OPE, Associativity, and Conformal Correlators

The previous page explained why conformal symmetry is more restrictive than scale symmetry. Inversion already forced a scalar two-point function to vanish unless the two operators have the same scaling dimension. This page pushes the same logic further. We derive the conformal form of three-point functions, explain why four-point functions contain genuine dynamical information, and introduce the operator product expansion as the local algebra of a critical theory.

The punchline is compact but profound: a conformal field theory is largely specified by its spectrum and OPE coefficients. The spectrum is the list of local primary operators and their dimensions and spins. The OPE coefficients tell us what appears when two local operators approach one another. Associativity of this local multiplication is not a decorative condition; it is the origin of crossing symmetry and, in two dimensions, of the algebraic constraints that later become Virasoro representation theory.

Required background. Lesson 14 supplies inversion, primary-field covariance, and the two-point-function argument used below.

Helpful background. Lesson 12 develops scaling operators and the Ising examples, while Lesson 13 explains how fixed-point correlators acquire homogeneous scaling laws.

Two-point functions as a warm-up

Distances and active primary transformations

We work in Euclidean signature. We distinguish the displacement vector

$$x_{ij}^\mu=x_i^\mu-x_j^\mu$$

from its Euclidean length $x_{ij}=|x_i-x_j|$; thus $x_{ij}^2=(x_i-x_j)^2$. Scalar primaries $O_i$ have scaling dimensions $\Delta_i$. Under a conformal transformation $x'=f(x)$ with local length scale $\Omega(x)$,

$$O_i'(x')=\Omega(x)^{-\Delta_i}O_i(x).$$

For inversion, $x'=x/x^2$ and $\Omega(x)=1/x^2$.

For scalar primaries, translation and rotation invariance imply that a two-point function depends only on $x_{12}$. Scale covariance permits

$$\langle O_i(x_1)O_j(x_2)\rangle ={C_{ij}\over x_{12}^{\Delta_i+\Delta_j}}.$$

But inversion covariance is stronger. Under inversion,

$$x_{12}'{}^2={x_{12}^2\over x_1^2x_2^2}.$$

The functional form at inverted points gives

$$\langle O_i(x_1')O_j(x_2')\rangle ={C_{ij}(x_1^2x_2^2)^{(\Delta_i+\Delta_j)/2} \over x_{12}^{\Delta_i+\Delta_j}}.$$

The primary transformation law gives instead

$$\langle O_i(x_1')O_j(x_2')\rangle =(x_1^2)^{\Delta_i}(x_2^2)^{\Delta_j} {C_{ij}\over x_{12}^{\Delta_i+\Delta_j}}.$$

For arbitrary $x_1$ and $x_2$, these agree only if either $C_{ij}=0$ or $\Delta_i=\Delta_j$. Thus a conformal basis can be chosen so that

$$\boxed{ \langle O_i(x)O_j(0)\rangle ={g_{ij}\over |x|^{2\Delta_i}}, \qquad g_{ij}=0\quad \text{unless }\Delta_i=\Delta_j. }$$

In a unitary theory one usually diagonalizes the positive two-point metric $g_{ij}$ within each sector of equal spin, dimension, and internal quantum numbers. For many formulas below we use an orthonormal scalar basis,

$$g_{ij}=\delta_{ij}.$$

This normalization is not physics; it is a basis choice. Once chosen, however, it fixes the normalization of OPE coefficients.

Three-point functions

Conformal symmetry fixes the coordinate dependence of scalar three-point functions completely. Start with the most general translation- and rotation-invariant product of powers,

$$\langle O_1(x_1)O_2(x_2)O_3(x_3)\rangle ={C_{123}\over x_{12}^{a}x_{23}^{b}x_{13}^{c}}.$$

Scale covariance requires

$$a+b+c=\Delta_1+\Delta_2+\Delta_3.$$

Inversion gives the remaining conditions. Since

$$x_{ij}'={x_{ij}\over (x_i^2x_j^2)^{1/2}},$$

the transformed ansatz is

$${C_{123} (x_1^2)^{(a+c)/2} (x_2^2)^{(a+b)/2} (x_3^2)^{(b+c)/2} \over x_{12}^{a}x_{23}^{b}x_{13}^{c}}.$$

