How can I obtain the DM DM -> SM SM annihilation cross section if the DM effective operator is given? It would be really helpful if someone can show me the detail derivation.
Dear Basabendu,
as I am actually updating my knowledge on cosmology, DM and gravitation wave detection I am following several experimental and theoretical approaches to identify DM. Amongst the publications I would kindly suggest two papers attached, whereof especially the dissertation of Eugenio Del Nobile gives a complete overview concerning the effective operators as well as annihilation cross section in chapter 5. The second paper (arXiv_1601.02181) is more related to fitting experimental data.
Reference of Thesis "Eugenio Del Nobile":
"Effective Operators for Dark Matter Detection"
"Dissertation submitted for the degree of Philosophiae Doctor"
November 2012
Good luck with your research. Kind regards Willi Hanstein
http://cp3-origins.dk/content/thesis/2012-11-delnobile.pdf
https://arxiv.org/abs/1601.02181
Thank you so much for the relevant documents. But I am still kind of in dark. What I need is how to obtain the scattering cross section given a DM operator.
Dear Basabendu,
as we are dealing with dark matter, it is not surprising that you feel "still in the dark".
But back to serious research: I have checked your problem more in detail and I suppose that need more direct connection for the calculation of the relevant cross sections. Please find attached some more links dealing with the calculation of cross sections based on effective operators (a file with the links and the links are attached to this message). The most promising papers are:
1) On the effective operators for Dark Matter annihilations and
2) Effective Operators for Dark Matter Interactions
I wish you good luck with the papers Willi Hanstein
https://arxiv.org/pdf/1410.4427.pdf
https://arxiv.org/pdf/1301.1486.pdf
https://arxiv.org/pdf/1311.6169.pdf
https://www.slac.stanford.edu/pubs/slacpubs/15250/slac-pub-15419.pdf
An effective operator is obtained when the effective interaction is sandwiched between initial and final states. Once you have a "model" the calculation proceeds as follows. By the term "model" I mean a field theoretic model describing visible sector and dark sector. Usually such interaction strength should be very weak.
To obtain scattering cross section one has to write two currents of the form Ψ̅ γμ Ψ , one from visible sector and one from dark sector, and insert a term for propagator. Then multiply with suitable interaction strengths to obtain the matrix element M, for the interaction. |M|2 is obtained as M*M. Now depending on the experiment one also sums over final spin states and average over initial spin states. Next step is to integrate spin averaged |M|2 over final state phase space. At this stage you will obtain a differential scattering cross section.
It is just as done in standard field theory. The difference is that it should be performed in the context of a new "model".
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