Three‐loop test of the dilatation operator and integrability in 𝒩 = 4 SYM

Abstract

We compute the three-loop anomalous dimension of the BMN operators with charges J = 0 (the Konishi multiplet) and J = 1 in the N = 4 super-Yang-Mills theory. We employ a method which effectively reduces the calculation to two loops. Instead of using the superconformal primary states, we consider the ratio of the two-point functions of suitable descendants of the corresponding multiplets. Our results unambiguously select the form of the N = 4 SYM dilatation operator which is compatible with BMN scaling. Thus, we provide evidence for BMN scaling at three loops.

Superspace calculation of the three-loop dilatation operator ofN=4supersymmetric Yang-Mills theory

“…The predictions have been tested by various field theory calculations. The three-loop anomalous dimension of the Konishi operator matches the conjectured eigenvalue from integrability [20,44]. At four loops and beyond, further field theory calculations tested the structure and various eigenvalues of the dilatation operator [22, 33-35, 41, 45, 46].…”

The AdS5× S5superstring in light-cone gauge and its Bethe equations

“…Here integrability secures the existence of a Bethe ansatz which enables one to reformulate the quantum spectral problem into the solution of a set of non-linear algebraic equations, the Bethe equations. These insights, at present firmly established up to the three-loop order in the 't Hooft coupling λ := g YM N 2 c [13,14] in certain closed subsectors of the full P SU(2, 2|4) symmetry group, led to the formulation of an exciting conjecture on the all-loop structure of these Bethe equations [15]. This conjecture has been by now extended to the full P SU (2, 2|4) case in the gauge theory [16] and is believed to hold in an asymptotic sense, where the classical scaling dimension of the operator in question determines the loop order to which a prediction is made by the Bethe equations.…”

“…To compare equations (8.16), (8.17), (8.19) and (8.20) with the finite-gap integral equations of [44,45] we should start with their unscaled form 14 2 − C dy ρ s (y) x − y = 2π(…”

Integrability and Maximally Helicity Violating Diagrams inN=4Supersymmetric Yang-Mills Theory

“…This has proved difficult for amplitudes, but addressing the calculation of just the UV-divergent part of the two-point correlation function may simplify this task enormously. At one loop the complete dilatation operator is known [17], while direct perturbative calculations at higher loops-without the assumption of integrability-have been performed only up to two [18][19][20], three [21][22][23], and four loops [24] in particular sectors. A simplified route to such a calculation would be greatly desirable, and would provide further verification of this crucial assumption.…”