Wilsonian RG

Goal

The Wilsonian renormalization group is the cleanest way to say what a quantum field theory is at long distances. It is also the cleanest way to understand why conformal field theories are the natural boundary theories in AdS/CFT.

The basic idea is simple: physics observed with finite resolution should not depend on microscopic variables that we cannot resolve. If the theory has a UV cutoff $\Lambda$, we integrate out modes with momenta near $\Lambda$, then rewrite the result as another theory with the same cutoff but different couplings. Repeating this step gives a flow in the space of theories.

The slogan is:

$$\boxed{ \text{Wilsonian RG}= \text{coarse grain}+ \text{rescale}+ \text{reparametrize couplings}. }$$

A CFT is a fixed point of this flow. Near a fixed point, the directions in theory space are classified as relevant, marginal, or irrelevant. In AdS/CFT language, this classification becomes the classification of bulk boundary conditions and radial falloffs.

This page is not about doing loop integrals efficiently. It is about the conceptual machine that turns microscopic theories into CFT data.

Cutoffs and effective actions

Start with a Euclidean QFT regulated at a momentum cutoff $\Lambda$. Schematically,

$$Z_\Lambda[J] = \int_{|p|<\Lambda} \mathcal D\phi\, \exp\left[-S_\Lambda[\phi]+\int d^d x\,J(x)\mathcal O(x)\right].$$

The cutoff is not necessarily a literal sharp momentum cutoff. It may be a lattice spacing $a$, a smooth regulator, Pauli-Villars fields, dimensional regularization plus a subtraction scale, or something else. The Wilsonian idea is independent of this choice:

$$\text{cutoff }\Lambda \quad\Longleftrightarrow\quad \text{resolution length }\ell_{\rm res}\sim \Lambda^{-1}.$$

The Wilsonian action $S_\Lambda$ is the action that reproduces observables at scales much longer than $\Lambda^{-1}$. Because every local interaction allowed by symmetries is generated by coarse graining, the correct Wilsonian action is not usually a short expression with two or three terms. It is an expansion in local operators:

$$S_\Lambda = \sum_i g_i(\Lambda)\int d^d x\,\mathcal O_i(x).$$

Here the index $i$ runs over all local operators compatible with the symmetries. For a scalar theory with a $\mathbb Z_2$ symmetry $\phi\mapsto -\phi$, this includes operators such as

$$1, \qquad \phi^2, \qquad (\partial\phi)^2, \qquad \phi^4, \qquad \phi^6, \qquad \phi^2(\partial\phi)^2, \qquad (\partial^2\phi)^2, \qquad \cdots.$$

This infinite expansion looks terrifying at first. Wilson’s key insight is that near a long-distance fixed point, most of these couplings become irrelevant. The infinite-dimensional theory space remains real, but the long-distance physics is controlled by a finite or small set of important directions.

One Wilsonian RG step

Let $b>1$. Split the field into slow and fast momentum modes,

$$\phi(p)=\phi_{<}(p)+\phi_{>}(p),$$

where

$$\phi_{<}(p): |p|<\Lambda/b, \qquad \phi_{>}(p): \Lambda/b<|p|<\Lambda.$$

The first step is to integrate out the fast modes:

$$\exp\left[-\widetilde S_{\Lambda/b}[\phi_{<}]\right] = \int_{\Lambda/b<|p|<\Lambda}\mathcal D\phi_{>}\, \exp\left[-S_\Lambda[\phi_{<}+\phi_{>}] \right].$$

This produces an effective action with a lower cutoff $\Lambda/b$. The second step is to rescale momenta and positions,

$$p'=bp, \qquad x'=x/b,$$

so that the cutoff is restored to $\Lambda$. Finally, rescale the fields so that the chosen normalization convention is preserved. The result is a new Wilsonian action $S'_\Lambda$ with new couplings $g_i'$.

