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\begin{document}
\begin{titlepage}

\title{Non-perturbative solution of metastable scalar models\\
Test of renormalization scheme independence}

\author{Vladim\'{\i}r \v{S}auli\footnote{{ \it Email address:} sauli@ujf.cas.cz}\\
{\footnotesize \it Department of Theoretical Nuclear Physics, 
Nuclear Institute,'Re\v{z} near Prague}}

\maketitle

\begin{abstract}
Schwinger-Dyson equations for propagators 
are solved for the scalar $\Phi^3$ theory and massive   Wick-Cutkosky model.
 With the help of integral representation  the results
are obtained directly in Minkowski space in and beyond bare vertex approximation.
Different by the finite strength field renormalization function  $Z$, the various  
renormalization scheme are employed. The $S-$matrix is puzzled from the Green's function
and the  convergence of the solutions is observed  when the vertex corrections
are added.
The numerical solutions break down for certain critical value of the coupling constant, for
which the on-shell renormalized propagator starts to develop the unphysical singularity at very high space-like
square of momenta.\\ \\
{\it{PACS:}} {11.10.Gh, 11.15.Tk}\\  \\ 
{\it Keywords:} {Dyson-Schwinger equation, Renormalization, Spectral
representation}
\end{abstract}

\end{titlepage}



\section{Introduction}

The Dyson-Schwinger equations (DSE) are an infinite tower of coupled integral 
equation relating Green functions of the quantum field theory. If solved 
exactly, they would provide solutions of the underlying quantum field theory. 
In practice, the system of equations is truncated and one hopes to get some 
information, in particular on the solution in the non-perturbative regime,    
from solving the simplest equation for the two-point Green functions- the 
propagators. The other vertex functions, which also enter the DSE for the propagator,
are either taken in their bare form or some physically motivated Ansatze is employed.

In most papers dealing with the solution of DSE's the Wick rotation from the Minkowski 
to Euclidean space is used to avoid singularities of the kernel inherent to the physical 
Green functions. To our knowledge, the only exception is the series of papers 
\cite{SALAM},\cite{DELB},\cite{DELB1},\cite{DELB2} employing so-called "Gauge technique" in quantum 
electrodynamic and its gauge invariant extension to quantum chromodynamic \cite{CORNWAL},
which work represents the born of the "Pinch Technique". Until now, the last mentioned 
approach  has been never used in its non-perturbative context. Although, not dealing with gauge 
theories, similarly to these techniques we instead solve the  directly in the momentum space,
 making use of the known analytical structure of the propagator, expressed
via the spectral decomposition. Spectral or dispersive technique write down the Green function
as spectral integral over certain weight function and denominator parameterizing known
or assumed analytical structure. The generic spectral decomposition of the renormalized
 propagator reads:

\be
 G(p^2) = \int d \alpha \frac{\tilde{\sigma}(\alpha)}{p^2- \alpha- i \epsilon} \  ,
\label{genspec}
\ee 
where $\tilde{\sigma}(\alpha)$ is called Lehmann weight or simply spectral function. 
If the threshold is situated above the particle mass as it is for the stable 
(and unconfined) particles then the spectral function typically look like 

\be
\tilde{\sigma}(\alpha)=r\delta(m^2-\alpha)+\sigma(\alpha)
\ee
where singular delta function corresponds with non-interacting fields
and  $\sigma $ is appeared due to the interaction. Finite parameters  $r$ then represents the propagator
residuum and is simply related to field renormalization.
 It is also supposed that  $\sigma$ is a 
positive regular function which is spread smoothly from the zero at the threshold. 
Note here, that the positivity of Lehmann weight is not required for our solution, 
but the models studied in this paper naturally embodied this property 
(see for instance \cite{SCHWEBER}, or any standard textbook).

Putting the spectral decomposition of the propagators and the expression for the vertex 
function into the DSE, allows one to derive the real integral equation for the      
weight function $\sigma(\alpha )$. This equation involves only one real principal value 
integration  and can be solved numerically by iterations. 
Our solutions are obtained both for space-like and time-like propagator momenta, getting
this in the Euclidean approach would require  tricky backward analytical continuation.
Since all momentum integration are performed analytically,
there is no numerical uncertainty following from the renormalization which is usually present
in Euclidean formalism \cite{ROBERT}. Here the renormalization procedure is performed
analytically with the help of the direct subtraction in momentum space. This perturbative perfectly known 
renormalization scheme (see \cite{COLLINS} for scalar models  and for instance
 \cite{SIVERS} for QED case, where also the comparison with other perturbative renormalization schemes were made) has been already
applied to the QED and Yukawa model \cite{SAULI} in its non-perturbative context. In this paper the off-shell
momentum subtraction renormalization scheme was introduced and used. In order to simplify the technique and 
order to compare various schemes we restrict ourself on the choice of on mass-shell
 subtraction point $\mu=m$.

In this work, we would like to present the certain solutions of rather obscure 
theories- $\Phi^3$ and $\Phi_i^2\Phi_j$ scalar models (the second one will be reffered as the (generalized) Wick-Cutkosky model (WCM)). 
In fact, not only models mentioned above but the all super-renormalizable four-dimensional scalar models  are not properly
defined since they have no true vacuum \cite{BAYM}, instead of this they have   only metastable vacua
 (here we assume  non-zero masses of all particle content, in the oposite case the appropriate
 classical potential would not posses a local minimum).
 Instead of discarding these type of models, as sometimes happens, we  look
 whether this 'inconsistency' can be captured  by the formalism of  DSE's, or whether
 the appropriate solutions 'behaves ordinary'.
The property of  super-renormalizability  makes our models particularly suitable for this purpose.
Actually, the super-renormalizability  here implies  the finiteness of the renormalization field constant $Z$ which 
therefore  can not be considered at all  (i.e.$Z=1$).
In the case of $\Phi^3$ theory we do not fully omit the field renormalization  
and within the help of the  appropriate choice of the constant $Z$ we choose  the given  renormalization scheme.
Making this explicitly and after the evalution of the scattering amplitude we look (in each scheme) whether 
the observables do converge (in all schemes) to some experimentally measurable values of the virtual scalar world .
As the suitable observables we choose
the amplitude $M$ for the scattering process  $\Phi\Phi\rightarrow \Phi\Phi$ 
and we have no find any pathology behavior (up to the certain large value of the coupling).
Instead of this, after when the approximation of the full solutions improve, 
we will see that the amplitudes calculated in the various renormalization scheme 
tends to converge to each other,  i.e. in this aspect, 
the $\Phi^3$ theory behaves as  the ordinary and physically meaningful one.
Here, this is the right place, to note that the models with the metastable ground state serve
as an useful methodological tool, the role in which they are often employed.
In fact, $\Phi^3$ theory  serves as a good ground for the study of the various phenomena 
 \cite{CORNWALL5}, \cite{CORNWALL6}, \cite{DELBOUN},\cite{KREIMER}
(including phenomena like non-perturbative asymptotic freedom  and non-perturbative renormalization). 
There also exist the number of papers dealing with WCM. The DSE's 
for propagators of WCM in their simple bare vertex approximation have been solved for purpose of
calculation relativistic bound states \cite{AHLIG} (for other recent work dealing the bound states problem  within the WCM see
\cite{TJON} and ref's. therein). For purpose of  the comparison with \cite{AHLIG}   we solve the
exactly analogical Minkowski problem. The obtained value of the critical coupling 
should  depends on the renormalization scheme. Having this slight dependence under the control it allows us to compare
with the other non-perturbative method \cite{ROSEN1},\cite{ROSEN2}. 
The comparison with conventional perturbation theory is  also made.

