Effect of color superconductivity on the mass of hybrid neutron stars in an effective model with pQCD asymptotics

The effective cold quark matter model by Alford, Braby, Paris and Reddy (ABPR) is used as a tool for discussing the effect of the size of the pairing gap in three-flavor (CFL) quark matter on the maximum mass of hybrid neutron stars (NSs). This equation of state (EOS) has three parameters which we suggest to determine by comparison with a nonlocal NJL model of quark matter in the nonperturbative domain. We show that due to the momentum dependence of the pairing which is induced by the nonlocality of the interaction, the effective gap parameter in the EOS model is well approximated by a constant value depending on the diquark coupling strength in the NJL model Lagrangian. For the parameter $a_4=1-2\alpha_s/\pi$ a constant value below about \num{0.4} is needed to explain hybrid stars with ${\rm M}_{\rm max} \gtrsim 2.0~{\rm M}_\odot$, which would translate to an effective constant $\alpha_s\sim 1$. The matching point with a running coupling at the 1-loop $\beta$ function level is found to lie outside the range of chemical potentials accessible in NS interiors. A dictionary is provided for translating the free parameters of the nlNJL model to those of the ABPR model. Both models are shown to be equivalent in the nonperturbative domain but the latter one allows to quantify the transition to the asymptotic behaviour in accordance with perturbative QCD. We provide constraints on parameter sets that fulfill the $2~{\rm M}_\odot$ mass constraint for hybrid NSs, as well as the low tidal deformability constraint from GW170817 by a softening of the EOS on the hybrid NS branch with an early onset of deconfinement at ${\rm M}_{\rm onset}<1.4~{\rm M}_\odot$. We find that the effective constant pairing gap should be around 100 MeV but not exceed values of about 130 MeV because a further increase of the gap would entail a softening of the EOS and contradict the $2~{\rm M}_\odot$ mass constraint.