Renormalization group procedure for potential −g/r2

Abstract

Schrödinger equation with potential −g/r 2 exhibits a limit cycle, described in the literature in a broad range of contexts using various regularizations of the singularity at r = 0. Instead, we use the renormalization group transformation based on Gaussian elimination, from the Hamiltonian eigenvalue problem. of high momentum modes above a finite, floating cutoff scale. The procedure identifies a richer structure than the one we found in the literature. Namely, it directly yields an equation that determines the renormalized Hamiltonians as functions of the floating cutoff: solutions to this equation exhibit, in addition to the limit-cycle, also the asymptotic-freedom, triviality, and fixed-point behaviors, the latter in vicinity of infinitely many separate pairs of fixed points in different partial waves for different values of g.

Momentum approach to the $1/r^2$ potential as a toy model of the Wilsonian renormalization

“…A number of authors [11][12][13][14][15], mostly with a high energy physics background, noticed that L α has a great pedagogical potential to illustrate the concept of renormalization group and Wilson's ideas. Typically, these authors look at the operator (2) in dimension 3 and the s-wave sector in the momentum representation.…”

“…A recent exposition of the "Wilsonian approach" to the inverse square potential can be found in the paper [11], which is the main inspiration for our work. [11] starts from the 3-dimensional operator (2) in the s-wave sector and develops its renormalization theory as a pedagogical illustration of Wilsonian ideas.…”

“…A recent exposition of the "Wilsonian approach" to the inverse square potential can be found in the paper [11], which is the main inspiration for our work. [11] starts from the 3-dimensional operator (2) in the s-wave sector and develops its renormalization theory as a pedagogical illustration of Wilsonian ideas. The exposition contained in [11] stays all the time on the momentum side, and the reader may find it difficult to connect it with the standard theory of self-adjoint realizations of L α , more transparent to describe in the position representation.…”

“…In Sec. 3 we recapitulate the content of [11]. On purpose, we stick to the original terminology and line of reasoning.…”

Asymptotic safety, the Higgs boson mass, and beyond the standard model physics

“…In the simplest case the couplings reach the fixed point (∀ i β i ({g j )}) = 0) and the running stops, making the theory scale invariant on the quantum level. However, this is not only possibility, since the coupling can also be attracted to a higher dimensional structure, like a limit cycle (see for example [3] for a limit cycle behaviour in 1/r 2 potential) or a chaotic attractor. Such theories can also be UV fundamental, yet they are not scale invariant.…”