Abstract
In a large class of models for Weakly Interacting Massive Particles (WIMPs), the WIMP mass $M$ lies far above the weak scale $m_W$. This work identifies universal Sudakov-type logarithms $\sim \alpha \log^2 (2\,M/m_W)$ that spoil the naive convergence of perturbation theory for annihilation processes. An effective field theory (EFT) framework is presented, allowing the systematic resummation of these logarithms. Another impact of the large separation of scales is that a long-distance wave-function distortion from electroweak boson exchange leads to observable modifications of the cross section. Careful accounting of momentum regions in the EFT allows the rigorously disentanglement of this so-called Sommerfeld enhancement from the short distance hard annihilation process. The WIMP is modeled as a heavy-particle field, while the light, energetic, final-state electroweak gauge bosons are treated as soft and collinear fields. Hard matching coefficients are computed at renormalization scale $\mu \sim 2\,M$, then evolved down to $\mu \sim m_W$, where electroweak symmetry breaking is incorporated and the matching onto the relevant quantum mechanical Hamiltonian is performed. The example of an $SU(2)_W$ triplet scalar dark matter candidate annihilating to line photons is used for concreteness, allowing the numerical exploration of the impact of next-to-leading order corrections and log resummation. For $M \simeq 3$ TeV, the resummed Sommerfeld enhanced cross section is reduced by a factor of $\sim 3$ with respect to the tree-level fixed order result.
Highlights
Temperatures of TeV until today,1 it is natural for a Weakly Interacting Massive Particle (WIMP) to freeze out with the measured dark matter abundance
We focus for simplicity on a self-conjugate scalar WIMP, necessarily a U(1)Y hypercharge singlet that transforms under a general integer isospin representation of SU(2)W, with generators ta
Note that we distinguish gand Zg in the full theory from g and Zg in the effective theory, which differ because the heavy WIMP has been integrated out below the scale M and as such no longer contributes to the running of the gauge coupling
Summary
Where (ta)bc = i bac are SU(2)W generators in the adjoint representation. In the basis of electric charge eigenstates we have iDμ i∂μ. Where Q ≡ t3 + Y is the electric charge in units of the proton charge, and t± = t1 ± it. From which it is straightforward to read off the Feynman rules. We neglect renormalizable self-couplings of the scalar field, ∼ φ4, and Higgs interactions, ∼ H†Hφ2. It would be straightforward to include these couplings in an extended analysis
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