Nuclear structure factors for general spin-independent WIMP-nucleus scattering

We present nuclear structure factors that describe the generalized spin-independent coupling of weakly interacting massive particles (WIMPs) to nuclei. Our results are based on state-of-the-art nuclear structure calculations using the large-scale nuclear shell model. Starting from quark- and gluon-level operators, we consider all possible coherently enhanced couplings of spin-$1/2$ and spin-0 WIMPs to one and two nucleons up to third order in chiral effective field theory. This includes a comprehensive discussion of the structure factors corresponding to the leading two-nucleon currents covering, for the first time, the contribution of spin-2 operators. We provide results for the most relevant nuclear targets considered in present and planned dark matter direct detection experiments: fluorine, silicon, argon, and germanium, complementing our previous work on xenon. All results are also publicly available in a Python notebook.

Figure 1

Calculated ${}^{19}\mathrm{F}$ and ${}^{29}\mathrm{Si}$ spectra compared to experiment.

Figure 2

Calculated ${}^{28,30}\mathrm{Si}$ spectra compared to experiment.

Figure 3

Calculated ${}^{40}\mathrm{Ar}$ spectrum compared to experiment.

Figure 4

Calculated ${}^{70,72}\mathrm{Ge}$ spectra compared to experiment.

Figure 5

Calculated ${}^{73}\mathrm{Ge}$ spectrum compared to experiment.

Figure 6

Calculated ${}^{74,76}\mathrm{Ge}$ spectra compared to experiment.

Figure 7

Structure factors for ${}^{132}\mathrm{Xe}$ from 1b and 2b contributions without interference terms: 1b ${\mathcal{O}}_1$ (black), ${\mathcal{O}}_{11}$ (green), radius corrections (blue), and ${\mathcal{O}}_3$ (red) contributions are shown together with 2b ${\mathcal{F}}_b$ (indigo) and ${\mathcal{F}}_{\pi }$ (orange) structure factors. Solid lines show isoscalar and 2b contributions, while dashed lines indicate isovector couplings. For ${\mathcal{O}}_{11}$ $m_{\chi }=2\mathrm{GeV}$ is assumed.

Figure 8

Structure factors for ${}^{132}\mathrm{Xe}$, 1b contributions only. Thick lines correspond to individual terms in Eq. (20), while interference terms are shown as thin lines. Description as in Fig. 7, with dotted-dashed lines representing interference terms involving isovector couplings.

Figure 9

Structure factors for ${}^{74}\mathrm{Ge}$, 1b contributions only including interference terms. The description is as in Fig. 8.

Figure 10

Structure factors for ${}^{40}\mathrm{Ar}$, 1b contributions only including interference terms. The description is as in Fig. 8.

Figure 11

Structure factors for ${}^{19}\mathrm{F}$, 1b contributions only including interference terms. The description is as in Fig. 8.

Figure 12

Structure factors for ${}^{132}\mathrm{Xe}$ from 2b contributions and 1b–2b interferences. Thick lines correspond to individual 2b terms in Eq. (1), while interference terms are shown as thin lines. The description is as in Fig. 8, with contributions involving ${\mathcal{F}}_b$ (${\mathcal{F}}_{\pi }$) structure factors in indigo (orange), except for the ${\mathcal{F}}_b$–${\mathcal{F}}_{\pi }$ interference, represented by the maroon dotted-double-dashed line.

Figure 13

Structure factors for ${}^{74}\mathrm{Ge}$ from contributions including 2b terms only. The description is as in Fig. 12.

Figure 14

Structure factors for ${}^{40}\mathrm{Ar}$ from contributions including 2b terms only. The description is as in Fig. 12.

Figure 15

Structure factors for ${}^{19}\mathrm{F}$ from contributions including 2b terms only. The description is as in Fig. 12.

Figure 16

Structure factors for ${}^{132}\mathrm{Xe}$ for the physical combinations of the response functions ${\mathcal{F}}_{\pi }$ and ${\mathcal{F}}_{\mathrm{b}}$ corresponding to the $SS$ (solid line), $\theta$ (dashed), and spin-2 (dotted-dashed) channels as in Eq. (25).

Figure 17

Structure factors for ${}^{40}\mathrm{Ar}$ for the physical combinations of the response functions ${\mathcal{F}}_{\pi }$ and ${\mathcal{F}}_{\mathrm{b}}$. The description is as in Fig. 16.

Figure 18

One-loop diagrams for the coupling of ${\theta }_{\mu }^{\mu }$ to one nucleon (the diagram for the wave function renormalization is omitted). Solid and dashed lines refer to nucleons and pions, respectively, and crosses represent the coupling of the external current.

Figure 19

Diagrams for the coupling of ${\theta }_{\mu }^{\mu }$ to two nucleons. The notation is as in Fig. 18.