Primary covariance demands the factor

$$(x_1^2)^{\Delta_1}(x_2^2)^{\Delta_2}(x_3^2)^{\Delta_3}.$$

Hence

$${a+c\over2}=\Delta_1, \qquad {a+b\over2}=\Delta_2, \qquad {b+c\over2}=\Delta_3.$$

Solving gives

$$a=\Delta_1+\Delta_2-\Delta_3,$$ $$b=\Delta_2+\Delta_3-\Delta_1,$$ $$c=\Delta_1+\Delta_3-\Delta_2.$$

Therefore

$$\boxed{ \langle O_1(x_1)O_2(x_2)O_3(x_3)\rangle ={C_{123}\over x_{12}^{\Delta_1+\Delta_2-\Delta_3} x_{23}^{\Delta_2+\Delta_3-\Delta_1} x_{13}^{\Delta_1+\Delta_3-\Delta_2}} .}$$

The three distances $x_{12}$, $x_{23}$, and $x_{13}$ carry powers fixed by inversion covariance. All dynamical information in a scalar three-point function is reduced to the coefficient $C_{123}$.

The constants $C_{123}$ are the first genuinely dynamical CFT data. Symmetries may force some of them to vanish. For example, in the Ising CFT the global spin-flip symmetry sends $\sigma\mapsto -\sigma$, so a correlator with an odd number of $\sigma$ insertions vanishes:

$$C_{\sigma\sigma\sigma}=0.$$

The three-point coefficient $C_{\sigma\sigma\varepsilon}$ is allowed and controls the first nontrivial term in the $\sigma\times\sigma$ operator product.

Four-point functions and cross ratios

For four points, conformal symmetry does not fix everything. There are conformally invariant shape variables. A convenient pair in $d\ge2$ is

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

These are dimensionless and invariant under translations, rotations, dilations, and inversion. The invariance under inversion follows because each $x_{ij}^2$ picks up one factor of $(x_i^2x_j^2)^{-1}$, and the factors cancel between numerator and denominator.

For four identical scalar primaries $O$ of dimension $\Delta$, a useful channel-adapted form is

$$\boxed{ \langle O(x_1)O(x_2)O(x_3)O(x_4)\rangle ={1\over x_{12}^{2\Delta}x_{34}^{2\Delta}}F(u,v). }$$

The prefactor has the correct scaling near the $12$ and $34$ pairs, while the function $F(u,v)$ contains the dynamical information. Another prefactor would give another function related to $F$ by a simple power of $u$ and $v$; the physics is not in that bookkeeping choice.

For four nonidentical scalar primaries one common convention is

$$\begin{aligned} \langle O_1(x_1)O_2(x_2)O_3(x_3)O_4(x_4)\rangle &={1\over x_{12}^{\Delta_1+\Delta_2}x_{34}^{\Delta_3+\Delta_4}} \left({x_{24}\over x_{14}}\right)^{\Delta_{12}} \\ &\quad\times \left({x_{14}\over x_{13}}\right)^{\Delta_{34}} F_{1234}(u,v). \end{aligned}$$

where

$$\Delta_{ij}=\Delta_i-\Delta_j.$$

All nontrivial dependence is again in a function of $u$ and $v$.

Four points retain two conformally invariant shape variables. In two-dimensional complex coordinates these are often packaged as $u=\eta\bar\eta$ and $v=(1-\eta)(1-\bar\eta)$.

In two dimensions one usually writes

$$\eta={z_{12}z_{34}\over z_{13}z_{24}}, \qquad \bar\eta={\bar z_{12}\bar z_{34}\over \bar z_{13}\bar z_{24}},$$

so that

$$u=\eta\bar\eta, \qquad v=(1-\eta)(1-\bar\eta).$$

The four-point function is where conformal field theory stops being pure kinematics. The possible functions $F(u,v)$ are heavily constrained, but not arbitrary powers fixed by symmetry alone. Their allowed form is determined by the spectrum, OPE coefficients, and associativity.

The operator product expansion

The operator product expansion, or OPE, is the statement that when two local operators approach one another, their product can be replaced inside correlation functions by a sum of local operators at a nearby point:

$$\boxed{ O_i(x)O_j(0) \sim \sum_k C_{ij}{}^k(x,\partial)O_k(0). }$$

The symbol $\sim$ means equality after expansion inside correlation functions; it is not an ordinary Taylor series because its coefficients can contain singular powers of $|x|$. In a unitary Euclidean CFT, radial quantization makes this expansion genuinely convergent whenever a sphere enclosing the two fused insertions contains no other insertion. In a general, nonconformal QFT the corresponding short-distance OPE may instead be only asymptotic or distributional.

