Non-planar form factors of generic local operators via on-shell unitarity and color-kinematics duality

Form factors, as quantities involving both local operators and asymptotic particle states, contain information of both the spectrum of operators and the on-shell amplitudes. So far the studies of form factors have been mostly focused on the large Nc planar limit, with a few exceptions of Sudakov form factors. In this paper, we discuss the systematical construction of full color dependent form factors with generic local operators. We study the color decomposition for form factors and discuss the general strategy of using on-shell unitarity cut method. As concrete applications, we compute the full two-loop non-planar minimal form factors for both half-BPS operators and non-BPS operators in the SU(2) sector in $$ \mathcal{N} $$ N = 4 SYM. Another important aspect is to investigate the color-kinematics (CK) duality for form factors of high-length operators. Explicit CK dual representation is found for the two-loop half-BPS minimal form factors with arbitrary number of external legs. The full-color two-loop form factor result provides an independent check of the infrared dipole formula for two-loop n-point amplitudes. By extracting the UV divergences, we also reproduce the known non-planar SU(2) dilatation operator at two loops. As for the finite remainder function, interestingly, the non-planar part is found to contain a new maximally transcendental part beyond the known planar result.

The integrable (hyper)eclectic spin chain

We refine the notion of eclectic spin chains introduced in [1] by including a maximal number of deformation parameters. These models are integrable, nearest-neighbor n-state spin chains with exceedingly simple non-hermitian Hamiltonians. They turn out to be non-diagonalizable in the multiparticle sector (n > 2), where their “spectrum” consists of an intricate collection of Jordan blocks of arbitrary size and multiplicity. We show how and why the quantum inverse scattering method, sought to be universally applicable to integrable nearest-neighbor spin chains, essentially fails to reproduce the details of this spectrum. We then provide, for n=3, detailed evidence by a variety of analytical and numerical techniques that the spectrum is not “random”, but instead shows surprisingly subtle and regular patterns that moreover exhibit universality for generic deformation parameters. We also introduce a new model, the hypereclectic spin chain, where all parameters are zero except for one. Despite the extreme simplicity of its Hamiltonian, it still seems to reproduce the above “generic” spectra as a subset of an even more intricate overall spectrum. Our models are inspired by parts of the one-loop dilatation operator of a strongly twisted, double-scaled deformation of $$ \mathcal{N} $$ N = 4 Super Yang-Mills Theory.

Nonrelativistic near-BPS corners of $$ \mathcal{N} $$ = 4 super-Yang-Mills with SU(1, 1) symmetry

We consider limits of $$ \mathcal{N} $$ N = 4 super Yang-Mills (SYM) theory that approach BPS bounds and for which an SU(1,1) structure is preserved. The resulting near-BPS theories become non-relativistic, with a U(1) symmetry emerging in the limit that implies the conservation of particle number. They are obtained by reducing $$ \mathcal{N} $$ N = 4 SYM on a three-sphere and subsequently integrating out fields that become non-dynamical as the bounds are approached. Upon quantization, and taking into account normal-ordering, they are consistent with taking the appropriate limits of the dilatation operator directly, thereby corresponding to Spin Matrix theories, found previously in the literature. In the particular case of the SU(1,1—1) near-BPS/Spin Matrix theory, we find a superfield formulation that applies to the full interacting theory. Moreover, for all the theories we find tantalizingly simple semi-local formulations as theories living on a circle. Finally, we find positive-definite expressions for the interactions in the classical limit for all the theories, which can be used to explore their strong coupling limits. This paper will have a companion paper in which we explore BPS bounds for which a SU(2,1) structure is preserved.

Scrambling in Yang-Mills

Acting on operators with a bare dimension ∆ ∼ N2 the dilatation operator of U(N) $$ \mathcal{N} $$ N = 4 super Yang-Mills theory defines a 2-local Hamiltonian acting on a graph. Degrees of freedom are associated with the vertices of the graph while edges correspond to terms in the Hamiltonian. The graph has p ∼ N vertices. Using this Hamiltonian, we study scrambling and equilibration in the large N Yang-Mills theory. We characterize the typical graph and thus the typical Hamiltonian. For the typical graph, the dynamics leads to scrambling in a time consistent with the fast scrambling conjecture. Further, the system exhibits a notion of equilibration with a relaxation time, at weak coupling, given by t ∼ $$ \frac{\rho }{\lambda } $$ ρ λ with λ the ’t Hooft coupling.

One-loop non-planar anomalous dimensions in super Yang-Mills theory

We consider non-planar one-loop anomalous dimensions in maximally supersymmetric Yang-Mills theory and its marginally deformed analogues. Using the basis of Bethe states, we compute matrix elements of the dilatation operator and find compact expressions in terms of off-shell scalar products and hexagon-like functions. We then use non-degenerate quantum-mechanical perturbation theory to compute the leading 1/N2 corrections to operator dimensions and as an example compute the large R-charge limit for two-excitation states through subleading order in the R-charge. Finally, we numerically study the distribution of level spacings for these theories and show that they transition from the Poisson distribution for integrable systems at infinite N to the GOE Wigner-Dyson distribution for quantum chaotic systems at finite N.

Rotating restricted Schur polynomials

Large [Formula: see text] but nonplanar limits of [Formula: see text] super-Yang–Mills theory can be described using restricted Schur polynomials. Previous investigations demonstrate that the action of the one loop dilatation operator on restricted Schur operators, with classical dimension of order [Formula: see text] and belonging to the [Formula: see text] sector, is largely determined by the [Formula: see text] [Formula: see text] symmetry algebra as well as structural features of perturbative field theory. Studies presented so far have used the form of [Formula: see text] symmetry generators when acting on small perturbations of half-BPS operators. In this paper as a first step towards going beyond small perturbations of the half-BPS operators, we explain how the exact action of symmetry generators on restricted Schur polynomials can be determined.