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

We show how to construct a holographic effective theory for the leading-twist operators in the $O(2)$ model in the $4-d=\epsilon$ expansion up to $O({\epsilon }^2)$, 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 $O({\epsilon }^2)$ 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 $\phi$, neutral scalar $s\sim \phi {\phi }^{\ast }$, and a `ghost’ field $c$, as well as a single local bulk interaction $(\phi {\phi }^{\ast }{)}^2$. We analyze various aspects of the spectrum and discuss their interpretation in light of the bulk description.