  Regardless of the facts mentioned above, we are far away to conclude that $\Phi^3$   model, 
is fully physically satisfactory one, since we do not know anything about the full solution.
 At this point, the study presented in this paper and the  studies
of $\Phi^3$ model in five \cite{CORNWALL5} and six \cite{CORNWALL6} 
dimensions are conclusive by the similarly cautious way. Probably more sophisticated conclusion
could be obtain by some lattice study, which are not yet done for this purpose.
    

In the next Section we present the DSE's for $\Phi^3$ for propagator and vertex function, subsequently we discuss 
the renormalization procedure and rewrite the propagator equation into its spectral form. Also the numerical results 
and limitations are discussed.
The Wick-Cutkosky model is dealt with in the Section 3. It is solved numerically in its  pure bare vertex approximation.
 The details of calculations are relegated
to the Appendices A and B.


\section{ $  \Phi^3  $ theory }

\subsection{Dyson-Schwinger equation  for  $  \Phi^3  $ theory} 

The Lagrangian density for this model  reads

\be \label{lagr}
{\cal L}=\frac{1}{2}\partial_{\mu}\phi_0(x)\partial^{\mu}\phi_0(x)-
\frac{1}{2}m_0^{2}\phi_0^{2}(x)-g_0\phi_0^{3}(x)
\ee

where index $0$ indicates the unrenormalized  quantities. With the help of the functional differentiation of generating functional
(for this procedure, see for instance \cite{ITZYKSON})
with classical action determined by (\ref{lagr}) one gets the following DSE (after transforming into the momentum space)
for the inverse propagator

\bea  \label{UDSE}
G_0(p^2)&=&p^2-m_0^2-\Pi_0(p^2)
\nn \\
\Pi_0(p^2)&=&i3g_0 \int\frac{d^4q}{(2\pi)^4}\Gamma_0(p-q,q)G_0(p-q)G_0(q)
\eea 


where $\Gamma_0$ is the full irreducible  three-point vertex function
which satisfies its own DSE (\ref{dseforrg}). The integral of $\Pi_0$ is divergent and requires
the mass renormalization. We will regularize $\Pi_0$ by on-mass-shell subtraction

\be
\Pi_{R1}(p^2)=\Pi_0(p^2)-\Pi_0(m^{2})
\ee
where $m$ is the pole "physical" mass, defined by $G^{-1}(m^2)=0$
Defining the mass counter-term

\be
m^2=m_0^2-\delta m^2, \quad \delta m^2=\Pi_o(m^2)
\ee
and introducing additional finite renormalization constant

\be\label{konvence}
\phi_0= \sqrt{Z}\phi ; \quad 
g_0=g\frac{Z_g}{Z^{\frac{3}{2}}}
\ee
we get the inverse of the full propagator in term of physical mass

\bea \label{DSE2}
G(p^2)&=&Z(p^2-m^2)-\Pi_1(p^2)
\nn \\
\Pi_1(p^2)&=&Z\left(\Pi_0(p^2)-\Pi_0(m^2)\right)
\nn \\
Z\Pi_0(p^2)&=&i3g^2 \int\frac{d^4q}{(2\pi)^4}\Gamma(p-q,q)G(p-q)G(q)
\eea 

where $g$ is a renormalized coupling and the constant $Z_g$ corresponds with the renormalization of  the vertex
function , and $G$ represents the  renormalized propagator with respect to the field strength
renormalization,i.e.

\be
\Gamma=Z_g\Gamma_0, \quad \quad G_0(p^2)=ZG(p^2)
\ee

We closed the system of DSE's already at the level of equation for proper vertex.
Instead of solving the full renormalized DSE for the vertex

\be   \label{dseforrg}
g\Gamma(p,l)=6g_0 i\int\frac{d^4q}{(2\pi)^4}\Gamma^{[3]}(p,q)G(q)G(l-q)M(q,l,p)
\ee
we approximate it by the first two terms of the appropriate skeleton expansion

\be\label{vert}
g\Gamma^{[3]}(p,l)=6g +i(6g)^3\int\frac{d^4q}{(2\pi)^4}G(q)G(p-q)G(l-q).
\ee
i.e., we approximate the vertex inside the loop by its bare value and 
the scattering matrix $M$ in Eq. (\ref{dseforrg})   is taken in its dressed tree approximation,i.e.,  $M=G$.

The equation for propagator is solved for bare vertex (first term in (\ref{vert}))and with the second term
included at each renormalization scheme separately. We will define them in the following section.


\subsection{Choosing the scheme}

We will assume (or rather we will neglect it) that there interaction does not create  
the bound states contributing to the weight function
$\sigma$. We will call the {\it minimal momentum subtraction renormalization scheme} (MMS) which is the scheme
 where only mass subtraction is used and where the field leaves unrenormalized, i.e. $Z=1$.
Therefore, we can write the spectral decomposition for the propagator  and for the self-energy $\Pi$ in the following form.

