Generalized Free Fields

Goal

The previous modules developed CFT as an operator algebra: local operators, Ward identities, correlation functions, radial quantization, conformal blocks, crossing, and the special machinery of two-dimensional CFT. We now turn to the extra structure that makes a CFT holographic.

The first large-$N$ idea is factorization. In a large-$N$ CFT, properly normalized single-particle operators have two-point functions of order one, while connected higher-point functions are suppressed. In the strict $N=\infty$ limit, such operators behave like generalized free fields.

This page explains what that means, why it is not the same thing as an ordinary free scalar field, and why it is the correct boundary language for a free quantum field in AdS.

The central picture is:

$$\boxed{ \text{large-}N\text{ factorization} \quad\Longrightarrow\quad \text{generalized free fields} \quad\Longrightarrow\quad \text{multi-particle states in AdS}. }$$

More concretely, a scalar single-trace operator $\mathcal O$ behaves at leading order as if all its correlation functions were generated by Wick contractions of its two-point function:

$$\langle \mathcal O(x_1)\cdots \mathcal O(x_{2m})\rangle_{N=\infty} = \sum_{\text{pairings}} \prod_{(i,j)}\langle \mathcal O(x_i)\mathcal O(x_j)\rangle.$$

But do not let the word “free” fool you. A generalized free field is generally not a fundamental Lagrangian field satisfying a local equation of motion. It is a universal large-$N$ limit of an operator sector.

Large-$N$ normalization and factorization

Let $C_T$ denote the normalization of the stress-tensor two-point function. In a matrix large-$N$ gauge theory, $C_T\sim N^2$. In a vector model, often $C_T\sim N$. For holography, $C_T$ is the natural measure of the number of CFT degrees of freedom and is proportional to an inverse bulk Newton constant:

$$C_T\sim \frac{R_{\rm AdS}^{d-1}}{G_N}.$$

Consider a scalar primary $\mathcal O$ normalized by

$$\langle \mathcal O(x)\mathcal O(0)\rangle = \frac{1}{x^{2\Delta}}.$$

Here $x^{2\Delta}$ means $(x^2)^\Delta$ in Euclidean signature. For a normalized single-particle operator in a large-$N$ CFT, connected correlators scale schematically as

$$\langle \mathcal O_1\cdots \mathcal O_k\rangle_{\rm conn} \sim C_T^{1-k/2}.$$

Thus

$$\langle \mathcal O\mathcal O\rangle\sim C_T^0, \qquad \langle \mathcal O\mathcal O\mathcal O\rangle\sim C_T^{-1/2}, \qquad \langle \mathcal O\mathcal O\mathcal O\mathcal O\rangle_{\rm conn} \sim C_T^{-1}.$$

In the strict $C_T\to\infty$ limit, connected correlators with three or more single-particle insertions vanish. Correlators then factorize into sums of products of two-point functions. This is the CFT version of the statement that the leading bulk theory is free.

For matrix theories, the same counting is often written as

$$\langle \mathcal O_1\cdots\mathcal O_k\rangle_{\rm conn} \sim N^{2-k},$$

for single-trace operators normalized to have order-one two-point functions. Since $C_T\sim N^2$, this is the same scaling.

Definition of a generalized free field

A scalar primary $\mathcal O$ of scaling dimension $\Delta$ is called a generalized free field if its correlation functions are completely determined by the two-point function and Wick-like factorization:

$$\langle \mathcal O(x)\mathcal O(0)\rangle = \frac{1}{x^{2\Delta}},$$ $$\langle \mathcal O(x_1)\cdots\mathcal O(x_{2m+1})\rangle=0,$$

and

$$\langle \mathcal O(x_1)\cdots\mathcal O(x_{2m})\rangle = \sum_{\text{pairings }P} \prod_{(i,j)\in P} \frac{1}{x_{ij}^{2\Delta}}.$$

Here

$$x_{ij}^2=(x_i-x_j)^2.$$

The adjective “generalized” is doing important work. For an ordinary free scalar field $\phi$ in $d$ dimensions, the scaling dimension is fixed to

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

and the field satisfies a local equation of motion

$$\partial^2\phi=0.$$

A generalized free scalar can instead have essentially any scalar-primary dimension consistent with unitarity,

$$\Delta\geq \frac{d-2}{2},$$

but for generic $\Delta$ it does not obey a local equation of motion. There is no null descendant $\partial^2\mathcal O=0$ unless $\Delta=(d-2)/2$ and the theory is truly free in the ordinary sense.

