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\begin{document}

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\begin{titlepage}

\title{
 {\bf
Renormalized interacting theories: 
redefining free parameters
}
           } 
\author{  {\bf F\'abio L. Braghin }\thanks{e-mail:
braghin@if.usp.br }   \\
{\normalsize FINPE, 
Instituto de F\'\i sica da Universidade de S\~ao Paulo } \\
{\normalsize C.P. 66.318,  C.E.P. 05315-970, S\~ao Paulo,    Brasil }
}

\maketitle
\begin{abstract}
The usual renormalization scheme 
for the variational 
approximation with a trial  Gaussian ansatz
for the $\lap$ model in 3+1 dimensions is re-analysed. 
The so called asymmetric phase of the model (where
$<vac|\phi |vac> \neq 0$) is considered
with the search for conditions that 
the parameters of the theory (mass and coupling constant)
would respect to obtain the  vacuum state.
The minimization of the energy
with relation to the renormalized  mass is done
and the resulting expression is faced 
as a new GAP equation.
The minimization of the energy with respect to the 
renormalized coupling constant is also done
resulting in expression for the check of reliability of 
the approximation.
A sort of ``scale invariant'' solutions for the renormalized 
mass and coupling constant values are found.
A non trivial solution for the model with
strong coupling constant is found.
Ranges of these values, mass and coupling constant, 
in which the approximation is 
expected to be more reliable as well as instabilities
 are found.
A different interpretation for the so called symmetry restoration
is found.
\end{abstract}

\vskip 0.4cm

  PACS numbers: 11.10.Gh; 11.15.Tk.
\vskip 0.4cm

 IFUSP-  2001


\end{titlepage}

\newpage
\setcounter{equation}{0}


\section{ Introduction}


The most developed approach to solve interacting 
field theories
is Perturbation theory which only works well for very small coupling 
constants
 as it occurs in Electrodynamics. 
Also in this approach  one has a 
systematic and direct way of 
dealing with ultraviolet (UV)
divergences, i.e.,  one knows precisely how to
renormalize. 
There are many motivations for the development of
non perturbative methods in Quantum Field Theories such as 
the description of Spontaneous Symmetry Breakings (SSB),
bound states and phase transitions.
One of quite well studied methods is the variational
approximation which, with the use of Gaussian wave functional,
has been showed to be powerful and useful in a wide
variety of situations. 
It corresponds to 
 a summation of  ``cactus'' type diagrams 
 for the energy \cite{BAGAN,STEVENSON,BARMOSHE,KMV}.
 It takes into account more non 
linearities than perturbation theory and it is equivalent
to the Hartree Bogoliubov approach.
In particular, the ultraviolet divergences are  present 
requiring the parameters of the model under 
consideration to be renormalized. 
This is not straightforwardly done.
In this approach the ground state of the system 
is determined by solving GAP equations derived by 
the minimization of the regularized energy density
with respect to a mass and a classical expected 
field, in particular  if there is any Spontaneous Symmetry 
Breaking (SSB). 
In spite of some limitations it may provide 
a frame for the study of non perturbative effects
mainly in the frame of many body quantum mechanics and 
eventually in quantum field theory.


The $\lambda \phi^4$ model has been extensively studied, for example,
to shed light on non perturbative effects in quantum field theory (QFT).
It's usually  considered for the study of Inflationary models
and it shares several properties with the linear 
sigma model which is an
effective model of QCD such as a 
SSB (generating a phase in which
the condensate $\bphi = <vac|\phi |vac> $ is non zero) and
asymptotic freedom \cite{BRANCHINA,KMV,ASYMPFREE}. 
The model nevertheless have  the intriguing
``triviality'' in the symmetric phase.

As it occurs for any theory, phenomenological or not, one
usually does not know how to predict values for the 
parameters (such as masses and couplings) from the theory itself.
Phenomenology with experimental data is always required to 
fix such parameters.
One of the aims of the present work is to suggest one reasoning
which may be used to constraint values for which the theory 
under investigation would be more appropriatedly used.
This is done by searching values of physical (renormalized)
couplings and masses which minimize the renormalized energy density.
This would be a first step toward a possible determination
of free parameters partially from theoretical grounds besides
imposing constraints for possible experimental values under 
investigation. 
This is a very general argument irrespective to
the model and approximative method used in this work.


