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Abstract Scalar field theories with quartic interactions are of central interest in the study of second-order phase transitions. For three-dimensional theories, numerous studies make use of the fixed-dimensional perturbative computation of [B. Nickel, D. Meiron, and G. Baker Jr, Compilation of 2-pt and 4-pt graphs for continuous spin model, University of Guelph report (1977)], unfortunately left unpublished. We independently verify the results of Nickel et al., and we extend the computation to the eighth order in the coupling constant. The results of our calculations, together with the tools developed, are made available in Phi4tools, a user-friendly package that allows displaying the information about the individual Feynman diagrams, including the numerical values for the diagrams for zero, two, and four-point functions. We also provide the perturbative series up to order eight for the renormalization-group functions for the O(N) and cubic anisotropic models.

Abstract In theories with renormalons the perturbative series is factorially divergent even after restricting to a given order in 1/N, making the 1/N expansion a natural testing ground for the theory of resurgence. We study in detail the interplay between resurgent properties and the 1/N expansion in various integrable field theories with renormalons. We focus on the free energy in the presence of a chemical potential coupled to a conserved charge, which can be computed exactly with the thermodynamic Bethe ansatz (TBA). In some examples, like the first 1/N correction to the free energy in the non-linear sigma model, the terms in the 1/N expansion can be fully decoded in terms of a resurgent trans-series in the coupling constant. In the principal chiral field we find a new, explicit solution for the large N free energy which can be written as the median resummation of a trans-series with infinitely many, analytically computable IR renormalon corrections. However, in other examples, like the Gross-Neveu model, each term in the 1/N expansion includes non-perturbative corrections which can not be predicted by a resurgent analysis of the corresponding perturbative series. We also study the properties of the series in 1/N. In the Gross-Neveu model, where this is convergent, we analytically continue the series beyond its radius of convergence and show how the continuation matches with known dualities with sine-Gordon theories.

Abstract In the classically unbroken phase, 3d O(N) symmetric ϕ 4 vector models admit two equivalent descriptions connected by a strong-weak duality closely related to the one found by Chang and Magruder long ago. We determine the exact analytic renormalization dependence of the critical couplings in the weak and strong branches as a function of the renormalization scheme (parametrized by κ) and for any N. It is shown that for κ = κ ∗ the two fixed points merge and then, for κ < κ ∗, they move into the complex plane in complex conjugate pairs, making the phase transition no longer visible from the classically unbroken phase. Similar considerations apply in 2d for the N = 1 ϕ 4 theory, where the role of classically broken and unbroken phases is inverted. We verify all these considerations by computing the perturbative series of the 3d O(N) models for the vacuum energy and for the mass gap up to order eight, and Borel resumming the series. In particular, we provide numerical evidence for the self-duality and verify that in renormalization schemes where the critical couplings are complex the theory is gapped. As a by-product of our analysis, we show how the non-perturbative mass gap at large N in 2d can be seen as the analytic continuation of the perturbative one in the classically unbroken phase.