So the right mental model is:

$$\text{ordinary free field} \subsetneq \text{generalized free field behavior}.$$

A generalized free field is free as an operator probability distribution, not necessarily as a local Lagrangian field.

At leading large $N$, normalized single-particle operators behave as generalized free fields. Correlators factorize, but the OPE still contains an infinite tower of double-trace primaries $[\mathcal O\mathcal O]_{n,\ell}$, which become two-particle states in global AdS. Interactions appear as $1/C_T$ corrections to OPE data.

The mean-field four-point function

The simplest nontrivial example is the four-point function of identical scalar generalized free fields. Wick factorization gives

$$\begin{aligned} \langle \mathcal O_1\mathcal O_2\mathcal O_3\mathcal O_4\rangle_{\rm GFF} &= \frac{1}{x_{12}^{2\Delta}x_{34}^{2\Delta}} + \frac{1}{x_{13}^{2\Delta}x_{24}^{2\Delta}} + \frac{1}{x_{14}^{2\Delta}x_{23}^{2\Delta}}. \end{aligned}$$

As usual for identical scalar four-point functions, define

$$\langle \mathcal O_1\mathcal O_2\mathcal O_3\mathcal O_4\rangle = \frac{1}{x_{12}^{2\Delta}x_{34}^{2\Delta}}\,\mathcal G(u,v),$$

where

$$u= \frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}, \qquad v= \frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}.$$

Then the generalized-free or mean-field four-point function is

$$\boxed{ \mathcal G_{\rm MFT}(u,v) = 1+u^\Delta+\left(\frac{u}{v}\right)^\Delta. }$$

This is often called mean field theory, abbreviated MFT, in the conformal-bootstrap literature. The name is slightly misleading but useful. It means the universal crossing-symmetric solution produced by generalized-free Wick contractions.

The identical-scalar crossing equation is

$$\mathcal G(u,v) = \left(\frac{u}{v}\right)^\Delta \mathcal G(v,u).$$

The MFT answer obeys it immediately:

$$\left(\frac{u}{v}\right)^\Delta \left[1+v^\Delta+\left(\frac{v}{u}\right)^\Delta\right] = \left(\frac{u}{v}\right)^\Delta+u^\Delta+1 = \mathcal G_{\rm MFT}(u,v).$$

This small equation contains a surprisingly deep point. The disconnected-looking four-point function is not empty. When expanded in a chosen OPE channel, it contains infinitely many conformal families.

The OPE is not trivial

The generalized-free four-point function has vanishing connected part, but its OPE is highly nontrivial. In the $12\to34$ channel, the product $\mathcal O(x_1)\mathcal O(x_2)$ contains the identity and an infinite tower of composite primaries:

$$\mathcal O\times\mathcal O \sim \mathbf 1 + \sum_{n=0}^{\infty} \sum_{\ell=0,2,4,\ldots} \lambda^{(0)}_{n,\ell}\,[\mathcal O\mathcal O]_{n,\ell}.$$

For identical bosonic scalars, only even spins appear. The schematic form of these operators is

$$[\mathcal O\mathcal O]_{n,\ell} \sim \mathcal O\,\partial_{\mu_1}\cdots\partial_{\mu_\ell}(\partial^2)^n\mathcal O + \text{subtractions}.$$

The subtractions are essential: the operator must be a conformal primary, so traces and descendants must be removed. At leading generalized-free order, the dimensions are

$$\boxed{ \Delta^{(0)}_{n,\ell} = 2\Delta+2n+\ell, \qquad n=0,1,2,\ldots, \qquad \ell=0,2,4,\ldots. }$$

Their twists are

$$\tau^{(0)}_{n,\ell} = \Delta^{(0)}_{n,\ell}-\ell = 2\Delta+2n.$$

Thus the MFT four-point function admits a conformal-block expansion

$$\mathcal G_{\rm MFT}(u,v) = 1+ \sum_{n=0}^{\infty} \sum_{\ell=0,2,4,\ldots} a^{(0)}_{n,\ell} G_{2\Delta+2n+\ell,\ell}(u,v),$$

with positive squared OPE coefficients

$$a^{(0)}_{n,\ell}=\left(\lambda^{(0)}_{n,\ell}\right)^2\geq0.$$

The precise closed-form expression for $a^{(0)}_{n,\ell}$ depends on the normalization convention for conformal blocks. The invariant statement is that crossing fixes all of them once the two-point normalization and dimension $\Delta$ are chosen.

