Abstract
We calculate the one-loop correction to the static potential induced by γ, W and Z-exchange at tree-level for arbitrary heavy standard model multiplets. We find that the result obeys a “Casimir-like” scaling, making the NLO correction to the potential a “low-energy” property of the electroweak gauge bosons. Furthermore, we discuss the phenomenology of the NLO potentials, the analytically known asymptotic limits and provide fitting functions in position space for easy use of the results.
Highlights
Next-to-leading order (NLO) electroweak corrections to the non-relativistic Sommerfeld effect expected to be the dominant source of theoretical uncertainty
Possible corrections to the Sommerfeld effect are the mass-splittings between the particles in the multiplet after electroweak symmetry breaking (EWSB) to two-loops, which are known for the simplest multiplets [15,16,17], and the next-to-leading order (NLO) EW potentials
At the one-loop order the universality of the potential correction is easiest seen by choosing a Coulomb gauge formulation in which only self-energies contribute to the NLO correction [22]
Summary
We follow the effective theory setup as outlined for non-relativistic heavy WIMPs in [5, 25, 26] that was constructed in analogy to the respective EFTs for QED and QCD [27,28,29,30,31]. We start from the non-relativistic effective theory. To stay close to the previous literature [4, 18, 19, 25] on non-relativistic effective theories for electroweak DM at the TeV scale, we use in the following the term DM synonymous to non-relativistic particles of mass mχ mZ, if not explicitly stated otherwise. The necessary Lagrangian pieces to obtain NLO non-relativistic accuracy are given by LPNRDM = χ†vi(x). For a detailed discussion of the power counting, the possible further terms in the Lagrangians, and other aspects, see [19]
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