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\begin{document}

\preprint{}

\title{The Lamb Shift and Ultra High Energy Cosmic Rays}

\author{She-Sheng Xue}

\email{xue@icra.it}

\affiliation{ICRA, INFN and
Physics Department, University of Rome ``La Sapienza", 00185 Rome, Italy}

\date{August, 2000}

%\date{\today}


\begin{abstract}
On the analogy with the Lamb shift, we study the vacuum effects that a proton 
interacts with virtual particles when it travels through the vacuum. We find 
that a moving proton is accelerated by gaining the zero-point energy from the vacuum 
($\sim 10^{-5}$ eV/cm).  
Such an effect possibly accounts for the mysterious origin and spectrum of ultra 
high-energy cosmic ray events above $10^{20}$eV, and explains the puzzle why 
the GZK cutoff is absent. The candidates of these events could be protons 
from early Universe. 
\end{abstract}

\pacs{12.20ds, 12.20fv, 98.70.S}

\maketitle


\vskip0.2cm
\noindent{\it Introduction.}\hskip0.3cm 
Since the first event of ultra high energy cosmic ray (UHECR) at $10^{20}$ 
eV was detected in 1962 \cite{lin63}, more than ten statistically relevant events 
above $10^{20}$ eV have been observed. The distribution of these events is isotropic and 
no any astrophysical sources are identified. The energy density of UHECR's particles is 
about $10^{-21}$erg/cm$^3$, which is 8 orders of magnitude less than CMB's. The energy
spectrum is a power-law $\Phi(E)\sim E^{-\gamma}$ with $1\leq\gamma\leq 2$. 
Apart from the mystery of 
energetic origin and no 
sources identified, these events evade the GZK cutoff\cite{gre66,zat66}, attributed 
to the energy threshold $\sim 10^{20}$eV of pion productions by UHECR particles colliding 
with CMB photons.

Although the phenomenon of ultra high energy cosmic rays has not been completely 
understood yet, there are many interesting ideas and models trying to explain the origin and 
characteristics of 
UHECR events\cite{nag00,bs,watson,olinto,ls,ellis}. These ideas and models can be 
classified into two 
major categories: astrophysical scenario and particle physics scenario. In the astrophysical 
scenario (bottom-up model), acceleration of UHECR occurs when large-scale macroscopic motion, 
such as shocks and turbulent flow, is transferred to individual particles. In the 
particle physics scenario (top-down model), UHECR events are possibly 
originated from 
the decays of super massive particles, topological defects and the quantum gravity 
effect. The absence of the GZK cutoff can be explained by near sources ($<50$Mpc) 
and the violation of the Lorentz invariance\cite{bs,ellis,lor}. 

On the basis of the quantum electromagnetic dynamics (QED) and analogy of the Lamb shift, we 
study the vacuum effects that a charged proton interacts with virtual particles when it travels 
through the vacuum. We find that protons gain 
the zero-point energy ($\sim 10^{-5}$eV/cm), while they are traveling through the vacuum. 
Such vacuum effects possibly provide an alternative scenario explaining the 
origin, spectrum and energy-density of UHECR events and the absence of the GZK cutoff.   

\vskip0.2cm
\noindent{\it The Lamb shift.}\hskip0.3cm
The vacuum has a very rich physical content in the context of relativistic 
quantum field theories. It consists of extremely large number of virtual 
particles and anti-particles. Pairs of virtual fermions and anti-fermions are created and 
annihilated by quantum-field fluctuations of virtual photons.  
These pairs can be considered as unstable excitations of bound states of virtual fermions 
and antifermions with a very short 
life-time.
In the case of no external field, the quantum-field fluctuations of these pairs and 
virtual photons are totally random in the spacetime. This contributes to the zero-point 
energy of the vacuum. In the case of external field, the quantum-field fluctuations 
of these pairs and virtual photons are impacted by the direction of external 
field, the vacuum behaves as a dielectric medium. 

