research35 min studyLesson 733 in Physics

Two- and three-point conformal correlators

Derive the coordinate dependence fixed by symmetry while keeping normalization and OPE data explicit.

Before you begin

Radial quantization and the state-operator map

On this page

The question for today

What you’ll understand

Derive scalar two- and three-point distance powers.
Track basis normalization and tensor structures.
Identify which four-point information remains dynamical.

Conformal symmetry is powerful enough to fix scalar two- and three-point coordinate dependence up to a small set of constants. Those constants still matter: rescaling an operator changes its two-point coefficient and its three-point coefficients together. Work with separated insertions in a flat Euclidean d-dimensional CFT, d at least three, and with real scalar primaries unless otherwise stated. Contact terms and coincident-point distributions require additional treatment beyond these ordinary functions of nonzero distances.

Use rotations, scaling and inversion together

Translation invariance makes a two-point correlator depend only on the separation. Rotation invariance reduces that dependence to its length. Dilatations fix a power equal to the sum of the dimensions, but that alone would allow operators of different dimensions. Special conformal covariance adds the condition that a nonzero scalar two-point function pairs equal dimensions. Internal charges can impose further zeros, and complex operators are ordinarily paired with their conjugates rather than with themselves.

Within a degenerate equal-dimension space, two-point coefficients form a matrix. Reflection positivity makes the nonnull physical sector positive, allowing an orthonormal basis. Choosing this basis is convenient rather than a prediction that every observable has unit strength. In particular, stress-tensor or current normalization is often retained because its coefficient carries useful dynamical information. Spinning two-point functions also have index structures; the scalar power by itself is not their complete correlator.

\langle\mathcal O_i(x)\mathcal O_j(0)\rangle=\frac{\delta_{ij}}{|x|^{2\Delta_i}}\quad\text{in an orthonormal real-scalar basis},\qquad I_{\mu\nu}(x)=\delta_{\mu\nu}-2\frac{x_\mu x_\nu}{x^2}

The orthonormal form includes vanishing correlators between inequivalent dimensions or quantum numbers. Iμν is the dimensionless inversion tensor entering spinning correlators, not an extra scalar normalization.↔ Scroll wide equations horizontally.

Solve the three-point exponent equations

For three scalar insertions there are no continuous conformal cross-ratios at separated generic points. Write a product of three pairwise distances with unknown exponents. Covariance at each point requires the two exponents meeting there to sum to twice that operator's dimension. Solving these three linear equations determines all three powers. The remaining coefficient is a genuine piece of CFT data once operator normalizations and any degeneracy basis have been fixed.

A negative distance exponent is not automatically an inconsistency: it places a positive power in the numerator, and the complete three-point expression still has the correct scaling. Internal symmetries may force the coefficient to vanish. For spinning insertions there can be several independent tensor structures and several coefficients, possibly with parity distinctions. Ward identities for conserved currents can relate some coefficients to charges or normalizations, but generic scalar triples do not become numerically determined by conformal covariance alone.

\langle\mathcal O_1\mathcal O_2\mathcal O_3\rangle=\frac{C_{123}}{r_{12}^{\Delta_1+\Delta_2-\Delta_3}r_{13}^{\Delta_1+\Delta_3-\Delta_2}r_{23}^{\Delta_2+\Delta_3-\Delta_1}},\qquad r_{ij}=|x_i-x_j|

Each distance has length units and the full correlator has length dimension −(Δ1+Δ2+Δ3). C123 is dimensionless after the stated primary normalizations.↔ Scroll wide equations horizontally.

Identify where functions of invariants first appear

Four separated scalar points have two conformal cross-ratios in generic dimensions. Symmetry fixes a convenient overall distance factor but leaves a reduced function of those ratios. The operator product expansion and crossing constrain that function; dimensional analysis alone does not fix it. The ratios are dimensionless, so a uniform rescaling of all positions changes the prefactor while leaving the reduced function unchanged. Permuting points can change both the ratios and the chosen prefactor.