\bea \label{vrku}
G(p^2)&=& \frac{r}{p^2-m^2}+\int d \alpha \frac{\sigma(\alpha)}{p^2- \alpha- i \epsilon}
\nn \\
&=&\left\{p^2-m^2-\Pi_1(p^2)\right\}^{-1}
\nn \\
\Pi_1(p^2)&=& \int_{thres.}^{\infty}d\alpha
\frac{\rho(\alpha)(p^2-m^2)}{(\alpha-m^2)(p^{2}-\alpha+i\epsilon)}
\eea

where $\pi\rho(s) $ represents the self-energy absorptive part. 
Obviously, in this MMS the propagator does not have the pole residue $r$ equal to unity

\bea \label{reziduum}
&&\lim_{p^2\rightarrow m^2}(p^2-m^2)G(p^2)=\left[1-\frac{d}{d p^2}\Pi_1(p^2)|_{p^2=m^2}\right]=r
\nn \\
&&\frac{d}{d p^2}\Pi_1(p^2)|_{p^2=m^2}=
\int d\alpha\frac{-\rho(\alpha)}{(\alpha-m^{2})^2}
\eea

After a simple algebra and taking the imaginary part of the Eq. (\ref{vrku}) we arrive 
to a relation between the spectral functions $\sigma$ and $\rho$

\be    \label{symb}
\sigma(\omega)=\frac{r\rho(\omega)}{(\omega - m^2)^2}+
\frac{1}{\omega - m^2}
 P.\int d\alpha \frac{\sigma(\omega)\rho(\alpha)\frac{\omega-m^2}{\alpha-m^2}
+\sigma(\alpha)\rho(\omega)}{\omega-\alpha}
\ee
where  $P.$ stands for principal value integration.

This is the first of equations which we actually solve. We are discussing it in some details
since its form depends only on the adopted renormalization procedure, not on the actual form of the interaction, 
neither on the approximation employed for the vertex function $\Gamma$ in the DSE for the propagator. 
The second equation connecting $\sigma$ and $\rho$ does depend on the form of vertex. Its derivation
is more complicated and we deal with it in the Appendix A.  

In some cases, the form (\ref{symb}) is not the most convenient one 
(for instance, when we want to look the bound state spectrum influence causes just by the self-energy effect (\cite{SAULI2})) .
Note the presence of the constant $r$ in the first term on the r.h.s.,
it has to be determined from the relations (\ref{reziduum}) after each iteration. To get rid of this we 
define the usual {\it on-shell renormalization scheme with unit residuum} (OSR) by  

\be
Z=1+\delta Z
; \quad \delta Z=\frac{d}{d p^2}\Pi_1(p^2)|_{p^2=m^2}
\ee
which gives the standard receipt how to calculate the OSR propagator

\bea  \label{prt}
G^{OSR}(p^2)&=&\left\{p^2-m^2-\Pi_2(p^2)\right\}^{-1}
\nn \\
\Pi_{2}(p^2)&=&\Pi(p^2)-\Pi(m^{2})-\frac{d}{d p^2}\Pi(p^2)|_{p^2=m^2}(p^2-m^2)
\nn \\
\Pi(p^2)&=&i3g_2^2 \int\frac{d^4q}{(2\pi)^4}\Gamma(p-q,q)G(p-q)G(q)
\eea

and subsequently implies the spectral decomposition for $G^{OSR}$ and $\Pi_{2}$

\bea \label{brku}
G_{OSR}(p^2)&=& \frac{1}{p^2-m^2}+\int d \alpha \frac{\sigma_2(\alpha)}{p^2- \alpha-i \epsilon}
\nn \\
\Pi_2(p^2)&=& \int_{thres.}^{\infty}d\alpha
\frac{\rho_2(\alpha)(p^2-m^2)^2}{(\alpha-m^2)^2(p^{2}-\alpha+i\epsilon)}.
\eea

The relation between $\sigma_2$ and $\rho_2$ is now derived by the same way as before and it reads

\be    \label{symb2}
\sigma_2(\omega)=\frac{\rho_2(\omega)}{(\omega-m^2)^2}+\frac{1}{\omega-m^2}
P.\int d\alpha\frac{\sigma_2(\omega)\rho_2(\alpha)\left[\frac{\omega-m^2}{\alpha-m^2}\right]^2
+\sigma_2(\alpha)\rho_2(\omega)}{\omega-\alpha}.
\ee

Note, that the Eqs. (\ref{symb}),(\ref{symb2}) are  inequivalent due to the scheme difference, the appropriate dependence   
of the weights $\rho$ and $\rho_2$  on the coupling constant $g $ and $g_2$ is explicitly written 
in the Appendix A and the Appendix B., respectively
( Two   inequivalent renormalization schemes should give the different Green function, but should give
the same S-matrix). 
 
In the end of this Section we very briefly discuss dimensional renormalization prescription 
\cite{DIMREG}, showing here that it is fully equivalent to MMS to all orders (Note that the perturbation theory 
is naturally generated by the coupling constant expansion of the DSE's solution). For this purpose we choose
 the {\it modified minimal subtraction} $\bar{MS}$ scheme, noting that any 
other sort of schemes based on the dimensional regularization method would be treated by the same way.
Since the only  infinite contribution are affected when this renormalization is applied, therefore
the contribution with the dressed vertex (master diagram and so that) satisfies the unsubtracted dispersion relation
while for instance the one loop skeleton self-energy  diagram 
(in a fact the only one irreducible contribution which is infinite in four dimension) looks
(for space-like momenta)

\be
\Pi_{\bar{MS}}^{[1]}(p^2)=\frac{18g^2}{(4\pi)^2}ln\left\{\frac{m^2-p^2x(1-x)}{\mu_{t'Hooft}}\right\}+\Pi(p^2)_{finite}
\ee
where $\Pi_{finite}$ represents the omitted the finite (high order) terms which are not affected by (any) 
dimensional renormalization. 