In the present article the usual renormalization
scheme as carried out, for example, in \cite{STEVENSON,KERVAU}
 of the Gaussian approach 
(and equivalently Hartree Bogoliubov approximation) 
for the $\lap$ model is analyzed differently from what has
been done previously.
We argue that
a more detailed and reasonable description of the ground state
can be found by minimizing the renormalized 
energy density instead of the regularized energy density.
This is claimed for any renormalization procedure even 
if the minimization of the regularized expressions
is done as it stands for the usual renormalization in the frame
of the variational approximation \cite{STEVENSON,KERVAU,KMV}.
We propose that the minimization of the renormalized
energy density with relation to the renormalized parameters
(coupling constant and mass) can yield an appropriate
calculation for finding suitable (physical) values for these 
parameters in the frame of a given approximation. 
The work is organized as follows. 
In the next section the Gaussian approximation is summarized:
the GAP equations obtained from the  regularized theory 
(with a cutoff) are 
derived and subsequently compared to the renormalized ones.
The usual renormalization procedure of the mass 
and coupling constant which was performed for example in 
 \cite{STEVENSON,KERVAU} is considered.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In sections 3, 4 and 5 new renormalized GAP equations are 
analyzed:  we seek for values of the renormalized
mass, condensate and coupling constant which minimize the 
energy density. 
They could correspond to values for which the 
approximation is more appropriated and 
they also define the vacuum of the system.
However in some cases 
instabilities are found.
The stability of the model for those values is 
therefore analyzed.
In the last section the results are summarized.


\section{ Gaussian approximation for the  $\lambda \phi^4$  model } 

The Lagrangian density 
 for a scalar field   $\phi (\bx)$ with bare mass   $m_0^2$
and coupling constant  $\lambda$ is given by:
\be \label{1}
\displaystyle{
{\cal L}(\bx) = \frac{1}{2}\left\{
\partial_{\mu} \phi(\bx) \partial^{\mu} \phi(\bx) - m_0^2 \phi^2(\bx) -
\frac{\lambda}{12} \phi^4(\bx) \right\}   }
\ee
The corresponding Hamiltonian density is written as: 
\begin{eqnarray} \label{2}
 H = \frac{1}{2} \left( \pi^{2}(\vx) +  (\nabla
\phi)^2 +  {m_0^2} \phi^{2}(\vx) + \frac{\lambda}{12}
\phi^{4}(\vx) \right).
\end{eqnarray}
The theory is quantized in the Schrodinger picture \cite{SCHRODINGER}
being the 
action of the field and momentum operators over a state 
$|\Psi>$ given respectively 
by:
\be \ba{ll} \label{quant}
\displaystyle{ \hat{\phi} |\Psi> = \phi |\Psi> \;\;\;\;\;\;\;
\hat{\pi} = - i \hbar \frac{\delta}{\delta \phi} |\Psi >
}
\ea
\ee

In the static Gaussian approximation at zero temperature, 
in the Schr\"odinger picture \cite{SCHRODINGER},
the ground state wave functional $\Psi$ is parametrized by:    
\be \label{4} \ba{ll}
\displaystyle{
\Psi\left[\phi(\vx )\right] = N  exp \left\{ -\frac{1}{4} \int
d \bx d \by \delta\phi(\vx ) G^{-1}(\vx ,\vy) 
 \delta \phi(\vy)
\right\}  , }
\ea
\ee
Where 
$ \delta\phi(\vx) = \phi(\vx)-\bar\phi(\vx) $ is the field 
shifted by a classical value where the wave function is centered; 
the normalization factor is $N$, 
the variational parameters are   
the classical expected value of the field $\phi$
(which will be called condensate)
 $ \bar \phi (\bx ) = < \Psi | \phi | \Psi > $ and 
quantum fluctuations represented by the width  of the Gaussian  
$ G(\vx,\vy) = <\Psi |\phi(\bx) \phi(\vy)  | \Psi>$. 
In variational calculations  the averaged energy calculated with 
$\Psi [ \phi(\bx ) ]$
is to be  minimized to obtain the GAP equations.
In principle it would yield a maximum bound for the ground 
state (averaged) energy. 
However there are subtleties associated to the Ultraviolet 
divergences from the local fields 
hindering an exact conclusion about this.

