---
title: 'Classical RG Flow Equations: Theory & Applications'
url: https://www.emergentmind.com/topics/classical-renormalisation-group-rg-flow-equation
type: topic
---

# Classical RG Flow Equations: Theory & Applications

The classical renormalisation group (RG) flow equation denotes a class of scale-evolution equations in which coarse-graining is represented as a continuous one-parameter semigroup acting on couplings, actions, or effective actions. In the literature, the term appears in several closely related senses: as the local flow generated by a beta function, as the tree-level or canonical part of exact RG equations of Polchinski or Wetterich type, as a gradient-flow-induced RG equation for the effective action, and, in recent gravitational work, as an exact non-perturbative flow for classical systems with no trace term and no $\hbar$ [1010.5174, 1707.09298, 1805.12094, 2605.22037]. Across these formulations, the common structure is a first-order evolution in RG time together with a precise distinction between semigroup composition, fixed points, and the loss of microscopic information under coarse-graining.

## 1. Continuous flow, semigroup structure, and local generators

A compact formulation treats the RG as a continuous one-parameter semigroup $f_t$ acting on couplings. The defining relations are
\[
f_{t+u}=f_t\circ f_u,\qquad f_0=\mathrm{id},
\]
with local generator
\[
\beta(g)=\left.\frac{d}{dt}f_t(g)\right|_{t=0}.
\]
In the standard normalization, this yields the local flow equation
\[
\frac{d g}{d\ln\mu}=\beta(g),
\]
while discrete rescaling by a factor $s>0$ is encoded by the step-scaling function $\sigma_s(g)=g(sL)$, with $\sigma_s=f_{\ln s}$ and composition law $\sigma_{s_1s_2}=\sigma_{s_1}\circ\sigma_{s_2}$ [1010.5174].

Functional conjugation makes this structure explicit. If $\Psi(g)$ linearizes the finite RG step through Schröder’s equation,
\[
\Psi(\sigma_s(g))=\lambda(s)\Psi(g),
\]
then the continuous flow is
\[
g(t)=\Psi^{-1}(e^{\alpha t}\Psi(g_0)),\qquad \beta(g)=\alpha \frac{\Psi(g)}{\Psi'(g)}.
\]
Equivalently, with the Abel function $A(g)=\alpha^{-1}\ln\Psi(g)$,
\[
A(\sigma_s(g))=A(g)+\ln s,\qquad g(t)=A^{-1}(A(g_0)+t),\qquad \beta(g)=\frac{1}{A'(g)}.
\]
These relations impose the exact compatibility condition
\[
\beta(\sigma_s(g))=\sigma_s'(g)\,\beta(g),
\]
which ties finite and infinitesimal rescalings globally rather than only perturbatively [1010.5174].

This framework also sharpens a common misconception. Zeroes of $\beta$ do not necessarily signal fixed points of the continuous flow, and fixed points of $\sigma_s$ are not always true fixed points of the continuous trajectory. In multi-branched flows reconstructed from non-invertible step-scaling maps, a zero of $\beta$ can instead mark a turning point of the trajectory [1010.5174].

At the level of theory space, the same semigroup logic appears in exact RG. For a Wilsonian action expanded as $I_\Lambda=\sum_A \bar g_A(\Lambda)\Phi_A[\phi]$ with dimensionless couplings defined by $\bar g_A(\Lambda)=\Lambda^{4-n_A}g_A(\Lambda)$, the flow takes the form
\[
\partial_t g_A+(4-n_A)g_A=\eta_A(\{g\}),
\]
where $(4-n_A)g_A$ is the classical scaling term and $\eta_A$ the fluctuation-induced contribution [1707.09298].

## 2. Classical and quantum pieces in exact renormalisation group equations

In Polchinski-type exact RG equations, the classical contribution is the term quadratic in first functional derivatives, while the quantum contribution is the second functional derivative term. For a scalar theory with scale-dependent cutoff covariance $C_\Lambda(x,y)$, the standard Polchinski equation is
\[
\partial_\Lambda S_\Lambda[\phi]
=
\frac{1}{2}\int d^d x\, d^d y\;
\partial_\Lambda C_\Lambda(x,y)
\left(
\frac{\delta S_\Lambda}{\delta \phi(x)}
\frac{\delta S_\Lambda}{\delta \phi(y)}
-
\frac{\delta^2 S_\Lambda}{\delta \phi(x)\delta \phi(y)}
\right).
\]
The first term implements tree-level coarse-graining, and the second term is the one-loop contribution from integrating out a scale shell [1707.09298].

