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Quarter BPS Operators in N=4 SYM

Anton V. Ryzhov

TL;DR

The paper investigates quarter-BPS chiral primaries in four-dimensional N=4 SYM by analyzing gauge-invariant scalar composites in SU(4) representations [p,q,p]. It computes order $g^2$ two-point functions, diagonalizes the dilatation operator within each representation, and demonstrates that certain linear combinations (pure operators) have protected dimensions $\Delta=2p+q$ and unrenormalized norms. The work provides explicit constructions for low-dimension cases at arbitrary $N$ and a large-$N$ framework for general $[p,q,p]$, revealing that quarter-BPS primaries are mixtures of single-, double-, and higher-trace operators. These results establish a concrete, perturbative footing for the non-renormalization properties of quarter-BPS operators and set the stage for higher-point function studies in this sector.

Abstract

Chiral primary operators annihilated by a quarter of the supercharges are constructed in the four dimensional N=4 Super-Yang-Mills theory with gauge group SU(N). These quarter-BPS operators share many non-renormalization properties with the previously studied half-BPS operators. However, they are much more involved, which renders their construction nontrivial in the fully interacting theory. In this paper we calculate order g^2 two-point functions of local, polynomial, scalar composite operators within a given representation of the SU(4) R-symmetry group. By studying these two-point functions, we identify the eigenstates of the dilatation operator, which turn out to be complicated mixtures of single and multiple trace operators. Given the elaborate combinatorics of this problem, we concentrate on two special cases. First, we present explicit computations for quarter-BPS operators with scaling dimension Delta < 8. In this case, the discussion applies to arbitrary N of the gauge group. Second, we carry out a leading plus subleading large N analysis for the particular class of operators built out of double and single trace operators only. The large N construction addresses quarter-BPS operators of general dimension.

Quarter BPS Operators in N=4 SYM

TL;DR

The paper investigates quarter-BPS chiral primaries in four-dimensional N=4 SYM by analyzing gauge-invariant scalar composites in SU(4) representations [p,q,p]. It computes order two-point functions, diagonalizes the dilatation operator within each representation, and demonstrates that certain linear combinations (pure operators) have protected dimensions and unrenormalized norms. The work provides explicit constructions for low-dimension cases at arbitrary and a large- framework for general , revealing that quarter-BPS primaries are mixtures of single-, double-, and higher-trace operators. These results establish a concrete, perturbative footing for the non-renormalization properties of quarter-BPS operators and set the stage for higher-point function studies in this sector.

Abstract

Chiral primary operators annihilated by a quarter of the supercharges are constructed in the four dimensional N=4 Super-Yang-Mills theory with gauge group SU(N). These quarter-BPS operators share many non-renormalization properties with the previously studied half-BPS operators. However, they are much more involved, which renders their construction nontrivial in the fully interacting theory. In this paper we calculate order g^2 two-point functions of local, polynomial, scalar composite operators within a given representation of the SU(4) R-symmetry group. By studying these two-point functions, we identify the eigenstates of the dilatation operator, which turn out to be complicated mixtures of single and multiple trace operators. Given the elaborate combinatorics of this problem, we concentrate on two special cases. First, we present explicit computations for quarter-BPS operators with scaling dimension Delta < 8. In this case, the discussion applies to arbitrary N of the gauge group. Second, we carry out a leading plus subleading large N analysis for the particular class of operators built out of double and single trace operators only. The large N construction addresses quarter-BPS operators of general dimension.

Paper Structure

This paper contains 27 sections, 96 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. $SU(2,2|4)$ group theory
  3. Gauge invariant scalar composite operators
  4. Contributing diagrams
  5. The simplest cases
  6. Scalar composites with weight [2,0,2]
  7. Scalar composites with weight [2,1,2]
  8. A Gauge Invariance Argument
  9. Operators of dimension 6 and higher
  10. Scalar composites with weight [2,2,2]
  11. Scalar composites with weight [3,1,3]
  12. Summary of $\Delta \le 7$ results
  13. Large $N$ analysis
  14. Operators $\hbox{$\cal O$}_{[p,q,p]}$ and $\hbox{$\cal K$}_{[p,q,p]}$
  15. General correlators $\langle {\hbox{$\cal O$}}_{[p,q,p]} \bar{\hbox{$\cal O$}}_{[p,q,p]} \rangle$ to order $g^2$
  16. ...and 12 more sections

Figures (2)

  • Figure 1: Structures contributing to two-point functions at order $g^2$. Thick lines correspond to exchanges of the gauge boson and auxiliary fields $F_i$ or $D$ in the $\cal N$=1 formulation.
  • Figure 2: Diagrams contributing to two-point functions of scalars at order $g^2$.