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\begin{document}
%
\title{Summing Non-Borel Tunneling Amplitudes
by Variational Perturbation Theory}
%
\author{B.~Hamprecht and H.~Kleinert}
%
\address{Institut fr Theoretische Physik, Freie Universitt Berlin,\\
Arnimallee 14, D-14195 Berlin, Germany\\
{\scriptsize e-mails: bodo.hamprecht@physik.fu-berlin.de}
{\scriptsize e-mails: hagen.kleinert@physik.fu-berlin.de}}
%
%
\begin{abstract}
We present a method for
evaluating divergent
Borel-nonsummable
 series
by
 an analytic continuation
 of variational perturbation
 theory. We demonstrate its validity
by an application to the exactly known partition function
of the anharmonic oscillator
in zero spacetime dimensions,
and derive
the full imaginary part of the
 ground state energy of
 the anharmonic oscillator for all negative
 values of
 the coupling constant $g$.
As a
highlight
 of the theory
we retrieve
from the divergent
perturbation expansion
the instanton action and the contribution
of the higher  loop
fluctuations
around the
instanton
in
the path integral.
\end{abstract}
%
\maketitle
%
%
\section{Introduction}
%
\noindent{\bf 1.}
None of
 the presently
known resummation schemes \cite{Resumm,Verena}
is able to deal with non-Borel series.
Such series arise
in
 the
 theoretical description
of many
important physical phenomena,
in particular tunneling
processes.
Any Borel-summable
series becomes non-Borel
if
 the expansion parameter, usually some coupling constant $g$,
is continued to negative values.
Here we show that
 the
Borel-nonsummable
series  can be evaluated with any desired precision
 by an extension of  variational perturbation
 theory \cite{Hagen,Verena} in
 to
 the complex $g$-plane. Variational perturbation
 theory
has a long history
\cite{refs,finiteg,KleinertJanke,Guida}. It is based on the introduction of a
dummy variational
 parameter $ \Omega $
on which
the full perturbation expansion does not depend,
while the truncated expansion
does. An optimal $ \Omega $ is then selected
by the principle of minimal sensitivity \cite{Stevenson},
 requiring the
quantity of interest to be stationary as function of the
variational parameter. The optimal
$ \Omega $
 is usually
taken from a zero of
 the derivative with
respect to $ \Omega $.
If the fist derivative has no zero,
a zero of the second derivative is chosen.
For Borel-summable series, these zeros are always real, in contrast
to statements
in
the literature
 \cite{Neveu,RULES,Braaten,Ramos}
which have proposed
the use of complex
zeros. Complex zeros, however, produce
in general  wrong results
for Borel-summable series, as was recently demonstrated
in Ref.~\cite{AntiBraaten}.

The purpose of this note is to show that there
does exist
 a wide range of applications
of complex zeros in the
previously untreatable
field of Borel-nonsummable series.
These arise typically
in tunneling problems, and we
shall
see
 that variational perturbation
theory provides us with an
efficient method for evaluating
these series and rendering
their real and imaginary parts
with any desirable accuracy
if only enough perturbation coefficients are available.
The choice of the
 complex zeros
is dictated by the
requirement
to achieve at
each order
the least oscillating
imaginary part when approaching
 the tip of
 the cut. We may call this selection rule the
{\em principle of minimal
sensitivity and oscillations\/}.

