# Exploring the hidden interior of the Earth with directional neutrino
measurements

Michael Leyton<EMAIL_ADDRESS>Institut de Física d’Altes Energies (IFAE),
Barcelona Institute of Science and Technology  Department of Physics, Royal
Holloway University of London  Laboratory for Nuclear Science, Massachusetts
Institute of Technology  Stephen Dye  Department of Physics and Astronomy,
University of Hawaii  Jocelyn Monroe  Department of Physics, Royal Holloway
University of London  Department of Physics, Massachusetts Institute of
Technology  High Energy Accelerator Research Organization (KEK)

###### Abstract

Roughly 40% of the Earth’s total heat flow is powered by radioactive decays in
the crust and mantle. Geo-neutrinos produced by these decays provide important
clues about the origin, formation and thermal evolution of our planet, as well
as the composition of its interior. Previous measurements of geo-neutrinos
have all relied on the detection of inverse beta decay reactions, which are
insensitive to the contribution from potassium and do not provide model-
independent information about the spatial distribution of geo-neutrino sources
within the Earth. Here we present a method for measuring previously unresolved
components of Earth’s radiogenic heating using neutrino-electron elastic
scattering and low-background, direction-sensitive tracking detectors. We
calculate the exposures needed to probe various contributions to the total
geo-neutrino flux, specifically those associated to potassium, the mantle and
the core. The measurements proposed here chart a course for pioneering
exploration of the veiled inner workings of the Earth.

## Introduction

The Earth’s surface heat flow Davies and Davies (2010) of 47 $\pm$ 2 TW is
fueled in part by the radioactive decay of uranium (U), thorium (Th), and
potassium (K) in the crust and mantle. The antineutrinos produced by these
decays, called geo-neutrinos (geo-$\overline{\nu}_{e}$s) due to their
geophysical origin, give us important clues about the composition of the
Earth’s interior and the size and sources of its internal heating due to
radioactivity, both in the current epoch and throughout its evolution.
Geo-$\overline{\nu}_{e}$s produced by decays from the radioactive cascades of
238U and 232Th have previously been measured by the KamLAND Araki  _et al._
(2005); Gando  _et al._ (2011, 2013) and Borexino Bellini  _et al._ (2010,
2013); Agostini  _et al._ (2015) experiments via the detection of inverse beta
decay on free protons, $\overline{\nu}_{e}p\rightarrow e^{+}n$, where
$\overline{\nu}_{e}$, $p$, $e^{+}$ and $n$ denote an electron antineutrino,
proton, positron and neutron, respectively.

While previous geo-$\overline{\nu}_{e}$ measurements Gando  _et al._ (2011)
have been used to constrain the portion of the Earth’s surface heat flow due
to radioactivity Usman  _et al._ (2015) (38% $\pm$ 23%), they are insensitive
to the contributions from 40K and 235U, whose geo-$\overline{\nu}_{e}$ spectra
fall entirely below the 1.8 MeV threshold of inverse beta decay on free
protons. Specifically, the 40K geo-$\overline{\nu}_{e}$ flux has never been
measured, although 40K decays may have been the dominant heat source for the
primordial Earth, due to the shorter half-life of 40K relative to 238U and
232Th. Measuring the 40K geo-$\overline{\nu}_{e}$ flux would provide a unique
view to the Earth’s origin and formation, as well as the composition of its
interior. In order to gain sensitivity to interactions of 40K
geo-$\overline{\nu}_{e}$s, a reaction with an energy threshold lower than the
1.3 MeV endpoint of the 40K beta ($\beta^{-}$) decay spectrum is required.

Moreover, existing measurements of geo-$\overline{\nu}_{e}$s from 238U and
232Th are non-directional and therefore do not provide model-independent
information about the location or spatial distribution of
geo-$\overline{\nu}_{e}$ sources in the crust and mantle. As a consequence,
resolution of crust and mantle contributions depends on geochemical modelling,
with the addition of associated uncertainties. Similarly, while the Earth’s
surface heat flow (47 $\pm$ 2 TW) is well measured Davies and Davies (2010),
the contributions from radiogenic heating Gando  _et al._ (2011); Usman  _et
al._ (2015), secular cooling of the mantle and heat flow out of the core Lay
_et al._ (2008) are relatively poorly constrained. Tightening these
constraints would provide better insight to the thermal evolution of our
planet.

Direction-sensitive measurements of geo-$\overline{\nu}_{e}$s could
additionally help unravel a long-standing mystery surrounding the source of
heat needed to power the geo-magnetic field, vital for protecting life on
Earth from the Sun’s radiation. In particular, constraints on the presence and
concentration of radioactive elements in the Earth’s core Rama Murthy  _et
al._ (2003); Pozzo  _et al._ (2012) are critical for understanding the geo-
dynamo and the evolution of the inner core. Recent measurements Wohlers and
Wood (2015) at high pressure (1.5 GPa) and high temperature ($\geq
1400\,^{\circ}$C) indicate that a sulfur-rich core would partition uranium
strongly and thorium slightly, resulting in U, Th concentrations of up to 10
p.p.b. in the core and producing 2.4 TW of heat, sufficient to power the geo-
dynamo, even without a contribution from 40K decay Rama Murthy  _et al._
(2003). Direction-sensitive measurements of geo-$\overline{\nu}_{e}$s could
shed light on this mystery by disentangling individual contributions from the
crust, mantle and core to the total geo-$\overline{\nu}_{e}$ flux.

We propose a method for measuring the geo-$\overline{\nu}_{e}$ flux from 40K,
the Earth’s mantle and the Earth’s core using neutrino-electron
($\nu_{\ell}\mathchar 45\relax e^{-}$) elastic scattering in low-background,
direction-sensitive tracking detectors. Unlike inverse beta decay, the elastic
scattering reaction discussed here,
$\nu_{\ell}e^{-}\rightarrow\nu_{\ell}e^{-}$, has no energy threshold and is
therefore sensitive to the 40K geo-$\overline{\nu}_{e}$ flux. Furthermore,
elastic scattering preserves directional information, since the direction of
the outgoing electron is well correlated with the direction of the incoming
neutrino, which can be exploited in order to reject backgrounds and
disentangle the various sources of geo-$\overline{\nu}_{e}$s. While
backgrounds from radioactivity can be quantified, controlled and minimized,
the dominant source of irreducible background to
$\overline{\nu}_{\ell}\mathchar 45\relax e^{-}$ elastic scattering comes from
elastic scattering of other neutrinos, primarily solar $\nu_{e}$s. However, a
detector with the ability to measure the direction of incident neutrinos has
additional rejection power against such events since the solar-$\nu_{e}$
background points back to the Sun, unlike much of the geo-$\overline{\nu}_{e}$
signal. Using the electron angle, energy and the position of the Sun at the
time of interaction, we show that a direction-sensitive detector can, in
principle, discriminate between the solar-$\nu_{e}$ background and the
geo-$\overline{\nu}_{e}$ signal.

In the following, we estimate the sensitivity of a direction-sensitive
detector to the flux of geo-$\overline{\nu}_{e}$s from 40K decays, the Earth’s
mantle and the Earth’s core. Detector resolution, efficiency and energy
thresholds are included in the analysis, based on the performance of previous
detectors Daraktchieva  _et al._ (2008); Dujmic  _et al._ (2008). We construct
detailed maps of the directions to geophysical structures Laske  _et al._
(2016); Dziewonski and Anderson (1981) for three underground sites where
large-scale neutrino observatories are currently operating and where such a
detector could be deployed: Kamioka Observatory, Japan; Laboratori Nazionale
del Gran Sasso (LNGS), Italy; and SNOLab, Canada. At each site, we compare the
geo-$\overline{\nu}_{e}$ signal derived from these maps with the calculated
background of $\nu_{e}$s from the Sun and $\overline{\nu}_{e}$s from nuclear
reactors around the world. Assuming ten live-years of operation, we find that
10-tonne-scale direction-sensitive detectors are capable of rejecting the
solar-$\nu_{e}$ background and resolving the predicted 40K
geo-$\overline{\nu}_{e}$ flux at 95% confidence level (CL). Similarly,
200-tonne-scale direction-sensitive detectors can identify the predicted
mantle geo-$\overline{\nu}_{e}$ flux at 95% CL, while 20-ktonne-scale
detectors can probe radioactivity in the core.

## Results

### Neutrino Flux Model

We consider the following neutrino fluxes: geo-$\overline{\nu}_{e}$s
111Although the majority of geo-neutrinos are antineutrinos
($\overline{\nu}_{e}$), a small fraction are neutrinos ($\nu_{e}$); however,
we will generically refer to them as $\overline{\nu}_{e}$ here. For 40K, the
neutrino spectrum is composed of 10.7% $\nu_{e}$ from electron capture and
89.3% $\overline{\nu}_{e}$ from beta ($\beta^{-}$) decay. from the Earth’s
crust, mantle and core; solar $\nu_{e}$s from the Sun; and reactor
$\overline{\nu}_{e}$s from worldwide nuclear power reactors. Interaction rates
for atmospheric and relic supernova neutrinos are estimated to be negligible.
A schematic illustrating these neutrino sources and their incident direction
with respect to a terrestrial detector is shown in Fig. 1.

