Gradient Flow Exact Renormalization (GFERG)

• GFERG is a framework where Wilsonian renormalization is reformulated using gradient-flow equations that equate flow time with the coarse-graining scale.
• It establishes an exact correspondence between flowed fields and elementary fields in a Wilson action, ensuring manifest gauge invariance, especially in Yang–Mills theory.
• The approach has been extended to include scalar fields, fermions, and matter sectors, reproducing key renormalization group functions and fixed-point behaviors.

Gradient Flow Exact Renormalization (GFERG) denotes a family of exact renormalization-group constructions in which Wilsonian coarse graining is defined through flowed fields solving diffusion or gradient-flow equations rather than through a conventional momentum cutoff alone. In the scalar setting, an exact correspondence was established between a diffused field in the bare theory and the elementary field in a Wilson action, with flow time identified with RG scale by $t=\Lambda^{-2}-\Lambda_0^{-2}$ or, in dimensionless conventions, $t=e^{2\tau}-1$ (Sonoda et al., 2019). Sonoda and Suzuki then used the same structural idea to define a Yang–Mills Wilson action directly from the flowed gauge field and to derive an ERG equation that preserves manifest gauge invariance; later work extended the framework to general scalar flows, vector-like gauge theories with fermions, scalar QED, and perturbative Yang–Mills renormalization (Sonoda et al., 2020).

1. Genealogy of the framework

The literature grouped under GFERG is relatively compact but internally differentiated. Its central papers organize naturally into scalar foundations, gauge-theory construction, matter extensions, and anomaly and perturbative checks.

Development	Core statement	arXiv
Scalar ERG-flow correspondence	Gradient flow derived from ERG with Gaussian cutoff	(Sonoda et al., 2019)
Yang–Mills GFERG	Wilson action defined through flowed gauge field; manifest gauge invariance	(Sonoda et al., 2020)
Scalar fixed points	Same fixed-point structure as Wilson–Polchinski under stated conditions	(Abe et al., 2022)
Fermion extension	Vector-like gauge theories with fermions; two chiral realizations	(Miyakawa et al., 2021)
Axial anomaly in GFERG	Correct $D=4$ anomaly reproduced	(Miyakawa, 2022)
Chiral anomaly as composite operator	Anomaly has scaling dimension zero in GFERG	(Miyakawa et al., 2023)
Pure Yang–Mills RG functions	One-loop $\beta$-function coefficients reproduced	(Nagao et al., 22 Aug 2025)

Within this line of work, the original GFERG of Sonoda and Suzuki was formulated for pure Yang–Mills theory, and the later fermionic extension generalized it to vector-like gauge theories while keeping manifest gauge invariance (Miyakawa et al., 2021). A recurrent structural theme is that the RG step is encoded by flowed fields at a flow time tied to the Wilsonian scale, while Gaussian functional differential operators convert the flowed-field identification into a genuine exact RG transformation rather than a mere change of variables (Sonoda et al., 2020).

2. Scalar origin: exact correspondence between diffusion and Wilsonian coarse graining

The scalar foundation of GFERG was established for a generic real scalar field theory in $D$-dimensional Euclidean space using an exact renormalization-group formalism with Boltzmann weight $e^{S_\Lambda[\phi]}$. The crucial regulator choice is the Gaussian cutoff

$$K\!\left(\frac{p}{\Lambda}\right)=e^{-p^2/\Lambda^2},$$

for which the ratio $K(p/\Lambda)/K(p/\Lambda_0)$ becomes the heat-kernel factor

$e^{-p^2(\Lambda^{-2}-\Lambda_0^{-2})}=e^{-tp^2}.$

This allows one to define the flowed field

$$\varphi(t,p)=e^{-tp^2}\phi(p), \qquad t=\frac{1}{\Lambda^2}-\frac{1}{\Lambda_0^2},$$

which satisfies the linear diffusion equation

$t=e^{2\tau}-1$0

The basic identification is therefore

$t=e^{2\tau}-1$1

meaning that correlators of flowed fields in the bare theory are exactly related to correlators of elementary fields in a Wilson action with finite cutoff $t=e^{2\tau}-1$2 (Sonoda et al., 2019).