The average value of the Hamiltonian 
 is calculated and expressed in terms of the variational
parameters by means of expressions (\ref{2}) and (\ref{4})
in the Schrodinger picture \cite{SCHRODINGER}:
\be \label{11a} \ba{ll}
{\cal H} & = \frac{1}{2} \left[ \frac{1}{4} G^{-1}(\bx,\bx) 
             - \Delta G(\bx,\bx) + m^2_0 G(\bx,\bx) + \frac{\lambda}{4} 
              G^2(\bx,\bx) + \right. \\
   & \left. + m^2_0 \bphi^2(\bx) + (\nabla \bphi(\bx))^2 
     + \frac{\lambda}{12}\bphi^4(\bx)
      + \frac{\lambda}{2} \bphi^2(\bx) G(\bx,\bx) \right].
\ea
\ee
Variations of the averaged energy density 
with respect to the variational parameters 
 yield the following GAP  equations which define the 
ground state of the model:
\be \label{7} \ba{ll}
\displaystyle{ \frac{\delta {\cal H}}{\delta G(\bx,\by)} \rightarrow
0 =  \left( -
\frac{1}{8}G^{-2}(\vx,\vy) \right)  +
 \left( \frac{\Gamma(\vx,\vy)}{2} +
\frac{ \lambda}{2} \bar \phi(\vx)^2 \right)  } \\
\displaystyle{
\frac{\delta {\cal H}}{\delta \bar{\phi}(\vx) } \rightarrow
0 = \Gamma(\vx,\vy)\bar \phi(\vy) +
\frac{ \lambda}{6}\bar \phi^2(\vx), }
\ea
\ee
Where
$\Gamma (\vx,\vy) = -\Delta  + \left( m_0^2 + 
\frac{\lambda}{2}G(\vx ,\vx )
\right)\delta (\vx-\vy)$ with implicit integrations in these
equations. 
The  quantum fluctuations  are therefore described by
the two point function $G$:
\be  \label{10}
G_0({\bx},{\by })=<{\bx}|\frac{1}{\sqrt{-\Delta + \mu^2_0}}|{\by}>
\ee
where  $\mu^2_0$  is given by the self consistent GAP equation
(expression (\ref{7})):
\be \label{11} \ba{ll}
\displaystyle{ \mu^2_0 = m^2_0 + \frac{\lambda}{2} Trace G(x,x,\mu^2) 
+ \frac{\lambda}{2} \bphi^2 .}
\ea
\ee
>From the above expressions we can see that
the non zero solution for the condensate, $\bphi$, is given by:
\be \label{cond7} \ba{ll}
\displaystyle{ \bar{\phi}^2  = - 6 \frac{m_0^2}{\lambda}  
- 3 G (\mu^2_0) = 
\frac{ 3 \mu^2_0}{\lambda}.}
\ea
\ee
Given a  mass, $\mu^2_0$, and the coupling
constant, $\lambda$,
the bare mass $m_0^2$ can be fixed by the vacuum of the chosen phase
(symmetric - $\bphi = 0$ - or asymmetric- $\bphi\neq 0$). 

The above expression for the Gaussian width (\ref{10}) (and its
inverse $G^{-1}_0$) can be calculated
in the momentum space with a regulator $\Lambda$ (cutoff) yielding:
\be \ba{ll}
\displaystyle{ G(\mu^2_0) = \frac{1}{8\pi^2} \left( \Lambda^2 - 
\mu^2_0
Ln \left(\frac{d \Lambda}{\mu_0} \right) \right), }\\
\displaystyle{ G^{-1}(\mu^2_0) 
= \frac{1}{8\pi^2} \left( 2 \Lambda^4
+ 2 \mu^2_0 \Lambda^2 - \frac{\mu^4_0}{4} - \mu^4_0 Ln \left(
\frac{d \Lambda}{\mu_0} \right) \right)
}
\ea
\ee
where $d= 2/\sqrt{e}$. 
In the (local) 
limit of infinite cutoff the average energy 
and other observables would diverge and the divergences must be eliminated.
These 
solutions have been studied 
in  three dimensions for example in \cite{BARMOSHE,STEVENSON,BRANCHINA,KMV} 
as well as a renormalization procedure which will be considered
in the present work.


