\documentstyle[12pt]{article}


\ifx\tocauthor\undefined\def\tocauthor#1{\relax}\fi
\ifx\toctitle\undefined\def\toctitle#1{\relax}\fi
\begin{document}
\tocauthor{A. V.~Kuznetsov, N. V.~Mikheev and 
  L. A.~Vassilevskaya}
\author{A.V.~Kuznetsov$^{1)}$, N.V.~Mikheev$^{1)}$ and 
L.A.~Vassilevskaya$^{1,2)}$\\ [5mm] 
       {\small\it 1) Department of Theoretical Physics, 
               Yaroslavl State University,} \\
       {\small\it    Sovietskaya 14, Yaroslavl 150000, 
               Russia} \\
       {\small\it  (E-mail addresses: avkuzn@uniyar.ac.ru, 
               mikheev@yars.free.net)} \\
       {\small\it 2) Moscow Lomonosov University, V-952, Moscow 117234, 
               Russia} \\
        {\small\it   (E-mail address: vassilev@yars.free.net)}}


\title{Neutrino-Photon Processes in Strong
Magnetic Field}

\date{}

\maketitle

\begin{abstract}
Neutrino-photon processes $\nu \to \nu \gamma$ and 
$\gamma \to \nu \bar \nu$ are investigated in a strong
magnetic field in the framework of the standard model.
Possible astrophysical manifestations of the processes considered
are analyzed.
\end{abstract}

\section{Introduction}

Nowadays, it is generally recognized that astrophysical objects and
processes inside them give us unique possibilities~\cite{Raffelt-book}
for investigations of particle pro\-per\-ties under extreme conditions of a
high density and/or temperature of matter, and also of a strong magnetic
field. A concept of the astrophysically strong magnetic field has been
changed in the recent years and now the field is considered as the strong
one if it is much greater than the known Schwinger value,
$B \gg B_e, B_e = m^2_e/e \simeq 4.41 \cdot 10^{13} G$.
Possible mechanisms are now discussed of a generation of such strong fields
($B \sim 10^{15} - 10^{17} G$) in ast\-ro\-phy\-si\-cal cataclysms like a 
supernova explosion or a coalescence of neutron stars~\cite{toroidal,poloidal}.

A strong magnetic field, like a medium, makes an active influence
on particle properties. First, it opens new channels forbidden in
vacuum, for example, $\gamma \to e^+ e^-$~\cite{Klepikov},
$\gamma \to \gamma \gamma $~\cite{Adler}. Second, the field can catalyze
essentially the decay processes as it takes with the massive neutrino
radiative decay $\nu \to \nu' \gamma$~\cite{GMV-96}. It was shown~\cite{GMV-96}
that the field-induced amplitude was not suppressed by smallness of neutrino
masses and did not vanish even in the case of massless neutrino as opposed
to the vacuum amplitude. The decay probability of the neutrino with
energy $E_\nu < 2 m_e$ was calculated under the assumption that the
dispersion relation of the final photon was close to the vacuum one $q^2 = 0$.
However the photon dispersion in a strong magnetic field differs
significantly from the vacuum dispersion with increasing photon
energy~\cite{Adler,Shabad1,Tsai}, so the real photon 4-momentum can appear
as a space-like and sufficiently large one ($\vert q^2 \vert \gg m^2_\nu$).
In this case the phase space for the neutrino transition $\nu \to \nu' \gamma$
with $m < m'$ is opened also. It means that the decay probability of
ultrarelativistic neutrino $\nu \to \nu' \gamma$ becomes insensitive
to the neutrino mass spectrum due to the photon dispersion relation in a
strong magnetic field. This phenomenon results in a strong suppression
($ \sim m^2_\nu / E^2_\nu$) of the neutrino transition with flavor
violation and the diagonal process $\nu_l \to \nu_l \gamma$
($l = e, \mu, \tau$) is realized only.
It means that in a strong magnetic field a question of neutrino mixing
is not relevant in the case of the ultrarelativistic
$\nu \to \nu \gamma $ transitions which can be considered in the
frame of the standard model without lepton mixing.

