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