This is one of the first lessons of large-$N$ CFT:

$$\text{factorized correlators} \neq \text{empty spectrum}.$$

Factorization says connected correlators vanish. It does not say the OPE contains only the identity.

Why double-trace dimensions are additive

The dimensions

$$\Delta^{(0)}_{n,\ell}=2\Delta+2n+\ell$$

have a simple explanation from radial quantization.

A scalar primary $\mathcal O$ creates a state on $S^{d-1}$ with cylinder energy

$$E=\Delta.$$

Descendants have energies $\Delta+m$, where $m$ is a nonnegative integer. In a generalized-free theory, two-particle energies add. The two-particle primary with spin $\ell$ and radial excitation number $n$ therefore has energy

$$E_{n,\ell}^{(0)} = \Delta+\Delta+\ell+2n.$$

By the state-operator map, cylinder energy is scaling dimension, so

$$E_{n,\ell}^{(0)} = \Delta_{n,\ell}^{(0)}.$$

The $\ell$ units correspond to angular momentum on the sphere. The $2n$ units correspond to radial excitation. This interpretation is exactly what one expects for two free particles in global AdS.

AdS interpretation

For a scalar primary $\mathcal O$ of dimension $\Delta$, the dual bulk scalar field $\phi$ in $\mathrm{AdS}_{d+1}$ has mass

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

At leading order in large $N$, the bulk action is quadratic:

$$S_{\rm bulk}[\phi] = \frac{1}{2}\int_{\mathrm{AdS}} d^{d+1}x\sqrt g \left[(\nabla\phi)^2+m^2\phi^2\right] +\text{boundary terms}.$$

A Gaussian bulk path integral produces a Gaussian boundary generating functional:

$$Z_{\rm bulk}[J] \sim \exp\left[-\frac{1}{2}\int d^d x\,d^d y\,J(x)K(x,y)J(y)\right],$$

where $K(x,y)$ is the boundary two-point kernel. Functional derivatives with respect to $J$ produce Wick factorization. Hence the CFT statement

$$\mathcal O\text{ is generalized free at }N=\infty$$

is the boundary version of the bulk statement

$$\phi\text{ is a free field in AdS at }G_N=0.$$

The double-trace operators are then interpreted as two-particle states:

$$[\mathcal O\mathcal O]_{n,\ell} \quad\longleftrightarrow\quad \text{two }\phi\text{ particles in global AdS with radial level }n\text{ and spin }\ell.$$

This is why generalized free fields are unavoidable in AdS/CFT. They are not a toy model. They are the zeroth-order approximation to any weakly coupled bulk theory.

Turning on interactions

A finite but large $C_T$ introduces connected correlators. In the four-point function this means

$$\mathcal G(u,v) = \mathcal G_{\rm MFT}(u,v) +\frac{1}{C_T}\mathcal G^{(1)}(u,v) +\frac{1}{C_T^2}\mathcal G^{(2)}(u,v) +\cdots.$$

Correspondingly, double-trace dimensions and OPE coefficients receive corrections:

$$\Delta_{n,\ell} = 2\Delta+2n+\ell + \frac{1}{C_T}\gamma^{(1)}_{n,\ell} + \frac{1}{C_T^2}\gamma^{(2)}_{n,\ell} + \cdots,$$ $$a_{n,\ell} = a^{(0)}_{n,\ell} + \frac{1}{C_T}a^{(1)}_{n,\ell} + \frac{1}{C_T^2}a^{(2)}_{n,\ell} + \cdots.$$

The anomalous dimensions $\gamma_{n,\ell}$ are especially important. In global AdS, they are binding energies of two-particle states. For example:

CFT effect	Bulk interpretation
nonzero connected four-point function	tree-level Witten diagram
$\gamma_{n,\ell}$	two-particle binding energy or phase shift
stress-tensor exchange	graviton exchange
conserved-current exchange	gauge-boson exchange
scalar single-trace exchange	exchange of another bulk scalar
$1/C_T^2$ corrections	loops and multi-particle effects

This is one of the cleanest bridges between the conformal bootstrap and AdS perturbation theory.

