At tree level, the spin independent (SI) direct-detection cross section includes effective coupling constant. How to calculate effective coupling constant? There is always a mandelstram variable in it. Do we need to write something for t mandelstram in case of direct detection?
The coupling constant in theoretical dark matter approaches is, to the best of my knowledge, a free parameter of the system as well as the mass of the particle. What is usually done by the experimentalist is to plot exclusion limits of the cross section with respect to the mass of the particle, depending on the parameters of the detector such as the exposure time, the detecting nucleus load in the detector (see arXiv:astro-ph/0507190 from the CDMS-II collaboration). More fundamentally, the freedom of the coupling constant resides in the intensity with the WIMP interacts with the standard matter (nuclei in most of the experiments). The intensity is governed by the mass of the exchanged boson between them and, of course, by the nature of the interaction. The last is supposed to be a interaction like the weak interaction due to it is the only one that satisfies the astrophysical measurements, nor the strong, neither the electromagnetic interaction. Finally, say that the coupling constant will be fixed once the cross section of the interaction and the mass is measured.