%Paper: 
%From: rubakov@inucres.msk.su (Valery Rubakov)
%Date: Wed, 12 Jan 1994 18:50:03 +0300

\documentstyle[12pt]{article}
\renewcommand{\baselinestretch} {1.20}
\topmargin -27pt
\textwidth   6in
\textheight  8.5in
\newcommand{\bra}[1]{\langle #1 |}
\newcommand{\ket}[1]{| #1 \rangle}
\newcommand{\sprod}[2]{\langle #1 , #2 \rangle}
\newcommand{\braket}[2]{\langle #1 | #2 \rangle}
\newcommand{\eq}[1]{eq.(\ref{#1})}
\newcommand{\dpar}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\vpar}[2]{\frac{\delta #1}{\delta #2}}
\newcommand{\ddpar}[2]{\frac{\partial^2 #1}{\partial #2^2}}
\newcommand{\vvpar}[2]{\frac{\delta^2 #1}{\delta #2^2}}
\newcommand{\bm}[1]{\mbox{\boldmath $#1$}}
\def\ltap{\raisebox{-.55ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}}
\def\gtap{\raisebox{-.55ex}{\rlap{$\sim$}} \raisebox{.4ex}{$>$}}
\def\gsim{\mathrel{\gtap}}
\def\lsim{\mathrel{\ltap}}
\def\ghost#1{\vrule height#1 depth#1 width0pt \displaystyle}
\def\const{\mbox{const}}
\def\e{\mbox{e}}
\def\ch{\mbox{ch}}
\def\sh{\mbox{sh}}
\def\arccosh{\mbox{arccosh}}
\def\half{{1 \over 2}}
%\def\theequation{\thesection.\arabic{equation}}
%addtoreset{equation}{section}

\begin{document}
\title{Instanton-like transitions at high energies in (1+1) dimensional
scalar models. II. Classically allowed induced vacuum decay.}
\author{
  V.~A.~Rubakov and D.~T.~Son\\
  {\small \em Institute for Nuclear Research of the Russian Academy of
  Sciences,}\\
  {\small \em 60th October Anniversary prospect 7a, Moscow 117312}\\
  }
\date{January 1994}

\maketitle
\begin{abstract}
  We consider classical Minkowskian solutions
 to the field equation in the (1+1)
dimensional  scalar model with the exponential interaction
that describe the unsuppressed false vacuum decay induced by $n$
initial particles. We find that there is a critical value of $n$ below
which there are no such solutions, i.e.,
 the vacuum decay is always suppressed. For the number of initial
particles larger than this value the vacuum decay is unsuppressed at high
enough energies.
\end{abstract}


\newpage

\section{Introduction}

Instanton--like processes induced by collisions of
highly energetic particles remain an
object of controversial discussion (for reviews see, e.g.,
\cite{Mrev,Trev}). One particular type of these processes is the induced
decay of the false vacuum in scalar theories
\cite{Affleck,VolSel,Vol,RSTind,Kisel,GorVol,Rutgers}. Being exponentially
suppressed at zero energy by the action of a bounce \cite{Coleman,CalCol},
the rate exponentially increases with the energy of incoming particles,
 and may or may not become large at high enough energies.
Similar behavior is inherent in the baryon number violating transitions in
the electroweak theory and analogous processes in other models with
instantons \cite{Ringwald,Espinosa}.

Although the two--particle initial states are of primary interest in this
problem, it has been argued that it is instructive to study the
multiparticle initial states and then consider the limit of small number of
initial particles \cite{RT,T,RST,Mueller}. If the
number of initial particles
is of order $1/g^2$ , where $g$ is small dimensionless coupling constant,
the induced instantion--like transition rate is calculable, at least
 in principle, in a semiclassical way. The corresponding classical
 solution is smooth on a particular contour in complex time plane and
 is determined by boundary conditions specifying the number of particles,
 as discussed in ref.\cite{RST}.

The classical boundary value problem relevant to the false vacuum decay
induced by multiparticle ''collisions'' has been studied in
our previous paper \cite{Rutgers} in
(1+1) dimensional theory with the Lagrangian
\begin{equation}
  L = {1\over 2}(\partial_{\mu}\phi)^2 - V(\phi),
  \label{Lagrangian}
\end{equation}
where the potential $V(\phi)$ has the following form,
\[
  V(\phi) =    {m^2\over 2}\phi^2 -
      {m^2v^2 \over 2}\exp\left[2\lambda\left({\phi\over v}-1\right)\right].
\]
 The dimensionless parameter $v$ plays the role of the inverse coupling
constant, while $\lambda$ is an additional dimensionless parameter of the
theory. For the semiclassical calculations to be reliable, one
requires $v\gg\lambda$, while the actual classical solutions can be
explicitly found at $\lambda\gg 1$. Thus, the convenient regime is
\[
  v\gg\lambda\gg 1.
\]
In this regime, the potential $V(\phi)$
is quadratic at $\phi<v$ and has steep cliff at
$\phi>v$, as shown in fig.1;
the false vacuum is $\phi=0$, while the true vacuum is $\phi=+\infty$.

When the energy of the initial state $E$ is roughly of order $mv^2$ (but
not too high), and the number of incoming particles is of order $v^2$, the
rate of the induced decay is suppressed by a semiclassical factor
\[
  \Gamma(E,n)\propto\e^{-F(E,n)}
\]
where
\[
  F(E,n)=v^2\tilde{F}\left({E\over E_{sph}},{n\over n_{sph}},\lambda\right)
\]
and
\[
  E_{sph}=mv^2
\]
\[
  n_{sph}={2\over\pi}v^2
\]
are the energy and number of particles for the sphaleron (critical bubble)
in this model.

At $n/n_{sph}\gg\lambda^{-1}$, the suppression factor has been calculated
\cite{Rutgers} in a wide interval of energies. It has been argued that
there exists some critical energy $E_{crit}(n)$ at which $F$ becomes equal
to zero, i.e., the exponential suppression disappears. The dependence
of the critial energy on the number of initial particles is particularly
simple at $1\gg n/n_{sph}\gg\lambda^{-1}$,
\begin{equation}
  E_{crit}={4\over\pi^3}\exp\left({\pi^2\over 4}{n_{sph}\over n}-1\right)
           \left({n\over n_{sph}}\right)^2E_{sph}
\label{3*}
\end{equation}
Notice that $E_{crit}$ is much larger than the sphaleron energy.

It remained unclear whether the exponential suppression of the induced
vacuum decay is exponentially suppressed or not at any energies for even
smaller number of incoming particles, $n/n_{sph}~\ltap~\lambda^{-1}$.
For these $n$, it has
been found only that  the exponent $F$ is substantially
different from its vacuum value  at exponentially high energies,
\begin{equation}
  \ln{E\over E_{sph}}\sim\lambda
\label{3**}
\end{equation}
Eq.(\ref{3**}) sets roughly the energy scale of interest in the case
$n/n_{sph}~\ltap~\lambda^{-1}$.