Generalized free fields versus ordinary free fields

It is worth being very explicit about the difference.

An ordinary free scalar in $d$ dimensions has

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

and satisfies the equation of motion

$$\partial^2\phi=0.$$

The equation of motion says that the descendant $P^2|\phi\rangle$ is null. This shortening is visible in the conformal multiplet.

A generalized free scalar of dimension $\Delta$ has no such equation for generic $\Delta$:

$$P^2|\mathcal O\rangle\neq0.$$

It is a long scalar conformal multiplet, not a shortened free-field multiplet.

There is also a stress-tensor issue. A complete local CFT in flat space has a stress tensor $T_{\mu\nu}$. The Ward identity fixes the existence of a nonzero three-point function

$$\langle \mathcal O\mathcal O T_{\mu\nu}\rangle.$$

With the standard normalization $\langle T T\rangle\sim C_T$, the OPE coefficient scales as

$$\lambda_{\mathcal O\mathcal O T} \sim \frac{\Delta}{\sqrt{C_T}}.$$

Thus $T_{\mu\nu}$ decouples from the $\mathcal O\times\mathcal O$ OPE at strict $C_T=\infty$, but it returns at finite $C_T$. This is exactly what should happen holographically: gravitons decouple when $G_N\to0$ and reappear when gravitational interactions are restored.

A standalone exact generalized free field is therefore usually not a complete local CFT with a finite stress tensor. It is best understood as either the leading sector of a large-$N$ CFT or a formal mean-field solution of crossing for a chosen operator. For AdS/CFT, the first interpretation is the important one.

Multiple generalized free fields

A large-$N$ CFT usually has many single-particle primaries $\mathcal O_A$. At leading order, one may choose a basis in which their two-point functions are diagonal:

$$\langle \mathcal O_A(x)\mathcal O_B(0)\rangle = \frac{\delta_{AB}}{x^{2\Delta_A}}.$$

At $N=\infty$, the multi-operator correlators factorize into pairings:

$$\langle \mathcal O_{A_1}(x_1)\cdots \mathcal O_{A_{2m}}(x_{2m})\rangle = \sum_{\text{pairings }P} \prod_{(i,j)\in P} \frac{\delta_{A_iA_j}}{x_{ij}^{2\Delta_{A_i}}},$$

assuming the basis has no two-point mixing. If operators have different quantum numbers, the corresponding pairings vanish.

Their OPEs contain mixed double-trace operators

$$[\mathcal O_A\mathcal O_B]_{n,\ell},$$

with leading dimensions

$$\Delta_{AB;n,\ell}^{(0)} = \Delta_A+ \Delta_B+2n+ \ell.$$

If $A=B$, Bose symmetry restricts the allowed spins. If $A\neq B$, both even and odd spins can appear, depending on the operator quantum numbers.

In AdS language, this is simply the spectrum of two-particle states made from different bulk fields.

Generalized free fields and the bootstrap

The mean-field four-point function is the universal starting point for large-$N$ bootstrap. One writes

$$\mathcal G(u,v) = \mathcal G_{\rm MFT}(u,v)+\epsilon\,\mathcal G^{(1)}(u,v)+O(\epsilon^2),$$

where

$$\epsilon\sim\frac{1}{C_T}.$$

Crossing then constrains $\mathcal G^{(1)}$. Its conformal block expansion determines the first anomalous dimensions and OPE-coefficient corrections of double-trace operators.

This perturbation theory is not a perturbation around a finite-dimensional spectrum. It is a perturbation around an infinite tower:

$$\left\{[\mathcal O\mathcal O]_{n,\ell}\right\}_{n\geq0,\ell\geq0}.$$

That infinite tower is not optional. It is the CFT signature of locality in the emergent bulk. Local bulk interactions produce highly constrained patterns of $\gamma_{n,\ell}$ and $a^{(1)}_{n,\ell}$.

