On the formalism and upper limits for spin-dependent cross sections in dark matter elastic scattering with nuclei

Abstract

We revise the spin-dependent neutralino-nucleus elastic scattering comparing the formalisms and approximations found in literature for the momentum transfer dependent structure functions. We argue that one of the normalized structure functions of Divari, Kosmas, Vergados and Skouras is all that one needs to correctly take into account the detailed nuclear physics information provided by shell-model calculations. The factorization of the particle physics degrees of freedom from the nuclear physics momentum dependent structure functions implied by this formalism allows for a better understanding of the so-called model independent method for setting upper limits. We further discuss the possibility of experiments with spin-dependent sensitivity like COUPP to test or set limits on the proton spin-dependent cross section in the framework of the stau co-annihilation region of the constrained minimal supersymmetric standard model. For this model with $A_0 =0$, we provide a fitting formula by which it is possible to convert an upper limit on the spin-independent cross section as a function of the neutralino mass directly into an exclusion plot in the ($m_{1/2}$, $\tan\beta$) plane.