= Auxiliary-field propagator
{title2=$D_\sigma(p)=-i/a$}
For a real <auxiliary field> with quadratic <Lagrangian density> $-a\sigma^2/2$, $a\ne0$, 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 $ig^2/a$. Completing the square in $-a\sigma^2/2+g\sigma B$ produces $g^2B^2/(2a)$, proving the equivalence to a local interaction by a <Gaussian integral>.
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