First-principle calculations of dark matter scattering off light nuclei

We study the scattering of dark matter particles off various light nuclei within the framework of chiral effective field theory. We focus on scalar interactions and include one- and two-nucleon scattering processes whose form and strength are dictated by chiral symmetry. The nuclear wave functions are calculated from chiral effective field theory interactions as well and we investigate the convergence pattern of the chiral expansion in the nuclear potential and the dark matter–nucleus currents. This allows us to provide a systematic uncertainty estimate of our calculations. We provide results for ${}^2\mathrm{H}$, ${}^3\mathrm{H}$, and ${}^3\mathrm{He}$ nuclei, which are theoretically interesting, and the latter is a potential target for experiments. We show that two-nucleon currents can be systematically included but are generally smaller than predicted by power counting and suffer from significant theoretical uncertainties even in light nuclei. We demonstrate that accurate high-order wave functions are necessary in order to incorporate two-nucleon currents.

Figure 1

Diagrams contribution to DM-nucleus scattering. Solid lines correspond to nucleons, single dashed lines correspond to pions, and double-dashed lines correspond to DM. Our analysis focuses on the diagrams in the first row, while those in the second row only appear at higher order.

Figure 2

The plots show different structure functions for isoscalar quark-DM interaction for the ${}^2\mathrm{H}$ system at three values of the momentum transfer $q$. From top to bottom the panels indicate, respectively, the leading, order (0), one-nucleon isoscalar structure functions; the full order (0+1) isoscalar structure functions including two-body and radius corrections; the modification in percent due to the radius corrections; and the modification in percent due to two-body corrections. Results are shown for different chiral wave functions (NLO up to $\mathrm{N}{}^4\mathrm{LO}$) and for the five different cutoffs given in Table 2. The colored bands, from outermost to innermost, correspond to averages over all cutoffs at ${\mathrm{N}}^2\mathrm{LO}$, ${\mathrm{N}}^3\mathrm{LO}$, and ${\mathrm{N}}^4\mathrm{LO}$ in the wave expansion. Results for LO wave functions are not shown because the uncertainty bands are too large.

Figure 3

The plots shows different structure functions for isoscalar quark-DM interaction for the ${}^3\mathrm{He}$ system at three values of the momentum transfer $q$. From top to bottom the panels indicate, respectively, the leading order (0) one-body isoscalar structure functions, the full order $(0+1)$ isoscalar structure functions including two-body and radius corrections, the modification in percent due to the radius corrections, and the modification in percent due to two-body corrections. Results are shown for different chiral wave functions (NLO up to $\mathrm{N}{}^4\mathrm{LO}$) and for the five different cut-offs given in Table 2. The colored bands, from outermost to innermost, correspond to averages over all cut-offs at ${\mathrm{N}}^2\mathrm{LO}$, ${\mathrm{N}}^3\mathrm{LO}$ and ${\mathrm{N}}^4\mathrm{LO}$ in the wave expansion.

Figure 4

The left column shows the full order (0+1) isoscalar-quark DM structure functions for ${}^2\mathrm{H}$ (top) and ${}^3\mathrm{He}$ (bottom). The second and third columns denote, respectively, the radius and two-nucleon corrections as defined in Eq. (36). The uncertainty bands correspond to chiral wave functions at ${\mathrm{N}}^2\mathrm{LO}$, ${\mathrm{N}}^3\mathrm{LO}$, and ${\mathrm{N}}^4\mathrm{LO}$ (from outer- to innermost: green, blue, red). Results from using phenomenological wave functions and for the fitted Helm form factor (left panel only) are also shown. The phenomenological wave functions for ${}^3\mathrm{He}$ are generated with and without three-nucleon forces (TNF).

Figure 5

Correlation between isoscalar two-nucleon matrix elements and the $D$-wave probability. Circles and squares correspond to ${}^2\mathrm{H}$ and ${}^3\mathrm{He}$, respectively. The sizes of the circles and squares correspond to the sizes off the cutoff; the smallest (largest) symbols represent ${\mathrm{\Lambda }}_1$ (${\mathrm{\Lambda }}_5$). Different colors correspond to different wave functions. A linear fit results in $\langle J_{q^{\left(\mathrm{is}\right)},2b}\rangle =4.9\times {10}^{-3}{P}_D-1.2\times {10}^{-2}$ for ${}^2\mathrm{H}$ and $\langle J_{q^{\left(\mathrm{is}\right)},2b}\rangle =8.2\times {10}^{-4}{P}_D-4.1\times {10}^{-3}$ for ${}^3\mathrm{He}$.

Figure 6

Hierarchy of contributions to $|\mathcal{F}\left({\mathit{q}}^2\right){|}^2$ for the choice $c_u=c_d$ and $c_s=c_G=0$ using ${\mathrm{N}}^2\mathrm{LO}$ chiral wave functions with cutoff ${\mathrm{\Lambda }}_2$. The diagram displays the absolute values of the various terms. For more explanation, we refer to the main text.

Figure 7

Plots corresponding to the scenario with nonzero isoscalar and gluonic DM interactions discussed in Sec. 4d. The coefficient $r$ has been chosen such that the order (1) currents provide a $50\%$ correction to the leading-order results. We plot the square of the response functions ${\left|\mathcal{F}\right|}^2\left({\mathit{q}}^2\right)$. The upper (lower) row represents ${}^2\mathrm{H}$ (${}^3\mathrm{He}$) results, while the left (right) panel corresponds to $\mathrm{N}{}^2\mathrm{LO}$ ($\mathrm{N}{}^3\mathrm{LO}$) chiral wave functions using cutoff ${\mathrm{\Lambda }}_2$. The green bands correspond to the leading contribution, while the blue and red bands include the order (1) and order (3) currents, respectively.