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arXiv:hep-ph/0305107v2 12 Sep 2003
DOI 10.1007/s1010508a123
EPJdirect A1, 1–11 (2008)
EPJdirect
electronic only
c ? Springer-Verlag 2008
Right-handed Dirac Neutrinos in νe−Scattering
and Azimuthal Asymmetry in Recoil Electron
Event Rates
S. Ciechanowicz1, M. Misiaszek2, W. Sobk´ ow1
1Institute of Theoretical Physics, University of Wroc? law, Pl. M. Borna 9,
PL-50-204 Wroc? law, Poland
e-mail: ciechano@rose.ift.uni.wroc.pl, sobkow@rose.ift.uni.wroc.pl
2M. Smoluchowski Institute of Physics, Jagiellonian University, ul. Reymonta 4,
PL-30-059 Krak´ ow, Poland
email: misiaszek@zefir.if.uj.edu.pl
Received:
Abstract. In this paper a scenario with the participation of the exotic scalar S, tensor
T and pseudoscalar P couplings of the right-handed neutrinos in addition to the stan-
dard vector V, axial A couplings of the left-handed neutrinos in the low-energy (νµe−)
and (νee−) scattering processes is considered. Neutrinos are assumed to be massive
Dirac fermions and to be polarized. Both reactions are studied at the level of the four-
fermion point interaction. The main goal is to show that the physical consequence of the
presence of the right-handed neutrinos is an appearance of the azimuthal asymmetry in
the angular distribution of the recoil electrons caused by the non-vanishing interference
terms between the standard and exotic couplings, proportional to the transverse neu-
trino polarization vector. The upper limits on the expected effect of this asymmetry for
the low-energy neutrinos (Eν < 1MeV ) are found. We also show that if the neutrino
helicity rotation (νL → νR) in the solar magnetic field takes place, the similar effect of
the azimuthal asymmetry of the recoil electrons scattered by the solar neutrinos should
be observed. This effect would also come from the interference terms between the stan-
dard (V,A)L and exotic (S,T,P)R couplings. New-type neutrino detectors with good
angular resolution could search for the azimuthal asymmetry in event number.
PACS: 13.15.+g, 13.88.+e
1 Introduction
The standard vector-axial (V − A) structure of the neutral and charged weak
interactions describes only what has been measured so far. We mean here the
measurement of the electron helicity [1], the indirect measurement of the neu-
trino helicity [2], the asymmetry in the distribution of the electrons from β-decay
[3] and the experiment with muon decay [4] which confirmed parity violation
[5]. Feynman, Gell-Mann and independently Sudarshan, Marshak [6] established
that only left-handed vector V , axial A couplings can take part in weak interac-
tions because this yields the maximum symmetry breaking under space inversion,
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EPJdirect A1, 1–11 (2008) Springer-Verlag2
under charge conjugation; the two-component neutrino theory of negative helic-
ity; the conservation of the combined symmetry CP and of the lepton number.
It means that produced neutrinos (antineutrinos) in V −A interaction can only
be left-handed (right-handed). However Wu [7] pointed out that exotic scalar S,
tensor T and pseudoscalar P weak interactions may be responsible for the neg-
ative electron helicity observed in β-decay. It would suggest that the generated
neutrinos (antineutrinos) in the (S,T,P) interactions may also be right-handed
(left-handed). The experimental precision of present measurements still does not
rule out the possible participation of the exotic (S,T,P) couplings of the right-
handed neutrinos beyond the the Standard Model (SM) [8, 9, 10].
So Sromicki at the PSI [11] searched for T-odd transverse electron polar-
ization in8Li β-decay. Armbruster et al. [12] measured the energy spectrum of
electron-neutrinos νe from µ-decay at rest in the KARMEN experiment using
the reaction12C(νe,e−)12Ng.s.. They gave the upper limit on the magnitude of
interference between scalar S and tensor T couplings. Shimizu et al. [13] deter-
mined the ratio of the strengths of scalar and tensor couplings to the standard
vector coupling in K+→ π0+ e++ νe decay at rest assuming the only left-
handed neutrinos for all interactions. Bodek et al. at the PSI [14] looked for the
evidence of the violation of time reversal invariance measuring T-odd transverse
positron polarization in µ+-decay. They also admitted the presence of the only
left-handed neutrinos produced in the scalar interaction. The emiT collabora-
tion presented new limits on the time reversal invariance violating D coefficient
using the polarized neutron beta-decay [15]. Presently at PSI, the experiment
with the decay of polarized neutrons is prepared to search for the time reversal
violating effects. A non-zero value of the T-odd transverse component of the
electron polarization would be a signal of the violation of this symmetry [16].
The transverse electron polarization for the electrons in the decay of polarized
muons was calculated by Shekhter and Okun [17] in 1958. The recent results
presented by the DELPHI Collaboration [18] concerning the measurement of
the Michel parameters and the neutrino helicity in τ lepton decays still admit
the deviation from the standard V − A structure of the charged current weak
interaction.
