The radiative decay of ultrarelativistic massive neutrino $\nu_i \rightarrow
\nu_j \gamma$ is investigated in electromagnetic fields in the framework of the
Standard Model with lepton mixing. Estimates of the decay probability and
``decay cross-section'' for accelerator neutrinos of high energies in the
electric field of nucleus permit one to discuss the general possibility of
carrying out the neutrino experiment. Such an experiment could give unique
information on mixing angles in the lepton sector of the Standard Model which
would be almost independent of the specific neutrino masses.Comment: 6 pages, LaTeX, Talk given by L.A.Vassilevskaya at the XXXth
  Rencontres de Moriond, Electroweak Interactions and Unified Theories, Les
  Arcs, Savoie, France, March 11-18, 199

Gvozdev, A. A.

Mikheev, N. V.

Vassilevskaya, L. A.

English

arXiv

The radiative decay of ultrarelativistic massive neutrino \nu_i \rightarrow \nu_j \gamma is investigated in electromagnetic fields in the framework of the Standard Model with lepton mixing. Estimates of the decay probability and ``decay cross-section'' for accelerator neutrinos of high energies in the electric field of nucleus permit one to discuss the general possibility of carrying out the neutrino experiment. Such an experiment could give unique information on mixing angles in the lepton sector of the Standard Model which would be almost independent of the specific neutrino masses

Gvozdev, A A

Mikheev, N V

Vasilievskaya, L A

CERN Document Server

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Yaroslavl State University
Preprint YARU-HE-95/03
hep-ph/9506429
Electromagnetic Catalysis of a Neutrino
Radiative decay
or
One More Source of Information on the
Lepton Mixing Angles?
A.A. Gvozdev, N.V. Mikheev and L.A. Vassilevskaya
Division of Theoretical Physics, Department of Physics,
Yaroslavl State University, Sovietskaya 14,
150000 Yaroslavl, Russian Federation
presented by L.A. Vassilevskaya
Abstract
The radiative decay of ultrarelativistic massive neutrino 
i
! 
j
 is investigated in elec-
tromagnetic fields in the framework of the Standard Model with lepton mixing. Estimates of
the decay probability and \decay cross-section" for accelerator neutrinos of high energies in
the electric field of nucleus permit one to discuss the general possibility of carrying out the
neutrino experiment. Such an experiment could give unique information on mixing angles in
the lepton sector of the Standard Model which would be almost independent of the specific
neutrino masses.
Talk given at the XXXth Rencontres de Moriond
Electroweak Interactions and Unied Theories
Les Arcs, Savoie, France, March 11-18, 1995
1
1It is known that neutrino oscillations are the main source of information on the
mixing angles and neutrino masses. However limitations on the mixing angles
thus obtained rigidly correlate with the neutrino's mass spectrum. In this work
we'd like to suggest one more source of information on the lepton mixing angles {
namely, the radiative decay of a high-energy neutrino in an external electromag-
netic eld of a nucleus which has virtually no correlation with the neutrino mass
spectrum.
All attempts to nd a possible manifestation of lepton mixing have given up
to now negative results. It is reasonable to suppose that, because of the in-
suciently high precision achieved in experimental studies of neutrino-involving
processes, the neutrino mass spectrum appears degenerate (the neutrinos mani-
fest themselves as massless particles). At the same time, with massive neutrino
the absence of lepton mixing seems unnatural and is virtually incompatible with
attempts to somehow extend the standard model (SM). On the other hand, lepton
mixing similar to quark mixing, in itself, does not go beyond the framework of
the standard model and may lead to some interesting physical phenomena, such
as 1) rare lepton decays with lepton number violation of type ! e, ! e,

i
! 
j
 
i
! 
j
, 2) neutrino oscillations.
It is known that the massive neutrino's properties are sensitive to the medium
it propagates through. It will suce to mention the well known problem of the
solar neutrino and the possibility of solving it, namely, the mechanism of resonance
enhancement of neutrino oscillations in substance. Substance is usually considered
as medium. We note, however, that medium can also be represented by an external
electromagnetic eld, which can signicantly inuence both the properties of the
massive neutrino itself and the process of its decay and even induce novel lepton
transitions with avour violation 
i
$ 
j
(i 6= j) [1] , forbidden in vacuum.
2
In the papers [2-5] we investigated the massive neutrino radiative decay 
i
!