Coming back to the case $n/n_{sph}\gg\lambda^{-1}$, the absence
 of the exponential suppression at $E>E_{crit}(n)$ signalizes
 that there
exist {\em classical} transitions from an initial state with $n$ particles
above the false vacuum, to the true vacuum. These transitions are described
by real Minkowskian solutions to the field equations
(the real-valuedness of the solutions at large time is ensured by
the summation over all final states \cite{RST}; then the solutions
are obviously real in the whole Minkowski space-time). At $t\to-\infty$,
these solutions should be the collections of plane waves above
the false vacuum, $\phi=0$,
\[
  \phi(x,t)=\int{dk\over 2\pi}\left[\e^{ikx-i\omega_kt}f_k+
            \e^{-ikx+i\omega_kt}f^*_k\right]
\]
They correspond to initial coherent states with
the energy and number of incoming particles
\[
  E={1\over\pi}\int dk\omega_k^2f^*_kf_k
\]
\[
  n={1\over\pi}\int dk\omega_kf^*_kf_k
\]
For this correspondence between the classical solutions and quantum
 states be exact, one requires that $E$ and $n$ are of order $v^2$,
up to arbitrary dependence on $\lambda$. It is this regime that will
be considered throughout this paper.
To describe the induced decay of the
 false vacuum, the solution should have a
singularity $\phi\to+\infty$ at finite time, because the (homogeneous)
field rolling down the cliff of $V(\phi)$ reaches
the "true vacuum" $\phi=\infty$ in finite
time.

At large enough, but fixed energy and number of initial particles,
there may exist a variety of initial coherent states that
induce the false vacuum decay without suppression.
 In other words, there may exist a variety
of the corresponding classical solutions. So, finding these solutions
 is not a well formulated mathematical problem, unlike the boundary
value problem, with expectedly unique solution at given $E$ and $n$,
relevant to the suppressed induced vacuum decay at relatively low
 energies and number of incoming particles. In reality this means that
 one searches for (Minkowskian) solutions by making use of a certain
Ansatz. The existence of a solution at given $E$ and $n$ within this
Ansatz guarantees that the false vacuum decay is classically allowed
at these $E$ and $n$, but the absence of such a solution within the
Ansatz tells essentially nothing. We will present one useful Ansatz
in the model (\ref{Lagrangian}), as well as the corresponding set of
explicit solutions, in sect.2.2. We will see that our solutions exist at
$n>\pi^2n_{sph}/\lambda$ and high enough energies, and, furthermore,
these solutions at $1\gg n/n_{sph}\gg\lambda^{-1}$ appear for
the first time at energies that coincide, up to a pre-exponential
numerical factor, with the critical energy, eq.(\ref{3*}). The latter
 result, combined with our previous results \cite{Rutgers}, gives a
 coherent picture of the induced vacuum decay at
 $1\gg n/n_{sph}\gg\lambda^{-1}$: the decay rate
is exponentially suppressed
at $E<E_{crit}(n)$, grows towards $O(1)$ as $E$ increases
 towards $E_{crit}(n)$,  and the decay
proceeds classically at $E>E_{crit}(n)$.

The absence, within the Ansatz of sect.2.2, of the relevant Minkowskian
solutions at $n<\pi^2n_{sph}/\lambda$ may be at best viewed as a loose
indication that the induced vacuum decay is classically forbidden
 at all energies for such $n$. To see that this is indeed the case,
one has to find arguments independent of any particular Ansatz. We present
two different arguments in sects.3 and 4; although we beleive that
each of these arguments is sufficient by itself, we discuss both of
them to illustrate slightly different techniques that may be of use in more
complicated models.

In sect.3 we show that at any energy, no real Minkowskian solutions
describing the induced vacuum decay exist at $n<n_{crit}$ where
 $n_{crit}=\pi^2n_{sph}/\lambda$.

In sect.4 we consider, instead of the number of incoming particles,
another operator,
\[
\hat{A}=\int dk \frac{1}{\omega_{k}}a^{\dagger}_ka_k
\]
to label initial states. The limit of a few highly energetic particles
corresponds to the limit $A/v^{2}\to 0$, while we show in sect.4 that
at given energy $E$, the classical solutions describing the induced vacuum
decay exist only at $A/v^{2}>B(E)$, where $B(E)$ is strictly positive
at any $E$. This again means that the vacuum decay induced by collisions
of a small number of particles is exponentially suppressed at all
energies.

Sect.5 contains concluding remarks.



\section{Classical scattering at high energies}

\subsection{Construction of classical solutions}

The Minkowskian field equation for the model with the Lagrangian
(\ref{Lagrangian}) has the following form,
\begin{equation}
  \partial_{\mu}^2\phi=-m^2\phi+\lambda m^2v\exp\left[2\lambda\left(
                      {\phi\over v}-1\right)\right]
  \label{fieldeq}
\end{equation}
The general solution to eq.(\ref{fieldeq}) is unknown.
 However, at large
$\lambda$ and high enough energy per an incoming particle,
$E/n\gg m$, the solutions of interest can be found explicitly.
Indeed, the incoming particles are ultra-relativistic, so one can
neglect their masses near the interaction region where the field
is non-linear. Due to the large value of $\lambda$, the size of the
interaction region, $r_{0}$, is small, $r_{0}\ll m^{-1}$.
 Inside the interaction region
(i.e. at $x$, $t~\ltap~r_0$), the mass term in eq.(\ref{fieldeq}) can be
neglected, so the field equation is reduced to the Liouville equation,
\begin{equation}
  \partial_{\mu}^2\phi=\lambda m^2v\exp\left[2\lambda\left(
                      {\phi\over v}-1\right)\right],
  \label{Liouveq}
\end{equation}
whose solutions are known explicitly.
 In contrast, at large distances, i.e.
at $x$, $t~\gtap~m^{-1}$, only the mass term is essential on the right
hand side of
eq.(\ref{fieldeq}), so the field is  a solution to the massive
Klein--Gordon equation. At intermediate distances, both the mass term and
the exponential interaction term in eq.(\ref{fieldeq}) are inessential, so
$\phi$ obeys the free massless field equation, $\partial_{\mu}^2\phi=0$.
The way to find a solution to eq.(\ref{fieldeq})
 is to solve it
separately in the two regions of small and large distances, $x$,
$t~\ltap~r_0$ and\\
 $x$, $t~\gtap~m^{-1}$, and then match the two field
configurations at intermediate values\footnote{This
approach, with appropriate modifications, has been used in
 ref.\cite{Rutgers} to find interesting solutions to euclidean
field equations in our model; in particular, the instanton (bounce)
and sphaleron (critical bubble) have been obtained in this way to the
leading order in $\lambda$.}\\ of $x$, $t$.

Let us first write down the general solution to the Liouville equation,
\begin{equation}
  \phi(x,t)={v\over 2\lambda}\ln{f'(x+t)g'(x-t)\over\left[1+{1\over a^2}
            f(x+t)g(x-t)\right]^2}
  \label{Liouvsol}
\end{equation}
where $f$ and $g$ are arbitrary real functions of the light "cone"
coordinates $x+t$ and $x-t$, respectively, $f'$ and $g'$ denote derivatives
of $f$ and $g$ with respect to their variables, and
\[
a={2\e^{\lambda}\over\lambda m}
\]
Note that the right
hand side of eq.(\ref{Liouvsol}) is invariant under the
following transformation,
\[
  f\to {Af+B\over Cf+D}
\]
\[
  g\to a^2{Dg-Ca^2\over Aa^2-Bg}
\]
where $A$, $B$, $C$, $D$ are arbitrary real constants.
 We make use of
this ambiguity to impose the following conditions on $f$ and $g$,
\begin{equation}
  f(-\infty)=g(+\infty)=0.
  \label{limit}
\end{equation}
Finally, we assume that $g$ and $f$ are related in a simple way,
\begin{equation}
  g(z)=-f(-z),
\label{5*}
\end{equation}
so the field configuration $\phi(x,t)$ is invariant under spatial reflection
$x\to-x$, $\phi(x,t)=\phi(-x,t)$. In other words, we consider the
classical scattering of two identical, up to the reflection, initial
wave packets; this ensures, in particular, that the total
spatial momentum is
zero.