For scalar primaries, conformal symmetry fixes the leading power in the contribution of a scalar primary $O_k$:

$$\boxed{ O_i(x)O_j(0) \sim \sum_k C_{ij}{}^k |x|^{\Delta_k-\Delta_i-\Delta_j} \left[O_k(0)+\text{descendants}\right]. }$$

The descendants are derivatives of $O_k$ and, for spinning exchanged operators, tensor structures built from $x^\mu$. Once the coefficient of the primary is known, conformal symmetry fixes the coefficients of all its descendants. The whole primary-plus-descendants contribution is called a conformal family.

The OPE replaces two nearby insertions by a sum of local operators at one point. The expansion is valid inside correlators when the fusion disk does not contain other insertions.

The relation between the three-point coefficient and the OPE coefficient is especially transparent in an orthonormal Hermitian scalar basis. Insert the OPE into a three-point function with $O_k(y)$:

$$\langle O_i(x)O_j(0)O_k(y)\rangle \sim C_{ij}{}^k |x|^{\Delta_k-\Delta_i-\Delta_j} \langle O_k(0)O_k(y)\rangle+\cdots.$$

Since

$$\langle O_k(0)O_k(y)\rangle={1\over |y|^{2\Delta_k}},$$

the leading OPE prediction is

$$\langle O_i(x)O_j(0)O_k(y)\rangle \sim {C_{ij}{}^k |x|^{\Delta_k-\Delta_i-\Delta_j} \over |y|^{2\Delta_k}}.$$

Now take the exact three-point function and let $x\to0$:

$$\langle O_i(x)O_j(0)O_k(y)\rangle ={C_{ijk}\over |x|^{\Delta_i+\Delta_j-\Delta_k} |y|^{\Delta_j+\Delta_k-\Delta_i} |y-x|^{\Delta_i+\Delta_k-\Delta_j}}$$ $$\sim {C_{ijk}|x|^{\Delta_k-\Delta_i-\Delta_j} \over |y|^{2\Delta_k}}.$$

Therefore, with orthonormal two-point functions,

$$\boxed{C_{ij}{}^k=C_{ijk}.}$$

With a general two-point metric $g_{ij}$, the raised-index coefficient is

$$C_{ij}{}^k=C_{ij\ell}g^{\ell k}, \qquad g^{\ell k}g_{km}=\delta^\ell{}_m.$$

The identity operator deserves special mention. For a Hermitian operator in a basis normalized as $\langle O_i(x)O_j(0)\rangle=\delta_{ij}/|x|^{2\Delta_i}$,

$$O_i(x)O_i(0)\sim {1\over |x|^{2\Delta_i}}\mathbf 1+\cdots.$$

The identity contribution is the most singular term in many OPEs. For a charged operator, the analogous identity term occurs in $O_i(x)O_i^\dagger(0)$ rather than necessarily in $O_i(x)O_i(0)$.

The first descendant coefficient

It is useful to see once how descendants are fixed. Consider the scalar contribution of $O_k$ in the OPE

$$O_i(x)O_j(0) \sim C_{ijk}|x|^{\Delta_k-\Delta_i-\Delta_j} \left[O_k(0)+a\,x^\mu\partial_\mu O_k(0)+\cdots\right].$$

Insert $O_k(y)$ and use

$$\partial_{0\mu}{1\over |y-0|^{2\Delta_k}} =2\Delta_k {y_\mu\over |y|^{2\Delta_k+2}}.$$

The OPE then predicts

$$\langle O_i(x)O_j(0)O_k(y)\rangle \sim {C_{ijk}|x|^{\Delta_k-\Delta_i-\Delta_j}\over |y|^{2\Delta_k}} \left[1+2a\Delta_k{x\cdot y\over y^2}+\cdots\right].$$

The exact three-point function gives, for small $x$,

$$|y-x|^{-(\Delta_i+\Delta_k-\Delta_j)} =|y|^{-(\Delta_i+\Delta_k-\Delta_j)} \left[1+(\Delta_i+\Delta_k-\Delta_j){x\cdot y\over y^2}+\cdots\right].$$

Matching the two expansions yields

$$\boxed{ a={\Delta_i-\Delta_j+\Delta_k\over 2\Delta_k} }$$

for $\Delta_k\ne0$. The identity family is a special case because the identity has no nonzero derivative descendant.