j
 (i 6= j) in external electromagnetic elds (an uniform magnetic eld, a eld
of intensive electromagnetic wave, a crossed eld, a Coulomb eld of nucleus).
The eect of gigantic enhancement of the decay probability by the external eld
(electromagnetic catalysis) was discovered. In particular, one of the results, we
obtained, is that the probability of the ultrarelativistic neutrino decay in external
eld is practically independent of the mass of the decaying neutrino, if only the
2
decay channel is open (m
2
i
> m
2
j
). Here we present the most interesting, in our
opinion, result of the high-energy neutrino radiative decay in the Coulomb eld
of nucleus.
At present the experimentally accessible strengths of electromagnetic elds are
signicantly below the critical strength (F=F
e
 1; F = B; E , F
e
= m
2
e
=e '
4:41  10
13
G). Because of this, eld-induced eects are especially marked in the
ultrarelativstic case with the dynamic parameter

2
= e
2
(pFFp)=m
6
(1)
being not small even for a relatively weak eld (F

is the external eld tensor,
p

is the 4-momentum, m is the mass of the particle). This is due to the fact
that in the relativistic particle rest frame the eld may turn out of order of the
critical one or even higher, appearing very close to the constant crossed eld
(
~
E ?
~
B, E = B). Thus, the calculation in the case of constant crossed eld is
relativistic limit of the calculation in an arbitrary weak eld and possesses a great
extent of generality. What is why the result we obtained for the amplitude of
the ultrarelativstic neutrino radiative decay in the crossed eld can be applied to
consider the high-energy neutrino radiative decay in the Coulomb eld of nucleus.
Here the non-uniformity of the electric eld
~
E of the nucleus isn't substantial,
because the loop process is \local" with the characteristic loop dimension x 
(E

eE)
 1=3
being signicantly less than the nucleus size at neutrino energy E


100GeV .
There is no need to give the total cumbersome expressions for the propagator
and amplitude of the charged fermion in the crossed eld. Here we present the
main part of the amplitude of neutrino radiative decay 
i
(p) ! 
j
(p
0
)(q) in a
physically more interesting case of the ultrarelativistic neutrinos
M '
e
2
G
F

2
("

~
Fp)
2
4
(1  x) +
m
2
j
m
2
i
(1 + x)
3
5
1=2
X
`
K
i`
K

j`
J(
`
);
J(
`
) = i
1
Z
0
dt z
`
1
Z
0
du exp
h
 i (z
`
u + u
3
=3)
i
; (2)
z
`
= 4
2
4
(1 + x)(1  t
2
)
0
@
1 
m
2
j
m
2
i
1
A

`
3
5
 2=3
;
where F

,
~
F

=
1
2
"

F

are the external electromagnetic field tensor and its
dual tensor, 
`
=
q
e
2
(pFFp)=m
3
`
is the so called dynamic parameter, m
l
is the
3
mass of the virtual charged lepton, K is the lepton mixing matrix of Kobayashi-
Maskawa type. x = cos ,  is the angle between the vector ~p (momentum of
the decaying ultrarelativistic neutrino) and ~q
0
(momentum of the photon in the
decaying neutrino rest frame). It is worth noting that the amplitude (2) is a sum
of three loop contributions (` = e; ;  ), each one being characterized by its \eld
form-factor" J(
`
).
In an electric eld the dynamic parameter can be represented as follows:

`
'
 
E

m
`
!
0
@
eE
m
2
`
1
A
sin; (3)
where  is the angle between the vector of the momentum ~p of the decaying neu-
trino and the electric eld strength
~
E . In a general way, cumbersome numerical
calculations are required to find the probability. In the limit of super-high neu-
trino energies (E

 1TeV ), however, the situation is drastically simplified, as
at such neutrino energies the conditions 
e
 

