Scattering of neutrinos on a polarized electron target as a test for new physics beyond the Standard Model

Abstract

In this paper, we analyze the scattering of the neutrino beam on the polarized electron target, and predict the effects of two theoretically possible scenarios beyond the Standard Model. In both scenarios, Dirac neutrinos are assumed to be massive. First, we consider how the existence of CP violation phase between the complex vector V and axial A couplings of the Left-handed neutrinos affects the azimuthal dependence of the differential cross section. The future superbeam and neutrino factory experiments will provide the unique opportunity for the leptonic CP violation studies, if the large magnetized sampling calorimeters with good event reconstruction capabilities are build. Next, we take into account a scenario with the participation of the exotic scalar S coupling of the Right-handed neutrinos in addition to the standard vector V and axial A couplings of the Left-handed neutrinos. The main goal is to show how the presence of the R-handed neutrinos, in the above process changes the spectrum of recoil electrons in relation to the expected Standard Model prediction, using the current limits on the non-standard couplings. The interference terms between the standard and exotic couplings in the differential cross section depend on the angle $\alpha$ between the transverse incoming neutrino polarization and the transverse electron polarization of the target, and do not vanish in the limit of massless neutrino. The detection of the dependence on this angle in the energy spectrum of recoil electrons would be a signature of the presence of the R-handed neutrinos in the neutrino-electron scattering. To make this test feasible, the polarized artificial neutrino source needs to be identified.

Full text

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arXiv:hep-ph/0309286v2 10 Jan 2005
Scattering of neutrinos on a polarized electron target as a test for new physics beyond
the Standard Model
S. Ciechanowicz∗and W. Sobk´ ow†
Institute of Theoretical Physics,
University of Wroc? law, Pl.
PL-50-204 Wroc? law, Poland
M. Born 9,
M. Misiaszek‡
M. Smoluchowski Institute of Physics,
Jagiellonian University, ul.
PL-30-059 Krak´ ow, Poland
(Dated: February 1, 2008)
Reymonta 4,
In this paper, we analyze the scattering of the neutrino beam on the polarized electron target,
and predict the effects of two theoretically possible scenarios beyond the Standard Model. In both
scenarios, Dirac neutrinos are assumed to be massive.
First, we consider how the existence of CP violation phase between the complex vector V and
axial A couplings of the Left-handed neutrinos affects the azimuthal dependence of the differential
cross section. This asymmetry does not vanish in the massless neutrino limit. The azimuthal angle
φe′ of outgoing electron momentum is measured with respect to the transverse component of the
initial electron polarization η⊥
to measure the CP violation in the νµe−scattering. The future superbeam and neutrino factory
experiments will provide the unique opportunity for the leptonic CP violation studies, if the large
magnetized sampling calorimeters with good event reconstruction capabilities are build.
Next, we take into account a scenario with the participation of the exotic scalar S coupling of
the Right-handed neutrinos in addition to the standard vector V and axial A couplings of the Left-
handed neutrinos. The main goal is to show how the presence of the R-handed neutrinos, in the
above process changes the spectrum of recoil electrons in relation to the expected Standard Model
prediction, using the current limits on the non-standard couplings. The interference terms between
the standard and exotic couplings in the differential cross section depend on the angle α between
the transverse incoming neutrino polarization and the transverse electron polarization of the target,
and do not vanish in the limit of massless neutrino. The detection of the dependence on this angle
in the energy spectrum of recoil electrons would be a signature of the presence of the R-handed
neutrinos in the neutrino-electron scattering. To make this test feasible, the polarized artificial
neutrino source needs to be identified.
e. We indicate the possibility of using the polarized electron target
PACS numbers: 13.15.+g, 13.88.+e, 14.60.St
I. INTRODUCTION
Standard Model (SM) of electro-weak interactions
[1, 2, 3] has a vector-axial (V-A) Lorentz structure [4], i.e.
only left-handed (L-handed chirality states) and massless
Dirac neutrinos may take part in the charged and neu-
tral weak interactions. The observed CP violation in the
decays of neutral kaons and B-mesons [5] is described by
a single phase of the CKM matrix.
The vector gL
pling constants are assumed to be real numbers, which
means that Im(gL
two couplings are derived from neutrino electron scat-
tering and from e+e−→ l+l−annihilation studies, but
in the fitting procedure the imaginary parts are fixed to
their Standard Model values.
Vand axial-vector gL
Aneutral current cou-
V) = Im(gL
A) = 0. The values of these
∗Electronic address: ciechano@rose.ift.uni.wroc.pl
†Electronic address: sobkow@rose.ift.uni.wroc.pl
‡Electronic address: misiaszek@zefir.if.uj.edu.pl
However, in the general case of complex gL
plings, we have one additional free parameter: the rela-
tive phase between these couplings denoted as βV A. The
CP-odd interference contribution enters the differential
cross section for the scattering of left-handed neutrinos
on the polarized electron target (PET), if |sin(βV A)| ?= 0.
The experimental measurement of the azimuthal angle
φe′ of outgoing electron momentum could be used to test
the CP symmetry in lepton sector of electroweak interac-
tions. The observation of asymmetry in the angular dis-
tribution of recoil electrons, caused by the interference
terms between the standard complex couplings would
give additional information about the coupling constants.
Vand gL
Acou-
The magnetized sampling calorimeters (e.g. MINOS
far detector) are composed of many steel leyers, which
are magnetized using a coil through a hole in the center
of the planes to an average field of about 1.5 T. In a piece
of magnetized iron, there are lots of unpaired electrons
all pointing the same direction. The Moeller polarime-
ters determine the polarization of the electron beam by
measuring the cross section asymmetry in the scattering

