Form factors for dark matter capture by the Sun in effective theories

In the effective theory of isoscalar and isovector dark matter-nucleon interactions mediated by a heavy spin-1 or spin-0 particle, 8 isotope-dependent nuclear response functions can be generated in the dark matter scattering by nuclei. We compute the 8 nuclear response functions for the 16 most abundant elements in the Sun, i.e. H, $^{3}$He, $^{4}$He, $^{12}$C, $^{14}$N, $^{16}$O, $^{20}$Ne, $^{23}$Na, $^{24}$Mg, $^{27}$Al, $^{28}$Si, $^{32}$S, $^{40}$Ar, $^{40}$Ca, $^{56}$Fe, and $^{58}$Ni, through numerical shell model calculations. We use our response functions to compute the rate of dark matter capture by the Sun for all isoscalar and isovector dark matter-nucleon effective interactions, including several operators previously considered for dark matter direct detection only. We study in detail the dependence of the capture rate on specific dark matter-nucleon interaction operators, and on the different elements in the Sun. We find that a so far neglected momentum dependent dark matter coupling to the nuclear vector charge gives a larger contribution to the capture rate than the constant spin-dependent interaction commonly included in dark matter searches at neutrino telescopes. Our investigation lays the foundations for model independent analyses of dark matter induced neutrino signals from the Sun. The nuclear response functions obtained in this study are listed in analytic form in an appendix, ready to be used in other projects.