 1, 

 1 are fulfilled in the
vicinity of the nucleus.
Therefore, it is sufficient for us to know only the asymptotics of the function
J():
J() = 1 +O(
2
);  1;
J() = O(
 2=3
);  1; (4)
so that the amplitude (2) is dominated by the contribution due to the virtual
 -lepton. The decay probability can be presented in the following form:
w '

4
G
2
F

3
E

e
2
E
2
sin
2

0
@
1 
m
4
j
m
4
i
1
A
jK
i
K

j
j
2
: (5)
We note that there is no suppression associated with the smallness of the mass
of the decaying neutrino. The comparison of this expression with the well-known
expression for the probability of the radiative decay 
i
! 
j
 of a high-energy
neutrino in vacuum (see, for example, [6]) demonstrates the enormous enhancing
inuence of the external eld on the radiative decay
R 
w
w
0
'
512
9
 
m
W
m

!
4
 
E

m
i
!
2
0
@
eE
m
2
i
1
A
2
sin
2
: (6)
As an illustration, let us give a numerical estimate of R in the case of the decay
of a neutrino of energy E

 1TeV in the vicinity of a nucleus with the atomic
number Z  20:
4
R ' 2  10
+61
 
1 eV
m
i
!
6
 
E

1TeV
!
2
: (7)
The neutrino radiative decay in substance must result in -quantum of energy
E

 E

being observed as the decay product. In experiment, this process would
appear as inelastic scattering of the neutrino on the nucleus. Using this expression
for the probability and taking the nucleus as a uniformly charged sphere of radius
r
N
we obtain the following expression for the "cross-section" of the decay of a
high energy neutrino in the electric eld of a nucleus with the atomic number Z:
 '
4
5
Z
2
 


!
3
G
2
F
E

r
N
0
@
1  
m
4
j
m
4
i
1
A
jK
i
K

j
j
2
' 10
 44
Z
2
0
@
10
 12
cm
r
N
1
A
 
E

1TeV
!
jK
i
K

j
j
2
(cm
2
): (8)
It is worthwhile noting that the \cross-section" we have presented is, of course,
numerically small, but not so small as not to allow discussion concerning the
possibility of such an experiment in the future. It is interesting to compare it
with the elastic 

e-scattering cross-section:

el
(

e) = G
2

M
m
e
E

Z
8
'
 
Z
100
!  
E

100GeV
!
10
 40
cm
2
;
 4  10
3
events of which were observed by Charm II Collab. The ratio of these
cross-sections does not depend on E

and is:

Z
(

! 
e
)

el
(

e)
=
1
70A
1=3
 
Z
100
!

1  m
4

e
=m
4



sin
2

12
 10
 3
sin
2

12
;
where it was put r
N
= r
0
A
1=3
, r
0
' 1=q
0
, q
0
' 0:15GeV , 
12
is the mixing angle
of 
1
and 
2
. This value of the ratio corresponds to a few events of a hard -ray
emission by neutrino beams in the conditions of Charm II. However, the number of
these events is proportional to E

, therefore at larger neutrino beam energy (e.g.
at LHC) and with larger detectors the number of emitted high energy photons
(with !

 E

) can be of order of hundreds of thousands per year.
5
References
[1] A.V.Borisov, I.M.Ternov and L.A.Vassilevskaya, Phys. Lett. B 273, 163
(1991).
[2] A.A.Gvozdev, N.V.Mikheev and L.A.Vassilevskaya, Phys. Lett. B 289, 103
(1992).
[3] A.A.Gvozdev, N.V.Mikheev and L.A.Vassilevskaya, Phys. Lett. B 321, 108
(1994).
[4] A.A.Gvozdev, N.V.Mikheev and L.A.Vassilevskaya, Phys. Lett. B 323, 179
(1994).
[5] A.A.Gvozdev, N.V.Mikheev and L.A.Vassilevskaya, Yad. Phys. 57, 124 (1994).
[6] B.W.Lee and R.E.Shrock, Phys. Rev. D 16, 1444 (1977).
6


Electromagnetic Catalysis of a Neutrino Radiative decay or One More Source of Information on the Lepton Mixing Angles?

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