Magnetic Moment of Neutrino in Statistical Background
Samina S.Masood
\ 85 Lomb Memorial Drive, Rochester, NY 14618
{Permanent Address:170 Danforth Crescent, Rochester, NY14618}
We calculate the magnetic moment of Dirac type of neutrinos in hot and dense background for different ranges of temperature and chemical potential. The properties of neutrinos are studied in the strong magnetic field where the chemical potential of particles is high enough to have more particles than the antiparticles. We show that in this situation, Weyl neutrino seems to explain the neutrino coupling with the magnetic field due to its effective mass which can couple with the magnetic field directly. We also investigate the electromagnetic properties of Weyl neutrino due to its effective mass developed in the strong magnetic field.
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\begin{document}

\author{\textit{Samina S.Masood} \\
%EndAName
\textit{Physics Department, Rochester Institute of Technology,}\\
\textit{\ 85 Lomb Memorial Drive, Rochester, NY 14618}\\
{\small Permanent Address:170 Danforth Crescent, Rochester, NY14618}}
\title{Magnetic Moment of Neutrino in Statistical Background}
\maketitle

\begin{abstract}
We calculate the magnetic moment of Dirac type of neutrinos in hot and dense
background for different ranges of temperature and chemical potential. The
properties of neutrinos are studied in the strong magnetic field where the
chemical potential of particles is high enough to have more particles than
the antiparticles. We show that in this situation, Weyl neutrino seems to
explain the neutrino coupling with the magnetic field due to its effective
mass which can couple with the magnetic field directly. We also investigate
the electromagnetic properties of Weyl neutrino due to its effective mass
developed in the strong magnetic field. 
\end{abstract}

Superkamiokande in June 1998 claimed the experimental evidence of nonzero
mass of neutrino which encouraged the phenomenologists to study the
properties of massive neutrinos even more carefully. The experimental bounds
on the mass of neutrino are measured to be as [1,2]

\begin{eqnarray*}
m_{\nu _{e}} &\sim &1.4\times 10^{-5}eV\hspace{1in}(1a) \\
m_{\nu _{\mu }} &\sim &2.8\times 10^{-3}eV\hspace{1in}(1b) \\
m_{\nu _{\tau }} &\sim &4.8\times 10^{-2}eV\hspace{1in}(1c)
\end{eqnarray*}

These tiny masses of neutrinos provide enough motivation to look at the
extensions of the standard electroweak model even more seriously, though
they are not expected to be high enough to change the physical results so
much. The massless neutrino of the standard electroweak model (SM), usually
called Weyl neutrino cannot acquire any mass in vacuum and have only one
degree of freedom which is conventionally the left handed neutrino. However,
if the neutrino mass is not exactly equal to zero, only then we can think
about the existence of right-handed neutrino. Now, to accommodate this right
handed neutrino in the SM, we have to look for different extensions of the
SM. The minimal extension of the standard model (MSM) would be the standard
model with an extra right-handed inert neutrino as a singlet. In MSM, the
individual lepton number conservation still holds, and practically the
framework of the SM is used for all sort of calculation. All the Feynman
diagrams of the SM contribute in the same way. On the other hand we can
always look for different extensions of the standard model which allow
neutrino as a doublet also. In that case, we have to extend the Higgs sector
to accommodate the neutrino doublet in the theory at the cost of the
individual lepton number. Total lepton number is always conserved. This new
charged Higgs can be added as a singlet [3] or as a doublet [4]. We would
refer the last one as the Higgs doublet model (HDM) throughout in this
paper. There are several other extensions of the standard model which we
don't include in our discussion because we mainly want to see that how the
quantum statistical background can effect the massive neutrinos, in general.
We will briefly mention about the supersymmetric models of the massive
neutrinos in this regard, for completeness though the comparison of
different model is not our prime goal here. We even know that the neutrino
masses are small enough to be ignored in most of the cases. However, we have
to see that what happens to the massive neutrinos in the astrophysical and
cosmological environments where they can show different kind of behavior
with this tiny mass. These special environments are usually hot and dense.
We would also study the behavior of Weyl neutrino in the constant magnetic
field. Our basic scheme of work is the real-time formalism [5] where we deal
with the real particles only and hence incorporate the 1-1 component of the
propagators. This is a valid approximation in the heat bath of real particle
gas at moderate energies.