The exact identities are especially transparent for connected correlators. For $t=e^{2\tau}-1$3, Wilsonian connected $t=e^{2\tau}-1$4-point functions are proportional to connected flowed-field correlators, while the two-point function contains an additional momentum-dependent shift term. In coordinate space and for $t=e^{2\tau}-1$5 with $t=e^{2\tau}-1$6, local products satisfy

$t=e^{2\tau}-1$7

together with the analogous two-point identity containing an additive contact term. For renormalizable theories this is reformulated on a renormalized trajectory, where the Wilson action at scale $t=e^{2\tau}-1$8 is represented as $t=e^{2\tau}-1$9 with $D=4$0 (Sonoda et al., 2019).

Two interpretive consequences follow directly from that correspondence. First, small flow time $D=4$1 is equivalent to large cutoff $D=4$2, so the small-$D=4$3 expansion of flowed local operators is just the short-distance/OPE expansion of the ultraviolet Wilsonian theory translated through the GFERG dictionary. Second, large $D=4$4 corresponds to small $D=4$5, so flowed observables at large diffusion time probe infrared fixed-point data. The framework is exact within the chosen ERG setup, but the identification with the standard linear heat equation depends crucially on the Gaussian regulator; for a different cutoff one still obtains a smeared field, but not necessarily one obeying $D=4$6 (Sonoda et al., 2019).

3. Yang–Mills construction and manifest gauge invariance

The Yang–Mills version of GFERG begins by imitating the scalar representation of the Wilson action. The flowed gauge field satisfies

$D=4$7

with flow time related to RG time by

$D=4$8

The Wilson action is then defined through a delta-functional constraint equating the coarse-grained field to a rescaled flowed gauge field, together with Gaussian functional differential operators before and after the constraint. This definition preserves the partition function and yields a scale-dependent Wilson action that is gauge invariant in the ordinary sense, not only through modified Ward identities (Sonoda et al., 2020).

A useful variable is the scaled gauge field

$D=4$9

under which the gauge transformation takes the standard form. Introducing the hatted operator

$\beta$0

the continuum Yang–Mills GFERG equation becomes

$\beta$1

The second-order functional-differential structure carried by the hat is what turns the flow-based construction into an exact RG equation rather than a deterministic field redefinition. The same idea admits a lattice formulation based on lattice gradient flow and Haar-measure-preserving differential operators, where partition-function preservation and gauge invariance are exact (Sonoda et al., 2020).

A later perturbative reformulation for pure Yang–Mills introduced a Reuter-equation-type integral representation for the Wilson action that couples Wilsonian gauge and ghost fields to flowed bare fields. In that formulation the flow time is written dimensionfully as

$\beta$2

the exact RG equation contains hatted differential operators for gauge and ghost variables, and the Wilson action satisfies a BRST Ward–Takahashi identity. The Wilsonian quadratic kernel is directly related to the flowed two-point function,

$\beta$3

so the renormalization problem of the Wilson action reduces to the renormalization of flowed-field correlators (Nagao et al., 22 Aug 2025).

Using that relation, the one-loop renormalization constants were extracted from the large-momentum asymptotics of the flowed self-energy integrand, yielding

$\beta$4

and therefore

$\beta$5

This reproduces the standard Yang–Mills one-loop $\beta$6-function coefficient. The same paper argues, from the correspondence with the Lüscher–Weisz gradient-flow formalism and equality of flowed correlators up to contact terms, that conventional RG functions are reproduced to all perturbative orders; that all-order statement is argued structurally rather than established by an explicit multiloop computation inside GFERG (Nagao et al., 22 Aug 2025).