Figure 1:  Schematic of signal and background neutrino sources. Neutrino
sources considered in this paper include geo-neutrinos
(geo-$\overline{\nu}_{e}$) from the Earth’s crust, mantle and core; solar
neutrinos (solar $\nu_{e}$) from the Sun; and reactor antineutrinos (reactor
$\overline{\nu}_{e}$) from man-made nuclear reactors.
$\theta_{\mathrm{zenith}}$ is the angle measured from the vertical axis, where
the vertical axis is defined opposite the direction pointing to the center of
the Earth. Figure is not drawn to scale. Figure 2:  Geophysical response of
seven silicate geochemical reservoirs. Geophysical response (g/cm2) calculated
at Gran Sasso (Italy) and shown here for the upper continental crust (UCC),
middle continental crust (MCC), lower continental crust (LCC), continental
lithospheric mantle (CLM), sediments (SED), oceanic crust (OCR) and mantle
(LID+MAN). The mantle is separated into a spherical shell (MAN) and a small
volume (LID) sitting atop that breaks the spherical symmetry at a given
location. The continental lithospheric mantle marks the transition between
continental crust and mantle, about 80 km on average beneath the surface of
the Earth. Values for SED (MAN) have been scaled up (down) by a factor of 10
(50) for the purposes of this figure. The azimuthal angle $\phi$ is measured
counterclockwise from due North, while $\theta_{\mathrm{nadir}}$ is measured
with respect to the direction pointing to the center of the Earth.

#### Geo-neutrinos

The flux of geo-$\overline{\nu}_{e}$s at the surface of the Earth depends on
the quantity and distribution of heat-producing elements, specifically U, Th
and K, in the Earth’s crust, mantle and core. The angular distribution of
geo-$\overline{\nu}_{e}$s is modeled by calculating the geophysical response
from eight potentially distinct geochemical reservoirs: sediments, upper,
middle and lower continental crust, continental lithospheric mantle, oceanic
crust, mantle and core. Seawater is not considered here, but estimated to
contribute negligibly ($<0.05$%) to the total geo-$\overline{\nu}_{e}$ flux.
The geophysical response, expressed in g/cm2, maps the spatial distribution
and density variations of a given reservoir and is shown in Fig. 2 for each of
the seven silicate (non-core) Earth reservoirs. The geo-$\overline{\nu}_{e}$
flux is then calculated from the geophysical response maps by assuming a
specified abundance of U, Th and K in each reservoir (see Methods for
details).

Table 1 lists the predicted fluxes of geo-$\overline{\nu}_{e}$s from U, Th and
K decay in the Earth’s crust, mantle and core, together with their respective
uncertainties, at the three underground sites. We consider two sub-crustal
Earth models, one with no radioactivity in the core and another with 10 p.p.b.
U, Th in the core, each of which corresponds to a total radiogenic heat of
$18\pm 11$ TW. The predicted incident angular distribution of
geo-$\overline{\nu}_{e}$s from the Earth’s crust and mantle at Gran Sasso is
shown in Fig. 3a and 3b, respectively.

#### Solar neutrinos

Solar-$\nu_{e}$ flux predictions, presented in Table 2, are taken from a
recent global analysis Bergstrom  _et al._ (2016) of solar and terrestrial
neutrino data in the framework of three-neutrino mixing. The model predictions
are all either in good agreement with recent measurements Bellini  _et al._
(2012, 2014a, 2014b) or reasonably well constrained Bellini  _et al._ (2012);
Gonzalez-Garcia  _et al._ (2010). Figure 3c shows the predicted incident
angular distribution of solar $\nu_{e}$s at Gran Sasso.

Site |  Reservoir |  $\Phi(^{238}$U) |  $\Phi(^{235}$U) |  $\Phi(^{232}$Th) |  $\Phi(^{40}$K)
---|---|---|---|---|---
($10^{6}$ cm-2 s-1) | ($10^{6}$ cm-2 s-1) | ($10^{6}$ cm-2 s-1) | ($10^{6}$ cm-2 s-1)
| Crust | 2.50 $\pm$ 0.65 | 0.08 $\pm$ 0.02 | 2.34 $\pm$ 0.35 | 10.0 $\pm$ 1.5
Kamioka | Mantle | 1.01 (0.71) $\pm$ 0.83 | 0.03 (0.02) $\pm$ 0.03 | 0.70 (0.63) $\pm$ 0.64 | 5.08 (3.56) $\pm$ 4.21
(Japan) | Core | 0.000 (0.300) | 0.000 (0.009) | 0.000 (0.066) | —
| Total | 3.51 $\pm$ 1.05 | 0.11 $\pm$ 0.03 | 3.04 $\pm$ 0.73 | 15.1 (13.6) $\pm$ 4.5
| Crust | 3.18 $\pm$ 0.79 | 0.10 $\pm$ 0.02 | 3.00 $\pm$ 0.43 | 12.7 $\pm$ 1.8
Gran Sasso | Mantle | 1.00 (0.70) $\pm$ 0.82 | 0.03 (0.02) $\pm$ 0.03 | 0.70 (0.63) $\pm$ 0.63 | 5.05 (3.54) $\pm$ 4.18
(Italy) | Core | 0.000 (0.301) | 0.000 (0.009) | 0.000 (0.066) | —
| Total | 4.19 $\pm$ 1.14 | 0.13 $\pm$ 0.04 | 3.70 $\pm$ 0.77 | 17.8 (16.3) $\pm$ 4.6
| Crust | 3.51 $\pm$ 0.91 | 0.11 $\pm$ 0.03 | 3.32 $\pm$ 0.48 | 14.4 $\pm$ 2.1
SNOLab | Mantle | 1.01 (0.71) $\pm$ 0.82 | 0.03 (0.02) $\pm$ 0.03 | 0.70 (0.63) $\pm$ 0.64 | 5.06 (3.54) $\pm$ 4.19
(Canada) | Core | 0.000 (0.301) | 0.000 (0.009) | 0.000 (0.066) | —
| Total | 4.52 $\pm$ 1.23 | 0.14 $\pm$ 0.04 | 4.02 $\pm$ 0.79 | 19.5 (18.0) $\pm$ 4.7
Table 1: Predicted flux of geo-neutrinos at three underground sites. Geo-neutrino flux ($\Phi$) at Kamioka (Japan), Gran Sasso (Italy) and SNOLab (Canada) associated to uranium (U), thorium (Th) and potassium (K) decays in the crust, mantle and core, assuming a sub-crustal Earth model with a homogeneous mantle and 0 (10) p.p.b. U, Th in the core. Values given here do not include the effect of neutrino oscillation. Uncertainties are calculated by propagating the uncertainties on the elemental abundances from Table 7. Source |  Predicted $\Phi$ |  Measured $\Phi$ |  Energy
---|---|---|---
(cm-2 s-1) |  (cm-2 s-1) | (MeV)
$pp$ |  $5.97$${}^{+0.04}_{-0.03}$ | $\times 10^{10}$ |  $6.6$${}^{+0.7}_{-0.7}$ | $\times 10^{10}$ | $<0.4$
7Be |  $4.80$${}^{+0.24}_{-0.22}$ | $\times 10^{9}~{}$ |  $4.75$${}^{+0.26}_{-0.22}$ | $\times 10^{9}~{}$ | 0.3, 0.8
13N |  $5.0$${}^{+8.6}_{-3.0}$ | $\times 10^{8}~{}$ |  $<7.7$ | $\times 10^{8}~{}$ | $<~{}2$
15O |  $1.3$${}^{+1.3}_{-0.9}$ | $\times 10^{8}~{}$ |  $<7.7$ | $\times 10^{8}~{}$ | $<~{}2$
17F |  $5.52$${}^{+79.48}_{-5.52}$ | $\times 10^{6}~{}$ |  – |  | $<~{}2$
8B |  $5.16$${}^{+0.13}_{-0.09}$ | $\times 10^{6}~{}$ |  $5.02$${}^{+0.17}_{-0.19}$ | $\times 10^{6}~{}$ | $<12$
$hep$ |  $1.9$${}^{+1.2}_{-0.9}$ | $\times 10^{4}~{}$ |  – |  | $<18$
$pep$ |  $1.45$${}^{+0.01}_{-0.01}$ | $\times 10^{8}~{}$ |  $1.6$${}^{+0.3}_{-0.3}$ | $\times 10^{8}~{}$ | 1.4
Total |  $6.53$${}^{+0.10}_{-0.05}$ | $\times 10^{10}$ |  – |  |
Table 2: Predicted and measured flux of solar neutrinos. Solar neutrino flux ($\Phi$) due to various nuclear fusion reactions, taken from the solar flux model of Bergstrom et al. Bergstrom  _et al._ (2016) and measurements by the Borexino collaboration Bellini  _et al._ (2012, 2014a, 2014b). Values do not include the effect of neutrino oscillation. The measured proton-proton ($pp$), 7Be, proton-electron-proton ($pep$) and 8B contributions are all in good agreement with model predictions. The CNO flux is constrained Bellini  _et al._ (2012) to $<7.7$$\times$$10^{8}$ cm-2 s-1 at 95% CL, also in agreement with the solar neutrino flux model. The remaining fluxes have not yet been directly observed, but the predictions are reasonably well constrained and have uncertainties at the level of 45% (CNO) and 77% ($hep$) Gonzalez-Garcia  _et al._ (2010).  Site |  $\Phi$
---|---
($10^{6}$ cm-2 s-1)
Kamioka (2010) |  $3.40$ p m 0.20
Kamioka |  $0.16$ p m 0.01
Gran Sasso |  $0.49$ p m 0.03
SNOLab |  $1.16$ p m 0.07
Table 3: Predicted flux of reactor antineutrinos at three underground sites.
Antineutrino flux ($\Phi$) from worldwide nuclear reactors, calculated at
Kamioka (Japan), Gran Sasso (Italy) and SNOLab (Canada) by averaging reactor
powers reported in a reference worldwide reactor model Baldoncini  _et al._
(2015) over 12 calendar months from 2014 (or 2010, where noted). The 2014
(2010) case at the Kamioka site corresponds to a scenario in which all
Japanese nuclear reactors have ceased (resumed) operation. Values given here
do not include the effect of neutrino oscillation. A total uncertainty of 6%
is estimated for all sites to account for uncertainties on the energy
spectrum, reported reactor powers and oscillation parameters (see Methods).