This calculation captures the general logic of conformal descendants: the three-point coefficient chooses the primary family, and conformal symmetry fills in the derivative tower.

Four-point factorization and conformal blocks

Apply the OPE to a four-point function in the $12$ channel:

$$O_1(x_1)O_2(x_2) \sim \sum_p C_{12}{}^p(x_{12},\partial_{x_2})O_p(x_2).$$

A second OPE can be applied to the $34$ pair. The result has the schematic form

$$\langle O_1O_2O_3O_4\rangle = \sum_p C_{12p}C_{34p}\,\mathcal G_p^{(12)(34)}(u,v),$$

up to the chosen external prefactor. Here $\mathcal G_p$ is the conformal block for the primary $O_p$ and all of its descendants in this channel. The block is kinematical once the dimension and spin of $O_p$ are known; the coefficients $C_{12p}C_{34p}$ are dynamical.

In the $12\to p$ and $34\to p$ channel, the four-point function factorizes into OPE coefficients multiplying the conformal block of the exchanged primary family $p$.

This statement should feel familiar from several earlier parts of the course. In statistical mechanics, it says that short-distance clusters can be replaced by effective local insertions. In QFT language, it resembles inserting a complete set of states. In two-dimensional radial quantization, these viewpoints become literally the same: the OPE is the short-distance version of Hilbert-space completeness on a circle surrounding the fused operators.

Associativity and crossing symmetry

The OPE is a local multiplication law. A multiplication law must be associative if it is to give unambiguous correlation functions. For three nearby operators, associativity says that fusing $O_i$ with $O_j$ first and then with $O_k$ must agree with fusing $O_j$ with $O_k$ first and then with $O_i$, wherever both expansions converge after analytic continuation.

For four-point functions this becomes crossing symmetry. Each channel expansion converges in its own radial domain, and analytic continuation must reconstruct one and the same correlator. In the equation below every channel block is expressed with one common external prefactor, so any channel-dependent crossing powers are included in the definition of $\mathcal G^{(ab)(cd)}$. The $(12)(34)$, $(13)(24)$, and $(14)(23)$ decompositions then obey

$$\begin{aligned} \sum_p C_{12p}C_{34p}\,\mathcal G_p^{(12)(34)}(u,v) &= \sum_q C_{13q}C_{24q}\,\mathcal G_q^{(13)(24)}(u,v) \\ &= \sum_r C_{14r}C_{23r}\,\mathcal G_r^{(14)(23)}(u,v). \end{aligned}$$

OPE associativity is crossing symmetry of four-point functions. Different fusion channels must reconstruct the same correlator.

For identical scalar primaries with

$$\langle O_1O_2O_3O_4\rangle ={1\over x_{12}^{2\Delta}x_{34}^{2\Delta}}F(u,v),$$

exchanging $x_1\leftrightarrow x_3$ gives

$$\boxed{ F(u,v)=\left({u\over v}\right)^\Delta F(v,u). }$$

This is a simple example of a bootstrap equation. It is simple to write and hard to satisfy. The equality compares two different infinite sums over exchanged primary families. In a consistent CFT, the spectrum and OPE coefficients must make all such equalities true.

The relation to Lie algebras is a useful preview. Suppose singular OPEs define modes $A_n$ by contour integrals. Then different ways of nesting OPEs become different ways of nesting commutators. Associativity of the OPE becomes the Jacobi identity,

$$[A_i,[A_j,A_k]]+[A_j,[A_k,A_i]]+[A_k,[A_i,A_j]]=0.$$

This is the mechanism behind the Virasoro algebra and current algebras in two-dimensional CFT. The algebra is not imposed from the outside; it is the short-distance consistency of local fields.