New high-precision low-energytests of the Lorentz structure using the electron-
neutrinos coming from the strong and polarized low-energy artificial neutrino
source or from the Sun would be sensitive to the effects caused by the interfer-
ence terms between the standard (V,A)Lcouplings of the left-handed neutrinos
and exotic (S,T,P)R couplings of the right-handed neutrinos in the neutrino-
electron scattering.
So far the neutrino-electron scattering was proposed to measure the az-
imuthal asymmetry in the recoil electron event rates produced by the non-zero
neutrino magnetic moments in the case of the solar neutrinos [19, 20]. This
asymmetry is caused by the non-vanishing interference between the weak and
electro-magnetic interaction amplitudes, proportional to µν, and depends on the
azimuthal angle between the transverse component of the neutrino polarization
and the momentum of the outgoing recoil electron.
The first concept of the use of the artificial neutrino source comes from
Alvarez who proposed a65Zn [21]. The51Cr and37Ar neutrino sources were
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proposed by Raghavan [22] in 1978 and Haxton [23] in 1988, respectively. The
idea of using the artificial neutrino source (reactor neutrinos) to search for the
neutrino magnetic moments was first proposed by Vogel and Engel [24]. The
strong51Cr source was used for the calibration of the GALLEX neutrino ex-
periment [25]. Miranda et al. [26] proposed the use of the51Cr source to probe
the gauge structure of the electroweak interaction. Currently at Gran Sasso, the
Borexino neutrino experiment [27] with the unpolarized51Cr source is designed
to search for the neutrino magnetic moment. There are also proposed the other
experiments to test the non-standard properties of neutrinos, in which both the
recoil electron scattering angle and the azimuthal angle would be measured with
good precision: the Hellaz [28], the Heron [29].
In this paper, we show that there is the other possible scenario of the ap-
pearance of the azimuthal asymmetry in the differential cross section for the
neutrino-electron scattering. The participation of the exotic (S,T,P)Rcouplings
in addition to the standard (V,A)Lcouplings can generate the azimuthal asym-
metry in the event number because in the final state (after scattering) all the
neutrinos are left-handed, and the interference terms between the standard and
exotic couplings do not depend on the neutrino mass. The main goal is to find
the upper limits on the expected magnitude of the azimuthal asymmetry in the
angular distribution of the recoil electrons for the low-energy (νµe−) and (νee−)
scattering processes (Eν= 0.746MeV,Eν= 0.863MeV), using the current lim-
its on the non-standard couplings [30]. This paper is also a generalization of the
considerations made in the [31]. The obtained results are analyzed in the context
of the future low-energy high-precision neutrino experiments.
In our considerations the system of natural units with ¯ h = c = 1, Dirac-Pauli
representation of the γ-matrices and the (+,−,−,−) metric are used [32].
2 Muon capture by proton as production process of
neutrinos
To show how the transverse components of the neutrino polarization, both T-
even and T-odd, may appear, we use the reaction of the muon capture by proton
(µ−+ p → n + νµ) as a production process of muon-neutrinos. The production
plane is spanned by the direction of the initial muon polarizationˆPµ and of
the outgoing neutrino momentum ˆ q, Fig. 1.ˆPµ and ˆ q are assumed to be per-
pendicular to each other because this leads to the unique conclusions as to the
possible presence of the right-handed neutrinos. Govaerts and Lucio-Martinez
[33] considered the nuclear muon capture on the proton and3He both within
and beyond SM admitting the general Lorentz invariant four-fermion contact
interaction and assuming the Dirac massless neutrino. However, they did not
calculate the neutrino observables. One assumes that the outgoing neutrino flux
is a mixture of the left-handed neutrinos produced in the standard V −A charged
weak interaction and the right-handed ones produced in the exotic scalar S, pseu-
doscalar P, tensor T charged weak interactions. In this scenario the interacting
muon is always left-handed. The complex fundamental coupling constants are
denoted as CL
Trespectively to the outgoing neutrino L- and
V,CL
Aand CR
S,CR
P,CR
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Fig.1. The production plane and reaction plane for νµneutrinos with the trans-
verse neutrino polarization vector η
ν.
′⊥
R-chirality:
Mµ−
=(CL
V+ 2MgM)(uνγλ(1 − γ5)uµ)(unγλup)
+ (CL
A+ mµ
2MgP)(uνiγ5γλ(1 − γ5)uµ)(uniγ5γλup)
+ CR
+ CR
(1)
q
S(uν(1 − γ5)uµ)(unup) + CR
T(uνσλ ρ(1 − γ5)uµ)(unσλ ρup),
P(uνγ5(1 − γ5)uµ)(unγ5up)
where gM,gP - the induced couplings of the left-handed neutrinos, i.e. the weak
magnetism and induced pseudoscalar, respectively; mµ,q,Eν,mν,M - the muon
mass, the value of the neutrino momentum, its energy, its mass and the nucleon
mass; up,un- the Dirac bispinors of initial proton and final neutron; uµ,uν- the
Dirac bispinors of initial muon and final neutrino.
