effects of polarized photon may provide another kinematical variable, like to the differential and total branching ratios for radiative decays [11].In this work we will investigate sensitivity of such “photon polarization asymmetry” inB
s
→
γ
ν
ν
- decay to the new Wilson coefficients in the family nonuniversal Z
′ model.The paper is organized as follows: in Section2, we present the family nonuniversal Z
′ model and form of the effective Hamiltonian and the parameterization of the hadronic matrix elements in terms of appropriate form factors. We then calculate the differential decay width and the photon polarization asymmetry for the B
s
→
γ
ν
ν
- decay. Section 3 is devoted to the numerical analysis and discussion of our results.


## 2. Matrix Element for theB
s
→
γ
ν
ν
- Decay

For the exclusiveB
s
→
γ
ν
ν
- transition, the b
→
s
ν
ν
- decay is described at quark level. In the SM, the effective Hamiltonian for the b
→
s
ν
ν
- transition is [12–14]
(1)
H
eff
SM
=
α
e
m
G
F
2
2
π
V
t
b
V
t
s
*
C
10
s
-
γ
μ
1
-
γ
5
b
ν
-
γ
μ
1
-
γ
5
ν
,
where V
t
b and V
t
s
* are the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Consider
(2)
C
10
(
x
t
)
=
x
t
8
x
t
+
2
x
t
-
1
+
3
x
t
-
2
x
t
-
1
2
ln
⁡
x
t
+
α
s
4
π
X
1
(
x
t
)
.
Here, x
t
=
m
t
2
/
m
W
2 and X
1
(
x
t
) is a term, which gives very small contribution. The explicit expression of X
1
(
x
t
) can be found in [15].If the mixing betweenZ and Z
′ is neglected, the contribution coming from Z
′ can be described by just modifying the Wilson coefficient without introducing any new operator structure. The expression of the effective Hamiltonian in this case can be written as follows [16–18]:
(3)
H
eff
Z
′
=
-
2
G
F
2
V
t
b
V
t
s
*
×
B
s
b
L
B
l
l
L
V
t
b
V
t
s
*
s
-
γ
μ
1
-
γ
5
b
ν
-
γ
μ
1
-
γ
5
ν
,
where B
s
b
L
=
B
s
b
L
e
i
φ
S
L and B
l
l
L
,
R correspond to the interaction vertex of Z
′ with quark and leptons. It follows from (1) and (2) that in order to take into account the contributions coming from the Z
′ boson it is enough to modify the Wilson coefficients C
10
(
x
t
) in the following way:
(4)
C
10
⟶
C
10
tot
=
C
10
+
4
π
α
s
28.82
B
s
b
L
V
t
b
V
t
s
*
(
B
l
l
L
-
B
l
l
R
)
,
where α
S is the strong coupling constant:
(5)
α
s
(
q
2
)
=
0.118
1
-
23
/
3
α
s
m
Z
/
2
π
ln
⁡
(
m
Z
/
q
2
)
.Using the effective Hamiltonian, the matrix elements for theB
s
→
γ
ν
ν
- decay at hadronic level can be calculated. Although the B
s
→
ν
ν
- rare decay is forbidden because of helicity conservation in case of massless neutrino, when a photon is emitted from initial quark lines, this process replaces it with the corresponding radiative one. Thus, we have eliminated the helicity suppression. After this operation, the rare decay has the following properties.(1)
