Pair Production and Angular Distribution of Helicity Flipped Neutrinos in a Left-Right Symmetric Model

16 Pages·0.17 MB·English

by  A. Gutiérrez-Rodríguez

Pair production and angular distribution of helicity flipped
neutrinos in a Left-Right symmetric model
A. Guti´errez-Rodr´ıguez 1, M. A. Hern´andez-Ru´ız 2, M. Maya 3 and M. L. Ortega 3
(1) Facultad de F´ısica, Universidad Auto´noma de Zacatecas
Apartado Postal C-580, 98060 Zacatecas, Zacatecas M´exico.
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0 (2) Facultad de Ciencias Qu´ımicas, Universidad Auto´noma de Zacatecas
0
2
Apartado Postal 585 Zacatecas, Zacatecas M´exico.

n
a (3) Facultad de Ciencias F´ısico Matema´ticas, Universidad Auto´noma de Puebla
J

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Apartado Postal 1364, 72000 Puebla, Puebla M´exico.
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1 (February 7, 2008)
v
0
3
2
1
Abstract
0
3
0
/
h
p
- We assume that a massive Dirac neutrino is characteized by two phe-
p
e
h nomenological parameters, a magnetic moment, and a charge radius, and
:
v
i we calculate the cross-section of the scattering e+e− νν¯ in a left-right
X →
r
symmetric model. We also analyze the angular distribution of the neutrino
a
(antineutrino) with respect to the original direction of the electron (positron)
to different state of helicity of the neutrino. We find that the favored direc-
tions for the neutrino (antineutrino) with respect to the electron (positron)
is forward (θ = 0) and backward (θ = π), and is not very probable in the
perpendicular direction (θ = π). The calculation is for φ = 0.005 and
2 −
M = 500 GeV, parameters of the Left-Right symmetric model.
Z2
PACS number(s): 13.10.+q, 14.60.St, 12.15.Mm, 13.40.Gp
Typeset using REVTEX
1
I. INTRODUCTION
Of all the particles of the Standard Model (SM) [1], neutrinos are the least known.
Because they aretreatedasmassless particles, the physical phenomena associated with them
are rather limited. On the other hand, in case of massive neutrinos, which are predicted by
some Grand Unified Theories [2], several new effects can occur. Massive neutrinos open up
the possibility of a variety of new physical phenomena.
Neutrinos seem to be likely candidates for carrying features of physics beyond the stan-
dard model. Not only masses and mixings, but also charge radius, magnetic moment and
electric dipole moment [3,4] are signs of new physics, and are of relevance in terrestrial
experiments, the solar neutrino problem [5,6], astrophysics and cosmology [7,8].
When the explosion of the Supernova 1987 A (SN 1987 A) occurred, the astrophysics of
neutrinos was born. The observation of neutrinos from SN 1987 A [9,10], in fair agreement
with predictions from supernova models, has been used by several authors to bound the
properties and interactions of various exotic and non-exotic particles [11]. The experimental
observation of SN 1987 A launched several new searches for supernova neutrinos. Besides
specially developed detectors, basically all new real-time solar neutrino detectors like Super-
Kamiokande, ICARUS and SNO will be able to see such neutrinos.
Thus we can conclude that, the study of the neutrino continues to be the subject of
current research, both theoretical and experimental.
At the present time, all the available experimental data for electroweak processes can
be well understood in the context of the Standard Model of the electroweak interactions
(SM) [1], except the results of the Super-Kamiokande experiment on the neutrino mass [12].
Hence, the SM is the starting point of all the extended gauge models. In other words,
any gauge group with physical sense must have as a subgroup the SU(2) U(1) group
L
×
of the standard model, in such a way that their predictions agree with those of the SM at
low energies. The purpose of the extended theories is to explain some fundamental aspects
which are not clarified in the framework of the SM. One of these aspects is the origin of the
2
parity violation at the current energies. The Left-Right Symmetric Model (LRSM) based
on the SU(2) SU(2) U(1) gauge group [13] gives an answer to that problem, since it
R L
× ×
restores the parity symmetry at high energies and gives its violations at low energies as a
result of the breaking of gauge symmetry. Detailed discussions on LRSM can be found in
the literature [13–16].
