Effects of symmetry energy on the properties of a hadron-quark mixed phase in hybrid stars

We study how the nuclear symmetry energy slope affects the properties of hadron-quark pasta phases and their sizes in massive hybrid stars ($>2{M}_{\ensuremath{\bigodot}}$). We utilize the relativistic mean-field model with a density-dependent isovector coupling constant fitted by varying values of the symmetry energy slope $L$ to describe the hadronic matter, while the quark matter is described by a modified MIT bag model with vector interactions. We construct the hadron-quark phase transition using the energy minimization method and compare with the Gibbs construction in the hybrid star. We find that the massive hybrid stars generally tend to have a hadron-quark pasta phase core rather than a pure quark core for most values of $L$, except in some special cases where they have a small pure quark core. Furthermore, we show that the mass and size of the pasta phase core are strongly influenced by the symmetry energy. The radius associated with the onset of pasta phases is also influenced by the symmetry energy. Additionally, we also evaluate the effects of the surface tension $\ensuremath{\sigma}$, the bag constant $B$, and the vector interaction ${G}_{V}$.