The Lamb shift\cite{lamb} shows that the degeneracy of the energy levels $2S_{1\over2}$ and 
$2P_{1\over2}$ in the spectrum of 
hydrogen atom obtained in the relativistic quantum mechanics is split. 
At a first and quick glimpse, this phenomenon could be easily explained by 
the vacuum polarization (charge screening) of quantum field theory QED.  
We would expect that the energy level $2S_{1\over2}$ should be shifted 
{\it downward}, comparing with the energy level $2P_{1\over2}$, 
for the reasons that 
an electron in the bound state $2S_{1\over2}$ is closer to the proton and feels less
charge screening than that in the bound state $2P_{1\over2}$, so that the 
electron in the state $2S_{1\over2}$ must gain larger negative Coulomb energy and 
its energy level must be shifted {\it downward} more than that in the 
state $2P_{1\over2}$. Indeed, if we take into account only the charge
screening effect of the vacuum polarization, the energy level of the state
$2S_{1\over2}$ is shifted {\it downward} $-27$MHz, compared with that of $2P_{1\over2}$.
 
While, the experimental measurement shows that the energy level $2S_{1\over2}$ is 
shifted {\it upward} $+1008$MHz, compared with $2P_{1\over2}$. This proves that other effects
of the QED vacuum largely overcompensate the vacuum polarization effect. The contributions of 
these vacuum effects to the $2S_{1\over2}-2P_{1\over2}$ splitting are accurately computed in the 
QED theory, precisely in accordance with the experimental measurement.
How these vacuum effects drain the zero-point energy to the system of proton and electron, 
bound by the Coulomb interaction. This is what we wish to understand together with the 
origin of UHECR events.

\vskip0.2cm
\noindent{\it The effective Lagrangian for a proton.}\hskip0.3cm
Considering a proton traveling through the vacuum and interacting with virtual particles, 
we introduce (i) $\Psi$ and $A_\mu$ describing a proton field and its electromagnetic potential;
(ii) $\psi_q$ and $A^q_\mu$ describing the quantum fields of virtual fermions and 
photons in the vacuum. 
To study this system, we start with the renormalized Lagrangian 
density ${\cal L}(x)$ and all necessary counterterms $({\rm c.t.})$,
\begin{eqnarray}
{\cal L}(x)&=& -{1\over4}F^2-{1\over4}F^2_q +\bar\Psi(i\gamma^\mu\partial_\mu - m_p +e
\gamma^\mu A^q_\mu)\Psi\nonumber\\
&\!+&\!\bar\psi_q\big[i\gamma^\mu\partial_\mu - m 
-e\gamma^\mu (A_\mu+ A^q_\mu)\big]\psi_q\!+\!({\rm c.t.}),
\label{l}
\end{eqnarray}
where $F, F_q$ are electromagnetic field tensors, $e$ and $m$ are electron's charge and mass,
$m_p$ the proton mass. This is a complex interacting system, the proton field $\Psi$, 
its electromagnetic field $A_\mu$, quantum fields of virtual fermions $\psi_q$ and photons 
$A^q_\mu$ in the vacuum are coupled together. 
In the absence of interactions with the quantum fields $\psi_q$ and $A^q_\mu$ 
in the vacuum, the equations of motion 
for the proton field $\Psi$ and its electromagnetic potential $A_\mu$ are given by:
\begin{eqnarray}
(i\gamma^\mu\partial_\mu - m_p)\Psi &=&0,\hskip0.3cm E= \sqrt{p^2+m_p^2};\nonumber\\
\Delta A_\mu &=&j_\mu,\hskip0.2cm j_\mu=-e\bar\Psi\gamma_\mu\Psi
\label{cl}
\end{eqnarray}
where the Lorentz gauge $\partial_\mu A^\mu=0$ is adopted. Whereas, in the presence of  
interactions with the quantum fields $\psi_q$ and $A^q_\mu$ in the vacuum, 
the proton field $\Psi$ and its electromagnetic 
potential $A_\mu$ are described by the effective action/Lagrangian:
\begin{equation}
iS_{\rm eff}=i\int d^4x {\cal L}_{\rm eff}(x)=\ln \int{\cal D}\psi_q{\cal D}
A^q_\mu e^{i\int d^4x {\cal L}(x)},
\label{q}
\end{equation}
where quantum field fluctuations of virtual fermions $\psi_q$ and photons 
$A^q_\mu$ are path-integrated. 

