Abstract
We present analytical formulas for the Sommerfeld corrections to the annihilation of massive colored particles into quarks and gluons through the strong interaction. These corrections are essential to accurately compute the dark matter relic density for coannihilation with colored partners. Our formulas allow us to compute the Sommerfeld effect, not only for the lowest term in the angular momentum expansion of the amplitude, but for all orders in the partial wave expansion. In particular, we carefully account for the effects of the spin of the annihilating particle on the symmetry of the two-particle wave function. This work focuses on strongly interacting particles of arbitrary spin in the triplet, sextet and octet color representations. For typical velocities during freeze-out, we find that including Sommerfeld corrections on the next-to-leading order partial wave leads to modifications of up to 10 to 20 percent on the total annihilation cross section. Complementary to QCD, we generalize our results to particles charged under an arbitrary unbroken SU(N) gauge group, as encountered in dark glueball models. In connection with this paper a Mathematica notebook is provided to compute the Sommerfeld corrections for colored particles up to arbitrary order in the angular momentum expansion.
Highlights
The strong interaction is short ranged at low energies, in the early universe the non-relativistic QCD potential can be approximated by a Coulomb potential at tree-level [16,17]
Aside from SU (3), Sommerfeld corrections for dark sector particles charged under a general SU (N ) gauge group have not been considered in the literature
In the previous section we have computed analytic expressions for the Sommerfeld corrections of processes with arbitrary partial waves and momentum dependence. This derivation is based on a Coulomb potential, while the interactions between colored particles are governed by a QCD potential
Summary
Computing the Sommerfeld corrections for an arbitrary process can prove a daunting task that often has to be performed numerically. Annihilations in the dark sector, involve heavy particles and can be studied in the non-relativistic limit In this limit, the tree-level amplitude for a given process can be reliably approximated by a partial wave expansion in the orbital angular momentum l and the spin s in either the initial or final state. Analytical formulas for the Sommerfeld-correction factors for higher waves have been computed in [21,23] assuming the amplitude is proportional to pl for the lth partial wave. This has been extended upon slightly in [22] allowing for a single term with a momentum dependence of pl+2n with n ≥ 0. In the rest of this section we consider a Coulomb potential and do not make assumptions as regards the spin of the initial-state particles
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