Fate of the topological susceptibility in two-color dense QCD

We explore the topological susceptibility at finite quark chemical potential and zero temperature in two-color QCD (QC$_2$D) with two flavors. Through the Ward-Takahashi identities of QC$_2$D, we find that the topological susceptibility in the vacuum solely depends on three observables: the pion decay constant, the pion mass, and the $\eta$ mass in the low-energy regime of QC$_2$D. Based on the identities, we numerically evaluate the topological susceptibility at finite quark chemical potential using the linear sigma model with the approximate Pauli-Gursey $SU(4)$ symmetry. Our findings indicate that, in the absence of $U(1)_A$ anomaly effects represented by the Kobayashi-Maskawa-'t Hooft-type determinant interaction, the topological susceptibility vanishes in both the hadronic and baryon superfluid phases. On the other hand, when the $U(1)_A$ anomaly effects are present, the constant and nonzero topological susceptibility is induced in the hadronic phase, reflecting the mass difference between the pion and $\eta$ meson. Meanwhile, in the superfluid phase it begins to decrease smoothly. The asymptotic behavior of the decrement is fitted by the continuous reduction of the chiral condensate in dense QC$_2$D, which is similar to the behavior observed in hot three-color QCD matter. In addition, effects from the finite diquark source on the topological susceptibility are discussed. We expect that the present study provides a clue to shed light on the role of the $U(1)_A$ anomaly in cold and dense QCD matter.