%
\noindent{\bf 2.}
 For an introduction
to the method
consider the exactly known partition function
of an  anharmonic oscillator
in zero spacetime dimensions
%\mn{check eqn}
, which is a simple integral
representation of a modified  Bessel function $K_ \nu (z)$:
\begin{align}
\label{FOKKER}
Z(g) &=& \frac{1}{\sqrt{2\pi}}
\int_{-\infty}^\infty dx\,\exp{(-x^2/2+g~x^4/4)} \nonumber \\
&=&\!\!\!\!\!\!\!\!{e^{-1/8g}~\sqrt{-4\pi g}}~{K_{1/4}(-1/8g)}~,~~~~~~~~~
\end{align}
For small $g>0$,
 the function $Z(g)$ and
its inverse $D(g)\equiv Z^{-1}(g)$ have a divergent Borel-nonsummable
 power
series. In the
strong-coupling regime, there exists a convergent expansion
%
\begin{align}
\label{FP-STRONG}
D(g) = g^\alpha~\sum_{l=0}^L\; b_l\;g^{-\omega l},
\end{align}
%
with $\alpha=1/4$ and
$\omega=1/2$. In the context of critical phenomena,
the exponent
$ \omega $ coincides with the the  Wegner exponent \cite{Wegner}
of approach to scaling \cite{strong}.
The $L$th variational approximation
depending on the variational parameter $ \Omega $
is given by the series
\cite{Hagen,Verena}
%
\begin{align}
\label{FP-VAR}
D_{\rm var}^{(L)}(g,\Omega) = \Omega^{pq} ~\sum_{j=0}^L \left(\frac{g}{\Omega^q}\right)^j \epsilon_j(\sigma)\,,
\end{align}
%
where $q=2/\omega=4$, $p=\alpha \omega /2=1/4$.
Introducing the
parameter
$\sigma=\Omega^{q-2}(\Omega^2-1)/g$, the re-expansion coefficients are
%
\begin{align}
\label{FP-EPS}
\epsilon_j(\sigma) = \sum_{l=0}^j c_l \binom{(p-l)q/2}{j-l} (-\sigma)^{j-l}\,.
\end{align}
%
%
Following
 the principle of minimal sensitivity,
we have to find  the zeros of
 the derivative of $\partial _ \Omega D_{\rm var}^{(L)}(g,\Omega)$,
which happen to coincide with the
zeros of the functions $\zeta ^{(L)}(\sigma)$:
%
\begin{align}
\label{FP-DERIV}
 \zeta ^{(L)}(\sigma)\! = \!\sum_{l=0}^L c_l\, (pq\!-\!lq\!+\!2l\!-\!2L)
 \binom{(p\!-\!l)q/2}{L\!-\!l} (-\sigma)^{L-l}\,,
\end{align}
%
as was
 shown
in
 \cite{JKsig},
and   explained in detail in
 the textbook  \cite{Hagen}
(p. 291).


%
%
As an example,
we take
 the weak-coupling expansion
to $L=16$th order and
calculate real and
imaginary parts for
 the non-Borel region $0.008<g<2$
selecting the zero
of $ \zeta ^{(16)}(\sigma)$
according to the principle
of minimal sensitivity and
oscillations.
The result is
 shown in
 Fig.~\ref{NB-IV}.
In order to exhibit
how well
the variational result
approximates  the
essential singularity
%at the origin
of the imaginary part
 $ \propto
\exp{1/4g}$ for small $g$,
 we have removed this factor.
%from
% the imaginary part, the removal being indicated by a prime.
If the instanton energy were unknown,
we would easily have deduced this factor from
the slope of $g$ times the logarithm
of the imaginary part.
The agreement of our result with the
exact curve is excellent down to very small
$g$.
%\mn{plot exact curve for $g\rightarrow 0$
%and show first deviating points at small $g$.
%Put second figure underneath the first.
%}
\begin{figure}[htp!]
%
\begin{center}
\setlength{\unitlength}{1cm}
\begin{picture}(18.5,12)
\put(0.5,6.4){\scalebox{.8}[.8]{\includegraphics*{nonbor-i.eps}}}
\put(0.5,.4){\scalebox{.8}[.8]{\includegraphics*{nonbor-r.eps}}}
\put(3.6,10.3){\small{Im $D(g)\exp{1/4g}$}}
\put(3.6,1.6){\small{Re $D(g)$}}
\put(.2,9.9){$\small{.7}$}
\put(.2,8.5){$\small{.6}$}
\put(.2,7.1){$\small{.5}$}
\put(-.05,3.7){$\small{-.9}$}
\put(-.05,2.){$\small{-1.}$}
\put(2.,6.1){$\small{-4}$}
\put(4.5,6.1){$\small{-2}$}
\put(7.25,6.1){$\small{0}$}
\put(2.,0.1){$\small{-4}$}
\put(4.5,0.1){$\small{-2}$}
\put(7.25,0.1){$\small{0}$}
\put(5.65,6.1){$\log g$}
\put(5.65,.1){$\log g$}
\end{picture}
\caption[NB-IV]{Imaginary and real parts
from variational perturbation theory
of 16th order $D_{\rm var}^{(16)}(g)$ as a function of $\log{g}$
(dots)
in comparison with exact curves (curves).
In the imaginary part
we have removed
the leading
essential singularity
by dividing out a  factor  $\exp{(-1/4g)}$
to see the amazing accuracy
with which this singularity is approximated. For very small $g$ the
onset of oscillations in the imaginary part can be seen
which moves
towards the origin for increasing order
$L$ of the variational approximation.
 }
\label{NB-IV}
\end{center}
\end{figure}