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TABLE I: Current limits on the non-standard couplings
Coupling constants
|gV
LL|
|gV
|gV
|gV
|gS
|gS
|gS
|gS
|gT
|gT
|gT
|gT
RR|
SM
1
0
0
0
0
0
0
0
0
0
0
0
Current limits
> 0.960
< 0.060
< 0.110
< 0.039
< 0.550
< 0.125
< 0.424
< 0.066
0
< 0.036
< 0.122
0
LR|
RL|
RR|
LL|
LR|
RL|
RR|
LL|
LR|
RL|
of polarized electrons by polarized electrons. Polarized
electrons are scattered off a polarized ferromagnetic foil,
and the foil polarization is determined by measurements
of the magnetization of the foil and its thickness. So,
there is the well-known and commonly used in accelera-
tor physics technique for developing PETs.
Although the SM agrees well with all experimental
data up to available energies, the experimental precision
of present measurements still does not rule out the pos-
sible participation of the exotic scalar S, tensor T and
pseudoscalar P couplings of the right-handed (R-handed
chirality states) Dirac neutrinos beyond the SM [6]. The
current upper limits on the all non-standard couplings,
obtained from the normal and inverse muon decay, are
presented in the Table I [7]. In the SM, only gV
zero value.
New effects due to the exotic right-handed weak inter-
actions (ERWI) could be detected by the measurement of
neutrino observables (NO) which consist only of the inter-
ferences between the standard V − A and ERWI, and do
not depend on the neutrino mass. The NO include the in-
formation on the transverse components of neutrino spin
polarization (TCNSP), both T-even and T-odd. These
quantities vanish in the SM, so the detection of the non-
zero values of the TCNSP would be a direct signature of
the R-handed neutrino presence in the weak interactions.
The scattering of intense and polarized neutrino beam,
coming from the artificial neutrino source, on the po-
larized electron target could detect the effects from the
ERWI. Presently the measurement of such observables
is only theoretically possible. We give an example how
the polarized neutrino flux can be produced if the ex-
otic scalar coupling CR
Sis present in the theory of muon
capture interaction.
The main goal of the first part of our paper (Section
II) is to show that the differential cross section for the
(νµe−) scattering of left-handed and longitudinally po-
larized muon neutrinos on the PET may be sensitive to
the CP-violating effects, if one assumes the complex stan-
LLis non-
dard couplings gL
(Section III) is to show how the presence of the R-handed
neutrinos, in the (νµe−) scattering process, changes the
energy spectrum of recoil electrons in relation to the ex-
pected SM prediction, using the current limits on the
non-standard couplings [7]. We concern the scattering of
transversely polarized muon neutrino beam on the PET
to probe ERWI effects and include a theoretical discus-
sion of the possibility of developing such a beam.
Our analysis is model-independent and is made in
the massless neutrino limit.
made in the limit of vanishing neutrino mass with the
Michel-Wightman density matrices [8] for the polarized
incoming neutrino beam (Appendix C) and for the
polarized electron target, respectively.
system of natural units with ¯ h = c = 1, Dirac-Pauli
representation of the γ-matrices and the (+,−,−,−)
metric [9].
V, gL
A. The main goal of the other part
All the calculations are
We use the
II.LEFT HANDED NEUTRINO SCATTERING
ON A POLARIZED ELECTRON TARGET
The Standard Model (SM) of electroweak interactions
is based on the gauge group SU(2)×(1). The left-handed
fermion fields ψi =
?
family transform as doublets under SU(2), where d′
?
are
νi
l−
i
?
and
?
ui
d′
i
?
of the ithfermion
i≡
jVijdjand V is the Cabibbo-Kobayashi-Maskawamix-
ing matrix. The vector and axial-vector couplings in SM
gL
gL
V(i) ≡ tL
A(i) ≡ tL
3(i) − 2q(i)sin2θW
3(i) (1)
where tL
ui and νi; −1/2 for di and li), qi is the charge of ψi
in units of e and θW is the weak angle. Because of the
model-dependent interpretation of the coupling constants
values, they are assumed to be real numbers. For exam-
ple, the total cross section for high energy neutral-current
(νµe−) scattering is
3(i) is the weak isospin of fermion i (+1/2 for
σSM(νµ+ e−→ νµ+ e−) ≃
2G2
FmeEνµ
3π
(gL2
V + gL2
A+ gL
VgL
A), (2)
but in the model-independent (MI) analysis we obtain:
σMI(νµ+ e−→ νµ+ e−) ≃
V|2+ |gL
2G2
FmeEνµ
3π
(|gL
A|2+ |gL
V||gL
A|cos(βV A)), (3)
where gL
coupling constants, Re(gL
βV A= βL
gL
Acouplings.
V= |gL
V|eiβL
V, gL
A= |gL
VgL∗
A|eiβL
A are the complex
V||gL
A) = |gL
A|cos(βV A) and
V−βL
Ais the relative phase between the gL
Vand