{}From eq.(\ref{Liouvsol}) it is clear that the final state of the field
configuration possesses the singularity,
$\phi=+\infty$, i.e., describes the transition to the true vacuum,
 provided there exist such $x$ and
$t$  that
\[
  f(x+t)g(x-t)=-a^2.
\]
Clearly, it is sufficient to require that the singlarity exists at $x=0$
at  some $t$, which means that we impose the
condition that,
\[
  f(t)g(-t)=-a^2.
\]
i.e.,
\begin{equation}
  f^{2}(t)=a^{2}
\label{6*}
\end{equation}
at some $t$.

Let us now match this solution to free massive field at negative $t$.
 Taking the formal limit $t\to-\infty$ in
eq.(\ref{Liouvsol}), and making use of the fact that the product\\
 $f(x+t)g(x-t)$ tends
to zero at all  $x$ due to eq.(\ref{limit}), one finds
that $\phi$ is a sum of two free waves moving along the light ``cones'',
\begin{equation}
  \phi(x,t)={v\over 2\lambda}(\ln f'(x+t)+\ln g'(x-t))
  \label{asympt}
\end{equation}
 This field configuration should be  matched to the massive linear
tail. Generally speaking,  one
 represents $\phi$ by the sum of plane waves,
\begin{equation}
  \phi(x,t)=\int dk\left(f_k\e^{ikx-i|k|t}+f^*_k\e^{-ikx+i|k|t}\right)
  \label{massless}
\end{equation}
The massive field configuration that is smoothly matched to the solution
to the Liouville equation can be obtained by replacing
$|k|$ by $\omega_k = (k^2 + m^2)^{1/2}$ in eq.(\ref{massless}),
\begin{equation}
  \phi(x,t)=\int dk\left(f_k\e^{ikx-i\omega_kt}+
            f^*_k\e^{-ikx+i\omega_kt}\right)
  \label{massive}
\end{equation}
In this way one obtains the field configuration both in the region of
non-linearity and in the asymptotics $t\to -\infty$. The conditions
(\ref{limit}) ensure that the field
at large negative time indeed describes linear excitations
above the false vacuum.

This procedure of obtaining the solutions to the field equations
is fairly general. The most non-trivial part of the problem is to
find a suitable function of one real argument, $f(z)$, which obeys
eqs.(\ref{limit}) and (\ref{6*}) and corresponds to a given number of
particles and energy.

\subsection{Examples}

Let us illustrate the above technique by the following choice of
the set of
functions $f$,
\begin{equation}
  f(z)=\int\limits_{-\infty}^{z}dz'{C\over(z'^2+\tau^2)^{\eta}}
  \label{ansatz}
\end{equation}
where $\eta$ and $\tau$ are real parameters and the constant
 $C$  will be defined below.
 At large negative time, eq.(\ref{asympt}) gives
\[
  \phi(x,t)={v\over 2\lambda}\left[-\eta\ln(x^2-(t-i\tau)^2)
            -\eta\ln(x^2-(t+i\tau)^2)+2\ln C\right]
\]
The solution to the massive Klein--Gordon equation that is
smoothly matched to this configuration is
\[
  \phi(x,t)={v\eta\over 2\lambda}\left[ K_0(m\sqrt{x^2-(t-i\tau)^2})
            + K_0(m\sqrt{x^2-(t+i\tau)^2})\right]
\]
provided the value of the constant $C$ is
\[
  C=\left({m\e^{\gamma}\over 2}\right)^{-2\eta}
\]
where $\gamma$ is the Euler constant. For the existence of singularities
ensuring the formation of the true vacuum, one requires (see eq.(\ref{6*}))
\[
  f(+\infty)=\int\limits^{+\infty}_{-\infty}dz'{C\over (z'^2+\tau^2)^{\eta}}
             > a = {2\e^{\lambda}\over\lambda m}
\]
This relation gives the following upper bound on the value of $\tau$,
\begin{equation}
  \tau < \tau_{crit} = \e^{-\lambda/2\eta-1}{2\over m\e^{\gamma}}
     \left[{\sqrt{\pi}\Gamma(\eta-1/2)\over\e^{\gamma}\Gamma(\eta)}\right]
     ^{1/(2\eta-1)}
  \label{taucrit}
\end{equation}

For calculating the energy and number of initial particles, one
evaluates the
Fourier components of $\phi$ in the initial asymptotics,
\[
  \phi(k) = {\eta v\over\lambda}{\pi\over\omega_k}\e^{-\omega_k\tau}
\]
The case of small enough $\tau$, such that
\[
  \ln{1\over m\tau}\gg 1
\]
 will be of special interest (note that this relation is valid for
$\tau \sim \tau_{crit}$). The number of particles and their energy
are then
\begin{equation}
  n = {1\over\pi}\int\limits^{+\infty}_{-\infty}\omega_k|\phi(k)|^2
    =  {2\pi\eta^2v^2\over\lambda^2}\left[\ln{1\over m\tau}
       -\gamma+\ln 2\right]
  \label{n}
\end{equation}
\begin{equation}
  E = {1\over\pi}\int\limits^{+\infty}_{-\infty}\omega_k^2|\phi(k)|^2
    =  {\pi\eta^2v^2\over\lambda^2}{1\over\tau}
  \label{E}
\end{equation}
Thus, the energy and number of incoming particles are expressed
through the two parameters of our solution, $\eta$ and $\tau$.
Alternatively, $\eta$ and $\tau$ may be viewed as functions of $E$
and $n$. Eq.(\ref{taucrit}) required for the solution to describe the
false vacuum decay, and not merely classical scattering above the
false vacuum, becomes then the constraint on $E$ and $n$, which we
now discuss in some detail.

The curve
\begin{equation}
  \tau(E,n) = \tau_{crit}(E,n)
\label{V9*}
\end{equation}
divides the $(E,n)$ plane into two parts, as shown in fig.2
($\tau_{crit}$ depends on $E$ and $n$ through $\eta(E,n)$). Above
this line (region II above the dashed line) eq.(\ref{taucrit}) is
valid, and our solution indeed describes the induced vacuum decay.

Let us consider first the case $n/n_{sph}\gg 1/\lambda$. It follows
from eqs.(\ref{n}) and (\ref{E}) that this case corresponds to
$\eta \gg 1$ (roughly speaking,
$\eta=O(\lambda)$).
Eqs.(\ref{n}),(\ref{E}) and (\ref{V9*}) determine the curve
parametrically, $\eta$ being the parameter along the curve,
\begin{equation}
  n = {\pi^2\eta\over 2\lambda}n_{sph}
\label{7*}
\end{equation}
\begin{equation}
  E = E_{sph}{\pi\eta^2\e^{\gamma}\over 2\lambda^2}
      \exp\left({\lambda\over 2\eta-1}\right)
\label{8*}
\end{equation}
It is clear from these equations that the critical energy at which
scattering leads to the vacuum decay increases as the number of
particles becomes smaller. The form of this curve is particularly
simple when $1\gg\ n/n_{sph}\gg \lambda^{-1}$: our
configuration describes vacuum decay when
\[
  E > E_{crit}(n) = {2\e^{\gamma}\over\pi^3}\left({n\over n_{sph}}\right)^2
                 \exp\left({\pi^2\over 4}{n_{sph}\over n}\right)E_{sph}
\]
Surprisingly enough, for a given number of initial particles $n$, the
minimal energy when the configuration describes the vacuum decay,
$E_{crit}(n)$, is larger than the one found in ref.\cite{Rutgers},
eq.(\ref{3*}), only by
a numerical factor  $e^{1+\gamma}/2$.