4. Matter fields, chiral structure, and anomaly sectors

The fermionic extension of GFERG treats vector-like gauge theories with flowed fermions $\beta$7 and $\beta$8 in addition to the flowed gauge field. The Wilson action at RG time $\beta$9 is defined by identifying its arguments with appropriately rescaled flowed fields at

$D$0

with running gauge coupling $D$1 and fermion wavefunction factor $D$2. The defining construction preserves the partition function and keeps ordinary gauge symmetry manifest because the delta constraints, the flowed equations, and the Gaussian functional operators are all gauge covariant or gauge invariant. In this framework the gauge transformation of the Wilsonian gauge field takes the form

$D$3

which encodes the normalization chosen for the gauge field (Miyakawa et al., 2021).

The inclusion of fermions introduces a chiral-symmetry choice. One option preserves the conventional chiral symmetry by replacing the simple fermionic Gaussian kernel with

$D$4

but this produces a more complicated ERG equation. The simpler main construction instead realizes chiral symmetry in a modified Ginsparg–Wilson-like form through

$D$5

If the fermion sector is bilinear,

$D$6

then $D$7 implies the continuum GW relation

$D$8

Within this setting a gauge-invariant local Wilson action was worked out in QED to the lowest nontrivial order $D$9, and the correct axial anomaly in $e^{S_\Lambda[\phi]}$0 was reproduced (Miyakawa et al., 2021).

The four-dimensional axial anomaly was then analyzed directly in GFERG. For a Wilsonian Dirac kernel $e^{S_\Lambda[\phi]}$1 satisfying the GW relation, the local chiral Ward identity acquires the contact term

$e^{S_\Lambda[\phi]}$2

and evaluating that term to order $e^{S_\Lambda[\phi]}$3 in QED gives

$e^{S_\Lambda[\phi]}$4

The derivation uses the perturbative GFERG Wilson action, flowed Gaussian factors such as $e^{S_\Lambda[\phi]}$5, and the identity $e^{S_\Lambda[\phi]}$6, which enforces the GW structure order by order (Miyakawa, 2022).

A closely related development reformulated the chiral anomaly as a composite operator in GFERG for massless Dirac fermions coupled to external chiral gauge fields. In that construction the anomaly is defined by

$e^{S_\Lambda[\phi]}$7

and the GFERG equation implies that $e^{S_\Lambda[\phi]}$8 is a composite operator of scaling dimension zero. In $e^{S_\Lambda[\phi]}$9, Wess–Zumino consistency together with manifest $$K\!\left(\frac{p}{\Lambda}\right)=e^{-p^2/\Lambda^2},$$0 invariance yields the Bardeen form of the anomaly in terms of the $$K\!\left(\frac{p}{\Lambda}\right)=e^{-p^2/\Lambda^2},$$1 variables, and the coefficient is found to be

$$K\!\left(\frac{p}{\Lambda}\right)=e^{-p^2/\Lambda^2},$$2

The same paper argues that the result extends to QCD, although the fully dynamical non-Abelian derivation is presented as incomplete (Miyakawa et al., 2023).

Matter-field extensions are not limited to fermions. In scalar QED, GF-ERG was built from a BRST-covariant diffusion equation for the charged scalar,

$$K\!\left(\frac{p}{\Lambda}\right)=e^{-p^2/\Lambda^2},$$3

with the other fields following simple diffusion. The resulting flow equation contains the ordinary Wilson–Polchinski terms plus extra terms generated by the gauge-covariant scalar flow. Solving the theory perturbatively around the Gaussian fixed point through $$K\!\left(\frac{p}{\Lambda}\right)=e^{-p^2/\Lambda^2},$$4, the anomalous dimension of the gauge field in $$K\!\left(\frac{p}{\Lambda}\right)=e^{-p^2/\Lambda^2},$$5 was found to be

$$K\!\left(\frac{p}{\Lambda}\right)=e^{-p^2/\Lambda^2},$$6

and the photon mass was shown to remain zero in general spacetime dimension,

$$K\!\left(\frac{p}{\Lambda}\right)=e^{-p^2/\Lambda^2},$$7

The vanishing of the photon mass contrasts with conventional cutoff ERG, where an artificial cutoff-scale photon mass is induced (Haruna et al., 2023).

