The $\theta$-angle and axion physics of two-color QCD at fixed baryon charge. Part I

We analyze the impact of the $\theta$-angle and axion dynamics for two-color (in fact any $Sp(2N)$) QCD at nonzero baryon charge and as a function of the number of matter fields on the vacuum properties, the pattern of chiral symmetry breaking as well as the spectrum of the theory. We show that the vacuum acquires a rich structure when the underlying $CP$ violating topological operator is added to the theory. We discover novel phases and analyse the order of their transitions characterizing the dynamics of the odd and even number of flavours. We further determine the critical chemical potential as function of the $\theta$ angle separating the normal from the superfluid phase of the theory. Our results will guide numerical simulations and novel tests of the model's dynamics. The results are also expected to better inform phenomenological applications of the model ranging from composite Higgs physics to strongly interacting massive dark matter models featuring number changing interactions. In the companion work \cite{PartII} we repurpose and upgrade the approach to determine the impact of the $\theta$-angle and axion physics on non-perturbative near conformal dynamics related to the fixed baryon charge sector.