The photon splitting into the neutrino-antineutrino pair 
$\gamma \to \nu \bar \nu$ is the other interesting manifestation of 
neutrino processes in the magnetic field. This process well studied in 
plasma~\cite{Raffelt-book,Raffelt-90}
becomes possible in the external field when the photon acquires 
the timelike 4-momentum $q$ ($q^2 > 0$). 

Here we present some results of the strong magnetic field influence
on the diagonal neutrino transition $\nu \to \nu \gamma$~\cite{GMV-97} and
the photon splitting $\gamma \to \nu \bar \nu$~\cite{KMV-98},
and discuss possible astrophysical consequences.
Below we consider a possible situation when
from both components of the active medium, a magnetic field and plasma,
presented in the most of astrophysical objects, the 
magnetic component dominates.
For example, in a supernova explosion or in a coalescence of neutron
stars a region could exist outside the neutrinosphere, of order of several
hundred kilometers in size, where plasma is rather rarefied, and the magnetic
field of the toroidal type~\cite{toroidal} could reach a value of
$10^{14} - 10^{16}$~G.

If the momentum transferred is relatively small, $|q^2| \ll M^2_W$,
the calculations can be performed 
using the effective electron--neutrino interaction:
%
\begin{eqnarray}
{\cal L} & = & \frac{G_F}{\sqrt 2}\; j_\alpha
\left ( \bar e \gamma_\alpha (g_V - g_A \gamma_5) e \right ).
\label{eq:Lagr}
\end{eqnarray}
%
Here $j_\alpha = \bar \nu \gamma_\alpha (1 - \gamma_5) \nu$ is the left 
neutrino current,
$g_V = \pm {1 \over 2} + 2 \sin^2 \theta_W$,
$g_A = \pm {1 \over 2}$, the upper signs correspond to the electron
neutrino ($\nu = \nu_e$) when both neutral and charged current interaction
takes part in a process. The lower signs correspond to $\mu$ and $\tau$
neutrinos ($\nu = \nu_\mu, \nu_\tau$), when the neutral current interaction
is only presented in the Lagrangian~(\ref{eq:Lagr}).

\section{Neutrino Bremsstrahlung
\lowercase{$\nu \to \nu \gamma$}}

Here we discuss the high energy diagonal neutrino transition
$\nu \to \nu \gamma$~\cite{GalNik,Skob,IR,GMV-97} 
taking into account the photon
dispersion in the magnetic field. Note that only two photon eigenmodes
with polarization vectors
%
\begin{eqnarray}
\varepsilon _{\mu}^{(\parallel)} =
\frac{ (q \varphi)_{\mu} }{ \sqrt{
 q^2_\perp  } }, \; \; \; \; \;
\varepsilon _{\mu}^{(\perp)} =
\frac{ (q \tilde \varphi)_{\mu} }{ \sqrt{ q^2_\parallel } },
\label{eq:EP}
\end{eqnarray}
%
are realized in the magnetic field (the so-called parallel ($\parallel$)
and perpendicular ($\perp$) polarizations according to Adler's 
notations~\cite{Adler}).
Here $ \varphi_{\alpha \beta} = F_{\alpha \beta} / B$ and

\noindent
${\tilde \varphi}_{\alpha \beta} = \frac{1}{2} \varepsilon_{\alpha \beta
\mu \nu} \varphi_{\mu \nu} \; $ are the dimensionless tensor and the dual
tensor of the external magnetic field with the strength $\vec B$;
$q_\mu$ is the photon 4-momentum;

\noindent
$q^2_{\parallel} = ( q \tilde \varphi \tilde \varphi q ) =
q_\alpha \tilde \varphi_{\alpha\beta} \tilde \varphi_{\beta\mu} q_\mu$,
$q^2_{\perp}=(q \varphi \varphi q)$.

