Elastic Neutrino–Atom Scattering as a Probe of Neutrino Millicharge and Magnetic Moment

Neutrino scattering on a target at low-energy transfer is one of the basic tools for searching for neutrino electromagnetic properties. We consider the effects of the neutrino millicharge and magnetic moment on the atomic recoil spectrum in elastic neutrino–atom scattering. The results of our calculations of differential cross sections for elastic collisions of tritium neutrinos with the H, ²H, ³He, and ⁴He atomic targets show that the corresponding experiments can achieve sensitivity to the indicated neutrino electromagnetic characteristics by orders of magnitude better than the available measurements of elastic neutrino–electron and neutrino–nucleus collisions.

The proposals are discussed in the literature to search for light dark matter particles using liquid and solid-state detectors, which make it possible to achieve sensitivity to low-energy signals down to energies of $\sim$1 meV (see, for example, [1–7]). Such detectors can also be used to study low-energy neutrino scattering [8, 9].

Studies of neutrino scattering on a target in the regime of low-energy transfer are of interest for a number of reasons. In 2017, the COHERENT collaboration reported the first experimental observation of coherent elastic neutrino–nucleus scattering (CE$\nu$NS) [10], which was theoretically predicted almost 50 years ago [11]. Ongoing measurements of CE$\nu$NS processes in the COHERENT experiment and a number of other experiments will allow not only testing the Standard Model of electroweak interactions at a new level of accuracy, but also exploring physics beyond the Standard Model, in particular, the electromagnetic properties of massive neutrinos [12]. For example, in [13], the first constraints on the neutrino transition charge radii were obtained.

However, the lower energy threshold of the detector in the same COHERENT experiment ($\sim$keV) does not allow reaching a new level of sensitivity to the most theoretically studied electromagnetic characteristic of a massive neutrino, namely, the magnetic moment, in comparison with experiments on elastic neutrino–electron scattering. In this regard, there are proposals in the literature for experiments on coherent elastic neutrino–atom scattering (CE$\nu$AS) using superfluid He-4 [8, 9], in which, due to the value of the detector’s lower energy threshold of $\sim$meV, one can achieve significantly better sensitivity to the neutrino magnetic moment than in the measurements of elastic neutrino–electron scattering [14–18].

For the first time, the CE$\nu$AS process was theoretically considered in the work of Gaponov and T-ikhonov [19]. In the framework of the $V{-} A$ theory of weak interaction, they showed that at neutrino energies $\lesssim {\kern 1pt} 10$ keV there is a region of coherent-optical neutrino phenomena, where the processes of elastic neutrino scattering on the atom as a whole dominate. Indeed, at these energies, the de Broglie wavelength of the neutrino is comparable to or even exceeds the atomic radius ${{R}_{{{\text{atom}}}}}$. This circumstance makes it possible to implement the regime of coherent elastic scattering $q{{R}_{{{\text{atom}}}}} \ll 1$, where $q = \left| {\mathbf{q}} \right|$ is the value of the three-dimensional momentum transfer, in a fairly wide range of neutrino scattering angles. In practice, neutrinos of the required energies can be obtained from a tritium source: electron antineutrinos produced in the beta decay of tritium have a continuous energy spectrum from 0 to 18.6 keV with an average value of 12.9 keV.

In a later work by other authors [20], within the Standard Model, it was predicted that for certain values of the atomic number and mass number in the elastic scattering of an electron (anti)neutrino by an atom, complete screening of the weak charge of the atomic nucleus by the electron cloud of the atom should be observed. Note that this screening effect is absent in the cases of muon and taon (anti)neutrinos. Complete screening is realized at a certain value of the recoil energy of the atom, which does not depend on the neutrino energy. In the case of the ⁴He atom, this energy is 9 meV [8], thus opening the door to testing the effects of physics beyond the Standard Model, which could give a non-zero contribution at recoil energies of about 9 meV. In this regard, of particular interest is the analysis of the sensitivity of the CE$\nu$AS process to the neutrino electromagnetic properties [12], the existence of which, particularly, the magnetic moment, is predicted already in the minimal extension of the Standard Model with right-handed massive Dirac neutrinos.