An usual renormalization procedure of the parameters of the
model is done as follows (which in principle can be called 
``precarious).
The divergent terms in the energy density of the 
symmetric phase
as well as in the GAP equation (\ref{11}) are subtracted
from the corresponding terms in the expressions of the asymmetric
phase. 
The resulting subtracted 
equations are written in terms of a renormalized
mass, coupling constant ($\lambda_R$) 
and a mass scale $\mu^2$ 
which eliminates the cutoff dependent
terms.
The expressions are the following:
\be \label{flow} \ba{ll}
\displaystyle{ m^2_R = \frac{ m^2_0 + 
\frac{\lambda \Lambda^2}{16 \pi^2} }{
1 + \frac{\lambda}{16 \pi^2} log\left( \frac{d \Lambda}{
\mu} 
\right)},
}\\
\displaystyle{ g_R = \frac{ - 
\frac{\lambda}{2} }{
1 + \frac{\lambda}{16 \pi^2} log\left( \frac{d \Lambda}{
\mu} 
\right)},
}
\ea
\ee
With which we can rewrite the (subtracted)
energy density  expression  ${\cH}_{sub} = {\cal H}(\bphi)
 - {\cal H}(\bphi = 0)$ as: 
\be \ba{ll}
\displaystyle{ {\cH}_{sub} = \frac{m_R^2}{2} \bphi^2 + 
\frac{1}{4 g_R} \left( m_R^2 - \mu^2 \right)^2 + 
\frac{1}{128 \pi^2} \left( m_R^4 Ln \left(
\frac{m_R^4}{\mu^4} \right) - m_R^4 + \mu^4
\right) , }
\ea
\ee
where the mass scale can be written from the GAP 
equation as defined in expression (\ref{11}):
\be \ba{ll} \label{MU2REN}
\displaystyle{ \mu^2 = m_R^2 + g_R \left( \bphi^2
+ \frac{m_R^2}{8 \pi^2} Ln \left( \frac{m_R}{\mu} 
\right) \right).}
\ea
\ee
It is seen from these expressions that 
in the limit of $\Lambda$ to infinity the bare
coupling constant would go to zero in order to keep
$g_R$ finite. 
This is related to the ``triviality problem but
it is possible to face this alternatively 
from the same equations by fixing
the value $\Lambda/\mu$ instead of only varying the cutoff
\cite{KMV}. 
$\mu$ is a mass scale to be fixed in the theory.
In fact the $\lap$ model is asymptotic free in the asymmetric phase
\cite{BRANCHINA,KMV,ASYMPFREE}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% de subsection to section

\section{ New Renormalized ``GAP equation}

In this subsection we derive new GAP equations by minimizing
the renormalized energy density ${\cH}_{sub}$
with relation to the renormalized mass:
$$ \frac{d {\cal H}_{sub}}{d m^2_R} = 0. $$

The derivative of the renormalized energy density with
relation to $m_R$ is therefore required
 to be zero. 
The roots of the resulting expression can be calculated
to find the minimum of the potential with relation to this
parameter.
The  new, renormalized,
GAP equation is given by:
\be \label{solutionmr} \ba{ll}
\displaystyle{ 0 = m_R^3 \left[ Ln 
\left(\frac{m_R}{\mu}\right)^2 a_1 + 
Ln \left(\frac{m_R}{\mu}\right) a_2 + a_3 
\right]
,}
\ea
\ee
where $a_i$ can be given in terms of 
$$J = 1 - \frac{g_R}{(8\pi)^2} = 1 - G_R, $$
 by:
\be \ba{ll}
\displaystyle{ a_1 = \frac{1}{g_R} J^2 
 + \frac{1}{32 \pi^2} 
,}\\
\displaystyle{ a_2 = \frac{2}{g_R} \left( -1 + J + 
\frac{J^2}{(32 \pi^2)} \right)+ \frac{1}{128 \pi^2}
\left( 1 + \frac{2 J^2}{(8 \pi)^2} \right)
,}\\
\displaystyle{ a_0 = -\frac{1}{g_R} - \frac{1}{32 \pi^2}
-\frac{1}{8. 128 \pi^4} \left( \frac{9}{8}  \right).
}
\ea
\ee
There are therefore 
five solutions for the renormalized mass $m_R^2$ which can be written
in the following form:
\be \ba{ll} \label{mpm}
\displaystyle{ m^3_R = 0, }\\
\displaystyle{ m^{\pm}_R = \mu exp( H^{\pm} ) ,}
\ea
\ee
where:
\be \ba{ll}
\displaystyle{ H^{\pm} = \frac{-a_2 \pm \sqrt{ a_2^2 - 4 a_1 a_0}
}{2 a_1}
.}
\ea
\ee
It is noted that there is a sort of ``scale invariance in these
expressions for $m_R^{\pm}$ for simultaneous changes in $\mu$.
The (degenerated) zero mass solutions correspond to a 
saddle point, they are not minima of the energy
density.
The stability of the others solutions are checked via the 
positiveness of the second derivative:
\be \ba{ll}
\displaystyle{\frac{d^2 \cH}{d m_R^2} = 
\frac{m_R^2}{(8\pi)^2} \left( 2 Ln
\left(\frac{m_R}{\mu}\right) a_1 + a_2 \right) > 0.
}
\ea
\ee
For the derivation of these expressions we have not used the 
completely self consistency of the Gaussian equations. 
There has been used a truncation on the dependence on $\mu$:
the dependence of $Ln(\mu)$ on $\mu$ (self consistency)
was considered only to first order.