A useful diagnostic is the large-spin behavior. For a holographic CFT, double-trace anomalous dimensions at large $\ell$ are controlled by low-twist single-trace exchanges:

$$\gamma_{n,\ell} \sim \frac{1}{\ell^{\tau_{\rm exch}}} \qquad (\ell\to\infty),$$

where $\tau_{\rm exch}=\Delta_{\rm exch}-\ell_{\rm exch}$ is the twist of the exchanged operator. This is the CFT version of long-distance forces in AdS.

What generalized free fields do and do not capture

Generalized free fields capture:

Captured by GFF	Meaning
two-point function of a single-particle operator	free bulk propagator
Wick factorization	no interactions at leading large $N$
double-trace tower	two-particle Fock states
additive dimensions	additive energies on the cylinder/global AdS
positive MFT OPE coefficients	unitary multi-particle Hilbert space

They do not capture:

Not captured at strict GFF level	Appears as
stress-tensor exchange	$1/\sqrt{C_T}$ OPE coefficient, $1/C_T$ effect in four-point data
binding energies	anomalous dimensions $\gamma_{n,\ell}$
bulk interactions	connected Witten diagrams
bulk loops	higher powers of $1/C_T$
finite-$N$ operator mixing	mixing of double- and multi-trace operators

So GFF is the correct zeroth order, but a holographic theory is not just a GFF. It is a controlled deformation of generalized free behavior, with special constraints from crossing, unitarity, sparse single-particle spectrum, and bulk locality.

AdS/CFT checkpoint

The core dictionary of this page is

$$\mathcal O \quad\longleftrightarrow\quad \text{one-particle bulk field }\phi,$$ $$[\mathcal O\mathcal O]_{n,\ell} \quad\longleftrightarrow\quad \text{two-particle state in global AdS},$$

and

$$\gamma_{n,\ell} \quad\longleftrightarrow\quad \text{bulk binding energy or scattering phase shift}.$$

The large-$N$ limit makes the CFT Hilbert space approximately a Fock space. Generalized free fields are the boundary operators that create the one-particle building blocks of this Fock space.

This is why a student of AdS/CFT must be comfortable with generalized free fields before learning Witten diagrams. The disconnected CFT correlator is the boundary imprint of the free bulk theory. The first connected correction is the boundary imprint of bulk interaction.

Summary

A generalized free field is a primary operator whose correlators are Wick-factorized but whose dimension need not be that of an ordinary free field. In a large-$N$ CFT, normalized single-particle operators become generalized free fields at leading order:

$$\langle \mathcal O_1\cdots\mathcal O_k\rangle_{\rm conn} \sim C_T^{1-k/2}.$$

The generalized-free four-point function is

$$\mathcal G_{\rm MFT}(u,v) = 1+u^\Delta+\left(\frac{u}{v}\right)^\Delta.$$

Its OPE contains the double-trace tower

$$[\mathcal O\mathcal O]_{n,\ell}, \qquad \Delta^{(0)}_{n,\ell}=2\Delta+2n+\ell.$$

In AdS, this is the free two-particle spectrum. Finite-$N$ corrections turn this into interacting bulk physics:

$$\Delta_{n,\ell} = 2\Delta+2n+\ell+ \gamma_{n,\ell}.$$

The anomalous dimensions $\gamma_{n,\ell}$ are where forces, binding energies, and Witten diagrams enter the CFT.

Exercises

Exercise 1. Derive the mean-field four-point function

Let $\mathcal O$ be a scalar generalized free field with

$$\langle \mathcal O(x)\mathcal O(0)\rangle=x^{-2\Delta}.$$

Show that

$$\mathcal G_{\rm MFT}(u,v)=1+u^\Delta+\left(\frac{u}{v}\right)^\Delta.$$

Solution

Wick factorization gives three pairings:

$$\langle \mathcal O_1\mathcal O_2\mathcal O_3\mathcal O_4\rangle = \frac{1}{x_{12}^{2\Delta}x_{34}^{2\Delta}} + \frac{1}{x_{13}^{2\Delta}x_{24}^{2\Delta}} + \frac{1}{x_{14}^{2\Delta}x_{23}^{2\Delta}}.$$

Factor out

$$\frac{1}{x_{12}^{2\Delta}x_{34}^{2\Delta}}.$$

The first term gives $1$. The second gives

$$\left(\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}\right)^\Delta =u^\Delta.$$