In this work, we analyze the sensitivity of CE$\nu$AS processes in the case of light atoms to such neutrino electromagnetic characteristics as electric charge ${{e}_{\nu }}$ (millicharge) and magnetic moment ${{\mu }_{\nu }}$ [12]. For this purpose, we take into account the neutrino millicharge and magnetic moment in the CE$\nu$AS cross section, which is differential with respect to the energy transfer, and on this basis we perform the corresponding numerical calculations.

We consider elastic neutrino–atom scattering in the following kinematical regime:

$${{E}_{\nu }} \ll m,\quad T \leqslant \frac{{2E_{\nu }^{2}}}{m} \ll {{E}_{\nu }},\quad {{E}_{\nu }} \ll \frac{1}{{{{R}_{{{\text{nuc}}}}}}},$$

where ${{E}_{\nu }}$ is the neutrino energy, $T$ is the energy transfer, $m$ is the atomic mass, and ${{R}_{{{\text{nuc}}}}}$ is the nuclear radius.

According to [8, 19–23], the differential cross section of the CE$\nu$AS process is given by

$$\frac{{d\sigma }}{{dT}} = \frac{{d{{\sigma }^{{(w,{{e}_{\nu }})}}}}}{{dT}} + \frac{{d{{\sigma }^{{({{\mu }_{\nu }})}}}}}{{dT}}.$$

(1)

The contribution from the weak interaction and neutrino millicharge has the form

$$\frac{{d{{\sigma }^{{(w,{{e}_{\nu }})}}}}}{{dT}} = \frac{{G_{F}^{2}m}}{\pi }\left[ {C_{V}^{2}\left( {1 - \frac{{mT}}{{2E_{\nu }^{2}}}} \right) + C_{A}^{2}\left( {1 + \frac{{mT}}{{2E_{\nu }^{2}}}} \right)} \right],$$

(2)

where

$${{C}_{V}} = Z\left( {\frac{1}{2} - 2{{{\sin }}^{2}}{{\theta }_{W}}} \right) - \frac{1}{2}N$$

$$ + \;Z\left( { \pm \frac{1}{2} + 2{{{\sin }}^{2}}{{\theta }_{W}}} \right){{F}_{{{\text{el}}}}}({{q}^{2}})$$

$$ + \;\frac{{\sqrt 2 \pi \alpha Z{{e}_{\nu }}}}{{{{G}_{F}}mT}}[1 - {{F}_{{{\text{el}}}}}({{q}^{2}})],$$

$$C_{A}^{2} = {{(C_{A}^{{{\text{nuc}}}})}^{2}} + \frac{1}{4}\sum\limits_{n,l} {{\left[ {\left( {L_{ + }^{{nl}} - L_{ - }^{{nl}}} \right)F_{{{\text{el}}}}^{{nl}}({{q}^{2}})} \right]}^{2}},$$

$${{(C_{A}^{{{\text{nuc}}}})}^{2}} = \frac{{g_{A}^{2}}}{4}{{\left[ {({{Z}_{ + }} - {{Z}_{ - }}) - ({{N}_{ + }} - {{N}_{ - }})} \right]}^{2}}.$$

Here $q$ is the momentum transfer, ${{q}^{2}} = 2mT$, the plus (minus) sign stands for the case of $\nu = {{\nu }_{e}}$ ($\nu = {{\nu }_{{\mu ,\tau }}}$), $Z$ ($N$) is the number of protons (neutrons) in the atomic nucleus, ${{F}_{{{\text{el}}}}}({{q}^{2}})$ is the Fourier-transform of the atomic electron density normalized to unity, or the atomic form factor, ${{g}_{A}} = 1.25$, ${{Z}_{ \pm }}$ and ${{N}_{ \pm }}$ are the numbers of protons and neutrons with the spin parallel (+) or antiparallel (–) to the nuclear spin, $L_{ \pm }^{{nl}}$ is the number of electrons on the $nl$ atomic orbital with the spin parallel (+) or antiparallel (–) to the total electron spin, and $F_{{{\text{el}}}}^{{nl}}({{q}^{2}})$ is the Fourier transform of the electron density on the $nl$ atomic orbital. The neutrino millicharge ${{e}_{\nu }}$ is in units of the elementary charge $e$, and $\alpha = {{e}^{2}}$ is the fine-structure constant.