The inverse of the full propagator reads in this scheme

\be
G_{\bar{MS}}^{-1}(p^2)=p^2-m^2(\mu_{t'Hooft})-\Pi_{\bar{MS}}^{[1]}(p^2)-\Pi(p^2)_{finite}.
\ee
Identifying the pole mass by equality  $G_{\bar{MS}}^{-1}(p^2=m^2_p)=0$
we simply arrive to the result

\be
G_{\bar{MS}}^{-1}(p^2)=p^2-m_p^2-\Pi_1(p^2)
\ee
where 

\bea \label{iden}
m_p^2=m^2(\mu_{t'Hooft})+\Pi_{\bar{MS}}^{[1]}(m_p^2)+\Pi(m_p^2)_{finite}
\nn \\
\Pi_1(p^2)=\Pi_{\bar{MS}}^{[1]}(p^2)-\Pi(p^2)_{finite}
-\Pi_{\bar{MS}}^{[1]}(m_p^2)-\Pi(m_p^2)_{finite}
\eea

Since the pole mass is renormgroup invariant quantity
we see that $\bar{MS}$ scheme exactly corresponds with the one subtraction renormalization scheme,i.e. the  MMS.
Note here, that in renormalizable models such identification is a not so straightforward but always possible
\cite{SIVERS}. Of course, the appropriate identification is then  rather complicated.
 To conclude this section, we see that the popular renormalization prescription like $MS$ or $\bar{MS}$ schemes
can be ordinarily used  in the non-perturbative context. At this point we disagree with the opposite statement of 
the paper \cite{TONIK}.

\subsection{Scattering matrix}

The physical observables should be invariant not only with respect to the choice 
of renormalization scale, but also with  respect of the choice of renormalization scheme. 
The first invariance is more then manifest in our approach, 
since all the quantity  used here are  the renormgroup invariants.
The second mentioned invariance is less obvious and in fact is clear only for some very simple cases 
(the most simple case is  the tree level  amplitude evaluation,
there the residua of the propagators may be exactly absorbed into the redefinitions of the coupling constants, but,  
of course in this case the renormalization is not required ).
In any reasonable renormalizable quantum field theory it is strongly
 believed that the any how obtained exact  Green functions must build the same $S$-matrix. 
In perturbation theory, one has usually several first term of perturbation expansion 
and one hope that the the perturbation series is sufficiently fast convergent when the 
"right" choice of renormalization scheme was made \cite{SIVERS},\cite{QCD}.
In DSE treatment we can talk about the level of their system truncation  
instead of about given order of the couplings. In this Section we compare the bare vertex solutions with the  improved vertex ones 
and then we look whether the 
physically measurable quantity calculated in different schemes  have or have not tendency to converge
to some common values.


To proceed this  explicitly we calculate the matrix element $M$ of the elastic scattering process 
$\phi\phi \rightarrow \phi\phi$

\be \label{emko}
M(s,t,u)=\sum_{a=s,t,u}\Gamma G(a)\Gamma+...=
\sum_{a=s,t,u}(6g)^2G(a)+...
\ee
in the both renormalization scheme. The letters $s,t,u $ in (\ref{emko}) represent the usual Mandelstam variables
that satisfies $s+t+u=4m^2$, since now, the external particles are on-shell and the dots means the neglected boxes and crossed boxes
contributions.

Using the notations introduced in the previous Section then the matrix $M$ in MMS scheme is calculated like

\be
M^{MMS}=\sum_{a=s,t,u}(6g)^2G(a), 
\ee
where the propagator is calculated through  (\ref{vrku}),(\ref{reziduum}),(\ref{symb}).
 For OSR, the scattering matrix is puzzled like

\be
M^{OSR}=\sum_{a=s,t,u}(6g_2)^2G_{OSR}(a) 
\ee
where the propagator is calculated through (\ref{prt}),(\ref{brku}),(\ref{symb2}) and the  relations for $\rho$'s
are reviewed in the Appendix A,B.

In ideal case we would obtain $M_{MMS}=M_{OSR}$ which should be true at each channel separately.
They would be equal (compare pole an absorptive parts)  if and only if 

\bea  
g_2^2&=&rg^2    
\nn \\
g_2^2\sigma_2(\omega)&=&g_2\sigma(\omega) \label{pisek}
\eea 

for all $\omega$. This exact scheme independence is broken due to truncation of DSE's and 
 the simple test of scheme dependence is to see the deviation from the exact equalities  given by Eqs..(\ref{pisek}) 
 The appropriate cook-book how to do it explicitly  is as follows:
Let us calculate the propagator   independently for both schemes
with the ratio of coupling $G-2/g$ adjusted  to be  $\sqrt{r}$ and check whether the ratio of the
propagators is given by $r$.
 
This what we actually did and the obtained result we describe in the next section, noting here,
that the next to leading order of $M$ within the constraint (\ref{pisek}) is exact scheme invariant 
and the only difference therefore follows from the remnant of the full DSE solution.

\subsection{Results}

The integral equation for Lehmann weights have been solved numerically by the 
method of iteration. These weight function solutions, obtained for several hundred of 
mesh-points and using  sophisticated integrator, have an accuracy of 
approximately 1 part of $10^4$ for reasonable value  ($\lambda<<\lambda_{crit}$) of the coupling strength 
$\lambda $ and   increase (up to several \%) when  $\lambda\simeq\lambda_{crit}$. 
The coupling strength  is defined to be a dimensionless quantity

\be \label{lambda}
\lambda=\frac{18g^2}{16\pi^2 m^2}.
\ee

The critical value of  $\lambda$ is simply defined by the collapse of ( numerically sophisticated)
solution of the imaginary part DSE's. Before making a 
comparison of physical quantities we present the numerical results for the
Green's functions. In Fig.1 the so called dynamical mass 

\be
M(p^2)=G^{-1}(p^2)-p^2
\ee
of phi-cube theory boson is presented for various coupling strengths
in both renormalization schemes. The infrared details are displayed in Fig.2.
The dynamical mass is not direct physical observables since it is scheme
dependent from the definition, the exception is the pole mass
which is scheme independent and renormgroup invariant as well. It is interesting
that there are time-like values of square of momenta where the 
propagators behave almost  like free one no matter how the coupling constant 
is strong. This happens somewhere when $p^2=6m^2$ for OSR scheme
and approximately at $p^2=20m^2$ for MMS scheme, which implies the physical irrelevance
of such a behavior (Of course, 
there are always differences in absorptive parts
$\pi\rho$ which are ordinary coupling constant dependent at these points). 
The appropriate relevance of propagator dressing is best seen when the dressed propagator
is compared with the free one $G=(p^2-m^2)^{-1}$. From the Fig.3 and Fig.4. 
we can see that  the propagator function is most sensitive with 
respect to the self-energy correction for the threshold momenta where 
these correction are enhanced about the one magnitude, while
they are largely suppressed for the above already  mentioned momenta. 
Note that nothing from these things can  be read from the purely Euclidean approach. 
Up to now presented results have been calculated in the bare vertex approximation,
the solution with vertex correction included will be discussed bellow. 
The appropriate bar vertex approximation  critical coupling value is  $\lambda_{crit}^{OSR}\simeq 3.5$ for OSR scheme 
and   $\lambda_{crit}^{MMS}\simeq 5$ for MMS scheme.
Their different values is not a discrepancy but the necessary consequence of 
the renormalization scheme dependence. 

