Towards Large-Spin Effective Theory II: $O(2)$ model in $d=4-ε$

We show how to construct a holographic effective theory for the leading-twist operators in the $O(2)$ model in the $4-d=ε$ expansion up to $O(ε^2)$, based on the separation of short-distance and long-distance effects that arises as a function of spin $J$. We obtain the Hamiltonian of the theory and show that it correctly reproduces all the dimensions at $O(ε^2)$ of the leading twist operators for all values of the charge $Q$ and spin $J$. The holographic Hamiltonian is given by the bulk exchange of a charged scalar $ϕ$, neutral scalar $s \sim ϕϕ^*$, and a `ghost' field $c$, as well as a single local bulk interaction $(ϕϕ^*)^2$. We analyze various aspects of the spectrum and discuss their interpretation in light of the bulk description.