Neutrino-nucleus coherent scattering as a probe of neutron density distributions

Neutrino-nucleus coherent elastic scattering provides a theoretically appealing way to measure the neutron part of nuclear form factors. Using an expansion of form factors into moments, we show that neutrinos from stopped pions can probe not only the second moment of the form factor (the neutron radius) but also the fourth moment. Using simple Monte Carlo techniques for argon, germanium, and xenon detectors of 3.5 tonnes, 1.5 tonnes, and 300 kg, respectively, we show that the neutron radii can be found with an uncertainty of a few percent when near a neutrino flux of $3\times {10}^7$ neutrinos/(cm${}^2$ s). If the normalization of the neutrino flux is known independently, one can determine the moments accurately enough to discriminate among the predictions of various nuclear energy functionals.

Figure 1

The neutron form factors $F_n\left(Q^2\right)$ for ${}^{40}$Ar (a), ${}^{74}$Ge (b), and ${}^{132}$Xe (c) predicted by the Skyrme functional SkM${}^{*}$ (solid black curve), and truncations of the expanded form factor at various orders of $Q$: $Q^0$ (dashed blue curve), $Q^2$ (dotted red curve), $Q^4$ (dot-dashed green curve), and, in the bottom panel, $Q^6$ (dot-dashed red curve). Terminating the expansion at $Q^4$ (with coefficient $\frac{1}{5!}\langle R_n^4\rangle$) gives good agreement with the full form factor in ${}^{40}$Ar and ${}^{74}$Ge over the range of $Q^2$ relevant for the scattering of neutrinos from stopped pion beams. Precise agreement in ${}^{132}$Xe requires the $Q^6$ term as well.

Figure 2

(a) Event rates in ${}^{40}$Ar as a function of recoil energy, with two different rms neutron radii. The red (solid) curve represents predictions of the Skyrme functional SkM${}^{*}$, while the blue (dotted) curve represents the same for an rms radius made $10\%$ larger, as described in the text. The flux at the detector is taken to be $3\times {10}^7$ neutrinos/(cm${}^2$ s) per flavor. (b) The difference between the two curves on top. The discontinuity in both panels is due to the monoenergetic muon neutrinos emitted promptly in pion decay.

Figure 3

Confidence regions in the ${\langle R_n^2\rangle }^{1/2}$–${\langle R_n^4\rangle }^{1/4}$ plane for an argon detector of mass 3.5 tonnes. The curves enclose confidence regions of 40$\%$, 91$\%$, and 97$\%$. The colored (shaded) vertical band shows the experimental result reported for the rms radius, obtained from argon-carbon scattering, in Ref. [44], and the black crosses are the predictions of some commonly used Skyrme functionals, including the functional SkM${}^{*}$ that we use to generate the “data.” Numerical results are summarized in Tables and . (a) The neutrino flux is allowed to vary by $\pm 10$$\%$. (b) The flux is assumed to be known exactly.

Figure 4

Same as Fig. 3, but for effective moments in germanium, and without experimental result.

Figure 5

Same as Fig. 4, but in xenon.