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The effective vector and axial-vector neutral coupling
constants obtained from the absolute neutrino-electron
scattering event rate are
gL
V≃ 0 , gL
gL
A≃ ±0.5 or
A≃ 0 .
V≃ ±0.5 , gL
(4)
However, from our MI expression (3) one can see that
the solution (with CP-violating phase):
|gL
V| = |gL
A| ≃ 0.35 and βV A= ±π
2
(5)
provides to the same total cross section value as the SM
fit (4). In the next subsection we present how the ex-
istence of non zero βV A phase is related to CP-odd in-
terference contribution in the differential cross section.
The fermion-antifermion pair production cross-sections
have only T-even contributions, but their experimental
observations are essential to determine a single soultion
from possible parameters (4). Even if βV A= 0 the scat-
tering of left-handed neutrinos on the PET provides a
new approach to decide which of the two coupling types,
(mainly) pure gL
Vcoupling, is realized in na-
ture. This approach is model independent in contrast to
e+e−experiments which make the assumption that the
neutral current is dominated by the exchange of a single
Z0.
As is well-known, CP violation has been observed only
in the decays of neutral kaons and B-mesons. The Stan-
dard Model describes the existing data by a single phase
of the CKM matrix, [10]. However, the baryon asymme-
try of the Universe can not be explained by the CKM
phase only, and at least one new source of CP violation
is required [11]. The first direct confirmation of a time
Aor pure gL
reversal violation has been published by CPLEAR Col-
laboration in 1998 [12]. Many non-standard models take
into account new CP-violating phases, and can be probed
in observables where the SM CP-violation is suppressed,
while alternative sources can generate a sizable effect, e.g.
the electric dipole moment of the neutron, the transverse
lepton polarization in three-body decays of charged kaons
K+[13, 14], transverse polarization of the electrons emit-
ted in the decay of polarized8Li nuclei [15]. There is no
direct evidence of CP violation in the leptonic processes,
i.e. a neutrino-electron scattering. However, the future
superbeam and neutrino factory experiments [16] will be
able to measure the CP violating effects in the lepton sec-
tor, where both neutrino and antineutrino oscillation will
be observed. We indicate that the scattering of neutrinos
on the PET has similar scientific possibilities.
A.CP violation in standard νe scattering
In this subsection, we consider the possibility of the
CP violation in the νµe−scattering, when the incoming
muon neutrino beam consists only of the L-handed and
longitudinally polarized neutrinos. We assume that these
neutrinos are detected in the standard V − A NC weak
interactions with the PET and both the recoil electron
scattering angle θ′
eand the azimuthal angle of outgoing
electron momentum φ′
eshown in Fig. 1 are measured
with a good angular resolution. Because we allow for
the non-conservation of the combined symmetry CP, the
amplitude includes the complex coupling constants de-
noted as gL
Arespectively to the initial neutrino of L-
chirality:
V,gL
Mνµe =
GF
√2
?
gL
V(ue′γαue)(uνµ′γα(1 − γ5)uνµ) + gL
A(ue′γ5γαue)(uνµ′γ5γα(1 − γ5)uνµ)
?
, (6)
where ueand ue′ (uνµand uνµ′) are the Dirac bispinors
of the initial and final electron (neutrino) respectively.
GF = 1.16639(1)×10−5GeV−2[7] is the Fermi constant.
The formula for the differential cross section includ-
ing the CP-odd contribution (ˆ q · (ˆ ηe× ˆ pe′) is T-odd
and Im(gL
magnitude of the transverse electron target spin polariza-
tion, with ˆ ηe⊥ ˆ q is of the form:
VgL∗
A) = |gL
V||gL
A|sin(βV A)), proportional to the
?
d2σ
dydφ′e
?
(V A)
=
Eνme
4π2
G2
2
F
(1 − ˆ ην· ˆ q)
?
|gL
A|2
?
−ˆ ηe· ˆ pe′
?2me
Eν
+ y(
?
y3− 2√y) +me
Eνy + (y − 2)y + 2
?
+ |gL
V|2
?
y2− ˆ ηe· ˆ pe′?
y3
?2me
Eν
+ y − y(me
Eν
+ 2) + 2
?

