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\begin{document}

\title{Spin light of neutrino in matter \\ and
electromagnetic fields}

\author{A.Lobanov, A.Studenikin\thanks{E-mail: studenik@srd.sinp.msu.ru}}
\date{}
\maketitle

\begin{center}
{\em
Department of Theoretical Physics, Moscow State University,
119992, Moscow, Russia}
\end{center}
%\rightline DTP-MSU/00-08

\begin{abstract}
A new type of electromagnetic radiation by a neutrino with
non-zero magnetic (and/or electric) moment moving in background
matter and electromagnetic field is considered. This radiation
originates from the quantum spin flip transitions and we have
named it as "spin light of neutrino". The neutrino initially
unpolarized beam (equal mixture of $\nu_{L}$ and $\nu_{R}$) can be
converted to the totally polarized beam composed of only $\nu_{R}$
by the neutrino spin light in matter and electromagnetic fields.
The considered radiation is important for environments with high
effective densities, $n$, because the total radiation power is
proportional to $n^{4}$. The spin light of neutrino, in contrast
to the Cherenkov or transition radiation of neutrino in matter,
does not vanish in the case of the refractive index of matter is
equal to unit.  The specific features of this new radiation are:
(i) the total power of the radiation is proportional to $\gamma
^{4}$, and (ii) the radiation is beamed within a small angle
$\delta \gamma \sim \gamma^{-1}$, where $\gamma$ is the neutrino
Lorentz factor. Applications of this new type of neutrino
radiation to astrophysics, in particular to gamma-ray bursts,
should be important.
\end{abstract}

 There exist at present convincing evidences
in favour of neutrino non-zero mass and mixing, obtained in the
solar and atmospheric experiments (see \cite{Bil02} for a review
on the status of neutrino oscillations). Apart from masses and
mixing non-trivial neutrino electromagnetic properties such as
non-vanishing magnetic, $\mu$, and electric, $\epsilon$, dipole
moments are carrying features of new physics.  It is  believed
that non-zero neutrino magnetic moment could have an important
impact on astrophysics and cosmology.


It is well known \cite{LSFS7780} that in the minimally extended
Standard Model with $SU(2)$-singlet right-handed neutrino the
one-loop radiative correction generates neutrino magnetic moment
which is proportional to neutrino mass
\begin{equation}
\mu_{\nu}={3 \over {8 \sqrt{2}\pi^{2}}}eG_{F}m_{\nu}=3 \times
10^{- 19}\mu_{0}\bigg({m_{\nu} \over {1 \mathrm{eV}}}\bigg),
\end{equation}
where $\mu_{0}=e/2m$ is the Bohr magneton, $m_{\nu}$ and $m$ are
the neutrino and electron masses. There are also models
\cite{KBMVVFY76-87} in which much large values for magnetic
moments of neutrinos are predicted. So far, the most stringent
laboratory constraints on the electron neutrino magnetic moment
come from elastic neutrino-electron scattering experiments:
$\mu_{\nu_{e}} \leqslant 1.5 \times 10^{-10} \mu_{0}$ \cite{PDG}.
More stringent constraints are obtained from astrophysical
considerations \cite{Raf96}.

In this paper we study a new mechanism for emission of photon by
the neutrino in presence of matter, assuming that the neutrino has
an intrinsic magnetic (and/or electric) dipole moment. This
phenomenon can be expressed as a process
\begin{equation}\label{b}
\nu \rightarrow \nu + \gamma
\end{equation}
that is the transition from a flavour neutrino in the initial
state to the same flavour neutrino plus a photon in the final
state. The mechanism under consideration can be also effective in
the case of neutrino transitions with change of flavour if
neutrino transition moment is not zero.

