# More general propagator of a real scalar field

I have some Lagrangian containing a real scalar field $$\phi$$ with mass $$m$$. Let $$A \in \mathbb{R}$$ be some constant. The Lagrangian takes the form:

$$$$\mathcal{L} = -\frac{A}{2} (\partial_\mu \phi)^2 - \frac{1}{2}m^2 \phi^2 + \mathcal{L}_{\phi \phi \phi} + \mathcal{L}_{\phi \phi \phi \phi},$$$$

where the last two terms indicate interaction terms. My question is whether it makes sense to compute the scattering amplitude for the case with $$A = 0$$?

1. On one hand, if $$A\equiv 0$$, then the field is non-propagating, and one cannot construct a scattering theory. This is OP's case.
2. On the other hand, if one makes a field-redefinition $$\phi^{\prime}~=~\sqrt{|A|}\phi, \quad m^{\prime}~=~\frac{m}{\sqrt{|A|}}, \quad g_3^{\prime}~=~\frac{g_3}{|A|^{3/2}}, \quad g_4^{\prime}~=~\frac{g_4}{|A|^2},$$ and takes the limit $$A\to 0$$, then the field becomes infinitely massive, and the coupling constants becomes infinitely large.
• $\uparrow$ Right. Apr 9, 2022 at 16:28