Furthermore, in order to see the effect of self-consistency of DSE treatment 
we compare the DSE result with the perturbation theory in OSR scheme. 
From the Fig.5 we can see that the perturbation theory is perfectly 
suited method when is applied somewhere bellow the critical value of the coupling
and the above mentioned self-consistency effect is not so large. Therefore the main goal of 
our solution is the information about and the domain of validity of given model.    

The solution of propagator DSE with the vertex improved by the one loop skeleton
diagram is  a next interest of us. To make this comparison more meaningful
we compare the observables $M$ in both scheme together.   
In the Fig.6 we compare the imaginary parts of scattering amplitudes 
$m$ at given kinematic channel. The comparison is
made by the approach described in the  section 1.3 . Therefore 
what is actually compared in this figures are the Lehmann weights
$\sigma$'s of the MMS scheme calculated for certain $\lambda_{MMS}$ 
and the rescaled Lehmann weights $r^2\sigma_2$ calculated for the OSR scheme
for the coupling strength $r^2$ times less then $\lambda_{ORS}$
,i.e.  $\lambda_{MMS}=\lambda_{ORS}/r^2$. Up to the infrared excess for a large 
coupling we see that the convergence of the observable results actually
appears when the truncation of DSE's system improves. 
Some infrared details for three choices of the  coupling constants are also displayed in Fig.7.
Not surprisingly, the critical values of the couplings somewhat decrease,
$\lambda_{crit}^{[2]}\simeq \lambda_{crit}^{bare}$ which is roughly
valid for the both renormalization scheme employed. We come back to the question of meaning
 $\lambda_{crit}$ when we will discuss Wick-Cutkosky model.

 
\section{DSE's for Wick-Cutkosky model}

The massive WCM is given by the following Lagrangian


\be \label{vcml}
 {\cal L} = \sum_i\frac{1}{2} \partial_{\mu} \Phi_{i}
 \partial^{\mu} \Phi_{i} - \sum_i \frac{1}{2} m_{i}^{2} \Phi_{i}^{2}
 +\left(\frac{g_{13}}{\sqrt{2}}  \Phi_{1}^{2} +\frac{g_{23}}{\sqrt{2}}
 \Phi_{2}^{2}\right) \Phi_{3} +C.P.
\ee

where $C.P.$ means the appropriate counter-term part.
Here we choose the second renormalization scheme employed in the previous section, i.e.  
propagators of all three particles have the unit residua.
All the definitions of counter-terms $\delta Z_i ,\delta m_i, \delta g_i$  correspond with the ORS defined above
but now for each particle separately. 
Furthermore, we adjust the couplings by
\be
 g_{i3}=\frac{Z _{g_{i3}}}{Z_i Z_3^{\frac{1}{2}}} g_{{i3}_0},  \quad \quad i=1,2
\ee
such that

\be
 g_{13}=g_{23} .
\ee

The equal mass case $m_1=m_2$ was already solved \cite{SAULI2}
for purpose to study the self-energy effect on the bound state spectrum.
Here we solved the unequal mass case

\be
 \frac{m_1}{m_2}=4 ;\quad  m_3=m_2
\ee
and compare the result with the Euclidean version of solution \cite{AHLIG}.
We restrict ourselves on the 
bare vertex approximation which is sufficient for comparison with \cite{AHLIG}.
Since all the derivation is rather straightforward we simply review the results
The renormalized DSE's in bare vertex approximation read

\bea  \label{WCEDSE}
  G_{Ri}^{-1}\left( p\right) &=& p^2-m_i^{2}-\Pi_{i(2)}(p^2) \quad \quad i=1,2
\nn  \\
  G_{R3}^{-1}\left( p\right) &=& p^2-m_3^{2}-\Pi_{3(2)}(p^2)
\nn \\  
\Pi_i(p^2)&=& i2g^{2}\int \frac{d^{4}q}{\left( 2\pi \right) ^{4}}
\,G_{3 }\left( p-q\right) G_{i}\left( q\right) \quad \quad  \quad i=1,2
\nn \\
\Pi_{3}(p^2)&=& ig^{2}\int \frac{d^{4}q}{\left( 2\pi \right) ^{4}}\,
  \sum_{i=1,2}
  G_{i}\left( p-q\right) G_{i}\left( q\right)
\eea
where the bracket index means the renormalization scheme employed , the second index 
label the particle associated with the appropriate field in the Lagrangian (\ref{vcml}).
All the propagators satisfy the Lehmann representation with unit residuum and all the 
proper function obeys the double subtracted dispersion relation  (\ref{brku})
and henceforth the appropriate spectral weights are related through the relations
 
\be    \label{symb3}
\sigma_{i}(\omega)=\frac{\rho_{i}(\omega)}{(\omega-m_i^2)^2}
 +\frac{1}{(\omega - m_i^2)}P.\int d\alpha
\frac{\sigma_i(\omega)\rho_{i}(\alpha)\left[\frac{\omega-m_i^2}{\alpha-m_i^2}\right]^2+
\sigma_i(\alpha)\rho_{i}(\omega)}{\omega-\alpha}  ,\quad i=1,2,3
\ee
and the expression for the absorptive parts read

 
\bea  
\rho_{\pi_i}(\omega)&=&\frac{2g^2}{(4\pi)^2}
\left[B(m_i^2,m_3^2;\omega)
+\int d\alpha \left(B(\alpha,m_i^2;\omega)\sigma_3(\alpha)
+B(\alpha,m_3^2;\omega)\sigma_i(\alpha)\right)
\right.
\nn \\
&+&\left.\int d\alpha d\beta
B(\alpha,\beta;\omega)\sigma_3(\alpha)
\sigma_i(\beta)\right]
,\quad \quad i=1,2
\nn \\
\rho_{\pi_3}(\omega)&=&\sum_{i=1,2}\frac{g^2}{(4\pi)^2}
\left[\sqrt{1-\frac{4m_i^2}{\omega}}
+2\int d\alpha B(\alpha,m_i^2;\omega)\sigma_i(\alpha)\right.
\nn \\
&+&\int\left. d\alpha d\beta
B(\alpha,\beta;\omega)\sigma_i(\alpha)\sigma_i(\beta)\right].
\eea