In figures 1a and 1b the solutions of the above equations
($m^{\pm}/\mu$ from (\ref{mpm})) 
are shown as a function of $G_R = g_R/(32 \pi^2)$.
All the solutions of figure 1a correspond to stable solutions 
($d^2 \cH/ d m_R^2 >0$). The solutions of figure 1b are stable
for $G_R$  nearly equal or smaller than $-1.45$ or  
equal or greater  than nearly $1.25$.

Furthermore, values between $ -1 < G_R < 0$ do not correspond to physical 
stable values of the condensate
as it will be shown below, expression (\ref{COUPLING1}).
The point $g_R=0$ is not plotted.
In the limits of $g_R \to \pm \infty$ we obtain analytically 
that 
either $ m_R = \mu$ or $m_R=0$.
For the case $\mu \to \infty$ the renormalized coupling constant
$G_R \to 0$.

While the $m^{-}_R$ solution in the weak coupling regime 
can be identified to the renormalization point usually considered
(for $\mu >> m_R$ and/or the cutoff going to infinite)
we found another stable solution $m^+$ for which $\mu \simeq m^+$.
This seems to be due to the asymptotic freedom and it is not a 
``trivial solution.

%%%%%%%%%%%%%% nova section instead of subsubsection

\section{ The condensate: $\bphi$}

Usually the variational equation for the condensate is obtained from the 
regularized energy density:
\be \ba{ll}
\displaystyle{ \frac{d {\cal H}_{reg}}{ d \bphi} = 0
}
\ea
\ee
The minimization of the renormalized energy,  ${\cH}$, 
with respect to the $\bphi$, the condensate,
yields the following solutions:
% of new GAP equations:
\be \label{phi0} \ba{ll}
\displaystyle{ \bphi = 0,}\\
\displaystyle{ \bphi^2 = - \frac{m_R^2}{g_R} 
\left( 1 + \frac{1}{8 \pi^2} Ln \left(\frac{m_R}{\mu}\right) \right) .}
\ea
\ee
This last expression may coincide with the expression of $\bphi_0$
obtained from the minimization of the regularized energy density
(expression (\ref{cond7}))
depending on the relation between 
$\lambda$ and mass scale $\mu$.





Also, from the  expression (\ref{phi0}) we can deduce the 
following conditions from the sign of the coupling constant
to obtain real values of $\bphi$:
\be \label{COUPLING1} \ba{ll}
\displaystyle{ if:  g_R > 0  \to 
Ln \left(\frac{m_R}{\mu}\right) < - 8 \pi^2 ,}\\
\displaystyle{ if:  g_R < 0  \to Ln 
\left(\frac{m_R}{\mu}\right) > - 8 \pi^2 .}
\ea
\ee


The energy density is expected be stable for 
the condensate values found in expression (\ref{phi0}).
This stability if found by calculating the second derivative
of the energy density with relation to $\bphi$.
Its positiveness  corresponds to the condition:
\be \label{COUPLPOS} \ba{ll}
\displaystyle{ g_R \left( 1 + \frac{g_R}{32 \pi^2} 
\right) > 0 .}
\ea
\ee
>From this we can state that for positive coupling constant
$g_R$ can assume any value (from this stability criterium)
whereas if $g_R < 0$ one has to consider $g_R < -32 \pi^2$.


%\subsection{ Possible interpretation for the symmetry restoration}


Expression (\ref{phi0}) can be written as:
\be \label{bphinovo} \ba{ll}
\displaystyle{ g_R \bphi^2 = - m_R^2 \left( 1 + \frac{1}{8 \pi^2}
Ln \left( \frac{m_R}{\mu} \right) \right).
}
\ea
\ee
When $\mu = m_R exp(8 \pi^2)$ we see that either $\bphi=0$ or $g_R=0$
in the asymmetric phase of the potential. 
This may correspond to the so called symmetry restoration 
when the condensate disappears at a particular
high excitations energy.