As discussed in sect.1, the critical energy found within the particular
Ansatz, eq.(\ref{ansatz}), might have been arbitrarily different from
the true critical energy. However, our solution gives the existence
proof that the induced vacuum decay is classically allowed at
$n/n_{sph}\gg\lambda^{-1}$ and sufficiently high energy.

Let us now consider our solution at $n/n_{sph}\sim \lambda^{-1}$.
This regime occurs when
 $\eta=O(1)$, and the critical curve has the following parametric form,
\[
  {n \over n_{sph}} = {\pi^2 \over \lambda} {\eta^2 \over 2\eta-1}
\]
\[
  {E \over E_{sph}} \sim \exp\left({\lambda \over 2\eta-1}\right)
\]
Note that in this case  $E$ is
exponential in $\lambda$, see fig.2.
Two properties of this curve are of particular interest. First, the
minimum value of $n$ is
\begin{equation}
n_{crit}={\pi^2\over\lambda}n_{sph}
\label{V13*}
\end{equation}
This means that the induced vacuum decay does not proceed
classically at any energy,
 within our Ansatz, at $n<n_{crit}$. We will see in
sect.3 that this property holds for all classical solutions, i.e.,
the vacuum decay is classically forbidden at any energy
 (independently of any
Ansatz) at  $n<n_{crit}$, where
  $n_{crit}$ is given by eq.(\ref{V13*}).

Second, at $\ln E/E_{sph}>\lambda$, the number of particles $n$ at
the curve increases with energy (dashed line in fig.2). This means
that at fixed $n$ greater than $n_{crit}$, our solution describes the
vacuum decay only in a finite interval of energies, i.e., the vacuum
decay appears  to be again suppressed at high energies. This
unexpected behavior is actually a peculiarity of our Ansatz: it is
clear that increasing the energy at fixed number of particles cannot
make the vacuum decay exponentially suppressed once it is classically
allowed at lower energies. Indeed, the initial state can spend some
fraction of its energy to radiate perturbatively a few highly
energetic particles, and the bulk of remaining particles would
classically induce the vacuum decay at effectively lower energy.
 Alternatively, the energy can be distributed among the initial
particles in such a way
 that  very few particles would carry most of the  energy while
the others have just the right amount of energy.
At the level of classical solutions the latter possibility
is realized by a trivial generalization of the Ansatz (\ref{ansatz}),
\[
  f(z) = \int\limits_{-\infty}^{z}dz'{C\over(z'^2+\tau_1^2)^{\eta_1}
         (z'^2+\tau_2^2)^{\eta_2}}
\]
where one assumes the following relations between the parameters $\tau_1$,
$\tau_2$, $\eta_1$, $\eta_2$,
\[
  \tau_1\ll\tau_2,~~~\eta_1\ll\eta_2
\]
\[
  \eta_1^2\ln{1\over m\tau_1} \ll \eta_2^2\ln{1\over m\tau_2},
\]
\[
  {\eta_1^2\over\tau_1} \gg {\eta_2^2\over\tau_2}.
\]
The singularity in the
 function $f'$ at $z=\pm i\tau_1$ corresponds to a few highly
energetic particles, while the other singularity, $z=\pm i\tau_2$,
 corresponds to
larger number of particles, each with much lower energy. The number of
particles and the energy of the initial state can be sraightforwardly
 calculated.
The number of particles depends on $\tau_2$ and $\eta_2$ only,
\[
  n = {2\pi v^2\eta_2^2\over\lambda^2}\ln{1\over m\tau_2}
\]
i.e. comes entirely from the soft component, while the expression for the
energy contains $\tau_1$ and $\eta_1$ only,
\[
  E = {\pi v^2\over\lambda^2}{\eta_1^2\over\tau_1}
\]
The condition $f(\infty)>a$, as one can easily see, puts a
restriction on $\tau_2$ and
$\eta_2$ only. Since there is no direct relationship between the energy and
the number of particles, these solutions describe the unsuppressed
induced vacuum decay at arbitrarily high energy for a given
number of particles (if $n>n_{crit}$). So, there is no suppression for
processes with $n>n_{crit}$ at very high energies and the actual
curve dividing the regions of the induced vacuum decay and classical
scattering in the false vacuum is the one shown in fig.2 by the solid
line.

\section{Minimum  number of incoming particles needed for inducing
vacuum decay}

The technique of sect.2.1 enables us to evaluate the minimum number
of incoming particles required for inducing the classically allowed
vacuum decay, no matter how high the energy is. Let us recall that right
before the collision,
the field is a sum of two wave packets moving along the two
light ``cones'',
\[
  \phi(x,t) = \phi_L(x+t) + \phi_R(x-t).
\]
where
\begin{equation}
  \phi_L(z) = {v\over 2\lambda}\ln f'(z),~~~
  \phi_R(z) = {v\over 2\lambda}\ln g'(z).
\label{10*}
\end{equation}
We consider parity symmetric initial states, eq.(\ref{5*}), so the
number of incoming particles is
\begin{equation}
  n = n_L + n_R =
       {2\over\pi}\int\limits_0^{\infty}dk\omega_k\phi_L(k)\phi_L(-k)
  \label{nphi}
\end{equation}
where
\[
  \phi_L(k) = \int dz\phi_L(z)\e^{-ikz}
\]
Our purpose is to minimize $n$ over all possible choices of $\phi_L$ (or
$f$). However, not all functions $f$ are possible: for the configuration to
describe the vacuum decay, one  requires that
\begin{equation}
  f(\infty) = \int\limits_{-\infty}^{\infty} dz f'(z)
            = \int\limits_{-\infty}^{\infty} dz\exp\left({2\lambda\over v}
              \phi_L(z)\right) > a
  \label{finf}
\end{equation}
Indeed, the incoming field (\ref{10*}) is non-singular only if
$f'(z)>0$ for any $z$. Therefore, if $f(\infty)<a$, then the basic
relation, eq.(\ref{6*}) is not valid at any time, and the solution
describes scattering above the false vacuum that does not lead to the
transition to the true one. Let us see that eq.(\ref{finf}) imposes
the energy--independent constraint
\begin{equation}
  n<n_{crit}
\label{V16*}
\end{equation}
where $n_{crit}$ is given by eq.(\ref{V13*}).

To this end, let us discuss slightly more general problem. Namely,
let us  find the minimum of $n$ for a given and fixed value of
$f(\infty)$. Introducing the Lagrange multiplier $\Lambda$, one writes
\begin{equation}
  \vpar{}{\phi_L(-k)}(n-\Lambda f(\infty)) = 0
  \label{condminn}
\end{equation}
Substuting eqs.(\ref{nphi}) and (\ref{finf}) into eq.(\ref{condminn}), one
obtains
\begin{equation}
  \omega_k\phi(k) = {\pi\lambda\over v}\Lambda\int dz
                    \exp\left({2\lambda\over v}\phi_L(z)\right)\e^{-ikz}
  \label{eqminn}
\end{equation}
At high energy per particle, $E/n\gg m$, the frequency
 $\omega_k$ may be replaced by $|k|$
in this equation.