A Wilsonian RG step integrates out fast modes in the shell $\Lambda/b<|p|<\Lambda$, rescales $p'=bp$ and $x'=x/b$, and produces new couplings $g_i'$. Near a fixed point $g_\star$, the flow is approximately linear: relevant directions have $y>0$ and move away under coarse graining, while irrelevant directions have $y<0$ and flow toward the fixed point.

The word “group” is slightly cheeky. Coarse graining throws away microscopic information, so the operation is usually not invertible. Mathematically it is closer to a semigroup. Physicists still say renormalization group because the infinitesimal flow is generated by beta functions and because the terminology is now immovable, like “canonical” in a context where nothing is canonical.

Dimensionless couplings

A coupling with dimensions is not a good coordinate on theory space. Suppose an operator $\mathcal O_i$ has scaling dimension $\Delta_i$ at some fixed point. The deformation

$$\delta S = g_i\int d^d x\,\mathcal O_i(x)$$

is dimensionless, so the engineering dimension of $g_i$ is

$$[g_i]=d-\Delta_i.$$

A dimensionless coupling at scale $\mu$ is therefore

$$\lambda_i(\mu)=g_i(\mu)\,\mu^{\Delta_i-d}.$$

Equivalently, in a Wilsonian step with scale factor $b>1$, the leading linearized transformation is

$$\lambda_i(b)=b^{d-\Delta_i}\lambda_i(1).$$

This equation is the first appearance of the relevance criterion:

$$\boxed{ \begin{array}{ccl} \Delta_i<d &\Longleftrightarrow& \mathcal O_i\text{ is relevant},\\ \Delta_i=d &\Longleftrightarrow& \mathcal O_i\text{ is marginal},\\ \Delta_i>d &\Longleftrightarrow& \mathcal O_i\text{ is irrelevant}. \end{array} }$$

Relevant deformations grow at long distances. Irrelevant deformations die away at long distances. Marginal deformations require more care: quantum corrections can make them marginally relevant, marginally irrelevant, or exactly marginal.

The language is physical. Relevant deformations matter in the IR. Irrelevant deformations become invisible in the IR. Marginal deformations sit at the border and make everyone work harder.

Beta functions

Let $g_i$ denote dimensionless coordinates on theory space. A Wilsonian flow can be written as

$$\frac{d g_i}{d\log b}=\beta_i(g),$$

where $b$ increases as we coarse grain toward the IR. This is the convention used in this page. Another common convention uses an energy scale $\mu$ and writes $\mu\,d g_i/d\mu$; the signs then look reversed because moving to the IR means decreasing $\mu$. When comparing formulas across books, always check the arrow of the RG flow.

A fixed point is a point $g_\star$ in theory space such that

$$\boxed{ \beta_i(g_\star)=0 \quad\text{for all }i. }$$

At a fixed point, coarse graining followed by rescaling gives back the same theory. This is scale invariance. Under suitable assumptions of locality, unitarity, Poincare invariance, and a well-behaved stress tensor, the fixed points of interest in this course are conformal field theories.

Near a fixed point, expand

$$g_i=g_{\star i}+\delta g_i.$$

The beta function becomes

$$\frac{d}{d\log b}\delta g_i = M_{ij}\delta g_j+O(\delta g^2), \qquad M_{ij}=\left.\frac{\partial\beta_i}{\partial g_j}\right|_{g=g_\star}.$$

Diagonalize $M$. If $u_a$ is an eigen-coordinate, then

$$\frac{d u_a}{d\log b}=y_a u_a, \qquad u_a(b)=b^{y_a}u_a(1).$$

The eigenvalue $y_a$ is the RG exponent. The corresponding operator dimension is

$$\boxed{ \Delta_a=d-y_a. }$$

Thus the RG classification can be restated as

$$y_a>0\text{ relevant}, \qquad y_a=0\text{ marginal}, \qquad y_a<0\text{ irrelevant}.$$

At an interacting fixed point, the eigen-operators are usually not the naive monomials in the microscopic fields. They are particular linear combinations that diagonalize the dilatation operator. This is one of the first conceptual shifts from Lagrangian QFT to CFT: the important objects are not bare terms in a Lagrangian but scaling operators.

