# Mass of the fields in quantum field theory

I understand that if I have an action $$S=\int \phi(\Box + m^2 )\phi$$ Then the field $$\phi$$ has mass $$m$$ since this is the pole of the propagator of $$\phi$$. Now If I have an action $$S=\int \phi_1 \Box \phi_2 + m^2 \phi_1^2$$ Then how do I interpret the mass of the field $$\phi_1$$ or $$\phi_2$$?

My thinking was that If I find the equations of motion for $$\phi_2$$ then I have $$\Box \phi_1 =0$$ Which is a Klein-Gordon equation with no-mass so we interpret the field $$\phi_1$$ to be massless??

Thank you

• Minor comment to the post (v2). 1. Where are the integration measures? 2. The mass terms has the wrong signs. 3. The kinetic term in the 2-boson theory is not positive definite. – Qmechanic Aug 15 '19 at 15:54

In order to calculate the mass you need to go to an eigenbasis. Unfortunately, your theory is ill-defined: the kinetic term has a negative eigenvalue: $$\begin{pmatrix}0&1/2\\1/2&0\end{pmatrix}\sim \begin{pmatrix}1/2&0\\0&-1/2\end{pmatrix}$$