Expressions (\ref{COUPLING1}) and (\ref{COUPLPOS}) are constraints for the 
values that the renormalized coupling may assume in order to 
yield stable real ground states.

%\centerline{*****}

The above expression for the condensate 
(\ref{phi0}) can therefore be equated to the 
previous (regularized) expression  (\ref{cond7}).
Taking into account the flow of the renormalized coupling constant
in terms of the bare one (expression (\ref{flow}))
this equality will be satisfied whenever the following expression 
holds:
\be \label{constraint} \ba{ll}
\displaystyle{ \lambda = 
\frac{16 \pi^2}{Ln \left(\frac{\Lambda d}{\mu}\right) }
\left( -1 + \frac{3}{2 \left( 1 + \frac{1}{8\pi^2} Ln \left(
\frac{m_R}{\mu} \right) \right) } \right) .
}
\ea
\ee
If the cutoff is sent to infinite the bare coupling constant would
assume different values depending on the values of $\mu$ and $m_R$.
We have assumed, as it usually is, that the minimum of the 
effective potential with relation to the condensate coincides
necessarily with its minimum in respect to the physical mass $m_R^2$.
For example, we can have a case in which $\lambda = 0$
if either $\Lambda \to \infty$ for finite $\mu$
or  $m_R/\mu = exp(4\pi^2)$.
However we see that varying $\mu$ together with $\Lambda$
there may have non zero  $\lambda$ solutions. 
Furthermore for $\Lambda/\mu$ finite, the coupling $\lambda$
may diverge when $m_R\mu = exp(-8 \pi^2)$.
This is the same point found above (for expression (\ref{phi0}) 
for the possible restoration
of the symmetry.

\section{ Analysis of the renormalized 
coupling constant values } 



One relevant subject for any approximation method is 
the understanding of the range of values of the parameters of the 
model (as mass and mainly coupling constants) 
for which the approximation is more appropriated. 
Let us check whether there are values for 
the renormalized coupling constant for which 
 the renormalized energy density is minimum. 
Besides that, this section can also be considered a preliminar
calculation complementary to the renormalization group 
of the model in the frame of the Gaussian approximation.

For this, we calculate:
$$d {\cal H}/ d g_R = 0.$$
It is  considered, in the following, a truncation
of the self consistency of the GAP equations.
This is done 
 by taking the scale parameter to be close
to the renormalized mass $\mu^2 = m_R^2 + \delta$, where $\delta << m_R^2$ 
is determined from the GAP equation self consistently.
>From the GAP equation (expression (\ref{MU2REN})) we have that:
\be \label{delta} \ba{ll}
\displaystyle{ \delta = \frac{g_R \bphi^2}{ 1 + \frac{g_R}{16 \pi^2}}
.}
\ea
\ee
The minimization of the renormalized energy yields the following 
equation:
\be \label{MINICOUPL} \ba{ll}
\displaystyle{ (G_R')^3 + (G_R')^2 \left( 3 + (1 + H)\frac{1}{16 \pi^2} 
\right) + G_R' \left( 3 + (1 + H)\frac{3}{2} + 
(1+ H)\frac{1}{32 \pi^2} \right) + 1 + \frac{(1+H)^2}{2} = 0,
}
\ea
\ee
where $H = Ln \left(\frac{m_R}{\mu}\right)/(8 \pi^2)$ 
and $G_R' = g_R/(16 \pi^2)$.

In figures 2a, 2b and 2c the solutions of expression (\ref{MINICOUPL})
are showed as  function of a limited range of 
$H$, i.e.,  $Ln(m_R/\mu)$. 
There are degenerated solutions in figures 2b and 2c.
It is plotted only the region in which the above truncation scheme
of the self consistency may be  expected 
to be reliable.

The stability of the solutions of the above equation can
also be verified. This is done by analyzing the 
positiveness of the second derivative:
\be \label{STABGR} \ba{ll}
\displaystyle{ \frac{d^2 {\cal H}_{sub}}{d g_R^2} > 0
.}
\ea
\ee
All the solutions found 
present a negative second derivative.
 These solutions will be studied at length in the frame of the 
renormalization group elsewhere.


It is very interesting to notice that the solutions for the 
coupling constant of expression (\ref{MINICOUPL}) 
depend only on the ratio $m_R/\mu$ and not 
on the absolute values of these parameters. 
This  corresponds
to a scale invariance for different 
physical processes at different
energy scales with different physical masses.