Universality

Universality is the reason the Wilsonian RG is powerful.

Different microscopic Hamiltonians can flow to the same fixed point. For example, many very different lattice systems can have the same long-distance Ising CFT. Their lattice spacings, short-distance interactions, and irrelevant couplings differ, but those differences are washed out under coarse graining.

Near a fixed point, theory space decomposes into directions:

$$\text{theory space near }g_\star = \text{relevant directions} \oplus \text{marginal directions} \oplus \text{irrelevant directions}.$$

The irrelevant directions flow into the fixed point. The relevant directions flow away from it. The set of points that flow into the fixed point under coarse graining is the critical surface. To reach the fixed point in the IR, one must tune the relevant couplings to land on this surface.

This explains why critical phenomena are robust. A microscopic model has many couplings, but near a fixed point the long-distance behavior is controlled by the relevant directions and the fixed-point data. Critical exponents, scaling dimensions, and universal ratios do not care about most microscopic details.

For a fixed point with one relevant temperature-like perturbation $t$, the flow is

$$t(b)=b^{y_t}t.$$

The correlation length obeys

$$\xi(t)=b\,\xi(t(b))$$

because one unit of length in the rescaled theory corresponds to $b$ units of length in the original theory. Choose $b$ so that $t(b)$ is order one:

$$b\sim |t|^{-1/y_t}.$$

Then

$$\boxed{ \xi(t)\sim |t|^{-\nu}, \qquad \nu=\frac{1}{y_t}. }$$

This is how an RG eigenvalue becomes a measurable critical exponent.

Example: the Gaussian fixed point

The simplest fixed point is the free massless scalar in $d$ dimensions:

$$S_\star[\phi] = \frac12\int d^d x\, \partial_\mu\phi\,\partial^\mu\phi.$$

At this fixed point, the scalar has dimension

$$\Delta_\phi=\frac{d-2}{2}.$$

The operator $\phi^n$ has engineering dimension

$$\Delta_{\phi^n}=n\frac{d-2}{2}$$

at the free fixed point, ignoring operator mixing and normal-ordering subtleties. The coupling of $\phi^n$ therefore has RG exponent

$$y_n=d-n\frac{d-2}{2}.$$

In $d=4$,

$$y_2=2, \qquad y_4=0, \qquad y_6=-2.$$

So $\phi^2$ is relevant, $\phi^4$ is classically marginal, and $\phi^6$ is irrelevant. Quantum corrections decide what happens to the marginal $\phi^4$ interaction.

In $d=3$,

$$y_2=2, \qquad y_4=1, \qquad y_6=0.$$

Now $\phi^4$ is relevant at the Gaussian fixed point. This is one reason the three-dimensional Ising critical point is not described by the free scalar fixed point. The long-distance fixed point is the interacting Wilson-Fisher fixed point, whose operator dimensions are not the Gaussian engineering dimensions.

Example: $\phi^4$ near four dimensions

A standard perturbative example is the scalar theory in $d=4-\epsilon$:

$$S=\int d^d x\left[ \frac12(\partial\phi)^2 +\frac12 r\phi^2 +\frac{g}{4!}\phi^4 \right].$$

Classically, the quartic coupling has dimension

$$[g]=4-d=\epsilon.$$

Thus $g$ is relevant near the Gaussian fixed point when $d<4$. Quantum corrections can produce a beta function of the schematic form

$$\frac{dg}{d\log b} = \epsilon g-A g^2+O(g^3), \qquad A>0.$$

This has two fixed points:

$$g=0, \qquad g_\star=\frac{\epsilon}{A}+O(\epsilon^2).$$

The first is the Gaussian fixed point. The second is the Wilson-Fisher fixed point. In $d=3$, this perturbative expansion becomes an approximation to the interacting fixed point governing the Ising universality class, after imposing the appropriate $\mathbb Z_2$ symmetry.

The important lesson is not the numerical value of $A$. The lesson is that a classically relevant coupling can grow until it reaches a new interacting fixed point. Many CFTs arise exactly this way.

