\section{Next-to-leading order correction to the decay amplitude}

In order to establish when the coupling constant $g$ can be regarded as small
such that the resummed one-loop approximation is justified, we calculate the
(upper limit of) the next-to-leading order diagram of the amplitude of the
decay process, see Fig. \ref{diagrams}. The decay amplitude results as the sum
of two terms
\begin{equation}
-i\mathcal{M}=-i\mathcal{M}_{1}-i\mathcal{M}_{2}\text{ ,
\end{equation}
where the tree-level and the triangle contributions depicted in Fig.
\ref{diagrams} read
\begin{equation}
-i\mathcal{M}_{1}=2ig\text{ ; }-i\mathcal{M}_{2}=4g^{3}I\text{ .}
\label{amplitudes
\end{equation}
The coefficients $2$ and $4$ in (\ref{amplitudes}) are the appropriate
symmetry factors. The quantity $I$ is the integra
\begin{equation}
I=I(m,M_{0})=-\int\frac{d^{4}q}{(2\pi)^{4}}\frac{\phi_{1}\phi_{2}\phi_{3
}{\left[  \left(  \frac{p}{2}-q\right)  ^{2}-m^{2}\right]  \left[  \left(
\frac{p}{2}+q\right)  ^{2}-m^{2}\right]  \left[  \left(  \frac{p}{2
+q-k_{1}\right)  ^{2}-M_{0}^{2}\right]  }\text{ .} \ ,\label{integral
\end{equation}
where $\phi_{i}$ is the cutoff function of the $i$-th vertex

\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.1819in,
width=5.3065in

{diagrams.eps
\caption{Tree-leve (left) and next-to-leading order (traingle diagram, right)
contributions to the decay amplitude.
\label{diagrams
\end{center}
\end{figure}


We have evaluated $I$ by using the free propagator of the $S$ virtual particle
exchanged by the two $\varphi$ fields. Thus, our results represent an upper
limit for $\left\vert I\right\vert $. Namely, $\left\vert I\right\vert $ would
be actually smaller by including the full propagator of $S$, i.e. taking into
account its finite width.

We define the coupling constant $g_{\max}$ as the value of $g$ for which
$\left\vert \mathcal{M}_{1}\right\vert =\left\vert \mathcal{M}_{2}\right\vert
.$ Namely, for $g=$ $g_{\max}=1/\sqrt{2\left\vert I\right\vert }$ the
next-to-leading order contribution equals the tree-level one. Thus, for
$g>g_{\max}$ one cannot neglect the role of higher order diagrams.

In particular, we obtain the following numerical results: For $M_{0}=1$ GeV,
$g_{\max}=19.503$ GeV; the critical value $g_{\ast},$ at which the new pole in
the I-Riemann sheet emerges, reads $3.379$ and the value $g_{E},$ for which
the stable state has a mass of $38$ MeV, reads $4.877$ GeV. Both values are
safely smaller than $g_{\max}$: the amplitude ratios read $\left[  \left\vert
\mathcal{M}_{2}\right\vert /\left\vert \mathcal{M}_{1}\right\vert \right]
_{g=g_{\ast}}=0.03$ and $\left[  \left\vert \mathcal{M}_{2}\right\vert
/\left\vert \mathcal{M}_{1}\right\vert \right]  _{g=g_{\ast}}=0.06,$ thus the
tree-level diagram is by far the dominant contribution for all the range of
coupling constants relevant for our study. For $M_{0}=0.45$ GeV one gets
$g_{\max}=3.666$ GeV, which should be compared with $g_{\ast}=1.399$ GeV and
$g_{E}=2.189$ GeV. The ratios read $\left[  \left\vert \mathcal{M
_{2}\right\vert /\left\vert \mathcal{M}_{1}\right\vert \right]  _{g=g_{\ast
}=0.14$ and $\left[  \left\vert \mathcal{M}_{2}\right\vert /\left\vert
\mathcal{M}_{1}\right\vert \right]  _{g=g_{\ast}}=0.36,$ which are still
safely small, although larger than in the previous example. It should once
more be stressed that our evaluation of $\left\vert I\right\vert $ represents
only an upper limit, and the value of $g_{\max}$ is indeed larger than the
estimate given here.

All these considerations enforce our point of view that the emergence of a
pole below threshold such as the one described in this manuscript is possible
in quantum field theory. Future evaluation of higher-loop contributions is planned.

\bigskip