#### Reactor antineutrinos

The intensity of $\overline{\nu}_{e}$s from nuclear power reactors is
calculated at each underground site using positions and average powers of
operating cores Baldoncini  _et al._ (2015) from 2014 (or 2010), assuming a
spherical Earth. Table 3 lists the flux of reactor $\overline{\nu}_{e}$s at
each underground site, along with the total estimated uncertainty due to
several potential sources of error (see Methods for details). Figure 3d shows
the predicted incident angular distribution at Gran Sasso of
$\overline{\nu}_{e}$s from worldwide nuclear power reactors.

Figure 3:  Angular distribution of incident neutrinos at Gran Sasso. Angular
distributions calculated at Gran Sasso (Italy) and shown here for incident
neutrinos from: (a) the Earth’s crust; (b) the Earth’s mantle; (c) the Sun;
and (d) worldwide nuclear power reactors. The azimuthal angle $\phi$ is
measured clockwise from due North, while the zenith angle
$\theta_{\mathrm{zenith}}$ is measured with respect to the vertical axis,
defined opposite the direction pointing to the center of the Earth. (a, b)
Crust and mantle geo-neutrino distributions are derived using CRUST 1.0 Laske
_et al._ (2016) and PREM Dziewonski and Anderson (1981), supplemented with
detailed topological information for cos $\theta_{\mathrm{zenith}}<0$. (c) The
solar neutrino distribution is integrated over one calendar year at a latitude
of 42.45∘. (d) The directions to nuclear power reactors Baldoncini  _et al._
(2015) are shown here in red, superimposed on the crust geo-neutrino map for
visualization. The cluster at $\phi=120^{\circ}$, $0.5<$ cos
$\theta_{\mathrm{zenith}}<0.6$ is due to nuclear power reactors in the eastern
half of the United States, while the cluster at
$100^{\circ}<\phi<180^{\circ}$, $0.05<$ cos $\theta_{\mathrm{zenith}}<0.15$,
and continuing to $-180^{\circ}<\phi<-100^{\circ}$, is due to nuclear power
reactors in Western Europe, particularly France. Nuclear power reactors in
South America, South Africa, Japan and India are also visible here. All plots
are normalized to unit volume.

Note that the fluxes presented in Tables 1, 2 and 3 do not include the effect
of neutrino oscillation, which reduces the number of neutrinos by a factor
Bellini  _et al._ (2014a) of $(1-0.5$ $\mathrm{sin}^{2}$ $2\theta_{12})=0.55$,
on average, in the energy region of interest 222Neutrinos created with a
specific lepton flavor (electron, muon, or tau) can later be measured to have
a different flavor—a phenomenon known as neutrino oscillation. The probability
of oscillation (or survival) varies periodically as the neutrino propagates
through space and depends on $L/E_{\nu}$, where $L$ is the baseline, or
distance it has traveled, and $E_{\nu}$ is its energy. In the baseline and
energy range of interest for this analysis, 500 km $<L<1$ A.U. and 200 keV
$<E_{\nu}<11$ MeV. $L/E_{\nu}$ is therefore $\gg 1/\Delta m_{12}^{2}$ and the
survival probability after oscillation averages to $(1-0.5$ $\mathrm{sin}^{2}$
$2\theta_{12})=0.55$. Here, $\Delta m_{12}$ is the mass difference between the
two neutrino mass eigenstates and $\theta_{12}$ parameterizes the rotation
between the flavor and mass bases..

### Direction-Sensitive Detectors

Gas-filled time projection chambers (TPCs) fulfil the detection requirements
posed by the geo-$\overline{\nu}_{e}$ measurements discussed here. Several
groups Nygren (2007); Daraktchieva  _et al._ (2008); Ahlen  _et al._ (2010)
have developed TPCs to study physics requiring low backgrounds, a number of
which Daraktchieva  _et al._ (2008); Ahlen  _et al._ (2010); Dujmic  _et al._
(2008); Ahlen  _et al._ (2011) are capable of reconstructing the direction of
low-energy recoils well below the 1.3 MeV maximum energy of the 40K
geo-$\overline{\nu}_{e}$ spectrum 333Water Cherenkov detectors, although
direction-sensitive, typically have energy thresholds too high for triggering
on geo-$\overline{\nu}_{e}$s, while liquid scintillating detectors require
further development of techniques for resolving direction.. The mechanism
underlying these detectors is illustrated in Fig. 4.

Figure 4:  Schematic of the detection principle in a direction-sensitive
detector. A detector vessel is filled with CF4 (or other) gas at low or high
pressure. A uniform electric field is created by applying a potential
difference across cathode (blue) and ground (yellow) planes that are
electrically joined via step-down field cage rings (gray) at the outer
extremity of the fiducial volume. (a) A target electron (orange) in the gas-
filled volume recoils when struck by an incoming neutrino (magenta),
depositing a substantial fraction of its kinetic energy as ionization (green)
along its path. (b) The electric field then drifts this primary ionization to
an amplification region, consisting of closely-spaced ground and anode (red)
planes with a strong electric field between them. (c) There, the ionization
electrons undergo proportional multiplication, emitting electroluminescence or
scintillation light (violet) and producing an image of the two-dimensional
projection of the electron recoil track onto the amplification plane. If the
amplification plane is read out with sufficient granularity and the diffusion
in the drift volume is sufficiently small, the axial angle of the recoil track
can be reconstructed, while the vector direction can be determined from the
variation of the ionization density along the track. The track extent in the
drift coordinate can additionally be reconstructed using fast read out of the
charge time distribution of the avalanche signal.

$\Longrightarrow$ $\Longrightarrow$

In the following analysis, we assume detector resolution, efficiency
444Electron tracks with energies as low as 5.9 keV have been successfully
reconstructed in small-scale detectors Billard  _et al._ (2012), with
simulations showing that the reconstruction efficiency reaches $\sim$100% at
10 keV. Several direction-sensitive dark matter search experiments have
demonstrated vector track reconstruction at low recoil energies using gas-
filled TPCs at $\sim$0.1 bar. The DMTPC collaboration measured an angular
resolution Ahlen  _et al._ (2011); Dujmic  _et al._ (2008) of 40∘ for nuclear
(F) recoils with kinetic energy above 50 keV and 15∘ above 100 keV in CF4. The
track-finding efficiency Lopez (2014) was measured to be 95% above 80 keV. The
NEWAGE collaboration has measured an angular resolution Miuchi  _et al._
(2010) of 55∘ for nuclear recoil tracks above 100 keV in a CF4 gas-filled TPC
with $\mu$-PIC strip and gas electron multiplier (GEM) readout. The track
detection efficiency was measured to be 80% at an energy threshold of 100
keV., and energy thresholds based on the performance of previous detectors
Daraktchieva  _et al._ (2008); Dujmic  _et al._ (2008). In a search for
anomalous neutrino magnetic moment using $\overline{\nu}_{e}\mathchar 45\relax
e^{-}$ elastic scattering, the MUNU collaboration Daraktchieva  _et al._
(2008) demonstrated electron energy and direction reconstruction performance
in precisely the energy range relevant for the geo-$\overline{\nu}_{e}$
measurements proposed here. The detector was a 1 m3 TPC, filled with CF4 gas
at 1 bar, instrumented with wire and strip readout at a pitch of 4.95 mm. The
electron energy ($T$) resolution Daraktchieva  _et al._ (2008) was determined
to be 10% at 200 keV and 6.8% at 478 keV, scaling as $T^{0.57}$. The measured
electron angular resolution Daraktchieva  _et al._ (2005, 2008) varied from
15∘ at 200 keV, to 12∘ at 400 keV, to 10∘ at 600 keV. A best fit to these data
points yields an energy dependence proportional to $T^{-0.361}$:

$\displaystyle\sigma_{\theta}~{}[^{\circ}]=102\,T^{-0.361},$ (1)

with electron energy, $T$, in keV.

In addition to requirements on detector response and resolution, the
geo-$\overline{\nu}_{e}$ measurements proposed here require detectors with
large target masses and ultra-low background rates. High-pressure time
projection chambers, operating at pressures up to 10–15 bar, offer a promising
solution. HPTPCs filled with gaseous xenon (Xe) are currently being used in
searches for neutrino-less double beta decay Martín-Albo  _et al._ (2016);
Álvarez  _et al._ (2012) and have successfully demonstrated excellent energy
resolution and background rates of $<\,$4$\times$10-4 counts/keV/kg/year.
Next-generation gaseous-Xe HPTPC experiments, such as PandaX-III Chen  _et
al._ (2016), will feature target masses at the tonne scale.

