Towards a Bootstrap approach to higher orders of epsilon expansion

TL;DR

This work develops analytic bootstrap methods to access higher-order ε-expansions for higher-spin operators. It blends large-spin analytic bootstrap with Mellin-space bootstrap to compute anomalous dimensions and OPE data at high orders for the Wilson-Fisher fixed point and for φ^3 theory in 6−ε, including systematic treatment of crossed-channel contributions. The results provide new predictions at $O(\epsilon^4)$–$O(\epsilon^5)$ and $O(\epsilon^2)$ in different contexts, offering cross-checks between approaches and highlighting technical challenges such as infinite operator sums and polynomial ambiguities in Witten-diagram bases. The findings advance analytic control of CFT data in perturbative regimes and suggest pathways to extend Mellin-bootstrap techniques to broader correlators and higher orders, with potential impact on understanding critical phenomena in diverse dimensions.

Abstract

We employ a hybrid approach in determining the anomalous dimension and OPE coefficient of higher spin operators in the Wilson-Fisher theory. First we do a large spin analysis for CFT data where we use results obtained from the usual and the Mellin Bootstrap and also from Feynman diagram literature. This gives new predictions at $O(ε^4)$ and $O(ε^5)$ for anomalous dimensions and OPE coefficients, and also provides a cross-check for the results from Mellin Bootstrap. These higher orders get contributions from all higher spin operators in the crossed channel. We also use the Bootstrap in Mellin space method for $φ^3$ in $d=6-ε$ CFT where we calculate general higher spin OPE data. We demonstrate a higher loop order calculation in this approach by summing over contributions from higher spin operators of the crossed channel in the same spirit as before.