If a photon is emitted from internal charged particles, then one gets a suppression factorm
b
2
/
m
W
2. Therefore the contributions of such diagrams can be safely neglected.
(2)
The Wilson coefficientC
tot is the same for the b
→
q
ν
ν
-
γ and b
→
q
ν
ν
- as a consequence of Law’s low energy theorem [19].As the last step for calculation, the matrix elements that we need for theB
s
→
γ
ν
ν
- decay are defined as follows (see [20–25]):
(6)
γ
k
s
-
γ
μ
1
-
γ
5
b
B
p
B
=
e
m
B
2
ϵ
μ
ν
λ
σ
ε
*
ν
q
λ
k
σ
g
(
q
2
)
+
i
ε
μ
*
k
q
-
ε
*
q
k
μ
f
q
2
,
where ε
μ
* and k
μ are the four-vector polarization and four momenta of the photon, respectively, q is the momentum transfer, and p
B is the momentum of the B meson. Using (1), the matrix element for B
s
→
γ
ν
ν
- decay can be written as follows:
(7)
M
=
α
G
F
4
2
π
V
t
b
V
t
s
*
sin
2
θ
W
e
m
B
2
×
+
i
A
2
ε
μ
*
k
q
-
ε
*
q
k
μ
ν
-
γ
μ
(
1
-
γ
5
)
ν
×
A
1
ϵ
μ
ν
α
β
ε
*
ν
q
α
k
β
+
i
A
2
ε
μ
*
k
q
-
ε
*
q
k
μ
,
where A
1
=
C
10
tot
g and A
2
=
C
10
tot
f.The differential decay rate of the rareB
s
→
γ
ν
ν
- decay was calculated as a function of dimensionless parameter x
=
2
E
γ
/
m
B, where E
γ is the photon energy.We have to examine the polarization of photons to obtain the final state photon in such a radiative decay. In fact, the main question is how sensitive is the branching ratio to theZ
′ model parameters when the photon is in the positive or negative helicity states.In the center of mass frame ofν
ν
- for the rare B
s
→
γ
ν
ν
- decay, we can prove d
Γ
(
ε
*
=
ε
1
)
/
d
x and d
Γ
(
ε
*
=
ε
2
)
/
d
x where four-momentum and polarization vectors, ε
1 and ε
2, are as follows:
(8)
p
B
=
E
B
,
0,0
,
E
k
,
k
=
E
k
,
0,0
,
E
k
,
p
1
=
p
,
0
,
p
1
-
z
2
,
-
p
z
,
p
2
=
(
p
,
0
,
-
p
1
-
z
2
,
p
z
)
,
ε
1
=
0,1
,
i
,
0
2
,
ε
2
=
0,1
,
-
i
,
0
2
,
where E
B
=
m
B
(
2
-
x
)
/
2
1
-
x, E
k
=
m
B
x
/
2
1
-
x, and p
=
m
B
1
-
x
/
2. We get the θ angle in (8) as z
=
cos
⁡
θ. θ is the angle between the momentum of the B meson and that of ν in the center of mass frame of ν
ν
-.With this information, we can obtain(9)
d
Γ
(
ε
*
=
ε
i
)
d
x
=
α
G
F
4
2
π
V
t
b
V
t
s
*
sin
2
θ
W
2
α
2
π
3
π
4
m
B
Δ
ε
i
,
where
(10)
Δ
(
ε
i
)
=
1
3
x
1
-
x
x
2
m
B
2
f
2
+
g
2
±
2
Re
g
f
*
with +
(
-
) for i
=
1
(
2
), respectively.In order to observe the effects of polarized photon, we have to calculate a variable “photon polarization asymmetry” [11]:
(11)
H
x
=
d
Γ
ε
*
=
ε
1
/
d
x
-
d
Γ
ε
*
=
ε
2
/
d
x
d
Γ
ε
*
=
ε
1
/
d
x
+
d
Γ
ε
*
=
ε
2
/
d
x
=
Δ
(
ε
1
)
-
Δ
(
ε
2
)
Δ
0
,
where
(12)
Δ
ε
1
-
Δ
ε
2
=
4
3
x
2
1
-
x
m
B
2
x
Re
g
f
*
C
10
2
,
(13)
Δ
0
=
x
3
2
3
m
B
2
1
-
x
f
2
+
g
2
C
10
2
.


