It has been a puzzle whether quarks may exist in the interior of massive neutron stars, since the hadron-quark phase transition softens the equation of state (EOS) and reduce the neutron star (NS) maximum mass very significantly. In this work, we consider the light U-boson that increases the NS maximum mass appreciably through its weak coupling to fermions. The inclusion of the U-boson may thus allow the existence of the quark degrees of freedom in the interior of large mass neutron stars. Unlike the consequence of the U-boson in hadronic matter, the stiffening role of the U-boson in the hybrid EOS is not sensitive to the choice of the hadron phase models. In addition, we have also investigated the effect of the effective QCD correction on the hybrid EOS. This correction may reduce the coupling strength of the U-boson that is needed to satisfy NS maximum mass constraint. While the inclusion of the U-boson also increases the NS radius significantly, we find that appropriate in-medium effects of the U-boson may reduce the NS radii significantly, satisfying both the NS radius and mass constraints well.
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