Gauge Invariant Functional Flow Equation

• The gauge invariant functional flow equation is a renormalization group tool that preserves gauge and diffeomorphism invariance by projecting out unphysical modes.
• It employs covariant projectors and tailored regulator terms to isolate physical degrees of freedom, enabling unbiased nonperturbative analysis in quantum field theories and gravity.
• The framework integrates algebraic and geometric methods, such as the BV formalism and gradient flow, offering robust computations of β-functions and fixed points.

A gauge invariant functional flow equation is a renormalization group (RG) flow equation for the effective (average) action in gauge or diffeomorphism-invariant theories, whose structure and regulator implementation guarantee manifest invariance under gauge transformations at every RG scale. Such formalism is essential for unbiased nonperturbative analysis in quantum gauge field theory and gravity, enabling computations of β-functions, fixed points, and critical phenomena without spurious gauge artifacts or the need for cumbersome gauge-variant identities. The modern development of gauge-invariant flow equations utilizes projection techniques, geometric field decompositions, algebraic formulations in the Batalin–Vilkovisky (BV) formalism, and carefully tailored regulator terms. Key constructions appear in the works of Wetterich, Sonoda, Suzuki, Lavrov, and others, with applications ranging from Yang–Mills theory to quantum gravity and QED.

1. The Principle of Physical and Gauge Mode Separation

Gauge invariant flow equations rely critically on the precise separation of “physical” (gauge-invariant) and “gauge” (pure gauge) components of fields. This is implemented by defining a covariant projector $P$ that annihilates gauge variations: for Yang–Mills, $P_\mu^\nu = \delta_\mu^\nu - D_\mu D^{-2} D^\nu$, where $D_\mu$ is the gauge-covariant derivative. Any gauge field fluctuation $h_\mu$ is decomposed as $h_\mu = f_\mu + a_\mu$ with $f_\mu = P_\mu^\nu h_\nu$ (physical, $D^\mu f_\mu=0$) and $a_\mu = (1-P)_\mu^\nu h_\nu$ (longitudinal, $a_\mu=D_\mu\lambda$) (Wetterich, 2017, Wetterich, 2016). The construction assigns to every gauge potential $A_\mu$ a unique "physical" representative $\hat{A}_\mu(A)$, characterized by (i) the response to infinitesimal changes being purely physical, and (ii) invariance under infinitesimal gauge transformations.

This principle extends structurally to gravity, where the York (TT) decomposition isolates the transverse-traceless and conformally-invariant scalar parts of the metric fluctuation as gauge-invariant variables, discarding the diffeomorphism fiber directions (Demmel et al., 2014).

2. Manifestly Gauge-Invariant Flow Equation Structure

Given this field decomposition, the effective average action $\Gamma_k$ is constructed to depend only on the physical sector—either on the explicit gauge-invariant representative $\hat{A}_\mu$, or on variables projected by $P$. The general form of a gauge-invariant flow equation is: $$\partial_t \Gamma_k = \frac{1}{2} \mathrm{Tr}\left[P^T ( \Gamma_k^{(2)} + R_k )^{-1} P\, \partial_t R_k \right] + \delta_k - \epsilon_k$$ where $R_k$ is an infrared (IR) regulator acting only on physical modes, and $\delta_k, \epsilon_k$ are measure contributions from the (decoupled) gauge sector (Wetterich, 2016, Wetterich, 2017, Demmel et al., 2014, Wetterich et al., 2019, Maitiniyazi et al., 16 Dec 2025).

By design, only transverse (genuine dynamical) fluctuations propagate inside the functional traces. For gravity, the same structure appears with projectors on transverse-traceless metric modes, enabling a manifestly diffeomorphism-invariant RG (Demmel et al., 2014, Maitiniyazi et al., 16 Dec 2025).

3. Gauge Fixing, Regulator Terms, and Ghost Decoupling

In traditional functional RG, gauge fixing introduces gauge artifacts, and ghosts are needed to cancel unphysical contributions. In the gauge-invariant formalism, "physical gauge fixing" is applied—effectively, a Landau-type gauge acting only on gauge (longitudinal) variables—which decouples in the limit $\alpha \to 0$ (Landau gauge), enforcing the gauge sector to $c'=0$. The Faddeev–Popov determinant is evaluated in terms of the physical background, but the essential outcome is that the functional measure and ghost contributions appear as a universal, regulator-determined "measure term" (e.g., $\delta_k$) that cancels all remaining gauge-variant artifacts (Wetterich, 2017, Wetterich, 2016, Demmel et al., 2014).