The partial amplitudes of the photon creation with the 
polarizations (\ref{eq:EP}) can be presented in the form:
%
\begin{eqnarray}
M_{\parallel} & \simeq & - \frac{e G_F }{\sqrt 2} \; g_V \; V_\parallel \;
\frac{(q \varphi j)}{\sqrt{q^2_\perp}}
\label{eq:Mpar} \\
M_{\perp} & \simeq & - \frac{e G_F}{\sqrt 2} \;
\frac{1}{\sqrt{q^2_\parallel}}
\left \lbrace g_V\; V_\perp (q \tilde \varphi j) + g_A\;
A (q \varphi \varphi j) \right \rbrace ,
\label{eq:Mperp}
\end{eqnarray}
%
where $M_\parallel$ and $M_\perp$ correspond to the creation of the
$\parallel$ and $\perp$ photon modes, respectively.
In general, the functions $V_\lambda$ and $A$
have cumbersome forms. Here we discuss their properties which are very
important for the analysis of this process. Namely, these functions
are singular with respect to the variable $q^2_\parallel$ in the
threshold points ${\cal E}^2_{nn'}=\left (\sqrt{m^2_e + 2 e B n} +
\sqrt{m^2_e + 2 e B n'}\right)^2$. The square-root singularities of the
functions $V_\lambda$ in the threshold points are known as the cyclotronic
resonances in the vacuum polarization~\cite{Shabad1}.

It is of interest for some astrophysical applications to investigate 
the case of relatively high neutrino energy
$E_\nu \simeq 10 \div 20 \, \mbox{MeV} \, \gg m_e$
and strong magnetic field $e B > E^2_\nu$ when a region
of the cyclotronic resonance on the ground Landau level dominates
in this process. As the analysis shows, the functions $V_\perp$ and
$A$ have the square-root singularity in this region:
%
\begin{eqnarray}
A \simeq - V_\perp \simeq
\frac{e B \; m_e \, e^{- q^2_\perp / 2 e B}}
     {2 \pi \sqrt{4 m^2_e - q^2_\parallel}}.
\label{eq:AVsing}
\end{eqnarray}
%
It means that only the amplitude $M_\perp$~(\ref{eq:Mperp})
which corresponds to the creation of $\perp$ photon mode
has the the enhancement due to the square-root singularity on the
ground Landau level $q^2_\parallel \rightarrow 4 m_e^2$.
As for $V_\parallel$, it has not such an enhancement and the
dispersion relation for $\parallel$ photon mode is close
to the vacuum one. So, in this region the amplitude
$M_\parallel \sim (q \varphi j) \sim q^2$
is small due to the collinear kinematics of the process
with creation of $\parallel$ photon mode.
We stress that the vacuum polarization ${\cal P}_\perp$ 
has also the singular behavior in the vicinity
of the threshold point $q^2_\parallel = 4 m_e^2$. 
It means that many-loops quantum corrections are of great importance in the
vicinity of the resonance.
These radiative corrections are reduced to the
renormalization of the wave function of $\perp$ photon mode:
%
\begin{eqnarray}
\varepsilon^{(\perp)}_\mu
\rightarrow \sqrt Z \; \varepsilon^{(\perp)}_\mu, \,\,\,\,
Z = \left ( 1 -
\frac{\partial {\cal P}_\perp}{\partial q^2_\|}
\right )^{-1},
\label{eq:renor}
\end{eqnarray}
%
and lead to the extra factor $\sqrt Z$ in the amplitude 
$M_\perp$~(\ref{eq:Mperp}).
The main contribution to the probability of $\nu \to \nu \gamma^{(\perp)}$
is determined from the vicinity of the resonance point
$q^2_\parallel = 4 m_e^2$ and 
in the limit $e B \gg E^2_\nu \sin^2 \theta$ has the form:
%
\begin{eqnarray}
W^{(\gamma)} & \simeq & \frac{\alpha G^2_F}{8 \pi^2} \, (g^2_V + g^2_A) \,
(e B)^2 \, E_\nu \, \sin^2 \theta.
\label{eq:W(gamma)}
\end{eqnarray}
%
Here $E_\nu$ is the initial neutrino energy,
$\theta$ is the angle between the vectors of the magnetic field
strength $\vec B$ and the momentum of the initial neutrino $\vec p$.

