Exact background-coupled RG comparison

Use natural units $c=\mathrm{\hslash }=1$. Fix an integer $d\ge 1$, a compact smooth oriented Euclidean $d$-manifold $M$ with no boundary, its tangential structure (including a spin structure when the fields require one), and the model's boundary conditions. Let $\mathrm{\Gamma }=(X,E_{\mathrm{\Gamma }})$ be the chosen finite regulator graph, with finite vertex set $X$. Fix a compact internal symmetry group $G$ with its actual global form and its action on every retained and integrated field multiplet. The symmetry is global, not a dynamical gauge redundancy. A background groupoid $\mathcal{B}$ is a category in which every arrow is invertible; its objects are the admitted nondynamical $G$-background data and its arrows are exactly the permitted background gauge transformations, including large transformations when the model admits them. The background data, local parameter charts on its object space, groupoid objects and arrows are part of the model input.

At each positive Wilsonian scale $\mu$ in a fixed interval $0<{\mu }_{\mathrm{I}\mathrm{R}}<{\mu }_{\mathrm{U}\mathrm{V}}$, specify a finite-dimensional field configuration space ${\mathrm{\Phi }}_{\mu }$ as a standard Borel space equipped with a model-chosen Polish topology ${\tau }_{\mu }$ that generates its standard Borel $\sigma$-algebra, and with a positive normalized Borel probability measure ${\nu }_{\mu }$ on ${\mathrm{\Phi }}_{\mu }$. Its finite retained coordinate set is denoted $L_{\mu }$; on each sector component the continuous field coordinates are ${\mathbb{R}}^{L_{\mu }}$ with their Euclidean topology, and any topology and measurable structure on admitted sector labels are part of the model data. The chosen topology ${\tau }_{\mu }$ is used whenever a low-field open set or continuity in low fields is specified. For every background object $b\in \mathrm{Ob}(\mathcal{B})$, supply a dimensionless Borel complex weight $w_{\mu }(-;b)\in L^1({\mathrm{\Phi }}_{\mu },{\nu }_{\mu })$. The weight is primary and may vanish. When a finite Borel action $S_{\mu }(-;b):{\mathrm{\Phi }}_{\mu }\to \mathbb{C}$ is supplied, its weight is $w_{\mu }=e^{-S_{\mu }}$; an integrated weight need not admit such an action globally. On each stated smooth local parameter chart of backgrounds, require $b\mapsto Z_{\mu }[b]$ below to be continuous in the chart topology. Define the finite partition functional by

$$Z_{\mu }[b]={\int }_{{\mathrm{\Phi }}_{\mu }}{w}_{\mu }(\varphi ;b)d{\nu }_{\mu }(\varphi )\in \mathbb{C}\mathrm{.}$$

Require $Z_{\mu }[b]\mathrm{\ne }0$ on every background patch where the effective functional is used. On each such patch choose a continuous branch ${\mathrm{Log}}_{\mu }$ and set $W_{\mu }[b]=-{\mathrm{Log}}_{\mu }{Z}_{\mu }[b]$; retain the transition phases when branches change on overlaps. If the action is real-valued, the integral is positive and this is the usual real-action case. For complex weights, the integral and background logarithm are the explicitly specified extension of the finite-regulator Wilsonian integral; the supplier definition for real Borel actions alone does not establish integrability or nonvanishing. All field representations, regulator data, measures, normalizations, actions, scale-dependent coordinate splits and background boundary conditions are fixed model inputs.

More precisely, a finite-regulator Wilsonian high/low integration is specified at each step of a finite scale chain ${\mu }_0={\mu }_{\mathrm{U}\mathrm{V}}>{\mu }_1>\cdots >{\mu }_n={\mu }_{\mathrm{I}\mathrm{R}}$ (a single step is allowed). At step $j$, the retained coordinate set at scale ${\mu }_j$ is split as $L_{{\mu }_j}=L_{{\mu }_{j+1}}\bigsqcup H_j$. On each sector component, the associated Borel product identification is ${\mathrm{\Phi }}_{{\mu }_j}\cong {\mathrm{\Phi }}_{{\mu }_{j+1}}\times {\mathbb{R}}^{H_j}$, and the scale measures factor as ${\nu }_{{\mu }_j}={\nu }_{{\mu }_{j+1}}\otimes {\nu }_{H,j}$, where ${\nu }_{H,j}$ is a normalized Borel probability measure on ${\mathbb{R}}^{H_j}$. Use the Wilsonian matching-scale label ${\mu }_W={\mu }_{j+1}$. The finite-regulator model-data supplier specifies retained and integrated spaces ${\mathbb{R}}^X$ and a real Borel ultraviolet action; its Wilsonian-action supplier defines the corresponding high-coordinate integral where it is finite and positive. Arbitrary finite coordinate sets $L_{{\mu }_{j+1}}$ and $H_j$, admitted sector labels, and complex actions here are additional supplied model data extending that real-action setting. When the optional Gaussian submodel is selected, its ${\mathbb{R}}^X$ field spaces and its $M>0$, $E\ge 0$, $\mathrm{\Lambda }>0$, $A$, and $J$ satisfy the supplier's stated spectral, range, and cutoff conditions. For each fixed background $b$, absolute integrability of $w_{{\mu }_j}(-;b)$ and the product-measure identification give, by Fubini, a Borel conull subset $F_{j,b}\subseteq {\mathrm{\Phi }}_{{\mu }_{j+1}}$ on which the high-coordinate integral is absolutely convergent. Define the integrated weight by

