One-loop contribution to dark matter-nucleon scattering in the pseudoscalar dark matter model

Abstract

Recent dark matter (DM) direct searches place very stringent constraints on the possible DM candidates proposed in extensions of the Standard Model. There are however models where these constraints are avoided. One of the simplest and most striking examples comes from a straightforward Higgs portal pseudoscalar DM model featured with a softly broken $U(1)$ symmetry. In this model the tree-level DM-nucleon scattering cross section vanishes in the limit of zero momentum-transfer. It has also been argued that the leading-order DM-nucleon cross-section appears at the one-loop level. %, which is too small to be constrained experimentally. In this work we have calculated the exact cross-section in the zero momentum-transfer at the leading-order i.e., at the one-loop level of perturbative expansion. We have concluded that, in agreement with expectations, the amplitude for the scattering process is UV finite and approaches zero in the limit of vanishing DM masses. Moreover, we made clear that the finite DM velocity correction at tree-level is subdominant with respect to the one-loop contribution. Based on the analytic formulae, our numerical studies show that, for a typical choice of model parameters, the DM nuclear recoiling cross section is well below ${\cal O}(10^{-50}~{\rm cm}^2)$, which indicates that the DM direct detection signal in this model naturally avoids the present strong experimental limits on the cross-section.