CP violation in cold dense quark matter and axion effects on the non-radial oscillations of neutron stars

Charge-conjugation and parity violation in strong interaction for cold dense quark matter is studied with axions of quantum chromodynamic within the three flavor Nambu--Jona-Lasinio model that includes the coupling of axions to quarks. We first calculate the effective potential for axions at finite baryon density and zero temperature including the effects of a first order chiral phase transition. Using the equation of state for quark matter with axions and a hadronic matter equation of state in the ambit of a relativistic mean field theory in quantum hadrodynamics, we discuss the hadron-quark phase transition. Inclusion of axions reduces the critical density for chiral transition. We use a Gibbs construct for the hadron-quark phase transition satisfying the constraints of beta equilibrium and charge neutrality as appropriate for the neutron star matter. The equation of state so obtained is used to investigate the structure of hybrid neutron stars. It is found that with the presence of axions, it is possible to have stable hybrid neutron stars having an inner core of quark matter both in pure quark matter phase as well as in a mixed phase with hyperonic matter along with a outer core of hyperonic matter and is in agreement with modern astrophysical constraints. We also discuss the properties of non-radial oscillations of such hybrid neutron stars. It is observed that the quadrupolar fundamental modes ($f$-modes) for such hybrid neutron stars get substantial enhancements both due to a larger quark core in the presence of axions and from the hyperons as compared to a canonical nucleonic neutron stars.