Wilsonian RG and the stress tensor

The local diagnostic of scale dependence is the trace of the stress tensor. In a theory deformed away from a fixed point by operators $\mathcal O_i$, one expects schematically

$$T^\mu{}_{\mu} = \sum_i \beta_i(g)\,\mathcal O_i +\text{contact terms} +\text{anomalies} +\text{improvement terms}.$$

This formula is schematic because the precise operator equation depends on conventions, curved-space counterterms, and operator mixing. But the message is essential:

$$\boxed{ \text{beta functions measure the failure of scale invariance.} }$$

At a fixed point, $\beta_i=0$. If the theory is a genuine CFT and the stress tensor is properly improved, then in flat space

$$T^\mu{}_{\mu}=0$$

up to possible anomalies in curved backgrounds. The next page will focus on this trace equation more carefully.

This is why RG fixed points and CFTs are tied together. The RG says the theory looks the same after changing scale. The stress tensor says the same thing locally.

Relevant, marginal, irrelevant: physical interpretation

The relevance classification is so important for AdS/CFT that it is worth saying three times in different languages.

First, in RG language:

$$\begin{array}{ccl} \text{relevant} &:& \text{grows in the IR},\\ \text{irrelevant} &:& \text{dies in the IR},\\ \text{marginal} &:& \text{requires nonlinear beta functions}. \end{array}$$

Second, in CFT language:

$$\begin{array}{ccl} \Delta<d &:& \int d^d x\,\mathcal O\text{ is relevant},\\ \Delta=d &:& \int d^d x\,\mathcal O\text{ is marginal},\\ \Delta>d &:& \int d^d x\,\mathcal O\text{ is irrelevant}. \end{array}$$

Third, in effective field theory language:

$$\begin{array}{ccl} \text{relevant} &:& \text{must usually be tuned to reach a critical point},\\ \text{irrelevant} &:& \text{encodes UV details suppressed at long distances},\\ \text{marginal} &:& \text{may generate logarithms or conformal manifolds}. \end{array}$$

The word “irrelevant” does not mean “unimportant for everything.” Irrelevant operators control corrections to scaling, encode microscopic details, and can be essential in UV completions. It means only that they become less important along the IR flow near the fixed point.

Scheme dependence and universal data

A Wilsonian RG step is not unique. We can choose different regulators, different field redefinitions, different normalizations of operators, and different coordinates on theory space. Therefore beta functions away from fixed points are scheme-dependent.

But the following are universal:

Quantity	Universal meaning
Fixed-point existence	There is a scale-invariant long-distance theory.
Scaling dimensions $\Delta_i$	Eigenvalues of the dilatation operator at the fixed point.
Critical exponents	RG eigenvalues rewritten as observable scaling laws.
OPE coefficients $C_{ijk}$	Local operator algebra of the fixed-point CFT.
Symmetry and anomaly data	Constraints shared by all descriptions in the universality class.

This distinction matters constantly in AdS/CFT. Coordinates in theory space are not sacred. CFT data are sacred.

The holographic checkpoint

The Wilsonian RG prepares one of the deepest pieces of the AdS/CFT dictionary:

$$\boxed{ \text{boundary RG scale} \quad\longleftrightarrow\quad \text{bulk radial direction}. }$$

In Poincare AdS,

$$ds^2 =\frac{L^2}{z^2}\left(dz^2+dx^\mu dx_\mu\right),$$

where the boundary is at $z\to0$. Small $z$ corresponds to UV boundary physics. Larger $z$ corresponds to longer-distance, more IR boundary physics. This is the UV/IR relation:

$$\boxed{ \text{UV in the CFT} \leftrightarrow \text{near the AdS boundary}, \qquad \text{IR in the CFT} \leftrightarrow \text{deeper into the bulk}. }$$

A scalar operator $\mathcal O$ of dimension $\Delta$ is dual to a bulk scalar $\phi$ with mass

$$m^2L^2=\Delta(\Delta-d).$$

Near the boundary, the bulk field behaves schematically as

$$\phi(z,x) \sim z^{d-\Delta}J(x)+z^\Delta A(x).$$

The coefficient $J(x)$ is the source for $\mathcal O$. If $\Delta<d$, then the source term $z^{d-\Delta}J$ grows as we move inward from the boundary. That is the bulk reflection of a relevant deformation growing toward the IR.

