$θ-$ Parameter in 2 Color QCD at Finite Baryon and Isospin Density

Abstract

We use 2-color QCD as a model to study the effects of simultaneous presence of the so-called $θ$ parameter, chemical potentials for baryon number, $μ_B$ and for isospin charge, $μ_I$. We pay special attention to $θ$, $μ_B$, $μ_I$ dependence of different vacuum condensates, including chiral and diquark condensates, as well as the gluon condensate, $ $, and the topological susceptibility. We find that two phase transitions of the second order will occur when $θ$ relaxes from $θ=2π$ to $θ=0$, if $μ$ is of order of the pion mass. We demonstrate that the transition to the superfluid phase at $θ= π$ occurs at a much lower chemical potential than at $θ= 0$. We also show that the strong $θ$ dependence present near $θ= π$ in vacuum (Dashen's phenomenon), becomes smoothed out in the superfluid phase. Finally, we comment on the relevance of this study for the real world with N_c=3.