For the geo-$\overline{\nu}_{e}$ sensitivity studies presented here, we assume
that the techniques employed by experiments searching for dark matter Akerib
_et al._ (2015); Aprile  _et al._ (2011); Ahlen  _et al._ (2010) or neutrino-
less double beta decay Martín-Albo  _et al._ (2016); Álvarez  _et al._ (2012);
Chen  _et al._ (2016) can be used to mitigate backgrounds arising from the
radioactivity of detector components, cosmogenic activation of the target gas
and any external neutron, photon and muon backgrounds. We consider the impact
of these backgrounds on the estimated sensitivity in the Discussion section.

### Event Rates

We calculate the $\nu_{\ell}\mathchar 45\relax e^{-}$ elastic scattering cross
section for both $\nu_{\ell}$ and $\overline{\nu}_{\ell}$ following a standard
example from the literature Vogel and Engel (1989); Bahcall  _et al._ (1995)
(see Methods). Using the differential cross section from equation (5),
combined with the geo-$\overline{\nu}_{e}$, solar-$\nu_{e}$ and
reactor-$\overline{\nu}_{e}$ flux values from Tables 1, 2 and 3, respectively,
we calculate the predicted number of events per tonne-year exposure as a
function of recoil electron kinetic energy, assuming a CF4 target. Our
calculation includes the effect of oscillation,
$\nu_{e}\rightarrow\nu_{\mu},\nu_{\tau}$ and subsequent $\nu_{\mu}\mathchar
45\relax e^{-}$ or $\nu_{\tau}\mathchar 45\relax e^{-}$ elastic scattering.

The predicted event rates, including the effect of oscillation, are shown in
Fig. 5 and listed in Tables 4 and 5. The geo-$\overline{\nu}_{\ell}$ signal to
solar-$\nu_{\ell}$ background ratio is maximum for an electron energy
threshold of 200 keV. In the following section, energy thresholds of 200, 250
and 800 keV are chosen to maximize the signal-to-background ratio and
sensitivity of each study.

Figure 5:  Predicted event rate over threshold per tonne-year exposure. Event rates over threshold are shown here versus electron energy threshold (keV) for solar neutrino (blue), geo-neutrino (red) and reactor antineutrino (orange) events, assuming a CF4 target and 45% probability of oscillation into $\nu_{\mu}$ or $\nu_{\tau}$. Geo-neutrino and reactor antineutrino rates are calculated using the normalizations at Gran Sasso from Tables 1 and 3, respectively. Error bands shown here correspond to uncertainties given in Tables 4 and 5. Event rates are approximately 0.4% higher for a SF6 target, or 13.8% lower for a Xe target. Source |  Total |  $>200$ keV |  $>250$ keV |  $>800$ keV
---|---|---|---|---
238U |  $61.26$ p m 16.68 |  $35.60$ p m 9.69 |  $31.73$ p m 8.64 |  $7.44$ p m 2.02
235U |  $1.36$ p m 0.37 |  $0.68$ p m 0.19 |  $0.59$ p m 0.16 |  $0.05$ p m 0.01
232Th |  $44.19$ p m 9.17 |  $24.18$ p m 5.02 |  $21.55$ p m 4.47 |  $4.65$ p m 0.96
40K |  $272.17$ p m 69.90 |  $126.66$ p m 32.53 |  $105.97$ p m 27.21 |  $4.69$ p m 1.21
geo-$\overline{\nu}_{\ell}$ total |  $378.98$ p m 72.44 |  $187.12$ p m 34.31 |  $159.83$ p m 28.90 |  $16.83$ p m 2.55
reactor $\overline{\nu}_{\ell}$ |  $17.69$ p m 1.06 |  $13.35$ p m 0.80 |  $12.50$ p m 0.75 |  $6.42$ p m 0.38
Table 4:  Predicted number of geo-neutrino and reactor antineutrino events. Event rates (per ktonne-year exposure) are calculated at Gran Sasso (Italy) and given here for three electron recoil energy thresholds, assuming a CF4 target and 45% probability of oscillation into $\nu_{\mu}$ or $\nu_{\tau}$. Rates (uncertainties) are calculated assuming the predicted flux normalizations (uncertainties) from Tables 1 and 3, respectively. Event rates are approximately 0.4% higher for a SF6 target, or 13.8% lower for a Xe target. Source |  Total |  $>200$ keV |  $>250$ keV |  $>800$ keV
---|---|---|---|---
$pp$ |  $433.97$${}^{+2.69}_{-2.40}$ |  $16.09$${}^{+0.10}_{-0.09}$ |  $0.50$${}^{+0.00}_{-0.00}$ |  $0.00$${}^{+0.00}_{-0.00}$
7Be (1) |  $156.78$${}^{+7.84}_{-7.19}$ |  $105.88$${}^{+5.29}_{-4.85}$ |  $93.59$${}^{+4.68}_{-4.29}$ |  $0.00$${}^{+0.00}_{-0.00}$
7Be (2) |  $3.57$${}^{+0.18}_{-0.16}$ |  $0.43$${}^{+0.02}_{-0.02}$ |  $0.00$${}^{+0.00}_{-0.00}$ |  $0.00$${}^{+0.00}_{-0.00}$
13N |  $13.88$${}^{+23.88}_{-8.33}$ |  $8.47$${}^{+14.57}_{-5.08}$ |  $7.30$${}^{+12.56}_{-4.38}$ |  $0.24$${}^{+0.41}_{-0.14}$
15O |  $5.64$${}^{+5.64}_{-3.90}$ |  $4.16$${}^{+4.16}_{-2.88}$ |  $3.79$${}^{+3.79}_{-2.62}$ |  $0.95$${}^{+0.95}_{-0.66}$
17F |  $0.23$${}^{+3.36}_{-0.23}$ |  $0.17$${}^{+2.48}_{-0.17}$ |  $0.16$${}^{+2.29}_{-0.16}$ |  $0.04$${}^{+0.56}_{-0.04}$
8B |  $1.80$${}^{+0.05}_{-0.03}$ |  $1.75$${}^{+0.04}_{-0.03}$ |  $1.73$${}^{+0.04}_{-0.03}$ |  $1.58$${}^{+0.04}_{-0.03}$
$hep$ |  $0.01$${}^{+0.01}_{-0.00}$ |  $0.01$${}^{+0.01}_{-0.00}$ |  $0.01$${}^{+0.01}_{-0.00}$ |  $0.01$${}^{+0.01}_{-0.00}$
$pep$ |  $9.39$${}^{+0.08}_{-0.08}$ |  $7.73$${}^{+0.07}_{-0.07}$ |  $7.33$${}^{+0.07}_{-0.07}$ |  $3.09$${}^{+0.03}_{-0.03}$
solar $\nu_{\ell}$ total |  $625.28$${}^{+26.12}_{-11.92}$ |  $144.70$${}^{+16.24}_{-7.60}$ |  $114.40$${}^{+14.12}_{-6.67}$ |  $5.90$${}^{+1.18}_{-0.68}$
Table 5:  Predicted number of solar neutrino events. Event rates (per tonne-
year exposure) are given here for three electron recoil energy thresholds,
assuming a CF4 target and 45% probability of oscillation into $\nu_{\mu}$ or
$\nu_{\tau}$. Rates (uncertainties) are calculated assuming the predicted flux
normalizations (uncertainties) from Table 2. Event rates are approximately
0.4% higher for a SF6 target, or 13.8% lower for a Xe target.

### Geo-neutrino Sensitivity Analysis

We estimate the sensitivity of multi-tonne-scale direction-sensitive detectors
to three sources of geo-$\overline{\nu}_{e}$s: 40K decays, the Earth’s mantle
and the Earth’s core. Our methodology is described in the Methods section.
Briefly, we simulate many pseudo-experiments and use a profile likelihood
statistic to assess the exposure required to either set an upper limit for
background-only pseudo-experiments, or exclude the null hypothesis for signal
+ background pseudo-experiments at 95%, or 90%, CL.

Figure 6: Zenith angle distribution of electron recoils induced by neutrinos.
Zenith angle distributions of electron recoils induced by signal geo-neutrinos
from: (a) potassium (40K) decays; (b) the mantle, assuming no radioactivity in
the core; and (c) the core, assuming 10 p.p.b. uranium and thorium in the
core. Signal-induced recoils (red solid) are shown together with recoils
induced by background geo-neutrinos (green hatched), solar neutrinos (blue
striped) and reactor antineutrinos (orange weave). Pseudo-experimental data
(black squares) are also shown, with their statistical errors. Recoil
distributions are calculated at Gran Sasso (Italy) and normalized to an
exposure of (a) 140 tonne-years, (b) 2.6 ktonne-years or (c) 200 ktonne-years,
including the effect of neutrino oscillation, with an applied electron energy
threshold of $T>$ (a) 200 keV, (b) 250 keV or (c) 800 keV. Reconstructed
angles have been smeared with a Gaussian distribution according to the angular
resolution parameterization given in equation (1). The zenith angle
($\theta_{\mathrm{zenith}}$) is measured with respect to the vertical axis,
defined opposite the direction pointing to the center of the Earth. A cut
requiring a minimum angular separation from the Sun ($\theta_{\mathrm{sun}}$)
has been applied in order to reject the dominant background from solar
neutrino events: cos $\theta_{\mathrm{sun}}<$ (a) -0.09, (b) 0.02 or (c) 0.54.