\subsection{ Fixing the energy density}

In \cite{FLB2001PRD} it was present one argument
for fixing the model parameters from a regularized expression of 
the total energy of a system. We propose here, analogously,
an expression (of  constraint) to the renormalized
coupling constant. 
For the analysis of a physical process
 whose energy density is given by $\cH_{sub}$,
a given energy scale  given by $\mu$ we 
find values for the renormalized coupling constant.
This corresponds to fix the renormalized mass and 
energy scale (${\cal H}_{sub}$) of a process and to calculate
the allowed physical coupling constants for the process involved
at a scale $\mu$.
We obtain a third degree algebraic equation which can
be writen as:
\be \label{CONDCOUPL} \ba{ll}
\displaystyle{ g^3_R \frac{H^2}{128 \pi^2} +
g_R^2 \left( \frac{H^2}{4} - \frac{2 H^2}{x} \right)
+ g_R \left( -\frac{H(H+1)}{2} + \frac{1}{x} \left(
\frac{H}{4} - 1 + H^2 \right) - \frac{{\cal H}_{sub}}{m_R^4}
\right) -\frac{1}{4} + \frac{H^2}{4} = 0.
}
\ea
\ee
where $x= 128 \pi^2$ and $H=Ln(m_R/\mu)/(8\pi^2)$.

This expression also presents a sort of  ``scale invariance for 
the parameters $m_R/\mu$ unless for the term which depends on the
energy density if ${\cal H}_{sub}$ scales differently from $m_R^4$.

In figures 3a, 3b and 3c we show the solutions of this algebraic
equation as function of $H$ for a fixed energy density 
$\cH = (100 MeV)^{-4}$ and $m_R = 100 MeV$. 
There is almost no deviation in the numerical 
results for a very large range of the ratio $\cH/m_R^4$.
 $g_R$ can be strong in the region of 
$\mu \simeq m_R$.
And in the limit of $\mu = m_R$ we found an unique
value which is given by:
\be \label{grHfix} \ba{ll} 
\displaystyle{ g_R = - \frac{1}{4 \left( \frac{\cH}{m_R^4} 
+ \frac{1}{128\pi^2} \right)
}.
}
\ea
\ee

Besides that, for $H \to - \infty$, which is 
equivalent to $\mu/m_R \to \infty$, we see that
$g_R \to 0$. 
However
solutions of figure 3b and 3c does not correspond to $g_R \to 0$, but
to a finite (quite strong) value close to 10 in agreement to the above
conclusions (previous sections).

\section{ Summary}

A further analysis of the usual 
renormalization scheme for the variational
Gaussian approximation were analyzed and 
some new consequences were found. 
New renormalized  
GAP equations were derived by minimizing the 
renormalized energy density for example with respect to 
the renormalized mass.
Five solutions were found which can correspond to stable
vacua in the some ranges of the renormalized coupling constant.
A ``scale invariant algebraic expression was derived in
this calculation.
New values for $\bphi$, in the vacuum, were found
by minimizing the effective potential with relation 
to it. From this expression we noted that either the 
``condensate'' or $g_R$ disappears when the mass scale 
(introduced in the renormalization procedure)
assumes the value $\mu = m_R exp(8 \pi^2)$. 
This can be seen
as a restoration of the spontaneously symmetry breaking.
Constraints for the possible values of the renormalized 
coupling constant were found.
We found, in particular, that $g_R$ can be positive or
negative and eventually 
very strong being the renormalization
mass scale $\mu$ a relevant parameter to 
fix its value.
The coupling constant fixes the values 
of the renormalized mass which yield a minimum of the 
effective potential.
A minimization of the effective potential with respect to 
the coupling constant was also performed in the limiting 
case that the mass scale $\mu$ is close to the physical mass. 
This is a way 
of truncating the self-consistency of the approximation.
A sort of ``scale invariant equation was obtained.
No stable solution for $g_R$ (considered without the whole
renormalization group equations) was found within the 
truncation scheme done for the self consistency.
With the present work we expect to have shown
that  a renormalized theory can provide 
some information on
the physical values of the coupling parameters
(renormalized mass and coupling constant).
The ground state in the frame of variational
approximation is found by the 
simultaneous minimization with respect to a physical mass
and to the classical field (condensate) $\bphi$ which are 
variational parameters (given by expressions (\ref{7}). 
However we still would like to point out that this may not be true
in the exact ground state, i.e., the energy must be minimum 
but it can be that it is minimum with respect to a 
particular combination of (physical) variables differently 
from what is usually considered. 
This may be a  guide for the development of other variational 
principles \cite{FLBcont}.