We compute the effective Lagrangian ${\cal L}_{\rm eff}(x)$ (\ref{q}) by the 
perturbation in terms of the 
electromagnetic coupling, and  obtain ($F^2$ term is dropped),
\begin{equation}
{\cal L}_{\rm eff}(x)=\bar\Psi(i\gamma^\mu\partial_\mu - m_p)\Psi\nonumber\\
\!+\!tr\ln\big[S_F^{-1}(x)-V(x)\big].
\label{effl}
\end{equation}
In Eq.(\ref{effl}), $V(x)$ is computed up to the first order of proton's gauge 
field $A_\mu$ and the 
second order of the quantum field $A^q_\mu$\cite{book},
\begin{equation} 
V(x)=e\gamma_\mu \big[1\!-\!
{\alpha\over3\pi m^2}\big(\ln{m\over\mu}\!-\!{7\over40}\big)\Delta\!+\!
{\alpha\over2\pi}{\sigma_{\nu\mu}\over2 m}\partial^\nu \big]A^\mu ,
\label{v}
\end{equation}
where the infrared cutoff $\mu$ will be treated properly. Whereas, $S_F(x)$ is 
computed to the zeroth order of proton's gauge field $A_\mu$ and all orders 
of quantum fields fluctuations $A^q_\mu$ the vacuum:
\begin{equation}
iS_F(x)=\int{\cal D}\psi_q{\cal D}A^q_\mu 
\psi_q(x)\bar\psi_q(0) e^{i\int d^4x {\cal L}_q(x)},
\label{pair}
\end{equation}
where ${\cal L}_q(x)$ is Eq.(\ref{l}) containing only quantum fields $\psi_q$ and $A^q_\mu$.

We are interested in the effective Lagrangian ${\cal L}_{\rm eff}(x)$ that is
bilinear in the proton field $\Psi$. On the basis of Eqs.(\ref{l}-\ref{pair}), up to
the leading order, straightforward computations lead us to
\begin{equation}
{\cal L}_{\rm eff}(x)= \bar\Psi(i\gamma^\mu\partial_\mu - m_p)\Psi + a_\mu 
\bar\Psi\gamma^\mu\Psi,
\label{proton}
\end{equation}
where
\begin{equation}
a_\mu={4\alpha^2\over3 m^2}\big(\ln{m\over\mu}
\!-\!{7\over40}\big)Tr\big[\gamma_\mu S_F(0)\big].
\label{efflp}
\end{equation}
In Eq.(\ref{proton}), the additional term $a_\mu \bar\Psi\gamma^\mu\Psi$ is in fact an 
interacting vertex of the 
proton and virtual particles, later are averaged over quantum-field fluctuations.

\vskip0.2cm
\noindent{\it Virtual pairs and energy gain.}\hskip0.3cm
As defined in Eq.(\ref{pair}), $S_F(0)$ represents close fermion loops of all possible 
one-particle irreducible functions in the representation of Feynman diagrams. 
By analyzing these Feynman diagrams contributing to $Tr\big[\gamma_\mu S_F(0)\big]$, 
we obtain the leading contribution $\sim\alpha^3m^3$.