Although in the framework of the SM, neutrinos are assumed to be electrically neutral.
Electromagnetic properties of the neutrino are discussed in many gauge theories beyond
the SM. Electromagnetic properties of the neutrino may manifest themselves in a magnetic
momentoftheneutrinoaswellasinanon-vanishingchargeradius, bothmakingtheneutrino
subject to the electromagnetic interaction.
In this paper, we start from a Left-Right Symmetric Model (LRSM) with massive Dirac
neutrinos left and right-handed, with an electromagnetic structure that consists of a charge
radius r2 and of a anomalous magnetic moment µ , and we calculated the total cross-
ν
h i
section of the scattering e+e− νν¯. We emphasize here the simultaneous contribution of
→
the charge radius, of the anomalous magnetic moment as well as of the additional Z heavy
2
gauge boson, and of the mixing angle φ parameters of the LRSM to the cross-section. The
Feynman diagrams which contribute to the process e+e− νν¯ are shown in Fig. 1. We
→
also analyzed the angular distribution of the neutrino (antineutrino) with respect to the
original direction of the electron (positron) to different state of helicity of the neutrino. We
find that the directions of the neutrino (antineutrino) with respect to the electron (positron)
is forward (θ = 0) and backward (θ = π), and is not very probable in the perpendicular
direction (θ = π).
2
This paper is organized as follows. In Sec. II we carry out the calculations of the process
e+e− νν¯. In Sec. III we present the expressions for the helicities. In Sec. IV we achieve
→
the numerical computations. Finally, we summarize our results in Sec. V.
3
II. THE ELECTRON POSITRON-NEUTRINO ANTINEUTRINO SCATTERING
In this section we obtain in the context of the LRSM the cross-section of the process
e−(p )+e+(p ) ν¯(k ,λ )+ν(k ,λ ), (1)
1 2 1 1 2 2
→
here p , p , k , and k are the particle momenta and λ (λ ) is the neutrino (antineutrino)
1 2 1 2 1 2
helicity.
We will assume that a massive Dirac neutrino is characterized by two phenomenological
parameters, a magnetic moment µ , expressed in units of the electron Bohr magnetons,
ν
and a charge radius r2 . Therefore, the expression for the amplitude of the process
h i M
e−e+ νν¯ Eq. (1) due only to γ and Z0 exchange, according to the diagrams depicted in
→
Fig. 1 is given by
Γµ
= ie2ν¯(k ,λ ) ν(k ,λ )e¯(p )γ e(p ), (2)
Mγ − 2 2 q2 1 1 2 µ 1
with
ie
Γµ = eF (q2)γµ F (q2)σµνq , (3)
1 2 ν
− 2m
ν
the neutrino electromagnetic vertex, where q is the momentum transfer and F (q2) are the
1,2
electromagnetic form factors of the neutrino. Explicitly [6]
1
F (q2) = q2 r2 ,
1
6 h i
m
F (q2) = µ ν,
2 ν
− m
e
where, as already mentioned, r2 is the neutrino mean-square charge radius and µ the
ν
h i
anomalous magnetic moment. Therefore
e2
= i [Fν¯(k ,λ )γµν(k ,λ )e¯(p )γ e(p )+GKµν¯(k ,λ )ν(k ,λ )e¯(p )γ e(p )], (4)
Mγ − q2 2 2 1 1 2 µ 1 2 2 1 1 2 µ 1
with
4
F
F = F +iF , G = i 2 , and Kµ = (k k )µ.