Different other processes characterized by the same signature
of Eq.(\ref{b}) have been considered  previously:

 i) the photon radiation by massless neutrino
$(\nu_{i} \rightarrow \nu_{j} + \gamma,\; i=j)$ due to the vacuum
polarization loop diagram  in presence of an external magnetic
field \cite{GalNik72,Sko76};

ii) the photon radiation by massive neutrino with non-vanishing
magnetic moment in constant magnetic and electromagnetic wave
fields \cite{BorZhuTer88,Sko91};

iii) the Cherenkov radiation due to the non-vanishing neutrino
magnetic moment in homogeneous and infinitely extended medium
which is only possible if the speed of neutrino is larger than the
speed of light in medium \cite{Rad75,GriNeu93};

iv) the transition radiation due to non-vanishing neutrino
magnetic moment which would be produced when the neutrino crosses
the interface of two media with different refractive indices
\cite{SakKur94-95,GriNeu95};

v) the Cherenkov radiation by  massless neutrino due to its
induced charge in medium \cite{OliNiePal96};

vi) the Cherenkov radiation by massive and massless  neutrino in
magnetized medium \cite{MohSam96, IoaRaf97};

vii) the neutrino radiative decay $(\nu_{i} \rightarrow \nu_{j} +
\gamma, \; i\not=j)$ in external fields and media or in vacuum
\cite{GvoMikVas92,Sko95,KacWun97}.

The process we are studying in this paper has never been
considered before. We discover a mechanism for electromagnetic
radiation generated by the neutrino magnetic (and/or electric)
moment rotation which occurs due to electroweak interaction with
the background environment. It should be noted that generalization
to the case of a photon emission by neutrino due to the neutrino
transition magnetic moment is straightforward. If neutrino is
moving in matter and an external electromagnetic field is also
superimposed, the total power of this radiation contains three
terms which originate from (i) neutrino interaction with particles
of matter, (ii) neutrino interactions with electromagnetic field,
(iii) interference of the mentioned above two types of
interactions. This radiation can be named as "spin light of
neutrino" in matter and electromagnetic field to manifest the
correspondence with the magnetic moment dependent term in the
radiation of an electron moving in a magnetic field. A review on
the spin light of electron can be found in \cite{BorTerBag95}.
Whereas the radiation of a neutral particle moving in external
electromagnetic field in the absence of matter has been considered
previously starting from \cite{TerBagKha65}, the mechanism of
radiation produced by interaction with matter is considered in
this our paper for the first time. It should be emphasize that the
neutrino spin light can not be described as the Cherenkov
radiation.

The spin light of neutrino in the background matter ( similar to
the radiation by neutrino moving in the magnetic field
\cite{BorZhuTer88}) originates from the quantum spin flip
transitions $\nu_{L} \rightarrow \nu_{R}$ . That is why the
initially unpolarized neutrino beam (equal mixture of active
left-handed and sterile right-handed neutrinos) can be converted
to the totally polarized beam composed of only $\nu_{R}$ due to
the spin light in contrast to the Cherenkov radiation which can
not produce the neutrino spin self-polarization effect.

Our approach is based on the quasi-classical Bargmann-Michel-Telegdi (BMT)
equation \cite{BMT59}
\begin{equation}\label{1}
{dS^{\mu} \over d\tau} =2\mu
 \Big\{ F^{\mu\nu}S_{\nu} -u^{\mu}(
u_{\nu}F^{\nu\lambda}S_{\lambda} ) \Big\} +2\epsilon \Big\{
{\tilde
F}^{\mu\nu}S_{\nu} -u^{\mu}(u_{\nu}{\tilde
F}^{\nu\lambda}S_{\lambda}) \Big\},
\end{equation}
that describes evolution of the spin $S_{\mu}$ of a neutral
particle with non-vanishing magnetic, $\mu$, and electric,
$\epsilon$, dipole moments in electromagnetic field, given by its
tensor $F_{\mu \nu}$ {\footnote{Within this approach neutrino spin
relaxation in stochastic electromagnetic fields without account
for matter effects was considered \cite{LoeSto89}.}}. This form of
the BMT equation corresponds to the case of the particle moving
with constant speed, $\vec \beta={\mathrm{const}}$,
$u^\mu=(\gamma,\gamma \vec \beta)$, in presence of an
electromagnetic field $F_{\mu\nu}$. The spin vector satisfies the
usual conditions, $S^2=-1$ and $S^{\mu}u_\mu =0$. Note that the
term proportional to $\epsilon$ violates $T$ invariance.

