On divergences in a four-derivative scalar field theory

We perform a detailed diagrammatic analysis of the renormalisation of a family of asymptotically free, shift-symmetric four-derivative scalar field theories introduced by Holdom in arXiv:2303.06723 and arXiv:2402.09223. We extend the renormalisation of the theory from one to three loops using both an $R^*$ method and an asymptotic expansion in momenta. We prove that the Euclidean correlators (or off-shell amplitudes) are IR finite, to all orders in perturbation theory, and derive a non-renormalisation theorem describing the all-order structure of the renormalisation constants. In particular, a purely cubic interaction is RG invariant and a perfect square Lagrangian density is preserved under renormalisation. The latter result is due to a Ward identity in a related $\textit{ gravitational}$ theory $-$ the conformally flat limit of quadratic gravity (CFQG). We show that the beta function for the perfect square theory maps exactly to that of an $O(2)$-symmetric, two-derivative, massless $φ^4$ theory at negative coupling. We verify this relationship explicitly up to three loops and thus determine the beta function and anomalous dimension for both the perfect square theory and CFQG to six loops.