The Lightcone OPE and Large-Spin Expansion

Crossing symmetry has a universal consequence in a unitary CFT with a scalar primary: a sufficiently singular crossed-channel contribution cannot be reproduced by finitely many direct-channel operators. It requires families whose spin becomes arbitrarily large and whose twists accumulate near sums of external twists. The conclusion is asymptotic; it does not by itself determine the spectrum at any fixed spin.

Required background. Lorentzian correlators and causal orderings fix the sheet and the two-scale null limit. Crossing equations and positivity fix the identical-scalar crossing convention.

Helpful background. Asymptotic scales and uniform remainders explain why a large-spin series need not be accurate at fixed spin.

Twist and the two-scale lightcone limit

Let $\phi$ be an identical real scalar of dimension $\Delta_\phi$ in a unitary CFT with $d>2$. Normalize

$$\langle\phi_1\phi_2\phi_3\phi_4\rangle =\frac{\mathcal G(u,v)}{(x_{12}^2x_{34}^2)^{\Delta_\phi}}, \qquad \mathcal G(u,v)=\left(\frac{u}{v}\right)^{\Delta_\phi}\mathcal G(v,u).$$

An exchanged primary of dimension $\Delta$ and spin $\ell$ has twist

$$\tau=\Delta-\ell.$$

In a lightcone OPE, powers are organized primarily by twist rather than dimension. The analytic-bootstrap regime is a hierarchical Lorentzian limit

$$0<z\ll 1-\bar z\ll1, \qquad u\simeq z, \qquad v\simeq1-\bar z,$$

reached on a declared sheet. Saying only $u,v\to0$ is insufficient: reversing the hierarchy changes which sum may be approximated and which channel singularity is dominant.

In the $t$ channel, the identity produces the leading singular behavior. In the crossed $s$-channel representation, every fixed-spin block is too soft to reproduce it as $v\to0$. The required enhancement comes from summing an unbounded range of spins, with the dominant scale roughly

$$\ell\sqrt z=O(1).$$

Thus the large-spin limit and the lightcone limit are linked. Taking $z\to0$ term by term before performing the spin sum destroys the effect one is trying to derive.

Double-twist accumulation

Crossing forces sequences customarily denoted $[\phi\phi]_{n,\ell}$ with

$$\tau_{n,\ell} =2\Delta_\phi+2n+\gamma_{n,\ell}, \qquad \lim_{\ell\to\infty}\gamma_{n,\ell}=0, \qquad n=0,1,2,\ldots .$$

The existence of twists approaching $2\Delta_\phi+2n$ follows nonperturbatively under the stated unitary-CFT assumptions; it is not an assumption of a weakly coupled Lagrangian Fitzpatrick et al. 2013, §§2.1–2.2, pp. 8–13 and Komargodski and Zhiboedov 2013, §§2–3, pp. 8–24.

The word “double-twist” describes the limiting quantum numbers. At finite spin the operator can mix with every primary of the same exact quantum numbers. In a theory with degeneracy, $\gamma_{n,\ell}$ is a matrix before one chooses an orthonormal basis. In a nonunitary theory, the positivity steps used in the standard existence argument need replacement.

It is often better to expand in conformal spin

$$J^2 =\left(\ell+\frac{\tau}{2}\right) \left(\ell+\frac{\tau}{2}-1\right),$$

rather than in $\ell$. The quadratic Casimir naturally produces $J$, and reciprocity properties frequently make the expansion an even series in $1/J$ after the leading exchanged twist is factored out Alday, Bissi, and Łukowski 2015, §§2–3.

How crossed-channel data enter

Suppose the crossed channel contains a primary $\mathcal O_m$ of twist $\tau_m$, spin $\ell_m$, and squared OPE coefficient $P_m$ in the declared block normalization. Its small-$v$ block contains powers and logarithms that must be matched by shifts in direct-channel dimensions and OPE coefficients. For the leading family,

$$\gamma_{0,\ell}=\frac{\gamma_0}{J^{\tau_m}} +O\!\left(J^{-\tau_m-\delta}\right),$$

where $\delta>0$ depends on the next available correction. The coefficient and the corresponding OPE correction are developed on Double-Twist Families and Anomalous Dimensions. The important structural point here is that the power $J^{-\tau_m}$ is fixed by the exchanged twist. Distinct control parameters—large spin, a small coupling, $1/N$, or a large spectral gap—must not be identified with one another.