In our previous studies \cite{EgoLobStu00,LobStu01} we have shown
that  the Lorentz invariant generalization of Eq.(\ref{1}) for the
case when effects of neutrino weak interactions are taken into
account can be obtained by the following substitution of the
electromagnetic field tensor $F_{\mu\nu}=(\vec E,\vec B)$:
\begin{equation}\label{2}
F_{\mu\nu} \rightarrow E_{\mu\nu}= F_{\mu\nu}+G_{\mu\nu},
\end{equation}
where the tensor $G_{\mu\nu}$ accounts for the neutrino
interactions with particles of the environment{\footnote{The
derivation of the quasi-classical Lorentz invariant neutrino spin
evolution equation taking into account general types of neutrino
non-derivative interactions with external fields is given in
\cite{DvoStu02}}. It follows, that  precession of the neutrino
spin can originate not only due to neutrino magnetic moment
interaction with external electromagnetic fields but also due to
the neutrino weak interaction with particles of the background
matter.

In evaluation of the tensor $G_{\mu\nu}$ we demand that the
neutrino evolution equation must be linear  over the neutrino
spin, electromagnetic field and such characteristics of matter
(which is composed of different fermions, $f=e,\ n,\ p...$) as
fermions currents
\begin{equation}
j_{f}^\mu=(n_f,n_f\vec v_f),
\label{3}
\end{equation}
and fermions polarizations
\begin{equation}
\lambda^{\mu}_f =\Bigg(n_f (\vec \zeta_f \vec v_f ),
n_f \vec \zeta _f \sqrt{1-v_f^2}+
{{n_f \vec v_f (\vec \zeta_f \vec v_f )} \over {1+\sqrt{1-
v_f^2}}}\Bigg).
\label{4}
\end{equation}
Here $n_f$, $\vec v_f$, and $\vec \zeta_f \ (0\leqslant |\vec
\zeta_f |^2 \leqslant 1)$ denote, respectively, the number
densities of the background fermions $f$, the speeds of the
reference frames in which the mean momenta of fermions $f$ are
zero, and the mean values of the polarization vectors of the
background fermions $f$ in the above mentioned reference frames.
The mean value of the background fermion $f$ polarization vector,
$\vec \zeta_f$, is determined by the two-step averaging of the
fermion relativistic spin operator over the fermion quantum state
in a given electromagnetic field and  over the fermion statistical
distribution density function. Thus, in general case of neutrino
interaction with different background fermions $f$ we introduce
for description of matter effects antisymmetric tensor
\begin{equation}
G^{\mu \nu}= \epsilon ^{\mu \nu \rho \lambda}
g^{(1)}_{\rho}u_{\lambda}- (g^{(2)\mu}u^\nu-u^\mu g^{(2)\nu}),
\label{7}
\end{equation}
where
\begin{equation}
g^{(1)\mu}=\sum_{f}^{} \rho ^{(1)}_f j_{f}^\mu
+\rho ^{(2)}_f \lambda _{f}^{\mu}, \ \
g^{(2)\mu}=\sum_{f}^{} \xi ^{(1)}_f j_{f}^\mu
+\xi ^{(2)}_f \lambda _{f}^{\mu},
\label{8}
\end{equation}
(summation is performed over the fermions $f$ of the background).
The explicit
expressions for the coefficients $\rho_{f}^{(1),(2)}$ and
$\xi_{f}^{(1),(2)}$
could be found if the particular
model of neutrino interaction is chosen.
For example, if one consider the electron neutrino propagation in moving and
polarized gas of electrons within the extended standard
model supplied with $SU(2)$-singlet right-handed neutrino
$\nu_{R}$, then the neutrino effective
interaction Lagrangian reads
\begin{equation}\label{Lag}
L_{eff}=-f^\mu \Big(\bar \nu \gamma_\mu {1+\gamma^5 \over 2} \nu
\Big),
\end{equation}
where
\begin{equation}
f^\mu={G_F \over \sqrt2}\Big((1+4\sin^2 \theta _W) j^\mu_e -
\lambda ^\mu _e\Big).
\end{equation}
In this case the coefficients $\rho_{e}^{(1),(2)}$  are
\begin{equation}\label{rho}
\rho^{(1)}_e={\tilde{G}_F \over {2\sqrt{2}\mu }}\,, \qquad
\rho^{(2)}_e =-{G_F \over {2\sqrt{2}\mu}}\,,
\end{equation}
where $\tilde{G}_{F}={G}_{F}(1+4\sin^2 \theta _W).$