The boson propagator in the real-time formalism can be written as [5,6]

\begin{center}
\[
D_{B}^{\beta }=\frac{1}{k^{2}+i\epsilon }-2\pi i\delta (k^{2})n_{B}(k)%
\hspace{1in}(1a) 
\]
\end{center}

and the fermion propagator as, 
\[
S_{F}^{\beta }=(\not{p}-m_{\ell })\left[ \frac{1}{p^{2}-m_{\ell
}^{2}+i\varepsilon }+2\pi i\delta (p^{2}-m_{\ell }^{2})n_{F}(p)\right] 
\hspace{1in}(1b) 
\]

with the Bose-Einstein distribution,

\[
n_{B}(k)=\frac{1}{e^{\beta k_{0}}-1}\hspace{1in}(2a) 
\]
for massless vector bosons. This automatically leads to the vanishing mass
density and hence the chemical potential [7]. The Fermi-Dirac distribution
is given by [8-10]

\[
n_{F}(p)=\frac{\theta (p_{0})}{e^{\beta (p_{0}-\mu )}+1}+\frac{\theta
(-p_{0})}{e^{\beta (p_{0}+\mu )}+1}\hspace{1in}(2b)
\]
such that the chemical potential of fermions is always equal and opposite to
the corresponding antifermions. in CP symmetric background. We will later
discuss the situation where CP symmetry breaks but Eq.(2b) will still hold.
Since all the particle species are in thermal equilibrium, it is convenient
to expand the distribution functions in powers of m$\beta $ , where m is the
mass of the corresponding particles and $\beta =$1/T. $\mu $ ,T and m are
expressed in the same units. We obtain [9,10]

\[
n_{F}\stackrel{\mu \prec T}{\rightarrow }\stackrel{\infty
}{\stackunder{n=1}{%
\sum }}(-1)^{n}e^{n\beta (\mu -p_{0})}\hspace{1in}(3a) 
\]

and in the limit of $\mu \succ T$ 
\[
n_{F}\stackrel{\mu >T}{\rightarrow }\theta (\mu -p_{0})\stackrel{\infty }{+%
\stackunder{n=1}{\sum }}(-1)^{n}e^{-n\beta (\mu -p_{0})}\hspace{1in}(3b) 
\]

for particles, and for antiparticle the distribution functions in the same
approximations give

\[
n_{F}\stackrel{\mu \prec T}{\rightarrow }\stackrel{\infty
}{\stackunder{n=1}{%
\sum }}(-1)^{n}e^{-n\beta (\mu +p_{0})}\hspace{1in}(3c) 
\]

and

\[
n_{F}\stackrel{\mu \succ T}{\rightarrow }\stackrel{\infty
}{\stackunder{n=0}{%
\sum }}(-1)^{n}e^{-n\beta (\mu +p_{0})}\hspace{1in}(3d) 
\]

p$_{0}$ correspond to the energy of the real particles, given by 
\[
p_{0}^{2}=p^{2}+m^{2}\hspace{1in}(4a) 
\]

where p is the magnitude of the three momentum of the particles.

. In the presence of the constant background magnetic field, it reads out to
be[11]

\[
p_{0}^{2}=p^{2}+m^{2}+eB(2n+1)\hspace{1in}(4b) 
\]

where n corresponds to the Landau Levels. We ignore the polarization effects
for the moment. The electromagnetic properties of neutrinos can be derived
from the radiative decay of neutrino $\nu \longrightarrow \nu \gamma $ or
the plasmon decay given as $\gamma \longrightarrow \gamma \overline{\nu }.$
These processes can only take place through the higher order processes. The
most general form of the decay rate of Dirac neutrino can be written in
terms of its form factors [12-13] as,

\[
\Gamma _{\mu }=\left[ F_{1}\overline{g}_{\mu \nu }\gamma ^{\nu }+F_{2}u_{\mu
}+iF_{3}(\gamma _{\mu }u_{\nu }-\gamma _{\nu }u_{\mu })q^{\nu
}+iF_{4}\epsilon _{\mu \nu \alpha \beta }\gamma ^{\nu }q^{\alpha }u^{\beta
}\right] L\hspace{0.5in}(5) 
\]

in usual notation with 
\begin{eqnarray*}
F_{1} &=&T_{T}+\frac{\omega }{Q^{2}}(T_{L}-T_{T})\hspace{1in}(6a) \\
F_{2} &=&\frac{1}{V^{2}}(T_{L}-T_{T})\hspace{1in}(6b) \\
iF_{3} &=&-\frac{\omega }{Q^{2}}(T_{L}-T_{T})\hspace{1in}(6c) \\
F_{4} &=&\frac{T_{P}}{Q}\hspace{1in}(6d)
\end{eqnarray*}

where L, T and P correspond to the Longitudinal, Transverse and the
Polarization components, respectively. F$_{1}$ here is in the form of
standard charge radius, F$_{2}$ yields an additional contribution to the
charge radius. F$_{3}$ and F$_{4}$ correspond to the electric and magnetic
form factors of neutrinos, respectively. It can also be seen that [14]