\vspace{1cm}

{ \Large{ \bf Acknowledgements}  }

This work has been supported by FAPESP, Brazil. 
The author would like to thank the hospitality of the 
Theory Group of Physics Department of BNL where a revision
of the work has been made.


\begin{thebibliography}{11}

\bibitem{BAGAN} T. Barnes and G.I. Ghandhour, 
Phys. Rev. {\bf D22}, 924 (1980).

\bibitem{STEVENSON} P.M. Stevenson, Phys. Rev. {\bf D32}, 1389 (1985).

\bibitem{BARMOSHE} W.A. Bardeen, M. Moshe, Phys. Rev. {\bf D 28}, 1372 (1983). 

\bibitem{KMV} A.K. Kerman, C. Martin and D. Vautherin, Phys. Rev. {\bf D47},
632 (1993).

\bibitem{BRANCHINA}  V. Branchina, P. Castorina, M. Consoli, 
D. Zappal\`a, Phys. Rev. {\bf D 42}, 3587 (1990).

\bibitem{ASYMPFREE} C.M. Bender, K.A. Milton, Van M. Savage, 
Phys. Rev. {\bf D 62}, 085001 (2000).

\bibitem{KERVAU} A.K. Kerman, D. Vautherin, Ann. Phys. (N.Y.) 
{\bf 192}, 408 (1989). 

\bibitem{SCHRODINGER}  R. Jackiw, {\it in} {\em Field Theory and 
Particle Physics-
V J.A.Swieca Summer School}, eds. O. \'Eboli, M. Gomes, A. Santoro (World
Scientific, Singapore, 1990). 


\bibitem{FLB2001PRD} F.L. Braghin, 
Phys. Rev. {\bf D 64} 125001 (2001).


\bibitem{FLBcont} F.L. Braghin, {\it in preparation}.

\end{thebibliography}

\vspace{2cm}

%\newpage

{\bf Figure captions}

\vspace{1cm}

\noindent{ \bf Figure 1a -} First solution of expression (\ref{solutionmr})
 - a new GAP equation - for 
the ratio of the renormalized mass to the mass scale $\mu$  as a 
function of $g_R/(8\pi^2)$.

\vspace{1cm}

\noindent{ \bf  Figure 1b -} Second solution of expression (\ref{solutionmr}) 
- a new GAP equation - for 
the ratio of the renormalized mass to the mass scale $\mu$  as a 
function of $g_R/(8\pi^2)$.

\vspace{1cm}

\noindent{ \bf Figure 2a -}  First solution of expression (\ref{MINICOUPL}) 
for the renormalized coupling constant -
from the minimization of the energy density with relation to the renormalized 
coupling constant - as a function of $H = Ln(m_R/\mu)/(8\pi^2)$.

\vspace{1cm}

\noindent{ \bf  Figure 2b -} Second solution of expression (\ref{MINICOUPL}) 
for the renormalized coupling constant -
from the minimization of the energy density with relation to the renormalized 
coupling constant - as a function of $H = Ln(m_R/\mu)/(8\pi^2)$.

\vspace{1cm}

\noindent{ \bf Figure 2c -} Third solution of expression (\ref{MINICOUPL}) 
for the renormalized coupling constant -
from the minimization of the energy density with relation to the renormalized 
coupling constant - as a function of $H = Ln(m_R/\mu)/(8\pi^2)$.

\vspace{1cm}

\noindent{ \bf  Figure 3a -} First solution for the 
renormalized coupling constant of  expression (\ref{CONDCOUPL}) -
fixing $\cH = (100 MeV)^4$ and $m_R= 100 MeV$ - as a function of
$H = Ln(m_R/\mu)/(8\pi^2)$. 


\vspace{1cm}

\noindent{ \bf Figure 3b -} Second solution for the 
renormalized coupling constant of  expression (\ref{CONDCOUPL}) -
fixing $\cH = (100 MeV)^4$ and $m_R= 100 MeV$ - as a function of
$H = Ln(m_R/\mu)/(8\pi^2)$. 

\vspace{1cm}

\noindent{ \bf Figure 3c -} Third solution for the 
renormalized coupling constant of  expression (\ref{CONDCOUPL}) -
fixing $\cH = (100 MeV)^4$ and $m_R= 100 MeV$ - as a function of
$H = Ln(m_R/\mu)/(8\pi^2)$. 


\end{document}

