= Gaussian gauge fixing with an auxiliary field
{title2=$\int Db\,e^{i\int(bF+\xi b^2/2)}\propto e^{-i\int F^2/(2\xi)}$}
Complete the square as $bF+\xi b^2/2=\xi(b+F/\xi)^2/2-F^2/(2\xi)$. Translation of the <Gaussian functional integral> gives the displayed identity for nonzero $\xi$, with the appropriate oscillatory contour prescription. At $\xi=0$, the <Nakanishi-Lautrup field> is a <Lagrange multiplier> imposing the gauge-condition delta functional. This field has no kinetic term and no independent propagating degree of freedom.
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