Musings

The situation in nonabelian gauge theories is more cloudy. Unlike in QED, $\mathcal{N}\coloneqq Z_g Z_A^{1/2}\neq 1$. Hence the BRST transformations need to be renormalized. Let $$\tilde{D}_\mu = \partial_\mu -i g\mathcal{N}A_\mu$$ be the renormalized covariant derivative and $$\tilde{F}_{\mu\nu} = \frac{i}{g\mathcal{N}}[\tilde{D}_\mu,\tilde{D}_\nu]= \partial_\mu A_\nu-\partial_\nu A_\mu -i g\mathcal{N}[A_\mu,A_\nu]$$ the renormalized field strength. The renormalized BRST transformations

explicitly involve both $\mathcal{N}$ and the ghost wave-function renormalization, $Z_{\text{gh}}$ and are corrected order-by-order in perturbation theory. Hence the relations which follow from $\delta_{\text{BRST}} \Gamma=0$ (called the Slavnov-Taylor Identities) are also corrected order-by-order in perturbation theory.

This is … awkward. The vertex functions are finite quantities. And yet the relations (naively) involve these infinite renormalization constants (which, moreover, are power-series in $g$).

But if we step up to the full-blown Batalin-Vilkovisky formalism, we can do better. Let’s introduce a new commuting adjoint-valued scalar field $\Phi$ with ghost number $-2$ and an anti-commuting adjoint-valued vector-field $S_\mu$ with ghost number $-1$ and posit that they transform trivially under BRST: $$\begin{split} \delta_{\text{BRST}} \Phi&=0\\ \delta_{\text{BRST}} S_\mu&=0 \end{split}$$ The renormalized Yang-Mills Lagrangian¹ is

where $$\begin{split} \mathcal{L}_{\text{GF}}+\mathcal{L}_{\text{gh}}&= \delta_{\text{BRST}}Tr\left(b(\partial\cdot A+\xi^{1/2}\chi)\right)\\ &=-\frac{1}{\xi} Tr(\partial\cdot A)^2+ Tr\chi^2 -2Z_{\text{gh}}Tr b\partial\cdot\tilde{D} c \end{split}$$ and $$\mathcal{L}_{\text{AF}} =Z_{\text{gh}}Tr(S^\mu\tilde{D}_\mu c) +\frac{i g}{2} Z_{\text{gh}}\mathcal{N} Tr(\Phi\{c,c\})$$ $\mathcal{L}_{\text{AF}}$ is explicitly BRST-invariant because what appears multiplying the anti-fields $S^\mu$ and $\Phi$ are BRST variations (respectively of $A_\mu$ and $c$). These were the “troublesome” BRST variations where the RHS of (1) were nonlinear in the fields (and hence subject to renormalization).

Now we can replace the “ugly” equation $\delta_{\text{BRST}}\Gamma=0$, which has explicit factors of $\mathcal{N}$ and $Z_{\text{gh}}$ and is corrected order-by-order, with

which is an exact (all-orders) relation among finite quantites. The price we pay is that the Zinn-Justin equation (3) is quadratic, rather than linear, in $\Gamma$.