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

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