The regulator term $R_k$ is always constructed from (covariant) Laplacians projected onto the physical subspace, ensuring commutation with gauge transformations. In advanced geometric settings, this is often realized using the heat kernel or diffusion kernel of the covariant Laplacian, as in the Gauge Flow Exact Renormalization Group (GFERG) (Sonoda et al., 30 Jun 2025, Sonoda et al., 2022).

4. Algebraic and Geometric Generalizations

Gauge-invariant flow equations have been generalized beyond Yang–Mills and gravity using several formalisms:

• Batalin–Vilkovisky (BV) Algebraic Approach: The effective average action $\Gamma_k[\Phi, \Phi^*]$ in the BV antifield formalism satisfies a modified Quantum Master Equation (QME) alongside the Wetterich-type flow, guaranteeing compatibility of the flow with BRST cohomology and Slavnov–Taylor (Zinn–Justin) identities. The regulator is inserted as part of a trivial pair in the cohomology, and the flow remains gauge-invariant up to the measure anomaly (D'Angelo et al., 2023, Echigo et al., 17 Jul 2025).
• Gradient Flow ERG (GFERG): Employs the Yang–Mills gradient flow to implement block-spinning, leading to an exact RG equation where the block-spin is covariant and gauge-invariant at every scale, with a unique ordering of functional derivatives fixing contact term ambiguities. Modified correlators of the Wilson action map to flowed-field correlators of the bare action; gauge invariance is manifest (Sonoda et al., 30 Jun 2025, Sonoda et al., 2022).
• Gauge-Invariant Cutoff Schemes: Deformed (star-product) gauge symmetries protect the invariance of the action under generalized gauge transformations with cutoff- or flow-dependent kernels; exact Ward or Slavnov–Taylor identities are valid along the full flow (Ardalan et al., 2011).
• Physics-Informed RG (PIRG): Splits the effective action into a gauge-invariant "quantum" part (function of the sum background + fluctuation) and a classical gauge-fixing part that only enters to ensure invertibility. The RG flow is constructed to preserve the gauge-invariant sector, and corrections are introduced—when necessary—to restore one-loop universality and background independence (Ihssen et al., 28 Mar 2025).

5. Applications: Yang–Mills, Gravity, Abelian Theories

Yang–Mills Theory:

The gauge-invariant flow equation enables computation of the nonperturbative running of the gauge coupling in terms of a single, gauge-invariant effective action. Standard results, such as the one-loop $\beta$-function and the flow of the propagator, are recovered exactly from the physical sector, with gauge artifacts eliminated by projectors and the universal measure term (Wetterich, 2017). Explicitly, for

$$\Gamma_k[A] = \frac{1}{4g^2(k)} \int \mathrm{Tr}\, F_{\mu\nu} F_{\mu\nu}$$

the running is determined by the trace over projected, cutoff-regularized propagators in covariantly constant backgrounds.

Quantum Gravity:

Similar constructions in geometric variables with the physical-metric decomposition yield fully geometric RG flows in the "linear-geometric approximation." The flow equations for the Newton coupling and cosmological constant, or higher-derivative $f(R)$ theories, retain manifest diffeomorphism invariance throughout, and reproduce phase diagrams and fixed-point structures essential for the Asymptotic Safety scenario (Demmel et al., 2014, Wetterich et al., 2019, Maitiniyazi et al., 16 Dec 2025).

QED and Abelian Theories:

The same logic extends via the BV formalism and in gradient flow-based approaches (GFERG), ensuring that the effective actions (in either the Wilsonian or 1PI formulation) are invariant under the conventional gauge transformation, facilitating construction of manifestly gauge-invariant truncations (Sonoda et al., 2022, Echigo et al., 17 Jul 2025).

