Conditional Slavnov–Taylor Identity and Counterterm Classification

In the axiom base inherited from the adopted model, fix the classical pure $SU(2)$ Yang–Mills model on oriented, time-oriented Minkowski spacetime $M={\mathbb{R}}^{1,3}$ with metric $\eta =\mathrm{diag}(-1,1,1,1)$, its selected smooth right-principal bundle sector $P\to M$, no matter fields, invariant form $\kappa (X,Y)=-2{\mathrm{Tr}}_2(XY)$ on $\mathfrak{s}\mathfrak{u}(2)$, and fixed $g_{\mathrm{Y}\mathrm{M}}>0$. Let $U\subset M$ be a nonempty local trivialization chart, use natural units $c=\mathrm{\hslash }=1$, and use the right-action curvature convention $F_A=dA+\frac{1}{2}[A\wedge A]$. The adopted local classical action density is $-\kappa (F_A\wedge {\ast }_{\eta }{F}_A)\mathrm{/}(2g_{\mathrm{Y}\mathrm{M}}^2)$. The local gauge potential $A_{\mu }$ is even with ghost number $0$ and engineering dimension $1$; the odd ghost $c$ has ghost number $1$ and dimension $0$. Define $D_{\mu }c={\mathrm{\partial }}_{\mu }c+[A_{\mu },c]$ and the minimal BRST transformations $s{A}_{\mu }=D_{\mu }c$, $sc=-\frac{1}{2}[c,c]$. When the nonminimal pair is included, let $\bar{c}$ be odd with ghost number $-1$ and dimension $2$, and $b$ even with ghost number $0$ and dimension $2$, with $s\bar{c}=b$ and $sb=0$. Antifields have the opposite field parity and ghost number $-1-\mathrm{gh}(\mathrm{\Phi })$, with engineering dimension $\dim {\mathrm{\Phi }}^{\ast }=4-\dim \mathrm{\Phi }$, so a nonzero ${\mathrm{\Phi }}^{\ast }s\mathrm{\Phi }$ has density dimension $4$. Use the extended finite-jet local BV/nonminimal horizontal complex of the Formal Perturbative Yang–Mills Renormalization Data definition: its generators are the jets of $A,c,\bar{c},b$ and their antifields, $d_H=d{x}^{\mu }{D}_{\mu }$, and $s=B_{{\mathrm{\Gamma }}_0}$ is the prolonged canonical BV differential, with $s^2=0$ and $s{d}_H+d_{H}s=0$ on the stated domain.

Use the canonical BV antibracket and let the classical BV functional be the pure-gauge extension ${\mathrm{\Gamma }}_0=S_0+{\int }_U{d}^{4}x(A_a^{\ast \mu }(D_{\mu }c{)}^a+\frac{1}{2}{c}_a^{\ast }[c,c{]}^a-{\bar{c}}_a^{\ast }b),$ where the nonminimal term is omitted if that pair is not included, and $S_0$ denotes the adopted classical action on the chosen domain. Assume ${\mathrm{\Gamma }}_0$ is even, has ghost number zero, satisfies $\frac{1}{2}({\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_0)=0$, and the chosen local BV functional domain has no nonzero central element of ghost number one. The model postulate and its classical gauge-invariance theorem are explicit inputs fixing and interpreting $S_0$; neither is proved by this theorem.

Choose the local Lorenz gauge ${\mathrm{\partial }}^{\mu }{A}_{\mu }=0$ with constant $\xi >0$, dimensional regularization $D=4-2ϵ$ with fixed positive reference mass $\mu$, $g_D={\mu }^{ϵ}g(\mu )$, $g_B={\mu }^{ϵ}{Z}_{g}g(\mu )$, $g(\mu )=g_{\mathrm{Y}\mathrm{M}}$ at the chosen scale, and modified minimal subtraction in the unrescaled-$\mu$ convention, removing pole parts and their associated $-{\gamma }_E+\log (4\pi )$ constants, and a finite engineering-dimension bound $d_{\ast }$. The geometric-connection gauge-action coefficient in the continued rules is ${\mu }^{-2ϵ}\mathrm{/}g(\mu {)}^2$, while all local cochain bounds below use four-dimensional engineering dimensions. For this adopted pure-gauge sector, treat dimensional regularization as BRST preserving in the formal analytic continuation, in the sense specified in the preceding formal-data definition; this is a regulator property, not a consequence of this theorem. The allowed counterterms are integrated Lorentz-scalar finite-jet polynomials of ghost number zero and dimension at most $d_{\ast }$ in $A,c,\bar{c},b$, their antifields, and their spacetime derivatives. The formal loop variable $\lambda$ is dimensionless; $\mathrm{\Gamma }(\lambda )={\mathrm{\Gamma }}_0+{\sum }_{n\ge 1}{\lambda }^n{\mathrm{\Gamma }}_n$ is a formal coefficient sequence with no convergence at nonzero $\lambda$ asserted. Write ${\mathrm{\Gamma }}^{(<n)}$ for a partial action whose coefficients below order $n$ already satisfy the Slavnov identity.

