Double-Twist Families and Anomalous Dimensions

Large-spin double-twist operators are not generally at their generalized-free dimensions. Crossed-channel primaries produce logarithms whose direct-channel interpretation is an anomalous dimension, while nonlogarithmic terms correct the OPE coefficients. The leading power is universal; its coefficient is meaningful only after the block and two-point normalizations are fixed.

Required background. The lightcone OPE and large-spin expansion establishes the double-twist families and the two-scale limit used here.

Helpful background. Current and stress-tensor CFT data fixes the Ward-identity conversion between a stress-tensor OPE coefficient and $C_T$.

Logarithms measure dimension shifts

For identical real scalars in a unitary CFT with $d>2$, write the leading family as

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

A block contribution contains a factor such as

$$v^{\frac12(2\Delta_\phi+\gamma_{0,\ell})} =v^{\Delta_\phi} \left[1+\frac12\gamma_{0,\ell}\log v +O(\gamma_{0,\ell}^2)\right].$$

The coefficient of $\log v$ in the crossed-channel matching therefore determines $\gamma_{0,\ell}$. The terms without $\log v$ determine the correction to the squared OPE coefficient, together with derivatives of the block with respect to twist. This separation is clean only after degeneracies have been resolved; otherwise the correlator measures weighted sums of an anomalous-dimension matrix.

Use conformal spin

$$J^2 =\left(\ell+\Delta_\phi+\frac12\gamma_{0,\ell}\right) \left(\ell+\Delta_\phi-1+\frac12\gamma_{0,\ell}\right).$$

Replacing $J$ by $\ell$ is harmless for the leading power but changes subleading coefficients.

One crossed-channel primary

Let $\mathcal O_m$ have twist $\tau_m$, spin $\ell_m$, and coefficient

$$P_m=\lambda_{\phi\phi\mathcal O_m}^2$$

when $\langle\mathcal O_m\mathcal O_m\rangle=1$ and the scalar blocks have the Fitzpatrick–Kaplan–Poland–Simmons-Duffin normalization. At fixed $n=0$ and large even spin,

$$\begin{aligned} \gamma_{0,\ell}={}&-P_m \frac{2\Gamma(\Delta_\phi)^2\Gamma(\tau_m+2\ell_m)} {\Gamma\!\left(\Delta_\phi-\frac{\tau_m}{2}\right)^2 \Gamma\!\left(\frac{\tau_m}{2}+\ell_m\right)^2}\\ &\times\frac{1}{J^{\tau_m}}+\cdots . \end{aligned}$$

This is the leading isolated contribution of $\mathcal O_m$. If several operators share the minimal twist, their convention-matched contributions must be summed. If $\Delta_\phi-\tau_m/2$ reaches a nonpositive integer, the apparent zero or singular limit must be taken together with other operators and mixing; substituting blindly into the isolated formula can be wrong.

At the same leading order, the relative OPE-coefficient correction in the large-$\ell$ convention of the original matching calculation is

$$\begin{aligned} \frac{\delta P_{0,\ell}}{P_{0,\ell}^{\mathrm{GFF}}} &=\frac{c_0}{\ell^{\tau_m}}+\cdots,\\ c_0&= \left[ \psi\!\left(\frac{\tau_m}{2}+\ell_m\right) +\gamma_E+\log2 \right]\gamma_0, \end{aligned}$$

where $\gamma_{0,\ell}=\gamma_0/\ell^{\tau_m}+\cdots$ in that convention. These formulas are obtained by matching the logarithmic and constant terms of the crossed-channel block Fitzpatrick et al. 2013, §2.2, eqs. (38)–(41), pp. 11–12.

Stress-tensor exchange

For the stress tensor,

$$\tau_T=d-2, \qquad \ell_T=2.$$

The leading contribution is therefore

$$\begin{aligned} \gamma_{0,\ell}^{(T)}={}&-P_T \frac{2\Gamma(\Delta_\phi)^2\Gamma(d+2)} {\Gamma\!\left(\Delta_\phi-\frac{d-2}{2}\right)^2 \Gamma\!\left(\frac{d+2}{2}\right)^2}\\ &\times\frac{1}{J^{d-2}}+\cdots . \end{aligned}$$

The Ward identity fixes $P_T$ in terms of $\Delta_\phi$ and the chosen $C_T$ normalization. Because conventions for $C_T$ differ by geometric factors, the safe invariant check is to convert both sources to the same normalized two- and three-point functions before using the displayed expression. For positive $P_T$ and a regular isolated contribution, the sign is negative. This agrees with the large-spin convexity analysis, which relates the approach to the limiting twist to the lowest nontrivial crossed-channel twist Komargodski and Zhiboedov 2013, §3 and appendix B.3.