The received formulas for the neutrino observables, in the case of non-vanishing
neutrino mass (mν?= 0), when the induced couplings are enclosed andˆPµ, ˆ q are
perpendicular to each other ((ˆPµ· ˆ q) = 0), are as follows:
T-even transverse component of the neutrino polarization:
< Sν·ˆPµ>f=| φµ(0) |2
q
Eν
4π
|Pµ|Re{(1 +
q
2MgP)CR∗
q
Eν
q
2M)(CL
q
2M(CL
V+ 2MgM)CR∗
S
(2)
+
q
2M(CL
q
Eν
mν
Eν(| CL
A+ mµ
P + 2q
Eν
V+ 2MgM)CR∗
T
+ 2(1 +
q
2M)(CL
A+ mµ
q
2MgP)CR∗
T
+1
2
V+ 2MgM|2− | CL
A+ mµ
q
2MgP|2+ | CR
S|2−2 | CR
T|2)}.
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T-odd transverse component of the neutrino polarization:
< Sν· (ˆPµ× ˆ q) >f=| φµ(0) |2
q
2M(CL
− 2(q
Eν
4π
|Pµ|Im{−(q
2M(CL
q
2MgP)CR∗
Eν
+
q
2M)(CL
V+ 2MgM)CR∗
S
(3)
−
A+ mµ
q
2M)(CL
q
2MgP)CR∗
P
− 2
q
V+ 2MgM)CR∗
+ 2mν
Eν
T
+
A+ mµ
T
q
2M(CR
SCR∗
T
− CR
TCR∗
P)},
where Sν - the operator of the neutrino spin; |Pµ| - the value of the muon
polarization in 1s state; φµ(0) - the value of the large radial component of the
muon Dirac bispinor for r = 0. The above neutrino observables are calculated
with the use of the density matrix of the final state.
It can be noticed that the neutrino observables consist only of the interfer-
ence terms between the standard (V,A)Lcouplings of the left-handed neutrinos
and exotic (S,T,P)Rones of the right-handed neutrinos in the limit of vanishing
neutrino mass. It can be understood as the interference between the neutrino
waves of negative and positive chirality. There is no contribution to these ob-
servables from the SM in which neutrinos are only left-handed and massless.
The mass terms in the above neutrino observables give a very small contribu-
tion in relation to the main one coming from the interference terms and they
are neglected in the considerations. If one assumes the production of the only
left-handed neutrinos in all the interactions, i.e. both for the standard V −A and
(S,T,P) interactions, there is no interference between the standard CL
CL
S,T,Pcouplings in the limit of vanishing neutrino mass. We see that the induced
couplings enter additively to the fundamental CL
ted in the considerations because their presence does not change qualitatively
the conclusions concerning the transverse neutrino polarization.
All the fundamental coupling constants CL
the couplings gγ
ǫµfor the normal and inverse muon decay [30], assuming the
universality of weak interactions. Here, γ = S,V,T indicates a scalar, vector,
tensor interaction; ǫ,µ = L,R indicate the chirality of the electron or muon
and the neutrino chiralities are uniquely determined for given γ,ǫ,µ. We get the
following relations:
V,Aand
V,Acouplings and they are omit-
V,A,CR
S,T,Pcan be expressed by
CL
V= A(gV
LL+ gV
RL), −CL
P= A(gS
A= A(gV
LL− gV
T= A(gT
RL), (4)
CR
S= A(gS
LL+ gS
RL), −CR
LL− gS
RL), CR
LL+ gT
RL),(5)
where A ≡ (4GF/√2)cosθc, GF = 1.16639(1) × 10−5GeV−2is the Fermi cou-
pling constant [30], θc is the Cabbibo angle. We calculate the lower limits on
the CL
0.850A, |CL
this way, we get the upper bound on the magnitude of the transverse neutrino
polarization vector proportional to the value of the muon polarization:
V,Aand upper limit on the CR
A| > 1.070A, |CR
S,T,P, using the current data [30]: |CL
S| < 0.974A, |CR
V| >
P| < 0.126A, |CR
T| < 0.122A. In
|η⊥
ν|=
?
< Sν· (ˆPµ× ˆ q) >
2
f+ < Sν·ˆPµ>
s < 1 >f
2
f
≤ 0.414|Pµ| (6)
|η
′⊥
ν|=
|η⊥
|Pµ|≤ 0.414,
ν|
(7)
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where s is the neutrino spin (s = 1/2) and the probability of muon capture
< 1 >f is of the form:
< 1 >f
=
| φµ(0) |2
4π
+ (3 + 2q
{(1 + 2q
q
2M) | CL
q
M(−(CL
Eν
q
2M) | CL
V+ 2MgM|2+ | CR
q
2MgP|2+(12 + 8q
S|2
(8)
Eν
A+ mµ
Eν
q
2M) | CR
T|2
+ 2Re[q
Eν
V+ 2MgM)(CL∗
A+ mµ
q
2Mg∗
P)
+ CR
TCR∗
P + CR
SCR∗
T) +mν
Eν((CL
V+ 2MgM)CR∗
S
− 6(CL
A+ mµ
q
2MgP)CR∗
T)]}.
