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On Operator Mixing in N=4 SYM

Massimo Bianchi, Burkhard Eden, Giancarlo Rossi, Yassen S. Stanev

TL;DR

This work resolves the mixing of scalar primary operators in the 20' representation of SU(4) in N=4 SYM by computing order g^2 anomalous dimensions and proving the absence of instanton contributions. Using orthogonalization of a basis of Δ0=4 operators and an OPE analysis of a four-point function, the authors extract the mixing matrix and demonstrate that, at large N, double-trace operators decouple and the remaining dimensions are generally irrational. A complementary second derivation confirms the results, with the anomalous dimensions linked to a cubic equation in O_i coefficients and, for N=2, to a simple ratio η=5/3. The results illuminate the structure of operator mixing beyond protected multiplets and outline connections to the Penrose pp-wave limit, hinting at how these mixing patterns may inform holographic descriptions and string-loop corrections.

Abstract

We resolve the mixing of the scalar operators of naive dimension 4 belonging to the representation 20' of the SU(4) R-symmetry in N=4 SYM. We compute the order g^2 corrections to their anomalous dimensions and show the absence of instantonic contributions thereof. Ratios of the resulting expressions are irrational numbers, even in the large N limit where, however, we observe the expected decoupling of double-trace operators from single-trace ones. We briefly comment on the generalizations of our results required in order to make contact with the double scaling limit of the theory conjectured to be holographically dual to type IIB superstring on a pp-wave.

On Operator Mixing in N=4 SYM

TL;DR

This work resolves the mixing of scalar primary operators in the 20' representation of SU(4) in N=4 SYM by computing order g^2 anomalous dimensions and proving the absence of instanton contributions. Using orthogonalization of a basis of Δ0=4 operators and an OPE analysis of a four-point function, the authors extract the mixing matrix and demonstrate that, at large N, double-trace operators decouple and the remaining dimensions are generally irrational. A complementary second derivation confirms the results, with the anomalous dimensions linked to a cubic equation in O_i coefficients and, for N=2, to a simple ratio η=5/3. The results illuminate the structure of operator mixing beyond protected multiplets and outline connections to the Penrose pp-wave limit, hinting at how these mixing patterns may inform holographic descriptions and string-loop corrections.

Abstract

We resolve the mixing of the scalar operators of naive dimension 4 belonging to the representation 20' of the SU(4) R-symmetry in N=4 SYM. We compute the order g^2 corrections to their anomalous dimensions and show the absence of instantonic contributions thereof. Ratios of the resulting expressions are irrational numbers, even in the large N limit where, however, we observe the expected decoupling of double-trace operators from single-trace ones. We briefly comment on the generalizations of our results required in order to make contact with the double scaling limit of the theory conjectured to be holographically dual to type IIB superstring on a pp-wave.

Paper Structure

This paper contains 10 sections, 110 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. Notation and conventions
  3. Comments on the UIR's of $SU(2,2|4)$
  4. Scalar operators in the ${\bf 20^{'}}$ representation
  5. Orthogonalization of the scalar operators
  6. The $N\rightarrow \infty$ limit
  7. The four point function $\langle {\cal C}^{11} {\cal C}_{11}^{\dagger} {\cal K}_1 {\cal K}_1 \rangle$ at order $g^4$
  8. A second derivation of the order $g^2$ anomalous dimension of the ${\bf 20^{'}}$ operators
  9. Relation with the Penrose double scaling limit
  10. Conclusions

Figures (1)

  • Figure 1: The ratios $\gamma_1^{{\cal O}_i}/\gamma_1^{\cal K}$, $i=1,2,3$, as functions of $N$. $\gamma_1^{\cal K}$ is the order $g^2$ coefficient of the Konishi multiplet anomalous dimension, displayed in eq. (\ref{['gammak11']}).