\[
F_{3}(q^{2}=0)=D_{E}=-\frac{i\omega }{q^{2}}(T_{L}-T_{T})\hspace{1in}(7a) 
\]

and

\[
F_{4}(q^{2}=0)=D_{M}=-\frac{T_{P}}{2q}\hspace{1in}(7b) 
\]

where q on right-hand side of the equation corresponds to the magnitude of
three-momentum vector \textbf{q, } and D$_{E}$ and D$_{M\text{ }}$are the
electric and magnetic dipole moment of Dirac neutrino, considering

\[
T_{\mu \nu }=T_{T}R_{\mu \nu }+T_{L}Q_{\mu \nu }+T_{P}P_{\mu \nu
}\hspace{1in%
}(8) 
\]

such that

\begin{eqnarray*}
R_{\mu \nu } &\equiv &\overline{g}_{\mu \nu }-Q_{\mu \nu }\hspace{1in}(9a)
\\
Q_{\mu \nu } &\equiv &\frac{\overline{u}_{\mu }\overline{u}_{\nu }}{%
\overline{u}}\hspace{1in}(9b) \\
P_{\mu \nu } &\equiv &\frac{1}{\left| q\right| }\epsilon _{\mu \nu \alpha
\beta }q^{\alpha }u^{\beta }\hspace{1in}(9c)
\end{eqnarray*}

with

\begin{eqnarray*}
\overline{g}_{\mu \nu } &=&g_{\mu \nu }-\frac{q_{\mu }q_{\nu }}{q^{2}}%
\hspace{1in}(10a) \\
\overline{u}_{\mu } &=&g_{\mu \nu }u^{\nu }\hspace{1in}(10b) \\
u_{\mu } &=&(1,0,0,0)\hspace{1in}(10c)
\end{eqnarray*}

for $\omega =q.u$ and q=($\omega ^{2}-\left| q\right| ^{2})^{1/2}$ and the
form factors can be written as

\begin{eqnarray*}
T_{T} &=&\frac{eg^{2}}{2M^{2}}\left( \alpha -\frac{\beta }{u^{2}}\right) 
\hspace{1in}(11a) \\
T_{L} &=&\frac{eg^{2}}{M^{2}}\frac{\beta }{u^{2}}\hspace{1in}(11b) \\
T_{P} &=&-\frac{eg^{2}}{M^{2}}\left| q\right| \kappa \hspace{1in}(11c)
\end{eqnarray*}

The integrals $\alpha ,\beta $ and $\kappa $ are given by

\begin{eqnarray*}
\alpha &=&\int \frac{d^{3}p}{(2\pi )^{3}2E}(n_{F}^{+}+n_{F}^{-})\left[
\frac{%
2m^{2}-2p.q}{q^{2}+2p.q}+(q\leftrightarrow -q)\right] \hspace{0.5in}(12a) \\
\beta &=&\int \frac{d^{3}p}{(2\pi )^{3}2E}(n_{F}^{+}+n_{F}^{-})\left[
\frac{%
2(p.u)^{2}-2(p.u)(q.u)-p.q}{q^{2}-2p.q}+(q\leftrightarrow -q)\right]
\hspace{%
0.25in}(12b) \\
\kappa &=&-\int \frac{d^{3}p}{(2\pi )^{3}2E}(n_{F}^{+}-n_{F}^{-})\left[ 
\frac{1}{q^{2}+2p.q}+(q\leftrightarrow -q)\right] \hspace{0.5in}(12c)
\end{eqnarray*}

We have evaluated these integrals at T $\geq \mu $ in Ref.[9], giving

\begin{eqnarray*}
\alpha  &\simeq &\frac{1}{\pi ^{2}}\left[ \frac{\widetilde{c}(m_{_{\ell
}}\beta ,\mu )}{\beta ^{2}}+\frac{m}{\beta }\widetilde{a}(m_{_{\ell }}\beta
,\mu )-\frac{m^{2}}{2}\widetilde{b}(m_{_{\ell }}\beta ,\mu )-\frac{%
m^{4}\beta ^{2}}{8}\widetilde{h}(m_{_{\ell }}\beta ,\mu )\right]
\hspace{1in}%
(13a) \\
\beta  &\simeq &\frac{1}{\pi ^{2}}\left[ (1+\frac{3}{8}\ln \frac{1-v}{1+v})%
\frac{\widetilde{c}(m_{_{\ell }}\beta ,\mu )}{\beta ^{2}}+\frac{m}{\beta }%
\widetilde{a}(m_{_{\ell }}\beta ,\mu )-\frac{m^{2}}{2}\widetilde{b}%
(m_{_{\ell }}\beta ,\mu )-\frac{m^{4}\beta ^{2}}{8}\widetilde{h}(m_{_{\ell
}}\beta ,\mu )\right] \hspace{0.3in}(13a) \\
\kappa  &\simeq &\frac{1}{\pi ^{2}}\left[ \overline{b}(m_{\ell }\beta ,\mu
)+m_{\ell }^{2}\overline{h}(m_{\ell }\beta ,\mu )\right] \ \hspace{0.5in}%
\hspace{1in}(13c)
\end{eqnarray*}

