Scattering of Leptons in Hot and Dense Media
Samina S. Masood
%EndAName Physics Department
85 Lomb Memorial Drive
Rochester NY14623
We study the propagation of leptons in hot and dense media and estimate the background corrections to the purely leptonic processes in the early universe and in the stellar cores. We show that the first order radiative corrections to purely leptonic processes in the real-time formalism can easily be evaluated without going into the resummation of thermal loops in these media. It is also possible to express the hot and dense corrections to the leptonic scattering cross-sections in terms of the corresponding corrections to the decay rates of the (heavy) intermediate particles in the real-time formalism upto the one loop level.
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%TCIDATA{Created=Samina S.Masood}
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\input tcilatex
\begin{document}

\title{Scattering of Leptons in Hot and Dense Media}
\author{Samina S. Masood \\
%EndAName
Physics Department, Rochester Institute of Technology, 85 Lomb Memorial
Drive, Rochester NY14623}
\maketitle

\begin{abstract}
We study the propagation of leptons in hot and dense media and estimate the
background corrections to the purely leptonic processes in the early
universe and in the stellar cores. We show that the first order radiative
corrections to purely leptonic processes in the real-time formalism can
easily be evaluated without going into the resummation of thermal loops in
these media. It is also possible to express the hot and dense corrections to
the leptonic scattering cross-sections in terms of the corresponding
corrections to the decay rates of the (heavy) intermediate particles in the
real-time formalism upto the one loop level.
\end{abstract}

\section{\textrm{Introduction}}

It is now well known that the stellar cores can have very high temperatures
and densities[1]. It therefore looks more reasonable to incorporate the
finite temperature and density (FTD) effects in the study of stellar media.
The physics of superdense collapsing stars is expected to be better
understood incorporating the statistical corrections due to the hot and
dense background in the stellar cores. The superdense collapsing stars like
supernovae have core temperatures around 30-70 Mev and the chemical
potential of the electrons $\mu _{e}\sim 250-300$ Mev [2], which is
sufficiently below the masses of W and Z. So the important contributions
from the background can be obtained from the hot and dense leptons in the
background heat-bath at thermal equilibrium. The standard big bang model of
cosmology [3] also predicts the creation and annihilation of leptons in the
very early universe. These processes were taking place during the first few
seconds of the creation of the universe and leptons were the first matter
particles. The cross sections of these reactions are also related to the
expansion rate of the universe[3]. The temperatures of the universe at that
time were high enough to show non-negligible contribution of the hot and
dense lepton background. The contributions due to the hot electroweak
massive bosons is not included because the background particle energies are
not sufficiently high to have real W's and Z's in the background. Moreover,
the lepton background effects will always dominate over these heavy mass
particles.

We discuss here the pure leptonic decays and cross-sections in the Standard
Electroweak Model (SEWM). We will not discuss background radiative
corrections to pure QED processes (Bhabha and Moller Scatterings) here,
because they deserve a detailed independent study of the topic, which is not
in the scope of this paper.. We mainly evaluate the first order radiative
corrections to the neutral current as well as the charge current processes
in hot and dense lepton background. Moreover, the first order radiative
corrections to the inelastic and to the elastic scatterings of leptons in
the standard model are discussed through the neutral current or the charged
current processes taking into account the thermal equilibrium of leptons and
photons in the background . Strictly speaking, we restrict ourselves to the
SEWM so the additional corrections to these processes from the extended
standard model [4,5] are not incorporated in the same statistical
background. For this purpose we will have to consider the massive neutrino
physics in such a medium. We restrict ourselves to the temperatures where
the light particles can only be found in the background heat-bath and the
first order QED type radiative corrections are significant. We use covariant
real time formalism [6,7] in which the renormalization of QED at finite
temperature and density (FTD) up to one loop level [7,8] have been discussed
in detail, for all possible ranges of T and $\mu .$ It has also been checked
that the renormalization scheme of QED in vacuum works perfectly at FTD and
there is no need of resummation [9] at the one loop level in QED for real
particle systems. It is mainly because of the reason that at the
sufficiently low temperatures the hard thermal loops cannot appear as the
background temperature is not high enough to generate real hadrons to
incorporate the statistical corrections from hadronic background. Moreover,
the box diagrams with the overlapping phases do not contribute[10].
Therefore, the 1-1 component of the propagator is important considering the
physical particles in the background [7,8]. The renormalization constants of
QED have been evaluated upto the two loop levels in literature[11]. We don't
need to incorporate the overlapping phases, because the overlapping phase
contribution is automatically approximated out. These additional thermal
contributions can then be estimated easily by applying the renormalization
techniques of QED. This procedure is not similar to Cutkowski Rules[12]. We
are not using any cutting of diagram as a rule, it is the approximation
under which we keep the leading order contributions and ignore the rest. We
could do it only because the real time formalism give thermal/statistical
correction term as the separate additive terms in all kind of Feynman
diagrams. We use the renormalization scheme of QED and since the angle of
scattering is independent of statistical background parameters, we can
simply express the cross-sections in the form of the decay rates. Using the
values of these renormalization constants evaluated in the real time
formalism we can easily evaluate the first order radiative corrections to
the decay rates and the scattering cross-sections. The first order radiative
corrections to the decay rates of H$^{0}\longrightarrow \ell \overline{\ell }
$ [7,8] and Z$^{0}\longrightarrow \ell \overline{\ell }$ [13] have already
been calculated for temperatures above the chemical potentials. We
generalize these techniques to evaluate the background corrections to the
leptonic decay processes of real particles and give a general discussion of
the QED type corrections to the electroweak vertices at FTD. Section 2
comprises the calculational scheme, whereas, we give the analytical forms of
the radiative corrections in different limits of temperatures and chemical
potentials in section 3.

We have made use of all the existing results of the QED renormalization
constants and filled up the gaps wherever they were. We had made use of the
renormalization constants to estimate the decay rates of the relevant
particles and then related the lepton scatterings to these decay rates. All
the calculations are done in the standard electroweak model. Some of the
possible applications of these results in astrophysics and cosmology are
discussed in Section 4.