6. Extensions, Limitations, and the Gauge Dependence Problem

While the methods above achieve gauge invariance for a wide class of RG flows, critical subtleties remain:

• For many conventional 1PI (and even 2PI) FRG equations with "non-covariant" regulator choices, gauge dependence persists at any finite scale $k>0$ due to explicit breaking of BRST invariance by the regulator. Exact gauge invariance is only achieved in the $k\to 0$ limit, i.e., for the full quantum effective action. Nielsen-type identities track this gauge dependence and its cancellation on shell (Lavrov, 2020).
• Genuinely nonperturbative, exactly gauge-invariant functional RG flows in the presence of generic IR regulators still present technical challenges, and fully BRST-exact, nonperturbative regulator constructions remain a subject of ongoing research (Lavrov, 2020, Ihssen et al., 28 Mar 2025).
• In geometric gravity truncations, the phase diagram and critical exponents are remarkably robust against changes in the coarse-graining scheme (e.g., regulator endomorphisms), supporting the universality of the physical predictions (Demmel et al., 2014, Maitiniyazi et al., 16 Dec 2025).
• Corrections arising from field-dependent cutoffs can be organized systematically as higher-loop operators, corresponding to field contractions in the flow equation, and can in principle be controlled via expansions (Wetterich, 26 Mar 2024).

7. Summary Table: Central Features of Gauge-Invariant Flow Equations

Feature	Description	Key References
Projector $P$	Covariant operator removing gauge modes from functional traces	(Wetterich, 2017, Wetterich, 2016)
Physical gauge fixing ($\alpha\to0$)	Gauge fixing only acts on gauge sector, decouples in flow	(Wetterich, 2017, Demmel et al., 2014)
Measure term ($\delta_k$)	Universal contribution cancelling gauge artifacts	(Wetterich, 2017, Wetterich et al., 2019)
BV formalism	Flow equation compatible with QME, Zinn-Justin (mSTI) master equation	(D'Angelo et al., 2023, Echigo et al., 17 Jul 2025)
GFERG/gradient flow	Gauge-invariant block-spin and RG evolution via covariant diffusion kernels	(Sonoda et al., 30 Jun 2025, Sonoda et al., 2022)
Geometric gravity FRG	Physical/metric split, TT and scalar modes, no ghosts, simple beta functions	(Demmel et al., 2014, Maitiniyazi et al., 16 Dec 2025)

• C. Wetterich, "Gauge-invariant fields and flow equations for Yang-Mills theories" (Wetterich, 2017).
• C. Wetterich, "Gauge invariant flow equation" (Wetterich, 2016).
• M. Demmel, F. Saueressig, O. Zanusso, "RG flows of Quantum Einstein Gravity in the linear-geometric approximation" (Demmel et al., 2014).
• A. Eichhorn, C. Wetterich, "Variable Planck mass from the gauge invariant flow equation" (Wetterich et al., 2019).
• H. Gies et al., "Scaling solutions for gauge invariant flow equations in dilaton quantum gravity" (Maitiniyazi et al., 16 Dec 2025).
• D. Sonoda, H. Suzuki, "A basis of the gradient flow exact renormalization group for gauge theory" (Sonoda et al., 30 Jun 2025); S. Aoki et al., "One-particle irreducible Wilson action in the gradient flow ERG formalism" (Sonoda et al., 2022).
• P. Lavrov, "Gauge dependence of alternative flow equation for the functional renormalization group" (Lavrov, 2020).
• L. Jans, J. Pawlowski, Y. Slade, "Physics-informed gauge theories" (Ihssen et al., 28 Mar 2025).
• E. D'Angelo, K. Rejzner, "A Lorentzian renormalisation group equation for gauge theories" (D'Angelo et al., 2023).
• A. Ardalan, M. Ghasemkhani, N. Sadooghi, "Gauge Invariant Cutoff QED" (Ardalan et al., 2011).
• L. Jans, J. Pawlowski, Y. Slade, "Functional Renormalization Group Flows and Gauge Consistency in QED" (Echigo et al., 17 Jul 2025).
• C. Wetterich, "Simplified functional flow equation" (Wetterich, 26 Mar 2024).

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In summary, gauge-invariant functional flow equations implement exact RG dynamics in gauge and gravity theories while preserving local symmetry at all scales. This is achieved via covariant mode projections, suitable gauge fixing, careful regulator choice, and structural compatibility with BRST cohomology. Such equations underpin nonperturbative investigations and have become standard in asymptotic safety, quantum gauge field theory, and quantum gravity research.