The obtained limit on the |η⊥
value of the |η
the value of the longitudinal neutrino polarization is equal to ˆ ην· ˆ q = −0.910.
The formula for the T-even longitudinal component of the neutrino polarization
is as follows:
ν| has to be divided by |Pµ| to have the physical
ν| generated by the exotic (S,T,P) interactions. It means that
′⊥
< Sν· ˆ q >f=| φµ(0) |2
− (1
2Eν
4π
q
2M) | CL
2M(CL
q
2M(1
{−(3
V+ 2MgM|2+1
2
q
Eν
+
q
2M) | CL
A+ mµ
q
2MgP|2
S|2+(6q
q
2MCR
q
2MgP)CR∗
(9)
q
+
2
q
Eν
q
2Mg∗
| CR
Eν
+ 8
q
2M) | CR
q
2MCR
T|2
+ 2Re[
q
V+ 2MgM)(CL∗
A+ mµ
P) +
SCR∗
T
+
TCR∗
P
−mν
Eν
+1
2(CL
2(CL
A+ mµ
q
2MgP)CR∗
P + (CL
A+ mµ
T
V+ 2MgM)CR∗
S
+ (CL
V+ 2MgM)CR∗
T)]}.
We see that in the longitudinal neutrino polarization and the probability of
muon capture, the occurrence of the interference terms between the standard
CL
S,T,Pones depends explicitly on the neutrino mass.
The dependence on the neutrino mass causes the ”conspiracy” of the interfer-
ence terms and makes the measurement of the relative phase between these
couplings impossible because term (mν/Eν)(q/2M) is very small and the stan-
dard CL
V,Acouplings of the left-handed neutrinos dominate in agreement with
the SM prediction. Therefore, the neutrino observables in which such difficulties
do not appear are proposed. In this way, the conclusions as to the existence of
the right-handed neutrinos can depend on the type of measured observables.
If mν→ 0, q/Eν→ 1 and the neutrino mass terms vanish in all the observ-
ables.
V,Acouplings and exotic CR
3Neutrino-electron scattering as detection process
The produced mixture of the muon-neutrinos is detected in the neutral current
weak interaction. We assume that the incoming left-handed neutrinos are de-
tected in the V −A neutral weak interaction, while the initial right-handed ones
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EPJdirect A1, 1–11 (2008) Springer-Verlag7
are detected in the exotic scalar S, tensor T and pseudoscalar P neutral weak
interactions. Then in the final state all the neutrinos are left-handed.
To describe (νµe−) scattering the following observables are used: ˆ ην - the
unit 3-vector of the initial neutrino polarization in its rest frame, q - the incom-
ing neutrino momentum, pe′ - the outgoing electron momentum. The coupling
constants are denoted as gL
neutrino L- and R-chirality:
V,gL
Aand gR
S, gR
T, gR
Prespectively to the incoming
Mνe
=
GF
√2{(ue′γα(gL
+1
2[(gR
+ (gR
P(ue′γ5ue)(uν′γ5(1 + γ5)uν)]},
where ueand ue′ (uν and uν′) are the Dirac bispinors of the initial and final
electron (neutrino) respectively.
V− gL
Aγ5)ue)(uν′γα(1 − γ5)uν)(10)
S(ue′ue)(uν′(1 + γ5)uν) + gR
T(ue′σαβue)(uν′σαβ(1 + γ5)uν)
3.1Laboratory differential cross section
The laboratory differential cross section for the νµe−scattering, in the limit of
vanishing neutrino mass, is of the form:
d2σ
dydφe′
=(
d2σ
dydφe′)(V,A)+ (
d2σ
dydφe′)(V S)+ (
Eνme
4π2
2
−mey
Eν
Eνme
4π2
2
+ ((2 − y)2−me
Eνme
4π2
2
d2σ
dydφe′)(S,T,P)
d2σ
dydφe′)(AT),
(11)
+ (
(
d2σ
dydφe′)(V,A)
=
G2
F
{(1 − ˆ ην· ˆ q)[(gL
V+ gL
A)2+ (gL
V− gL
A)2(1 − y)2
(12)
((gL
V)2− (gL
(1 + ˆ ην· ˆ q){1
A)2)]},
(
d2σ
dydφe′)(S,T,P)
=
G2
F
8y(y + 2me
T|2+ y(y − 2)1
Eν)|gR
S|2+1
8y2|gR
P|2
(13)
Eνy)|gR
y(y + 2me
2[Re(gR
Sg∗R
T) + Re(gR
Pg∗R
T)]},
(
d2σ
dydφe′)(V S)
=
G2
F
{
?