\section{Calculational Scheme}

\ \qquad The statistical background corrections to the physical processes
are calculated using the Feynman rules of vacuum theories. However, the
particle propagators are modified in hot and dense medium using the
distributions of the background particles. The boson propagator in the
real-time formalism can be written as

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

and the fermion propagator as,

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

with the Bose-Einstein distribution,

\[
n_{B}(k)=\frac{1}{e^{\beta k_{0}}-1}\hspace{1in}(3) 
\]
for massless vector bosons. This automatically leads to the vanishing mass
density and hence the chemical potential [14]. The Fermi-Dirac distribution
is given by [15]

\[
n_{F}(p)=[\frac{\theta (p_{0})}{e^{\beta (p_{0}-\mu )}+1}+\frac{\theta
(-p_{0})}{e^{\beta (p_{0}+\mu )}+1}]\hspace{1in}(4) 
\]
such that the chemical potential of fermions is always equal and opposite to
the corresponding antifermions. Since all the particle species are in
thermal equilibrium, it is convenient to expand the distribution functions
in powers of $\beta p_{0}$, where $\beta $ is the inverse temperature and p$%
_{0}$ is the energy of the corresponding particles. T, $\mu $ and m are all
expressed in the same units.We obtain[16]

\[
n_{F}\stackrel{\mu \prec T}{\rightarrow }\stackrel{\infty }{\stackunder{n=1}{%
\sum }}(-1)^{n}e^{n\beta (\mu -p_{0})}\hspace{1in}(5a) 
\]

and

\[
n_{F}\stackrel{\mu \succ T}{\rightarrow }\stackrel{\infty }{\stackunder{n=1}{%
\sum }}(-1)^{n}e^{-n\beta (\mu -p_{0})}+\theta (\mu -p_{0})\hspace{1in}(5b) 
\]

for particles, and

\[
n_{F}\stackrel{\mu \prec T}{\rightarrow }\stackrel{\infty }{\stackunder{n=1}{%
\sum }}(-1)^{n}e^{-n\beta (\mu +p_{0})}\hspace{1in}(5c) 
\]

and

\[
n_{F}\stackrel{\mu \succ T}{\rightarrow }\stackrel{\infty }{\stackunder{n=0}{%
\sum }}(-1)^{n}e^{-n\beta (\mu +p_{0})}\hspace{1in}(5d) 
\]
for antiparticles in the corresponding temperature and density regime. We do
all our calculations by the corresponding substitution of the distribution
functions from Eqs.(5). We have explicitly checked the validity of the
renormalization scheme of QED by reproducing the physically measurable
results of the first order background corrections to the decay rates by
adding all the one loop diagrams with those by replacing the vacuum
parameters with the corresponding renormalized parameters at FTD upto the
one loop level.

\section{Calculational Details}

We discuss the renormalization constants of QED in hot and dense medium. We
mention here the results of the first order radiative corrections to the
self mass, wavefunction and the charge of electron (QED\ coupling). We have
used the renormalization scheme of QED in the calculation of radiative
corrections to the gauge vertices of electroweak theory. We have explicitely
demonstrated the order by order cancellation of singularities upto the two
loop level in QED. We have calculated the QED type of hot and dense
background corrections to the $Z^{0}\rightarrow \ell \overline{\ell }$
electroweak vertex to explicitely check the renormalizability of the
corresponding gauge theories and to evaluate the radiative corrections. The
renormalization constants of QED have already been calculated upto the
two-loop level in our scheme[11]. We will be using this renormalization
scheme in our calculations to estimate the statistical background
corrections.We briefly mention the previously calculated renormalization
constants of QED and tried to put everything together in the convenient form
and have tried to fill the gaps if there were any. We refer to the existing
results for the details, wherever it is relevant. Then using all these
results we go to the calculations of the decay rates. Then we discuss, how
we can use the radiative corrections to the decay rates to evaluate the
first order radiative corrections to the crossections. We have to rewrite
the already existing results sometimes, to develop a uniform way of writing
everything for the comparison purposes.

\subsection{Renormalization of QED}

We calculate the QED type radiative corrections here so we use the
renormalization of QED and make use of the renormalization constants given
in literature [4-18] in the relevant ranges of T and $\mu .$ We only rewrite
them in the convenient form. The FTD radiative corrections to the electron
mass $\frac{\delta m}{m}$ correspond to the renormalization constant of
electron at T $\geq \mu $ and can be expressed as [7]

\[
\frac{\delta m}{m}\simeq \frac{\alpha \pi T^{2}}{3m^{2}}\left[ 1-\frac{6}{%
\pi ^{2}}\widetilde{c}(m\beta ,\mu )\right] +\frac{2\alpha }{\pi }\frac{T}{m}%
\widetilde{a}(m\beta ,\mu )-\frac{3\alpha }{\pi }\widetilde{b}(m\beta ,\mu )%
\hspace{1in}(6) 
\]

with

\[
\widetilde{a}(m_{_\ell }\beta ,\mu )\simeq \frac 12\ell n\left[ \left(
1+e^{-\beta (m_{_\ell }-\mu )}\right) \left( 1+e^{-\beta (m_{_\ell }+\mu
)}\right) \right] \hspace{1.0in}\left( 7a\right) 
\]

\[
\widetilde{b}(m_{_\ell }\beta ,\mu )\simeq \stackrel{\infty }{\stackunder{n=1%
}{\sum }}(-1)^n\cosh (n\beta \mu )\limfunc{Ei}(-nm_\ell \beta )\hspace{1.0in}%
\left( 7b\right) 
\]

\[
\widetilde{c}(m_{_\ell }\beta ,\mu )\simeq \stackrel{\infty }{\stackunder{n=1%
}{\sum }}(-1)^n\cosh (n\beta \mu )\frac{e^{-n\beta m_{_\ell }}}{n^2}\hspace{%
1.0in}\left( 7c\right) 
\]

and, in the limit T $\leq \mu $ , $\frac{\delta m}{m}$ appears [17] to be

\begin{eqnarray*}
\frac{\delta m}{m} &\simeq &\ \frac{\alpha \pi T^{2}}{3m^{2}}-\frac{2\alpha 
}{\pi m^{2}}\left( \frac{\mu ^{2}}{2}(1-\frac{m^{2}}{\mu ^{2}})-\frac{%
A(m\beta ,\mu )}{\beta }+\frac{C(m\beta ,\mu )}{\beta ^{2}}\right) -\frac{%
3\alpha }{\pi }[\ell n\frac{\mu }{m}+B(m\beta ,\mu )]- \\
&&-\frac{\alpha }{8\pi }\left[ \left( 1-\frac{m^{2}}{\mu ^{2}}\right) -\frac{%
m^{2}}{2\mu ^{2}}+n_{F}\left( \mu -m\right) +m\beta \frac{e^{-\beta (\mu -m)}%
}{\left[ 1+e^{-\beta (\mu -m)}\right] ^{2}}+m^{2}D(m\beta ,\mu )\right] 
\hspace{0.5in}(8)
\end{eqnarray*}

with

\[
A(m_{_\ell }\beta ,\mu )=\mu \ell n2-m\ell n[1+e^{-\beta (\mu -m)}]\hspace{%
1.0in}(9a) 
\]

\[
B(m_{_\ell }\beta ,\mu )=\stackunder{n=1}{\stackrel{\infty }{\sum }}\left(
-1\right) ^ne^{-n\beta \mu }\limfunc{Ei}(nm\beta )\hspace{1.0in}(9b) 
\]

