The spectrum of critical exponents in $(\vec\phi^2)^2$-theory in $d=4-\eps$ dimensions -- Resolution of degeneracies and hierarchical structures

TL;DR

This paper investigates the critical exponents of the N-vector model in 4-ε dimensions up to second order in ε, revealing a hierarchical structure of the spectrum with limit points formed by sums of anomalous dimensions.

Contribution

It resolves degeneracies in the spectrum at two-loop order and uncovers a hierarchical structure of limit points in the critical exponents spectrum.

Findings

Degeneracies are lifted at two-loop order.

The spectrum exhibits a hierarchical structure with limit points.

Sums of anomalous dimensions generate a hierarchy of limit points.

Abstract

The spectrum of critical exponents of the $N$--vector model in $4-\eps$~dimensions is investigated to the second order in~$\eps$. A generic class of one--loop degeneracies that has been reported in a previous work is lifted in two--loop order. One-- and two--loop results lead to the conjecture that the spectrum possesses a remarkable hierarchical structure: The naive sum of any two anomalous dimensions generates a limit point in the spectrum, an anomalous dimension plus a limit point generates a limit point of limit points and so on. An infinite hierarchy of such limit points can be observed in the spectrum.