Effective theory of nuclear scattering for a WIMP of arbitrary spin

We introduce a systematic approach to characterize the most general nonrelativistic weakly interacting massive particle (WIMP)-nucleus interaction allowed by Galilean invariance for a WIMP of arbitrary spin $j_{\chi }$ in the approximation of one-nucleon currents and for a WIMP-nucleon effective potential at most linear in the velocity. Under these assumptions our framework can be matched to any high-energy model of particle dark matter, including elementary particles and composite states. Five nucleon currents arise from the nonrelativistic limit of the free nucleon Dirac bilinears. Our procedure consists in (1) organizing the WIMP currents according to the rank of the $2j_{\chi }+1$ irreducible operator products of up to $2j_{\chi }$ WIMP spin vectors, and (2) coupling each of the WIMP currents to each of the five nucleon currents. The transferred momentum $q$ appears to a power fixed by rotational invariance. For a WIMP of spin $j_{\chi }$ we find a basis of $4+20j_{\chi }$ independent operators that exhaust all the possible operators that drive elastic WIMP-nucleus scattering in the approximation of one-nucleon currents. By comparing our operator basis, which is complete, to the operators already introduced in the literature we show that some of the latter for $j_{\chi }=1$ were not independent and some were missing. We provide explicit formulas for the squared scattering amplitudes in terms of the nuclear response functions, which are available in the literature for most of the targets used in WIMP direct detection experiments.

Figure 1

Expected differential scattering rate (1.5) normalized to 1 event for a 10 GeV mass WIMP with a xenon target and for the 10 irreducible effective operators ${\mathcal{O}}_{X,2,l}$ defined in Eqs. (3.22), assuming a WIMP of spin $j_{\chi }\ge 1$.

Figure 2

Same as Fig. 1 but for the 10 irreducible effective operators ${\mathcal{O}}_{X,3,l}$ and assuming a WIMP of spin $j_{\chi }\ge 3/2$.

Figure 3

Same as Fig. 1 but for a fluorine target.

Figure 4

Same as in Fig. 2 but for a fluorine target.