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We study the ${}^{1}S_0$ proton pairing gap in beta-equilibrated neutron starmatter within the framework of chiral effective field theory. We focus on therole of three-body forces, which strongly modify the effective proton-protonspin-singlet interaction in dense matter. We find that three-body forcesgenerically reduce both the size of the pairing gap and the maximum density atwhich proton pairing may occur. The pairing gap is computed within BCS theory,and model uncertainties are estimated by varying the nuclear potential and thechoice of single-particle spectrum in the gap equation. We find that asecond-order perturbative treatment of the single-particle spectrum suppressesthe proton ${}^{1}S_0$ pairing gap relative to the use of a free spectrum. Weestimate the critical temperature for the onset of proton superconductivity tobe $T_c = (3.7 - 6.0)\times 10^{9} $ K, which is consistent with previoustheoretical results in the literature and marginally within the range deducedfrom a recent Bayesian analysis of neutron star cooling observations.

Proton pairing in neutron stars from chiral effective field theory