A generalized background-field version, designed for gauge and diffeomorphism invariant theories, was proposed in the form
\[
\Lambda \frac{d}{d\Lambda} I_\Lambda[\phi_c]
=
\mathrm{Tr}\exp\left\{-\frac{1}{\Lambda^2}\big(I^{(2)}_\Lambda[\phi_c]+I^{(2),\mathrm{GF}}_\Lambda[\phi_c,\alpha]\big)\right\}
-
\mathrm{Tr}\exp\left\{-\frac{1}{\Lambda^2} I^{(2),\mathrm{ghost}}_\Lambda[\phi_c,\alpha]\right\}.
\]
This version preserves background gauge or diffeomorphism invariance, uses a single set of background fields, and is well-defined at the UV cutoff when combined with proper-time regularization [1707.09298].

The same classical-versus-quantum split appears in the Wetterich formulation,
\[
\partial_k \Gamma_k[\Phi]
=
\frac{1}{2}\,\mathrm{Tr}\,\big[(\Gamma_k^{(2)}[\Phi]+R_k)^{-1}\,\partial_k R_k\big].
\]
When the flow is projected onto dimensionless couplings, one obtains equations of the form
\[
\partial_t \tilde g_i=d_i\tilde g_i + [\text{quantum terms}],
\]
so the canonical dimensions enter as the classical part, while the trace supplies the fluctuation-induced contribution [1707.09298, 1012.3081].

In BRST-compatible Wilsonian exact RG, the classical limit is obtained by dropping the loop term. The resulting Hamilton–Jacobi-type equation is
\[
-\Lambda \partial_\Lambda S_\Lambda[\phi]
=
\frac{1}{2}
\left(\frac{\delta S_\Lambda}{\delta\phi}\right)\!\cdot\!\dot C_\Lambda\!\cdot\!
\left(\frac{\delta S_\Lambda}{\delta\phi}\right),
\]
whereas on the 1PI side the classical effective action is $\Lambda$-independent, $\partial_\Lambda \Gamma_{\mathrm{cl}}=0$ [1904.08231]. This establishes that “classical RG flow equation” can refer either to the canonical/tree-level part of an exact RG equation or to an autonomous classical flow obtained after taking the $\hbar\to0$ limit.

## 3. Gradient-flow realisations of the classical RG equation

A distinct construction uses the effective action itself to generate a gradient flow of fields,
\[
\partial_\tau \phi_\tau(x)
=
-\frac{\delta S_\tau}{\delta\phi(x)}[\phi_\tau],
\qquad
\phi_{\tau=0}(x)=\phi_0(x),
\]
together with the self-consistency condition
\[
e^{-S_\tau[\phi]}
\equiv
\int[d\phi_0]\,
\delta[\phi-\phi_\tau(\phi_0)]\,e^{-S_0[\phi_0]}.
\]
Differentiation yields the basic evolution equation
\[
\partial_\tau S_\tau[\phi]
=
\int_x
\left[
-
\frac{\delta^2 S_\tau[\phi]}{\delta\phi(x)^2}
+
\frac{\delta S_\tau[\phi]}{\delta\phi(x)}
\frac{\delta S_\tau[\phi]}{\delta\phi(x)}
\right],
\]
and, after introducing the heat kernel $K_\tau(x-y)$ to implement coarse-graining, the kernel-regularized flow becomes
\[
\partial_\tau S_\tau[\phi]
=
\int_{x,y} K_\tau(x-y)
\left[
\frac{\delta S_\tau[\phi]}{\delta\phi(x)}
\frac{\delta S_\tau[\phi]}{\delta\phi(y)}
-
\frac{\delta^2 S_\tau[\phi]}{\delta\phi(x)\delta\phi(y)}
\right].
\]
This has the same algebraic structure as the Wilson–Polchinski equation, with a classical term quadratic in first derivatives and a quantum term given by the second derivative [1805.12094].