The above set of equations have been actually solved numerically. The main result for us is the appearance of the critical
coupling strength $\alpha_c\equiv g_c^2/(4\pi m_2^2)=0.12$ which rather accurately corresponds with the point where the renormalization 
constant $Z_2$ turns to be negative. The appropriate dependence of the  renormalization constants $
Z_i$  is  presented in the Fig.8 for all three particle. The obtained critical value is in reasonable agreement with the one obtained by 
the Euclidean solution of DSE's system \cite{AHLIG}, where $\lambda_c=0.086$, as well as with the 
critical value $\lambda_c=0.063$ which was found using
a variational approach \cite{ROSEN1},\cite{ROSEN2}.


Furthermore, the existence of the critical coupling of OSR scheme can be seen from the
analytic formula for the inverse of propagator   

\be
G_i^{-1}=p^2-m_i^2-
\int d\alpha
\frac{\rho_i(\alpha)(p^2-m_i^2)^2}{(\alpha-m_i^2)^2(p^{2}-\alpha+i\epsilon)}
\ee
which implies that for the strong coupling enough the Landau pole should appear, which  must arise when
the factor $L$ 
\be \label{factor}
L=[1-\int\frac {\rho(\alpha)}{(\alpha-m^2)^2}]
\ee
is negative (when it is just zero then the Landau pole is situated in space-like infinite, and for the positive 
$L$ this singularity never appears due to finiteness of the appropriate integral in (ref{factor})).
For negative $L$ the propagator can not be described by  the Lehmann representation at all and at least 
the Minkowskian treatment used in this work must fails. Comparing (ref{factor}) with the definition of renormalization constant $Z$
we clearly have the identification $L=Z$. 
As we have mentioned, the numerical solution start to fail when the condition $Z=0$ is fulfilled.  This statement is justified
with the 10 \% numerical accuracy. 

(We have no similar guidance for MMS scheme but we expect the similar appearance of the critical coupling
$\lambda_{MMS}$ for this scheme as it was happened  in $\Phi^3$ theory, from this point of view the  correspondence
of numerical failure and the condition $Z_{OSR}=0$ is a bit surprising (at least for the author).).
 
\section{Conclusion}

We have obtained numerical solutions of the   DSE's in  Minkowski space for simple
$ \Phi^3$ theory and Wick-Cutkosky model. It suggest, that the expansion of the theory
 around the metastable vacuum leads to the predicative result.
Our technique allows us
to extract propagator spectral function $\rho(s)$ with reasonably high numerical accuracy.
Since the renormalization procedure is performed analytically, it has no effect
on the precision of solution.  When the coupling do not exceed certain critical value
then the  domain of analyticity of the propagator is the all real axis of  $p^2$. 
The attempt to clarified the meaning of critical coupling value was made. This suggest 
that it  corresponds with  appearance of  unphysical singularity in the on-shell renormalized propagator.  
Consequently, the field renormalization constant (in on shell scheme) turns to be negative
for $\lambda>\lambda_{crit}$. 



\appendix
\section{ Dispersion relations for self-energies in bare vertex approximation }

In this Appendix we derive DR's for self-energies in the both renormalization schemes
for bare vertex. The calculation is very straightforward, and in fact it represents nothing 
else but evaluation of the one loop scalar 
Feynman diagram with different masses in internal lines. 
 
Substituting the Lehmann representation for MMS propagators (\ref{vrku})
the unrenormalized $\Pi$ can be split like   

\bea  \label{SELF}
\Pi(p^2)&=&\Pi_{(b,b)}(p^2)+2\Pi_{(b,s)}(p^2)+\Pi_{(s,s)}(p^2)
\nn  \\
\Pi_{(b,b)}(p^2)&=&\int d\bar{q}
\frac{18r^2g^{2} }{((p+q)^{2}-m^{2}+i\epsilon)(q^{2}-m^{2}+i\epsilon)}
\nn\\
\Pi_{(b,s)}(p^2)&=&\int d\bar{q}
\int d\alpha\frac{18rg^{2}\sigma(\alpha)}
{(q^{2}-\alpha+i\epsilon)((p+q)^{2}-m^{2}+i\epsilon)}
\nn \\
\Pi_{(s,s)}(p^2)&=&\int d\bar{q}
\int d\alpha d\beta
\frac{18g^{2}\sigma(\alpha)\sigma(\beta)}
{((p+q)^{2}-\alpha+i\epsilon)(q^{2}-\beta+i\epsilon)},
\eea
where we have  used shorthand notation for the measure  $id^4q/(2\pi)^4\equiv d\bar{q}$.
Making the subtraction we immediately arrive for the pure perturbative contribution
(up to the presence of the square of residuum):
 
\bea   \label{PP}
\Pi_{1(b,b)}&=&\int_{4m^2}^{\infty} d\omega \frac {\rho_{1(b,b)}(p^2-m^2)}
{(p^2-\omega +i\epsilon)(\omega-m^2)}
\nn \\
\rho_{1(b,b)} (\omega)&=&\frac{18r^2g^2}{(4\pi)^2}
\sqrt{1-\frac{4m^{2}}{\omega}}
\eea

The most general integral to be solved is just the above case where the physical masses are replaced
by their continuous partners. 
 