1 2 2 1
− 2m −
ν
Furthermore
g2
MZ0 = −i8c2 (q2 M2 )[Pν¯(k2,λ2)γµν(k1,λ1)e¯(p2)γµe(p1)
W − Z1
+Qν¯(k ,λ )γµγ ν(k ,λ )e¯(p )γ e(p )
2 2 5 1 1 2 µ 1
+Rν¯(k ,λ )γµν(k ,λ )e¯(p )γ γ e(p ) (5)
2 2 1 1 2 µ 5 1
+Sν¯(k ,λ )γµγ ν(k ,λ )e¯(p )γ γ γ e(p )],
2 2 5 1 1 2 µ 5 5 1
where
P = (A+2B +C)g ,
V
Q = ( A+C)g , (6)
A
−
R = ( A+C)g ,
V
−
S = (A 2B +C)g ,
A
−
the constants A, B and C depend only on the LRSM, and are given by [17]
s2 s2
A = a2 +Γc2 = (c Ws )2 +Γ( Wc +s )2,
φ φ φ φ
− r r
W W
s2 c2 s2 c2
B = ab+Γcd = (c Ws )( W s )+Γ( Wc +s )( Wc ),
φ φ φ φ φ φ
− r −r r r
W W W W
c2 c2
C = b2 +Γd2 = ( Ws )2 +Γ( W c )2,
φ φ
r r
W W
with
q2 M2
Γ = − Z10.
q2 M2
− Z20
While g = 1 +2sin2θ and g = 1, according to the experimental data [18].
V −2 W A −2
The square of the amplitude is obtained by sum over spin states of the final fermions, so
X|MT|2 = X|Mγ +MZ0|2 = X(|Mγ|2 +|MZ0|2 +MZ0M†γ +M†Z0Mγ), (7)
sp sp sp
where:
5
X γ 2 = 4H1E4 (F12 +F22)(1+x2)(1 λν¯λν)
|M | { −
sp
E2
2F2(x2 1 λ λ )+ F2(1 x2)(1+λ λ ) , (8)
− 2 − − ν¯ ν m2 2 − ν¯ ν }
ν
X Z0 2 = 4H2E4 (P2+Q2 +R2 +S2)(1+x2)(1 λν¯λν)
|M | { −
sp
+4x(PS +QR)(1 λ λ )+2(PQ+RS)(1+x2)(λ λ )
ν¯ ν ν ν¯
− −
+4x(PR+QS)(λ λ ) , (9)
ν ν¯
− }
X(MZ0M†γ +M†Z0Mγ) = 8H3E4{F1[P(1+x2)(1−λν¯λν)+Q(1+x2)(λν −λν¯)
sp
+2xR(λ λ )+2xS(1 λ λ )]
ν ν¯ ν¯ ν
− −
1
+F [P(2 λ λ x2)+Q(λ λ )(1 x2)
2 ν¯ ν ν ν¯
− − − 2
3
+ xR(λ λ )+xS(2 λ λ )] , (10)
ν ν¯ ν¯ ν
2 − − }
with
e4 g4 e2g2
H = , H = , H = ,
1 q4 2 64c4 (s M2 )2 3 8c2 q2(q2 M2 )
W − Z10 W − Z10
and x = cosθ, where θ is the scattering angle.
In the expressions (8), (9), and (10) the simultaneous contribution of the anomalous
magnetic moment, of the charge radius electroweak, of the heavy gauge boson Z0 and of
R
the mixing angle φ are observed.
The scattering cross-section in the center of mass system (where s is the square of the
center-of-mass energy) is given by
dσ 1
dΩ = 64π2s X|MT|2, (11)
sp
where the square of the total amplitude of transition 2 is given in the Eq. (7).
Psp T
|M |
The differential cross-section to each contribution is
dσ α2
( ) = (F ,F ,E,m ,λ ,λ ,x), (12)
γ 1 1 2 ν ν¯ ν
dΩ 16sF
dσ α2
( )Z0 = R12(s) 2(P,Q,R,S,λν¯,λν,x), (13)
dΩ 64s F
dσ α2
(dΩ)γZ0 = 16sR1(s)F3(F1,F2,P,Q,R,S,λν¯,λν,x), (14)
6
so that the total cross-section is
dσ α2
( ) = (F ,F ,E,m ,λ ,λ ,x)
T 1 1 2 ν ν¯ ν
dΩ 16sF
α2
+ R2(s) (P,Q,R,S,λ ,λ ,x)
64s 1 F2 ν¯ ν
α2
+ R (s) (F ,F ,P,Q,R,S,λ ,λ ,x), (15)
1 3 1 2 ν¯ ν
16s F
where
s
R (s) = , (16)
1 sin22θ (s M2 )
W − Z10
is the factor of resonance.