Example: Ising fusion rules

The two-dimensional critical Ising model has three primary families in the minimal local theory:

$$\mathbf 1, \qquad \sigma, \qquad \varepsilon,$$

with dimensions

$$\Delta_{\mathbf 1}=0, \qquad \Delta_\sigma={1\over8}, \qquad \Delta_\varepsilon=1.$$

The fusion rules are

$$\sigma\times\sigma=\mathbf 1+\varepsilon,$$ $$\sigma\times\varepsilon=\sigma,$$ $$\varepsilon\times\varepsilon=\mathbf 1.$$

With the standard normalization

$$\langle \sigma(x)\sigma(0)\rangle={1\over |x|^{1/4}}, \qquad \langle \varepsilon(x)\varepsilon(0)\rangle={1\over |x|^{2}},$$

the first OPE begins as

$$\sigma(x)\sigma(0) \sim {1\over |x|^{1/4}} \left[\mathbf 1+{1\over2}|x|\,\varepsilon(0)+\cdots\right].$$

The power of the identity term is fixed by

$$\Delta_{\mathbf 1}-2\Delta_\sigma=-{1\over4}.$$

The power of the energy term is fixed by

$$\Delta_\varepsilon-2\Delta_\sigma=1-{1\over4}={3\over4},$$

which is the same as the factored form $|x|^{-1/4}\cdot |x|$. The coefficient $1/2$ is not fixed by dimensional analysis; it is an OPE coefficient. In the minimal Ising CFT, crossing symmetry and the finite operator content determine it, once the two-point normalizations are chosen.

The disorder operator $\mu$ belongs to the same Virasoro representation as $\sigma$ and has the same dimension,

$$\Delta_\mu={1\over8},$$

and it obeys the same family-level rule $\mu\times\mu=\mathbf 1+\varepsilon$. It is not an additional mutually local primary of the diagonal three-primary theory: it is defined at the endpoint of a disorder line. Its mutual locality with $\sigma$ therefore differs, and the mixed OPE $\sigma\times\mu$ contains fermionic fields with branch-cut behavior. Conformal dimensions and fusion powers do not erase the topological information carried by defect lines.

Two-dimensional notation

In two Euclidean dimensions, write

$$z=x^1+i x^2, \qquad \bar z=x^1-i x^2.$$

A local conformal map is

$$z'=f(z), \qquad \bar z'=\bar f(\bar z),$$

and the metric transforms as

$$dz' d\bar z'=f'(z)\bar f'(\bar z)\,dz d\bar z.$$

A primary field with holomorphic and antiholomorphic weights $(h,\bar h)$ transforms as

$$O'(z',\bar z') =\left(f'(z)\right)^{-h} \left(\bar f'(\bar z)\right)^{-\bar h} O(z,\bar z).$$

Its scaling dimension and spin are

$$\Delta=h+\bar h, \qquad s=h-\bar h.$$

For spinless fields $h=\bar h=\Delta/2$. The two-point function becomes

$$\langle O_i(z_1,\bar z_1)O_j(z_2,\bar z_2)\rangle ={\delta_{ij}\over z_{12}^{2h_i}\bar z_{12}^{2\bar h_i}}$$

in an orthonormal basis. The three-point function factorizes similarly into a holomorphic power law times an antiholomorphic power law. The next pages exploit this holomorphic structure and eventually turn the stress tensor into an infinite set of generators.

Summary

Conformal symmetry fixes scalar two- and three-point functions up to constants. In an orthonormal conformal basis,

$$\langle O_i(x)O_j(0)\rangle={\delta_{ij}\over |x|^{2\Delta_i}},$$

and

$$\langle O_1(x_1)O_2(x_2)O_3(x_3)\rangle ={C_{123}\over x_{12}^{\Delta_1+\Delta_2-\Delta_3} x_{23}^{\Delta_2+\Delta_3-\Delta_1} x_{13}^{\Delta_1+\Delta_3-\Delta_2}}.$$

Four-point functions contain arbitrary functions of cross ratios, but these functions are not arbitrary in a consistent theory. The OPE decomposes a four-point function into conformal blocks with coefficients determined by three-point data. Associativity of the OPE requires equality of different channel decompositions. This is crossing symmetry, the central consistency condition of conformal bootstrap.

Common pitfalls

Confusing dimensions with OPE coefficients. Symmetry fixes the powers of $x_{ij}$, but it does not in general fix the constants $C_{ijk}$. Dimensions and OPE coefficients are independent pieces of CFT data constrained jointly by crossing.

Calling every OPE merely asymptotic. In a unitary Euclidean CFT, radial quantization gives a convergent expansion when the fusion sphere excludes all other insertions. The more cautious asymptotic statement is appropriate for generic QFTs, not for the CFT setting developed on this page.

