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

TL;DR

This paper refines calculations of nuclear spin matrix elements for dark matter detection, incorporating recent chiral effective field theory corrections, and demonstrates that simplified formulas can accurately reproduce complex 2-body current effects in WIMP-nucleus scattering analyses.

Contribution

It introduces a method to include recent EFT corrections into existing nuclear spin matrix element calculations and simplifies the treatment of interference functions in spin-dependent WIMP scattering.

Findings

Recalculated WIMP-nucleus cross sections with new corrections.

Showed equivalence of simplified and exact form factor methods.

Provided practical formulas for nuclear spin matrix elements.

Abstract

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.