A closed clockwork theory: $\mathbb{Z}_2$ parity and more

We develop a new class of clockwork theories with an augmented structure of the near-neighbour interactions along a one-dimensional closed chain. Such a topology leads to new and attractive features in addition to generating light states with hierarchical couplings via the usual clockwork mechanism. For one, there emerges a $\mathbb{Z}_2$ symmetry under the exchange of fields resulting in a physical spectrum consisting of states, respectively even and odd under the exchange parity with a two-fold degeneracy at each level. The lightest odd particle, being absolutely stable, could be envisaged as a potential dark matter candidate. The theory can also be obtained as a deconstruction of a five-dimensional theory embedded in a geometry generated by a linear dilaton theory on a $S^1/\mathbb{Z}_2$ orbifold with three equidistant 3-branes. Analogous to the discrete picture, the $\mathbb{Z}_2$ symmetry in the bulk theory necessitates the existence of a KK spectrum of even and odd states, with doubly degenerate modes at each KK level when subject to certain boundary conditions.