Dropping the identity family. The identity contribution often dominates the short-distance limit. Forgetting it gives the wrong leading singularity and usually spoils the disconnected part of a four-point function.

Treating the reduced correlator as convention independent. A four-point function can be written with many different kinematic prefactors. The function called $F(u,v)$ changes with that choice, while the full correlator and its crossing content do not.

Reducing crossing to a relabeling. Crossing equates different OPE sums, generally organized around different limits and convergence regions. The equality therefore imposes dynamical constraints on the spectrum and OPE coefficients.

Exercises

Exercise 1: Three-point exponents from inversion

Derive the conformal form of a scalar three-point function by starting from

$$G_3={C_{123}\over x_{12}^{a}x_{23}^{b}x_{13}^{c}}$$

and imposing inversion covariance.

Solution

Under inversion,

$$x_{ij}'={x_{ij}\over (x_i^2x_j^2)^{1/2}}.$$

Thus

$$G_3(x_i') ={C_{123} (x_1^2)^{(a+c)/2} (x_2^2)^{(a+b)/2} (x_3^2)^{(b+c)/2} \over x_{12}^{a}x_{23}^{b}x_{13}^{c}}.$$

Primary covariance requires

$$G_3(x_i') =(x_1^2)^{\Delta_1}(x_2^2)^{\Delta_2}(x_3^2)^{\Delta_3}G_3(x_i).$$

Therefore

$${a+c\over2}=\Delta_1, \qquad {a+b\over2}=\Delta_2, \qquad {b+c\over2}=\Delta_3.$$

Solving gives

$$a=\Delta_1+\Delta_2-\Delta_3,$$ $$b=\Delta_2+\Delta_3-\Delta_1,$$ $$c=\Delta_1+\Delta_3-\Delta_2.$$

Substitution gives the three-point formula in the main text.

Exercise 2: Invariance of the cross ratios

Show that

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

are invariant under inversion.

Solution

Inversion sends

$$x_{ij}^2\mapsto x_{ij}'{}^2={x_{ij}^2\over x_i^2x_j^2}.$$

For $u$ this gives

$$u' ={x_{12}^2(x_1^2x_2^2)^{-1}x_{34}^2(x_3^2x_4^2)^{-1} \over x_{13}^2(x_1^2x_3^2)^{-1}x_{24}^2(x_2^2x_4^2)^{-1}}.$$

The factors $x_1^2x_2^2x_3^2x_4^2$ cancel, leaving

$$u'=u.$$

The calculation for $v$ is identical:

$$v' ={x_{14}^2(x_1^2x_4^2)^{-1}x_{23}^2(x_2^2x_3^2)^{-1} \over x_{13}^2(x_1^2x_3^2)^{-1}x_{24}^2(x_2^2x_4^2)^{-1}} =v.$$

Exercise 3: Three-point data from the OPE

Assume an orthonormal Hermitian scalar basis. Use the OPE to show that the leading coefficient in

$$O_i(x)O_j(0)\sim\sum_k C_{ij}{}^k |x|^{\Delta_k-\Delta_i-\Delta_j}O_k(0)+\cdots$$

is equal to the three-point coefficient $C_{ijk}$.

Solution

Insert both sides into a correlator with $O_k(y)$. The OPE gives

$$\langle O_i(x)O_j(0)O_k(y)\rangle \sim C_{ij}{}^k |x|^{\Delta_k-\Delta_i-\Delta_j} \langle O_k(0)O_k(y)\rangle.$$

Using the orthonormal two-point function,

$$\langle O_k(0)O_k(y)\rangle={1\over |y|^{2\Delta_k}},$$

we get

$$\langle O_i(x)O_j(0)O_k(y)\rangle \sim {C_{ij}{}^k |x|^{\Delta_k-\Delta_i-\Delta_j}\over |y|^{2\Delta_k}}.$$

The exact three-point function is

$${C_{ijk}\over |x|^{\Delta_i+\Delta_j-\Delta_k} |y|^{\Delta_j+\Delta_k-\Delta_i} |y-x|^{\Delta_i+\Delta_k-\Delta_j}}.$$

Taking $x\to0$ gives

$${C_{ijk}|x|^{\Delta_k-\Delta_i-\Delta_j}\over |y|^{2\Delta_k}}.$$

Matching the two expressions gives

$$C_{ij}{}^k=C_{ijk}.$$

Exercise 4: The first scalar descendant

Find the coefficient $a$ of the first derivative descendant in

$$O_i(x)O_j(0) \sim C_{ijk}|x|^{\Delta_k-\Delta_i-\Delta_j} \left[O_k(0)+a x^\mu\partial_\mu O_k(0)+\cdots\right].$$

Assume $\Delta_k\ne0$.

