Neutrinos with a magnetic dipole moment propagating in a medium with a velocity larger than the phase velocity of light emit photons by the Cerenkov process. The Cerenkov radiation is a helicity flip process via which a left-handed neutrino in a supernova core may change into a sterile right-handed one and freestream out of the core. Assuming that the luminosity of the sterile right-handed neutrinos is less than 10^{53} ergs/sec gives an upper bound on the neutrino magnetic dipole moment \mu_\nu < 0.5 \times 10^{-13} \mu_B. This is two orders of magnitude more stringent than the previously established bounds on \mu_\nu from considerations of supernova cooling rate by right-handed neutrinos

Mohanty, S

Samal, M K

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Cerenkov radiation by neutrinos in a supernova core
Subhendra Mohanty
y
and Manoj K. Samal
#
Theory Group, Physical Research Laboratory,
Ahmedabad - 380 009, India
Abstract
Neutrinos with a magnetic dipole moment propagating in a medium with a
velocity larger than the phase velocity of light emit photons by the

Cerenkov
process. The

Cerenkov radiation is a helicity ip process via which a left-
handed neutrino in a supernova core may change into a sterile right-handed
one and free-stream out of the core. Assuming that the luminosity of the
sterile right-handed neutrinos is less than 10
53
ergs/sec gives an upper bound
on the neutrino magnetic dipole moment 

< 0:5  10
 13

B
. This is two
orders of magnitude more stringent than the previously established bounds on


from considerations of supernova cooling rate by right-handed neutrinos.
y
E-mail:mohanty@prl.ernet.in
#
E-mail:mks@prl.ernet.in
PRL-TH-95/10
June 1995
1
The observation of neutrinos [1] from the supernova SN1987A has provided a number of
constraints on the properties of neutrinos [2{4]. Most of the mechanisms for the constraints
on the mass and magnetic moment of the neutrinos depend upon the the helicity ip of a
left-handed neutrino into a sterile right-handed one which can free-stream out of the core
and hence deplete the energy of the supernova core within a timescale of  1 sec. Since the
observed time-scale of neutrino emission [1] is of the order of 1-10 secs, it is expected that
the luminosity of the right-handed neutrinos is less than 10
53
ergs/ sec, which is the total
neutrino luminosity of the supernova. The mechanism for helicity ip caused by a neutrino
magnetic moment that have been considered so far are (i) helicity ip in an external magnetic
eld of the neutron star in the supernova core [2] and (ii) helicity ip by scattering with
charged fermions i.e. the proccesses 
L
e
 
! 
R
e
 
; 
L
p ! 
L
p [3]. The proccess (i) leads
to an upper bound 

< 10
 14

B
(
B
= e=2m
e
, the Bohr magneton), but is unreliable since
it relies on a high magnetic eld ( 10
14
Gauss) in a supernova core which has not been
observed. The scattering proccess (ii) leads to an upper bound 

< (0:2   0:8)  10
 11

B
[3].
In this letter we propose a third mechanism for the neutrino helicity ip which occurs
via a

Cerenkov radiation proccess in the medium of the supernova core. In the supernova
core the refractive index of photons is determined by the electric permitivity of the e
 
, p
+
plasma and the paramagnetic susceptibility of the degenerate neutron gas. We nd that

Cerenkov emission of a photon from a neutrino is allowed in the photon frequency range
!
p
<

! < 2E=(n+1), (where !
p
is the plasma frequency, E is the initial neutrino energy and
n is the refractive index of the medium). Since the

Cerenkov emission proccess is due to the
magnetic dipole operator 



k



, it is a helicity ipping proccess 
L
! 
R
. The helicity
ipping is more ecient in the

Cerenkov proccess because unlike the proccess (i) there is no
dependence on external magnetic eld and unlike (ii) it is a single vertex proccess, so the rate
is larger than the scattering rate 
L
e
 