\[
C(m_{_\ell }\beta ,\mu )=\frac{\pi ^2}{12}+\stackunder{n=1}{\stackrel{\infty 
}{\sum }}\frac{\left( -1\right) ^n}{n^2}e^{n\beta \left( m-\mu \right) }%
\hspace{1.0in}(9c) 
\]

\[
D(m\beta ,\mu )=\stackunder{n=1}{\stackrel{\infty }{\sum }}(-1)^n\frac{%
(n\beta )^2}2\left[ \ell n\frac \mu m-\frac 1{2n\beta \mu }+e^{-n\beta \mu }%
\limfunc{Ei}(nm\beta )\right] \hspace{1.0in}(9d) 
\]

for electron. It is also worth-mentioning that for T $\geq \mu $ the
electrons and positrons get equal contribution from the background. However
when T $\leq \mu $ the situation is different. In the classical limit T $\ll
m<\mu $ and A,B,C,and D functions vanish for electrons and the chemical
potential correspond to the Fermi energy. In this case the there will be no
antiparticles in the background and we don't have to bother so much about
the self-mass corrections to positrons. However, just for completenes we
evaluate the background contribution to the positron mass at T $\leq \mu $ as

\begin{eqnarray*}
\ \ \frac{\delta m}{m} &\simeq &\frac{\alpha \pi T^{2}}{3m^{2}}-\frac{%
2\alpha }{\pi m^{2}}\left( \frac{\overline{A}(m\beta ,\mu )}{\beta }-\frac{%
\overline{C}(m\beta ,\mu )}{\beta ^{2}}\right) -\frac{3\alpha }{\pi }%
\overline{B}(m\beta ,\mu )- \\
&&\ -\frac{\alpha }{8\pi }\left[ n_{F}\left( \mu -m\right) +m\beta \frac{%
e^{-\beta (\mu +m)}}{\left[ 1+e^{-\beta (\mu +m)}\right] ^{2}}+m^{2}%
\overline{D}(m\beta ,\mu )\right] \\
&&\hspace{1in}\hspace{1in}\hspace{1in}\hspace{1in}(10)
\end{eqnarray*}

with

\[
\overline{A}(m\beta ,\mu )=m\ell n[1+e^{-\beta (m+\mu )}]\hspace{1.0in}(11a) 
\]

\[
\overline{B}(m\beta ,\mu )=\stackunder{n=1}{\stackrel{\infty }{\sum }}\left(
-1\right) ^ne^{-n\beta \mu }\limfunc{Ei}(-nm\beta )\hspace{1.0in}(11b) 
\]

\[
\overline{C}(m\beta ,\mu )=\stackunder{n=1}{\stackrel{\infty }{\sum }}\frac{%
\left( -1\right) ^n}{n^2}e^{-n\beta (m+\mu )}\hspace{1.0in}(11c) 
\]

\[
\overline{D}(m\beta ,\mu )=\stackunder{n=1}{\stackrel{\infty }{\sum }}(-1)^n%
\frac{(n\beta )^2}2\limfunc{Ei}(-nm\beta )e^{-n\beta \mu }\hspace{1.0in}%
(11d) 
\]

The wavefunction renormalization constants can be obtained from the selfmass
of the corresponding particle using Ward identities[18]. We have obtained
the renormalization constant of the e$^{-}$ and e$^{+}$wavefunction at T $%
\geq \mu ,$as 
\begin{eqnarray*}
Z_{2}^{-1} &\simeq &Z_{2}^{-1}(T=0)-\frac{2\alpha }{\pi }\int \frac{dk}{k}%
n_{B}(k)-\frac{5\alpha }{\pi }\widetilde{b}(m_{_{\ell }}\beta ,\mu )-\frac{%
\alpha }{\pi v}\frac{1}{E^{2}\beta ^{2}}\ell n\frac{1-v}{1+v}\times \\
&&\times [\frac{\pi ^{2}}{6}-\widetilde{c}(m_{\ell }\beta ,\mu )+m\beta 
\widetilde{a}(m_{_{\ell }}\beta ,\mu )]\hspace{1in}(12)
\end{eqnarray*}
whereas, in the limit T $\leq \mu $ one obtains different forms of the
wavefunction renormalization for particle and antiprticle. The e$^{-}$and e$%
^{+}$ wavefunction in the high chemical potential limit come out to be[17]

\begin{eqnarray*}
Z_{2}^{-1}(e^{-}) &\simeq &Z_{2}^{-1}(T=0)-\frac{2\alpha }{\pi }\left( \int 
\frac{dk}{k}n_{B}(k)-\frac{2}{3}\frac{\pi ^{3}T^{2}}{E^{2}}\frac{1}{v}\ell n%
\frac{1+v}{1-v}\right) - \\
&&\frac{\alpha }{2\pi E^{2}}\ell n\frac{1+v}{1-v}\left[ \frac{\mu ^{2}}{2}%
\left( 1-\frac{m^{2}}{\mu ^{2}}\right) -\frac{C(m_{\ell }\beta ,\mu )}{\beta
^{2}}+\frac{A(m_{_{\ell }}\beta ,\mu )}{\beta }\right] - \\
&&-\frac{\alpha E^{2}}{8\pi m^{2}}\left[ \left( 1-\frac{m^{2}}{\mu ^{2}}%
\right) -\frac{m^{2}}{2\mu ^{2}}+n_{F}\left( \mu -m\right) +m\beta \frac{%
e^{-\beta (\mu -m)}}{\left[ 1+e^{-\beta (\mu -m)}\right] ^{2}}\right] \\
&&+\frac{\alpha E^{2}}{8\pi }D(m\beta ,\mu )-\frac{3\alpha }{\pi }\left[
\ell n\frac{\mu }{m}+B(m_{_{\ell }}\beta ,\mu )\right] \hspace{1in}\left(
13a\right)
\end{eqnarray*}
The positron wavefunction renormalization can easily be evaluated by
replacing the distribution function of electron by that of positron in
highly dense systems. It comes out to be 
\begin{eqnarray*}
Z_{2}^{-1}(e^{+}) &\simeq &Z_{2}^{-1}(T=0)-\frac{2\alpha }{\pi }\left( \int 
\frac{dk}{k}n_{B}(k)-\frac{2}{3}\frac{\pi ^{3}T^{2}}{E^{2}}\frac{1}{v}\ell n%
\frac{1+v}{1-v}\right) - \\
&&-\frac{\alpha }{2\pi E^{2}}\ell n\frac{1+v}{1-v}\left[ \frac{\overline{A}%
(m\beta ,\mu )}{\beta }-\frac{\overline{C}(m\beta ,\mu )}{\beta ^{2}}\right]
-\frac{3\alpha }{\pi }\left[ \overline{B}(m\beta ,\mu )\right] \\
&&-\frac{\alpha E^{2}}{8\pi m^{2}}\left[ \left( 1-\frac{m^{2}}{\mu ^{2}}%
\right) -\frac{m^{2}}{2\mu ^{2}}+n_{F}\left( \mu -m\right) +m\beta \frac{%
e^{-\beta (\mu -m)}}{\left[ 1+e^{-\beta (\mu -m)}\right] ^{2}}\right] \\
&&+\frac{\alpha E^{2}}{8\pi }\overline{D}(m\beta ,\mu )\hspace{1in}\left(
13b\right)
\end{eqnarray*}