\noindent{\bf 3.}
 Let us now turn to the nontrivial problem
of summing the
 instanton region of
 the anharmonic oscillator for $g<0$.
The divergent weak-coupling perturbation expansion for
 the ground state energy of
 the anharmonic oscillator
 with
 the potential $V(x)=x^2/2+g\,x^4$ is, to order $L$;
%
\begin{align}
\label{WEAK}
E_{0,\rm weak}^{(L)}(g) = \sum_{l=0}^L\; c_l\;g^l\,,
\end{align}
%
where $c_l=(1/2,~3/4,~-21/8,~333/16,~-30885/128,~\dots)$.
The expansion is obviously
not Borel-summable
 for $g<0$, but will now be evaluated
with our new technique,
proceeding in the
 same way as
before
via
Eqs.~(\ref{FP-VAR}) through (\ref{FP-DERIV}).
The known strong-coupling growth parameters
are
 $\alpha=1$ and
$\omega=2/3$, so that $q=3$ and
$p=1/3$ in Eq.~(\ref{FP-DERIV}) which will guarantee
 the correct scaling properties for $g \to \infty$.
To order $L=64$ we obtain
from
 the optimal  zero
of $ \zeta ^{(64)}( \sigma )$
 the logarithm of the
imaginary part.
A plot allows us to read off the
leading exponential behavior
for $g\rightarrow 0-$
which determines the action of the instanton.
Removing this exponential
we obtain
%
\begin{align}
\label{IM64}
f(g):=
\log\left[ {\sqrt{-\pi g/2}~E_{0,\rm var}^{(64)}(g)}\right] -1/3g\,,
\end{align}
%
 shown
 in Fig.~\ref{I} for $-0.2<g<-0.006$. The point $g=-0.006$ is
 for $L=64$
  the closest approach to
 the singularity at $g=0$
before
 the onset of oscillations.

%

Let us compare
 our curve with
the expansion
derived
 from
 instanton calculations,
shown as thin curve in Fig.~\ref{I}.
It has a divergent expansion for which Ref.~\cite{ZINNJ}
calculates the first 10 coefficients, starting with
%
%
\begin{align}
\label{ZJ}
f(g)&=b_1 g-b_2 g^2+b_3 g^3-b_4 g^4+\dots ~,
\end{align}
%
where
$b_1 =3.95833,~b_2=19.344,~b_3=174.21,~b_4=2177$.
%
This
expansion
contains the information
on the fluctuations
around the instanton up to 10 loops.
It is divergent and
Borel-nonsummable for $g<0$.
Remarkably, our theory allows us
 to retrieve
the first three
terms of this expansion
from our resummation of the
perturbation expansion.
Since our result
contains a regular approximation
of
the essential singularity,
 the fitting procedure will
 depend somewhat on
 the
 interval over which we fit
our curve by a power series.
A compromise between a sufficiently long
 interval and
the runaway of
 the divergent
 instanton expansion is obtained for
a lower limit
 $g>-.0229\pm .0003$
and an
 upper limit $g=-0.006$. Fitting a polynomial to
 the data, we extract
 the following
 first three coefficients:
%
\begin{eqnarray}
\label{INST}\!\!\!\!
b_1 \!=\!3.9586\pm .0003,~b_2\!=\!19.4\pm .12,~b_3\!=\!135\pm 18.
\label{@agree}\end{eqnarray}
%
The agreement
of these numbers
 with those in
(\ref{ZJ})
demonstrates
 that
our method is capable of probing
 deeply into
 the
 instanton region of the coupling constant.
\\

