Towards Large-Spin Effective Theory I: Three-Particle States in AdS $\phi^4$ Theory

We describe how to construct an effective Hamiltonian for leading twist states in $d\ge 3$ CFTs based on the separation of scales that emerges at large spin $J$ between the AdS radius $\ell_{\rm AdS}$ and the characteristic distance $\sim \ell_{\rm AdS} \log J$ between particles rotating in AdS with angular momentum $J$. As a controlled example, we work specifically with the toy model of a bulk complex scalar field $\phi$ with a $\lambda |\phi|^4$ coupling in AdS, up to $O(\lambda^2)$. For a given choice of twist cutoff $\Lambda_\tau$ in the effective theory, interactions are separated into long-distance nonlocal potential terms, arising from $t$-channel exchange of states with twist $\le \Lambda_\tau$, and short-distance local terms fixed by matching to low spin CFT data. At $O(\lambda^2)$, the effective Hamiltonian for the toy model has two-body nonlocal potential terms from one-loop bulk diagrams as well as three-body nonlocal potential terms from tree-level exchange of $\phi$. We describe in detail how these contributions are evaluated and how they are related to the CFT data entering in the large spin expansion. We discuss how to apply the construction of such effective Hamiltonians for models which do not have a large central charge or a sparse spectrum and are not typically considered holographic.