Critical indices and the conformally invariant bootstrap method

Abstract

The anomalous dimensions are calculated within the Migdal-Polyakov bootstrap approach to conformalinvariant field theory by means of an expansion around the canonical dimensions. In contrast to the € expansion, this procedure works directly in three-dimensional space. The corresponding values for the critical indices are almost identical with the Wilson result.

"Old" Conformal Bootstrap in the AdS/CFT Context

“…Here g φψ(ψφ) is a bulk coupling constant of two primary fields φ(X), ψ(X) and bulk scalar field corresponding to composite conformal operator O ∆ (ψφ) ; it will reduce in final expressions since Γ ∆ ψ ,∆ φ ,∆ (ψφ) ( x 1 , x 2 , x 3 ) in the LHS of (51) includes it (see (11)).…”

“…Eq-s (2) mean that only planar diagrams are taken into account in (1). Planar expansion is traditionally used in the self-consistent calculations of the anomalous conformal dimensions, beginning from the pioneer Migdal and Polyakov works on the "old" conformal bootstrap [3], [4] which were developed in [5] - [10], see for example [11] and references therein. In some models the choice of only planar diagrams may be justified as "most divergent" ones or in frames of the 1/N expansion.…”

“Old” conformal bootstrap on AdS: O(N) symmetric scalar model

“…In the "old" conformal bootstrap pioneered in [5], [6] and developed in [7] - [12], see e.g. [13] and references therein, the planar approximation Σ ∼ G 2 is traditionally used in (1) in self-consistent calculations of the anomalous conformal dimensions. In this case first line of (1) takes a form (triple interaction is supposed, g is the coupling constant):…”

UV-finite “old” conformal bootstrap on AdS: scalar case

“…In papers [2] it was shown that "old" conformal bootstrap (proposed about 50 years ago in pioneer papers [3], [4] and developed in [5] - [10], see e.g. [11] and references therein) considered in the AdS/CFT context permits to calculate conformal dimensions, that is bulk masses of the fields. The simplest "old" conformal bootstrap equation for Green function G(X, Y ) traditionally written in planar approximation looks as:…”