For all studies, a flat $\pm$7% systematic uncertainty is applied on the
pseudo-data, intended to cover measurement uncertainties on the global
acceptance, track reconstruction efficiency, cross section, number of targets
in the fiducial volume, energy resolution and angular resolution, in addition
to uncertainties on oscillation parameters. This value is consistent with the
systematic uncertainty reported by the MUNU collaboration Daraktchieva  _et
al._ (2005). A ${}^{+1.5\%}_{-0.8\%}$ systematic uncertainty on the
normalization of the total solar-$\nu_{e}$ flux is also applied (Table 2).
This results in a ${}^{+11.2\%}_{-~{}\,5.3\%}$, ${}^{+12.3\%}_{-~{}\,5.8\%}$,
or ${}^{+20.0\%}_{-11.5\%}$ change in the solar-$\nu_{\ell}$ event rate for an
energy threshold of 200, 250 or 800 keV, respectively (Table 5). Similarly, a
$\pm 6$% uncertainty on the reactor-$\overline{\nu}_{e}$ flux and
corresponding event rate is also applied to cover uncertainties on the energy
spectrum, reactor powers and oscillation parameters (Methods, Tables 3 and 4).
The remaining systematic uncertainty on the geo-$\overline{\nu}_{e}$
background varies according to each study and is discussed in the respective
sections below. Energy thresholds have been chosen to optimize the sensitivity
of each analysis (i.e. minimize the required exposure), assuming the angular
resolution parameterization given in equation (1).

####  40K geo-neutrino flux

We first estimate the sensitivity of a direction-sensitive detector to a non-
zero 40K geo-$\overline{\nu}_{e}$ flux. For electron recoil energies below the
1.3 MeV endpoint energy of 40K $\beta^{-}$ decay, the detector proposed here
would be unable to distinguish, on an event-by-event basis, the radioactive
isotope from which an observed geo-$\overline{\nu}_{\ell}$-induced event was
produced 555A spectral analysis of the electron recoil events pointing anti-
parallel to the Sun (cos $\theta_{\mathrm{sun}}<0$) at the time of interaction
should, in principle, exhibit a unique signature (namely, a kink) at 1.3 MeV
due to the fall-off of the 40K energy spectrum. This spectral feature can aid
considerably in justifying the claim of a detected signal associated to 40K
geo-$\overline{\nu}_{e}$s.. A measurement of the 40K geo-$\overline{\nu}_{e}$
flux therefore benefits from prior experimental knowledge of the contributions
from U and Th decays.

We use the most recently published measurements at Kamioka Gando  _et al._
(2013) and Gran Sasso Agostini  _et al._ (2015) to constrain our model
prediction and estimate the background from U and Th geo-$\overline{\nu}_{e}$s
(Table 1). The uncertainties on the predicted geo-$\overline{\nu}_{e}$ flux
result in a $\pm$20%, $\pm$18% and $\pm$18% change in the predicted
geo-$\overline{\nu}_{\ell}$ background event rate at Kamioka, Gran Sasso and
SNOLab, respectively, for an energy threshold of 200 keV (Table 4). Although
the uncertainties on these geo-$\overline{\nu}_{e}$ flux measurements may
decrease between now and the completed construction of a multi-tonne-scale
detector, we have instead opted for a conservative scenario using current
measurement uncertainties.

To reject the background from solar-$\nu_{\ell}$ events, a cut on the angular
separation from the Sun at the time of interaction, $\theta_{\mathrm{sun}}$,
is applied by fitting the measured spectrum (following scattering and angular
smearing) for all recoil electrons with energy $T>$ 200 keV with the
functional form given in equation (9) (see Methods for details). The cut on
electron energy has been optimized for this study to enhance the signal-to-
background ratio (Fig. 5). Using the results of the fit, cos
$\theta_{\mathrm{sun}}$ is required to lie outside of the range $\mu\pm
4\sigma$, or cos $\theta_{\mathrm{sun}}<$ -0.09. Placement of the cut at
4$\sigma$ has been optimized for this particular study and energy threshold.
For an energy threshold of 200 keV and assuming the angular resolution
parameterization given in equation (1), this cut accepts, on average, 44% of
geo-$\overline{\nu}_{\ell}$ and reactor-$\overline{\nu}_{\ell}$ events (both
signal and background) and 0.009% of solar-$\nu_{\ell}$ events. The acceptance
was studied as a function of electron recoil energy, over the kinematic range
$T<$ 2.0 MeV, and seen to vary by at most 1.3% for geo-$\overline{\nu}_{\ell}$
events and reactor-$\overline{\nu}_{\ell}$ events, safely within the allotted
$\pm$7% measurement uncertainty.

For this study, we assume an Earth model with no radioactivity in the core. An
example pseudo-experiment at Gran Sasso is shown in Fig. 6a. The results of
the sensitivity analysis are presented in Table 6 and shown in Fig. 7a, b. The
background-only measurement is statistics-limited for exposures $<200$ tonne-
years, while the signal + background measurement is statistics-limited for
exposures $\lesssim 500$ tonne-years.

#### Mantle geo-neutrino flux

We next estimate the sensitivity of a direction-sensitive detector to the flux
of geo-$\overline{\nu}_{e}$s coming from the Earth’s mantle by treating all
other sources, including geo-$\overline{\nu}_{e}$s from the Earth’s crust, as
background. From an optimization study, an energy threshold of 250 keV is
chosen to enhance the signal-to-background ratio (Fig. 5), but also remove
lower energy tracks that scatter significantly and do not point along the
incident neutrino direction. Similar to the previous study, a cut on cos
$\theta_{\mathrm{sun}}$ is applied in order to remove the solar-$\nu_{\ell}$
background. For this study and energy threshold, cos $\theta_{\mathrm{sun}}$
is required to lie outside of the range $\mu\pm 4\sigma$, or cos
$\theta_{\mathrm{sun}}<0.02$. Assuming the angular resolution parameterization
given in equation (1), this cut accepts, on average, 49% of crust and mantle
geo-$\overline{\nu}_{\ell}$ events, 49% of reactor $\overline{\nu}_{\ell}$
events and 0.008% of solar-$\nu_{\ell}$ events. Uncertainties on the predicted
geo-$\overline{\nu}_{e}$ flux are also applied (Table 1), resulting in a
$\pm$11% change in the predicted background geo-$\overline{\nu}_{\ell}$ event
rate at all sites (Table 4).

For this study, we assume an Earth model with no radioactivity in the core. An
example pseudo-experiment at Gran Sasso is shown in Fig. 6b. The results of
the sensitivity analysis are presented in Table 6 and shown in Fig. 7c, d. The
background-only measurement is statistics-limited for exposures $<1500$ tonne-
years. The signal + background measurement is statistics-limited for exposures
$\lesssim 3000$ tonne-years.

#### Core geo-neutrino flux

Finally, we estimate the sensitivity of a direction-sensitive detector to the
flux of geo-$\overline{\nu}_{e}$s coming from the Earth’s core, assuming an
abundance of 10 p.p.b. U and Th, but no K. All other sources, including
geo-$\overline{\nu}_{e}$s from the Earth’s crust and mantle, are treated as
background. Based on optimization results, a higher energy threshold of 800
keV is used for this study in order to select a sample of electron recoils
with good pointing resolution to the Earth’s core, after the effects of
scattering and angular smearing have been applied.

Similar to before, a cut on cos $\theta_{\mathrm{sun}}$ is applied in order to
reject the solar-$\nu_{\ell}$ background. For this study and energy threshold,
cos $\theta_{\mathrm{sun}}$ is required to lie outside of the range $\mu\pm
4.5\sigma$, or cos $\theta_{\mathrm{sun}}<0.54$. Assuming the angular
resolution parameterization given in equation (1), this cut accepts, on
average, 78% of core geo-$\overline{\nu}_{\ell}$ events, 75% of crust
geo-$\overline{\nu}_{\ell}$ events, 77% of mantle geo-$\overline{\nu}_{\ell}$
events, 74% of reactor $\overline{\nu}_{\ell}$ events and 0.03% of
solar-$\nu_{\ell}$ events. A $\pm$7% systematic uncertainty is applied to the
geo-$\overline{\nu}_{\ell}$ background, assuming that the
geo-$\overline{\nu}_{e}$ flux from 40K decays and the mantle have previously
been measured, as described above.

An example pseudo-experiment at Gran Sasso is shown in Fig. 6c. The results of
the sensitivity analysis are presented in Table 6 and shown in Fig. 7e, f. The
background-only measurement is statistics-limited for exposures $<90$ ktonne-
years. The signal + background measurement is statistics-limited for exposures
$\lesssim 180$ ktonne-years.