We consider $\kappa =0$ because we were just considering the CP symmetric
background there. The a,b,c functions (we call them a$_{i}$-functions for
simplicity) for particles and antiparticles are evaluated in the same way.
At large T, the particle and antiparticles have same chemical potential with
different sign and so give the equal contribution. We stick to the notation
of Ref. [9] and define

\begin{eqnarray*}
\widetilde{a_{i}}(m_{\ell }\beta ,\mu ) &=&\frac{1}{2}[a_{i}(m_{\ell }\beta
,\mu )+a_{i}(m_{\ell }\beta ,-\mu )] \\
\overline{a_{i}}(m_{\ell }\beta ,\mu ) &=&\frac{1}{2}[a_{i}(m_{\ell }\beta
,\mu )-a_{i}(m_{\ell }\beta ,-\mu )]
\end{eqnarray*}

giving

such that , in the limit $T\gg \mu $

\[
\overline{a_{i}}(m_{\ell }\beta ,\mu )\longrightarrow 0 
\]

or in other words the function $\kappa \longrightarrow 0$ indicating the
vanishing contributions to the magnetic moment of neutrino from the tadpole
diagram in this limit.

The calculational detail of these a$_{i}$ functions can be found in Ref.[9].
These functions always appear in this scheme of calculations. In the limit
$%
T\geq \mu ,$ these functions read out to be as 
\[
\widetilde{a}(m_{_{\ell }}\beta ,\mu )\simeq \frac{1}{2}\ell n\left[ \left(
1+e^{-\beta (m_{_{\ell }}-\mu )}\right) \left( 1+e^{-\beta (m_{_{\ell }}+\mu
)}\right) \right] \hspace{1in}\left( 14a\right) 
\]

\[
\widetilde{b}(m_{_{\ell }}\beta ,\mu )\simeq \stackrel{\infty
}{\stackunder{%
n=1}{\sum }}(-1)^{n}\cosh (n\beta \mu )\limfunc{Ei}(-nm_{\ell }\beta )%
\hspace{1in}\left( 14b\right) 
\]

\begin{eqnarray*}
\widetilde{c}(m_{_{\ell }}\beta ,\mu ) &\simeq &\stackrel{\infty }{%
\stackunder{n=1}{\sum }}(-1)^{n}\cosh (n\beta \mu )\frac{e^{-n\beta
m_{_{\ell }}}}{n^{2}}\hspace{1in}\left( 14c\right) \\
\widetilde{h}(m_{_{\ell }}\beta ,\mu ) &\simeq &\stackrel{\infty }{%
\stackunder{n=1}{\sum }}(-1)^{1+n}\left( \frac{\beta
^{2}n^{2}}{2}\limfunc{Ei%
}(-nm_{\ell }\beta )+n\beta \frac{e^{-n\beta m_{_{\ell }}}}{m_{\ell }}%
\right) \cosh (n\beta \mu )\hspace{1in}(14d)
\end{eqnarray*}

whereas,

\begin{eqnarray*}
\overline{b}(m_{\ell }\beta ,\mu ) &\longrightarrow
&0\hspace{1in}\overline{%
(14a)} \\
\overline{h}(m_{\ell }\beta ,\mu ) &\longrightarrow
&0\hspace{1in}\overline{%
(14b)}
\end{eqnarray*}

In the high density regime, at $\mu \geq T$ however, these functions are
evaluated as

