Analysis of p-Wave Sommerfeld Resonance Behavior in Finite-Size Dark Matter

Different stages of cosmic evolution impose different requirements on the dark matter annihilation cross section. Sommerfeld enhancement provides a possible way to accommodate them. However, in the point-like dark matter scenario, satisfying these requirements with either $s$-wave or $p$-wave Sommerfeld enhancement can strongly constrain the dark matter and mediator masses and their coupling. A finite dark matter size introduces an additional parameter and may relax these constraints. Motivated by this possibility, in this work we study the $p$-wave Sommerfeld resonances of finite-size dark matter (like proton or neutron) without specifying its internal structure. We find that the $p$-wave contribution can exceed the $s$-wave contribution. Nevertheless, the finite-size $p$-wave resonances are weaker than those in the point-like case. Despite this suppression, their velocity dependence remains similar to that of point-like dark matter. We then extend our analysis to nugget dark matter composed of a small number of constituents. In this case, the $p$-wave Sommerfeld enhancement exhibits behavior similar to that of point-like dark matter.