Functional perturbative RG and CFT data in the $$\epsilon $$ ϵ -expansion

Abstract

We show how the use of standard perturbative RG in dimensional regularization allows for a renormalization group-based computation of both the spectrum and a family of coefficients of the operator product expansion (OPE) for a given universality class. The task is greatly simplified by a straightforward generalization of perturbation theory to a functional perturbative RG approach. We illustrate our procedure in the -expansion by obtaining the nextto-leading corrections for the spectrum and the leading corrections for the OPE coefficients of Ising and Lee-Yang universality classes and then give several results for the whole family of renormalizable multi-critical models φ 2n . Whenever comparison is possible our RG results explicitly match the ones recently derived in CFT frameworks.

A functional perspective on emergent supersymmetry

“…Its negative is sometimes denoted ω = −θ 3 = β λ (λ ) = and its value differs from what one would naively obtain from setting n = 3 in the first terms of the first line of (5.7) because of the presence of the anomalous dimension in the beta function (5.4). The ellipsis denotes subsequent irrelevant operators for n > 3 which are subject to further mixing with derivative operators [46]. Therefore, our formula is not expected to give the correct result for their leading expansion in .…”

“…The effective action used in the following can be thought of in the spirit of Landau-Ginzburg-Wilson actions, obtained from a suitable coarse graining. Within the non-perturbative functional RG used and described in the appendices, a precise connection exists to the full 1PI effective action [44][45][46].…”

“…3 Perturbative vs. non-perturbative RG We investigate the scale dependence of all the systems (2.4), (2.6) and (2.7) by means of two different and rather complementary RG methods, one based on perturbation theory [47] which was dubbed functional perturbative RG in [46], and another based on a non-perturbative RG equation that goes under the name of Wetterich equation and is known as functional RG [44]. We illustrate the most important results of the paper by means of functional perturbation theory from section 4 to section 8, because it makes our discussion much more transparent.…”

“…All perturbative results given in the main text of the paper can be derived from a standard application of perturbation theory and dimensional regularization, but are also fully contained as the logarithmic terms of the non-perturbative results given in the appendices. Both procedures are described in more detail in [46].…”

“…We refer the Reader to appendix C for a more detailed explanation on how to derive functional perturbative RG flows and their relation with non-perturbative flows. The RG flow (5.2) can be understood as the generating functional of the beta functions of the couplings to the local operators that are obtained from raising the field ϕ to an arbitrary power [46]. To clarify the relation between (5.2) and the beta function presented in [21], let us introduce the critical coupling λ, which is canonically dimensionless in four dimensions, and insert the following parametrization for the critical potential w(ϕ) = …”