\begin{eqnarray*}
\widetilde{a}(m_{\ell }\beta ,\mu ) &\simeq &\mu -m_{\ell }-\frac{1}{\beta
}%
\sum_{r=1}^{\infty }\sum_{n=1}^{\infty }\frac{(-1)^{n}}{(n\beta \mu
)^{1-r}}%
e^{-n\beta m_{\ell }}\cosh (n\beta \mu )\hspace{0.5in}(15a) \\
\widetilde{b}(m_{\ell }\beta ,\mu ) &\simeq &\ln \frac{\mu }{m_{\ell }}%
\hspace{0.5in}(15b) \\
\widetilde{c}(m_{\ell }\beta ,\mu ) &\simeq &\frac{\mu ^{2}-m_{\ell
}^{2}}{2}%
-\mu ^{2}\sum_{r=1}^{\infty }(-1)^{r}\sum_{n=1}^{\infty }\frac{(-1)^{n}}{%
(n\beta \mu )^{2-r}}e^{-n\beta m_{\ell }}\cosh (n\beta \mu )\hspace{0.5in}%
(15c) \\
\widetilde{h}(m_{\ell }\beta ,\mu ) &\simeq &\frac{\mu ^{-4}-m_{\ell
}^{-4}}{%
2}-\sum_{r=1}^{\infty }\frac{(-1)^{r}}{\mu ^{r}}\sum_{n=1}^{\infty }\frac{%
(-1)^{n}}{(n\beta )^{4+r}}e^{-n\beta m_{\ell }}\cosh (n\beta \mu )\hspace{%
0.5in}(15d)
\end{eqnarray*}

and

\begin{eqnarray*}
\overline{b}(m_{\ell }\beta ,\mu ) &\simeq &2\ln \frac{\mu }{m_{\ell }}%
\hspace{0.5in}\overline{(15a)} \\
\overline{h}(m_{\ell }\beta ,\mu ) &\simeq &\frac{\mu ^{-4}-m_{\ell
}^{-4}}{2%
}+\stackrel{\infty }{\stackunder{n=1}{\sum }}(-1)^{1+n}\left( \frac{\beta
^{2}n^{2}}{2}\limfunc{Ei}(-nm_{\ell }\beta )+n\beta \frac{e^{-n\beta
m_{_{\ell }}}}{m_{\ell }}\right) \sinh (n\beta \mu
)\hspace{0.5in}\overline{%
(15b)}
\end{eqnarray*}

These integrals of Eqs.(12) help to understand the nature of the background
contributions. At high temperatures, the background contributions due to
fermion and antifermion distributions are approximately equal and hence they
cancel each other such that

\[
\kappa \longrightarrow 0\text{ as }\mu \longrightarrow 0 
\]
At high densities, however, particle and antiparticle contribution need not
to be same all the time which implies that

\[
\kappa \neq 0\hspace{1in}(16) 
\]
In the CP symmetric dense background, the chemical potential of the fermions
is equal and opposite to the chemical potential of the antifermions. This is
the situation where again the contribution due to fermions would not be the
same as the antifermion contribution because of the different sign of the
chemical potential so the relation (16) holds true.

giving 
\[
\kappa \simeq -\frac{1}{8\pi ^{2}}\ln \frac{\mu }{m_{\ell }}\hspace{0.5in}%
(17) 
\]

The magnetic moment of neutrino can simply be obtained form the bubble
diagram of the plasmon decay in the MSM. This contribution comes out to be

\begin{eqnarray*}
\Lambda _{0}(p_{1},p_{2}) &=&-\frac{g^{2}}{m_{W}^{2}}\int \frac{d^{4}k}{%
(2\pi )^{4}}\gamma _{\alpha }L(p_{2}-k+m_{\ell })\gamma _{\mu
}(p_{1}-k+m_{\ell })\gamma ^{\alpha }L \\
&&x\frac{2\pi i\delta \left[ (p_{2}-k)^{2}-m_{\ell }^{2}\right] }{%
(p_{2}-k)^{2}-m_{\ell }^{2}}n_{F}(p_{2}-k)+ \\
&&+\left( p_{1}\leftrightarrows p_{2}\right) \hspace{1in}(18)
\end{eqnarray*}

in standard notation of particle physics. We approximate the W propagator as

$\frac{1}{m_{W}^{2}}$ with k$^{2}\ll m_{W}^{2}$, where $m_{W}$ is the mass
of W boson. The magnetic moment of neutrino contribution from this diagram
comes out to be equal to [14]

\[
\alpha _{\nu _{\ell }}^{\beta }\approx \frac{G_{F}m_{\ell }}{4\pi ^{2}}%
m_{\upsilon _{\ell }}\left[ \widetilde{b}(m_{_{\ell }}\beta ,\mu
)+\frac{4}{%
M^{2}}\left( m_{_{\ell }}\widetilde{a}(m_{_{\ell }}\beta ,\mu
)-\widetilde{c}%
(m_{_{\ell }}\beta ,\mu )\right) \mu _{B}\right] \hspace{0.36in}(19) 
\]

where $\mu _{B}$ is the Bohr Magneton. Eq.(19) gives the total background
contribution at T$\succ \mu $. In the limit $\mu \succ T$ the magnetic
moment contribution is coming from the tadpole diagram also. Comparing
Eq.(3b) with that of (3d), one can easily see the difference of behavior
between the particle and antiparticle distribution functions in this regime.
So, we have to evaluate the integral $\kappa $ to find the contribution from
the tadpole diagram also. It can easily be seen. It happens because the CP
symmetry breaks at low temperatures, however, we are not considering the
cold system so the statistical distribution function is still considered to
incorporate the particle background effects. We first evaluate Eq.(18) to
find the contribution of bubble diagram and then $\kappa $ is evaluated to
find the contribution of the tadpole diagram. Both of these contributions
are added up together to give the statistical background contribution to the
magnetic moment of Dirac neutrino of MESM in the high density regime as.

