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Loops in AdS from Conformal Field Theory

Ofer Aharony, Luis F. Alday, Agnese Bissi, Eric Perlmutter

TL;DR

This paper develops a framework to compute AdS loop amplitudes from the conformal bootstrap by two complementary routes: a systematic subleading $1/N$ spin expansion solving crossing equations and a Mellin-space reconstruction that fixes loop amplitudes from leading-order data. It introduces a Casimir-based method to solve one-loop crossing, revealing a harmonic-polylogarithm basis that governs the finite-spin structure and enabling explicit results for $\phi^4$ in AdS and $\phi^3+\phi^4$ in AdS. The work confirms loop-level AdS results with bulk calculations, analyzes UV divergences and the role of local counterterms, and discusses extensions to more complex theories including $\mathcal{N}=4$ SYM. Together, these results illuminate the non-planar, loop-level structure of holographic CFTs and provide practical tools for constructing AdS loop amplitudes from CFT data.

Abstract

We propose and demonstrate a new use for conformal field theory (CFT) crossing equations in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to non-planar correlators in holographic CFTs. Loops in AdS are largely unexplored, mostly due to technical difficulties in direct calculations. We revisit this problem, and the dual $1/N$ expansion of CFTs, in two independent ways. The first is to show how to explicitly solve the crossing equations to the first subleading order in $1/N^2$, given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for finite values of the spin. Our second approach involves Mellin space. We show how the polar part of the four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order data. The anomalous dimensions computed with both methods agree. In the case of $φ^4$ theory in AdS, our crossing solution reproduces a previous computation of the one-loop bubble diagram. We can go further, deriving part of the four-point function in $φ^3+φ^4$ theory in AdS which had never been computed. In the process, we show how to analytically derive anomalous dimensions from Mellin amplitudes with an infinite series of poles, and discuss applications to more complicated cases such as the ${\cal N}=4$ super-Yang-Mills theory.

Loops in AdS from Conformal Field Theory

TL;DR

This paper develops a framework to compute AdS loop amplitudes from the conformal bootstrap by two complementary routes: a systematic subleading spin expansion solving crossing equations and a Mellin-space reconstruction that fixes loop amplitudes from leading-order data. It introduces a Casimir-based method to solve one-loop crossing, revealing a harmonic-polylogarithm basis that governs the finite-spin structure and enabling explicit results for in AdS and in AdS. The work confirms loop-level AdS results with bulk calculations, analyzes UV divergences and the role of local counterterms, and discusses extensions to more complex theories including SYM. Together, these results illuminate the non-planar, loop-level structure of holographic CFTs and provide practical tools for constructing AdS loop amplitudes from CFT data.

Abstract

We propose and demonstrate a new use for conformal field theory (CFT) crossing equations in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to non-planar correlators in holographic CFTs. Loops in AdS are largely unexplored, mostly due to technical difficulties in direct calculations. We revisit this problem, and the dual expansion of CFTs, in two independent ways. The first is to show how to explicitly solve the crossing equations to the first subleading order in , given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for finite values of the spin. Our second approach involves Mellin space. We show how the polar part of the four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order data. The anomalous dimensions computed with both methods agree. In the case of theory in AdS, our crossing solution reproduces a previous computation of the one-loop bubble diagram. We can go further, deriving part of the four-point function in theory in AdS which had never been computed. In the process, we show how to analytically derive anomalous dimensions from Mellin amplitudes with an infinite series of poles, and discuss applications to more complicated cases such as the super-Yang-Mills theory.

Paper Structure

This paper contains 43 sections, 198 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. Summary of results
  4. Crossing symmetry in the $1/N$ expansion
  5. Setup
  6. Implications from crossing at order $1/N^4$
  7. Truncated solutions at order $1/N^2$
  8. Solutions with infinite support at order $1/N^2$
  9. Loop amplitudes in $AdS$
  10. Mellin amplitudes
  11. Tree-level
  12. One-loop
  13. UV divergences and $f_{{\rm reg}}$
  14. Examples
  15. Large spin and the lightcone bootstrap
  16. ...and 28 more sections

Figures (3)

  • Figure 1: The schematic form of the loop expansion of AdS four-point amplitudes ${\cal G}$, shown only in a single channel for simplicity. The one-loop diagrams are holographically dual to the leading non-planar corrections to four-point correlation functions in holographic, large $N$ CFTs.
  • Figure 2: The one-loop bubble diagram in AdS $\phi^4$ theory. The corresponding Mellin amplitude is given in \ref{['m1loopads']}.
  • Figure 3: The one-loop triangle diagram in AdS $\phi^3+\phi^4$ theory. Together with the $\phi^4$ renormalization of $\phi^3$ tree-level diagrams, the corresponding Mellin amplitude for a $m^2=-4$ scalar in AdS$_5$ is given in \ref{['mlooptri2']}. The amplitude for a massless scalar in AdS$_3$ is given in \ref{['mlooptrid2']}.