Reanalysis of nuclear spin matrix elements for dark matter spin-dependent scattering

We show how to include in the existing calculations for nuclei other than $^{129}$Xe, $^{131}$Xe, the corrections to the isovector coupling arising in chiral effective field theory recently found in Ref. \cite{Menendez1}. The dominant, momentum independent, 2-body currents effect can be taken into account by formally redefining the static spin matrix elements $< \mathbf{S}_{p,n} >$. By further using the normalized form factor at $q\neq 0$ built with the 1-body level structure functions, we show that the WIMP-nucleus cross section and the upper limits on the WIMP-nucleon cross sections coincide with the ones derived using the exact functions at the 2-body level. We explicitly show it in the case of XENON100 limits on the WIMP-neutron cross section and we recalculate the limits on the WIMP-proton spin dependent cross section set by COUPP. We also give practical formulas to obtain $< \mathbf{S}_{p,n} >$ given the structure functions in the various formalisms/notations existing in literature. We argue that the standard treatment of the spin-dependent cross section in terms of three independent isospin functions, $S_{00}(q)$, $S_{11}(q)$, $S_{01}(q)$, is redundant in the sense that the interference function $S_{01}(q)$ is the double product $|S_{01}(q)|=2\sqrt{S_{00}(q)}\sqrt{S_{11}(q)}$ even when including the new effective field theory corrections.