Free and Generalized Free CFTs

Free fields and generalized free fields give exact crossing-symmetric correlators in any dimension. They are indispensable block and normalization benchmarks, but they are not interchangeable: a local free field has an equation of motion and a conserved stress tensor, while a generic generalized free field has Wick-like correlators without a local stress tensor in its operator algebra.

Required background. Crossing symmetry and positivity fix the four-point conventions used below. Unitarity bounds and null states explain the shortening of free scalar and fermion multiplets.

Helpful background. Scalar propagators and ordered correlators derive the Green functions whose Euclidean conformal form is used here.

Local free fields

Let

$$S_d=\frac{2\pi^{d/2}}{\Gamma(d/2)}$$

be the area of the unit $(d-1)$-sphere. For a canonically normalized massless scalar with action $S=\frac12\int d^dx\,(\partial\phi)^2$,

$$\langle\phi(x)\phi(0)\rangle =\frac{1}{(d-2)S_d|x|^{d-2}}, \qquad d>2.$$

The unit-normalized conformal primary

$$\widehat\phi=\sqrt{(d-2)S_d}\,\phi$$

has

$$\Delta_\phi=\frac{d-2}{2}, \qquad \langle\widehat\phi(x)\widehat\phi(0)\rangle=|x|^{-2\Delta_\phi}.$$

The equation of motion $\partial^2\phi=0$ removes the level-two scalar descendant. This is precisely the shortening that occurs when a unitary scalar saturates $\Delta\geq(d-2)/2$. The conformal stress tensor is the improved tensor

$$\begin{aligned} T_{\mu\nu}={}&\partial_\mu\phi\partial_\nu\phi -\frac12\delta_{\mu\nu}(\partial\phi)^2\\ &+\frac{d-2}{4(d-1)} \left(\delta_{\mu\nu}\partial^2-\partial_\mu\partial_\nu\right)\phi^2, \end{aligned}$$

which is conserved and traceless using the equation of motion, up to contact terms at insertions.

For a canonically normalized massless Dirac fermion,

$$\langle\psi(x)\bar\psi(0)\rangle =\frac{1}{S_d}\frac{\gamma^\mu x_\mu}{(x^2)^{d/2}}, \qquad \Delta_\psi=\frac{d-1}{2}.$$

The shortening condition is the Dirac equation $\gamma^\mu\partial_\mu\psi=0$. The dimension of the spinor representation, the reality condition, and chirality depend on $d$ and the chosen Euclidean spin cover; formulas involving traces of gamma matrices must state those choices.

Generalized-free scalar crossing

Now take a scalar primary $\mathcal O$ normalized by

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

and define all higher correlators by Gaussian pairings. With

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

the four-point function is

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

It obeys identical-scalar crossing exactly:

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

For $\Delta\geq(d-2)/2$, its block coefficients are nonnegative in the standard unitary block normalization. At generic $\Delta>(d-2)/2$, however, the resulting operator algebra has no local conserved stress tensor. Generalized-free theories therefore solve crossing and reflection positivity without satisfying the locality requirement of an ordinary stress-tensor CFT. This distinction is emphasized in Poland, Rychkov, and Vichi 2019, §III.I.1, pp. 24–25.

The bilinear spectrum and its leading coefficients

The $\mathcal O\times\mathcal O$ OPE contains the identity and symmetric-traceless bilinear primaries

$$[\mathcal O\mathcal O]_{n,\ell} \sim\mathcal O\,\partial_{(\mu_1}\cdots\partial_{\mu_\ell)} (\partial^2)^n\mathcal O-\text{descendants and traces},$$

with

$$\Delta_{n,\ell}=2\Delta+2n+\ell, \qquad n=0,1,2,\ldots, \qquad \ell=0,2,4,\ldots.$$