The kinematics to each contribution to be contained in the functions
γ : = (F2 +F2)(1+x2)(1 λ λ ) 2F2(x2 1 λ λ )
1 1 2 ν¯ ν 2 ν¯ ν
F − − − −
E2
+ F2(1 x2)(1+λ λ ), (17)
m2 2 − ν¯ ν
ν
Z0 : = (P2 +Q2 +R2 +S2)(1+x2)(1 λ λ )
2 ν¯ ν
F −
+4x(PS +QR)(1 λ λ )+2(PQ+RS)(1+x2)(λ λ )
ν¯ ν ν ν¯
− −
+4x(PR+QS)(λ λ ), (18)
ν ν¯
−
γZ0 : = F [P(1+x2)(1 λ λ )+Q(1+x2)(λ λ )
3 1 ν¯ ν ν ν¯
F − −
+2xR(λ λ )+2xS(1 λ λ )]
ν ν¯ ν¯ ν
− −
1
+F [P(2 λ λ x2)+Q(λ λ )(1 x2)
2 ν¯ ν ν ν¯
− − − 2
3
+ xR(λ λ )+xS(2 λ λ )], (19)
ν ν¯ ν¯ ν
2 − −
where explicitly P, Q, R, and S are
s c
P = [(c φ )2 +Γ(s + φ )2]g ,
φ φ V
− r r
W W
s2
Q = (c Ws )(Γ 1)g ,
2φ 2φ A
− r −
W
s2
R = (c Ws )(Γ 1)g , (20)
2φ 2φ V
− r −
W
S = [(c +r s )2 +Γ(s r c )2]g .
φ W φ φ W φ A
−
7
III. FORMULAS FOR THE HELICITIES
A. Formulas with right currents
We only take the part of interference Eq. (14) for the analysis. To simplify, we define
the functions and from the following manner
1 2
H H
= F1 1(φ,MZ0,x,λν¯,λν)LRSM +F2 2(φ,MZ0,x,λν¯,λν)LRSM, (21)
F H 2 H 2
where
1(φ,MZ0,x,λν¯,λν)LRSM = R1(s)[P(1+x2)(1 λν¯λν)+Q(1+x2)(λν λν¯)
H 2 − −
+2xR(λ λ )+2xS(1 λ λ )], (22)
ν ν¯ ν¯ ν
− −
x2
2(φ,MZ0,x,λν¯,λν)LRSM = R1(s)[P(2 λν¯λνx2)+Q(1 )(λν λν¯)
H 2 − − 2 −
3
+ xR(λ λ )+xS(2 λ λ )], (23)
ν ν¯ ν¯ ν
2 − −
these functions depend on the parameters of the LRSM, on the helicities of the neutrino and
on the scattering angle.