\section{The Neutrino Energy and Momentum Losses}

It should be noted that a practical significance of these processes
for astrophysics could be in the mean values of the neutrino energy
and momentum losses rather than in the probabilities.
These mean values could be found from the four-vector
%
\begin{equation}
Q^{\alpha} =  E_\nu\, \int d W\;q^\alpha =
\frac{1}{16 \pi^3} \, \int
\frac{d^3 k}{\omega} \, q^{\alpha}
\, Im \,{\cal M}(\nu \bar \nu \to \nu \bar \nu) .
\label{eq:Q}
\end{equation}
%
Its zero component is connected with the mean neutrino energy
loss in a unit time, ${\cal I} = dE/dt$.
The space components of the four-vector~(\ref{eq:Q}) are connected
similarly with the neutrino momentum
loss in unit time, $\vec {\cal F} = d\vec p/dt$.
Here we present the expression for $Q^\alpha$ in the limiting case
when $e B \gg E^2 \sin^2 \theta$:

\vspace{-3mm}

\begin{eqnarray}
{\cal I} & = & \frac{1}{4} E W \, 
\left ( 1 + \frac{2 g_V g_A}{g_V^2 + g_A^2} \cos \theta \right ),
\label{eq:I} \\
{\cal F}_z & =  & \frac{1}{4} E W \, 
\left ( \cos \theta  + \frac{2 g_V g_A}{g_V^2 + g_A^2} \right ),
\quad
{\cal F}_\perp = \frac{1}{2} E W \, \sin \theta,
\label{eq:calF}
\end{eqnarray}
%
where the $z$ axis is directed along the field, the vector
$\vec {\cal F}_\perp$, transverse to the field, lies in the plane of
the vectors $\vec B$ and $\vec p$.

\section{Possible Astrophysical Consequences}

To illustrate a possible application of our results, below we estimate
the neutrino energy and momentum losses in an astrophysical
cataclysm of the type of a supernova explosion or a merger of
neutron stars. We assume that for some reasons a compact remnant
has a strong poloidal magnetic field of order
$B \sim 10^{15} \div 10^{17}$~G. Objects of this type, the
so-called ``magnetars'', were investigated in~\cite{DT}.
According to the standard astrophysical models~\cite{Imshennik},
the neutrinos of all species with the typical mean energy
$\bar E_\nu \sim 20$~MeV are radiated from a neutrinosphere
in above mentioned astrophysical cataclysm. 
In this case the neutrino propagating through the
magnetic field will lose the energy and the momentum in accordance
with our formulae. The main contribution to the total energy lost by
neutrinos in the field could be estimated from Eq.~(\ref{eq:I}):
%
\begin{eqnarray}
\frac{\Delta {\cal E}}{{\cal E}_{tot}}
 \simeq 0.6 \cdot 10^{-2}\,
\left (\frac{B}{10^{17} \, \mbox{G}} \right )\,
%\nonumber \\
%& \times & 
\left (\frac{\bar E}{20 \, \mbox{MeV}} \right )^3\,
\left (\frac{ \Delta \ell}{10 \, \mbox{km}} \right ).
\label{eq:DE1}
\end{eqnarray}
%
Here $\Delta \ell$ is a characteristic size of the region
where the field strength varies in\-signi\-fi\-cant\-ly,
${\cal E}_{tot}$ is the total energy carried off by
neutrinos in a supernova explosion,
$\bar E$ is the neutrino energy averaged over the neutrino spectrum.
Here we take the energy scales which are believed to be typical for
supernova explosions~\cite{Imshennik}.

The asymmetry of outgoing neutrinos
$A={|\sum_i {\bf p}_i|}/{\sum_i |{\bf p}_i|}$
is another interesting manifestation.
In the same limit of the strong field we obtain
%
\begin{equation}
A \sim  10^{-2}\,
\left (\frac{B}{10^{17} \, \mbox{G}} \right )
\left (\frac{\bar E}{20 \, \mbox{MeV}} \right )^3
\left (\frac{ \Delta \ell}{10 \, \mbox{km}} \right ).
\label{eq:A2}
\end{equation}
%
One can see from Eqs.~(\ref{eq:DE1}) and~(\ref{eq:A2}) that the effect
could manifest itself at a level of about percent. In principle,
it could be essential in a detailed theoretical description of the
process of supernova explosion.

