Primary Operator Correlators: Two and Three-Point Functions

Two-Point Functions: A Step-by-Step Derivation

Let's begin by considering the two-point function of two scalar primary operators, ${\mathcal{O}}_1(x_1)$ and ${\mathcal{O}}_2(x_2)$, with scaling dimensions ${\Delta }_1$ and ${\Delta }_2$. We can deduce its form by systematically imposing the symmetries of the conformal group.

1. Translational Invariance: The correlator cannot depend on the absolute position of the operators, only on their relative separation vector $x_{12}=x_1-x_2$.

   $$⟨{\mathcal{O}}_1(x_1){\mathcal{O}}_2(x_2)⟩=G(x_1-x_2)$$

2. Rotational Invariance: Since the operators are scalars, the function $G$ must be invariant under rotations of its argument. This means it can only depend on the magnitude of the separation vector, $|x_{12}|$.

   $$⟨{\mathcal{O}}_1(x_1){\mathcal{O}}_2(x_2)⟩=F(|x_{12}{|}^2)$$

3. Scale Invariance: Under a scaling $x\to \lambda x$, the correlator must transform as:

   $$⟨{\mathcal{O}}_1(\lambda x_1){\mathcal{O}}_2(\lambda x_2)⟩={\lambda }^{-{\Delta }_1}{\lambda }^{-{\Delta }_2}⟨{\mathcal{O}}_1(x_1){\mathcal{O}}_2(x_2)⟩$$

   Applying this to our form $F(|\lambda x_{12}{|}^2)=F({\lambda }^2|x_{12}{|}^2)$, we get a functional equation for $F$ which is solved by a power law:

   $$F(|x_{12}{|}^2)=\frac{C_{12}}{(|x_{12}{|}^2{)}^{({\Delta }_1+{\Delta }_2)/2}}=\frac{C_{12}}{|x_{12}{|}^{{\Delta }_1+{\Delta }_2}}$$

   Here, $C_{12}$ is an arbitrary constant.

4. Special Conformal Transformation (SCT) Invariance: This is the crucial final constraint. While scale invariance alone allows for two operators of different dimensions to have a non-zero two-point function, full conformal invariance does not. Applying the Ward identity for SCTs leads to the condition that $C_{12}=0$ unless ${\Delta }_1={\Delta }_2$.

Combining these constraints, we arrive at the universal form for the two-point function of scalar primary operators:

This is a remarkable result. It implies that primary operators of different scaling dimensions are orthogonal. By rescaling the operators, we can typically set the normalization constant $C_{12}$ to 1 when $i=j$. So, for a set of orthonormal primary operators, we have $⟨{\mathcal{O}}_i(x_1){\mathcal{O}}_j(x_2)⟩=\frac{{\delta }_{ij}}{|x_{12}{|}^{2{\Delta }_i}}$.

The Elegance of the Embedding Space Formalism

The step-by-step derivation is instructive, but it can become cumbersome, especially for higher-point functions or operators with spin. There is a more elegant and powerful method that makes conformal invariance manifest from the start. This is the embedding space (or projective null cone) formalism.

The key idea is to represent points $x^{\mu }$ in $d$-dimensional Euclidean space ${\mathbb{R}}^d$ as light-rays on the future null cone of a larger, $(d+2)$-dimensional Minkowski space ${\mathbb{R}}^{d+1,1}$. A point $x\in {\mathbb{R}}^d$ is mapped to a null vector $X\in {\mathbb{R}}^{d+1,1}$ via:

You can check that $X^2={\eta }_{AB}{X}^A{X}^B=-(X^{+}{X}^{-})+X^{\mu }{X}_{\mu }=-x^2+x^2=0$, so these points indeed lie on the null cone.

The magic of this formalism is that the conformal group $SO(d+1,1)$ acts as the standard linear Lorentz group on these embedding space vectors $X$.
A primary operator $\mathcal{O}(x)$ with dimension $\Delta$ is lifted to a field $\Phi (X)$ on the null cone which is required to be homogeneous of degree $-\Delta$:

Correlation functions of primary operators in ${\mathbb{R}}^d$ become correlation functions of these homogeneous fields in ${\mathbb{R}}^{d+1,1}$ that must be invariant under the $(d+2)$-dimensional Lorentz group.

Two- and Three-Point Functions in the Embedding Space

As you saw in the notes, this formalism makes our previous derivations almost trivial and immediately generalizes.

Two-Point Function Revisited:
We want to find the form of $⟨{\Phi }_1(X_1){\Phi }_2(X_2)⟩$.

• Lorentz Invariance: The only scalar we can form from two vectors $X_1,X_2$ is their inner product, $X_1\cdot X_2$. So the correlator must be a function of this, $f(X_1\cdot X_2)$.
• Homogeneity: The requirement that the function scales as ${\lambda }_1^{-{\Delta }_1}{\lambda }_2^{-{\Delta }_2}$ under $X_1\to {\lambda }_1{X}_1,X_2\to {\lambda }_2{X}_2$ forces $f(z)\propto z^{-k}$ and requires ${\Delta }_1={\Delta }_2=k$. If the dimensions are not equal, no such function exists, and the correlator must be zero.
• Projection: We project back to physical space using the relation from the notes: $X_1\cdot X_2=-\frac{1}{2}|x_1-x_2{|}^2$.