 In the usual notations the antisymmetric tensor $G_{\mu
\nu}$ can be written in the form,
\begin{equation}
G_{\mu \nu}= \big(-\vec P,\ \vec M),
\label{9}
\end{equation}
where
\begin{equation}
\vec M= \gamma \big\{(g^{(1)}_0 \vec \beta-\vec g^{(1)})
- [\vec \beta \times \vec g^{(2)}]\big\}, \
\vec P=- \gamma \big\{(g^{(2)}_0 \vec \beta-\vec g^{(2)})
+ [\vec \beta \times \vec g^{(1)}]\big\}.
\label{10}
\end{equation}
It worth to note that the substitution (\ref{2}) implies that the
magnetic $\vec B$
and electric $\vec E$ fields are shifted by the vectors $\vec M$
and $\vec P$,
respectively:
\begin{equation}
\vec B \rightarrow \vec B +\vec M, \ \ \vec E \rightarrow \vec E -
\vec P.
\label{11}
\end{equation}

We have also recently derived the total radiation power of a
neutral unpolarized fermion with anomalous magnetic moment
\cite{LobPav00} {\footnote {The generalization to the case of
neutrino with non-zero electric dipole moment in electromagnetic
field can be found in \cite{LobPav_Vest00}.}}. In that derivation
we have supposed that the spin dynamics of  a neutral particle is
governed by the Bargmann-Michel-Telegdi equation and that the
energy of the radiated photons is much less than the particle
energy. Now in order to treat the electromagnetic radiation by the
neutrino moving in background matter and electromagnetic field we
use the substitution prescription of Eq.(\ref{2}) and for the
total radiation power get (for simplicity the case of
$T$-invariant model of neutrino interaction is considered
hereafter)
\begin{equation}\label{power} I={16 \over
3}\mu^2 \Big[4({\mu^2}{u_\rho}{\tilde E^{\rho \lambda}} {\tilde
E_{\lambda\sigma}}{u^\sigma})^2+ {\mu^2}{u_\rho}{{\dot { \tilde
E}}^{\rho\lambda}}{\dot {\tilde E}_{\lambda \sigma}} {u^\sigma}
\Big].
\end{equation}
We also get for the solid angle distribution of the radiation
power :
\begin{equation}\label{ra11}
\begin{array}{c}
\!\!\!\displaystyle\frac{dI}{d\Omega}=\displaystyle\frac{\mu^2}{\pi
\gamma (ul)^5} \Big\{\big[ 4({\mu^2}{u_\rho}{\tilde E
^{\rho\lambda}}{\tilde E_{\lambda\sigma}}{u^\sigma})^2+
({\mu^2}{u_\rho}{{\dot {\tilde E} }^{\rho \lambda}}{{\dot {\tilde
E} }_{\lambda \sigma}}
{u^\sigma})\big] (ul)^2 +\\[14pt]\quad
+4({\mu^2}{u_\rho}{\tilde E ^{\rho\lambda}}{\tilde E
_{\lambda\sigma}} {u^\sigma}) ({\mu}{u_\rho}{\tilde E
^{\rho\lambda}}l_{\lambda})^2+
({\mu}{u_\rho}{{\dot {\tilde E }}^{\rho\lambda}}l_{\lambda})^2 +\\[14pt]
\qquad+4\mu^2({\mu}{u_{\rho}}{\tilde E ^{\rho\lambda}}l_{\lambda})
e^{\mu\nu\rho\lambda} u_{\mu}{\dot {\tilde E
}}_{\nu\sigma}{u^{\sigma}} {\tilde E _{\rho\delta}}{u^{\delta}}
{l_{\lambda}}\Big\},
\end{array}
\end{equation}
here $\tilde E_{\mu\nu}=-\oks{1}{2} \epsilon
_{\mu\nu\alpha\beta}E^{\alpha\beta}$ is the dual tensor
$E_{\mu\nu},\,$ $l^{\mu}=(1, \vec l)$ and   $\vec l$ is the unit
vector pointing the direction of radiation.  The derivatives in
the right-hand side of Eqs.(\ref{power}) and (\ref{ra11}) are
taken with respect of proper time $\tau$ in the rest frame of the
neutrino.