The electron charge renormalization in QED\ is\ determined through the
self-energy of photons. So, the fermion charge renormalization constant in
the limit T $\geq \mu ,$it comes out to be [18]

\begin{eqnarray*}
Z_{3} &\sim &Z_{3}(T=0)+\left[ \frac{8\alpha }{m^{2}\pi }\left\{ \frac{m%
\widetilde{a}(m\beta ,\mu )}{\beta }+\frac{1}{4}\left( m^{2}+\frac{\omega
^{2}}{3}\right) \widetilde{b}(m\beta ,\mu )-\frac{\widetilde{c}(m\beta ,\mu )%
}{\beta ^{2}}\right\} \right] \\
&&\hspace{1in}\hspace{1in}\hspace{1in}\hspace{1in}(14)
\end{eqnarray*}
and at T $\leq \mu $ it appears to be

\[
Z_3\sim Z_3(T=0)+\frac{8\alpha }{m^2\pi }\left\{ I_1+\frac 14\left( m^2+%
\frac{\varpi ^2}3\right) I_2-\frac 14m^4I_3\right\} \hspace{1.0in}(15) 
\]

with

\begin{eqnarray*}
I_{1} &=&\frac{1}{2}\frac{\mu ^{2}}{2}\left( 1-\frac{m^{2}}{\mu ^{2}}\right)
+\frac{1}{2\beta }\left( A(m\beta ,\mu )-\overline{A}(m\beta ,\mu )\right) -
\\
&&-\frac{1}{2\beta ^{2}}\left( C(m\beta ,\mu )-\overline{C}(m\beta ,\mu
)\right) \hspace{1in}(16a)
\end{eqnarray*}

\[
I_{2}=\frac{1}{2}\left[ \ell n\frac{m}{\mu }+\left( B(m\beta ,\mu )-%
\overline{B}(m\beta ,\mu )\right) \right] \hspace{1in}(16b) 
\]

\begin{eqnarray*}
I_{3} &=&\frac{1}{4m^{2}}\left( 1-\frac{m^{2}}{\mu ^{2}}\right) -\frac{1}{%
8\mu ^{2}}+\frac{1}{2m^{2}}\left[ n_{f}(\mu +\beta )+n_{f}(\mu -\beta
)\right] \\
&&+\frac{\beta }{2m}\left( \frac{e^{-\beta (\mu +m)}}{\left[ 1+e^{-\beta
(\mu +m)}\right] ^{2}}+\frac{e^{-\beta (\mu -m)}}{\left[ 1+e^{-\beta (\mu
-m)}\right] ^{2}}\right) +D(m\beta ,\mu )-\overline{D}(m\beta ,\mu ) \\
&&\hspace{1in}\hspace{1in}\hspace{1in}(16c)
\end{eqnarray*}

The fact that QED is renormalizable in this regime and also there is no
doubt about the renormalizability of electroweak theory because no extra hot
infrared singularities enter into the calculations due to the massive
intermediate vector bosons. All the background corrections to the
electroweak processes are suppressed by the square of the electroweak gauge
boson masses as compared to the lepton masses so we don't have to
incorporate them here.

It has been explicitely checked that the QED\ renormalization scheme works
perfectly in the quantum statistical medium. The calculations of the
leptonic decays, using the renormalization constants at FTD reproduces all
the existing results obtained by adding all the first order diagrams in QED.
In the next subsection we discuss the leptonic decay rates in hot and dense
background in a little bit more detail to see that how can we use these
decay rates to evaluate the leptonic crossections.

\subsection{Leptonic Decays}

The radiative corrections to the decay rates of charged leptons can be
estimated from the renormalization constants of QED . The decay rate of Z$%
^{0}$ (i.e.e;the Z width) is an important gauge vertex of the electroweak
theory. We have evaluated it in the limit T$\geq \mu $ in the rest frame of Z%
$^{0}.$ It comes out to be equal to [14]

\[
\frac{\Gamma ^\beta (Z^0\rightarrow \ell \overline{\ell })}{\Gamma
_0(Z^0\rightarrow \ell \overline{\ell })}\simeq \frac{4\alpha T}{\pi m_Zv}%
\left[ \left( 1-\frac{2m_\ell }{m_Z}+\frac{4m_\ell ^2}{m_Z^2v^2}\right) \ell
n\frac{1+v}{1-v}-\frac{2m_\ell }{m_Zv}\left( 3+\frac{4m_\ell ^2}{m_Z^2v^3}%
\right) \right] \widetilde{a}(m_\ell \beta ,\mu ) 
\]

\[
\frac{4\alpha }\pi \left[ 1+\frac{2m_\ell ^2}{m_Z^2v^2}-\frac 1{2v}\left( 1-%
\frac{5m_\ell ^2}{4m_Z^2v^2}+\frac{m_\ell ^2}{4m_Z^2v^4}\right) \ell n\frac{%
1+v}{1-v}\right] \widetilde{b}(m_\ell \beta ,\mu ) 
\]

\[
\frac{4\alpha }{\pi }\left[ \frac{2T^{2}}{m_{Z}^{2}v}\left\{ \left( 1+\frac{%
m_{\ell }^{2}}{m_{Z}^{2}v^{3}}\right) \ell n\frac{1+v}{1-v}+\frac{1}{v}%
\left( 3-\frac{2m_{\ell }^{2}}{m_{Z}^{2}v^{2}}\right) \right\} \widetilde{c}%
(m_{\ell }\beta ,\mu )\right] \hspace{1in}\left( 17\right) 
\]
and the scalar Higgs decay rate is calculated in the rest frame of Higgs
which comes out to be equal to [17]

