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Four-point correlators of BPS operators in N=4 SYM at order g^4

G. Arutyunov, S. Penati, A. Santambrogio, E. Sokatchev

TL;DR

This work investigates large-$N$ degeneracy in the four-point functions of 1/2-BPS operators in perturbative ${\cal N}=4$ SYM, showing that one-loop degeneracy among conformal invariants is lifted at two loops for weights $k=3$ and $k=4$, but persists for $k>4$.Using an ${\cal N}=2$ harmonic-superspace formulation and the HM projection, the authors perform a controlled two-loop calculation of the four-point amplitude, identifying the conformal invariant functions that survive in the large-$N$ limit and expressing them in terms of scalar box integrals. An explicit OPE-based analysis complements the diagrammatic result: for weight $k=3$ the two-loop amplitude is completely fixed by known one-loop anomalous dimensions, while weight $k=4$ receives additional constraints from twist-2 and twist-4 primaries; in both cases the OPE data agree with the perturbative results, supporting the AdS/CFT expectations that perturbative amplitudes capture the maximal set of conformal invariants available at a given order. Overall, the study demonstrates that perturbative corrections progressively lift degeneracy as higher orders are included, consistent with dual gravity expectations and offering a precise map between large-$N$ CFT data and AdS/CFT predictions.

Abstract

We study the large N degeneracy in the structure of the four-point amplitudes of 1/2-BPS operators of arbitrary weight k in perturbative N=4 SYM theory. At one loop (order g^2) this degeneracy manifests itself in a smaller number of independent conformal invariant functions describing the amplitude, compared to AdS_5 supergravity results. To study this phenomenon at the two-loop level (order g^4) we consider a particular N=2 hypermultiplet projection of the general N=4 amplitude. Using the formalism of N=2 harmonic superspace we then explicitly compute this four-point correlator at two loops and identify the corresponding conformal invariant functions. In the cases of 1/2-BPS operators of weight k=3 and k=4 the one-loop large N degeneracy is lifted by the two-loop corrections. However, for weight k > 4 the degeneracy is still there at the two-loop level. This behavior suggests that for a given weight k the degeneracy will be removed if perturbative corrections of sufficiently high order are taken into account. These results are in accord with the AdS/CFT duality conjecture.

Four-point correlators of BPS operators in N=4 SYM at order g^4

TL;DR

This work investigates large-$N$ degeneracy in the four-point functions of 1/2-BPS operators in perturbative ${\cal N}=4$ SYM, showing that one-loop degeneracy among conformal invariants is lifted at two loops for weights $k=3$ and $k=4$, but persists for $k>4$.Using an ${\cal N}=2$ harmonic-superspace formulation and the HM projection, the authors perform a controlled two-loop calculation of the four-point amplitude, identifying the conformal invariant functions that survive in the large-$N$ limit and expressing them in terms of scalar box integrals. An explicit OPE-based analysis complements the diagrammatic result: for weight $k=3$ the two-loop amplitude is completely fixed by known one-loop anomalous dimensions, while weight $k=4$ receives additional constraints from twist-2 and twist-4 primaries; in both cases the OPE data agree with the perturbative results, supporting the AdS/CFT expectations that perturbative amplitudes capture the maximal set of conformal invariants available at a given order. Overall, the study demonstrates that perturbative corrections progressively lift degeneracy as higher orders are included, consistent with dual gravity expectations and offering a precise map between large-$N$ CFT data and AdS/CFT predictions.

Abstract

We study the large N degeneracy in the structure of the four-point amplitudes of 1/2-BPS operators of arbitrary weight k in perturbative N=4 SYM theory. At one loop (order g^2) this degeneracy manifests itself in a smaller number of independent conformal invariant functions describing the amplitude, compared to AdS_5 supergravity results. To study this phenomenon at the two-loop level (order g^4) we consider a particular N=2 hypermultiplet projection of the general N=4 amplitude. Using the formalism of N=2 harmonic superspace we then explicitly compute this four-point correlator at two loops and identify the corresponding conformal invariant functions. In the cases of 1/2-BPS operators of weight k=3 and k=4 the one-loop large N degeneracy is lifted by the two-loop corrections. However, for weight k > 4 the degeneracy is still there at the two-loop level. This behavior suggests that for a given weight k the degeneracy will be removed if perturbative corrections of sufficiently high order are taken into account. These results are in accord with the AdS/CFT duality conjecture.

Paper Structure

This paper contains 12 sections, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Generalities and main result
  3. Mixing and large $N$ degeneracy
  4. The ${\cal N}=2$ insertion procedure and Feynman rules
  5. Two-loop four-point amplitudes
  6. Two-loop four-point amplitudes from the OPE
  7. Weight 3
  8. Weight 4
  9. Harmonic analyticity constraints
  10. Sample calculations: $k=3$ and $k=4$
  11. The case $k=3$
  12. The case $k=4$

Figures (2)

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