The contribution from the neutrino magnetic moment is

$$\frac{{d{{\sigma }^{{({{\mu }_{\nu }})}}}}}{{dT}} = \frac{{\pi {{\alpha }^{2}}{{Z}^{2}}}}{{m_{e}^{2}}}{{\left| {{{\mu }_{\nu }}} \right|}^{2}}\left( {\frac{1}{T} - \frac{1}{{{{E}_{\nu }}}}} \right){{\left[ {1 - {{F}_{{{\text{el}}}}}({{q}^{2}})} \right]}^{2}},$$

(3)

where the neutrino magnetic moment ${{\mu }_{\nu }}$ is in units of the Bohr magneton ${{\mu }_{{\text{B}}}}$. Unlike the case of neutrino millicharge, the interaction of a neutrino due to its magnetic moment flips the neutrino helicity, and, hence, it does not interfere with the weak-interaction contribution to the cross section (1).

Below we present the numerical results for the differential cross sections of elastic scattering of electron antineutrinos with energy ${{E}_{\nu }} = 10$ keV, which is typical for the tritium source, on the hydrogen, deuterium, helium-3 and helium-4 atoms. In the numerical calculations, the effects due to nonzero neutrino millicharge $\left| {{{e}_{\nu }}} \right| = {{10}^{{ - 15}}}$ (in units of $e$) and magnetic moment ${{\mu }_{\nu }}{{ = 10}^{{ - 12}}}$ (in units of ${{\mu }_{{\text{B}}}}$) are taken into account. It should be noted that the most stringent laboratory bounds on the ${{\nu }_{e}}$ millicharge and magnetic moment are obtained from the analysis of the recent data of the XENONnT [17] and LUX-ZEPLIN [18] experiments on elastic scattering of solar neutrinos on electrons. The 90% C.L. bounds in the XENONnT case are ${{\mu }_{{{{\nu }_{e}}}}} < 0.85 \times {{10}^{{ - 11}}}$ and $- 2.5 \times {{10}^{{ - 13}}}$ < ${{e}_{{{{\nu }_{e}}}}}$ < $9.0 \times {{10}^{{ - 13}}}$ [24], and in the LUX-ZEPLIN case: ${{\mu }_{{{{\nu }_{e}}}}} < 1.5 \times {{10}^{{ - 11}}}$ and $- 2.1 \times {{10}^{{ - 13}}}$ < ${{e}_{{{{\nu }_{e}}}}}$ < $2.0 \times {{10}^{{ - 13}}}$ [25]. For comparison, the most stringent bounds derived from the analysis of the data on elastic scattering of reactor ${{\bar {\nu }}_{e}}$’s on electrons are ${{\mu }_{{{{\nu }_{e}}}}} < 2.9 \times {{10}^{{ - 11}}}$ [14] and $\left| {{{e}_{{{{\nu }_{e}}}}}} \right| < 1.5 \times {{10}^{{ - 12}}}$ [26].

For calculating the atomic form factors of hydrogen and deuterium we use the analytical expression

$${{F}_{{{\text{el}}}}}({{q}^{2}}) = {{\left( {1 + \frac{{{{q}^{2}}}}{{m_{e}^{2}\alpha }}} \right)}^{{ - 2}}}.$$

The form factor $F_{{{\text{el}}}}^{{(1s)}}({{q}^{2}})$ in these cases is given by the same expression.

As pointed out in [8], the atomic form factor of helium practically does not depend on the method of its calculation. We employ the ³He and ⁴He atomic form factors obtained using the relativistic Hartree–Fock approximation [27]. The electron contribution to ${{C}_{A}}$ in the case of the ³He and ⁴He atoms is absent.