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FIG. 1: Figure shows the reaction plane for the νµe−scattering, ˆ ην - the unit 3-vector of the initial neutrino polarization in its
rest frame, η⊥
of µ−+ p → n + νµ. In Section II, the scattering of left handed neutrinos is considered, which have no transverse polarization
η⊥
ν = 0. For the considerations described in Section III, the muon capture reaction is used as a source of transversely polarized
neutrinos.
e - the transverse electron polarization vector of target and the production plane of νµ-neutrinos for the reaction
+ Im(gL
VgL∗
A)ˆ q · (ˆ ηe× ˆ pe′)
?
y(2me
Eν
?
+ y) (7)
+ Re(gL
VgL∗
A)
?
ˆ ηe· ˆ pe′(y − 1)
y(2me
Eν
+ y) + (2 − y)y
??
where ˆ ην· ˆ q = −1 is the longitudinal polarization of the
incoming L-handed neutrino, q - the incoming neutrino
momentum, pe′ - the outgoing electron momentum, ˆ ηe-
the unit 3-vector of the initial electron polarization in its
rest frame, see Fig.1. The measurement of the azimuthal
angle of outgoing electron momentum φe′ is only possible
when the electron target polarization is known. The po-
larization vector for electrons is parallel to the magnetic
field vector. The variable y is the ratio of the kinetic
energy of the recoil electron Teto the incoming neutrino
energy Eν:
y ≡Te
Eν
=me
Eν
2cos2θe′
Eν)2− cos2θe′.
(1 +me
(8)
It varies from 0 to 2/(2+me/Eν). θe′ - the polar angle be-
tween the direction of the outgoing electron momentum
ˆ pe′ and the direction of the incoming neutrino momen-
tum ˆ q (recoil electron scattering angle), me- the electron
mass.
After the simplification of the vector products and us-
ing the complex number identities in the formula (7) for
the cross section, we obtain with ˆ ηe⊥ ˆ q the new form:

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-1,0 -0,8-0,6 -0,4-0,20,00,2 0,4 0,60,8 1,0
2,10
2,11
2,12
2,13
2,14
2,15
2,16
2,17
2,18
2,19
d?/d?e
' [ 10
-43cm
2rad
-1 ]
Azimuthal angle ?e
' [ ? rad ]
FIG. 2: Plot of the
dσ
dφ′e (V A)as a function of the azimuthal angle φe′ for the (νµe−) scattering, Eν = 1 GeV, y = 0.5, ˆ ην·ˆ q = −1
e| = 1: a) the case of the pure and real axial-vector coupling i.e. gL
and real vector coupling i.e. gL
|gL
2(dashed line).
and |ˆ η⊥
A= 0.5 and gL
V= 0 (solid line), b) the case of the pure
A= 0 and gL
V= 0.5 (dotted line), c) CP violation, the case of the complex coupling constants
A| = 0.354 with the relative phase βV A =π
V| = |gL
?
d2σ
dydφ′e
?
(V A)
=Eνme
4π2
G2
2
F
(1 − ˆ ην· ˆ q)
?
|η⊥
e|
?me
Eνy[2 − y(2 +me
Eν)] (9)
·
?
cos(φe′)?2|gL
??|gL
V||gL
A|cos(βV A)y + (2 − y)|gL
A|2− y|gL
V|2?− 2|gL
V||gL
A|cos(φe′ + βV A)
?
+
V|2+ |gL
A|2??y2− 2y + 2?+ 2|gL
V||gL
A|cos(βV A)y(2 − y) −me
Eνy?|gL
V|2− |gL
A|2???
.
It can be noticed that the interference terms between
the standard gL
V,Acouplings depend on the value of the
βV A phase. However, the angular asymmetry of recoil
electrons is not vanishing even if βV A = 0. The CP-
violating phase enters the cross section and changes the
angle at which the number of recoil electrons will be max-
imal (φmax
e′
).
III.SCATTERING OF TRANSVERSELY
POLARIZED MUON NEUTRINO BEAM ON A
POLARIZED ELECTRON TARGET
So far the scattering of left-handed and longitudinally
polarized neutrino beam on a polarized electron target
(SLoPET) was proposed to probe the neutrino mag-
netic moments [17, 18] and the flavor composition of a
(anti)neutrino beam [19].
There were also the ideas of using the scattering of
transversely polarized neutrino beam on the unpolar-
ized electron target to probe the nonstandard properties