Mooses, Topology and Higgs

TL;DR

Addresses the electroweak hierarchy problem by exploring weakly coupled little-Higgs theories realized in theory-space moose models. Shows low-energy content is fixed by the topology of theory space, encoded in the fundamental group $\pi_1$, enabling reverse engineering of spaces to yield desired Higgs sectors. Develops a practical procedure to extract the moduli space by gauge fixing and plaquette minimization, linking it to the topology of the space and to a tree-level potential $\text{Tr } U V U^{\dagger} V^{\dagger}$. Derives mild one-loop quadratic-divergence constraints: no link begins and ends on the same site and no plaquette contains the same link twice, yielding $m_{\text{LH}}^2 \sim \frac{g^4}{(4\pi)^4} \Lambda^2$ up to IR contributions; shows radiative corrections do not force a large hierarchy. Describes Yukawa coupling implementations using Wilson lines, introduces generalized plaquettes to lift unwanted adjoint states, and presents a minimal two-site model that yields two light Higgs doublets near $100\,\text{GeV}$ with TeV-scale triplets/singlets, plus TeV-scale vectors and fermions and potential collider signatures.

Abstract

New theories of electroweak symmetry breaking have recently been constructed that stabilize the weak scale and do not rely upon supersymmetry. In these theories the Higgs boson is a weakly coupled pseudo-Goldstone boson. In this note we study the class of theories that can be described by theory spaces and show that the fundamental group of theory space describes all the relevant classical physics in the low energy theory. The relationship between the low energy physics and the topological properties of theory space allow a systematic method for constructing theory spaces that give any desired low energy particle content and potential. This provides us with tools for analyzing and constructing new theories of electroweak symmetry breaking.

Figures (11)

• Figure 1: The geometry of theory space being built up from points, lines, and faces. These geometrical objects are identified as gauge groups, fields, potentials in the action.
• Figure 2: A deconstruction of a disk with two holes.
• Figure 3: Gauge fixing of the torus where crossed lines are gauged to the identity. The plaquettes are then minimized. Plaquette $W_{3,3}$ forces $U V U^{-1} V^{-1} = 1$.
• Figure 4: Gauge fixing of the torus where crossed lines are gauged to the identity. The plaquettes are then minimized. Since plaquette $W_{3,3}$ is absent, there is no relation and the moduli space is arbitrary $U$ and $V$ and there is no low energy potential for the little Higgs.
• Figure 5: The minimal theory space with three cycles, $X$, $Y$, $Z$ and the potential $V = \text{ Tr } X Y Z X^{-1} Y^{-1}Z^{-1}$. The arrows along the links indicate whether the fields are $X_i$ or $X_i^{-1}$.