5. Fixed points, universal data, and asymptotic flow-time structure

A general scalar GFERG equation was later formulated for flows of the form

$$K\!\left(\frac{p}{\Lambda}\right)=e^{-p^2/\Lambda^2},$$8

where $$K\!\left(\frac{p}{\Lambda}\right)=e^{-p^2/\Lambda^2},$$9 consists of a diffusion term plus polynomial nonlinearities. The Wilson action is defined through flowed fields and a Gaussian functional operator built from

$K(p/\Lambda)/K(p/\Lambda_0)$0

Differentiation yields the compact GFERG equation

$K(p/\Lambda)/K(p/\Lambda_0)$1

where the first two lines reproduce Wilson–Polchinski structure and the remaining terms encode the nonlinear part of the chosen gradient flow (Abe et al., 2022).

The central fixed-point result is asymptotic. Writing

$K(p/\Lambda)/K(p/\Lambda_0)$2

one has asymptotically

$K(p/\Lambda)/K(p/\Lambda_0)$3

If

$K(p/\Lambda)/K(p/\Lambda_0)$4

then $K(p/\Lambda)/K(p/\Lambda_0)$5, so all nonlinear GFERG-specific terms vanish at large $K(p/\Lambda)/K(p/\Lambda_0)$6. The fixed-point equation therefore reduces exactly to the Wilson–Polchinski fixed-point equation. Under that condition, GFERG and Wilson–Polchinski have the same fixed points; relevant critical exponents also coincide, while irrelevant exponents can differ because finite-$K(p/\Lambda)/K(p/\Lambda_0)$7 nonlinear source terms can change asymptotic decay rates in irrelevant directions. For the cutoff choice used in that paper, there is no fixed point at finite flow time (Abe et al., 2022).

This structure was illustrated in the $K(p/\Lambda)/K(p/\Lambda_0)$8 nonlinear sigma model in $K(p/\Lambda)/K(p/\Lambda_0)$9 dimensions. The flow equation

$e^{-p^2(\Lambda^{-2}-\Lambda_0^{-2})}=e^{-tp^2}.$0

leads asymptotically to the Wilson–Fisher fixed point of the $e^{-p^2(\Lambda^{-2}-\Lambda_0^{-2})}=e^{-tp^2}.$1 model, with

$e^{-p^2(\Lambda^{-2}-\Lambda_0^{-2})}=e^{-tp^2}.$2

In the local-potential approximation, the relevant operator $e^{-p^2(\Lambda^{-2}-\Lambda_0^{-2})}=e^{-tp^2}.$3 has the same exponent as in Wilson–Polchinski, and derivative-free irrelevant operators can also retain the same exponent when the inhomogeneous GFERG source vanishes in those directions (Abe et al., 2022).

The scalar correspondence of the original ERG derivation gives a complementary asymptotic interpretation. Small $e^{-p^2(\Lambda^{-2}-\Lambda_0^{-2})}=e^{-tp^2}.$4 is the ultraviolet/OPE regime because $e^{-p^2(\Lambda^{-2}-\Lambda_0^{-2})}=e^{-tp^2}.$5, whereas large $e^{-p^2(\Lambda^{-2}-\Lambda_0^{-2})}=e^{-tp^2}.$6 is the infrared regime controlled by fixed points. The framework therefore ties three notions together without truncation: diffusion time, Wilsonian scale, and universal long-distance data (Sonoda et al., 2019).