## 3. Results and Discussion

In this part, we will indicate our numerical analysis about the branching ratio (BR) and the photon polarization asymmetryH for the rare B
s
→
γ
ν
ν
- decay. To make some numerical estimates, the explicit forms of the form factors g, f, g
1, and f
1 are necessary in (9) and (12). In the framework of light-cone QCD sum rules, the form factors were calculated in [20, 21], in terms of two parameters F
(
0
) and m
F as
(14)
F
q
2
=
F
0
1
-
1
-
x
m
B
2
/
m
F
2
2
,
where the values F
(
0
) and m
F for the B
s
→
γ transition are listed in Table 1.Table 1
B meson decay form factors in the light-cone QCD sum rule.
F
(
0
)
m
F
g
1 GeV
5.6 GeV
f
0.8 GeV
6.5 GeVWe have performed the numerical analysis about the branching ratio (BR) and the photon polarization asymmetryH for B
s
→
γ
ν
ν
- decay in the family nonuniversal Z
′ model. In this study, we have been used to the input parameters as follows:
(15)
m
B
s
=
5.28
GeV
,
τ
(
B
s
)
=
1.61
×
1
0
-
12
s
,
V
t
b
V
t
s
*
=
0.045
,
α
-
1
=
137
,
G
F
=
1.17
×
1
0
-
5
GeV
-
2
.Besides the remaining input parameters of the family nonuniversalZ
′ model, using the latest improvement measurements on B meson decays in [26, 27] for the Z-coupling parameters B
s
b
L, B
l
l
L and B
l
l
R are obtained under two circumstances:
(16)
B
s
b
L
≤
0.96
×
1
0
-
3
Scenario
1
means
that
B
s
b
R
=
0
is
assumed
,
B
s
b
L
≤
0.42
×
1
0
-
3
in
Scenario
2
it
is
assumed
that
B
s
b
L
=
B
s
b
R
with
ϕ
s
L
=
-
92
±
3
0
°
.Using the bound on the mass of theZ
′ boson with what follows from analysis of B
→
μ
+
μ
- decay (see, e.g., [26, 27] and LHC data [1]) for the B
μ
μ
L and B
μ
μ
R parameters we get what is presented in Table 2.Table 2
The values of theZ
′ model parameters for two different scenarios. For mass of Z
′ boson we put m
Z
′
=
3 TeV.
B
s
b
L
×
1
0
-
3
φ
S
L
(
0
)
B
μ
μ
L
×
1
0
-
2
B
μ
μ
R
×
1
0
-
2
S1
0.96
-
72
-
1.4
0.6
S2
0.42
-
92
±
30
-
0.6
0.2In Figure1, we indicate the dependence of the BR
(
1
) and BR
(
2
) for B
s
→
γ
ν
ν
- decay in the family nonuniversal Z
′ model. The superscripts (
1
) and (
2
) represent the positive and negative helicity states of photon, respectively. We reach the information from these figures that the branching ratio in both cases is very sensitive to the Z
′ model parameters. For S1 scenario, BR
(
1
) is larger about 3 times and for S2 scenario BR
(
1
) is larger about 1.3 times compared to that of the SM prediction. For S1 scenario, BR
(
2
) gets even larger enhancement, which is about 3 times, and for S2 scenario BR
(
2
) is larger about 2.5 times compared to the SM.Figure 1
The dependence of the integrated branching ratiosBR
(
1
) and BR
(
2
) for the B
s
→
γ
ν
ν
- decay. The superscripts (
1
) and (
2
) indicate the photon in positive and negative helicity state, respectively.In Figure2, we show the dependence of the integrated photon polarization asymmetry H as a function of x for B
s
→
γ
ν
ν
- decay in the family nonuniversal Z
′ model. From this figure, we see that, in scenario S2, at low photon energies for H predicts are larger four times than that one in SM.Figure 2
The dependence of the integrated photon polarization asymmetry of theB
s
→
γ
ν
ν
- decay.As a conclusion, the branching ratio of the rareB
s
→
γ
ν
ν
- decay was examined when photon has positive and negative helicities. By the same token, the photon polarization asymmetry of this decay was calculated by using the family nonuniversal Z
′ model parameters. One can conclude that measurement of H at low photon energies can give beneficial information about new physics. It would be possible to detect this rare process in the LHC. At the LHC-B and B TeV hadronic machines 1
0
11
–
1
0
12
b
b pair per year [26, 27] will be produced. Therefore, the number of expected events is N
≈
1
0
2
–
1
0
3 which detect this rare decay. The signature of this rare decay will be single photon and missing energy.



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*Source: 298604-2015-02-03.xml*