Eq.(\ref{eqminn}) optimizes the form of the incoming wave packet
$\phi_L(z)$ at given $f(\infty)$ in such a way
 that the number of particles
is minimal.
Let us check that  the
 solution to
 eq.(\ref{eqminn}) is given by the function $f(z)$ considered in sect.2.2,
eq.(\ref{ansatz}), with $\eta=1$.
At $\eta=1$ eq.(\ref{ansatz}) reads, in terms of $\phi_L$,
\[
  \phi_L(z) = -{v\over 2\lambda}\ln\left[{m^2\e^{2\gamma}\over 4}
               (z^2+\tau^2)\right]
\]
One has
\[
  \phi_L(k) = \int dz\phi_L(z)\e^{-ikz} =
 \frac{\pi v}{\lambda |k|}\e^{-|k|\tau}
\]
\[
  \int dz\exp\left({2\lambda\over v}\phi_L(z)\right)\e^{-ikz} =
  {4\pi\over m^2\e^{2\gamma}}{1\over\tau}\e^{-|k|\tau}
\]
It is now clear that  eq.(\ref{eqminn}) is satisfied by our
ansatz if $\Lambda$ is related to $\tau$ as follows,
\[
  \Lambda = {v\over\pi\lambda^2}{m^2\e^{2\gamma}\tau\over 4}
\]
So, we have found the solution with the
 minimal number of particles among
all solutions with fixed $f(\infty)$.
The value of $f(\infty)$ for this configuration is
\begin{equation}
  f(\infty)=\frac{4\pi}{m^{2}\e^{2\gamma}}\frac{1}{\tau}
\label{V18*}
\end{equation}
This relates the only parameter of the solution, $\tau$, to
$f(\infty)$. The number of incoming particles is given by eq.(\ref{n})
with $\eta = 1$ and increases as $f(\infty)$ grows. Eq.(\ref{V16*}) is
now a straightforward consequence of the constraint (\ref{finf}) and
eq.(\ref{V18*}).



\section{Another argument for suppression of decay rate  at
small number of incoming particles}

In the previous section we have seen that
  the vacuum decay is always suppressed at small enough number of initial
particles. Now we  present
another argument for the suppression of the false vacuum decay in the limit
of a small number of incoming
 particles. Our discussion is based on
the arguments of ref.\cite{RT} that the rate of the decay
induced by a few particles can
be estimated by considering the maximum probability among all initial states
in the common eigenspace
 of the Hamiltonian and some other operator $\hat{A}$,
\[
  \hat{A}=\int dk A(k)a^{\dagger}_ka_k
\]
The usual choice is $A(k)=1$ for all
$k$, so the operator $\hat{A}$ is just
 the number of particles.
However, other choices of $A(k)$ are equally suitable, the only
restrictions being that $A(k)$ is non-negative at any $k$,
 does not depend explicitly on
the coupling constant ($v$ in our model) and does not grow like $|k|$
at large $k$ (otherwise $\hat{A}$ coincides with the Hamiltonian, up to
an unimporatnt piece; we assume for definiteness that $A(k)$ does not
grow at all at large momenta). If the eigenvalue $A$ of $\hat{A}$ is of
order $v^{2}$, the induced decay rate is semiclassically calculable
and gives the upper bound for the two-particle cross
 section\footnote{In fact, it has been conjectured
 in refs.\cite{RT,T} that
the leading exponent for the two-particle cross section can be
obtained by considering the limit $A/v^{2} \to 0$; we will not make
use of this conjecture in this paper.}.
So, we argue that the vacuum decay induced by a few highly energetic
particles is not exponentially suppressed only if there exist
real classical Minkowskian solutions with arbitrarily small $A/v^{2}$
which end up in the true vacuum, i.e., have a singularity
 $\phi = \infty$ at $x=0$ and finite $t$.