Odd spin is absent because the external scalars are identical. Normalize conformal blocks so that their leading OPE term has unit coefficient in the symmetric-traceless polarization basis. Then the squared OPE coefficient of the leading-twist family is

$$a_{0,\ell} =\bigl(1+(-1)^\ell\bigr) \frac{(\Delta)_\ell^2} {\ell!\,(2\Delta+\ell-1)_\ell},$$

where $(a)_n=\Gamma(a+n)/\Gamma(a)$. In particular,

$$[\mathcal O\mathcal O]_{0,0} =\frac{:{\mathcal O^2}:}{\sqrt2}, \qquad \lambda_{\mathcal O\mathcal O[\mathcal O\mathcal O]_{0,0}} =\sqrt2, \qquad a_{0,0}=2.$$

The scalar coefficient follows directly from

$$\langle:{\mathcal O^2}:(x):{\mathcal O^2}:(0)\rangle =2|x|^{-4\Delta}.$$

Higher-$n$ coefficients are fixed by the same Wick correlator after subtracting descendants. General-dimensional double-trace construction and normalization are developed in Fitzpatrick and Kaplan 2012, §§2.1–2.3.

At the free-scalar value $\Delta=(d-2)/2$, equations of motion create additional null relations among these formal bilinears. The generic generalized-free decomposition must be reduced accordingly. This is a decisive test: substituting the free value into a generic spectrum without quotienting null descendants overcounts operators.

Free fermions and statistics

Wick contraction also solves correlators of free fermions, but spinor indices and signs carry physical information. The four-point function is a sum of three pairings with a minus sign for odd permutations. Its OPE contains bosonic bilinears such as

$$\bar\psi\gamma_{(\mu_1}\partial_{\mu_2}\cdots \partial_{\mu_s)}\psi-\text{traces},$$

subject to the Dirac equation and dimension-specific gamma identities. A tensor basis valid for formal Dirac spinors in generic $d$ can become redundant after imposing Weyl, Majorana, or parity conditions in an integer dimension. Consequently, the number of fermionic structures must be checked after fixing dimension and spinor type.

What the exact solution does and does not establish

Check	Free local field	Generic generalized-free field
Crossing	Exact by Wick contraction	Exact by Wick-like pairing
Positive block coefficients	Yes in the unitary theory	Yes when $\Delta$ satisfies the scalar unitarity bound
Shortening	Scalar or Dirac equation	None at generic $\Delta$
Local stress tensor	Present after improvement	Absent from the algebra generated by $\mathcal O$
Spectrum	Identity plus local composite families with null quotients	Identity plus exact bilinear towers $2\Delta+2n+\ell$
Use as benchmark	Propagators, shortening, conserved data, blocks	Crossing, double-trace data, large-$N$ factorization

Passing crossing is therefore not a proof of locality or of spectrum completeness for a proposed QFT. One must also identify a stress tensor with the correct Ward identity, account for every shortening relation, and state whether the chosen operator algebra is intended as a full theory or only a subsector.

Continue to The embedding-space formalism to encode the tensor and spinor structures needed beyond scalar correlators.

Exercises

Verify generalized-free crossing

Insert $\mathcal G(u,v)=1+u^\Delta+(u/v)^\Delta$ into the identical-scalar crossing equation.

Solution

$$\left(\frac uv\right)^\Delta\mathcal G(v,u) =\left(\frac uv\right)^\Delta+u^\Delta+1 =\mathcal G(u,v).$$

Normalize the first scalar bilinear

Use Wick contractions to normalize $:{\mathcal O^2}:$ and determine its OPE coefficient in $\mathcal O\times\mathcal O$.

Solution

There are two contractions between the two normal-ordered bilinears, so their two-point coefficient is $2$. Hence $[\mathcal O\mathcal O]_{0,0}=:{\mathcal O^2}:/\sqrt2$. The OPE contains $:{\mathcal O^2}:$ with coefficient one, and therefore the coefficient of the unit-normalized primary is $\sqrt2$.

References

• Fitzpatrick, A. Liam, and Jared Kaplan. “AdS Field Theory from Conformal Field Theory.” Journal of High Energy Physics 2012, no. 10 (2012): 032. DOI. Open PDF
• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI. Open PDF