Using Eqs. (\ref{11}) and (\ref{power}) we find the total
radiation power as a function of the magnetic field strength in
the rest frame of the neutrino, $\vec B_{0}$, and the vector $\vec
M_{0}$ which accounts for effects of  the neutrino interaction
with moving and polarized matter:
\begin{equation}\label{0ra13}
I=\frac{16}{3}\mu^4 \Big[\big(2{\mu}(\vec B_{0}+\vec
M_{0}\big)^{2}\big)^2 +(\dot{\vec {B}_{0}} +\dot{\vec
{M}_{0}})^{2}\Big],
\end{equation}
where
\begin{equation}
\vec B_0=\gamma\Big(\vec
B_{\perp}
+{1 \over \gamma} \vec B_{\parallel} + \sqrt{1-\gamma^{-2}}
\Big[{\vec E_{\perp} \times \vec n}\Big]\Big),
\end{equation}

\begin{equation}
\vec {M_0}=\vec {M}{_{0_{\parallel}}}+\vec {M_{0_{\perp}}},
\label{M_0}
\end{equation}
\begin{equation}
\begin{array}{c}
\displaystyle \vec {M}_{0_{\parallel}}=\gamma\vec\beta{n_{0} \over
\sqrt {1- v_{e}^{2}}}\left\{ \rho^{(1)}\left(1-{{\vec v}_e
\vec\beta \over {1- {\gamma^{-2}}}} \right)\right.
-\\- \displaystyle\rho^{(2)}\left.
\left(\vec\zeta_{e}\vec\beta \sqrt{1-v^2_e}+ {(\vec \zeta_{e}{\vec
v}_e)(\vec\beta{\vec v}_e) \over 1+\sqrt{1-v^2_e} }\right){1 \over
{1- {\gamma^{-2}}}} \right\}, \label{M_0_parallel}
\end{array}
\end{equation}
\begin{equation}\label{M_0_perp}
\begin{array}{c}
\displaystyle \vec {M}_{0_{\perp}}=-\frac{n_{0}}{\sqrt {1-
v_{e}^{2}}}\Bigg\{ \vec{v}_{e_{\perp}}\Big(
\rho^{(1)}+\displaystyle\rho^{(2)}\frac
{(\vec{\zeta}_{e}{\vec{v}_e})}
{1+\sqrt{1-v^2_e}}\Big)+{\vec{\zeta}_{e_{\perp}}}\rho^{(2)}\sqrt{1-v^2_e}\Bigg\},
\end{array}
\end{equation}
and $n_0$ is the invariant number density of matter given in the
reference frame for which the total speed of matter is zero.

It is evident that the total radiation power, Eq.(\ref{0ra13}),
 is composed of the three contributions,
\begin{equation}
I=I_{F}+I_{G}+I_{FG},
\end{equation}
where $I_{F}$ is the radiation power due to the neutrino magnetic
moment interaction with the external electromagnetic field,
$I_{G}$ is the radiation power due to the neutrino weak
interaction with particles of the background matter, and $I_{FG}$
stands for the interference effect of electromagnetic and weak
interactions. It should be pointed out that  the electromagnetic
contribution $I_{F}$ to the radiation of a neutrino (or a neutral
fermion) in different field configurations has been considered in
literature (see, for example,
(\cite{TerBagKha65,BorZhuTer88,Sko91}), whereas the contribution
to radiation by neutrino moving in matter, $I_{G}$, and also the
interference term $I_{FG}$ have never been considered before.

These three types of neutrino radiation have common nature: they
originates as effects of the neutrino interactions with the
external electromagnetic field and background matter under which
the neutrino spin is rotating. As it has been pointed out above,
the discussed here radiation cannot be treated as the neutrino
Cherenkov \cite{Rad75,GriNeu93,OliNiePal96,MohSam96,IoaRaf97} and
transition \cite{SakKur94-95,GriNeu95} radiations. Contrary to the
Cherenkov and transition radiations the considered new type of
neutrino radiation is  not forbidden when the refractive index of
photons in the background environment is equal to
$n^{ref}_{\gamma}=1$. In order to highlight this distinction we
consider here the particular case of $n^{ref}_{\gamma}=1$, however
generalization to the case $n^{ref}_{\gamma}>1$ is
straightforward.