$$q_{j,b}(\mathrm{\ell })={\int }_{{\mathbb{R}}^{H_j}}{w}_{{\mu }_j}(\mathrm{\ell },h;b)d{\nu }_{H,j}(h)(\mathrm{\ell }\in F_{j,b}),w_{{\mu }_{j+1}}(\mathrm{\ell };b)=e^{c_j[b]}{q}_{j,b}(\mathrm{\ell }),$$

where $c_j[b]$ is the declared dimensionless local background counterterm assigned to that step (zero is allowed). On the exceptional Borel null set choose a finite Borel extension of $q_{j,b}$, for example zero; its value does not affect any partition functional. These assertions are for each fixed $b$; any additional regularity jointly in backgrounds is supplied model data. The resulting low weight is integrable because $\mathrm{\mid }{q}_{j,b}(\mathrm{\ell })\mathrm{\mid }\le \int \mathrm{\mid }{w}_{{\mu }_j}(\mathrm{\ell },h;b)\mathrm{\mid }d{\nu }_{H,j}(h)$ almost everywhere. Its zero locus is allowed, including a set of positive low-field measure.

A field-action description is additional local data. On an open patch in the product of the declared low-field topology and a background-chart topology where a chosen representative of $q_{j,b}(\mathrm{\ell })$ is continuous and nonzero and admits a continuous logarithm, choose such a branch and define only on that patch

$$S_{{\mu }_{j+1}}(\mathrm{\ell };b)=-\mathrm{Log}{q}_{j,b}(\mathrm{\ell })-c_j[b]\mathrm{.}$$

Where an ultraviolet action is supplied, $q_{j,b}(\mathrm{\ell })=\int e^{-S_{{\mu }_j}(\mathrm{\ell },h;b)}d{\nu }_{H,j}(h)$, so this is the original local Wilsonian logarithm formula. For real actions and $c_j=0$, with the real logarithm and field data in the suppliers' domain, it agrees with the supplied Wilsonian-action definition. No local-action patches are required to cover a zero locus. For fixed $b$, a global finite Borel action representing a chosen Borel low weight exists exactly when that representative is nowhere zero: necessity follows from $e^{-S}\mathrm{\ne }0$, and sufficiency from the Borel logarithm $\log \mathrm{\mid }w\mathrm{\mid }+i\mathrm{Arg}(w)$ with $\mathrm{Arg}\in (-\pi ,\pi ]$. A finite Borel action representing the weight almost everywhere exists exactly when it is nonzero almost everywhere, with arbitrary finite action on the exceptional null set. A continuous global action requires, in addition, a continuous logarithm of the nonvanishing weight; nonvanishing alone does not guarantee this. Scale measures are compatible products, so every partition functional is the corresponding iterated finite integral, whether or not a global action exists. Every retained field and every retained topological sector is included at ${\mu }_{\mathrm{I}\mathrm{R}}$.

With these product measures, Fubini and the integrated-weight definition give the global identity $Z_{{\mu }_{j+1}}[b]=e^{c_j[b]}{Z}_{{\mu }_j}[b]$, including all low-field zero loci. On nonvanishing background patches, compatible logarithm branches then give $W_{{\mu }_j}[b]-W_{{\mu }_{j+1}}[b]=c_j[b]$. Choose the logarithm branches compatibly along the chain; any transitions between local background patches retain their transition phases.

For each background-groupoid arrow, the model supplies the Borel identifications of the field spaces and the induced pullbacks on local partition branches. Each high-coordinate weight-integration map must intertwine those pullbacks (as equalities of integrable weights modulo null sets), including the induced branch-transition phases wherever logarithmic descriptions exist; this is equivariance of the integration map and does not assert that each local scalar partition branch is gauge invariant. An exact background-coupled RG comparison is such a finite sequence of integrations for which the sum $C[b]={\sum }_{j=0}^{n-1}{c}_j[b]$ is a single-valued admissible dimensionless local background functional and

$$W_{{\mu }_{\mathrm{U}\mathrm{V}}}[b]-W_{{\mu }_{\mathrm{I}\mathrm{R}}}[b]=C[b]$$

on each common local branch and compatibly on overlaps. Here local means an integral or finite-regulator sum of densities depending on background data and finitely many derivatives or neighboring cochains; the allowed counterterm space is fixed as part of the model. This definition records a finite-regulator flow interface and its locality hypothesis. It does not assert that such a flow exists for an arbitrary continuum QFT or that any continuum limit exists.

Finite-regulator EFT model and scale data

Finite-regulator Wilsonian action

Fubini theorem for integrable complex weights

Tong, Gauge Theory, §§2.4 and 3.5

Harvey, TASI 2003 Lectures on Anomalies, §2.3