We note that $\gamma$-quanta produced are captured by the strong magnetic 
field and propagate along the field. At the first glance, it seems photons 
are confined. However the magnetosphere of a ``magnetar'' has a ``polar cap''
region which is defined as a narrow cone along the magnetic field axis
with open lines of the magnetic field strength.
So the particles which are created in the mentioned above neutrino
reactions within a narrow cone can escape outside. Below we estimate the
neutrino energy loss in the ``polar cap'' regions of a millisecond
``magnetar'' taking into account that both processes are comparable for
the values of the physical parameters:
%
\begin{eqnarray}
{\cal E} & \sim & 10^{48} \, \mbox{erg} \,
\left ( \frac{{\cal E}_{tot}}{3 \cdot 10^{53} \, \mbox{erg}} \right )
\left ( \frac{B}{10^{17} \, \mbox{G}} \right )^2
\label{eq:E} \\
& \times &
\left ( \frac{\bar E}{20 \, \mbox{MeV}} \right )
\left ( \frac{R}{10 \, \mbox{km}} \right )^3 \;
\left ( \frac{10^{-3} \, \mbox{sec}}{P} \right )^2,
\nonumber
\end{eqnarray}
%
where ${\cal E}_{tot} \sim 10^{53}$~erg is the typical total neutrino
radiation energy; $P$ is the ``magnetar'' rotation period~\cite{poloidal};
$B$ is the magnetic field strength in the vicinity of
the neutrinosphere of radius $R$.

We point out that the energy loss~(\ref{eq:E}) in terms
of $4 \pi$-geometry is close to energy
deposition observed as $\gamma$-ray bursts (GRB's).
In the standard ``fireball'' model energy of order of $10^{51}$~erg
is deposited in a small volume and results in an
ultrarelativistic ejecta. A collision of the ejecta with
the intergalactic medium can be a source of the GRB~\cite{DS}.
It is interesting that rapid rotation of the remnant combined
with the strong magnetic field becomes popular for understanding
GRB's afterglow~\cite{Paszynski}.

\section{Photon Splitting
\lowercase{$\gamma \to \nu \bar \nu$}}

The photon splitting $\gamma \to \nu \bar \nu$ is the crossed channel
of the gamma radiation by a neutrino $\nu \to  \nu \gamma $.
Due to the photon dispersion in a magnetic field~\cite{Shabad2},
the necessary condition of the splitting, $q^2 > 0$, is
realized for the perpendicular ($\perp$) polarization in the region
$q^2_\parallel > 4 m_e^2$ and for the parallel
($\parallel$) polarization in the region
$q^2_\parallel > ( m_e + \sqrt{m^2_e + 2 e B})^2$.
However, due to the collinearity of the kinematics,
$j_{\alpha} \sim q_{\alpha} \sim p_{\alpha} \sim p'_{\alpha}$,
the amplitude of the $\parallel$ mode splitting is suppressed.

The total probability of $\gamma^{(\perp)} \to \nu \bar \nu$
into all neutrino species is defined by the expression
%
\begin{equation}
W_{\perp} = {1 \over 16 \pi \omega} \,
\int\limits^1_0 dx \; | M_\perp |^2 =
\frac{\alpha \; G_F^2}{16 \pi^4 \omega} \;
e^2 (q F F q) \; | J |^2.
\label{eq:W1}
\end{equation}
%
Here we assume that all neutrino masses are much smaller than the
field-induced ``photon mass''. In the case when the field strength $B$
appears to be the largest physical
parameter, $e B \gg q^2_\parallel$, the integral $J$ is:
%
\begin{equation}
J \simeq  \frac{1 - v^2 }{2 v}
\left (
\ln \frac{1 + v}{1 - v} - i \pi
\right ) + 1,
\label{eq:J2}
\end{equation}
%
where $v = \sqrt{1 - 4 m^2_e/q^2_\parallel}$.