Figure 1 presents the numerical results for the differential cross section (1) in the Standard Model and taking into account the neutrino magnetic moment. In all considered cases of an atomic target, taking into account the magnetic moment leads to a noticeable smoothing of the effect of screening of the weak charge of the nucleus by the electron cloud of the atom, which is predicted by the Standard Model. The smoothing is especially significant in the case of the helium-4 atom, where the screening effect is most pronounced. In addition, due to the neutrino magnetic moment, the absolute value of the cross section also noticeably increases: by almost an order of magnitude, for example, in the deuterium case.

Fig. 1.

(Color online) CE$\nu$AS differential cross sections within the Standard Model (${{\mu }_{\nu }} = 0$) and with account for neutrino magnetic moment (${{\mu }_{\nu }}{{ = 10}^{{ - 12}}}{{\mu }_{{\text{B}}}}$).

The results for the differential cross section, taking into account the neutrino millicharge, are shown in Fig. 2. In the case of atomic hydrogen, the millicharge contribution to the cross section is significant in absolute terms for both negative and positive values of the millicharge. In the case of deuterium, a noticeable dip in the cross section appears at $T \simeq 5$ meV, when ${{e}_{\nu }} > 0$. The interference of the contributions of the neutrino millicharge and the weak interaction to the cross section for neutrino scattering on the helium-4 atom is reflected in the position of the pronounced dip in the cross section predicted by the complete-screening effect in the Standard Model.

Fig. 2.

(Color online) CE$\nu$AS differential cross sections within the Standard Model (${{e}_{\nu }} = 0$) and with account for the neutrino millicharge (${{e}_{\nu }} = \pm {{10}^{{ - 15}}}$e).

The above results point to a large potential for the search for the neutrino magnetic moment and millicharge in the processes of elastic neutrino–atom scattering. The differential cross section (1) determines the shape of the atomic recoil spectrum, which, in principle, can be measured experimentally. From this point of view, the choice of light atoms as a target seems to be the most natural, since it increases the range of recoil energies available for measurement. In particular, the most promising at the moment is the use of a tritium neutrino source with a superfluid helium-4 detector. The corresponding experiment is already under preparation, and it can achieve the sensitivity to the neutrino magnetic moment at the level of $\sim {\kern 1pt} (2{-} 4) \times {{10}^{{ - 13}}}{{\mu }_{{\text{B}}}}$ (see [28] for details). It is supposed to involve a cylindrical tritium source with an initial activity of at least 10 MCi that will be surrounded by a cylindrically shaped 1-m³ volume of liquid helium-4 at temperatures as low as few tens of millikelvins. The flux of the tritium ${{\bar {\nu }}_{e}}$ in the liquid helium-4 volume will be at the level of  ~10¹³–10¹⁴ cm^–2 s^–1.

Tables 1 and 2 show the expected numbers of CE$\nu$AS events in the helium-4 detector after a 5-year data-taking in the case of the 10-MCi tritium activity for different values of the neutrino magnetic moment and millicharge. It should be noted that the amount of tritium in the upcoming experiment [28] can potentially be increased to reach the activity of 40 MCi. In such a case, the expected number of CE$\nu$AS events scales by a factor of approximately 3.3.¹

Table 1. Average number of expected CE$\nu$AS events ${{N}^{{{\text{CE}\nu \text{AS}}}}}$ in a superfluid He-4 detector after 5 years of data collection depending on the ${{\mu }_{\nu }}$ value (second row, in units of ${{\mu }_{{\text{B}}}}$)

Table 2. Same as in Table 1, but depending on the ${{e}_{\nu }}$ value (second row, in units of $e$)

Finally, some comments should be made about the potential of the experiment [28] to probe the neutrino millicharge. As can be seen from Table 2, the ${{e}_{\nu }}$ contribution to the CE$\nu$AS events in the helium-4 detector can be significant, especially in the ${{e}_{\nu }} < 0$ case, even if the $\left| {{{e}_{\nu }}} \right|$ value does not exceed ${{10}^{{ - 14}}}$ in units of $e$. This should be contrasted with the prospected combined limits $- 1.8 \times {{10}^{{ - 14}}}$ < ${{e}_{\nu }}$ < $1.8 \times {{10}^{{ - 14}}}$ [29] based on the current and future experiments on elastic neutrino–electron and neutrino–nucleus scattering.