The third gives

$$\left(\frac{x_{12}^2x_{34}^2}{x_{14}^2x_{23}^2}\right)^\Delta = \left(\frac{u}{v}\right)^\Delta.$$

Therefore

$$\mathcal G_{\rm MFT}(u,v) = 1+u^\Delta+\left(\frac{u}{v}\right)^\Delta.$$

Exercise 2. Check crossing symmetry

For identical scalars, crossing requires

$$\mathcal G(u,v)=\left(\frac{u}{v}\right)^\Delta\mathcal G(v,u).$$

Verify this for $\mathcal G_{\rm MFT}(u,v)$.

Solution

Start from

$$\mathcal G_{\rm MFT}(v,u) = 1+v^\Delta+\left(\frac{v}{u}\right)^\Delta.$$

Then

$$\left(\frac{u}{v}\right)^\Delta\mathcal G_{\rm MFT}(v,u) = \left(\frac{u}{v}\right)^\Delta +u^\Delta +1.$$

Reordering the terms gives

$$1+u^\Delta+\left(\frac{u}{v}\right)^\Delta = \mathcal G_{\rm MFT}(u,v).$$

So the generalized-free four-point function is crossing-symmetric.

Exercise 3. Connected four-point function

Show that the connected four-point function of an exact generalized free field vanishes.

Solution

For any operator distribution, the connected four-point function is

$$\begin{aligned} \langle 1234\rangle_{\rm conn} &= \langle 1234\rangle -\langle 12\rangle\langle 34\rangle -\langle 13\rangle\langle 24\rangle -\langle 14\rangle\langle 23\rangle, \end{aligned}$$

where $i$ denotes $\mathcal O(x_i)$. For a generalized free field,

$$\langle 1234\rangle = \langle 12\rangle\langle 34\rangle +\langle 13\rangle\langle 24\rangle +\langle 14\rangle\langle 23\rangle.$$

Substitution gives

$$\langle 1234\rangle_{\rm conn}=0.$$

At finite large $N$, this equality is corrected by terms of order $1/C_T$ for normalized single-particle operators.

Exercise 4. Double-trace dimensions from radial quantization

Use radial quantization to explain why the leading dimensions of double-trace operators are

$$\Delta_{n,\ell}^{(0)}=2\Delta+2n+\ell.$$

Solution

A scalar primary $\mathcal O$ creates a one-particle state on $S^{d-1}$ with energy

$$E=\Delta.$$

In the generalized-free limit, two-particle energies add. A two-particle state with orbital angular momentum $\ell$ and radial excitation number $n$ has energy

$$E_{n,\ell}=\Delta+\Delta+\ell+2n.$$

The state-operator map identifies cylinder energy with scaling dimension. Therefore

$$\Delta_{n,\ell}^{(0)}=E_{n,\ell}=2\Delta+2n+\ell.$$

The same statement is the global-AdS energy spectrum of two noninteracting particles.

Exercise 5. Why exact GFF is not usually a complete finite-$C_T$ CFT

Explain why a scalar generalized free field with generic $\Delta$ is not the same as an ordinary free scalar field, and why a complete finite-$C_T$ CFT cannot make $\langle \mathcal O\mathcal O T\rangle$ vanish for a nontrivial scalar primary $\mathcal O$.

Solution

An ordinary free scalar has

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

and satisfies the equation of motion

$$\partial^2\phi=0.$$

In representation-theory language, the descendant $P^2|\phi\rangle$ is null. A generalized free field with generic $\Delta$ has no such null descendant, so it is not an ordinary free scalar.

For the stress tensor, the Ward identity fixes how $T_{\mu\nu}$ couples to any primary with nonzero scaling dimension. With the standard normalization $\langle TT\rangle\sim C_T$, the OPE coefficient behaves schematically as

$$\lambda_{\mathcal O\mathcal O T} \sim \frac{\Delta}{\sqrt{C_T}}.$$

Thus this coupling vanishes only in the strict $C_T\to\infty$ limit. At finite $C_T$, the stress tensor cannot be removed from a complete local CFT. This is why exact generalized-free behavior should be understood as a leading large-$N$ limit or as a formal mean-field sector, not usually as a full finite-$C_T$ CFT.