\noindent{\bf 4.}
Further evidence
for the quality
of our theory
comes from a comparison
with  the analytically continued strong-coupling result
plotted to order $L=22$ as a fat curve in Fig.~\ref{I}.
This expansion was derived by a
procedure
of summing non-Borel series  developed
in Chapter 17
of the textbook \cite{Hagen}.
It was based on a two-step process:
the derivation of a strong-coupling
expansion of the type
(\ref{FP-STRONG}) from the divergent weak-coupling expansion, and an
analytic
continuation of the strong-coupling expansion to negative $g$.
This method was applicable
only for large enough
coupling strength
where the
strong-coupling expansion converges,
the so-called {\em sliding regime\/}. It could
  not invade into the
 tunneling regime
at
small $g$
governed
by instantons.
The instanton regime was treated
in \cite{Hagen}
 by a separate variational procedure.
The present
work fills the missing gap
by
extending variational perturbation theory
to {\em all\/} $g$ arbitrarily close to zero, without the need
 for a separate treatment of the tunneling regime.

There exist, of course,
 a wealth of
 possible applications  of this
simple theory, in particular to quantum field theory
where variational perturbation theory has so far yielded
the most accurate
critical exponents
from Borel-summable
series \cite{strong,Verena,Lipa}.


%
%
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%
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%

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%
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%
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%
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%
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%%



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%
%\bibitem{HaPe}
%W. Janke, ......
%{{\it Fluctuating Paths and
%Fields}, (World Scientific,Singapore, 2001), pg. 347}
%
%


%
%
\end{thebibliography}
%

\begin{figure}[htp!]
%
\begin{center}
\setlength{\unitlength}{1cm}
\begin{picture}(15,8)
\put(.5,.4){\scalebox{.8}[.8]{\includegraphics*{anh-1.eps}}}
\put(-.1,1.25){$\small {-.8}$}
\put(-.1,3.1){$\small {-.4}$}
\put(4.5,3.5){$l(g)$}
\put(.15,4.9){$\small {0}$}
\put(2.03,.1){$\small {-5}$}
\put(3.7,.1){$\small {-4}$}
\put(5.4,.1){$\small {-3}$}
\put(7.05,.1){$\small {-2}$}
\put(4.55,.1){$\log  g$}
\end{picture}
\caption[I]{Logarithm of
 the imaginary part of
 the ground state energy of
 the anharmonic oscillator with the essential singularity
factored out for better visualization,
$
l(g)=\log\left[ {\sqrt{-\pi g/2}~E_{0,\rm var}^{(64)}(g)}\right] -1/3g$,
plotted against
 small negative values of
 the coupling constant $-0.2<g<-.006$
where the series is
 Borel-nonsummable.
The thin curve represents
the divergent expansion around an instanton
of Ref.~\cite{ZINNJ}.
The fat curve is the
 $22$nd order approximation of the
 strong-coupling expansion,
 analytically continued
to negative $g$ in the sliding regime calculated
 in
Chapter 17 of the textbook \cite{Hagen}.
 }
\label{I}
\end{center}
\end{figure}
%
%

\end{document}

