Vanishing beta function curves from the functional renormalization group

Abstract

In this paper we will discuss the derivation of the so-called vanishing beta function curves which can be used to explore the fixed point structure of the theory under consideration. This can be applied to the O(N ) symmetric theories, essentially, for arbitrary dimensions (D) and field component (N ). We will show the restoration of the Mermin-Wagner theorem for theories defined in D ≤ 2 and the presence of the Wilson-Fisher fixed point in 2 < D < 4. Triviality is found in D > 4. Interestingly, one needs to make an excursion to the complex plane to see the triviality of the four-dimensional O(N ) theories. The large -N analysis shows a new fixed point candidate in 4 < D < 6 dimensions which turns out to define an unbounded fixed point potential supporting the recent results by R. Percacci and G. P. Vacca in: "Are there scaling solutions in the O(N )-models for large -N in D > 4?" [Phys. Rev. D 90, 107702 (2014)].

Spheres, charges, instantons, and bootstrap: A five-dimensional odyssey

“…Each superconformal multiplet can be labeled by the charges ∆, J ± , J R of its highest weight state under the Cartan of so(2) × so(5) × su(2) R , where J ± are the Cartan generators of the su(2) + ×su(2) − ⊂ so(4) ⊂ so (5). All the charges are real for unitary representations of the Lorentzian conformal algebra so(2, 5) × su(2) R .…”

“…The short representations are classified into A, B, D types, satisfying the following conditions [158,159,143,144] A : ∆ = 2J + + 3J where d 1 = 2J − and d 2 = 2J + − 2J − are the so(5) Dynkin labels. Due to OPE selection rules, in the bootstrap analysis, we only have to consider multiplets whose superconformal primaries are in the symmetric rank-representation of so (5). We denote such representations by…”

“…The fixed point was shown to be unitary to all orders in 1 N , but it remained unclear whether unitarity holds non-perturbatively. In fact, numerical studies by the exact renormalization group equations suggested that the fixed point is metastable [5,6]. 2 These conditions are often referred to as the Intriligator-Morrison-Seiberg conditions.…”

Critical scaling in the large-NO(N)model in higher dimensions and its possible connection to quantum gravity

“…Focus now shifts to the author's previous results [12] where a different technique, based upon polynomial expansion, was used to find the fixed point solutions of the flow equation. The potential is assumed to be analytic in this case:…”

“…It can be obtained as the tensionless limit of string theory, where the infinite tower of higher-spin string modes are massless, and since there is no energy scale it can be considered as a toy model describing physics beyond the Planck scale [7]. Studies related to the existence of the UV fixed point in the large-N O(N ) model, using conformal bootstrap approach and exact (or functional) renormalization group (ERG or FRG) methods, can be found in [8][9][10] and [11,12], respectively. The structure of the paper is the following.…”