Eν)[−ˆ ην· (ˆ pe′ × ˆ q)Im(gL
S)] − y(1 +me
y(y + 2me
Eν)[−ˆ ην· (ˆ pe′ × ˆ q)Im(gL
T)] − 2y(1 +me
VgR∗
S)(14)
+ (ˆ ην· ˆ pe′)Re(gL
Eνme
4π2
2
VgR∗
Eν)(ˆ ην· ˆ q)Re(gL
VgR∗
S)},
(
d2σ
dydφe′)(AT)
=
G2
F
{2
?
AgR∗
T)(15)
+ (ˆ ην· ˆ pe′)Re(gL
AgR∗
Eν)(ˆ ην· ˆ q)Re(gL
AgR∗
T)},
where:
y≡
Te
Eν
=me
Eν
2cos2θe′
Eν)2− cos2θe′,
(1 +me
(16)
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EPJdirect A1, 1–11 (2008) Springer-Verlag8
where y - the ratio of the kinetic energy of the recoil electron Teto the incoming
neutrino energy Eν, θe′ - the angle between the direction of the outgoing electron
momentum ˆ pe′ and the direction of the incoming neutrino momentum ˆ q (recoil
electron scattering angle), me- the electron mass, ˆ ην·ˆ q - the longitudinal polar-
ization of the incoming neutrino, φe′ - the angle between the production plane
and the reaction plane spanned by the ˆ pe′ and ˆ q, Fig. 1. All the calculations are
made with the Michel-Wightman density matrix [34] for the polarized incoming
neutrinos in the limit of vanishing neutrino mass (see Appendix). The interfer-
ence terms between the standard and exotic couplings, Eqs. (14, 15), include
only the contributions from the transverse components of the initial neutrino
polarization, both T-even and T-odd:
(
d2σ
dydφe′)(V S)+ (
×{|gL
d2σ
dydφe′)(AT)
=B|η
2|gL
′⊥
ν|
?me
Eνy[2 − (2 +me
Eν)y] (17)
V||gR
S|cos(φ − αSV)+
A||gR
T|cos(φ − αTA)},
where αSV ≡ αR
and gR
the transverse neutrino polarization vector and is connected with the φe′ in the
following way; φ = φ0−φe′, where φ0- the angle between the production plane
and the transverse neutrino polarization vector, Fig. 1.
The presence of the interference terms between the standard and exotic cou-
plings in the cross section depending on the φe′ generates the azimuthal asymme-
try in the angular distribution of the recoil electrons. Because the right-handed
neutrinos are produced and detected in the exotic (S,T,P) interactions, one uses
the same upper limits on the gR
versality of weak interactions. We take the values |η
for the muon-neutrinos to get the upper limit on the expected effect of the az-
imuthal asymmetry in the cross section. The situation is illustrated in the Fig.
2. The plot for the SM is made with the use of the present experimental values
for gL
A= −0.507 ± 0.014 [30], when ˆ ην· ˆ q = −1, Fig. 2
(solid line). If one integrates over the φe′, both interference terms vanish and
the cross section dσ/dy consists of only two terms.
If one assumes the production of only left-handed neutrinos in the standard
(V − A) and non-standard (S,T,P) weak interactions, there is no interference
between the gL
S,T,Pcouplings in the differential cross section, when
mν→ 0, and the azimuthal distribution of the recoil electrons is symmetric. We
do not consider this scenario.
Considering the low-energyintense artificial (51Cr) and natural (Sun) electron-
neutrino sources, we show the upper limit on the expected magnitude of the az-
imuthal asymmetry in the cross section for the electron-neutrinos, Fig. 3. It can
be noticed that the possible effect is much larger than for the muon-neutrinos at
the same neutrino energy Eν= 0.746MeV. In the case of the (νee−) scattering,
the charged current weak interaction must be included, i. e. gL
use the same upper limits on the exotic couplings as for the gR
ing the universality of weak interactions. We also take the values |η
ˆ ην· ˆ q = −0.910. The plot for the SM is made with the same values of the stan-
S−αL
V, αTA≡ αR
T−αL
A- the relative phases between the gR
S, gL
V
T, gL
Acouplings respectively, φ - the angle between the reaction plane and
S,gR
T,gR
Pas for the CR
S,CR
′⊥
ν| = 0.414, ˆ ην·ˆ q = −0.910
T,CR
P, assuming the uni-
V= −0.040 ± 0.015, gL
V,Aand gL
V+1, gL
S,gR
A+1. We
T,gR
′⊥
ν| = 0.414,
P, assum-
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EPJdirect A1, 1–11 (2008) Springer-Verlag9
0.00.10.20.3 0.4
y
0.50.60.7
0.215
0.220
0.225
0.230
0.235
0.240
0.245
0.250
0.255
0.260
0.265
0.270
d
2σ/dy dφe' [ 10
-45 cm
2 ]
Fig.2. Plot of the
0.746MeV; a) SM with the left-handed neutrino (solid line), b) the case of the
exotic S, T, P couplings of the right-handed neutrinos for φ − αSV = 0,φ −
αTA= 0 (long-dashed line), φ − αSV = π,φ − αTA= π (short-dashed line) and
φ − αSV = π/2,φ − αTA= π/2 (dotted line), respectively.
d2σ
dydφe′as a function of y for the (νµe−) scattering, Eν =
dard coupling constants as for the (νµe−) process, i. e. −0.040+ 1, −0.507+ 1,
when ˆ ην· ˆ q = −1, Fig. 3 (solid line).