The RG interpretation becomes precise only after a field-variable transformation is performed at each step so that the kinetic term remains canonical. In the second-order local-potential truncation
\[
I_\tau[\varphi]
=
\int_x
\left[
U_\tau(\varphi_x)
+
\frac{1}{2}W_\tau(\varphi_x)(\partial_\mu\varphi_x)^2
\right],
\]
the field redefinition is chosen so that $W_{\tau+\epsilon}\to1$ for the canonically normalized field. In dimensionless variables, the resulting LPA flow is
\[
\begin{aligned}
\tau\,\partial_\tau V_\tau(\phi)
&=
\frac{d}{2}V_\tau(\phi)
-
\frac{d-2}{4}\phi V_\tau'(\phi)
-
V_\tau'(\phi)^2
+
B_d V_\tau''(\phi)
-
B_d V_\tau''(\phi)^2 \\
&\quad
+
V_\tau'(\phi)\int_0^\phi d\phi\,V_\tau''(\phi)^2
-
\frac{d}{2}B_d,
\qquad
B_d=\frac{1}{(4\pi)^{d/2}}.
\end{aligned}
\]
Within the diagrammatic interpretation given for this truncation, the term $-V'^2$ corresponds to a 1PR contraction, $+B_dV''$ to the 1PI one-loop contraction, $-B_dV''^2$ to local two-propagator contractions, and the integral term to 2PR contractions [1805.12094].

The same gradient-flow logic was extended to a manifestly gauge-invariant ERG equation for Yang–Mills theory by defining the Wilson action through the flowed gauge field. The resulting ERG equation is
\[
\frac{\partial}{\partial\tau}e^{S_\tau[A]}
=
\int d^D x\,
\frac{\delta}{\delta\widetilde{A_\mu^a}(x)}
\Bigg[
-2\,\widehat{\widetilde{D_\nu F_{\nu\mu}^a}(x)}
-2\alpha_0\,\widehat{\widetilde{D_\mu\partial_\nu A_\nu^a}(x)}
-
\left(\frac{D-2}{2}+x_\nu\partial_\nu\right)\widehat{\widetilde{A_\mu^a}(x)}
\Bigg]
e^{S_\tau[A]},
\]
with the hat operator generating the quantum functional-derivative insertions [2012.03568].

## 4. Dissipation, irreversibility, and entropy production

A major reformulation of functional RG flow identifies the flow equation, in appropriate variables, with a non-linear diffusion equation in field space. For the zero-dimensional $O(1)$ model in local potential approximation, the exact potential flow
\[
\partial_t U(t,\sigma)
=
\frac{\tfrac12\,\partial_t r(t)}{r(t)+\partial_\sigma^2 U(t,\sigma)}
\]
induces, for $u(t,x)=\partial_x U(t,x)$ and $x\equiv \sigma$, the conservative non-linear diffusion equation
\[
\partial_t u(t,x)
=
\frac{d}{dx}
\left(
\big[\tfrac12\,\partial_t r(t)\big]\,
\frac{1}{r(t)+\partial_x u(t,x)}
\right).
\]
In this formulation, field space plays the role of the diffusion space, and the RG flow becomes a dissipative partial differential equation [2108.10085].

The corresponding entropy analysis is exact in this setting. For any convex $s\in C^2(\mathbb{R})$, define
\[
S[f(x)] \equiv -\int_{-\infty}^{\infty} dx\, s(f(x)).
\]
Then for $f=\partial_x u(t,x)$ one has
\[
\frac{d}{dt}S[\partial_x u(t,x)]\ge 0.
\]
With the choice $s(y)=y^2$,
\[
S[\partial_x u(t,x)]
=
-
\int_{-\infty}^{\infty}dx\,[\partial_x u(t,x)]^2,
\]
and the normalized entropy
\[
\mathcal{C}[\partial_x u(t,x)]
=
S[\partial_x u(t,x)]-S[\partial_x u(t=0,x)]
\]
is monotone under RG time. This makes irreversibility, entropy production, and the semigroup character of RG transformations explicit at the level of the flow equation itself [2108.10085].

The dissipative picture is reflected numerically. In finite-volume discretizations of the conservative PDE, the semi-discrete entropy $\mathcal{C}$ is non-decreasing and the total variation is non-increasing, consistent with the general TVNI property of parabolic diffusion equations. The Kurganov–Tadmor central scheme was used precisely because the conservative reformulation admits stable finite-volume methods that preserve these entropy and TV properties [2108.10085].