In general, we can express $S_F(0)$ in terms of $(v^\circ,v_\mu^1,v_{\mu\nu}^2,v_\mu^3,v^4)$
on the basis $(I,\gamma^\mu,\sigma^{\mu\nu},\gamma^\mu\gamma_5,\gamma_5)$ of the Dirac spinor 
space. $Tr\big[\gamma_\mu S_F(0)\big]=4v^1_\mu$. We attempt to 
make an estimation of the leading contribution to $S_F(0)$ ($v^1_\mu$) and discuss its 
physical content, rather than giving a precise calculation. In fact, 
$S_F(0)$ describes the density of the creations and annihilations of virtual 
fermion and antifermion pairs in the vacuum, which we call {\it virtual pairs}.  
We consider these virtual pairs as unstable excitations of bound states of virtual fermions 
and antifermions. We approximately estimate their binding energy, size and life-time. 
The energy scale of quantum-field fluctuations of virtual fermions and antifermions 
must be much smaller than the electron mass $m$, otherwise real electrons and positrons 
would be created. We thus adopt a non-relativistic description for these virtual pairs. 
Their size is about ${1\over\alpha m}$, binding energy $\sim\alpha^2 m$. 
Given the size of these virtual pairs, we can estimate that the spatial 
density of these virtual pairs in the vacuum is about ${3\over4\pi}\alpha^3 m^3$.    
We approximately have $v^1_\mu={\alpha^3m^3\over4\pi}(1,-\vec v)$, where $\vec v$ is the 
velocity of the proton. 

The cross-section (probability) of the annihilation and creation of such a virtual pair
is about $\pi({\alpha\over m})^2$. The life time of such a virtual pair is 
then $\delta\tau\sim {1\over\alpha^5 m}$, which is a rather long time comparing with the 
scale of its binding energy. This shows that these virtual pairs have a life-time
$\delta\tau \sim 3.1\cdot 10^{-11}$sec., living as electric dipoles 
and interacting with the external electric field of the proton. The large wavelength 
modes of proton's gauge field $A_\mu$ are sensitive to the low-lying states of 
virtual pairs. This suggests the infrared cutoff of the order $\mu\simeq\alpha m$. 
 
The effective Lagrangian (\ref{proton},\ref{efflp}) and estimate value of 
$Tr\big[\gamma_\mu S_F(0)\big]$ determine the modified dispersion relation of the proton, 
\begin{eqnarray}
E+\delta\epsilon &=& \sqrt{(\vec p+\delta \vec p)^2+m_p^2},\nonumber\\
\delta\epsilon &\!=\!&{4\alpha^5\over3\pi}m\big(\ln{m\over\mu}
\!-\!{7\over40}\big)\simeq 2.1\cdot 10^{-5}{\rm eV},
\label{pe}
\end{eqnarray}
where $\delta \vec p=\vec v \delta\epsilon$ is the momentum gain.
This shows that the proton gains the zero-point energy from the vacuum. 
We turn to the discussions why such an energy-gain process for the proton 
must be achieved.
  
\vskip0.2cm
\noindent{\it Discussions.}\hskip0.3cm
In the absence of any external field, the quantum-field fluctuations 
of virtual photons and virtual pairs are entirely random in the spacetime. This determines 
the maximum value of the zero-point energy. However, in the presence of an external field that
couples to virtual pairs,
the quantum-field fluctuations of virtual pairs and photons are re-oriented towards the 
direction of the external field, so that the zero-point energy is reduced. 
And the difference of the zero-point energies in the presence and absence of the 
external field must drain to the system of the external field and particle as a recoiling effect. 
At a spatial point, the re-orientation of quantum-field fluctuations of virtual pairs towards external 
field's direction and the variation of the zero-point energy take place during life-time 
$\delta\tau\sim {1\over\alpha^5 m}= 3.1\cdot 10^{-11}$sec.~ of these virtual pairs. The zero-point 
energy variation (reducing/draining) $\delta\epsilon\sim\alpha^5 m= 1.04\cdot 10^{-5}$eV, given 
by the Heisenberg uncertainty relationship. 

