Planar plane-wave matrix theory at the four loop order: Integrability without BMN scaling

TL;DR

This work presents a four-loop perturbative analysis of planar SU(N) plane-wave matrix theory in its SU(2) subsector, obtaining the planar dilatation operator via a large-scale symbolic computation and revealing integrability beyond three loops. It develops a perturbative asymptotic Bethe Ansatz to characterize the two-magnon S-matrix, finding a structure similar to, yet distinct from, Beisert-Dippel-Staudacher and Arutyunov-Frolov-Staudacher forms, including an exponential phase tied to higher charges. The key result is the explicit demonstration that BMN scaling breaks down at four loops due to contact interactions (wrapping effects), even though integrability persists and degeneracies of parity-paired states are observed. These findings illuminate the nuanced relationship between PWMT and N=4 SYM, highlight the role of wrapping in long-range spin chains, and showcase a computational framework (gemstone) capable of tackling high-order planar perturbation theory. The work suggests a generic long-range integrable spin-chain S-matrix with exponential factors and paves the way for further exploration of PWMT’s integrable structure and its connection to gauge/string dualities.

Abstract

We study SU(N) plane-wave matrix theory up to fourth perturbative order in its large N planar limit. The effective Hamiltonian in the closed su(2) subsector of the model is explicitly computed through a specially tailored computer program to perform large scale distributed symbolic algebra and generation of planar graphs. The number of graphs here was in the deep billions. The outcome of our computation establishes the four-loop integrability of the planar plane-wave matrix model. To elucidate the integrable structure we apply the recent technology of the perturbative asymptotic Bethe Ansatz to our model. The resulting S-matrix turns out to be structurally similar but nevertheless distinct to the so far considered long-range spin-chain S-matrices of Inozemtsev, Beisert-Dippel-Staudacher and Arutyunov-Frolov-Staudacher in the AdS/CFT context. In particular our result displays a breakdown of BMN scaling at the four-loop order. That is, while there exists an appropriate identification of the matrix theory mass parameter with the coupling constant of the N=4 superconformal Yang-Mills theory which yields an eigth order lattice derivative for well seperated impurities (naively implying BMN scaling) the detailed impurity contact interactions ruin this scaling property at the four-loop order. Moreover we study the issue of ``wrapping'' interactions, which show up for the first time at this loop-order through a Konishi descendant length four operator.

Figures (5)

• Figure 1: Forming all Graphs: One particular choice where the last two vertices to be added are of the same type.
• Figure 2: Fusing legs
• Figure 3: Calculations on terms without dynamic memory management
• Figure 4: On the value of lazy evaluation: Every node corresponds to a partially constructed graph and its associated (lazy) term. The root node is just the initial domain. Every link leads from a graph to a graph which has been extended by one more vertex. In (#1), no algebraic calculations are performed, as graph generation gets stuck with partial graphs that cannot be completed to final planar graphs. In (#2), a planar graph could be generated, and term algebra forces all the promises for calculations down to the root to be fulfilled (bold lines). However, the result is zero, coming from a zero at level two that is discovered in this step only. All the other graphs and partial graphs that would also derive from this node are then not even generated, as the 'known zero value' information is used in graph generation. In (#3), we get final nonzero contributions. Note furthermore that intermediary results down the path to the root are only calculated once!
• Figure 5: Accumulated calculation time vs. part of sequences for $ZZ$ at three-loop order