A related information-theoretic formulation recasts Polchinski and generalized Wegner–Morris flows as optimal-transport gradient flows of relative entropy. For the regulated probability functional $P_\Lambda[\phi]$ and Gaussian reference $Q_\Lambda[\phi]$, the exact statement is
\[
-\Lambda \frac{d}{d\Lambda} P_\Lambda[\phi]
=
-
\nabla_{\mathcal{W}_2}
S\big(P_\Lambda[\phi]\|Q_\Lambda[\phi]\big).
\]
Within this framework, a regularized relative entropy is an RG monotone:
\[
-\Lambda \frac{d}{d\Lambda}M_\Lambda(P_\Lambda)\ge0.
\]
This places entropy production, monotonicity, and coarse-graining in a unified Wasserstein-$2$ geometry of probability measures over fields [2202.11737].

## 5. Exact classical renormalisation group equations in General Relativity

In recent work on the conservative two-body problem in General Relativity, the phrase “classical RG flow equation” is used in a stricter sense: an exact non-perturbative flow equation for classical gravitational systems that contains no loop trace and no $\hbar$. On the average-action side, the central equation is
\[
\partial_k S_k[g]
=
-
\frac{\kappa}{2}\,
S_k^{(1)}[g]\cdot
\partial_k\big(S_g^{(2)}[g]+R_k\big)^{-1}
\cdot
S_k^{(1)}[g],
\]
where $\kappa=32\pi G_N$, $R_k$ is an IR regulator, $S_g^{(2)}[g]$ is the Hessian of the pure-gravity action, and $S_k^{(1)}[g]=\delta S_k/\delta g$ is the running classical source [2605.22037].

The dual Wilsonian formulation is a classical Polchinski-type flow for the metric fluctuation $h$ around a background $\bar g$:
\[
\partial_k S_k[h]
=
-
\frac{\kappa}{2}\,
S_k^{(1)}[h]\cdot
\partial_k\frac{1}{\Delta+R_k}\cdot
S_k^{(1)}[h],
\]
with $\Delta=S_g^{(2)}[\bar g]$. A Legendre transform of the pure gravity sector maps this Wilsonian flow exactly to the average-action flow, establishing a classical duality between two exact RG equations for GR [2605.22037].

The physical interpretation is shell-by-shell classical coarse-graining of metric modes. The UV boundary condition is
\[
\lim_{k\to\infty}S_k[g,x_a]=S_{\mathrm{pp}}[g,x_a],
\]
while the IR limit satisfies
\[
\lim_{k\to0}S_k[\eta,x_a]=S_{\mathrm{eff}}[x_a].
\]
Here the worldline variables $x_a^\mu$ remain explicit, and the metric is the field being coarse-grained [2510.27676].

The flow reproduces the post-Minkowskian expansion. Writing
\[
S_k=S_{\mathrm{pp}}+\kappa S_1+\kappa^2 S_2+\kappa^3 S_3+\cdots,
\]
one obtains at 1PM
\[
S_1(k=0)
=
-\frac{1}{2}S_{\mathrm{pp}}^{(1)}\cdot \Delta^{-1}\cdot S_{\mathrm{pp}}^{(1)},
\]
and the higher PM orders reproduce the standard 2PM and 3PM topologies built from regularized propagators and pure-gravity vertices [2510.27676, 2605.22037].

A practical consequence is the recovery of the 1PN two-body action from the exact flow without an explicit three-graviton vertex calculation. For the instantaneous 1PN ansatz, integrating the projected flow from $k=\infty$ to $k=0$ yields
\[
N_{k=0}=1,\quad
A_{k=0}=\frac{3}{2},\quad
B_{k=0}=-\frac{7}{2},\quad
C_{k=0}=-\frac{1}{2},\quad
D_{k=0}=0,\quad
F_{k=0}=0,\quad
H_{k=0}=-\frac{1}{2},
\]
and hence the harmonic-gauge 1PN Lagrangian
\[
\begin{aligned}
L_{1PN}
&=
\frac{1}{2}m_1 v_1^2
+
\frac{1}{2}m_2 v_2^2
+
\frac{G_N m_1 m_2}{R} \\
&\quad
+
\frac{1}{c^2}
\Bigg\{
\frac{m_1}{8}v_1^4
+
\frac{m_2}{8}v_2^4
+
\frac{G_N m_1 m_2}{R}
\left[
\frac{3}{2}(v_1^2+v_2^2)
-\frac{7}{2}v_1\!\cdot\! v_2
-\frac{1}{2}(n\!\cdot\! v_1)(n\!\cdot\! v_2)
\right]
-
\frac{1}{2}\frac{G_N^2 m_1 m_2(m_1+m_2)}{R^2}
\Bigg\},
\end{aligned}
\]
which matches the Einstein–Infeld–Hoffmann Lagrangian in harmonic gauge [2510.27676].