\begin{eqnarray*}
\frac{\Gamma ^{\beta }(H^{0}\rightarrow \ell \overline{\ell })}{\Gamma
_{0}(H^{0}\rightarrow \ell \overline{\ell })} &\simeq &\frac{4\alpha T\text{ 
}}{\pi m_{H}^{2}v^{2}}\left[ -m_{\ell }+\frac{1}{v}\left( m_{\ell }-m_{H}%
\frac{1+v^{2}}{4}\right) \text{ }\ell n\frac{1-v}{1+v}\right] \widetilde{a}%
(m_{\ell }\beta ,\mu )+ \\
&&\frac{2\alpha }{\pi v^{2}}\left[ 3v^{2}-v\ell n\frac{1-v}{1+v}+22\frac{%
m_{\ell }^{2}}{m_{H}^{2}}-\frac{1}{2}\right] \widetilde{b}(m_{\ell }\beta
,\mu )- \\
&&\frac{8\alpha T^{2}}{\pi v^{3}m_{H}^{2}}\left[ \ell n\frac{1-v}{1+v}%
+3v\right] \widetilde{c}(m_{\ell }\beta ,\mu )\hspace{1in}\hspace{1in}(18)
\end{eqnarray*}
$m_{H}$ is the mass of the neutral Higgs of SEWM. We generalize the existing
results of Ref.[17] to evaluate the Higgs decay rate at high densities at
non-negligible temperatures. We use the approximation $m_{H}^{2}>>m_{\ell
}^{2}$ to ignore the higher order terms in $\frac{m_{\ell }^{2}}{m_{H}^{2}}.$
The calculational techniques are same as are used in Ref.[17]. This can be
considered as a straight forward generalization because all the techniques
are same. The neutral Higgs decay rate for $\mu \succeq T$ , then appears to
be equal to 
\begin{eqnarray*}
\frac{\Gamma ^{\mu }(H^{0}\rightarrow \ell \overline{\ell })}{\Gamma
_{0}(H^{0}\rightarrow \ell \overline{\ell })} &\simeq &\frac{4\alpha }{\pi
v^{2}}\frac{f_{1}(E)}{2E}A(m_{_{\ell }}\beta ,\mu )+\frac{4\alpha }{\pi v^{2}%
}\left( \frac{9m_{\ell }^{2}}{\pi }-vf_{2}(E)\right) \frac{B(m_{_{\ell
}}\beta ,\mu )}{2E^{2}}+ \\
&&+\frac{4\alpha }{\pi v^{2}}\left( \frac{vf_{5}(E)}{6m_{\ell }^{3}E^{3}}%
\right) D(m_{_{\ell }}\beta ,\mu )+\frac{4\alpha }{\pi v^{2}}\left( \frac{%
vf_{3}(E)}{2m_{\ell }E^{2}}\right) G(m_{_{\ell }}\beta ,\mu )+ \\
&&+\frac{4\alpha }{\pi v^{2}}\left( \frac{3m_{\ell }^{2}}{2\mu _{\ell }^{2}}%
\left[ \frac{3m_{\ell }^{2}}{\mu _{\ell }^{2}}+\frac{2p^{2}}{\mu _{\ell }^{2}%
}-1\right] -\frac{f_{4}(E)}{4m_{\ell }^{3}E^{3}}\right) H(m_{_{\ell }}\beta
,\mu )\hspace{1in}\left( 19\right)
\end{eqnarray*}

with 
\[
f_{1}(E)=E\frac{1+v^{2}}{1-v^{2}}\ln \frac{1-v}{1+v}\qquad \qquad \qquad
(20a) 
\]

\[
f_{2}(E)=\left( m^{2}+\left( 1-v^{2}\right) E^{2}\right) \frac{1}{v^{2}}\ln 
\frac{1-v}{1+v}+2pE\frac{1+v^{2}}{v^{2}}\qquad \qquad \qquad (20b) 
\]

\begin{eqnarray*}
f_{3}(E) &=&\left( m^{2}+\left( 1+v^{2}\right) E^{2}\right) \frac{E}{v^{2}}%
\ln \frac{1-v}{1+v}+3pE^{2}\frac{\left( 1+v^{2}\right) }{v^{2}}+ \\
&&+\frac{2p}{v}m^{2}+4E^{2}p\qquad \qquad \qquad \qquad \qquad (20c)
\end{eqnarray*}

\begin{eqnarray*}
f_{4}(E) &=&\left( m^{2}+\left( 1+v^{2}\right) E^{2}\right) \frac{E^{3}}{%
v^{2}}\ln \frac{1-v}{1+v}+7pE^{4}\frac{\left( 1+v^{2}\right) }{v^{2}}+\frac{%
7pE^{2}m^{2}}{v}+ \\
&&+\frac{10p^{2}-m^{2}}{v^{2}}p\left( m^{2}+\left( 1+v^{2}\right)
E^{2}\right) +4E^{4}p\qquad \qquad \qquad \qquad \qquad (20d)
\end{eqnarray*}

\begin{eqnarray*}
f_{5}(E) &=&\left( m^{2}+\left( 1+v^{2}\right) E^{2}\right) \frac{E^{4}}{%
v^{2}}\ln \frac{1-v}{1+v}+\left( 2p^{2}+5E^{2}\right) E^{4}\frac{\left(
1+v^{2}\right) }{v^{2}}+\frac{7pE^{3}m^{2}}{v}+ \\
&&+\frac{m^{2}+\left( 1+v^{2}\right) E^{2}}{v^{2}}\left(
10p^{2}-m^{2}\right) pE+\frac{\left( 1+v^{2}\right) }{v}pm^{2}E^{3}+4p\left(
p^{2}+m^{2}\right) E^{3}\qquad (20e)
\end{eqnarray*}
and the functions G\&H are given as

\[
G(m\beta ,\mu )=\stackunder{n=1}{\stackrel{\infty }{\sum }}(-1)^{n}\left[
\beta n\limfunc{Ei}(-nm\beta )+\frac{e^{-nm\beta }(n\beta )^{2}}{m}\right]
e^{-n\beta \mu }\hspace{1in}(9e) 
\]
and

\[
H(m\beta ,\mu )=\stackunder{n=1}{\stackrel{\infty }{\sum }}(-1)^{n}\left[ 
\frac{(\beta n)^{2}}{2}\limfunc{Ei}(-nm\beta )+\frac{n\beta e^{-nm\beta }}{m}%
\right] e^{-n\beta \mu }\hspace{1in}(9f) 
\]

We use the rest frame of Z$^{0}$ and also the energy of Z$^{0}$ is high
enough to ignore higher order contribution from the distribution function.
We include all the one loop Z diagrams including the bremstrahlung. The
calculational scheme is very similar to the one adopted previously in
Ref.[13,17]. We conclude that the leading order contributions due to the
highly dense background comes out to be some complicated function of several
parameters given below. They are even more complicated than the results of
Eqs.(19 \& 20). We postpone the detail discussion of these results for now.

\[
\frac{\Gamma ^{\mu }(Z^{0}\rightarrow \ell \overline{\ell })}{\Gamma
_{0}(Z^{0}\rightarrow \ell \overline{\ell })}\sim F\left( E,M_{Z},\mu ,\ell n%
\frac{1-v}{1+v}\right) \hspace{1in}(21) 
\]

The calculational scheme which we have been using throughout can actually
help us to evaluate the statistical corrections to any leptonic decay
processes. We, therefore, evaluate the QED type radiative corrections to the
meson decay in the statistical background. We make use of the
renormalization constants of QED upto the one loop level and replace all the
relevant bare parameters with the corresponding FTD corrected values.

