  The structured hadron-quark mixed phase, known as the pasta phase, is
expected to appear in the core of massive neutron stars. Motivated by the
recent advances in astrophysical observations, we explore the possibility of
the appearance of quarks inside neutron stars and check its compatibility with
current constraints. We investigate the properties of the hadron-quark pasta
phases and their influences on the equation of state (EOS) for neutron stars.
In this work, we extend the energy minimization (EM) method to describe the
hadron-quark pasta phase, where the surface and Coulomb contributions are
included in the minimization procedure. By allowing different electron
densities in the hadronic and quark matter phases, the total electron chemical
potential with the electric potential remains constant, and local ? equilibrium
is achieved inside the Wigner-Seitz cell. The mixed phase described in the EM
method shows the features lying between the Gibbs and Maxwell constructions,
which is helpful for understanding the transition from the Gibbs construction
(GC) to the Maxwell construction (MC) with increasing surface tension. We
employ the relativistic mean-field model to describe the hadronic matter, while
the quark matter is described by the MIT bag model with vector interactions. It
is found that the vector interactions among quarks can significantly stiffen
the EOS at high densities and help enhance the maximum mass of neutron stars.
Other parameters like the bag constant can also affect the deconfinement phase
transition in neutron stars. Our results show that hadron-quark pasta phases
may appear in the core of massive neutron stars that can be compatible with
current observational constraints.

Nuclear Theory;High Energy Astrophysical Phenomena