At the first, we discuss these vacuum effects leading to the positive (upward) Lamb 
shift of a hydrogen atom. The electric field, which binds a proton and an electron, interacts with 
quantum-field fluctuations of virtual pairs and photons. Those quantum-field fluctuations exposed 
to the electric field align along the direction of the electric field, so that the zero-point
energy becomes smaller and drains to the bound state of the proton and electron. This happens in 
particular for the quantum-field fluctuations near to the proton, where the electric field is the 
strongest. The electric field strength is larger, 
the more zero-point energy is reduced and drains to the bound state of proton and electron.
As a result, the proton-electron bound state gains this drain of the zero-point 
energy from the vacuum, giving rise to the splitting of energy levels $2S_{1\over2}$ and $2P_{1\over2}$. 
This energy splitting is indeed proportional to $\alpha^5 m$. 
This in turn justifies that the life time of quantum-field fluctuations of virtual pairs and photons 
in the vacuum is $O({1\over\alpha^5 m})$.

The total amount of the zero-point energy drain to the bound 
state of the proton and electron is limited. The reason is that 
the size of the neutral bound state of proton and electron 
is finite, so that the number of degree of freedoms of quantum-field fluctuations 
exposed to the external field is limited. 
The maximum of the zero-point energy drain is achieved, when all of these virtual 
pairs, as dipoles in the electric field, have orientated towards the direction of the external field.
Although the proton-electron bound state gains the zero-point energy 
from the vacuum to be in a higher energy state, the vacuum reduces its zero-point energy. As a result, 
the coupling system of the vacuum and the proton-electron bound state is in 
the lower energy state and thus such a process of the zero-point energy drain is energetically
favourable. 

We turn to the case of a proton passing through the vacuum. The coupling system 
composes of the vacuum and the proton in motion. As the proton arrives at a point ($x,t$) 
of the spacetime, quantum-field fluctuations of virtual pairs and photons at that point oriente towards
the direction of proton's electric field. Analogous to the vacuum effects for the Lamb shift, 
the zero-point energy becomes smaller than that before the proton arrival and the difference drains 
to the system of the proton and its electromagnetic field.  
As the proton passes through the vacuum from one spacetime point to another, more and more quantum-field 
fluctuation modes of virtual photons and pairs are involved and re-oriented towards the direction of 
proton's electric field. 
As a consequence, more and more the zero-point energy drains into the proton, 
as if the proton gets a continuous recoil from the vacuum when it travels from one spacetime point to another. 
This indicates that the proton gains energy and is accelerated while it is traveling, as shown in 
Eqs.(\ref{proton},\ref{efflp},\ref{pe}).

Although the proton gains the zero-point energy while it is traveling through the vacuum, 
whereas the vacuum reduces its zero-point energy so that the whole interacting system of the vacuum and the 
moving proton minimizes its interacting energy. This shows that it is energetically favourable
for the proton gaining the zero-point energy while it traveling 
through the vacuum from one spacetime point to another, as if it rolls 
down along a potential with a very small slop. However,
this potential is not macroscopic, created by an external field. Instead, it 
is microscopic and local at the point $x$, attributed to the fact that the proton arrives 
at the point $x$ and interacts with the quantum-field fluctuations of the vacuum at the point 
$x$, so that the zero-point energy of the vacuum is reduced. As a consequence, the 
interacting system of the vacuum and proton must undergo the process that the proton keeps 
in gaining the zero-point energy and moving forwards. Such a driving force is rather 
analogous to the Casimir force.  
 
As discussed, the zero-point energy varitation $\delta\epsilon\simeq 2.1\cdot 10^{-5}{\rm eV}$ 
(\ref{pe}) in the life time of virtual pairs $\delta\tau\simeq 3.1\cdot 10^{-11}$sec.~ and  
the corresponding distance $\delta x\simeq 0.93$cm. 
Thus, the rate of a proton gaining the zero-point energy $\delta E$ from the vacuum is:
\begin{equation}
\delta E = \delta \vec p \cdot \vec v ;\hskip0.3cm
{\delta E\over\delta x}= {\delta p\over\delta \tau}\simeq 2.25\cdot 10^{-5}
{v\over c}({\rm eV/cm}),
\label{rate}
\end{equation}
where straight-line motion is assumed in the second equation. A proton 
travels distance $D$, it gains 
the total kinetic energy $E$: 
\begin{equation}
E=m_pc^2\left[\sqrt{1+\left({2.25\cdot 10^{-5}\over m_pc^2}D+\beta\right)^2}-1\right],
\label{total}
\end{equation}
where $\beta^2={E_\circ\over m_pc^2}({E_\circ\over m_pc^2}+2)$ and $E_\circ$ 
is proton's initial kinetic energy.
It is worthwhile to see if such an effect could 
experimentally be tested in a ground laboratory. An analogous study for 
neutrinos is in progress.
 