The correspondence is not “RG is literally the radial coordinate” in a regulator-independent way. Wilsonian RG has scheme dependence, and radial coordinates have gauge dependence. The invariant statement is more subtle and more powerful: holographic radial evolution geometrizes the scale dependence of CFT data.

At a CFT fixed point, the dual geometry is exactly AdS, at least in the simplest semiclassical cases. An RG flow between fixed points is often represented by a domain-wall geometry that approaches one AdS space in the UV and another in the IR:

$$\text{CFT}_{\rm UV} \longrightarrow \text{CFT}_{\rm IR} \qquad \leftrightarrow \qquad \text{asymptotic AdS}_{\rm UV} \longrightarrow \text{asymptotic AdS}_{\rm IR}.$$

This is why we begin the CFT course with Wilsonian RG. The radial direction of holography is already hiding inside the question: what remains after coarse graining?

Common pitfalls

Pitfall 1: thinking that a QFT is only its classical Lagrangian

A classical Lagrangian is a coordinate chart on theory space, not the whole theory. Quantum corrections generate all operators allowed by symmetries. A Wilsonian action should be understood as an effective action at a scale.

Pitfall 2: confusing cutoff dependence with physical dependence

Bare couplings depend on the cutoff. Physical observables should not depend on arbitrary regulator choices after parameters are fixed. RG flow describes how descriptions change with scale, not how nature changes because we changed our notation.

Pitfall 3: saying irrelevant means negligible in the UV

Irrelevant operators are suppressed in the IR near a fixed point. They may dominate UV sensitivity and encode the microscopic completion.

Pitfall 4: assuming every scale-invariant theory is automatically conformal

For the unitary, local, Poincare-invariant theories relevant in this course, fixed points of interest are CFTs. But the implication from scale invariance to conformal invariance has assumptions and subtleties. The stress tensor and its possible improvement terms are the right language for those subtleties.

Summary

A Wilsonian RG transformation integrates out short-distance degrees of freedom, rescales, and produces new couplings. A fixed point is a theory unchanged by this operation. Near a fixed point, perturbations are eigen-directions with exponents $y_a$, or equivalently operator dimensions $\Delta_a=d-y_a$.

The main chain is:

$$\boxed{ \text{Wilsonian RG} \to \text{fixed point} \to \text{scaling operators} \to \text{CFT data} \to \text{holographic radial physics}. }$$

This is the first bridge from ordinary QFT to the CFT language of AdS/CFT.

Exercises

Exercise 1 — Relevance from rescaling

Let $\mathcal O$ be a scalar operator of dimension $\Delta$ at a fixed point in $d$ dimensions. Consider

$$\delta S=g\int d^d x\,\mathcal O(x).$$

Under the Wilsonian rescaling $x'=x/b$ with $b>1$, show that

$$g'=b^{d-\Delta}g.$$

Classify the deformation for $\Delta<d$, $\Delta=d$, and $\Delta>d$.

Solution

Under $x=bx'$, the measure transforms as

$$d^d x=b^d d^d x'.$$

A scaling operator of dimension $\Delta$ transforms as

$$\mathcal O(x)=\mathcal O(bx')=b^{-\Delta}\mathcal O'(x').$$

Therefore

$$\delta S =g\int d^d x\,\mathcal O(x) =g\,b^{d-\Delta}\int d^d x'\,\mathcal O'(x').$$

Thus

$$g'=b^{d-\Delta}g.$$

If $\Delta<d$, then $g'$ grows under coarse graining, so the deformation is relevant. If $\Delta=d$, it is marginal at linear order. If $\Delta>d$, it is irrelevant.