Figure 7:  Mean geo-neutrino flux sensitivity at Gran Sasso. Flux sensitivity is calculated at Gran Sasso (Italy) and shown here versus exposure for the geo-neutrino flux from: (a, b) potassium (40K) decays; (c, d) the mantle, assuming no radioactivity in the core; and (e, f) the core, assuming 10 p.p.b. uranium and thorium in the core. Error bars denote s.e.m. Each curve has been fit with a line plus exponential. (a, c, e) show the mean 95% (90%) confidence level (CL) upper limits for background-only pseudo-experiments; the star-shaped markers represent the exposures needed to achieve an upper limit equal to the predicted signal flux. (b, d, f) show the mean 95% (90%) confidence interval for signal + background pseudo-experiments; the star-shaped markers represent the exposures at which 95% (90%) of pseudo-experiments have a non-zero lower limit. A CF4 target and angular resolution given by equation (1) have been assumed, with an applied electron energy threshold of $T>$ (a, b) 200 keV, (c, d) 250 keV, or (e, f) 800 keV. $\Phi$ |  Site (configuration) |  Case (i) |  Case (ii)
---|---|---|---
background only |  signal + background
(tonne-yr) |  (tonne-yr) |  ($10^{6}$ cm-2 s-1)
40K | Kamioka | 122 ( 85) | 157 (107) | 2.84 (3.28)
Kamioka (2010) | 200 (130) | 279 (190) | 3.31 (3.67)
Gran Sasso | 109 ( 75) | 140 ( 99) | 3.51 (4.01)
Gran Sasso (high res.) |  70 ( 48) |  85 ( 58) | 3.38 (3.88)
Gran Sasso (no syst.) | 106 ( 74) | 132 ( 94) | 3.63 (4.07)
Gran Sasso (w/bkgnd) | 125 ( 86) | 161 (113) | 3.51 (4.12)
SNOLab | 105 ( 72) | 132 ( 93) | 3.62 (4.18)
| Kamioka | 635 (437) | 1,980 (1,050) | 2.57 (2.51)
| Kamioka (2010) | 829 (567) | 2,630 (1,490) | 2.79 (2.80)
Mantle (no | Gran Sasso | 767 (523) | 2,560 (1,420) | 2.74 (2.73)
radioactivity | Gran Sasso (high res.) | 525 (356) | 1,630 ( 933) | 2.71 (2.68)
in core) | Gran Sasso (no syst.) | 705 (488) | 1,850 (1,050) | 2.50 (2.51)
| Gran Sasso (w/bkgnd) | 915 (608) | 3,010 (1,680) | 2.76 (2.77)
| SNOLab | 826 (564) | 2,680 (1,560) | 2.79 (2.83)
| Kamioka | 73,600 (53,300) | 215,000 (136,000) | 0.176 (0.185)
Core | Kamioka (2010) | 87,300 (59,000) | 343,000 (200,000) | 0.170 (0.176)
(10 p.p.b. | Gran Sasso | 64,900 (47,600) | 202,000 (131,000) | 0.173 (0.183)
U, Th | Gran Sasso (high res.) | 54,400 (40,300) | 165,000 (102,000) | 0.172 (0.180)
in core) | Gran Sasso (no syst.) | 62,000 (46,200) | 192,000 (126,000) | 0.180 (0.188)
| SNOLab | 64,000 (46,600) | 223,000 (131,000) | 0.171 (0.175)
Table 6: Mean geo-neutrino flux sensitivity at three underground sites. Mean
sensitivity to the geo-neutrino flux ($\Phi$) from potassium (40K) decays, the
mantle and the core, calculated at Kamioka (Japan), Gran Sasso (Italy) and
SNOLab (Canada), assuming a CF4 target and angular resolution given by
equation (1). Case (i): exposure needed to achieve a mean 95% (90%) confidence
level upper limit equal to the predicted flux, for background-only pseudo-
experiments. Case (ii): exposure and mean lower limit at which 95% (90%) of
signal + background pseudo-experiments have a non-zero 95% (90%) confidence
level lower limit. An electron energy threshold of 200 keV (40K), 250 keV
(mantle) or 800 keV (core) has been applied. Systematic uncertainties for each
study are discussed in the main text. Four variations of the baseline
configuration are also presented here: (2010) uses reactor positions and
powers from 2010, rather than 2014; (high res.) denotes improved angular
resolution, equal to half that given by equation (1); (no syst.) denotes no
systematic uncertainties; and (w/bkgnd) denotes inclusion of isotropic
backgrounds, normalized to $20\%\pm 2\%$ ($62\%\pm 6\%$) of the 40K (mantle)
geo-neutrino signal.

## Discussion

This paper introduces a method for measuring previously unresolved sources of
radiogenic heating within the Earth associated to the geo-$\overline{\nu}_{e}$
flux from 40K, the mantle and the core. The technique exploits the directional
information of $\nu_{\ell}\mathchar 45\relax e^{-}$ elastic scattering in
order to positively identify rare signal interactions over significant
backgrounds commonly thought of as irreducible. Gas-filled time projection
chambers fulfil the detection requirement of tracking the direction of low-
energy recoil electrons. The analysis presented here incorporates original,
detailed mapping of geophysical structure, realistic assessments of neutrino
background and the effects of detector response, in order to estimate the
exposures needed to probe the geo-$\overline{\nu}_{e}$ flux from 40K, the
mantle and the core. In each case, direction reconstruction, with modest
angular resolution previously demonstrated by past experiments, mitigates the
dominant background to the geo-$\overline{\nu}_{e}$ signal coming from solar
$\nu_{e}$s.

Assuming ten live-years of operation and nuclear reactor positions and powers
from 2014, we find that 10-tonne-scale direction-sensitive detectors have
sensitivity to the predicted 40K geo-$\overline{\nu}_{e}$ flux at 95%
confidence level (CL) at the three underground laboratories studied here:
Kamioka (Japan), Gran Sasso (Italy) and SNOLab (Canada). With an exposure of
105–122 (72–85) tonne-years, a direction-sensitive detector with a CF4 target
and angular resolution parameterized by equation (1) can set a 95% (90%) CL
upper limit on the predicted 40K geo-$\overline{\nu}_{e}$ flux of
15.1–19.5$\times$$10^{6}$ cm-2 s-1 under the assumption that there is no
signal. An exposure of 132–157 (93–107) tonne-years is needed to exclude the
null hypothesis and set a non-zero 95% (90%) CL lower limit 95% (90%) of the
time. For this study, the calculated exposures depend strongly on the angular
resolution of the detector since it plays an important role in discriminating
signal (geo-$\overline{\nu}_{\ell}$) vs. background (solar-$\nu_{\ell}$)
events, particularly for low-energy events that scatter considerably with
respect to the incident neutrino direction. Enhancing the angular resolution
of the detector by a factor of 2 results in a significant reduction in the
required exposures: 39–41% reduction for the signal + background case and
36–37% reduction in the background-only case. In contrast, removing all
systematic uncertainties reduces the required exposure only marginally, by
4.8–5.8% for the signal + background result and by 2.1–2.9% for the
background-only result.

Under the same set of assumptions, we find that a 200-tonne-scale direction-
sensitive detector has sensitivity to the predicted geo-$\overline{\nu}_{e}$
flux from the Earth’s mantle at 95% CL, assuming no radioactivity in the core.
With an exposure of 635–826 (437–564) tonne-years, a direction-sensitive
detector can set a 95% (90%) CL upper limit on the predicted mantle
geo-$\overline{\nu}_{e}$ flux of 6.8$\times$$10^{6}$ cm-2 s-1 assuming that
there is no signal, or set a non-zero 95% (90%) CL lower limit 95% (90%) of
the time with an exposure of 1980–2680 (1050–1560) tonne-years. For this
particular study, a detector at Kamioka requires the smallest exposure of the
sites considered, due to Kamioka having the largest predicted ratio of mantle
geo-$\overline{\nu}_{e}$s (signal) to crust geo-$\overline{\nu}_{e}$s
(background). The higher predicted ratio of mantle-to-crust flux at Kamioka is
a consequence of combining the assumption of a homogeneous mantle underneath
all sites with the lower total geo-$\overline{\nu}_{e}$ flux measured by
KamLAND, relative to Borexino. As before, the results depend strongly on the
assumed angular resolution since it is a crucial ingredient in the
discrimination between signal geo-$\overline{\nu}_{e}$s from the mantle and
background geo-$\overline{\nu}_{e}$s from the crust. Enhancing the angular
resolution of the detector by a factor of 2 results in a 34–36% reduction in
the required exposure for the signal + background case, or a 32% reduction for
that of the background-only case. Systematic uncertainties play a larger role
here than in the previous study, particularly in the signal + background case.
Removing all considered uncertainties reduces the required exposure by 26–28%
in the signal + background case and by 6.7–8.1% in the background-only case.

Finally, we find that 20-ktonne-scale direction-sensitive detectors have
sensitivity to the predicted geo-$\overline{\nu}_{e}$ flux from the Earth’s
core at 95% CL, assuming a concentration of 10 p.p.b. of U and Th in the core.
Although this target mass is perhaps beyond the feasibility of current gaseous
detectors, the results are included here for completeness. A direction-
sensitive detector can set a 95% (90%) CL upper limit on the predicted core
geo-$\overline{\nu}_{e}$ flux of 0.38$\times$$10^{6}$ cm-2 s-1 with an
exposure of 64–74 (47–53) ktonne-years, assuming that there is no signal. To
set a non-zero 95% (90%) CL lower limit 95% (90%) of the time, an exposure of
202–223 (131–136) ktonne-years is needed. Angular resolution plays a less
critical role for this study since the higher energy threshold of 800 keV
ensures smaller angular scattering of the electron recoil with respect to the
incident neutrino. Improving the angular resolution by a factor of 2 reduces
the exposures required by 18–23% for the signal + background case and by
15–16% for the background-only case. Removing all systematic uncertainties
reduces the required exposure by 4.1–4.7% for the signal + background case and
by 2.9–4.4% for the background-only case.

