Abstract
Three related analyses of ϕ4 theory with O(N) symmetry are presented. In the first, we review the O(N) model over the p-adic numbers and the discrete renormalization group transformations which can be understood as spin blocking in an ultrametric context. We demonstrate the existence of a Wilson-Fisher fixed point using an ϵ expansion, and we show how to obtain leading order results for the anomalous dimensions of low dimension operators near the fixed point. Along the way, we note an important aspect of ultrametric field theories, which is a non-renormalization theorem for kinetic terms. In the second analysis, we employ large N methods to establish formulas for anomalous dimensions which are valid equally for field theories over the p-adic numbers and field theories on ℝn. Results for anomalous dimensions agree between the first and second analyses when they can be meaningfully compared. In the third analysis, we consider higher derivative versions of the O(N) model on ℝn, the simplest of which has been studied in connection with spatially modulated phases. Our general formula for anomalous dimensions can still be applied. Analogies with two-derivative theories hint at the existence of some interesting unconventional field theories in four real Euclidean dimensions.
Highlights
Subsequent work, including [6, 7] and reviewed in [8], established rigorous results on the solvability and fixed points of the renormalization group for hierarchical models as realized by the recursion relations
Methods based on the Hubbard-Stratonovich transformation have been developed, notably in [14, 15], which resum an infinite set of diagrams of the O(N ) model at fixed order in large N and allow a determination of critical exponents at the Wilson-Fisher fixed point which are known exactly as functions of and to a few orders in large N
Our main technical result, summarized in (4.2)-(4.3), is the expression of anomalous dimensions γφ and γσ as residues at δ = 0 of meromorphic functions gφ(δ) and gσ(δ) of a quantity δ, understood as a shift in the dimension of the Hubbard-Stratonovich field σ that we impose as a regulator and remove at the end of the calculation
Summary
To renormalize φ4 theory we typically need to handle divergences in the two-point and four-point functions. These Green’s functions take the following forms: G(ij2)(k). In (2.4) and below, we omit the momentum-conserving delta functions from the Green’s functions. I4(T ) and I4(U) are defined like I4(S), but with k1 +k2 replaced by k1 +k3 for I4(T ) and by k1 +k4 for I4(U). A diagrammatic account of the formulas (2.4) is summarized in figure 1. The standard challenge of perturbative renormalization group analysis is to tame divergences at large | | (the ultraviolet) arising in the integrals (2.5)
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