\begin{eqnarray*}
D_{M}^{stat} &=&\frac{eg^{2}}{2M^{2}}\kappa +\alpha _{\nu _{\ell }}^{stat}%
\hspace{1in}(20a) \\
D_{E}^{stat} &=&-\frac{i\omega }{q^{2}}\frac{eg}{2M^{2}}(3\beta -\alpha )%
\hspace{1in}(20b)
\end{eqnarray*}

whereas, 
\[
a_{\nu }^{stat}\simeq \frac{G_{F}m_{\ell }}{8\pi ^{2}}m_{\nu _{\ell }}\ln 
\frac{\mu }{m_{\ell }}\hspace{0.5in}\left( 21\right) 
\]

and

giving the magnetic moment in the units of the Bohr magneton as, 
\[
D_{M}^{stat}\simeq \left( \frac{G_{F}m_{\ell }}{8\pi ^{2}}-\frac{m_{\ell
}g_{Z}^{2}}{16M^{2}}\right) m_{\nu _{\ell }}\ln \frac{\mu }{m_{\ell }}%
\hspace{0.5in}(22) 
\]

M is this mass of Z and $g_{Z}$ is the coupling constant of $%
Z\longrightarrow \ell \overline{\ell }$ vertex. The relative sign difference
between the bubble diagram indicates the decrease in the background
contribution. The two terms in bracket cannot cancel each other because they
still have the same type of contribution and the difference appears as a
constant factor only. This factor is comparable to each other so provides a
strong decreasing factor. We can safely say that this contribution is
negligible for large chemical potentials. As a quick estimate of the
magnetic moment value shows that the contribution of Eq.(22) at $\mu
_{e}\sim 10m_{e}$ is around 10$^{-19}$ Bohr magneton. which is even smaller
than the vacuum value of the magnetic moment of neutrino which is around
10$%
^{-18}$ Bohr megneton[13]. In this situation, it seems as if the nonzero
neutrino mass does not help to get the desired larger value of the magnetic
moment of neutrino.

In the extended standard models like HDM the individual lepton no. does not
conserve so the FTD corrections to the magnetic moment of neutrino could be
different because we have to add new individual lepton number violation
diagrams. The additional background FTD corrections to the magnetic moment
of neutrino comes out to be [15]

\[
a_{\nu _{\ell }}^{stat}\simeq \stackrel{\infty }{\stackunder{\ell ^{^{\prime
}}=e,\mu ,\tau }{\sum }}\frac{f_{\ell \ell ^{^{\prime }}}h_{\ell \ell
^{^{\prime }}}}{16\pi ^{2}m_{h}^{2}}m_{_{\ell }}^{2}\widetilde{b}(m_{_{\ell
}}\beta ,\mu )\mu _{B}\hspace{1in}(23) 
\]

with

\begin{eqnarray*}
f_{e\mu }h_{e\mu }/m_{h}^{2} &\precsim &2.8\times
10^{-6}GeV^{-2}\hspace{1in}%
(24a) \\
f_{e\tau }h_{e\tau }/m_{h}^{2} &\precsim &2.8\times 10^{-6}GeV^{-2}\hspace{%
1in}(24b)
\end{eqnarray*}

where m$_{h}$is the mass of the charged higgs in the new higgs doublet in
this model. Eq.(23) leads to 
\[
a_{\nu _{\ell }}^{stat}\simeq \stackrel{\infty }{\stackunder{\ell ^{^{\prime
}}=e,\mu ,\tau }{\sum }}\frac{f_{\ell \ell ^{^{\prime }}}h_{\ell \ell
^{^{\prime }}}}{16\pi ^{2}m_{h}^{2}}m_{_{\ell }}^{2}\ln \frac{\mu }{m_{\ell
}%
}\mu _{B}\hspace{1in}(25) 
\]

at $\mu \succ T.$

It is well-known that the properties of Majorana neutrinos are different
from that of Dirac neutrino. So, it cannot have the dipole moment as we
describe for Dirac neutrino. However, it can have a transition magnetic
dipole moment. Similarly the neutrinos of the SUSY models also can only have
the transition magnetic moment. Authors of Ref.[12-14] have noticed that the
Majorana neutrino can have almost comparable value of the magnetic moment.
Actually it comes out to be different from each other just by an integral
factor. Therefore, we should expect the same order of magnetic moment
contributions in both cases. Riotto[16] has claimed that a large class of
models where charged scalar boson couple to leptons can provide the magnetic
moment of electron type neutrino as large as 10$^{-12}$ Bohr Magnetons,
which can play a relevant role in different astrophysical phenomenon.
However, these extended standard models are constructed to get large
magnetic moment values. This is the same thing which works for the HDM also.
So, it is not really so much exciting, if we can get the right amount of the
magnetic moment value in these models.