One of the most important properties of the neutrino spin light in
the background matter is the strong dependence of the total
radiation power on the density of matter,
\begin{equation} \label{G}
I_{G} \sim {n_{0}^{4}}.
\end{equation}

The total radiation power, Eq.(\ref{0ra13}),  contains different
terms in respect to dependence on the neutrino magnetic moment
$\mu$. If we consider non-derivative terms and compare
contributions to the neutrino spin light radiation power from the
interaction with matter and electromagnetic field we come to the
conclusion that, as it follows from Eq.(\ref{rho}), the ratio
$I_{G}/I_{F}$ is proportional to $\mu^{-4}$. From Eq.(\ref{0ra13})
it is also obvious that the ratios of the radiation power in
matter with varying density (in varying electromagnetic fields) to
the radiation power in matter with constant density (in constant
electromagnetic fields) are proportional to $\mu^{2}$. Accounting
for smallness of the neutrino magnetic moment, it follows that the
proposed new mechanism of neutrino spin light radiation could be
efficient in the environments with varying densities of matter
(with varying electromagnetic fields).

We consider at first the spin light of neutrino in presence of
constant density matter and constant magnetic field. For
definiteness we assume that the spin light radiation is produced
by the electron neutrino $\nu_{e}$ moving in unpolarized
($\zeta_e=0$) matter composed of only electrons. Then for the
total spin light radiation power we get
\begin{equation}\label{BG}
I={64 \over 3} \mu^{6} \gamma^{4}\Big[ \big( \vec B_{ \perp} + {1
\over \gamma} \vec B_{
\parallel} +  n\rho^{(1)} \vec \beta (1- \vec \beta \vec
v_e) - {1 \over \gamma} n \rho^{(1)} \vec v_{e_\perp}\big)^{2}
\Big]^{2},
\end{equation}
where the terms proportional to ${\gamma^{-2}}$ in the brackets
are neglected. Here we use the notation,
\begin{equation}
n={n_0 \over \sqrt{1-v_{e}^{2}}},
\end{equation}
$\vec B_{ \perp , \parallel}$ are the transversal and longitudinal
magnetic field components in respect to the neutrino motion, $\vec
v_{\perp}$ is the transversal component of the matter speed in the
laboratory frame of reference. From Eq.(\ref{BG}) it follows that
the correlation term $I_{FG}$ is suppressed  in respect to the
terms $I_F$ and $I_G$ by the presence of additional neutrino
Lorentz factor. That is why there is a reason to compare
contributions to the radiation power produced by the neutrino
interaction with matter which is moving longitudinally with
non-relativistic speed, $v \ll 1$,
\begin{equation}
I_{G}={64 \over 3} \mu^{6}\gamma^{4}\big(  n\rho^{(1)} \beta
\big)^{4},
\end{equation}
and the contribution to the radiation power produced by the
interaction with the transversal magnetic field{\footnote{This our
result is smaller by a factor $1/2$ relative to the result of
\cite{TerBagKha65} for the radiation power of the polarized
neutral particle.}},
\begin{equation}
I_{F}={64 \over 3} \mu^{6}\gamma^{4}B_{ \perp}^{4}.
\end{equation}
If we take $B_{\perp} \sim 3\times 10^{5}\,\mathrm{G}$  and $n
\sim 10^{23}\, \mathrm{cm}^{-3}$ that corresponds to the case of
the solar convective zone and $\mu_{\nu_e} \sim 10^{-18} \mu_{0}$,
then we get that the matter term in the spin light exceeds the
transversal magnetic field term by a factor of $\sim 2 \times
10^{26}$. With increasing of the neutrino magnetic moment due to
the inverse proportionality $\rho^{(1)} \sim \mu_{\nu_e}^{-1}$ the
ratio $ I_{G}/ I_{F}$ decreases and becomes equal to unit only for
$\mu_{\nu_e} \sim 3 \times 10^{-12}\mu_{0}$.