In the baseline configuration described above, nuclear reactor positions and
powers from 2014 have been assumed, reflecting a scenario in which all
Japanese reactors remain shut down throughout operation and data-taking. The
impact of this assumption on the results of these studies is estimated by
instead using reactor positions and powers from 2010, reflecting an
alternative scenario in which all Japanese reactors have resumed operation at
the start of data-taking. The impact of this 2010 scenario is largest at the
Kamioka site due to the proximity of Japanese reactors and is presented in
Table 6. For the signal + background case, the calculated exposures increase
by 77–83%, 33–42%, and 47–59% for the 40K, mantle, and core
geo-$\overline{\nu}_{e}$ studies, respectively. For the background-only case,
the required exposures increase by 52–64%, 30–31% and 11–19%, respectively.
These results could likely be improved by imposing additional cuts on energy
(since the reactor energy spectrum is harder than that of
geo-$\overline{\nu}_{e}$s) or on the angular separation to the nearest
reactor(s).

The exposures quoted above assume perfect rejection of non-neutrino
backgrounds, namely those due to radioactive contamination of the detector or
shield materials, intrinsic radioactivity or cosmogenic activation of the gas
target, or other backgrounds coming from the cavern walls. To assess the
impact of these non-neutrino backgrounds, we include an isotropic background
at the level of 20% $\pm$ 2% (62% $\pm$ 6%) of the 40K (mantle)
geo-$\overline{\nu}_{e}$ signal. This background level corresponds to a factor
of $\sim$600 lower than that achieved by the NEXT-100 experiment Martín-Albo
_et al._ (2016), after scaling to the energy region of interest (200 keV to 1
MeV) using the shape of the electronic recoil spectrum measured by the
XENON100 collaboration Aprile  _et al._ (2011) and removing the contribution
from photomultiplier tubes (PMTs), which account for approximately 25% of the
background rate Martín-Albo  _et al._ (2016). The presence of this background
increases the required exposures by approximately 15% (16–19%) for the 40K
(mantle) geo-$\overline{\nu}_{e}$ case.

While challenging, an improvement in background levels at the scale assumed
here is plausible, given extrapolations from current experiments searching for
dark matter and neutrino-less double beta decay. Relative to the background
rate projected by the NEXT-100 experiment, a substantial portion of this
improvement would be attributed to the scale-up in fiducial volume, which
reduces the relative background contribution coming from the vessel walls (due
to the increase in the ratio of fiducial volume to surface area) and further
suppresses the probability of single-scatter events in the fiducial volume. A
scale-up of a similar size has been proposed by the LUX-ZEPLIN (LZ) experiment
Akerib  _et al._ (2015), relative to its predecessor Akerib  _et al._ (2014),
with a projected improvement in the electronic recoil background rate of a
factor of $\sim$1000\. Additional background reduction can be achieved by
using lower-background material for the detector vessel, such as the high-
purity copper used to construct the pressure vessel for the TREX-DM experiment
Iguaz  _et al._ (2016), a high-pressure TPC searching for low-mass dark
matter. The radioactive contamination of the TREX-DM copper pressure vessel
due to 238U, 232Th, and 40K is a factor of 8, 3 and 9 lower, respectively,
than that of the LZ titanium cryostat vessel Akerib  _et al._ (2015). The
background rate can be reduced further by surrounding the detector vessel with
an active liquid scintillator veto layer to act as an anti-Compton detector,
or optimizing the energy region of interest taking into account the complete
background model.

Assuming a CF4 (Xe) target and an operating pressure of 10 bar, a 15-tonne
detector would have a fiducial volume of 403 (260) m3, or approximately 7.4
(6.4) m on one side. Such a detector would be capable of making the first
measurement of the 40K geo-$\overline{\nu}_{e}$ flux at 95% CL with ten live-
years of data. Our results show that the most important experimental parameter
to optimize is the angular resolution. The choice of target gas and operating
pressure is therefore a trade-off between density and angular resolution,
since higher density gives a greater event rate per unit volume, while also
increasing the probability of multiple scattering along the track. To limit
the effect of multiple scattering (diffusion) in the drift region of the TPC,
the detector should be segmented into many TPC units, each with a drift length
of $\sim$1 m, housed within a single pressure vessel. This segmentation would
also serve to suppress backgrounds coming from the vessel walls, via
identification of coincident scattering interactions in multiple TPCs. To
design the optimal detector, new measurements of angular resolution as a
function of pressure in target gases of interest are needed.

In summary, we have proposed a method for measuring previously unresolved
components of Earth’s radiogenic heating using neutrino-electron elastic
scattering in low-background direction-sensitive tracking detectors. According
to the results of this study, the exposures needed to access 40K, mantle and
core geo-neutrinos at 95% CL are 140 tonne-years, 2.6 ktonne-years and 200
ktonne-years, respectively. Realization of these exposures faces the challenge
of constructing gas-filled TPCs with large target mass that are capable of
stable operation for many years. Experiments employing high-pressure TPCs have
demonstrated the possibility of combining large target masses with ultra-low
background rates, suggesting feasibility.

The measurements proposed here chart a course for pioneering exploration of
the veiled inner workings of the Earth, with the potential to unlock important
clues about the radiogenic heating within our planet. Moreover, the required
detectors, which are capable of reconstructing the direction of low-energy
particles, open sensitivity to a wide range of topics and applications outside
of geophysics, including searches for CP violation in the neutrino sector,
astrophysical point sources, searches for dark matter, nuclear reactor
monitoring and nuclear non-proliferation.

## Methods

### Detector sites

The geographical coordinates and elevations used for each of the three
underground sites considered here are: Kamioka Observatory, Japan (36.427∘ N,
137.300∘ E), 358 m above sea level; Laboratori Nazionale del Gran Sasso
(LNGS), Italy (42.450∘ N, 13.567∘ E), 936 m above sea level; and SNOLab,
Canada (46.475∘ N, -81.201∘ E), 1761 m below sea level.

### Geo-neutrino flux calculation

Geophysical response maps for the near-surface reservoirs (i.e. all reservoirs
other than the continental lithospheric mantle, mantle and core) are
constructed at a given location using the thicknesses and densities given by
CRUST 1.0 Laske  _et al._ (2016). CRUST 1.0 provides a constant average
elevation for each $1^{\circ}$$\times$$1^{\circ}$ unit of surface area. While
this granularity is generally sufficient for the studies presented here, the
angular distribution of geophysical response for the upper continental crust
does not accurately represent observation sites located in mountainous
terranes, such as Gran Sasso and Kamioka. In these cases, detailed topographic
information is used to supplement CRUST 1.0. The mountainous regions
surrounding the experimental site at Gran Sasso can be seen in Fig. 3a for cos
$\theta_{\mathrm{zenith}}<0$.

Calculation of the geophysical response of the mantle uses data from the
Preliminary Reference Earth Model (PREM) Dziewonski and Anderson (1981),
supplemented by data from CRUST 1.0. The main mantle volume is modeled as a
spherical shell, as defined by PREM, extending outward from the core-mantle
boundary ($R=$ 3480 km) to about 80 km, on average, beneath the surface. Shell
density decreases radially outward, except for a slight increase over the
outermost 140 km. Sitting atop the shell and beneath oceanic crust is a small
mantle volume, which breaks the spherical symmetry of the geophysical response
at the considered observation sites. The slight difference in elevations of
each of the sites also contributes a small amount to this asymmetry.

The geophysical response of the mantle is calculated assuming that all element
concentrations are homogeneously distributed. There is, however, experimental
evidence of inhomogeneities in the deep mantle. Seismologists have resolved
two large structures, called large low-shear-velocity provinces (LLSVPs)
Garnero and McNamara (2008), off the west coast of Africa and near Tahiti in
the South Pacific, with indication that they are thermally and compositionally
different from the surrounding mantle. While these deep mantle structures
could result in anisotropies Sramek  _et al._ (2013) in the
geo-$\overline{\nu}_{e}$ signal, particularly interesting for a direction-
sensitive detector, their size and location are not currently agreed upon and
are therefore not considered further in this paper.

The geo-$\overline{\nu}_{e}$ flux at a given point on the surface of the Earth
is calculated by summing each of the geophysical response maps and multiplying
by well-established values for neutrino luminosities per unit mass (g-1 s-1),
loosely constrained geochemical estimates of element abundances in each
reservoir (Table 7) and natural abundances of the isotopes (Table 8). The
estimated average abundances used here are generally consistent with other
published abundances Rudnick and Gao (2003); Huang  _et al._ (2013) within the
same reservoir. Geo-$\overline{\nu}_{e}$ energy spectra are taken from Enomoto
Enomoto (2005) and shown in Fig. 8.

