Deconstructing Six-Dimensional Gauge Theories with Strongly Coupled Moose Meshes
7/12/2002
12 citations. Extended the deconstruction framework from 5D to 6D using two-dimensional moose meshes.
The Problem
Arkani-Hamed, Cohen, and Georgi had shown that five-dimensional gauge theories could be “deconstructed” — reproduced as the low-energy limit of purely four-dimensional theories arranged in a chain (a moose or quiver diagram). The chain of gauge groups with link fields between them, in the limit of many sites, becomes indistinguishable from a continuous extra dimension. The Minimal Moose and the broader Little Higgs program exploited this idea with short chains to build realistic models of electroweak symmetry breaking. But the real world might have more than one extra dimension. Six-dimensional gauge theories were particularly interesting because they naturally produce self-couplings among the higher-dimensional components of gauge fields, and these self-couplings are exactly what generate the Higgs quartic coupling in Little Higgs models. The question was whether deconstruction could be extended from a one-dimensional chain to a two-dimensional mesh, and whether the resulting 4D theory would correctly reproduce 6D physics.
The Key Idea
The paper constructs two-dimensional moose meshes with alternating weak and strong gauge groups. The strong groups confine, producing the link fields dynamically rather than inserting them by hand. A new ingredient appears in 6D that is absent in 5D: the commutator of gauge field components in the two extra dimensions (the F562 term) generates a potential for the pseudo-Goldstone bosons in the deconstructed picture. The paper shows that finite one-loop corrections in the weakly coupled 4D theory generate this potential with the right structure and size, matching what the continuum 6D theory predicts. This was the technical foundation for understanding why the Higgs quartic coupling in Little Higgs models, which arises from exactly this F562 structure, is calculable and radiatively stable. The construction also provided the mathematical framework that the Mooses, Topology and Higgs paper generalized: understanding which topologies of moose lattices produce which low-energy spectra.
Impact
The 6D deconstruction provided the conceptual origin of the Higgs quartic coupling in the Minimal Moose and related models: the quartic arises from the gauge self-coupling in the extra dimensions, which deconstruction turns into a calculable one-loop potential. The connection between 6D gauge theories and 4D Little Higgs models, running through this paper, is also what underlies the observation noted in the Minimal Moose recollections that many later researchers built models using Randall-Sundrum extra-dimensional constructs without recognizing that these were nearly isomorphic to the Little Higgs models via deconstruction. The Higher Dimensional SUSY in 4D Superspace paper provided the supersymmetric version of the same dictionary.
Recollections
Nima Arkani-Hamed asked Thomas Gregoire and me to look at this. It was early in the Little Higgs era when we were still thinking very extra-dimensionally about the physics and were looking at UV-completing extra dimensions into ordinary 4D theories. Six-dimensional gauge theories have bad UV behavior in quantum field theory, so if we wanted to solve the “big hierarchy problem,” the one that extends up to the GUT or Planck scale, having a UV completion would be important. Understanding the origin of the plaquette operators also required working from a UV theory. So we set out to extend the methods that Arkani-Hamed, Cohen, and Georgi had developed for deconstructing dimensions to the 6D case.
Ultimately, this ended up being a lot of work without a lot of impact. I don’t think we ended up with a superconvincing story. It was still influential for me personally in terms of learning that the “if you write they will come” mentality doesn’t actually work. Not every technically correct paper finds an audience, and effort invested doesn’t correlate with citations received.