A BRST-Exact Lorenz-Gauge Functional

Let $U\subseteq {\mathbb{R}}^{1,3}$ be a nonempty open region with compact support or boundary conditions that remove total-derivative terms. Use ${\eta }_{\mu \nu }=\mathrm{diag}(-1,1,1,1)$ and ${\mathrm{\partial }}^{\mu }:={\eta }^{\mu \nu }{\mathrm{\partial }}_{\nu }$. Work in the adopted pure $SU(2)$ model, with Lie algebra $\mathfrak{s}\mathfrak{u}(2)$ and normalized positive $\mathrm{Ad}$-invariant form $\kappa (X,Y)=-2{\mathrm{Tr}}_2(XY)$, extended bilinearly to the graded coefficient algebra. In the finite-jet algebra take $A_{\mu }$ to be an even $\mathfrak{s}\mathfrak{u}(2)$-valued field of ghost number $0$ and dimension $1$, $c$ an odd $\mathfrak{s}\mathfrak{u}(2)$-valued field of ghost number $1$ and dimension $0$, $\bar{c}$ an odd $\mathfrak{s}\mathfrak{u}(2)$-valued field of ghost number $-1$ and dimension $2$, and $b$ an even $\mathfrak{s}\mathfrak{u}(2)$-valued field of ghost number $0$ and dimension $2$. Let $\xi >0$ be dimensionless. Use the Lie-algebra bracket and odd left BRST derivation

$s{A}_{\mu }={\mathrm{\partial }}_{\mu }c+[A_{\mu },c],sc=-\frac{1}{2}[c,c],s\bar{c}=b,sb=0,$

extended to jets by commuting with total derivatives. Define the Lorenz-gauge fermion and its BRST-exact functional by

${\mathrm{\Psi }}_{\xi }={\int }_U{d}^{4}x\kappa (\bar{c},{\mathrm{\partial }}^{\mu }{A}_{\mu }+\frac{\xi }{2}b),S_{\mathrm{g}\mathrm{f}}:=s{\mathrm{\Psi }}_{\xi }\mathrm{.}$

Then

$S_{\mathrm{g}\mathrm{f}}={\int }_U{d}^{4}x(\kappa (b,{\mathrm{\partial }}^{\mu }{A}_{\mu })+\frac{\xi }{2}\kappa (b,b)-\kappa (\bar{c},{\mathrm{\partial }}^{\mu }{D}_{\mu }c)),D_{\mu }c:={\mathrm{\partial }}_{\mu }c+[A_{\mu },c],$

is a local, even, ghost-number-zero functional of density dimension $4$, and it is BRST-closed. It is an exact gauge-fixing term in the formal local algebra; this example does not claim a global representative for every gauge orbit or a continuum functional measure.

Barnich, Brandt, and Henneaux, §§2.2–2.3 and Appendix 2.A

Maas, §§4.5 and 4.7.2