6. Related gradient formulations and conceptual boundaries

Several adjacent programs sharpen the conceptual meaning of GFERG, but they are not identical to the Sonoda–Suzuki construction. Abe and Fukuma proposed a self-consistent scalar flow

$e^{-p^2(\Lambda^{-2}-\Lambda_0^{-2})}=e^{-tp^2}.$7

with $e^{-p^2(\Lambda^{-2}-\Lambda_0^{-2})}=e^{-tp^2}.$8 defined as the distribution of flowed fields and obeying the exact functional equation

$e^{-p^2(\Lambda^{-2}-\Lambda_0^{-2})}=e^{-tp^2}.$9

They argued that this becomes RG-like only after a field redefinition that keeps the kinetic term canonical; in LPA and the $$\varphi(t,p)=e^{-tp^2}\phi(p), \qquad t=\frac{1}{\Lambda^2}-\frac{1}{\Lambda_0^2},$$0-expansion it reproduces the Gaussian and Wilson–Fisher linearized eigenvalues to $$\varphi(t,p)=e^{-tp^2}\phi(p), \qquad t=\frac{1}{\Lambda^2}-\frac{1}{\Lambda_0^2},$$1 (Abe et al., 2018).

Other works recast ERG as diffusion or transport on configuration space. An entropic-dynamics formulation derives exact RG as a functional Fokker–Planck equation with drift, generated by maximum-entropy transition probabilities for field configurations; this gives ERG an inferential interpretation but not a modern geometric gradient-flow structure (Pessoa et al., 2017). Tokar’s lattice Ginzburg–Landau formalism evolves the interaction functional $$\varphi(t,p)=e^{-tp^2}\phi(p), \qquad t=\frac{1}{\Lambda^2}-\frac{1}{\Lambda_0^2},$$2 by an exact diffusion equation, with the transformed equation for $$\varphi(t,p)=e^{-tp^2}\phi(p), \qquad t=\frac{1}{\Lambda^2}-\frac{1}{\Lambda_0^2},$$3 taking a viscous Hamilton–Jacobi/Burgers form and admitting a transparent semigroup structure; the paper is adjacent to GFERG in its diffusion viewpoint rather than in the Yang–Mills flowed-field sense (Tokar, 2021).

A different but mathematically sharper bridge is provided by the optimal-transport formulation of exact RG. There Polchinski’s equation, and more generally Wegner–Morris flows, are shown to be equivalent to a Wasserstein-2 gradient flow of a field-theoretic relative entropy,

$$\varphi(t,p)=e^{-tp^2}\phi(p), \qquad t=\frac{1}{\Lambda^2}-\frac{1}{\Lambda_0^2},$$4

with a nonperturbative monotone

$$\varphi(t,p)=e^{-tp^2}\phi(p), \qquad t=\frac{1}{\Lambda^2}-\frac{1}{\Lambda_0^2},$$5

This is a genuine gradient-flow formulation of exact RG on probability space, but it is conceptually distinct from the flowed-field, manifestly gauge-invariant GFERG program (Cotler et al., 2022).

These related constructions clarify several recurring misconceptions. GFERG is not simply the statement that ordinary gradient flow is RG; the Wilsonian content resides in the exact action representation and the accompanying second-order functional operators. Nor is every diffusion-based ERG automatically the same framework: the scalar heat-kernel correspondence with standard gradient flow is exact only for the special Gaussian cutoff $$\varphi(t,p)=e^{-tp^2}\phi(p), \qquad t=\frac{1}{\Lambda^2}-\frac{1}{\Lambda_0^2},$$6, while the gauge-theory program is distinguished by the preservation of ordinary gauge invariance rather than modified cutoff Ward identities (Sonoda et al., 2019, Sonoda et al., 2020). At the same time, the present literature leaves open questions. In the fermionic gauge-theory extension locality is demonstrated perturbatively in QED only to the lowest nontrivial order, in scalar QED the photon sector is controlled but higher matter-sector renormalization remains unfinished, and in pure Yang–Mills the all-order agreement with conventional RG functions is argued from the gradient-flow correspondence rather than established by a complete multiloop solution (Miyakawa et al., 2021, Haruna et al., 2023, Nagao et al., 22 Aug 2025).

Within those boundaries, GFERG has a precise core meaning: Wilsonian exact renormalization is reformulated through flowed fields, with flow time acting as the coarse-graining parameter and, in gauge theory, with manifest gauge invariance retained throughout the RG evolution.