As usual, the eigenvalue of $\hat{A}$ in terms of the classical field
is
\[
    \hat{A}=\int dk \omega (k) A(k) \phi(k) \phi(-k)
\]
where $\phi$ is the asymptotics of the field at large negative time.
Let us choose
\[
  A(k) = \omega_k^{-1}
\]
We now show that at given energy $E$ there are no relevant classical solutions
 for
sufficiently small $A$. We again make use of the technique of
sect.2.1, and also use the notations of sect.3.
In complete analogy to eq.(\ref{nphi})
one has
\[
   A = {2\over\pi}\int\limits_0^{\infty}dk\phi_L(k)\phi_L(-k)
\]
We will minimize this expression for all functions $\phi_L$
under the condition that
the energy
\[
  E = {2\over\pi}\int\limits_0^{\infty}dk \omega^{2}_{k}\phi_L(k)\phi_L(-k)
\]
and $f(\infty)$ (\eq{finf}) are fixed. Introducing two Lagrange multipliers,
 $\Lambda^2$ and $\Lambda'^2$ we obtain the
following equation,
\begin{equation}
  \omega_k^2\phi_L(k) = \Lambda^2\phi_L(k)-\Lambda'^2\int dz\e^{-ikz}
  \exp\left({2\lambda\over v}\phi_L(z)\right)
  \label{momrepr}
\end{equation}
 For
further convenience let us introduce a new Lagrange parameter
 $\phi_0$ instead of
$\Lambda'$,
\[
  \Lambda'^2={\lambda\Lambda^2\phi_0^2\over v}
   \exp\left(-\frac{2\lambda}{v}\phi_{0}\right)
\]
Let us  assume that $\phi_0~\gtap~v$ and $\Lambda\gg m$; this assumption
will be justified a posteriori. Eq.(\ref{momrepr})
can be rewritten in the coordinate representation,
\[
  \partial_z^2\phi_L(z) = \Lambda^2\phi_L(z)-
  {\lambda\Lambda^2\phi_0^2\over v}
  \exp\left({2\lambda\over v}(\phi_L(z)-\phi_0)\right)
\]
The latter
 equation has the same form as the equation for the sphaleron in our
model. The solution in two different (and overlapping)
 regions  is as follows \cite{Rutgers}
\[
  \phi(z) = \phi_0 - {v\over\lambda}\ln\left[\cosh\left(
            {\lambda\Lambda\phi_0\over v}z\right)\right],
  ~~~\mbox{at } z\ll\Lambda^{-1}
\]
\[
  \phi(z) = \phi_0\e^{-\Lambda |z|},
            ~~~\mbox{at } z\gg (\lambda\Lambda)^{-1}
\]
For this field configuration, the value of $A$ is
\[
  A = {2\phi_0^2\over\Lambda}
\]
while the energy and $f(\infty)$ are
\begin{equation}
  E = 2\phi_0^2\Lambda
\label{13*}
\end{equation}
\begin{equation}
  f(\infty) = {2v\over\lambda\Lambda\phi_0}
              \exp\left({2\lambda\over v}\phi_0\right)
\label{13**}
\end{equation}
It is clear that the condition $f(\infty)>a$ imposes a lower bound on
 $A$
for a given value of energy $E$.  In the most interesting case of
 exponentially high energies, $\ln E/E_{sph} \sim \lambda$, one has
\[
  \frac{A}{v^{2}}>B(E)
\]
where
\[
   B(E)\sim \frac{E_{sph}}{mE}
\]
is strictly positive. Thus, at arbitrary but fixed energy, the
induced vacuum decay is classically forbidden in the limit
$A/v^{2}\to0$, i.e., in the limit of small number of incoming
particles.

Finally, let us justify the above assumptions on the Lagrange
parameters $\phi_{0}$ and $\Lambda$, again for simplicity
at exponentially high
energies. At $f(\infty)\sim a$ one finds from eqs.(\ref{13*}) and
(\ref{13**})
\[
  \Lambda \sim \frac{mE}{E_{sph}},~~~~~~~~
 \phi_{0}
 \sim \frac{v}{2}\left( 1 + \frac{1}{\lambda}\ln{\frac{E}{E_{sph}}}\right)
\]
so one indeed has $\Lambda \gg m$, $\phi_{0} \sim v$. This completes
the argument.


\section{Conclusions}

In this paper we have considered real classical solutions that
describe the unsuppressed vacuum decay. We have shown that for $n>n_{crit}$
these solutions exist when the energy is larger than some
 $n$--dependent
 critical
value. In the case  $\lambda^{-1}\ll n/n_{sph}\ll 1$, this critical
energy is essentially the same as
 the one found in our
previous study of the decay rate in the
suppressed regime \cite{Rutgers}.
 For $n<n_{crit}$, there are no such solutions,
so we  conclude that the decay rate is always suppressed if the number
of initial particles is small enough. We have presented
 an independent argument showing
that the induced vacuum decay is suppressed in the limit of small number of
initial particles. Our results,
 in particular, imply that the exponential
suppression of the decay of the false vacuum induced by two colliding
 particles persists at arbitrarily high energies. In other words, the
function $F(E)=v^{-2}\ln\Gamma(E)$, where $\Gamma(E)$ is the decay rate
induced by two particles with the
center-of-mass energy $E$, is always negative.

The absence of the classical Minkowskian solutions tells nothing
about the actual behavior of the suppression factor. The calculation
of $F(E)$ in our model
by the technique of the boundary value problem in complex
time also met technical problems at small $n$ and very high
energies \cite{Rutgers}. So, the (exponentially suppressed)
two-particle cross section of the
induced vacuum decay is not known in the interesting energy region.



There are two
possibilities of
how the function $F(E)$ may behave at high energies. The
first one
 is that $F$ tends to some finite value $F(\infty)>0$. In this case,
the exponential suppression is always strong (stronger than
$\e^{-v^2F(\infty)}$). The other possibility is that $F$ asymptotically
tends to zero at as $E\to\infty$. In that
 case, the induced false vacuum decay
may become "observable" at very high energies
(much higher than the one-instanton estimate), where the function $F$
actually becomes
small. Which of these possibilities is realized in our model is still an open
question.

The authors are indebted to T.~Banks and P.~Tinyakov for stimulating
disscussions. V.R. thanks Rutgers University, where part of this work
has been done, for hospitality.
 The work of D.T.S is supported in part by the Russian Foundation
for Fundamental Research (project 93-02-3812) and by the Weingart
Foundation through a cooperative agreement with the Department of Physics
at UCLA.


\begin{thebibliography}{99}
\bibitem{Mrev}
    M.Mattis, {\em Phys.Rep.} {\bf 214} (1992) 159.
\bibitem{Trev}
    P.G.Tinyakov, {\em Int.J.Mod.Phys.} {\bf A8} (1993) 1823.
\bibitem{Affleck}
    I.K.Affleck and F. De Luccia, {\em Phys.Rev.} {\bf D20} (1979) 3168.
\bibitem{VolSel}
    M.B.Voloshin and K.G.Selivanov, {\em JETP Lett}. {\bf 42} (1985) 422;
