Scalar Two- and Three-Point Functions

At separated points, conformal covariance fixes scalar two- and three-point functions up to constant tensors. Those constants become OPE coefficients only relative to a two-point metric and an operator basis. The derivation below is Euclidean in $d>2$; Lorentzian Wightman or time-ordered correlators are boundary values with an ordering-dependent $i\epsilon$ prescription, not the same real powers with signs ignored.

Required background. Primaries, Descendants, and Conformal Multiplets supplies primary transformation laws. Localized Transformations and Ward–Takahashi Identities supplies distributional Ward identities. Helpful background. Coincident Products and Contact Terms separates separated-point functions from their distributional extensions.

The scalar two-point metric

Translations and rotations imply that two scalar primaries have

$$\langle\mathcal O_i(x_1)\mathcal O_j(x_2)\rangle =f_{ij}(x_{12}^2).$$

Dilatations give $f_{ij}(s)\propto s^{-(\Delta_i+\Delta_j)/2}$. Inversion supplies the remaining condition. Since

$$x_{12}'{}^2=\frac{x_{12}^2}{x_1^2x_2^2}, \qquad \mathcal O_i'(x_i')=(x_i^2)^{\Delta_i}\mathcal O_i(x_i),$$

covariance is possible for nonzero separated-point support only when $\Delta_i=\Delta_j$. Thus, within a set of scalar primaries with the same dimension and conjugate internal quantum numbers,

$$\boxed{ \langle\mathcal O_i(x)\mathcal O_j(0)\rangle =\frac{G_{ij}}{(x^2)^\Delta}. }$$

$G_{ij}$ is a basis-dependent two-point metric. In a reflection-positive theory, a Hermitian basis can be chosen so that $G$ is positive definite and then orthonormalized to $\delta_{ij}$. For complex operators the nonzero pairing is generally between a representation and its conjugate; writing $\langle\mathcal O\mathcal O\rangle=1$ can be false even when $\langle\mathcal O\mathcal O^\dagger\rangle$ is positive. A change of degenerate basis $\mathcal O_i'=M_i{}^j\mathcal O_j$ sends

$$G'\,=\,MGM^{\mathsf T}$$

in a real basis, or $G'=MGM^\dagger$ in a complex Hermitian basis. Only basis-covariant contractions are physical.

Three points and their independent constant

For scalar primaries, translation and rotation invariance allow a function of $x_{12}^2$, $x_{23}^2$, and $x_{31}^2$. Write a monomial ansatz

$$\langle\mathcal O_i(x_1)\mathcal O_j(x_2)\mathcal O_k(x_3)\rangle =\frac{C_{ijk}} {(x_{12}^2)^a(x_{23}^2)^b(x_{31}^2)^c}.$$

Inversion contributes a factor $(x_r^2)^{\Delta_r}$ at each point. Matching the powers of $x_1^2,x_2^2,x_3^2$ gives

$$a+c=\Delta_i,\qquad a+b=\Delta_j,\qquad b+c=\Delta_k,$$

and hence

$$\boxed{ \langle\mathcal O_i(x_1)\mathcal O_j(x_2)\mathcal O_k(x_3)\rangle = \frac{C_{ijk}} {(x_{12}^2)^{(\Delta_i+\Delta_j-\Delta_k)/2} (x_{23}^2)^{(\Delta_j+\Delta_k-\Delta_i)/2} (x_{31}^2)^{(\Delta_k+\Delta_i-\Delta_j)/2}}. }$$

No continuous cross ratio exists for three generic points, so $C_{ijk}$ is the only parity-even scalar structure. For mutually local bosonic scalars it is symmetric under simultaneous exchange of labels and points. In a unitary relativistic theory obeying spin–statistics, a fermionic operator cannot transform as a spacetime scalar; internal indices can nevertheless make $C_{ijk}$ an invariant tensor with nontrivial permutation symmetry. The general separated-point derivation and conserved-operator extensions are given in Osborn and Petkou 1994, §§ 2–3.

Extracting an OPE coefficient

Let scalar primaries $\mathcal O_a$ of dimension $\Delta_a$ span a possibly degenerate exchanged sector. The leading primary part of the Euclidean OPE is

$$\mathcal O_i(x)\mathcal O_j(0) \sim \sum_a \lambda_{ij}{}^a (x^2)^{(\Delta_a-\Delta_i-\Delta_j)/2} \mathcal O_a(0)+\text{descendants}.$$

Insert $\mathcal O_k(y)$ with $\lvert x\rvert<\lvert y\rvert$ and use the two-point function:

$$\langle\mathcal O_i(x)\mathcal O_j(0)\mathcal O_k(y)\rangle \sim \sum_a\lambda_{ij}{}^aG_{ak} (x^2)^{(\Delta_a-\Delta_i-\Delta_j)/2} (y^2)^{-\Delta_a}.$$

Taking the same limit in the exact three-point function shows

$$\boxed{C_{ijk}=\lambda_{ij}{}^aG_{ak}.}$$

Therefore $C_{ijk}=\lambda_{ij k}$ only in an orthonormal basis with the exchanged index lowered. Under a degenerate basis change, $C$ and $G$ transform together. The invariant four-point weight is $\lambda_{12}{}^aG_{ab}\lambda_{34}{}^b$, not a coordinate-dependent component.

