Semiclassics for the QCD vacuum structure through $T^2$-compactification with the baryon-'t Hooft flux

We study QCD vacuum structure with the topological $\theta$ angle using a recently proposed semiclassical approach on $\mathbb{R}^2 \times T^2$ with the 't Hooft and baryon magnetic fluxes. Under the assumption of adiabatic continuity in this setup, the confining vacuum can be described by the dilute gas of center vortices. With this semiclassical approach, we derive the 2d effective description at small $T^2$ and successfully explain the reasonable theta dependence of the QCD vacuum: In the one-flavor QCD at $\theta = \pi$, the $CP$ symmetry is spontaneously broken for quark mass above a critical value and restored for a subcritical mass, while the $CP$ symmetry is always spontaneously broken in the multi-flavor QCD at $\theta = \pi$. From our semiclassical description, we discuss implications to the $4$d chiral Lagrangian and propose how the $\eta'$ meson should be incorporated in consistent with known global structures: The periodicity of the $\eta'$ should be extended from the naive one $2\pi$ to $2\pi N$. Additionally, we revisit the phase diagram of $N_f = 1+1$ and $N_f = 1+1+1$ QCD on the up and down quark mass plane, confirming and refining the existence of the $CP$-broken Dashen phase.