    {\em Yad.Fiz.} {\bf 44} (1986) 1336
    ({\em Sov.J.Nucl.Phys.} {\bf 44} (1986) 868).
\bibitem{Vol}
  M.B.Voloshin {\em Nucl.Phys.} {\bf B363} (1991) 425.
\bibitem{RSTind}
  V.A.Rubakov, D.T.Son and P.G.Tinyakov,
  {\em Phys.Lett.} {\bf 278B} (1992) 279.
\bibitem{Kisel}
  V.G.Kiselev, {\em Phys.Rev.} {\bf D45} (1992) 2929.
\bibitem{GorVol}
  A.S.Gorsky and M.B.Voloshin, {\em Non--Perturbative Production of
Multi--Boson States and Quantum Bubbles}, Minnesota preprint,
TPI-MINN-93/20-T,  1993.
\bibitem{Rutgers}
  D.T.Son and V.A.Rubakov, {\em Instanton--Like Transitions at High
Energies in (1+1) Dimensional Scalar Models}, Rutgers preprint, RU-93-48,
1993.
\bibitem{Coleman}
    S.Coleman, {\em Phys.Rev.} {\bf D15} (1977) 2929.
\bibitem{CalCol}
    C.Callan and S.Coleman {\em Phys.Rev.} {\bf D16} (1977) 1762.
\bibitem{Ringwald}
     A.Ringwald, {\em Nucl.Phys.} {\bf B330} (1990) 1.
\bibitem{Espinosa}
     O.Espinosa, {\em Nucl.Phys.} {\bf B334} (1990) 310.
\bibitem{RT}
  V.A.Rubakov and P.G.Tinyakov, {\em Phys.Lett.} {\bf B279} (1992) 165.
\bibitem{T}
  P.G.Tinyakov, {\em Phys.Lett.} {\bf B284} (1992) 410.
\bibitem{RST}
  V.A.Rubakov, D.T.Son and P.G.Tinyakov,
  {\em Phys.Lett.} {\bf 287B} (1992) 342.
\bibitem{Mueller}
    A.H.Mueller, {\em Nucl.Phys.} {\bf B401} (1993) 93.

\end{thebibliography}
\newpage

FIGURE CAPTIONS

1. The potential $V(\phi)$.

2. The $(E,n)$ plane, $n_{crit}=\pi^{2}n_{sph}/\lambda$.
The regions II and I above and below the solid line correspond to
unsuppressed and suppressed induced vacuum decay, respectively. The
dashed line is the (unphysical) border between the regions of
suppressed and unsuppressed vacuum decay within the Ansatz of
 eq.(\ref{ansatz}).



\end{document}
gsave
300 300 scale
0.3 1 translate
0.002 setlinewidth
0 0.32002 moveto
1.1 0.32002 lineto
stroke
0.49529 0 moveto
0.49529 1 lineto
stroke
/Helvetica findfont 0.04 scalefont setfont
0.519 1 moveto (V) show
0.543 1 moveto (\() show
0.583 1 moveto (\)) show
0.462 0.287 moveto (0) show
0.934 0.287 moveto (v) show
0.43 -0.4 moveto (Fig.1) show
/Symbol findfont 0.04 scalefont setfont
1.067 0.287 moveto (\146) show
0.559 1 moveto (\146) show
0.003 setlinewidth
0.02381 0.97619 moveto
0.06349 0.87038 lineto
0.10317 0.77387 lineto
0.14286 0.68666 lineto
0.18254 0.60874 lineto
0.22222 0.54012 lineto
0.2619 0.4808 lineto
0.30159 0.43077 lineto
0.34127 0.39004 lineto
0.38095 0.35861 lineto
0.40079 0.34638 lineto
0.42063 0.33647 lineto
0.43056 0.33239 lineto
0.44048 0.32889 lineto
0.4504 0.32597 lineto
0.46032 0.32363 lineto
0.46528 0.32268 lineto
0.47024 0.32187 lineto
0.4752 0.32121 lineto
0.48016 0.3207 lineto
0.48264 0.32049 lineto
0.48512 0.32033 lineto
0.4876 0.3202 lineto
0.48884 0.32014 lineto
0.49008 0.3201 lineto
0.49132 0.32007 lineto
0.49256 0.32004 lineto
0.4938 0.32003 lineto
0.49504 0.32002 lineto
0.49628 0.32002 lineto
0.49752 0.32004 lineto
0.49876 0.32006 lineto
0.5 0.32009 lineto
0.50248 0.32017 lineto
0.50496 0.3203 lineto
0.50744 0.32046 lineto
0.50992 0.32065 lineto
0.51488 0.32115 lineto
0.51984 0.3218 lineto
0.52976 0.32353 lineto
0.53968 0.32584 lineto
0.55952 0.3322 lineto
0.57937 0.34089 lineto
0.61905 0.36523 lineto
0.65873 0.39888 lineto
0.69841 0.44181 lineto
0.7381 0.49401 lineto
0.77778 0.55537 lineto
0.81746 0.62525 lineto
0.85714 0.70027 lineto
0.87698 0.73553 lineto
0.8869 0.75056 lineto
0.89187 0.75691 lineto
0.89683 0.76219 lineto
0.89931 0.76434 lineto
0.90179 0.76611 lineto
0.90303 0.76684 lineto
0.90427 0.76746 lineto
0.90551 0.76795 lineto
0.90675 0.76832 lineto
0.90799 0.76856 lineto
0.90923 0.76866 lineto
0.91047 0.76861 lineto
0.91171 0.7684 lineto
0.91295 0.76802 lineto
0.91419 0.76747 lineto
0.91543 0.76673 lineto
0.91667 0.76579 lineto
0.91915 0.76329 lineto
0.92163 0.75985 lineto
0.92411 0.75538 lineto
0.92659 0.74976 lineto
0.93155 0.73451 lineto
0.93651 0.71285 lineto
0.94147 0.68326 lineto
0.94643 0.64384 lineto
0.95139 0.59226 lineto
0.95635 0.52564 lineto
0.96131 0.44041 lineto
0.96627 0.33218 lineto
0.97123 0.19555 lineto
0.97619 0.02381 lineto
stroke
showpage grestore
%!PS-Adobe-2.0
%%Creator: James R. Van Zandt
%%Title:   GRAPH PostScript Output
%%BoundingBox: 0 0 612 792
%%Pages: 1
%%EndComments
%
% GRAPH PostScript driver version 1.45
%
% This file is a printout from GRAPH intended for a PostScript printer.
% The PostScript code is largely due to Thomas B. Passin.
% The adaptation for GRAPH was written by James R. Van Zandt.
% Thanks are also due to David Paul Feldman.  His program GLOPS,
% which converts an HPGL file into PostScript, inspired the
% original PostScript version of GRAPH.
%
% This file includes structure comments required by Tom Passin's POSTOGRF,
% which can be used to add labels.
%
% ------------------------ Prologue  -------------------------------
%   DVIALW by Nelson H. F. Beebe (Beebe@Science.Utah.Edu)
%   is a program to convert a TeX DVI file into PostScript for the
%   Apple LaserWriter.  This structure comment tells DVIALW which
%   part of a GRAPHPS plot to incorporate into a TeX document...
%begin(plot)

%Abbreviations
/cpt  /currentpoint   load   def
/l    /lineto         load   def
/m    /moveto         load   def
/sd   /setdash        load   def
/sf   /setfont        load   def
/sg   /setgray        load   def
/st   /stroke         load   def

/inch { 72 mul } def
/linethick 10 def     %default line thickness

/setorigin { 1.500 inch 3.625 inch translate
             0 0 m  } def

% portrait orientation

%   left margin: 1.500 in =  3.81 cm
%    plot width: 6.247 in = 15.87 cm =  1874 pixels
%  right margin: 0.750 in =  1.91 cm

%    top margin: 1.125 in =  2.86 cm
%   plot height: 6.247 in = 15.87 cm =  1874 pixels
% bottom margin: 3.625 in =  9.21 cm

%FontDefinitions
/font0 /Helvetica findfont 180 scalefont def
/font1 /Symbol    findfont 180 scalefont def
/font2 /Helvetica findfont 120 scalefont def
/font3 /Times-Roman findfont 360 scalefont def
%EndFonts
%%EndProlog
%%BeginSetup
gsave
statusdict /jobname (GRAPH) put
statusdict /waittimeout 30 put
newpath
setorigin
0.072 0.072 scale                % set scale to 1/1000 inch