For identical Hermitian $\phi$ with $\langle\phi(x)\phi(0)\rangle=(x^2)^{-\Delta_\phi}$ and an orthonormal Hermitian exchanged basis, $C_{\phi\phi a}=\lambda_{\phi\phi a}$ is real. Reflection positivity then makes its identical-pair four-point contribution $\lambda_{\phi\phi a}^2\geq0$. Neither reality nor positivity follows for a generic ordering $\langle\phi_1\phi_2\phi_3\phi_4\rangle$. The radial-state derivation of this normalization and positivity statement is given in Simmons-Duffin 2017, §§ 5–7.

Selection rules and normalization checks

Check	Consequence	Failure if omitted
Equal scaling dimension and conjugate quantum numbers	A nonzero scalar two-point pairing is allowed	One writes a metric between inequivalent representations
Positive reflection form	$G$ can be orthonormalized in a physical Hermitian sector	An indefinite theory is incorrectly assigned positive squares
Internal singlet in $R_i\otimes R_j\otimes R_k$	$C_{ijk}$ may be nonzero	A forbidden invariant tensor is retained
Identical-boson permutation	The coefficient tensor has the required symmetry	Odd exchange sectors contaminate the OPE
Declared rescaling $\mathcal O_i\to a_i\mathcal O_i$	$G_{ij}\to a_ia_jG_{ij}$ and $C_{ijk}\to a_ia_ja_kC_{ijk}$	Numbers from two sources are compared directly
Separated points	The displayed powers are ordinary functions	Contact terms are mistaken for new conformal structures

As a quick consistency test, send $x_3$ to infinity by defining

$$\mathcal O_k(\infty) =\lim_{x_3^2\to\infty}(x_3^2)^{\Delta_k}\mathcal O_k(x_3).$$

The three-point function becomes

$$\langle\mathcal O_i(x)\mathcal O_j(0)\mathcal O_k(\infty)\rangle =\frac{C_{ijk}} {(x^2)^{(\Delta_i+\Delta_j-\Delta_k)/2}},$$

which reproduces the leading OPE power and coefficient contraction above.

Lorentzian boundary values and contact terms

In Lorentzian signature with the site’s $(+---)$ convention, $x_{ij}^2$ can vanish or change sign. A Wightman function is defined by a boundary value such as

$$x_i^0\longmapsto x_i^0-i\epsilon_i, \qquad \epsilon_1>\epsilon_2>\cdots,$$

with the inequalities chosen for the operator ordering. Different orderings need not lie on the same analytic sheet. The Euclidean expressions determine these boundary values only when the required analyticity and continuation path are available. The theorem-level relation among Wightman domains, Jost points, and locality is treated separately in Jost Points, Edge of the Wedge, and Locality.

At coincident points, renormalized correlators can contain delta functions and derivatives of delta functions. Such terms are invisible in the separated-point classification yet contribute to Ward identities and Fourier transforms. Their coefficients can be scheme dependent unless fixed by a charge normalization or anomaly. Momentum-Space Correlators and Conformal Ward Identities keeps these terms explicit.

Common pitfalls

A unit two-point coefficient is automatic. It is a basis choice, possible only in a nondegenerate positive sector. Complex and degenerate representations require their full pairing.

Every three-point coefficient is an OPE coefficient. The upper-index OPE coefficient is obtained by contracting the three-point tensor with $G^{-1}$. They coincide componentwise only in an orthonormal convention.

Euclidean powers define every Lorentzian ordering. Complex powers need a boundary-value and branch prescription. Changing the ordering can move the correlator to another sheet.

Exercises

Show that a scalar two-point function vanishes for $\Delta_i\neq\Delta_j$ at separated points.

Solution

Dilatations allow $f(x^2)=C(x^2)^{-(\Delta_i+\Delta_j)/2}$. Under inversion, covariance requires one factor $(x_1^2)^{\Delta_i}$ and one factor $(x_2^2)^{\Delta_j}$, whereas the transformed separation supplies equal powers $(x_1^2x_2^2)^{(\Delta_i+\Delta_j)/2}$. Equality for arbitrary $x_1,x_2$ therefore requires $\Delta_i=\Delta_j$, unless $C=0$.

References

• Osborn, Hugh, and Andreas Petkou. “Implications of Conformal Invariance in Field Theories for General Dimensions.” Annals of Physics 231 (1994): 311–362. DOI; Open PDF
• Simmons-Duffin, David. “TASI Lectures on the Conformal Bootstrap.” In New Frontiers in Fields and Strings, 1–74. Singapore: World Scientific, 2017. DOI; Open PDF