This immediately recovers our result: $⟨{\mathcal{O}}_1(x_1){\mathcal{O}}_2(x_2)⟩\propto \frac{{\delta }_{{\Delta }_1,{\Delta }_2}}{|x_{12}{|}^{2{\Delta }_1}}$.

Three-Point Function:
Now for the three-point function $⟨{\mathcal{O}}_1(x_1){\mathcal{O}}_2(x_2){\mathcal{O}}_3(x_3)⟩$.

• Lorentz Invariance: In the embedding space, we can form three invariants: $X_1\cdot X_2$, $X_2\cdot X_3$, and $X_3\cdot X_1$. The most general form of the correlator is a product of powers of these: $$⟨{\Phi }_1(X_1){\Phi }_2(X_2){\Phi }_3(X_3)⟩=C_{123}(X_1\cdot X_2{)}^{-{\gamma }_{12}}(X_2\cdot X_3{)}^{-{\gamma }_{23}}(X_3\cdot X_1{)}^{-{\gamma }_{31}}$$
• Homogeneity: Imposing homogeneity for each field (e.g., scaling $X_1\to {\lambda }_1{X}_1$ must produce a factor ${\lambda }_1^{-{\Delta }_1}$) gives a system of three linear equations for the three unknown exponents: $$\begin{gathered} {\gamma }_{12}+{\gamma }_{31}={\Delta }_1 \\ {\gamma }_{12}+{\gamma }_{23}={\Delta }_2 \\ {\gamma }_{23}+{\gamma }_{31}={\Delta }_3 \end{gathered}$$
• Solution and Projection: This system has a unique solution, given in the notes you read: ${\gamma }_{12}=({\Delta }_1+{\Delta }_2-{\Delta }_3)/2$, and so on cyclically. Projecting back to ${\mathbb{R}}^d$ gives the final, celebrated result:

This form is fixed entirely by symmetry, up to a single constant $C_{123}$. These constants, often called OPE coefficients, are part of the fundamental data that define a specific CFT.

Extension to Spinning Operators

The same logic applies to primary operators with spin (i.e., those that transform in non-trivial representations of the rotation group $SO(d)$). The embedding space formalism is particularly suited for this.

For example, for a vector primary operator $J_{\mu }$ (spin-1) of dimension $\Delta$, the two-point function is fixed to be:

The tensor $I_{\mu \nu }(x)$ is the unique structure that ensures the correlator transforms correctly under SCTs. It emerges naturally from the projection from embedding space, as shown in Rychkov's notes.

Similarly, three-point functions involving spinning operators, like $⟨{\mathcal{O}}_1(x_1){\mathcal{O}}_2(x_2)J_{\mu }(x_3)⟩$, are also fixed up to a single coefficient.

The crucial takeaway is that the low-point correlation functions of a CFT are not complicated functions that need to be calculated perturbatively. They are simple, universal power-laws, entirely fixed by symmetry. The non-trivial information about the theory's dynamics is boiled down to two sets of numbers:

1. The Spectrum of Dimensions: $\{{\Delta }_i\}$
2. The OPE Coefficients: $\{C_{ijk}\}$

These two sets of numbers are often called the CFT data. The grand challenge of solving a CFT is to determine its CFT data.

Conclusion

In this lesson, we have seen one of the most powerful consequences of conformal symmetry. By systematically applying the constraints from the conformal group, we derived the universal form of two-point and three-point functions.

Key Takeaways:

• Conformal symmetry dictates that the two-point function of scalar primary operators $⟨{\mathcal{O}}_1(x_1){\mathcal{O}}_2(x_2)⟩$ is non-zero only if their scaling dimensions are equal, in which case it is a simple power law: $\frac{C_{12}}{|x_{12}{|}^{2{\Delta }_1}}$.
• The three-point function of scalar primaries is also fixed to a specific power-law form, determined by the three scaling dimensions, up to an overall constant $C_{123}$.
• The embedding space formalism provides an elegant and efficient way to derive these results by mapping the problem to one of Lorentz invariance in a higher-dimensional space.
• These results generalize to operators with spin, where low-point correlators are again fixed up to a small number of constants and tensor structures.
• A CFT is fundamentally defined by its "data": the spectrum of operator dimensions $\{{\Delta }_i\}$ and the structure constants $\{C_{ijk}\}$.

So far, we have considered correlation functions with up to three points. What happens when we have four? As you might guess, the form is no longer completely fixed. In our next lesson, we will see that the four-point function can depend on an arbitrary function of conformally invariant cross-ratios, opening the door to the rich structure of the Operator Product Expansion and the conformal bootstrap.