\be  \label{gener}
I(p^2)=\int d\bar{q}
\frac{1}{((p+q)^{2}-\alpha)(q^{2}-\beta)}
\ee
which after the subtraction (\ref{firstline}) and integrating over the Feynman parameter $x$
leads to the appropriate one subtracted DR (\ref{gener2})


\bea    \label{firstline}
I_{1s}(p^2)&=&I(p^2)-I(m^2)= \frac{1}{(4\pi)^2}\int_{0}^{[1]}dx
\int_{\frac{m^{2}x+\alpha(1-x)}{x(1-x)}}^{\infty}
 d\omega\frac{(p^2-m^2)}
{(\omega-m^2)(p^{2}-\omega+i\epsilon)}
\nn \\
&=&\int_{0}^{\infty}d\omega \frac{p^2-m^2}{(p^2-\omega+i\epsilon)}  
\frac{B(\alpha,\beta;\omega)}
{(4\pi )^{2}(\omega-m^{2})} \label{gener2},
\eea
where the function $B(u,v;\omega)$ is defined through the  Khallen triangle function $\lambda$ 
like

\bea
B(u,v,\omega)&=&\frac{\lambda^{1/2}(u,\omega,v)}{\omega}
\Theta\left(\omega-(\alpha^{\frac{1}{2}}+\beta^{\frac{1}{2}})^2\right)
\nn \\
\lambda(u,\omega,v)&=&(u-\omega-v)^2-4\omega v
\nn \\
&=&\omega^2+u^2+v^2-2\omega v -2\omega u -2u v.
\eea
It can be easy checked that $B(m^2,m^2;\omega)=\sqrt{1-\frac{4m^2}{\omega}}\Theta(\omega-4m^2)$
which was already introduced in (\ref{PP}).

The OSR scheme requires additional subtraction which is finite and henceforth  
can be proceed by making a simple algebra

\bea \label{gener3}
I_{2s}(p^2)&=&I_{1s}(p^2)-\frac{d}{d p^2}|_{p^2=m^2}I_{1s}(p^2)
\nn \\
&=&\frac{1}{(4\pi)^2}\int_{0}^{\infty}d\omega \frac{(p^2-m^2)^2}{(\omega-m^{2})^2(p^2-\omega+i\epsilon)}
B(\alpha,\beta;\omega).
\eea

To summarize the results we see that MMS self-energy satisfies one subtracted DR
with the absorptive part $\pi\rho_1$ given like 

\bea  \label{fresult}
\rho_{\pi_1}(\omega)&=&\rho_{1(b,b)}(\omega) +2\rho_{1(b,s)}(\omega)+\rho_{1(s,s)}(\omega)
\nn \\
\rho_{1(b,b)}(\omega)&=&\frac{18r^2g^2}{(4\pi)^2}
\sqrt{1-\frac{4m^{2}}{\omega}}
\nn \\
\rho_{1(b,s)}(\omega)&=&\int_{9m^2}^{\infty}d\alpha
\frac{18rg^{2}}{(4\pi )^{2}}B(\alpha,m^2;\omega)\sigma(\alpha)
\nn \\
\rho_{1(s,s)}(\omega)&=&\int_{16m^2}^{\infty}d\alpha d\beta
\frac{18g^2}{(4\pi )^{2}} B(\alpha,\beta;\omega)\sigma(\alpha)\sigma(\beta)
\eea

while the self-energy in OSR scheme satisfies double subtracted DR with
the absorptive part $\pi\rho_2$

\bea  \label{sresult}
\rho_{\pi_2}(\omega)&=&\rho_{2(b,b)}(\omega) +2\rho_{2(b,s)}(\omega)+\rho_{2(s,s)}(\omega)
 \nn \\
\rho_{2(b,b)}(\omega)&=&\frac{18g^2}{(4\pi)^2}
\sqrt{1-\frac{4m^{2}}{\omega}}
\nn \\
\rho_{2(b,s)}(\omega)&=&\int_{9m^2}^{\infty}d\alpha
\frac{18g^{2}}{(4\pi )^{2}}B(\alpha,m^2;\omega)\sigma_2(\alpha)
\nn \\
\rho_{2(s,s)}(\omega)&=&\int_{16m^2}^{\infty}d\alpha d\beta
\frac{18g^2}{(4\pi )^{2}} B(\alpha,\beta;\omega)\sigma_2(\alpha)\sigma_2(\beta).
\eea

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Two-loop skeleton self-energy DR}

The finite  two-loop integral appears after the substitution of the vertex (\ref{vert})
to the self-energy formula

\bea \label{twopi}
\int d\bar{q}d\bar{k}
&&\left[ (k^2-\alpha_1+i\epsilon)((p+k)^2-\alpha_2+i\epsilon)
((k-q)^2-\alpha_3+i\epsilon)\right.
\nn \\
&&\left.(q^2-\alpha_4+i\epsilon)
((p+q)^2-\alpha_5+i\epsilon)\right]^{-1}
\eea
where all the irrelevant pre-factors are omitted for purpose of the brevity. They will be correctly add at the end
of  calculation for both renormalization schemes separately.
The contribution is UV finite therefore we first calculate the unrenormalized result.
Firstly,  we parameterize the  off-shell vertex  by matching the first three denominators 
and consequently we integrate over the  triangle  loop momentum $k$

\bea \label{vertex}
&&\int \bar{k}
\left[ (k^2-\alpha_1+i\epsilon)((p+k)^2-\alpha_2+i\epsilon)
((k-q)^2-\alpha_3+i\epsilon)\right]^{-1}=
\nn\\
&&\int \bar{k}
\int_0^1 dx dy 2y \left[k^2xy+(p+k)^2(1-x)y+(k-q)^2(1-y)\right.
\nn \\
&&-\left.\alpha_1 xy-\alpha_2(1-x)y-\alpha_3(1-y)+i\epsilon\right]^{-3}=
 \\
&&\int_0^1 \frac{dx dy y}{(4\pi)^2}
\left[p^2(1-x)y(1-(1-x)y)+q^2y(1-y)+2p.q(1-x)y(1-y)\right.
\nn \\
&&\left.-\alpha_1 xy-\alpha_2(1-x)y-\alpha_3(1-y)+i\epsilon\right]^{-1}.
\nn
\eea
At the next we substitute $x\rightarrow 1-x$ and  
after a little algebra we obtain for (\ref{vertex})

\be \label{form}
\int_0^1 \frac{dx dy }{(4\pi)^2(1-y)}
\left[q^2+2p.qx+p^2\frac{x(1-xy)}{(1-y)}-O_{1-3}+i\epsilon\right]^{-1}
\ee
where we used short notation $O_{1-3}=\alpha_1 \frac{1-x}{1-y}
+\alpha_2\frac{x}{1-y}+\alpha_3\frac{1}{y}$.
Continuing by matching of (\ref{form}) with   two spare denominators in (\ref{twopi})  by using Feynman  
variables $z $ and $u$ for denominators with $\alpha_4$ and $\alpha_5$, respectively,
Then we can write for (\ref{twopi}) 