The thermal corrections to the decay rates of neutral mesons like pions and
kaons can also be calculated in a similar way. They simply comes out to be
very similar to the thermal corrections to the Higgs decay rates. This
happens only when we cannot consider any corrections to the vertices other
than QED type radiative corrections. It appears to be an over-simplification
of the mathematical calculations but the results are acceptable in the
statistical theory where we need to assume real highly relativistic hot
particles in the background. This is the beauty of thermodynamics that we
have to be in the systems where we can deal with the real particles on mass
shell in the heat bath. Now the heavy particles exchanging contributions can
easily be ignored as compared to the lighter ones based on the expected
relativistic behavior

\begin{eqnarray*}
\frac{\Gamma ^{\beta }(K^{0}\rightarrow \ell \overline{\ell })}{\Gamma
_{0}(K^{0}\rightarrow \ell \overline{\ell })} &\simeq &\frac{\alpha T\text{ }%
}{\pi E^{2}v^{2}}\digamma _{1}\left( m_{\ell ,}E,\ell n\frac{1-v}{1+v}%
\right) \widetilde{a}(m_{\ell }\beta ,\mu )+ \\
&&\frac{2\alpha }{\pi v^{2}}\digamma _{2}\left( m_{\ell ,}E,\ell n\frac{1-v}{%
1+v}\right) \widetilde{b}(m_{\ell }\beta ,\mu )- \\
&&\frac{2\alpha T^{2}}{\pi v^{3}E^{2}}\digamma _{3}\left( m_{\ell ,}E,\ell n%
\frac{1-v}{1+v}\right) \widetilde{c}(m_{\ell }\beta ,\mu )\hspace{1in}%
\hspace{1in}(22)
\end{eqnarray*}

$\digamma _{1},\digamma _{2,}$ and $\digamma _{3}$ are very similar to the
corresponding coefficients in Eq.17. Main difference occurs in the bare
decay rate, but the radiative corrections (especially the leading order QED
type contributions) are fairly similar and so are easy to evaluate for
leptonic decays of mesons also. We have always preferred to use the rest
frame of the decaying particles to evaluate these contributions.

However, $4E^{2}=m_{meson}^{2}$, whereas, $m_{meson}$ correspond to the mass
of the decaying meson and $\ell $ correspond to the lighter leptons(m$_{\ell
}\prec m_{meson}$). For heavy mesons the above relations can also hold for
taons. In the leptonic decays of charged leptons, the vertex corrections
terms do not get any background contribution. So, we can simply replace rest
masses and the vacuum couplings by the corresponding physical masses and the
background renormalized couplings. The radiative decays of the charged
mesons should however appear like

\begin{eqnarray*}
\frac{\Gamma ^{\beta }(K^{\pm }\rightarrow \gamma \ell ^{\pm }\nu _{\ell })}{%
\Gamma _{0}(K^{\pm }\rightarrow \gamma \ell ^{\pm }\nu _{\ell })} &\cong &%
\frac{\alpha }{\pi }\int_{0}^{\infty }dkn_{F}\left( p\mp \mu \right) x \\
&&\left\{ \left[ 2+\left[ 1+v^{2}-\frac{2m_{\ell }^{2}}{m_{K}^{2}}\right] 
\frac{1}{v}\ln \frac{1+v}{1-v}\right] \frac{1}{k}+\frac{k}{m_{K}^{2}v}\left[
1+\frac{1}{v}\ln \frac{1+v}{1-v}\right] \right\} \hspace{0.2in}\ (23)
\end{eqnarray*}

In these kind of three body decay processes, the phase space corrections due
to the absorption and emission of photon at the electron leg of this vertex
contribute. We can evaluate Eq.23 for all the interesting ranges of
temperature and chemical potential using the corresponding expressions of $%
n_{F}\left( p\mp \mu \right) .$ All the physical parameters, like electron
charge mass and the wavefunction are needed to be replaced by the
corresponding renormalized (background corrected) values. These relations
are mentioned in the next section in Eqs.24. We can evaluate the radiative
corrections to different leptonic processes upto the one loop level in
almost all the ranges of temperature and chemical potential in our frame of
calculations. However, leptonic scatterings are very important in the early
universe when the temperatures were sufficiently smaller than the
electroweak gauge bosons mass and still high enough to use the limit T$\gg
m_{e}$, for m$_{e}$ to be the electron mass in the electron distribution
function. We can obviously incorporate the FTD corrections using our frame
of calculation. Since the high density corrections, where densities are much
larger than the masses, we have to be careful to ignore FTD corrections
which are not QED type. Also, in those dense systems, we might need to
incorporate the magnetic field effects also. So, we postpone the discussion
of lepton scatterings in the highly dense systems at the moment. In the next
section, we evaluate the lepton scatterings at high temperature and low
densities and try to investigate that how the background corrections can
effect the lepton scattering cross-sections in the early universe and also
their implications to the big bang cosmology.

\subsection{ Scatterings of Leptons in Electroweak Theory}

In the calculation of the cross-sections, one can easily make use of the
renormalization constants of QED evaluated in hot and dense media.The
leptonic scatterings can either take place in QED or in the electroweak
theory. QED\ is important for charged particle processes of $\ell \overline{%
\ell }\rightarrow \ell \overline{\ell \text{ }}$whereas $\ell $ correspond
to the charged leptons only. In the standard model additional leptonic
processes of the type $\ell ^{\pm }\nu _{\ell }\rightarrow \ell ^{\pm }\nu
_{\ell }$or $\ell \overline{\ell }\rightarrow \nu \overline{\nu }.$would
also be expected. The individual lepton number conservation would be
required in the standard model. A very quick estimate of the background
corrections can be made by simply replacing the bare lepton masses, charge
of electron(QED couplings) and the wavefunctions by the background
renormalized mass, charge, and wavefunctions correspondingly. So, we can
directly replace the bare mass of lepton be the corresponding renormalized
mass of leptons given as

\[
m_{\ell }\rightarrow m_{\ell }[1+\frac{\delta m_{\ell }}{m_{\ell }}]\hspace{%
1in}(24a) 
\]

The wavefunction of charged fermions by the corresponding renormalized
wavefunction given as

\[
\Psi _{\ell }\rightarrow Z_{2}\Psi _{\ell }\hspace{1in}(24b) 
\]

and the coupling constant or the charge of electron can be rewritten as

\[
e\rightarrow Z_{3}^{-1}e\hspace{1in}(24c) 
\]

The values of all the coupling constants can be used for the given limits of
temperature and the chemical potential from previous sections. The phase
space corrections can also be obtained from the cross-sections of the
corresponding processes directly. Just to keep ourself limited to the
electroweak scatterings of particles in the relevant temperature and density
regime, we discuss in a little more detail, the electroweak scattering of
leptons in the physical conditions of the early universe.

