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Tailoring Non-Compact Spin Chains

Pedro Vieira, Tianheng Wang

Abstract

We study three-point correlation functions of local operators in planar $\mathcal{N}=4$ SYM at weak coupling using integrability. We consider correlation functions involving two scalar BPS operators and an operator with spin, in the so called SL(2) sector. At tree level we derive the corresponding structure constant for any such operator. We also conjecture its one loop correction. To check our proposals we analyze the conformal partial wave decomposition of known four-point correlation functions of BPS operators. In perturbation theory, we extract from this decomposition sums of structure constants involving all primaries of a given spin and twist. On the other hand, in our integrable setup these sum rules are computed by summing over all solutions to the Bethe equations. A perfect match is found between the two approaches.

Tailoring Non-Compact Spin Chains

Abstract

We study three-point correlation functions of local operators in planar SYM at weak coupling using integrability. We consider correlation functions involving two scalar BPS operators and an operator with spin, in the so called SL(2) sector. At tree level we derive the corresponding structure constant for any such operator. We also conjecture its one loop correction. To check our proposals we analyze the conformal partial wave decomposition of known four-point correlation functions of BPS operators. In perturbation theory, we extract from this decomposition sums of structure constants involving all primaries of a given spin and twist. On the other hand, in our integrable setup these sum rules are computed by summing over all solutions to the Bethe equations. A perfect match is found between the two approaches.

Paper Structure

This paper contains 19 sections, 75 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction and Main Result
  2. Conjecture versus Data
  3. Twist 2 operators
  4. Higher twist operators
  5. Sum Rules and the OPE Decomposition
  6. Derivation (tree level) and educated guess (1 loop)
  7. The (ultra-local) wavefunctions for the operator $\mathcal{O}_S$
  8. Tree-level derivation
  9. One-loop conjecture
  10. Further Comments and Speculative Remarks
  11. More details on the twist $4$ analysis
  12. Explicit expressions for the sums $\mathcal{P}_S^{(n,m)}$
  13. Other examples: Twist $3$, Twist $6$ and a special Twist $4$
  14. Sums from Integrability and Comparison with One Loop Correlators
  15. Two Loop Sums from 4pt Correlation Functions
  16. ...and 4 more sections

Figures (2)

  • Figure 1: In this paper we consider a correlation function of two $SU(2)$ BPS operators and one non-protected $SL(2)$ primary operator of spin $S$, twist $L$ and dimension $\Delta$. The BPS operators are given by the sum over all the positions of inserting some complex scalars $X$ or $\bar{X}$ in a sea of complex scalars $\bar{Z}$, see (\ref{['BPSop']}). The non-BPS spin-$S$ operator is more interesting and its form is governed by a non-trivial wave function, see (\ref{['generalForm']}).
  • Figure 2: In tailoringI and tailoringIV the structure constant involving three $SU(2)$ operators as depicted in this figure was computed at tree level and at one loop respectively.