\bea
&&\int d\bar{q} \int_0^1 \frac{dx dy dz du 2u }{4\pi)^2(1-y)}
\left[q^2+2p.qxzu +2p.q(1-u)\right.
\nn \\
&&+\left.p^2\frac{(1-xy)xzu}{(1-y)}+p^2(1-u)-O_{1-5}+i\epsilon\right]^{-3}
\eea
where we used shorthand notation $O_{1-5}=O_{1-3}zu+\alpha_4(1-z)u+
\alpha_5(1-u)$. Shifting $q+p(xzu+1-u) \rightarrow q$ and integrating
over new $q$ it yields:

\bea
&&\int_0^1 \frac{dx dy dz du u }{(4\pi)^4(1-y)F(x,y,z)}
\frac{1}{\left[p^2-\frac{O_{1-5}}{F(x,y,z)}+i\epsilon\right]}
\nn \\
&&F(x,y,z)=1-u+\frac{(1-xy)xzu}{(1-y)}-(xzu+(1-u))^2
\eea 
At the next we make substitution $u\rightarrow \omega$
where $\omega= O_{1-5}/F(x,y,z)$. Using the notation 

\bea \label{blabla}
\omega&=&\frac{u a_1+a_2}{u^2b_1+ub_2}
\\
a_1&=&(\alpha_1 (1-x)y
+\alpha_2 xy+\alpha_3 (1-y))z+(\alpha_4(1-z)-\alpha_5) y(1-y)
\nn \\
a_2&=&\alpha_5 y(1-y)
\nn \\
b_1&=&-(1-xz)^2y(1-y)
\nn \\
b_2&=&(1-2xz)y(1-y)+(1-xy)xyz.
\eea
we can write down the appropriate DR for (\ref{twopi}) 

\bea  \label{rezek}
\Omega(\omega;\alpha_1,..\alpha_5)&=&\int_0^{\infty} \frac{d\omega}{p^2-\omega+i\epsilon}
\int_0^1\frac{dx dy dz}{(4\pi)^4(1-y)}\frac{\Theta\left(\omega-\frac{a_1+a_2}{b_1+b_2}\right)
\Theta(D)}
{\left[\frac{\alpha_5}{U^2}-\omega(1-xz)^2\right]}
\\
U&=&\frac{-B+\sqrt{D}}{2A}; \quad D=B^2-4AC
 \nn \\
A&=&\omega b_1; \quad B=\omega b_2-a_1; \quad C=-a_2
\nn
\eea

Note here, that spectral function  (everything after the first fraction in (\ref{rezek})
is always positive for allowed values of $\alpha's$
and it is regular function of it's argument $\omega$. 
The various sub-thresholds are then given by the values of Lehmann variables $\alpha$'s
 in accordance with the step function presented, noting that the perturbative threshold is given
again by $4m^2$ and in that case case the result partaly simplified.
  For completeness we reviewed the associated simplifications, namely:
 $a_1=m^2z(1-y(1-y)); \quad a_2=m^2y(1-y) $.
Making one subtraction for the MMS and two subtraction for OSR scheme
we can recognize that the appropriate skeleton DR for master diagram
has the absorptive part  



\be  \label{uno}
\rho_{1}^{[2]}(\omega)=\frac{(6g)^4}{2}\prod_{i=1}^5 \int d\alpha_i 
\tilde{\sigma}(\alpha_i)\Omega(\omega,\alpha_1,..\alpha_2)
\ee
for MMS scheme and

\be
\label{duo}
\rho_{2}^{[2]}(\omega)=\frac{(6g_2)^4}{2}\prod_{i=1}^5 \int d\alpha_i 
\tilde{\sigma}_2(\alpha_i)\Omega(\omega,\alpha_1,..\alpha_2)
\ee
for OSR scheme, respectively.
In fact it  gives rise 28 of various
contributions to $\rho^{[2]}$  (only 12 are actually topologically independent,
 distinguished by the number of continuous Lehmann weights with the appropriate position of spectral variable 
in $\Omega$. All of them have been found numerically for the purpose of DSE's solution. 
 


 
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\newpage

\begin{figure}
\centerline{ \mbox{\psfig{figure=Fig1.eps,height=12truecm,angle=270}} }
\caption{Dynamical mass of scalar particle in $\Phi^3$ theory calculated 
in bare vertex approximation in the both renormalization schemes.
The lines are labeled by the value of
$\lambda_{MMS}$ for MMS scheme and $\lambda_{OSR}$ for OSR renormalization scheme.}
\end{figure} 

\begin{figure}
\centerline{ \mbox{\psfig{figure=Fig2.eps,height=12truecm,angle=270}} }
\caption{ Infrared (threshould) details of the Fig.1.  }
\end{figure}

\begin{figure}
\centerline{ \mbox{\psfig{figure=Fig3.eps,height=12truecm,angle=270}} }
\caption{ The propagators deviations from free theory. The propagator is calculated 
in minimal momentum  renormalization scheme   for various $\lambda_{MMS}$ }
\end{figure}

\begin{figure}
\centerline{ \mbox{\psfig{figure=Fig4.eps,height=12truecm,angle=270}} }
\caption{ The propagator deviation from free theory. The propagator is calculated 
in on mass-shell renormalization scheme with unit residuum  for various $\lambda_{OSR}$. }
\end{figure}


\begin{figure}
\centerline{ \mbox{\psfig{figure=Fig5.eps,height=12truecm,angle=270}} }
\caption{ Comparison of DSE results in bare vertex approximation with the  perturbation theory result. DSE  and
bubble summation is obtained for OSR scheme for various $\lambda_{OSR}$ .}
\end{figure}

\begin{figure}
\centerline{ \mbox{\psfig{figure=Fig6.eps,height=12truecm,angle=270}} }
\caption{ Imaginary parts of scattering matrix calculated with propagator
which have been obtained in MMS and OSR scheme with (dv) and without (bv) improved vertex.
Each set of lines corresponding to the same model is labeled by the  coupling strength $\lambda_{MMS}$.}
\end{figure}

\begin{figure}
\centerline{ \mbox{\psfig{figure=Fig7.eps,height=12truecm,angle=270}} }
\caption{ The low frequency details of the Fig.6.}
\end{figure}

\begin{figure}
\centerline{ \mbox{\psfig{figure=Fig8.eps,height=12truecm,angle=270}} }
\caption{ The dependence of field strength renormalization constants on the coupling strength of Wick-Cutkosky model.
The index 1-3 labels the particle. }
\end{figure}

\end{document}