From Eqs. (22) and (23) we consider four combinations of helicities for the neutrino
(antineutrino), obtaining the following:
Case 1
Neutrino and antineutrino with positive helicity
1(φ,MZ0,x,λν¯ = λν = 1)LRSM = 0,
H 2
2(φ,MZ0,x,λν¯ = λν = 1)LRSM = R1(s)[P(2 x2)+Sx]. (24)
H 2 −
Case 2
Neutrino and antineutrino with negative helicity
1(φ,MZ0,x,λν¯ = λν = 1)LRSM = 0,
H 2 −
2(φ,MZ0,x,λν¯ = λν = 1)LRSM = R1(s)[P(2 x2)+Sx]. (25)
H 2 − −
8
Case 3
Neutrino with positive helicity and antineutrino with negative helicity
1(φ,MZ0,x,λν¯ = 1,λν = 1)LRSM = R1(s)[2(P +Q)(1+x2)+4x(R+S)],
H 2 −
2(φ,MZ0,x,λν¯ = 1,λν = 1)LRSM = R1(s)[P(2+x2)+Q(2 x2)+3x(R+S)]. (26)
H 2 − −
Case 4
Neutrino with negative helicity and antineutrino with positive helicity
1(φ,MZ0,x,λν¯ = 1,λν = 1)LRSM = R1(s)[2(P Q)(1+x2) 4x(R S)],
H 2 − − − −
2(φ,MZ0,x,λν¯ = 1,λν = 1)LRSM = R1(s)[P(2+x2) Q(2 x2) 3x(R S)]. (27)
H 2 − − − − −
B. Formulas without right currents
In this case we calculate the functions and in the absence of right currents. This
1 2
H H
is obtained taking the limit when the mixing angle φ = 0 and MZ0 so that Γ 0. In
2 → ∞ →
this limit P = g , Q = g , R = g , S = g and the functions and Eqs. (22) and
V A V A 1 2
− − H H
(23) take the form
(x,λ ,λ ) = R (s)[ g (1+x2)+2xg (1 λ λ )
1 ν¯ ν 1 V A ν¯ ν
H { } −
g (1+x2)+2xg (λ λ )], (28)
A V ν ν¯
−{ } −
x2
(x,λ ,λ ) = R (s)[g (2 λ λ x2) g (1 )(λ λ )
2 ν¯ ν 1 V ν¯ ν A ν ν¯
H − − − 2 −
3
g x(λ λ )+g x(2 λ λ )]. (29)
V ν ν¯ A ν ν¯
−2 − −
In a similar manner as in the previous section, we consider the following states of helicity
of the neutrino (antineutrino):
Case 1
Neutrino and antineutrino with positive helicity
(x,λ = λ = 1) = 0,
1 ν¯ ν
H
(x,λ = λ = 1) = R (s)[g (2 x2)+g x]. (30)
2 ν¯ ν 1 V A
H −
9
Case 2
Neutrino and antineutrino with negative helicity
(x,λ = λ = 1) = 0,
1 ν¯ ν
H −
(x,λ = λ = 1) = R (s)[g (2 x2)+g x]. (31)
2 ν¯ ν 1 V A
H − −
Case 3
Neutrino with positive helicity and antineutrino with negative helicity
(x,λ = 1,λ = 1) = 2R (s)(g g )(1 x)2,
1 ν¯ ν 1 V A
H − − −
(x,λ = 1,λ = 1) = R (s)[g (2+x2) g (2 x2)+3x( g +g )]. (32)
2 ν¯ ν 1 V A V A
H − − − −
Case 4
Neutrino with negative helicity and antineutrino with positive helicity
(x,λ = 1,λ = 1) = 2R (s)(g +g )(1+x)2,
1 ν¯ ν 1 V A
H −
(x,λ = 1,λ = 1) = R (s)[g (2+x2)+g (2 x2)+3x(g +g )]. (33)
2 ν¯ ν 1 V A V A
H − −
Inthefollowingsectionweanalyzetheangulardistributionoftheneutrino(antineutrino),
and interpret the cases obtained for the four combinations of helicity.
IV. RESULTS
The experiments of collision in the accelerators give results that depend on the collision
energy E between the electron and the positron. We consider energies available in the actual
accelerators, that is, √s = 100 GeV [18]. This energy is distributed between the particles
that collide; then the center-of mass energy varies by a few GeV and up to E = 50 GeV.
The mass of the Z10 is MZ0 = 91.2 GeV [18], therefore resonance exists when E = 45.6 GeV,
1
that is, when √s = 2E = 91.2 GeV. This is manifest in the factor of resonance R (s), Eq.
1
(16).
We first analyze the different states of helicity of the neutrino (antineutrino) and subse-
quently the angular distribution of the pair production of neutrinos.
10