Solution

Insert $O_k(y)$. The derivative acts on the two-point function as

$$\partial_{0\mu}|y|^{-2\Delta_k} =2\Delta_k {y_\mu\over |y|^{2\Delta_k+2}}.$$

The OPE therefore predicts

$${C_{ijk}|x|^{\Delta_k-\Delta_i-\Delta_j}\over |y|^{2\Delta_k}} \left[1+2a\Delta_k{x\cdot y\over y^2}+\cdots\right].$$

The exact three-point function contains the factor

$$|y-x|^{-(\Delta_i+\Delta_k-\Delta_j)}.$$

For small $x$,

$$|y-x|^{-A}=|y|^{-A} \left[1+A{x\cdot y\over y^2}+\cdots\right], \qquad A=\Delta_i+\Delta_k-\Delta_j.$$

Matching the coefficient of $x\cdot y/y^2$ gives

$$2a\Delta_k=\Delta_i+\Delta_k-\Delta_j.$$

Thus

$$\boxed{ a={\Delta_i-\Delta_j+\Delta_k\over 2\Delta_k}. }$$

Exercise 5: Identical-scalar crossing equation

For identical scalar primaries, derive the crossing relation

$$F(u,v)=\left({u\over v}\right)^\Delta F(v,u)$$

from the equality of the four-point function under $x_1\leftrightarrow x_3$.

Solution

Write

$$G_4(x_1,x_2,x_3,x_4) ={1\over x_{12}^{2\Delta}x_{34}^{2\Delta}}F(u,v).$$

Exchanging $x_1$ and $x_3$ sends $u\leftrightarrow v$ and gives

$$G_4(x_3,x_2,x_1,x_4) ={1\over x_{23}^{2\Delta}x_{14}^{2\Delta}}F(v,u).$$

For identical operators the two correlators are equal. Hence

$${F(u,v)\over x_{12}^{2\Delta}x_{34}^{2\Delta}} = {F(v,u)\over x_{23}^{2\Delta}x_{14}^{2\Delta}}.$$

Multiplying through gives

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

Using

$${u\over v}={x_{12}^2x_{34}^2\over x_{14}^2x_{23}^2},$$

we obtain

$$F(u,v)=\left({u\over v}\right)^\Delta F(v,u).$$

Exercise 6: Ising short-distance powers

Use the Ising dimensions $\Delta_\sigma=1/8$ and $\Delta_\varepsilon=1$ to determine the powers of $|x|$ in the identity and energy contributions to $\sigma(x)\sigma(0)$.

Solution

The scalar OPE power for an exchanged operator $O_k$ is

$$|x|^{\Delta_k-\Delta_i-\Delta_j}.$$

For $\sigma(x)\sigma(0)$, $\Delta_i=\Delta_j=\Delta_\sigma=1/8$.

For the identity, $\Delta_{\mathbf 1}=0$, so

$$\Delta_{\mathbf 1}-2\Delta_\sigma=0-{1\over4}=-{1\over4}.$$

Thus the identity contribution scales as

$$|x|^{-1/4}\mathbf 1.$$

For the energy operator,

$$\Delta_\varepsilon-2\Delta_\sigma=1-{1\over4}={3\over4}.$$

Thus the energy contribution scales as

$$|x|^{3/4}\varepsilon(0).$$

Factoring out the leading identity singularity gives

$$\sigma(x)\sigma(0) \sim |x|^{-1/4}\left[\mathbf 1+C_{\sigma\sigma\varepsilon}|x|\varepsilon(0)+\cdots\right].$$

With standard Ising normalization, $C_{\sigma\sigma\varepsilon}=1/2$.

References

• A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory,” Nuclear Physics B 241 (1984) 333–380.
• J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics 5, Cambridge University Press (1996).
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer (1997).
• A. M. Polyakov, Gauge Fields and Strings, Contemporary Concepts in Physics 3, Harwood Academic Publishers (1987).
• S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, SpringerBriefs in Physics, Springer (2017).
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 5th ed., Oxford University Press (2021).