Nakanishi-Lautrup field ($B^a$)

The Nakanishi-Lautrup field is an auxiliary bosonic field used to write gauge fixing as a BRST-exact term. Its algebraic field equation produces the usual quadratic gauge-fixing term.

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.