4Astrophysical sources of right-handed
neutrinos - neutrino spin flip
If a neutrino has a large magnetic moment, the helicity of a neutrino can be
flipped when it passes through a region with magnetic field perpendicular to the
direction of propagation. The spin flip would change the left-handed neutrino
that is active in SM (V, A left-couplings) into a right-handed neutrino (ˆ ην·ˆ q = 1)
that is sterile in SM:
(d2σ
dydϕ)(V,A)= (1 − ˆ ην· ˆ q) · f(Eν,y)= 0.(18)
The mechanism of neutrino “spin flip” in the Sun’s convectionzone is proposed to
explain the observed depletion of the solar neutrinos [35]. The most restrictable
bound on the neutrino magnetic moment arrives from astrophysical considera-
tion of a supernova explosion. The scattering due to the photon exchange be-
tween a neutrino and a charged particle in plasma leads to neutrino spin flip. The
energy released in supernova implosion is taken partly away by sterile neutri-
nos without further interactions. In this scenario the neutrino magnetic moment
should be bounded because of the observed neutrino signal of SN 1987A[36]. Our
paper shows that the participation of the exotic couplings of the right-handed
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EPJdirect A1, 1–11 (2008) Springer-Verlag10
0.00.00.10.10.20.20.30.30.40.40.50.50.60.60.70.7
0.850.85
0.90 0.90
0.95 0.95
1.001.00
1.051.05
1.101.10
1.151.15
1.201.20
dd
2σ/dy dφe' [ 10
-45 cm
2σ/dy dφe' [ 10
2 ]
-45 cm
yy
2 ]
Fig.3. Plot of the
0.746MeV; a) SM with the left-handed neutrino (solid line), b) the case of the
exotic S, T, P couplings of the right-handed neutrinos for φ − αSV = 0,φ −
αTA= 0 (long-dashed line), φ − αSV = π,φ − αTA= π (short-dashed line) and
φ − αSA= π/2,φ − αTA= π/2 (dotted line), respectively.
d2σ
dydφe′as a function of y for the (νee−) scattering, Eν =
neutrinos can modify the both astrophysical considerations. The right-handed
neutrino is no longer “sterile”. The total cross section for νee−scattering with
the coupling constants from the current data (Section 2) can be calculated from
our general formulas (see Fig. 4). In this scenario the right-handed neutrinos can
be detected by neutrino detectors and could help simultaneously to transfer the
energy to presupernova envelope.
If the conversions νeL→ νeRin the Sun are possible, the azimuthal asymme-
try in the angular distribution of the recoil electrons generated by the interference
terms between the standard (V,A)Land exotic (S,T,P)Rcouplings should oc-
cur. If one assumes that a survival probability for the left-handed7Be-neutrinos
is equal to PeL = 0.5, the value of the transverse neutrino polarization as a
function of this PeL will be large, |η
ˆ ην· ˆ q = 1 − 2 · PeL= 0), see Eq. (9) in [19]. The equation on the |η
from the density matrix for the relativistic neutrino chirality. The situation is
illustrated in the Fig. 5 for the same limits on the exotic couplings as for the
51Cr-neutrinos and Eν = 0.863MeV. In this way, the expected effect of the
azimuthal asymmetry would be much stronger than for the51Cr-neutrinos.
′⊥
ν| = 2?PeL(1 − PeL) = 1, (for this case
′⊥
ν| arises
5Conclusions
In this paper, we show that the production of the R-handed neutrinos in the ex-
otic (S,T,P) weak interactions in addition to the L-handed ones in the standard
V −A weak interaction should manifest in the observation of the azimuthal asym-
metry of the electrons recoiled after the subsequent neutrino scattering. This
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EPJdirect A1, 1–11 (2008) Springer-Verlag11
0.111020
0.0
0.5
1.0
1.5
2.0
2.5
3.0
σ ( E ) [ 10
-45 cm
2 ]
E [ M e V ]
ν
Fig.4. Plot of the total cross section σ(E) as a function of right-handed (ˆ ην·ˆ q =
1) neutrino energy Eνfor the (νee−) scattering.
