Cutting and gluing with running couplings in $\mathcal{N}=2$ QCD

TL;DR

This paper investigates the behavior of the order parameter in $ ext{SU}(2)$ $ ext{N}=2$ QCD with up to three flavors, revealing how its modular properties depend on the mass parameters and the structure of the fundamental domain in the upper half-plane.

Contribution

It demonstrates the modular nature of the order parameter u in $ ext{N}=2$ QCD and describes how the fundamental domain is constructed and altered by mass parameters, including the emergence of branch points and cuts.

Findings

u is one-to-one on a restricted fundamental domain

Special mass choices lead to u being a modular function without branch points

Mass variations cause the fundamental domain to be cut and glued, affecting singularity structure

Abstract

We consider the order parameter $u=\left<{\rm Tr}\phi^2\right>$ as function of the running coupling constant $\tau \in \mathbb{H}$ of asymptotically free $\mathcal{N}=2$ QCD with gauge group $SU(2)$ and $N_f\leq 3$ massive hypermultiplets. If the domain for $\tau$ is restricted to an appropriate fundamental domain $\mathcal{F}_{N_f}$, the function $u$ is one-to-one. We demonstrate that these domains consist of six or less images of an ${\rm SL}(2,\mathbb{Z})$ keyhole fundamental domain, with appropriate identifications of the boundaries. For special choices of the masses, $u$ does not give rise to branch points and cuts, such that $u$ is a modular function for a congruence subgroup $\Gamma$ of ${\rm SL}(2,\mathbb{Z})$ and the fundamental domain is $\Gamma\backslash\mathbb{H}$. For generic masses, however, branch points and cuts are present, and subsets of $\mathcal{F}_{N_f}$ are being cut and glued upon varying the mass. We study this mechanism for various phenomena, such as decoupling of hypermultiplets, merging of local singularities, as well as merging of non-local singularities which give rise to superconformal Argyres-Douglas theories.