! 
R
e
 
by 
em
e
 ~
e
=T
, where the exponetial factor
is due to the Pauli blocking of the outgoing charged fermion. We compute the luminosity
Q

R
of the right-handed neutrinos produced by the

Cerenkov proccess. The constraint that
2
Q
R
< 10
53
ergs/sec (the total observed luminosity) leads to the bound on the neutrino
magnetic moment 

< 0:5  10
 13

B
. This is two order of magnitude improvement over
the previously established bound [3] owing to to the fact that the

Cerenkov radiation is a
single vertex proccess unlike the scattering proccesses considered in ref [3].
The amplitude for the

Cerenkov radiation proccess 
L
(p)! 
R
(p
0
)(k) is given by
M =


n
u(p
0
; s
0
)

k

u(p; s)

(k; ); (1)
where 

is the magnetic dipole moment of neutrino and n is the refractive index of the
medium. The transition rate of the

Cerenkov proccess is
  =
1
2E
Z
d
3
p
0
(2)
3
2E
0
d
3
k
(2)
3
2!
(2)
4

(4)
(p  p
0
  k)jMj
2
; (2)
where p = (E; p), p
0
= (E
0
; p
0
) and k = (!; k) are the four momenta of the incoming neutrino,
outgoing neutrino and the emitted photon respectively. Using the identity
Z
d
3
p
0
2E
0
=
Z
d
4
p
0
(E
0
) (p
02
 m
2

);
and integrating over the  function in (2) we obtain
  =
1
16
Z
k
2
dk
E
2
!
2
n
d(cos ) (
2!E   k
2
2jkjjpj
  cos )jMj
2
; (3)
where cos  is the angle between the emitted photon and the incoming neutrino.
In a medium with the refractive index n(= jkj=!), the  function in (3) constrains cos 
to have the value
cos  =
1
nv
[1 +
(n
2
  1)!
2E
]; (4)
where v = jpj=E is the particle velocity and k
2
=  (n
2
  1)!
2
. It is clear that the kinemat-
ically allowed region for the

Cerenkov proccess is where j cos j  1. This leads to an upper
limit on the range of the allowed photon frequency
! <
2E
n+ 1
: (5)
3
where we have taken jpj = E(1   m
2
=E
2
)
1=2
' E since we are dealing with extremely
relativistic neutrinos with m
2
=E
2
< 10
 6
. The refractive index of photons in a medium is
n
2
= , where  and  are the electric permitivity and magnetic permiability of the medium.
For the case of supernova the medium is a plasma consisting of degenerate electrons, protons
and neutrons at temperature T  60 MeV [5]. The permitivity  is given by
 = (1 
!
2
p
!
2
); (6)
where !
p
is the plasma frequency that is determined by the chemical potential of the electrons
~
e
 280 MeV and !
p
= (4=3)
1=2
~
e
 15 MeV [6]. The neutrons do not contribute to
, but they do contribute towards  due to the paramagnetic susceptibility of the neutron
gas which can be treated as a degenerate Fermi gas of magnetic dipoles with para-magnetic
susceptibility  given by [7]
 = 1 + 4;
with  equals to
 =
(2m
n
)
3=2

n
2
~
1=2
n
2
2
; (7)
where m
n
, 
n
(=  1:91e=2m
p
) and ~
n
being the mass, magnetic moment and chemical
potential of neutron respectively. In the supernova core the neutron density is 
n
 410
14
gm/ cm
3
and ~
n
 400 MeV [5]. The paramagnetic susceptibility is  = 0:084 and the
magnetic permiability  = 1 + 4 = 2:055. Thus the refractive index of the medium
becomes
n
2
= 1 + 4 
!
2
p
!
2
: (8)
For  = 0:084, we have n = 1:433 at ! >> !
p
. The refractive index is positive denite and
the

Cerenkov condition (5) is satised in the frequency range
0:70 !
p
 !  0:82 E; (9)
which is the parameter space of the allowed

