https://doi.org/10.1140/epjc/s10052-016-3991-2
Regular Article - Theoretical Physics
Relativistic and nonrelativistic annihilation of dark matter: a sanity check using an effective field theory approach
Departamento de Física Aplicada, Facultad de Ciencias Experimentales, Universidad de Huelva, 21071, Huelva, Spain
* e-mail: mirco.cannoni@dfa.uhu.es
Received:
29
June
2015
Accepted:
1
March
2016
Published online:
11
March
2016
We find an exact formula for the thermally averaged cross section times the relative velocity 
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 
directly gives the nonrelativistic limit of 
, 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 
in powers of the nonrelativistic relative velocity 
. We show the correct invariant procedure that gives the nonrelativistic average 
coinciding with the large x expansion of 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 
, showing the uselessness of the Møller velocity and further elucidating the conceptual and numerical inconsistencies related with its use.
© The Author(s), 2016