Let us note that the photon splitting probability~(\ref{eq:W1})
with~(\ref{eq:J2}) contains at first sight the pole-type singularity at
$q^2_\parallel \to 4 m^2_e$ ($v \to 0$). However, the solution
of the equation of the photon dispersion in this limit shows that:
%
\begin{equation}
| q^2_\parallel - 4 m^2_e |_{min} = \omega \;
\Gamma_{\gamma \to e^- e^+}.
\label{eq:disp}
\end{equation}
%
\noindent It is known that the similar seeming singularity, but of the
square-root-type, takes place in the process of the photon splitting into
the electron-positron pair $\gamma \to e^+ e^-$~\cite{Klepikov}. As was shown
in Ref.~\cite{Shabad2}, taking account of the photon dispersion in the process
$\gamma \to e^+ e^-$ leads to a finite value for the decay width, which is
maximal at the point $q^2_\parallel = 4 m^2_e$
%
\begin{equation}
\left ( \Gamma_{\gamma \to e^+ e^-} \right)_{max} =
\frac{\sqrt 3}{2} \;
\left ( \frac{2 \alpha e B}{m^2_e} \right)^{2/3} \;
\frac{m^2_e}{\omega}.
\label{eq:max}
\end{equation}
%
The probability of the splitting $\gamma \to \nu \bar \nu$ is also finite,
in view of Eqs.~(\ref{eq:disp}) and~(\ref{eq:max}), and amounts up to the
maximal value:
%
\begin{equation}
(W_\perp)_{max} = \frac{(G_F \;m^2_e)^2}{4 \sqrt 3 \pi^2} \,
\left ( \frac{2 \alpha e B}{m^2_e} \right )^{1/3} \;
\frac{e B}{\omega} .
\label{eq:W2}
\end{equation}

It is evident that the probability~(\ref{eq:W2}) of the electroweak process
$\gamma \to \nu \bar \nu$ is suppressed by the factor $(G_F \;m^2_e)^2$
as compared with the probability~(\ref{eq:max}) of the pure electromagnetic
process $\gamma \to e^+ e^-$. However, the process with neutrinos could
play a role of an additional channel of stellar energy-loss.

The energy-loss rate per unit volume of the photon gas due to the splitting
$\gamma \to \nu \bar \nu$ is defined by
%
\begin{equation}
Q =  \int \frac{d^3 k}{(2 \pi)^3} \; \frac{1}{ e^{\omega/T} - 1} \;
\omega \; W_{\perp}.
\label{eq:Q1}
\end{equation}
%
Substitution of the probability~(\ref{eq:W1}) into Eq.~(\ref{eq:Q1}) gives
%
\begin{equation}
Q = \frac{\alpha \,G_F^2}{8 \pi^4}\;m_e^5 \; (e B)^2 \;
{\cal F}(T)
\simeq 0.96 \cdot 10^{18} \; \frac{erg}{cm^3\;s} \;
\left (\frac{B}{B_e} \right )^2 \;{\cal F}(T).
\label{eq:Qspl}
\end{equation}
%
The temperature function ${\cal F}(T)$ in the strong
field limit, $e B \gg T^2 \gg m_e^2$, 
takes the form 
%
\begin{equation}
{\cal F}(T)  \simeq  \frac{T}{2 m_e}\;
\left ( \ln \frac{T}{m_e} \ln\frac{4 T}{\Gamma_\gamma} - 0.187
 \right ).
\label{eq:F2}
\end{equation}

\noindent Here the main contribution arises from the vicinity of the resonance
$q^2_\parallel  \sim 4 m^2_e$. The width $\Gamma_\gamma$ should be taken from
Eq.~(\ref{eq:max}) at $\omega = 2 m_e$.

In conclusion, we estimate the contribution of the photon splitting
$\gamma \to \nu \bar \nu$ into the neutrino luminosity in a supernova
explosion, from a region of order of a hundred kilometers in size
outside the neutrinosphere, where a rather strong magnetic field of the
toroidal type could exist:
%
\begin{equation}
\frac{d E}{d t} \sim 10^{45} \, \frac{erg}{s}
\left (\frac{B}{10^{15} G} \right )^2
 \left (\frac{T}{2 MeV} \right )^5
 \left (\frac{R}{100 km} \right )^3.
\label{eq:lum}
\end{equation}
%
It is obviously much smaller than the total neutrino luminosity
from the neutrinosphere $ \sim 10^{52} erg/s$. It is interesting,
however, that the process $\gamma \to \nu \bar \nu$ could give an
appreciable contribution, equal for all neutrino species, to the low-energy
part of the neutrino spectrum.

%\section{Acknowledgements}
\vspace{5mm}

This work was supported in part by the INTAS Grant N~96-0659  
and by the Russian Foundation for Basic Research Grant N~98-02-16694.
The work of A.K. and N.M. was supported in part by the
International Soros Science Education Program under the Grants 
N~d98-179 and N~d98-181.

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