\vskip0.2cm
\noindent{\it UHECRs.}\hskip0.3cm
For $D\gg 10^{14}$cm, Eq.(\ref{total}) shows $E\simeq 2.25\cdot 10^{-5}D$(eV/cm). 
With the present size of the Universe $\sim 10^{28}$cm, protons can reach 
the energy more than $10^{21}$eV, if they travel a distance of $10^{27}-10^{28}$cm 
without energy loss. In this scenario, {\it primary protons}, 
the candidates of UHECR events, 
could be originated from the astrophysical sources of large redshift 
$z$, like Quasars, or from the early Universe, and no particular arrival 
direction can be identified. 

The GZK cutoff does not apply to such a process of protons gaining 
energy bit by bit on their way to us. The reason is that protons, beyond $\sim 50$Mpc from us,
has an energy much smaller than the energy threshold $10^{20}$eV. This explains the absence of
the GZK cutoff in UHECR events. However, when the proton near us reaches the 
energy $10^{21}$eV, the GZK effect acts and average energy loss $\sim 10^{-5}$eV/cm\cite{aloss}, 
which is the same order of the energy gain (\ref{rate}). This implies that ultra high energy 
protons would not have much chance to exceed the energy $10^{22}$eV. 

On the other hand, since a traveling proton is accelerated bit by bit, 
though acceleration is very small, we have to 
estimate the energy loss in the process of the Bremsstrahlung radiation:
\begin{equation}
{\delta E_{loss}\over\delta x}= {2e^2\over 3 m_p^2}\left({\delta E\over\delta x}\right)^2\simeq
1.88\cdot 10^{-24}\left({\delta E\over\delta x}\right),
\label{lrate}
\end{equation}
which is much smaller than the proton energy gain and thus can be neglected. The energy
loss due to the pair production and synchrotron radiation effects can be neglected in the proton
propagation\cite{bs,ls}.
 
We set the origin of a spherical coordinate at the center of a primary proton's source, whose size
is $R_\circ$, the number density $n_\circ$ and mean outgoing velocity $v_\circ$. 
The total flux out of the source is  $4\pi R_\circ^2 n_\circ v_\circ$.  
The Earth is located at $R$ distance away from the source. The total flux passing through 
the spherical surface $4\pi R^2$ is $4\pi R^2 n v$, where $n$ is the 
number-density of UHECR protons and $v$ the mean velocity. We have the conservation of 
total numbers of UHECR protons:
\begin{equation}
4\pi R^2n v= 4\pi R_\circ^2n_\circ v_\circ (1+z)^{-3},
\label{conf}
\end{equation}
where the factor $(1+z)^{-3}$ is due to the effect of expanding Universe. Thus we obtain 
the flux of UHECRs measured on the Earth,
\begin{equation}
\Phi=4\pi R_\circ^2 n_\circ v_\circ (1+z)^{-3}{1\over4\pi R^2}\sim {1\over R^2}.
\label{fluxe}
\end{equation}
Due to the distribution of intergalactic magnetic field and/or galactic wind etc, 
protons normally travel in a zigzag way with a mean-free path 
$\lambda_p$. The distance $D$ that protons travel is larger than $R$, $D>R$. 
In one extreme case, protons travel to us in a straightforward line, $D=R$. 
While in another extreme case, protons travel in a way of random 
walk, $D={R\over\lambda_p}R$. This gives rise to the spectrum of UHECR flux observed on the Earth: 
\begin{equation}
\Phi(E)\sim {1\over R^2}\sim {1\over E^\gamma}\hskip0.5cm 1\le\gamma\le 2,
\label{spec}
\end{equation}
where $\gamma=2$ is for proton traveling in a straightforward line ($\lambda_p=R$) and 
$\gamma=1$ in random walk.
   