## 6. Geometric and cosmological manifestations of classical RG flow

In two-dimensional sigma models, the one-loop RG flow of the target-space metric is the Ricci flow
\[
\frac{d g_{\mu\nu}(t)}{dt}=-R_{\mu\nu}(g(t)),
\qquad
t=-\log\mu.
\]
Thus the classical RG equation is realized geometrically as an evolution equation for the metric on the target manifold [2509.13092].

This geometric incarnation introduces singularity theory into RG analysis. For a finite-time singularity at $T$, a type I singularity obeys
\[
\sup_{M\times[0,T)}\big((T-t)\,|R(x,t)|\big)\le C<\infty,
\]
with $|R|=(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma})^{1/2}$. The blow-up limit near such a singularity is a shrinking Ricci soliton, satisfying
\[
R_{\mu\nu}
=
\Lambda g_{\mu\nu}
+
\nabla_\mu X_\nu
+
\nabla_\nu X_\mu.
\]
Although the original sigma-model RG flow does not contain an explicit cosmological-constant term, the soliton limit carries an effective $\Lambda$; in $n>2$ dimensions it is related to Einstein’s cosmological constant by
\[
\Lambda=\frac{2\Lambda_c}{n-2}.
\]
This gives a precise sense in which an effective cosmological constant emerges from the singularity structure of the RG flow [2509.13092].

De Sitter space provides a direct example. For an Einstein metric with $\mathrm{Ric}(g_{dS})=\Lambda g_{dS}$, the unnormalized RG/Ricci flow yields
\[
g_{dS}(t)=(1-\Lambda t)\,g_{dS},
\qquad
\Lambda_{\mathrm{eff}}(t)=\frac{\Lambda}{1-\Lambda t},
\]
so de Sitter space remains a solution of the flow for $t<1/\Lambda$ and develops an IR singularity at $t=1/\Lambda$ [2509.13092].

A different cosmological use of exact RG appears in RG-improved $f(R)$ gravity. Starting from the Einstein–Hilbert truncation
\[
\Gamma_k[g,\psi]
=
\int d^4x\sqrt{-g}\,
\left[\frac{R(g)-2\Lambda_k}{16\pi G_k}\right]
+
\int d^4x\sqrt{-g}\,\mathcal{L}_{\mathrm{matter}}(\psi,g),
\]
with dimensionless couplings
\[
g(k)=\frac{k^2 G(k)}{24\pi},
\qquad
\lambda(k)=\frac{\Lambda(k)}{k^2},
\]
one identifies the scale covariantly through
\[
k^2=\rho R.
\]
This reorganizes the action into
\[
f(R)=R^2 h(R),
\qquad
h(R)=\rho\,\frac{1-2\rho\,\lambda(R)}{g(R)}.
\]
At any RG fixed point, $h(R)$ is constant and the action becomes effectively $R^2$ gravity, reflecting scale invariance [1203.3957].

In the Einstein–Hilbert truncation used there, the flow possesses the Gaussian fixed point $(g^\ast,\lambda^\ast)=(0,0)$ and a nontrivial UV fixed point $(g^\ast,\lambda^\ast)=(1/64,1/4)$. The resulting cosmology exhibits an unstable UV de Sitter phase, a long classical General Relativity regime when the trajectory passes close to the Gaussian fixed point, and a stable IR de Sitter phase [1203.3957]. This connects classical gravitational dynamics, exact RG trajectories, and fixed-point structure in a form that is conceptually distinct from, but formally related to, the strictly classical two-body flows of General Relativity.

The various formulations therefore converge on a common picture. Classical RG flow equations define a semigroup rather than a reversible group, separate canonical from fluctuation-induced running when embedded in exact RG, admit realizations as gradient or diffusion equations, possess entropy-like monotones, and can be implemented as exact classical coarse-graining equations in gravity and geometry. A plausible implication is that “classical RG flow equation” is best understood not as a single formula but as a family of mathematically equivalent or complementary structures whose shared content is semigroup evolution, irreversible coarse-graining, and scale-dependent reorganization of effective dynamics [1010.5174, 2108.10085, 2605.22037].

Source: https://www.emergentmind.com/topics/classical-renormalisation-group-rg-flow-equation