In this situation we assume that the neutrino mass does not help to
sufficiently increase the magnetic moment value at this point and we will
like to go back again in the standard model with Weyl neutrino and see what
happens there. We have noticed that the Weyl neutrino in the strong magnetic
field acquires a Dirac type of effective mass, which can couple with the
strong magnetic field in the background. In the constant magnetic field,
this effective mass comes out to be[17]

\[
m_{eff}=\frac{g^{2}e\left| B\right| \mu _{e}}{(2\pi )^{2}(m_{W}^{2}-e\left|
B\right| )}\hspace{1in}(26) 
\]

This effective mass has a pole at $m_{W}^{2}-e\left| B\right| $ which gives
a natural limit on the magnetic field of highly dense objects as

\[
\left| B\right| \prec \frac{m_{W}^{2}}{e}=10^{24}Gauss\hspace{1in}(27)
\]

So, B should always be sufficiently below the 10$^{24}$Gauss. This limit
seems to allow the presence of very high magnetic field, as it  is expected
in magnetars[18]. However, the effective mass of neutrino with the rise of
the magnetic field could be high enough to cause gravitational collapse
also. So, the stability of highly magnetic stars could still be
questionable.

It is interesting to note that this effective mass of neutrino is the Dirac
type of mass acquired by neutrino in the strong magnetic field and does not
depend on the extension of the standard model. However, it can simply be
treated as the Dirac type of mass of neutrino and its electromagnetic
properties can be investigated using the above treatment starting from
Eq.(5). The form factors of this type of neutrino and the magnetic moment
values are not constant. These quantities depend on the magnetic field and
the chemical potential. We will discuss here the magnetic moment of this
Weyl neutrino with the large effective mass in detail and see how we can
succeed to obtain the desired values of the magnetic moment of neutrino,
using this effective mass.

If we just go back to the calculation of the magnetic moment of neutrino in
vacuum, it is always proportional to the mass of neutrino itself. Now, if we
replace the mass of neutrino m$_{\nu _{\ell }}$ in the standard relation of
the magnetic moment[13] by the effective mass of neutrino from Eq.(26), we
get the magnetic field background contribution to be equal to

\[
a_{\nu _{\ell }}^{B}=\frac{3G_{F}^{2}}{4\sqrt{2}\pi ^{4}}\frac{e\left|
B\right| \mu _{e}}{(1-\frac{e\left| B\right| }{m_{W}^{2}})}\mu _{B}\hspace{%
1in}(27)
\]

where $a_{\nu _{\ell }}^{B}$ correspond to the magnetic moment of neutrino
in the strong magnetic field. It shows that the desired value of the
magnetic moment of neutrino can be obtained by imposing new limits on the
magnetic field of the medium and the chemical potential of electron. A quick
estimate of Eq.(27) gives the magnetic moment due to this effective mass of
the neutrino in the magnetic field 

\[
eB\simeq 0.9m_{W}^{2}
\]

and the chemical potential of electron as

\[
\mu _{e}\simeq 100m_{e}
\]

\[
a_{\nu _{\ell }}^{B}\simeq 2\times 10^{-15}\mu _{B}\hspace{1in}(28)
\]

which is much larger than the vacuum value. However, for the magnetic field
values, expected to be in neutron stars or supernova(10$^{12}-10^{14}$%
Gauss), the magnetic moment of neutrino is still around 10$^{-18}\mu _{B}$,
which is no improvement, as compared to the vacuum values of 1ev
neutrino[13]. 