Free-scalar limit. If $\Delta_\phi=(d-2)/2$, the denominator contains $\Gamma(0)$. The isolated stress-tensor coefficient then vanishes in this formula, while the free theory also has an infinite tower of conserved higher-spin currents. The full answer is obtained only after treating the enhanced higher-spin structure; the isolated-exchange approximation has reached a singular corner of its assumptions.

Mixing and finite-spin continuation

At large spin, families can be distinguished by their limiting twist and global quantum numbers. At finite spin, several effects intervene:

• primaries with the same spin and internal representation mix;
• descendants of lower-twist families contribute at subleading powers;
• a second crossed-channel operator can contribute at the same order in $1/J$;
• terms exponentially small in $J$ are invisible to a formal power series;
• analytic continuation from even integer spin can acquire low-spin additions.

For a degenerate family, let $\Gamma_{ab}(J)$ be the anomalous-dimension matrix in an orthonormal generalized-free basis. A single four-point function generally measures combinations

$$\sum_{a,b}\lambda_{12,a}\,\Gamma_{ab}(J)\,\lambda_{34,b},$$

not every matrix element. Mixed correlators or other independent observables are needed to diagonalize the problem.

The Lorentzian inversion formula improves finite-$J$ control because it yields a coefficient function analytic in spin in a stated half-plane, with a convergent large-$J$ expansion under its Regge assumptions Caron-Huot 2017, §§3.4–4.3. It still does not license evaluation below its spin threshold without the missing arc or subtraction data.

Solvable checks

Generalized free field. Only the crossed identity is needed to reproduce the exact double-twist dimensions, so $\gamma_{n,\ell}=0$. A calculation that assigns a nonzero anomalous dimension to identity exchange has mixed the mean-field baseline with the perturbation.

Single scalar exchange. Setting $\ell_m=0$ gives

$$\begin{aligned} \gamma_{0,\ell}^{(m)}={}&-P_m \frac{2\Gamma(\Delta_\phi)^2\Gamma(\tau_m)} {\Gamma\!\left(\Delta_\phi-\frac{\tau_m}{2}\right)^2 \Gamma\!\left(\frac{\tau_m}{2}\right)^2}\\ &\times J^{-\tau_m}+\cdots . \end{aligned}$$

The dimensions and sign can be checked directly: Gamma functions are dimensionless, the only large-spin scale is $J$, and a positive $P_m$ gives the expected negative leading correction in the regular unitary case.

Common pitfalls

Dropping the block normalization. $P_m$ is not a universal number independent of the two-point and block conventions. Convert the three-point coefficient before applying the formula.

Treating $n$ as uniformly small. The displayed result is at fixed $n$ as $J\to\infty$. A simultaneous large-$n$, large-$J$ limit is a different expansion.

Extrapolating through a singular Gamma factor. A zero can signal enhanced symmetry or cancellations, not the absence of all crossed-channel effects.

Inferring a unique operator from one correlator. Degenerate trajectories require a matrix treatment and enough independent correlators.

Exercises

Insert $\tau_T=d-2$ and $\ell_T=2$ into the single-exchange formula and show that the power of conformal spin is fixed by the conserved stress tensor’s twist.

Solution

The power is $J^{-\tau_T}=J^{-(d-2)}$. The Gamma-function arguments become $\tau_T+2\ell_T=d+2$ and $\tau_T/2+\ell_T=(d+2)/2$, which gives the stress-tensor expression displayed above.

References

• Caron-Huot, Simon. “Analyticity in Spin in Conformal Theories.” Journal of High Energy Physics 2017, 078 (2017). doi:10.1007/JHEP09(2017)078.
• 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.