Simplifying large spin bootstrap in Mellin space

TL;DR

This work advances analytic conformal bootstrap by formulating and exploiting Mellin-space techniques to obtain all-order expressions for the large-spin double-trace sector. The authors derive closed-form, all-order expansions for anomalous dimensions and OPE coefficients in terms of generalized Bernoulli polynomials and Mack polynomials, establishing equivalence with crossing-symmetric Witten-diagram bootstrap at leading order in the anomalous dimension and clarifying the subleading polynomial-ambiguity constraints. They also uncover universal asymptotics in inverse conformal spin, showing that key ratios scale linearly with the loop order and hinting at analyticity-in-spin structures and resummation prospects. The results provide a clean, algebraic handle on the large-spin regime, with implications for epsilon-expansion, finite-spin corrections, and the systematic reconciliation of different bootstrap formulations.

Abstract

We set up the conventional conformal bootstrap equations in Mellin space and analyse the anomalous dimensions and OPE coefficients of large spin double trace operators. By decomposing the equations in terms of continuous Hahn polynomials, we derive explicit expressions as an asymptotic expansion in inverse conformal spin to any order, reproducing the contribution of any primary operator and its descendants in the crossed channel. The expressions are in terms of known mathematical functions and involve generalized Bernoulli (Norlund) polynomials and the Mack polynomials and enable us to derive certain universal properties. Comparing with the recently introduced reformulated equations in terms of crossing symmetric tree level exchange Witten diagrams, we show that to leading order in anomalous dimension but to all orders in inverse conformal spin, the equations are the same as in the conventional formulation. At the next order, the polynomial ambiguity in the Witten diagram basis is needed for the equivalence and we derive the necessary constraints for the same.

Figures (2)

• Figure 1: Plots of $\log {r_\gamma}$ vs $\log k$ for various twists and spins. The solid black line is $r_\gamma=k/\pi$. The dashed lines are for $d=3,\Delta_\phi=0.518$, the dotted lines are for $d=4,\Delta_\phi=1.28$ and the dot-dashed lines for $d=5,\Delta_\phi=1.88$.
• Figure 2: Plots of $\log {r_{OPE}}$ vs $\log k$ for various twists and spins. The solid black line is $r_{OPE}=k/\pi$. The dashed lines are for $d=3,\Delta_\phi=0.518$, the dotted lines are for $d=4,\Delta_\phi=1.28$ and the dot-dashed lines for $d=5,\Delta_\phi=1.88$. The OPE for the $\tau=1.41, d=3$ exhibits an interesting feature which needed us to go to a higher number of points to see the asymptotic behaviour.