0.00.10.20.3 0.40.50.60.7
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
d
2σ/dy dφe' [ 10
- 45 cm
2 ]
y
Fig.5. Plot of the
0.863MeV; a) SM with the left-handed neutrino (solid line) for ˆ ην· ˆ q = −1,
b) the case of the exotic S, T, P couplings of the right-handed neutrinos for
φ−αSV = 0,φ−αTA= 0 (long-dashed line), φ−αSV = π,φ−αTA= π (short-
dashed line) and φ−αSA= π/2,φ−αTA= π/2 (dotted line), respectively, when
ˆ ην· ˆ q = 0, |η
d2σ
dydφe′as a function of y for the (νee−) scattering, Eν =
′⊥
ν| = 1.
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EPJdirect A1, 1–11 (2008) Springer-Verlag12
asymmetry would be due to the terms with the interference between (V,A)L
and (S,T,P)Rweak interactions, which stay present even in the limit of mass-
less neutrino. The scenario with interfering L- and R-handed neutrinos could be
tested with the intense electron-neutrino beams, e.g. from the artificial polarized
51Cr source, Fig. 3. If the neutrino helicity flip in the solar magnetic field takes
place, the similar effect of the azimuthal asymmetry in the event number for
the solar neutrinos should appear, Fig. 5 (7Be-neutrinos). It would indicate the
neutrino spin flip scenario (νL→ νR) as a possible solution of the observed solar
neutrino deficit. In both cases, the azimuthal asymmetry would arise from the
interference terms between the standard (V,A)Land (S,T,P)Rexotic couplings,
proportional to the transverse neutrino polarization vector.
It is well-known that according to the SM the angular distribution of the
recoil electrons does not depend on the azimuthal angle φe′, i.e. is the azimuthally
symmetric. The detection of the azimuthal asymmetry would be a signature of
the R-handed neutrinos. It can be noticed that the expected effect would be
much stronger for the low-energy neutrino-electron scattering (Eν < 1MeV)
than for the high-energy one. The artificial neutrino source has to be polarized
to have the assigned direction of the transverse neutrino polarization vector with
respect to the production plane because it would allow to measure the φe′. In the
case of the solar neutrinos, the η
ν
would be directed along the solar magnetic
field. The neutrino detectors with the good angular resolution have to observe
the direction of the recoil electrons and to analyze all the possible reaction planes
corresponding to the given recoil electron scattering angle in order to verify if
the azimuthal asymmetry in the cross section appears.
′⊥
This work was supported in part by the grant 2P03B 15522 of The Polish
Committee for Scientific Research and by The Foundation for Polish Science.
6 Appendix
The formulas for the 4-vector initial neutrino polarization S in its rest frame
and for the initial neutrino moving with the momentum q, respectively, are as
follows:
S=(0, ˆ ην),
ˆ ην· q
Eν
|q|
mν(ˆ ην· ˆ q),
Eν
mν(ˆ ην· ˆ q)ˆ q + ˆ ην− (ˆ ην· ˆ q)ˆ q,
(19)
S′
=
·
1
mν
?
Eν
q
?
+
?
0
ˆ ην
?
−
ˆ ην· q
Eν(Eν+ mν)
?
0
q
?
, (20)
S0′
=
(21)
S′
=(22)
where ˆ ην - the unit vector of the initial neutrino polarization in its rest frame.
The formulas for the Michel-Wightman density matrix [34] in the case of the
polarized neutrino with the non-zero mass,
Λ(s)
ν
=
?
r=1,2
urur∼ [1 + γ5(S
′µγµ)][(qµγµ) + mν](23)
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EPJdirect A1, 1–11 (2008) Springer-Verlag13
=[(qµγµ) + mν+ γ5(S
ˆ ην· q
Eνmν(qµγµ) − (ˆ ην−
mν
Eν
ˆ ην· q
Eν
′µγµ)(qµγµ) + γ5(S
(ˆ ην· q)q
Eν(Eν+ mν)) · γ,
(ˆ ην· q)q
Eν(Eν+ mν)) · γ(qµγµ),
(ˆ ην· q)q
Eν(Eν+ mν)) · γ,
′µγµ)mν],
(S
′µγµ)=
(24)
(S
′µγµ)(qµγµ)=
ˆ ην· q − (ˆ ην−(25)
(S
′µγµ)mν
=
(qµγµ) − mν(ˆ ην−(26)
and in the limit of vanishing neutrino mass, we have
lim
mν→0Λ(s)
ν
=[1 + γ5[ˆ ην· q
|q|
− (ˆ ην−(ˆ ην· q)q
|q|2
) · γ]](qµγµ).(27)
We see that in spite of the singularities m−1
ˆ q), the Michel-Wightman density matrix, in the limit of vanishing neutrino mass
mν, remains finite including the transverse component of neutrino polarization.