The purely leptonic scatterings can either take place in QED or in the
electroweak theory. In QED, the elastic or inelastic scatterings of charged
leptons, namely $\ell \overline{\ell }\rightarrow \ell \overline{\ell \text{ 
}}$can take place for $\ell =\overline{\ell }\ $or $\ell \neq \overline{\ell 
}.$ We have explicitely checked that the QED renormalization scheme works
perfectly well in hot and dense medium upto the one loop level. Due to this
reason, it makes no difference if we calculate each and every first order
diagram separately and add them together or calculate the matrix elements
directly from the renormalization constants we reach the same result. It has
also been noticed that the radiative corrections are obtained in a power
series of $\frac{m_{\ell }^{2}}{m_{X}^{2}}$ where $m_{X}$ correspond to the
mass of exchanging heavy particle. We use this approximation because the
heavy particle exchange in the standard basically allows the exchange of W$%
^{\pm }$ , Z$^{0}$ , and the standard model Higgs H$^{0}.$ These particles
have such a heavy mass values that we can easily ignore the higher order
terms in the power series of $\frac{m_{\ell }^{2}}{m_{X}^{2}}.$ Due to the
presence of this small parameter, we have to restrict to the first order
radiative corrections. Higher order in the coupling constant makes the
radiative corrections fairly negligible. Moreover, the temperature of our
system is small enough to make the terms proportional to $\frac{T^{2}}{%
m_{X}^{2}},$ even more negligible. Starting from the matrix element for $%
\ell \overline{\ell }\rightarrow \ell \overline{\ell \text{ }},$we have to
replace the ordinary vacuum fermion and boson propagators by the
corresponding FTD propagators from Eqs.1\&2. Now, using the properties of
Dirac $\delta ,$ we can only get the nonzero contributions which are of the
same form which we obtain in the decay rates because the radiative
corrections from one vertex can be considered at a time to be in the lowest
loop level. .In this situation, the statistical contributions to the
cross-section can directly be related to the decay rate upto the first order
in $\alpha .$ We can write it as 
\[
\frac{d\sigma ^{\beta }(\ell \overline{\ell }\rightarrow \ell \overline{\ell 
\text{ }})}{d\sigma _{0}(\ell \overline{\ell }\rightarrow \ell \overline{%
\ell \text{ }})}\longrightarrow 2\frac{\Gamma ^{\beta }(Z^{0}\rightarrow
\ell \overline{\ell })}{\Gamma _{0}(Z^{0}\rightarrow \ell \overline{\ell })}%
+2\frac{\Gamma ^{\beta }(H^{0}\rightarrow \ell \overline{\ell })}{\Gamma
_{0}(H^{0}\rightarrow \ell \overline{\ell })}\hspace{1in}(25)
\]

At the very low energies, where exchanging particles have very low momenta,
the contribution of all of these diagrams is added up together to give

\begin{eqnarray*}
\frac{d\sigma ^{\beta }(\ell \overline{\ell }\rightarrow \ell \overline{\ell 
\text{ }})}{d\sigma _{0}(\ell \overline{\ell }\rightarrow \ell \overline{%
\ell \text{ }})} &\approx &\frac{4\alpha T}{\pi m_{Z}v}\left[ \left( 1-\frac{%
2m_{\ell }}{m_{Z}}+\frac{4m_{\ell }^{2}}{m_{Z}^{2}v^{2}}\right) \ell n\frac{%
1+v}{1-v}-\frac{2m_{\ell }}{m_{Z}v}\left( 3+\frac{4m_{\ell }^{2}}{%
m_{Z}^{2}v^{3}}\right) \right] \widetilde{a}(m_{\ell }\beta ,\mu )+ \\
&&\frac{4\alpha }{\pi }\left[ 1+\frac{2m_{\ell }^{2}}{m_{Z}^{2}v^{2}}-\frac{1%
}{2v}\left( 1-\frac{5m_{\ell }^{2}}{4m_{Z}^{2}v^{2}}+\frac{m_{\ell }^{2}}{%
4m_{Z}^{2}v^{4}}\right) \ell n\frac{1+v}{1-v}\right] \widetilde{b}(m_{\ell
}\beta ,\mu )+ \\
&&\frac{4\alpha }{\pi }\left[ \frac{2T^{2}}{m_{Z}^{2}v}\left\{ \left( 1+%
\frac{m_{\ell }^{2}}{m_{Z}^{2}v^{3}}\right) \ell n\frac{1+v}{1-v}+\frac{1}{v}%
\left( 3-\frac{2m_{\ell }^{2}}{m_{Z}^{2}v^{2}}\right) \right\} \widetilde{c}%
(m_{\ell }\beta ,\mu )\right] + \\
&&\frac{4\alpha T\text{ }}{\pi m_{H}^{2}v^{2}}\left[ -m_{\ell }+\frac{1}{v}%
\left( m_{\ell }-m_{H}\frac{1+v^{2}}{4}\right) \text{ }\ell n\frac{1-v}{1+v}%
\right] \widetilde{a}(m_{\ell }\beta ,\mu )+ \\
&&\frac{2\alpha }{\pi v^{2}}\left[ 3v^{2}-v\ell n\frac{1-v}{1+v}+22\frac{%
m_{\ell }^{2}}{m_{H}^{2}}-\frac{1}{2}\right] \widetilde{b}(m_{\ell }\beta
,\mu )- \\
&&\frac{8\alpha T^{2}}{\pi v^{3}m_{H}^{2}}\left[ \ell n\frac{1-v}{1+v}%
+3v\right] \widetilde{c}(m_{\ell }\beta ,\mu )\hspace{1in}(26)
\end{eqnarray*}
which gives the leading order contribution at very high temperatures as 
\[
\frac{d\sigma ^{\beta }(\ell \overline{\ell }\rightarrow \ell \overline{\ell 
\text{ }})}{d\sigma _{0}(\ell \overline{\ell }\rightarrow \ell \overline{%
\ell \text{ }})}\symbol{126}\frac{2\alpha \pi T^{2}}{v^{2}}\left[ \frac{1}{%
3m_{Z}^{2}}+\frac{1}{m_{H}^{2}}\right] \hspace{1in}(27) 
\]
for the photon energy $E^{2}>>m_{\ell }^{2}$, and T $\geq \mu .$

The radiative corrections to the scattering cross-sections of charge current
processes can be evaluated in terms of the decay rates of W mainly.

\[
\frac{d\sigma ^{\beta }(\ell ^{\pm }\nu _{\ell }\rightarrow \ell ^{\pm }\nu
_{\ell })}{d\sigma _{0}(\ell ^{\pm }\nu _{\ell }\rightarrow \ell ^{\pm }\nu
_{\ell })}\longrightarrow 2\frac{\Gamma ^{\beta }(W^{\pm }\rightarrow \ell
^{\pm }\nu _{\ell })}{\Gamma _{0}(W^{\pm }\rightarrow \ell ^{\pm }\nu _{\ell
})}\hspace{1in}(28) 
\]

This contribution is always proportional to $\frac{T^{2}}{m_{\ell }^{2}}.%
\frac{1}{m_{W}^{2}}.$So, larger contributions are obtained for larger
temperatures.

