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