Assume the local quantum action principle (QAP): at each first failing order $n$ there are a local ghost-number-one four-form ${\delta }_n\in {\mathcal{A}}_{\mathrm{B}\mathrm{V},\mathrm{n}\mathrm{m}}^{1,4}(U)$ of engineering dimension at most $d_{\ast }$ and local ghost-number-two three-form ${\rho }_n\in {\mathcal{A}}_{\mathrm{B}\mathrm{V},\mathrm{n}\mathrm{m}}^{2,3}(U)$ of dimension at most $d_{\ast }-1$ such that $\mathfrak{S}({\mathrm{\Gamma }}^{(<n)})={\lambda }^n{\int }_U{\delta }_n+O({\lambda }^{n+1}),B_{{\mathrm{\Gamma }}_0}{\delta }_n+d_H{\rho }_n=0,$ where $\mathfrak{S}(\mathrm{\Gamma })=\frac{1}{2}(\mathrm{\Gamma },\mathrm{\Gamma })$, $B_{{\mathrm{\Gamma }}_0}X=({\mathrm{\Gamma }}_0,X)$, and $d_H=d{x}^{\mu }{D}_{\mu }$ is the horizontal differential on finite-jet local forms. Assume local anomaly cancellation in the explicit form of a supplied sequence of allowed ghost-number-zero four-forms ${\mathrm{\Xi }}_n\in {\mathcal{A}}_{\mathrm{B}\mathrm{V},\mathrm{n}\mathrm{m}}^{0,4}(U)$ and ghost-number-one three-forms ${\mathrm{\Theta }}_n\in {\mathcal{A}}_{\mathrm{B}\mathrm{V},\mathrm{n}\mathrm{m}}^{1,3}(U)$ satisfying ${\delta }_n=B_{{\mathrm{\Gamma }}_0}{\mathrm{\Xi }}_n+d_H{\mathrm{\Theta }}_n$ at every first failing order; put ${\mathrm{\Xi }}_n={\mathrm{\Theta }}_n=0$ at nonfailing orders. Compact support or the model's boundary conditions make the integral of each $d_H$-exact four-form vanish. Then the counterterm series $C(\lambda )=-{\sum }_{n\ge 1}{\lambda }^n{\int }_U{\mathrm{\Xi }}_n$ yields $\mathfrak{S}(\mathrm{\Gamma }(\lambda )+C(\lambda ))=0$ coefficientwise.

Use the formal-data definition's Lorentz-covariant subspaces ${\mathcal{L}}_{\le d_{\ast }}^{g,p}$ of the full BV/nonminimal complex: coefficients have dimension at most $d_{\ast }-4+p$, top forms have Lorentz-scalar densities, and three-forms are covariant currents. Assume the QAP representatives and supplied exactness sequence lie in these subspaces: ${\delta }_n\in {\mathcal{L}}_{\le d_{\ast }}^{1,4}$, ${\rho }_n\in {\mathcal{L}}_{\le d_{\ast }}^{2,3}$, ${\mathrm{\Xi }}_n\in {\mathcal{L}}_{\le d_{\ast }}^{0,4}$, and ${\mathrm{\Theta }}_n\in {\mathcal{L}}_{\le d_{\ast }}^{1,3}$. For $g\in \{0,1\}$ use $H_{\mathrm{L}\mathrm{o}\mathrm{r},\le d_{\ast }}^{g,4}(s\mid d_H)$: its cocycles are $a\in {\mathcal{L}}_{\le d_{\ast }}^{g,4}$ with $sa+d_{H}b=0$ for some $b\in {\mathcal{L}}_{\le d_{\ast }}^{g+1,3}$, and equivalences are $a\sim a+sm+d_{H}n$ with $m\in {\mathcal{L}}_{\le d_{\ast }}^{g-1,4}$ and $n\in {\mathcal{L}}_{\le d_{\ast }}^{g,3}$. Thus both cochains and primitives obey covariance and the stated bounds; unrestricted BV exactness alone is insufficient. The resulting ghost-number-zero quotient is exactly the infinitesimally Slavnov-invariant local counterterm freedom modulo the stated allowed trivial counterterms and total derivatives. The ghost-number-one quotient classifies local obstruction classes in that space; a nonzero class cannot be removed by an allowed local counterterm. This is a quotient statement and asserts no finite basis of representatives. The conclusion is conditional on the local QAP and supplied anomaly-cancellation sequence; it proves neither premise and asserts no empirical result, continuum functional measure, convergence, or nonperturbative existence. For supplied finite-dimensional regulated gauge-slice data, the Faddeev–Popov Jacobian at every point $x$ of the gauge-condition domain is the density determinant ${\mathrm{\Delta }}_{\mathrm{F}\mathrm{P}}(x)=\mathrm{\mid }\det (d{\chi }_x\circ R_x)\mathrm{\mid }$ of the specified regulated right action and gauge condition, relative to the specified positive densities. It equals zero when $d{\chi }_x\circ R_x$ is singular and is positive at a slice point exactly when the slice is transverse; it is not a continuum functional determinant.

Barnich, Brandt, and Henneaux, §§2.3–2.4 and 12.2–12.3

Maas, §4.7.2 and §§5.2.7–5.2.8