2015/06/30 by Mirco Cannoni · 1 citation
Physics and Astronomy · #Annihilation #Cosmology and Gravitation Theories #Cross section (physics) #Dark Matter and Cosmic Phenomena #Dark matter #Effective field theory #Invariant (physics) #Invariant mass #Limit (mathematics) #Particle physics theoretical and experimental studies #Relative velocity #Relativistic particle #astro-ph.CO #gr-qc #hep-ph #hep-th
paper · pdf · doi:10.1140/epjc/s10052-016-3991-2
published as Eur. Phys. J. C 76, no. 3, 137 (2016) · 10 pages, two columns, 3 figures. v3: published version
openalex publication_date 2016/03/01 · arxiv created 2016/03/19 · arxiv updated 2016/03/22 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
We find an exact formula for the thermally averaged cross section times the relative velocity ⟨ σ v_\text rel ⟩ with relativistic Maxwell–Boltzmann statistics. The formula is valid in the effective field theory approach when the masses of the annihilation products can be neglected compared with the dark matter mass and cut-off scale. The expansion at x=m/T≫ 1 directly gives the nonrelativistic limit of ⟨ σ v_\text rel⟩ , which is usually used to compute the relic abundance for heavy particles that decouple when they are nonrelativistic. We compare this expansion with the one obtained by expanding the total cross section σ (s) in powers of the nonrelativistic relative velocity vr . We show the correct invariant procedure that gives the nonrelativistic average ⟨ σ _\mathrmnr vr ⟩ _\mathrmnr coinciding with the large x expansion of ⟨ σ v_\text rel⟩ in the comoving frame. We explicitly formulate flux, cross section, thermal average, collision integral of the Boltzmann equation in an invariant way using the true relativistic relative v_\text rel , showing the uselessness of the Møller velocity and further elucidating the conceptual and numerical inconsistencies related with its use.