Interpret constraints on scattering amplitude

[Valeriia Lukashenko]

Homework Statement

For the following theory: $\mathcal{L}=\frac{1}{2}[(\partial \phi)^2-m^2\phi^2+(\partial\Phi)^2-M^2\Phi^2]+g\phi^2 \Phi^2$

Compute s-channel amplitude for process $\phi\phi \rightarrow \phi\phi$. Interpret result for $M>2m$.

Homework Equations

Scattering amplitude: $\mathcal{iA}=(ig)^2\frac{i}{s-M^2}$
$s=(p_1+p_2)^2$

The Attempt at a Solution

Choosing to work in center-of-mass frame: $\vec{p_1}+\vec{p_2}=0$.
$s=(E_1+E_2)^2$ in CoM.
$$E_1=E_2$$, because $|\vec{p_1}|=|\vec{p_2}|$ and masses are same.
Then $s=4E^2$

We get in CoM:$\mathcal{A}=(ig)^2\frac{1}{4E^2-M^2}=(ig)^2\frac{1}{4(\vec{p}^2+m^2)-M^2}$
Applying $M>2m$ we get $\mathcal{A}> (ig)^2\frac{1}{4|\vec{p}|^2}$

So far it is hard for me to interpret this result. Could anyone give me some hints how to think about this constraint?

 

[mfb]

What can happen with the denominator that cannot happen for M<2m?