= Gauge-fixed Yang-Mills Lagrangian in a covariant gauge
{title2=$\mathcal L=-\tfrac14F^a_{\mu\nu}F^{a\mu\nu}-(\partial^\mu A^a_\mu)^2/(2\xi)-\bar c^a\partial^\mu(D_\mu c)^a$}
Use $F^a_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu+g f^{abc}A^b_\mu A^c_\nu$ and $(D_\mu c)^a=\partial_\mu c^a+g f^{abc}A^b_\mu c^c$. The <Faddeev-Popov gauge-orbit identity> with <Lorenz gauge> function $G^a=\partial^\mu A^a_\mu$, followed by Gaussian averaging of $G^a=f^a$, supplies the quadratic gauge-fixing term. Represent the determinant of $-\partial^\mu D_\mu$ using a <Grassmann Gaussian integral> over a <Faddeev-Popov ghost field> and <antighost field> to obtain the ghost term. The minus sign in this operator differs by a field-independent determinant factor from the direct gauge-condition Jacobian; it is a consistent antighost convention.
The ghost term equals $(\partial^\mu\bar c^a)(D_\mu c)^a$ up to a boundary term. Thus it contains a <ghost-gluon vertex> with the sign matching this kinetic convention. An equivalent auxiliary-field form replaces the gauge-fixing term by $B^aG^a+\xi B^aB^a/2$. The <Nakanishi-Lautrup field> equation gives $B^a=-G^a/\xi$ and recovers the displayed Lagrangian. With left-acting <BRST transformations> $sA=Dc$, $sc=-g[c,c]/2$, $s\bar c=B$, and $sB=0$, the gauge-fixing and ghost terms together are $s[\bar c^a(G^a+\xi B^a/2)]$.
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