The identity contribution determines the leading mean-field asymptotics of the OPE coefficients. Nonidentity exchanges perturb those data. Descendants and neighboring twist families enter at subleading order, and an expansion that is uniform in $n$ need not follow from one proven at fixed $n$.

Exact check: generalized free field

For generalized free field,

$$\mathcal G_{\mathrm{GFF}}(u,v) =1+u^{\Delta_\phi} +\left(\frac{u}{v}\right)^{\Delta_\phi}.$$

The last term is the crossed contraction. Its $s$-channel conformal-block decomposition contains even-spin primaries with

$$\Delta_{n,\ell}=2\Delta_\phi+2n+\ell, \qquad \gamma_{n,\ell}=0.$$

This is the limiting spectrum predicted by the lightcone argument, realized exactly at every allowed spin in this solvable theory. It also shows what the argument does not say: crossing permits the anomalous dimensions to vanish identically. Their nonzero values are determined by additional crossed-channel operators, not by the identity alone.

To see the nonuniformity explicitly, approximate the large-$\ell$ collinear block by a Bessel kernel. The spin sum is dominated by fixed $\ell^2 z$; the approximation fails for $\ell\gg z^{-1/2}$ term by term even though the properly summed asymptotics remain controlled. Fitzpatrick et al. state this limitation directly in their matching calculation Fitzpatrick et al. 2013, §2.2, eqs. (37)–(41), pp. 11–12.

What is universal, and what is not

The lightcone analysis supports the following conclusion:

Under the declared unitarity, OPE, ordering, and crossed-channel assumptions, there are large-spin families whose twists approach the indicated double-twist values, and crossed-channel operators determine an asymptotic expansion of their averaged or resolved CFT data.

It does not, without further work, establish:

• convergence of the $1/J$ expansion;
• accuracy at spin $0$, $1$, or $2$;
• absence of exponentially small or otherwise nonperturbative terms in $J$;
• resolution of degenerate mixing from a single correlator;
• positivity in a nonunitary or logarithmic theory;
• existence of a local bulk dual.

The Lorentzian inversion formula strengthens the analysis by packaging the data into a function analytic in spin in a controlled half-plane. Even there, low-spin contributions can require separate terms.

Common pitfalls

Taking limits in the wrong order. The identity singularity is reproduced by a collective large-spin sum. Keep the scaling variable $\ell\sqrt z$ before interchanging the limit and sum.

Calling an asymptotic family an exact operator formula. The label $[\phi\phi]_{n,\ell}$ identifies a trajectory at large spin. At finite spin, mixing and level crossings can obstruct a unique continuation.

Using $\ell$ where the Casimir fixes $J$. This can create spurious odd powers and shift subleading coefficients. State which variable organizes the expansion.

Forgetting the sheet. The same algebraic limit of $z,\bar z$ on another sheet can be a Regge rather than a lightcone limit.

Exercises

Show that the generalized-free spectrum has the required double-twist accumulation points.

Solution

For $[\phi\phi]_{n,\ell}$, subtract the spin from the exact generalized-free dimension:

$$\tau_{n,\ell} =\Delta_{n,\ell}-\ell =2\Delta_\phi+2n.$$

It is independent of $\ell$, so it equals—and therefore trivially approaches—the crossing-predicted accumulation value.

References

• Alday, Luis F., Agnese Bissi, and Tomasz Łukowski. “Large Spin Systematics in CFT.” Journal of High Energy Physics 2015, 101 (2015). doi:10.1007/JHEP11(2015)101.
• Fitzpatrick, A. Liam, Jared Kaplan, David Poland, and David Simmons-Duffin. “The Analytic Bootstrap and AdS Superhorizon Locality.” Journal of High Energy Physics 2013, 004 (2013). doi:10.1007/JHEP12(2013)004.
• Komargodski, Zohar, and Alexander Zhiboedov. “Convexity and Liberation at Large Spin.” Journal of High Energy Physics 2013, 140 (2013). doi:10.1007/JHEP11(2013)140.