Let us postulate that these primary protons are originated from early Universe, possibly
inbetween the baryogenesis epoch and the last scattering epoch. Using the variation ranges of
the redshift $z$, temperature $T_\circ$, mean velocity of protons $v_\circ$, 
the number-density of protons $n_\circ$ and the ratio $R_\circ/R$ of the horizon sizes of today 
and early Universes, as well as the ionization rate, we
find that Eq.(\ref{fluxe}) gives the variation range of the flux and energy density of UHECR protons 
to be consistent with observation data. Yet, many properties of such a scenario 
need to be analyzed to see if they are in accordance with all observation data of ultra high 
energy cosmic rays.    
\vskip0.1cm
\noindent{\it Acknowledgments.}\hskip0.3cm 
I thanks Pascal Chardonnet for many discussions on the phenomenon of 
ultra high energy cosmic rays.    
 
        
\begin{thebibliography}{99}

\bibitem{lin63} 
J.~Linsley, Phys.~Rev.~Lett.~10 (1963) 146.

\bibitem{gre66} 
K.~Greisen, Phys.~Rev.~Lett.~16 (1966) 748.

\bibitem{zat66} G.T.~Zatsepin and  V.A.~Kuzmin, JETP Lett.~4 (1966) 78.

\bibitem{nag00} M.~Nagano and A.A.~Watson, Rev.  
Mod.~Phys.~72 (2000) 689, references therein.

\bibitem{bs}
P.~Bhattacharjee and G\"unter Sigl, Phys.~Rep.~327 (2000) 109-247, 
references therein.

\bibitem{watson}
A.~A.~Watson, Phys.~Rep.~333-334 (2000) 309-327, references therein.

\bibitem{olinto}
A.~V.~Olinto, Phys.~Rep.~333-334 (2000) 329-348, references therein.

\bibitem{ls}
A.~Letessier-Selvon,  references therein.

\bibitem{ellis}
J.~Ellis, Nuovo Cim. 24C (2001) 483-496, references therein.

\bibitem{lor}
S.~Coleman and S.~Glashow, Phys.~Rev.~D59 (1999) 116008

\bibitem{lamb}
W.~E.~Lamb and R.~C.~Retherford, Phys.~Rev.~vol.~72 (1947) 241. 

\bibitem{book}
C.~Itzykson and J.-B.~Zuber, ``Quantum Field Theory'' 1980 McGraw-Hill Inc.

\bibitem{aloss}
J.W.~Cronin, Nucl.~Phys.~(Proc.~Suppl.) B28, 213 (1992).


\end{thebibliography}


\newpage

%\begin{figure}
%\vspace{5cm}
%\caption{The rate of energy release $d {\cal E}/d t$ (erg/sec) as a 
%function of the radius $R$
%in unit of $2M$.} 
%\label{rate}
%\end{figure}

%\begin{figure}
%\vspace{5cm}
%\caption{The spatial density of energy release (erg/cm$^3$) as a 
%function of radius $r$ (cm).}
%\label{density}
%\end{figure}
%\bibitem{grb} T.~Piran, Phys.~Rep.{\bf 314}, 575 (1999).
%The redshift $10^3\le z\le 10^{13}$,
%the temperature 1eV$\le T_\circ\le$1GeV, the mean velocity of protons 
%$10^{-5}\le {v_\circ\over c}\le 10^{-1}$, the number-density of protons 
%$10^3/cm^3\le n_\circ\le 10^{33}/cm^3$
%and the ratio of the horizon sizes of today and early Universes 
%$10^{-5}\le R_\circ/R\le 10^{-23}$.
%The ionization rate is $\sim 10^{-3}$.
\end{document}