For the limits of T $\leq \mu ,$ in the relativistic limit, it turns out to
be a comparatively complicated expression. We don't want to involve in the
detailed discussion of these contributions because in the early universe and
in the stable stars we usually come across the limit $T\succeq m\succeq \mu
. $ Just to have an idea of the expression, we calculate the leading order
contribution, at large chemical potentials due to the Higgs exchanging
vertex, which comes out to be like,

\[
\frac{d\sigma ^{\mu }(\ell \overline{\ell }\rightarrow \ell \overline{\ell 
\text{ }})}{d\sigma _{0}(\ell \overline{\ell }\rightarrow \ell \overline{%
\ell \text{ }})}\approx \frac{2\alpha \mu }{\pi v^{2}E}\ell n\frac{1+v}{1-v}+%
\frac{\alpha E^{3}}{6\pi vm^{3}}\left( 1+\ell n\frac{1+v}{1-v}\right) 
\hspace{1in}(29)\qquad 
\]

T $\succ \mu $ could be important for the superdense collapsing stars like
supernova. It can be clearly seen that each term can have infrared
divergences in certain limit. We can play with the comparative limits to
find physically interesting limits. The Z$^{0}$ vertex will not be any
different. Only the coefficients of mass divergent terms like $\ell n$ $%
\frac{1+v}{1-v}$ would be different. This thing may give some help to
address the stability of the superdense stars also. But, in such collapsing
systems the densities are high enough to worry about the magnetic fields. We
therefore, postpone this discussion for the moment and move over to the
conclusion of our present results.

\section{Results and Discussions}

The results calculated in the previous section are mostly relevant from the
point of view of cosmology. In the beginning of the universe, the
temperature of the universe was close to the decoupling temperature, and the
leptonic processes were taking place. Eq.27 shows the radiative corrections
to the cross-section are simply proportional to T$^{2}$ which is the
standard expected results in literature Ref.[18,19]. It is also worth
mentioning that the above mentioned results are correct in the standard
model. In the extended standard model, we have to incorporate massive
neutrino effects and several other contributions.

The present experimental limit [20] on the standard model Higgs mass is m$%
_{H}\succ 107GeV.$ In this situation, we cannot further simplify Eq.27,
because we cannot ignore one term as compared to the other in bracket due to
comparable contributions. Just to have a very quick estimate of the
magnitude of this contribution, we can assume the mass of Higgs to be 100
GeV, T =2MeV and the standard values of the gauge boson masses to use them
in Eq.27. The radiative corrections from the background, then appears to be, 
\[
\frac{d\sigma ^{\beta }(\ell \overline{\ell }\rightarrow \ell \overline{\ell 
\text{ }})}{d\sigma _{0}(\ell \overline{\ell }\rightarrow \ell \overline{%
\ell \text{ }})}\rightarrow 10^{-10}\ \ \hspace{0.4in}\ (30)
\]
This contribution would be increased with increasing temperature but cannot
go beyond a couple of orders higher, otherwise, the validity of the
calculational approximations would be questionable after T $\succ $ a few
tens of MeV. It happens because we deal with the exponential functions in
the statistical distribution functions, and the approximations used for them
are usually for a small range.

Therefore, we reach the conclusion that FTD corrections to the scattering
cross-sections are not so much significant in the early universe. But, the
energy density of the universe goes with the fourth power of these
scattering cross-sections [3], so we still need to evaluate it to improve
the precision in estimating the energy density of the early universe. On the
other hand, the FTD corrections are not so insignificant after the universe
was at a little lower temperature and the nucleosynthesis started taking
place[22].

\section{References and Footnotes}

\begin{enumerate}
\item  Georg G.Raffelt,`Stars as Laboratories for Fundamental
Physics',Chicago University Press,1996.

\item  R.Barbieri and P.Mohapatra, Phys.Lett. B61,23(1988)

\item  See for example, S.Weinberg `Gravitation and Cosmology' (Wiley, New
York, 1972) and references therein

\item  K.S.Babu and V.S.Mathur, Phys.Lett. B196, 218 (1987); M.Fukujita and
T.Yanagida, Phys.Rev.Lett.58,1807(1987)

\item  Pal and Mohapatra, Massive Neutrinos in Physics and Astrophysics'
(World Scientific publication, 1991)

\item  L.Dolan and R.Jackiw, Phys.Rev. D9,3320(1974)

\item  P.Landsman and Ch. G. Weert, Phys.Rep.145,141(1987) and references
therein

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

\item  E.Braaten and R.D.Pisarski, Nucl. Phys. B339,310(1990); ibid, B337,
569(1990)

\item  Private communication with prof.W.Hollick

\item  Mahnaz Qader, Samina S.Masood and K.Ahmed, Phys.Rev.D44,3322(1991);
ibid,D46,5633(1992)and references therein. See also Ref.[7].

\item  R.Kobes and G.W.Semenoff, Nucl.Phys. B260,714(1985); ibid,
B272,329(1986)

\item  Samina S.Masood and Mahnaz Qader,Phys.Rev.D46,5110(1992).

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

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

\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) and
references therein.

\item  See for Example, A.Weldon, Phys.Rev.D26,1394(1992),ibid;2789(1982),
and Ref.[8]

\item  A.Niegava Phys.Rev.Lett.,71,3055(1993), Naoki Ashida,et.al.;
Phys.Rev. D45, 2066(1992),;A.Niegawa, Phys.Lett.B247,351(1990) and
references therein

\item  R.D.Pisraski, Phys.Rev.Lett.63:1129,1989, ibid; Physica
A158:246-250,1989

\item  ALEPH Collaboration, Phys.Lett. B499 (2001) 53-66

\item  Samina Saleem (Masood), Phys. Rev. D36 , 2602 (1987).
\end{enumerate}

\end{document}



