Sigma models with $A_k$ singularities in Euclidean spacetime of dimension 0<=D<4 and in the limit N->infinity

TL;DR

This paper analyzes the critical behavior of single-O(N)-vector linear sigma models with $A_k$ singularities in the action, in the double scaling limit, across dimensions 0 to 4, revealing dimension-dependent constraints.

Contribution

It provides an exact analysis of critical objects for all $A_k$ singularities in various spacetime dimensions, highlighting special dimensions with stronger constraints.

Findings

Derived critical coupling constants, indices, and susceptibility matrices for all $A_k$ and dimensions.

Identified exceptional spacetime dimensions with enhanced constraints on singularity degree.

Extended understanding of sigma models' critical behavior in non-integer dimensions.

Abstract

For the case of the single-O($N$)-vector linear sigma models the critical behaviour following from any $A_k$ singularity in the action is worked out in the double scaling limit $N \rightarrow \infty$, $f_r \rightarrow f_r^c$, $2 \leq r \leq k$. After an exact elimination of Gaussian degrees of freedom, the critical objects such as coupling constants, indices and susceptibility matrix are derived for all $A_k$ and spacetime dimensions $0 \leq D < 4$. There appear exceptional spacetime dimensions where the degree $k$ of the singularity $A_k$ is more strongly constrained than by the renormalizability requirement.