Exercise 2 — Gaussian power counting

At the free massless scalar fixed point in $d$ dimensions,

$$\Delta_\phi=\frac{d-2}{2}.$$

For the deformation

$$\delta S_n=g_n\int d^d x\,\phi^n,$$

compute the RG exponent $y_n$ at the Gaussian fixed point. Classify $\phi^2$, $\phi^4$, and $\phi^6$ in $d=4$ and $d=3$.

Solution

At the Gaussian fixed point,

$$\Delta_{\phi^n}=n\Delta_\phi=n\frac{d-2}{2}.$$

The RG exponent is

$$y_n=d-\Delta_{\phi^n} =d-n\frac{d-2}{2}.$$

In $d=4$,

$$y_n=4-n.$$

Thus

$$y_2=2, \qquad y_4=0, \qquad y_6=-2.$$

So $\phi^2$ is relevant, $\phi^4$ is marginal at tree level, and $\phi^6$ is irrelevant.

In $d=3$,

$$y_n=3-\frac n2.$$

Thus

$$y_2=2, \qquad y_4=1, \qquad y_6=0.$$

So $\phi^2$ and $\phi^4$ are relevant at the Gaussian fixed point in $d=3$, while $\phi^6$ is marginal at tree level.

Exercise 3 — Correlation length exponent

Suppose a fixed point has one relevant temperature-like perturbation $t$ with

$$t(b)=b^{y_t}t, \qquad y_t>0.$$

Use RG to derive

$$\xi(t)\sim |t|^{-\nu}, \qquad \nu=\frac1{y_t}.$$

Solution

After a coarse-graining step by $b$, lengths measured in the original units scale by $b$. The correlation length therefore satisfies

$$\xi(t)=b\,\xi(t(b))$$

if $\xi(t(b))$ is measured in the rescaled theory’s microscopic units. Choose $b$ so that the renormalized perturbation is order one:

$$b^{y_t}|t|\sim1.$$

This gives

$$b\sim |t|^{-1/y_t}.$$

At this scale, $\xi(t(b))$ is order one, so

$$\xi(t) \sim b \sim |t|^{-1/y_t}.$$

Hence

$$\nu=\frac1{y_t}.$$

Exercise 4 — A simple interacting fixed point

Consider a one-coupling beta function in the IR-oriented convention

$$\frac{dg}{d\log b}=\epsilon g-A g^2, \qquad \epsilon>0, \qquad A>0.$$

Find the fixed points and determine the linearized RG exponent at each fixed point.

Solution

The fixed points obey

$$\epsilon g-A g^2=0.$$

Thus

$$g_0=0, \qquad g_\star=\frac{\epsilon}{A}.$$

The linearized exponent is

$$y(g_*)=\left.\frac{d\beta}{dg}\right|_{g=g_*} =\epsilon-2Ag_*.$$

At the Gaussian fixed point,

$$y(g_0)=\epsilon>0,$$

so $g$ is relevant. At the interacting fixed point,

$$y(g_\star) = \epsilon-2A\frac{\epsilon}{A} =-\epsilon<0,$$

so this direction is irrelevant there. The coupling grows away from the Gaussian fixed point and approaches the interacting fixed point in the IR.

Exercise 5 — Relevant deformations and AdS falloff

A scalar operator $\mathcal O$ in a $d$-dimensional CFT is dual to a bulk scalar with near-boundary behavior

$$\phi(z,x) \sim z^{d-\Delta}J(x)+z^\Delta A(x).$$

Assume $\Delta<d$. Explain why the source term represents a relevant deformation from the radial viewpoint.

Solution

For $\Delta<d$, the exponent $d-\Delta$ is positive. Near the AdS boundary, $z\to0$, the source term behaves as

$$z^{d-\Delta}J(x)\to0$$

for fixed $J(x)$, but as one moves inward to larger $z$, the same term grows like $z^{d-\Delta}$. This mirrors the Wilsonian statement that a relevant coupling grows toward the IR.

The precise map between $z$ and an RG scale is scheme-dependent, but the scaling is the important point:

$$\Delta<d \quad\Longleftrightarrow\quad \text{source grows toward the IR radial direction}.$$