ν
in the longitudinal component (ˆ ην·
References
1. H. Frauenfelder et al.: Phys. Rev. 106 (1957) 386
2. M. Goldhaber, L. Grodzins, A.W. Sunyar: Phys. Rev. 109 (1958) 1015
3. C.S. Wu et al.: Phys. Rev. 105 (1957) 1413; 106 (1957) 1361
4. R.L. Garwin, L.M. Lederman, M. Weinrich: Phys. Rev. 105 (1957) 1415
5. T. D. Lee, C. N. Yang: Phys. Rev. 104 (1956) 254
6. R. P. Feynman, M. Gell-Mann: Phys. Rev. 109 (1958) 193
E. C. G. Sudarshan, R. E. Marshak: Phys. Rev. 109 (1958) 1860
7. C.S. Wu, S.A. Moszkowski: Beta decay. Wiley, New York 1966
8. S. L. Glashow: Nucl. Phys. 22 (1961) 579
9. S. Weinberg: Phys. Rev. Lett. 19 (1967) 1264
10. A. Salam: Elementary Particle Theory. N. Svartholm (Almquist and Wiksells),
Stockholm 1969
11. J. Sromicki: Search For Time Reversal Violation With Nuclei And Particles. In-
stitut f¨ ur Teilchenphysik, ETH Z¨ urich, 1994
12. B. Armbruster et al.: Phys. Rev. Lett. 81 (1998) 520
13. S. Shimizu et al.: Phys. Lett. B 495 (2000) 33
14. K. Bodek et al.: in Talk given at ETH Z¨ urich, 25 April 2000
I. Barnett et al.: PSI proposal R-94-10.1, 1995
15. L.J. Lising et al.: Phys. Rev C 62 (2000) 055501
16. I.C. Barnett, K. Bodek, P. Boeni, D. Conti, W. Fetscher, M. Hadri, W. Haeberli,
St. Kistryn, J. Lang, M. Markiewicz, O. Naviliat-Cuncic, A. Serebrov, J. Sromicki,
E. Stephan, J. Zejma: Fundamental Neutron Physics Research at SINQ. Search for
Time Reversal Violation in the Decay of Free, Polarized Neutrons, PSI Proposal,
June 1997
17. L.B. Okun, V.M. Shekhter: JETP 7 (1958) 864; Nuovo Cimento 10 (1958) 359
18. P. Abreu et al., DELPHI Coll.: Eur. Phys. J. C 16 (2000) 229
19. R. Barbieri, G. Fiorentini: Nucl. Phys. B 304 (1988) 909
20. S. Pastor, J. Segura, V. B. Semikoz, J. W. F. Valle: Phys. Rev. D 59 (1998) 013004
21. L. W. Alvarez: Lawrence Radiation Laboratory Physics Note (1973) 767
http://link.springer.de/link/service/journals/10105/index.html
Page 14
EPJdirect A1, 1–11 (2008) Springer-Verlag14
22. R. S. Raghavan: Proc. Conf. on Status and Future of Solar Neutrino Research,
BNL Report 50879 (1978), Vol. 2, p. 270
23. W. C. Haxton: Phys. Rev. C 38 (1988) 2474
24. P. Vogel, J. Engel: Phys. Rev. D 39 (1989) 3378
25. P. Anselmann et al., GALLEX Coll.: Phys. Lett. B 327 (1994) 377
P. Anselmann et al., GALLEX Coll.: Phys. Lett. B 342 (1995) 440
M. Cribier et al.: Nucl. Inst. Methods A 378 (1996) 233
W. Hampel et al., Gallex Coll.: Phys. Lett. B 420 (1998) 114
R. Bernabei: Nucl. Phys. B (Proc. Suppl.) 48 (1996) 304
26. O. G. Miranda, V. Semikoz, Jos´ e W. F. Valle: Nucl. Phys. Proc. Suppl. 66 (1998)
261
27. J. B. Benziger et al.: A proposal for participating in the Borexino solar neutrino
experiment, October 30, 1996
28. F. Arzarello et al.: Report No. CERN-LAA/94-19, College de France LPC/94-28,
1994
J. Seguinot et al.: Report No. LPC 95 08, College de France, Laboratoire de
Physique Corpusculaire, 1995
29. R. E. Lanou et al.: The Heron project, Abstracts of Papers of the American Chem-
ical Society 2(217), 021-NUCL 1999
30. Review of Particle Physics, K. Hagiwara et al.: Phys. Rev. D 66 (2002) 010001
31. W. Sobk´ ow: Phys. Lett. B 555 (2003) 215
32. F. Mandl, G. Shaw: Quantum field theory. John Wiley and Sons Ltd, Chichester,
New York, Brisbane, Toronto, Singapore 1984
33. J. Govaerts, Jose-Luis Lucio-Martinez, Nucl. Phys. A 678 (2000) 110
34. L. Michel, A. S. Wightman: Phys. Rev. 98 (1955) 1190
35. L. B. Okun, M. B. Voloshin, M.I. Vysotsky: Sov. J. Nucl. Phys. 44 (1986) 440
36. I. Goldman et al.: Phys. Rev. Lett. 60 (1988) 1789
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