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