Let us turn to consideration the contribution to the spin light of
neutrino that is generated by the neutrino interaction with matter
of varying density. We again assume that matter is composed of
electrons and neutrino interaction with matter is given by
Lagrangian of Eq.(\ref{Lag}). Since $\dot n_e = \gamma \vec \beta
\vec {\nabla} n_e$, we get for the total power of the neutrino
spin light in this case
\begin{equation}\label{0ra}
I_{G}=\frac{2}{3}\mu^{2}\gamma^{4}
\left\{\frac{1}{2}{\tilde{G}_F^{4}n_e}^{4}+\tilde{G}_F^{2}
\left[({\vec{\beta}}{\vec{\nabla}})n_e\right]^{2} \right\}\!.
\end{equation}
If we consider the case of moving and polarized matter then the
effective number density depends on the value of the total mater
speed $\vec v_{e}$ and polarization $\vec \zeta _{e}$, as well as
on neutrino speed $\vec \beta$ and correlations between these
three vectors (see \cite{LobStu01,GriLobStuplb_02}):
\begin{equation}\label{2ra}
\begin{array}{c}
n_e= \displaystyle{n_{0} \over {\sqrt{1-v_{e}^2}}}\bigg\{ \big(
1-\left(1+4\sin^2 \theta _W\right)^{-1}{{\vec {\zeta}}_{e}}{{\vec
{v}}}_e\big ) \big( 1-{{\vec{\beta}}}{\vec{v}}_e \big)-\\-
\displaystyle\left(1+4\sin^2 \theta _W\right)^{-1}\sqrt{1-v^2_e}
\bigg[ { ({\vec{ \zeta}}_{e}{\vec{v}}_e)({\vec{\beta}}{\vec{v}}_e)
\over 1+\sqrt{1-v^2_e} }-{\vec{\zeta}}_{e}{\vec{\beta}} \bigg] +
O\big({1 \over \gamma}\big)
 \bigg\}.
\end{array}
\end{equation}

Using Eqs. (\ref{ra11}) and (\ref{0ra}) for the solid angle
distribution we get
\begin{equation}\label{1ra}
\frac{dI_{G}}{d\Omega}=\frac{3}{8\pi}I_{G}
 \bigg\{\!\gamma^{-4}(1-\beta\cos\vartheta)^{-3}\!
-\gamma^{-6}(1-\beta\cos\vartheta)^{-4}\!+
\frac{1}{2}\,\gamma^{-8}(1-\beta\cos\vartheta)^{-5}\!\bigg\}.
\end{equation}
From the last formula it follows that the neutrino spin light is
strongly beamed and is confined within the cone given by $\delta
\theta \sim \gamma ^{-1}$. It should be noted here that this is a
common feature of different contributions to the neutrino spin
light radiation which is the inherent property of radiation by
ultra-relativistic particles.

In conclusion, we have considered a new mechanism of the
electromagnetic radiation emitted by a neutrino  with
non-vanishing magnetic moment ("spin light of neutrino") moving in
the background matter and external electromagnetic field. The
generalization to the case of a neutrino with non-zero electric
dipole moment is just straightforward. Our general new result is
the prediction of a new type of electromagnetic radiation that is
emitted by a neutral particle with non-vanishing magnetic (and/or
electric) moment moving in the background matter. The considered
radiation must be important for environments with high effective
densities, $n$, because the total radiation power is proportional
to $n^{4}$. The neutrino spin light originates from the quantum
spin flip transitions. Thus, the initially unpolarized neutrino
beam can be converted to the totally polarized beam composed of
only right-handed neutrinos $\nu_{R}$ in the considered radiation
process if the right-handed neutrinos are sterile states, i.e. do
not interact with the background environment. In contrast to the
Cherenkov and transition radiations by neutrino, the spin light is
produced by neutrino even in the case when the refractive index of
photons in the background matter is equal to $n^{ref}_{\gamma}=1$.
It is also shown that the neutrino spin light is strongly beamed
in the direction of neutrino propagation and is confined within a
small cone given by $\delta\theta\sim\gamma^{-1}$. The total power
of the spin light of neutrino is increasing with the neutrino
energy increase and is proportional to the fourth power of the
neutrino Lorentz factor, $I \sim {\gamma }^{4}$. It is also
possible to show that the average energy of photons of the spin
light in matter is $\omega_{\gamma} \sim G_{F} n_{e} \gamma^{2}$.
Thus, the spin light emission rate is proportional to
\begin{equation}
W_{\gamma} \sim \gamma^{2}.
\end{equation}
From these estimations we predict that the spin light of neutrino,
together with the synchrotron mechanism of radiation by charged
particles, should be important  for understanding of astrophysical
phenomena where powerful beams of gamma-rays are produced.  One of
the possible examples could be the gamma-ray bursts.

We would like to thank Leo Stodolsky for helpful discussions.

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\end{thebibliography}
\end{document}

