 Auxiliary-field propagator

For a real auxiliary field with quadratic Lagrangian density $-a{\sigma }^2/2$, $a\ne 0$, the Feynman propagator is the inverse quadratic kernel multiplied by $i$: $D_{\sigma }(p)=-i/a$. It is independent of momentum; in position space it is the contact distribution $-i{\delta }^{(d)}(x-y)/a$. Two interaction vertices $ig$ joined by this auxiliary-field propagator give $i{g}^2/a$. Completing the square in $-a{\sigma }^2/2+g\sigma B$ produces $g^2{B}^2/(2a)$, proving the equivalence to a local interaction by a Gaussian integral.