linethick setlinewidth
%%EndSetup
%StartLabels
%EndLabels

%StartGraph
[] 0 sd
0 sg
% initialization complete
cpt st m
[] 0 sd  % solid line
966 313 m
966 313 l
966 1796 l
1046 1796 l
1126 1796 l
966 1796 l
966 3280 l
1046 3280 l
1126 3280 l
966 3280 l
966 4763 l
1046 4763 l
1126 4763 l
966 4763 l
966 6243 l
%1046 6243 l
%1126 6243 l
966 313 m
966 313 l
2023 313 l
2023 393 l
2023 473 l
2023 313 l
3076 313 l
3076 393 l
3076 473 l
3076 313 l
4133 313 l
4133 393 l
4133 473 l
4133 313 l
5186 313 l
5186 393 l
5186 473 l
5186 313 l
6240 313 l
%6240 393 l
%6240 473 l
%540 6173 m
%cpt st m
%font0 sf (2.0) show
1120 6170 m
font0 sf (n/n) show
1390 6140 m
font2 sf (crit) show
540 4690 m
cpt st m
font0 sf (1.5) show
753 3206 m
cpt st m
font0 sf (1) show
540 1723 m
cpt st m
font0 sf (0.5) show
540 243 m
cpt st m
font0 sf (0.0) show
913 80 m
cpt st m
font0 sf (0) show
1970 80 m
cpt st m
font0 sf (1) show
3023 80 m
cpt st m
font0 sf (2) show
4080 80 m
cpt st m
font0 sf (3) show
5133 80 m
cpt st m
font0 sf (4) show
%6190 80 m
%cpt st m
%font0 sf (5) show
5600 30 m
font1 sf (l) show
5672 150 m
font2 sf (-1) show
5780 30 m
font0 sf (ln\(E/E    \)) show
6290 0 m
font2 sf (sph) show
3100 -900 m
font0 sf (Fig.2) show
2023 1796 m
font3 sf (I) show
2023 4763 m
font3 sf (II) show
4133 3450 m
font3 sf (II) show
1176 5653 m
1176 5653 l
1190 5483 l
1200 5330 l
1210 5190 l
1220 5063 l
1230 4946 l
1240 4840 l
1253 4743 l
1263 4653 l
1273 4566 l
1283 4490 l
1293 4416 l
1303 4350 l
1316 4286 l
1326 4230 l
1336 4173 l
1346 4123 l
1356 4073 l
1366 4030 l
1380 3986 l
1390 3946 l
1400 3910 l
1410 3873 l
1420 3840 l
1430 3806 l
1443 3776 l
1453 3750 l
1463 3723 l
1473 3696 l
1483 3673 l
1493 3650 l
1506 3630 l
1516 3606 l
1526 3590 l
1536 3570 l
1546 3553 l
1556 3536 l
1570 3520 l
1580 3503 l
1590 3490 l
1600 3476 l
1610 3463 l
1620 3453 l
1633 3440 l
1643 3430 l
1653 3420 l
1663 3410 l
1673 3400 l
1683 3390 l
1696 3383 l
1706 3373 l
1716 3366 l
1726 3360 l
1736 3353 l
1746 3346 l
1756 3340 l
1770 3336 l
1780 3330 l
1790 3326 l
1800 3320 l
1810 3316 l
1820 3313 l
1833 3310 l
1843 3306 l
cpt st m
1853 3303 l
1863 3300 l
1873 3296 l
1883 3293 l
1896 3290 l
1906 3290 l
1916 3286 l
1926 3286 l
1936 3283 l
1946 3283 l
1960 3283 l
1970 3280 l
1980 3280 l
1990 3280 l
2000 3280 l
2010 3280 l
2023 3280 l
[120 30 ] 0 sd
2033 3280 l
2043 3280 l
2053 3280 l
2063 3280 l
2073 3280 l
2086 3283 l
2096 3283 l
2106 3283 l
2116 3283 l
2126 3286 l
2136 3286 l
2150 3290 l
2160 3290 l
2170 3293 l
2180 3293 l
2190 3296 l
2200 3296 l
2213 3300 l
2223 3303 l
2233 3303 l
2243 3306 l
2253 3310 l
2263 3310 l
2276 3313 l
2286 3316 l
2296 3320 l
2306 3323 l
2316 3323 l
2326 3326 l
2340 3330 l
2350 3333 l
2360 3336 l
2370 3340 l
2380 3343 l
2390 3346 l
2403 3350 l
2413 3353 l
2423 3356 l
2433 3360 l
2443 3363 l
2453 3366 l
2466 3370 l
2476 3376 l
2486 3380 l
2496 3383 l
2506 3386 l
2516 3390 l
2530 3393 l
2540 3400 l
2550 3403 l
2560 3406 l
2570 3410 l
2580 3416 l
2593 3420 l
2603 3423 l
2613 3430 l
2623 3433 l
2633 3436 l
2643 3443 l
2653 3446 l
2666 3450 l
2676 3456 l
2686 3460 l
2696 3463 l
2706 3470 l
2716 3473 l
2730 3480 l
2740 3483 l
2750 3486 l
2760 3493 l
2770 3496 l
2780 3503 l
2793 3506 l
2803 3513 l
2813 3516 l
2823 3523 l
2833 3526 l
2843 3533 l
2856 3536 l
2866 3543 l
2876 3546 l
2886 3553 l
2896 3560 l
2906 3563 l
cpt st m
2920 3570 l
2930 3573 l
2940 3580 l
2950 3583 l
2960 3590 l
2970 3596 l
2983 3600 l
2993 3606 l
3003 3610 l
3013 3616 l
3023 3623 l
3033 3626 l
3046 3633 l
3056 3640 l
3066 3643 l
3076 3650 l
3086 3656 l
3096 3660 l
3110 3666 l
3120 3673 l
3130 3676 l
3140 3683 l
3150 3690 l
3160 3696 l
3173 3700 l
3183 3706 l
3193 3713 l
3203 3716 l
3213 3723 l
3223 3730 l
3236 3736 l
3246 3740 l
3256 3746 l
3266 3753 l
3276 3760 l
3286 3763 l
3300 3770 l
3310 3776 l
3320 3783 l
3330 3786 l
3340 3793 l
3350 3800 l
3363 3806 l
3373 3813 l
3383 3816 l
3393 3823 l
3403 3830 l
3413 3836 l
3426 3843 l
3436 3846 l
3446 3853 l
3456 3860 l
3466 3866 l
3476 3873 l
3490 3880 l
3500 3883 l
3510 3890 l
3520 3896 l
3530 3903 l
3540 3910 l
3550 3916 l
3563 3923 l
3573 3926 l
3583 3933 l
3593 3940 l
3603 3946 l
3613 3953 l
3626 3960 l
3636 3966 l
3646 3970 l
3656 3976 l
3666 3983 l
3676 3990 l
3690 3996 l
3700 4003 l
3710 4010 l
3720 4016 l
3730 4023 l
3740 4030 l
3753 4033 l
3763 4040 l
3773 4046 l
3783 4053 l
3793 4060 l
3803 4066 l
3816 4073 l
3826 4080 l
3836 4086 l
3846 4093 l
3856 4100 l
3866 4106 l
3880 4110 l
3890 4116 l
3900 4123 l
3910 4130 l
3920 4136 l
3930 4143 l
3943 4150 l
3953 4156 l
3963 4163 l
3973 4170 l
cpt st m
3983 4176 l
3993 4183 l
4006 4190 l
4016 4196 l
4026 4203 l
4036 4210 l
4046 4216 l
4056 4223 l
4070 4230 l
4080 4236 l
4090 4240 l
4100 4246 l
4110 4253 l
4120 4260 l
4133 4266 l
4143 4273 l
4153 4280 l
4163 4286 l
4173 4293 l
4183 4300 l
4196 4306 l
4206 4313 l
4216 4320 l
4226 4326 l
4236 4333 l
4246 4340 l
4260 4346 l
4270 4353 l
4280 4360 l
4290 4366 l
4300 4373 l
4310 4380 l
4323 4386 l
4333 4393 l
4343 4400 l
4353 4406 l
4363 4413 l
4373 4420 l
4386 4426 l
4396 4433 l
4406 4440 l
4416 4446 l
4426 4453 l
4436 4460 l
4446 4466 l
4460 4473 l
4470 4480 l
4480 4486 l
4490 4493 l
4500 4500 l
4510 4510 l
4523 4516 l
4533 4523 l
4543 4530 l
4553 4536 l
4563 4543 l
4573 4550 l
4586 4556 l
4596 4563 l
4606 4570 l
4616 4576 l
4626 4583 l
4636 4590 l
4650 4596 l
4660 4603 l
4670 4610 l
4680 4616 l
4690 4623 l
4700 4630 l
4713 4636 l
4723 4643 l
4733 4650 l
4743 4656 l
4753 4663 l
4763 4670 l
4776 4680 l
4786 4686 l
4796 4693 l
4806 4700 l
4816 4706 l
4826 4713 l
4840 4720 l
4850 4726 l
4860 4733 l
4870 4740 l
4880 4746 l
4890 4753 l
4903 4760 l
4913 4766 l
4923 4773 l
4933 4780 l
4943 4790 l
4953 4796 l
4966 4803 l
4976 4810 l
4986 4816 l
4996 4823 l
5006 4830 l
5016 4836 l
5030 4843 l
5040 4850 l
cpt st m
5050 4856 l
5060 4863 l
5070 4870 l
5080 4876 l
5093 4886 l
5103 4893 l
5113 4900 l
5123 4906 l
5133 4913 l
5143 4920 l
5156 4926 l
5166 4933 l
5176 4940 l
5186 4946 l
[] 0 sd
2023 3280 m
5186 3280 l
966 313 m
966 313 l
5713 313 l
966 313 m
966 313 l
966 6243 l
stroke
%EndGraph
%end(plot)
 showpage grestore
%%Trailer