Eq.(28) shows that we can get [19] the desired value of the magnetic moment
of neutrino for larger value of the magnetic field or the chemical potential
or both of them. In the large magnetic field with high chemical potential,
we have to consider relatively cold systems where temperature of a system
has to be sufficiently below its chemical potential. So the desired values
of the magnetic moment of neutrino is obtained directly from the very large
magnetic field background in superdense stars. We can clearly see that the
magnetic collapse is expected at $eB\simeq m_{W}^{2}$, which provides a
natural limit on the magnetic field value, i.e; $eB\prec m_{W}^{2}$, which
corresponds to B=10$^{24}$Gauss. At this stage, we can reach a very
interesting result. The rate of change of this effective mass with the
magnetic field comes out to be

\[
\frac{dm_{eff}}{dB}=\frac{g^{2}e\mu _{e}m_{W}^{2}}{(2\pi
)^{2}(m_{W}^{2}-e\left| B\right| )^{2}}\hspace{1in}(29)
\]

The rate of change of this mass with the magnetic field decreases rapidly
with the increasing magnetic field. Eq.(29) correspond to the rate of change
of the effective mass of neutrino with the magnetic field due to the
coupling of the effective mass of neutrino with the magnetic field. In this
sense it corresponds to something similar to the form factor of neutrinos at
low energy. We can call it effective magnetic moment of neutrino and define
it as 

\[
(a_{\nu _{\ell }}^{B})_{eff}\equiv \frac{dm_{eff}}{dB}\hspace{1in}(30)
\]

This quantity $\frac{dm_{eff}}{dB}$ has been studied in detail in Ref.(17).
At B=0, $(a_{\nu _{\ell }}^{B})_{eff}$ $\longrightarrow 10^{-12}$, and in
fact we can obtain the desired value of $(a_{\nu _{\ell }}^{B})$ from single
relation in different limits of B and $\mu _{e}$

It is also worth-mentioning that the magnetic moment of neutrino plays an
important role to estimate the lepton scattering crossections. in the early
universe and in the hot and dense stellar media. The results obtained in
Ref.[20] in the standard model can change a lot with incorporating the
magnetic moment of neutrino in the evaluation of lepton scatterings [13,
21]. It also helps to understand the expansion rate of the universe and the
solar neutrino problem. 

\section{References and Footnotes}

\begin{enumerate}
\item  T.Kajita, `Atmospheric Neutrino Results from Superkamiokande-Evidence
for $\nu _{m}u$ oscillations', Neutrino'98, Tokayama, Japan, (June 1998)

\item  M.Koshiba, `Kamiokande and Superkamiokande' , Fifth School on
Non-Accelerator Particle Astrophysics, Abdus Salam ICTP, Trieste, Italy
(July 1998)

\item  K.S.Babu and V.S.Mathur, Phys.Lett. B196, 218 (1987);

\item  M.Fukujita and T.Yanagida, Phys.Rev.Lett.58,1807(1987)

\item  P.Landsman and Ch. G. Weert, Phys.Rep.145,141(1987) and references
therein, and L.Dolan and R.Jackiw, Phys.Rev. D9,3320(1974)

\item  E.J.Levinson and D.H.Boal, Phys.Rev. D31,3280(1985)

\item  See for example,L.D.Landau and E.M.Lifshitz,`Statistical
Physics'(Pergamon Press Ltd.London,1968)

\item  See for Example, K.Ahmed and Samina Saleem (Masood) Phys.Rev
D35,1861(1987);ibid, 4020(1987); ibid,Ann.Phys.164,460(1991)

\item  .Samina S.Masood,Phys.Rev.D48, 3250(1993) and references therein

\item  Samina S.Masood,Phys.Rev.D44,3943(1991),;ibid,D47,648(1993)

\item  H.Nunokawa,et.al;Nucl.Phys.B501, 17(1997) and references therein

\item  J.C Olivo, J.F.Nieves, and P.B.Pal, D40,3679(1989) and references
therein.

\item  R.N.Mohapatra and P.B.Pal, Massive Neutrinos in Physics and
Astrophysics, (1991) (World Scientific Publication)

\item  J.F.Nieves, and P.B.Pal, D40,1693(1989)

\item  Samina S.Masood, Astroparticle Physics, 4,189(1995)

\item  A.Riotto, Phys.Rev. D50, 2139(1994)

\item  See for details, Samina S.Masood et.al.`: Effective Magnetic Moment
of Neutrinos in Strong Magnetic Fields'hep-ph/9902269(1999)

\item  See for Example; D.L.Kaplan et.al; `HST Observations of SGR 0526-66:
New Constraints on Accretion and Magnetar Models'astro-ph/0103179 and
references therein

\item  Due to the absence of the research facilities, we have to postpone
the detailed quantitive analysis of this equation at this stage.

\item  J.A.Morgan, Phys.Lett. B102,247(1981) and Samina S. Masood,
`Scattering of Leptons in Hot and dense Media', hep-ph/0108126, ibid;
`Statistical Corrections to the Scattering Amplitudes in the Extended
Standard Models' (Under Preparation)and references therein

\item  See for example;M.Parkash, et.al; `Neutrino Propagation in Dense
Astrophysical Systems.'astr-ph/0103095
\end{enumerate}

\end{document}