Cerenkov radiation frequency.
4
Evaluating from (1) we have
jMj
2
=

2

n
2
[ 16(p:k)
2
+ 12(p:k)k
2
+ 16m
2

k
2
]: (10)
Substituting the above expression in (3) and performing the integral over  function and
using (4) for cos , we have the expression for transition rate for the

Cerenkov process [8,9]
  =

2

16E
2
Z
!
2
!
1
d![2(n
2
  1)
2
!
4
  16m
2

(n
2
  1)!
2
]; (11)
where the limits of the integral are from (9), !
1
= 0:7!
p
and !
2
= 0:8E. The

Cerenkov
proccess 
L
(p) ! 
R
(p
0
)(k) changes the 
L
's to sterile 
R
's which can free stream out of
the supernova core. Neglecting the second term (since m

 1 eV and !  60 MeV) in the
expression for  , the luminosity of the sterile 
R
's is
Q

R
=
V 
2

8E
2
Z
1
0
dE[f

(E)  f

(E)]E
2
Z
!
2
!
1
d!(E   !)(n
2
  1)
2
!
4
; (12)
where f

(E) = [e
(E ~

)=T
+1]
 1
and f

(E) = [e
(E+~

)=T
+1]
 1
are the statistical distribution
function of the 
L
and 
L
in the supernova. Performing the integrals over E, the luminosity
of right handed neutrinos is
Q

R
= V 
2

(n
2
  1)
2
(0:002)[
~
7

7
+ 
2
T
2
~
5

+
7
3

4
T
4
~
3

)]: (13)
We take the volume V  4 10
18
cm
3
, ~

 160 MeV, T = 60 MeV [3,5] for the supernova
core parameters within 1 second after collapse. The luminosity turns out to be
Q

R
= 0:164  10
53

2

(GeV
4
); (14)
in terms of the magnetic moment 

. Assuming that the entire energy of the core collapse
is not carried out by the right handed sterile neutrinos, i.e. Q

R
< 10
53
ergs=sec, we have
from (13) the upper bound on the neutrino magnetic dipole moment


< 0:5 10
 13

B
:
This is two orders of magnitude better than the previously established [3] upper bound
from the 
R
luminosity of supernova. The proccess for generating 
R
in the supernova core
5
considered in ref. [3] is via the helicity ip scattering 
L
e
 
! e
 

R
and 
L
p
 
! p
 

R
etc. This proccess has an extra electromagnetic vertex and a Pauli blocking factor for the
outgoing charged fermion compared to the proccess that we have considered. That accounts
for the more stringent bound we established compared to ref. [3].
6
REFERENCES
[1] K. Hirata et. al., Phys. Rev. Letts. 58 (1987) 1490; R. Bionta et. al, Phys. Rev. Letts.
58 (1987) 1494.
[2] S. Nussinov and Y. Rephaeli, Phys. Rev. D 36 (1987) 2278.
[3] R. Barbieri and R. N. Mohapatra, Phys. Rev. Letts. 61 (1988) 27; J. M. Lattimer and
J. Cooperstein, Phys. Rev. Letts. 61 (1988) 23.
[4] D. Notzold, Phys. Rev. D 38 (1988) 1658; I. Goldman, G. Alexander, S. Nussinov and
Y. Aharonov, Phys. Rev. Lett. 60 (1988) 1789.
[5] R. P. Brinkman and M. S. Turner, Phy. Rev. D 38 (1992) 2338.
[6] G. Beaudet, V. Petrosian, and E. E. Salpeter, Astrophys. J. 150, 979 (1967); E. Braaten,
Phys. Rev. Letts. 66 (1991) 1655.
[7] L. Landau and H. Lifshitz, Statistical Physics, p.173, Pergamon Press, (1980), London.
[8] J. S. Bell, Nucl. Phys. 112, 461 (1976).
[9] V. L. Ginzburg, Theoretical Physics and Astrophysics, Pergamon Press (1979), London.
7


Cerenkov radiation by neutrinos in a supernova core

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