These formulas give quick consistency tests for data and calculations. Check the power at each insertion, internal symmetry selection rules, the two-point basis and any omitted tensor structures before interpreting a fitted constant. A sine-wave interference lab can illustrate that an overall pattern depends on relative parameters, but it cannot generate Euclidean conformal correlators or impose their distance powers. A numerical similarity between a plotted curve and a power law is not a Ward-identity derivation.

u=\frac{r_{12}^2r_{34}^2}{r_{13}^2r_{24}^2},\qquad v=\frac{r_{14}^2r_{23}^2}{r_{13}^2r_{24}^2},\qquad \langle\phi_1\phi_2\phi_3\phi_4\rangle=\frac{\mathcal G(u,v)}{r_{12}^{2\Delta_\phi}r_{34}^{2\Delta_\phi}}

The final expression is for four identical real scalar primaries of dimension Δφ. This prefactor convention will be maintained in the crossing lessons.↔ Scroll wide equations horizontally.

Work it through

Determine all three powers

Take scalar dimensions $\Delta_1=1$ , $\Delta_2=2$ , $\Delta_3=2$ and $C_{123}=6$ . For a Euclidean triangle with distances $r_{12}=2$ , $r_{13}=3$ , $r_{23}=4$ in one fixed reference unit, find the three-point value and its change under doubling every distance.

The three exponents are $1+2-2=1$ , $1+2-2=1$ and $2+2-1=3$ . Their sum is five, the sum of the operator dimensions.
The correlator is $6/(2\cdot3\cdot4^3)=1/64$ in the corresponding inverse-fifth-power reference units. The distances obey the triangle inequalities, so this is a valid separated configuration.
Doubling all distances multiplies the denominator by $2^5=32$ . The new value is $1/2048$ , while the dimensionless coefficient remains six. Changing the geometry uniformly does not renormalize that CFT datum.

The result

The correlator is 1/64 in fixed reference units and becomes 1/2048 after a uniform doubling.

The living notebook

Explore the model

Interactive lab

Interactive lab · waves

Parameters matter, but this is not a correlator

Reset and set BOTH Phase difference = 0° and Amplitude ratio = 1; predict reinforcement with total wave amplitude 2. Reset and set Phase difference = 180° and Amplitude ratio = 1; predict cancellation. These equal-frequency sine waves illustrate parameter dependence only, not conformal distance powers, tensor structures or cross-ratios.

SumFirst componentSecond component

Sum at x = 0: 0.866
Resultant amplitude: 1.732
Phase difference: 60°

Phase difference60°Amplitude ratio1

t = 0 s

Two waves share the same frequency and wave number. Playback advances t in sin(2x − t) + a sin(2x − t + phase). Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA curve must satisfy the relevant symmetry relations before it can represent a correlator.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Covariance fixes the distance exponents. The normalized three-point coefficient is dynamical data unless additional symmetry or Ward identities determine it.

Your turn

Try it yourself

An unnormalized real scalar has two-point coefficient 4 and three-point coefficient 24 for three copies of itself. Rescale it to unit two-point coefficient. What is the new three-point coefficient, and why is dividing only by four incorrect?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Each correlator receives one rescaling factor for every operator insertion.

Compare with the solution

Define $\widehat{\mathcal O}=\mathcal O/2$ . Its two-point coefficient becomes $4/2^2=1$ . Its three-point coefficient becomes $24/2^3=3$ . Dividing by four would normalize two insertions rather than three and would mix incompatible conventions for the CFT data. The same bookkeeping must be used when relating normalized three-point coefficients to operator-product coefficients and their squared contributions in a four-point block expansion.

Keep the idea

Take it with you

Special conformal covariance supplies constraints beyond scaling alone.
Three-point exponents are kinematic; normalized coefficients are data.
Four-point functions retain nontrivial dependence on cross-ratios.

Read further

Original sources and deeper explanations to follow at your own pace.

Optional progress, saved in this browser when available. Revisit any lesson at any time.

Lesson not marked complete.