Reservoir/Isotope |  U ($\mu$g/g) |  Th ($\mu$g/g) |  K (wt. %)
---|---|---|---
Upper continental crust |  $2.7$ p m 0.6 |  $10.5$ p m 1.1 |  $2.3$ p m 0.2
Middle continental crust |  $1.3$ p m 0.4 |  $6.5$ p m 0.5 |  $1.9$ p m 0.3
Lower continental crust |  $0.20$ p m 0.11 |  $1.3$ p m 0.9 |  $0.71$ p m 0.28
Continental lithospheric mantle |  $0.045$ p m 0.035 |  $0.24$ p m 0.19 |  $0.04$ p m 0.03
Sediment |  $1.7$ p m 0.1 |  $8.1$ p m 0.6 |  $1.8$ p m 0.1
Oceanic crust |  $0.07$ p m 0.02 |  $0.21$ p m 0.06 |  $0.07$ p m 0.02
Mantle (no radioactivity in core) |  $0.011$ p m 0.009 |  $0.036$ p m 0.033 |  $0.016$ p m 0.013
Mantle (10 p.p.b. U, Th in core) |  $0.008$ p m 0.009 |  $0.033$ p m 0.033 |  $0.011$ p m 0.013
Core | $0.010$ | $0.010$ | $0.000$
Table 7: Element abundances in eight geochemical reservoirs. Abundance of uranium (U), thorium (Th) and potassium (K) in eight geochemical reservoirs, assuming a sub-crustal Earth model with a homogeneous mantle and either no radioactivity or 10 p.p.b. U, Th in the core. Values given here are used to calculate the predicted geo-neutrino flux from the geophysical response (see Methods). Mantle values and uncertainties are constrained using measurements by the KamLAND Gando  _et al._ (2013) and Borexino Agostini  _et al._ (2015) collaborations. Isotope |  Neutrino luminosity | Natural abundance
---|---|---
(g-1 s-1) | (%)
238U | $7.46\times 10^{4}$ | $99.27$
235U | $3.20\times 10^{5}$ | $0.7204$
232Th | $1.62\times 10^{4}$ | $100.0$
40K | $2.31\times 10^{5}$ | $0.0117$
Table 8: Isotopic neutrino luminosities and natural abundances. Neutrino
luminosity (g-1 s-1) and natural abundance (%) of uranium (U), thorium (Th)
and potassium (K) radioactive isotopes. Values given here are used to
calculate the predicted geo-neutrino flux from the geophysical response (see
Methods). Figure 8:  Predicted flux of geo-neutrinos at Kamioka. The predicted
geo-neutrino flux at Kamioka (Japan) is shown here for decays of 238U (red
long-dashed), 235U (green solid), 232Th (black dash-dotted) and 40K (blue
short-dashed) as a function of neutrino energy (MeV), not including the effect
of neutrino oscillation. Flux spectra are taken from Enomoto Enomoto (2005),
with normalization according to Table 1.

### Reactor antineutrino flux calculation

Reactor positions and powers are taken from a reference worldwide reactor
model Baldoncini  _et al._ (2015). Reactor powers are averaged over 12
calendar months of data from 2014 (or 2010). Intensities of each of the 439
(or 444) contributing reactor cores are calculated at the three underground
sites assuming a spherical Earth. The total reactor-$\overline{\nu}_{e}$ flux
is calculated by scaling the intensities by the average number of
$\overline{\nu}_{e}$s released per unit energy, then summing over all reactor
cores. Assuming a flux of 6 $\overline{\nu}_{e}$s and 205 MeV per fission, the
average number of $\overline{\nu}_{e}$s released per unit energy Usman  _et
al._ (2015) is $1.83\times 10^{11}$ J-1.

The total estimated uncertainty on the reactor-$\overline{\nu}_{e}$ flux
includes several potential sources of error: uncertainties in the power value
reported by plant operators, oscillation parameters, conversion of reactor
power to flux and seasonal changes in the reactor power output due to load
requirements on the user side, refueling or long-term shutdown. Assuming that
the power output of all reactors is known exactly, then the uncertainty
Baldoncini  _et al._ (2015) in the flux above the inverse beta decay threshold
of 1.8 MeV is approximately 2.2-2.5%, dominated mostly by the uncertainty on
oscillation parameters. The added uncertainty due to power reporting is
difficult to estimate reliably and should be studied in more detail.

While the shape of the reactor-$\overline{\nu}_{e}$ spectrum is well
understood above the inverse beta decay threshold, it is less certain below
this energy. Here we parameterize the spectral probability density function of
reactor $\overline{\nu}_{e}$s using the functional form Usman  _et al._
(2015):

$\displaystyle
N(E_{\overline{\nu}})=\frac{6}{2.0523}e^{-(0.3125E_{\overline{\nu}}+0.25)^{2}},$
(2)

where $E_{\overline{\nu}}$ is in MeV. This parameterization is close to that
given in the literature Vogel and Engel (1989). A 5.5% uncertainty on the
reactor-$\overline{\nu}_{e}$ energy spectrum is absorbed as a flat systematic
uncertainty on the reactor-$\overline{\nu}_{e}$ flux, giving a total
systematic uncertainty of 6%.

### Neutrino scattering cross-section

Many experiments detect neutrinos using $\nu_{e}\mathchar 45\relax e^{-}$
elastic scattering Bahcall  _et al._ (1995); Daraktchieva  _et al._ (2005,
2008); Bellini  _et al._ (2014a). This reaction is advantageous because it has
no threshold, the maximum recoil electron kinetic energy can be of order MeV
and the direction of the outgoing electron is well correlated with the
direction of the incident neutrino (Fig. 9). The theoretical uncertainties on
the scattering cross section are at the few-percent level; this process is
therefore used to search for new physics, e.g. anomalous neutrino magnetic
moment Daraktchieva  _et al._ (2008). The magnitude of the cross section
Bahcall  _et al._ (1995) for momentum transfer $q$ is roughly $\sigma(q)$
$\sim$ 9.2 $\times$ ($q$/10 MeV) $\times$ 10-44 cm2.

Following Vogel & Engel Vogel and Engel (1989), the cross section for
$\nu_{\ell}\mathchar 45\relax e^{-}$ elastic scattering is calculated in terms
of the recoil electron kinetic energy, $T$, and scattering angle relative to
the incident neutrino direction, $\chi$. The scattering angle depends on the
incident neutrino energy $E_{\nu}$ as

$\displaystyle\mathrm{cos\,}\chi\ =\
\frac{E_{\nu}+m_{e}}{E_{\nu}}\biggl{[}\frac{T}{T+2m_{e}}\biggr{]}^{1/2},$ (3)

with a maximum recoil kinetic energy given by

$\displaystyle T_{\mathrm{max}}\ =\ \frac{2E_{\nu}^{2}}{2E_{\nu}+m_{e}},$ (4)

where $m_{e}$ is the mass of the target electron. The differential cross
section is

$\frac{d\sigma}{dT}\ =\
\frac{G_{F}^{2}m_{e}}{2\pi}\biggl{[}(g_{V}+x+g_{A})^{2}+(g_{V}+x-g_{A})^{2}\biggr{(}1-\frac{T}{E_{\nu}}\biggl{)}^{2}+\\\
(g_{A}^{2}-(g_{V}+x)^{2})\frac{m_{e}T}{E_{\nu}^{2}}\biggr{]}+\frac{\pi\alpha^{2}\mu_{\nu}^{2}}{m_{e}^{2}}\biggl{(}\frac{1}{T}-\frac{1}{E_{\nu}}\biggr{)},$
(5)

where

$\displaystyle g_{V}=\begin{cases}2\textrm{ sin}^{2}\textrm{
}\theta_{W}+\frac{1}{2},&\text{for }\nu_{e}\\\ 2\textrm{ sin}^{2}\textrm{
}\theta_{W}-\frac{1}{2},&\text{for }\nu_{\mu},\nu_{\tau}\end{cases}$ (6)
$\displaystyle g_{A}=\begin{cases}+\frac{1}{2},&\text{for }\nu_{e}\\\
-\frac{1}{2},&\text{for }\nu_{\mu},\nu_{\tau}\end{cases}$ (7) $\displaystyle
x=\frac{\sqrt{2}\pi\alpha\langle r^{2}\rangle}{3G_{F}},$ (8)

$\alpha$ is the strong coupling constant, $G_{F}$ is the Fermi coupling
constant, $\theta_{W}$ is the weak mixing angle, $\langle r^{2}\rangle$ is the
neutrino charge radius and $\mu_{\nu}$ is the neutrino magnetic moment. We set
the latter two constants to zero in our calculation, consistent with
experimental results Bahcall  _et al._ (1995); Barranco  _et al._ (2008);
Cañas  _et al._ (2016) thus far. For $\overline{\nu}$, $g_{A}\rightarrow-
g_{A}$ in equation (5). This interference between charged and neutral currents
lowers the cross section by a factor of approximately 2.4 at
geo-$\overline{\nu}_{e}$ energies.

Figure 9:  Differential neutrino-electron elastic scattering cross section.
Cross sections are shown here versus scattering angle ($\chi$) for three
representative neutrino energies, $E_{\nu}=$ 0.5, 1.5 and 5.0 MeV, integrated
over the allowed kinematic range of electron recoil kinetic energies,
$0<T<T_{\mathrm{max}}$. Solid (dashed) lines represent incident neutrinos
(antineutrinos). Each curve has been normalized to unit area.

Figure 10:  Angular distribution of electron recoils induced by neutrinos.
Angular distributions are calculated at Gran Sasso (Italy) and shown here for
electron recoils induced by neutrinos from 10 the Earth’s crust, 10 the
Earth’s mantle, 10 the Sun and 10 worldwide nuclear reactors. The azimuthal
angle $\phi$ is measured clockwise from due North, while the zenith angle
$\theta_{\mathrm{zenith}}$ is measured with respect to the vertical axis,
defined opposite the direction pointing to the center of the Earth.
Reconstructed angles have been smeared with a Gaussian distribution according
to the angular resolution parameterization given in equation (1). An electron
energy threshold of 250 keV has been applied to all distributions. All plots
are normalized to unit volume.

Figure 9 shows the differential $\nu_{\ell}\mathchar 45\relax e^{-}$ elastic
scattering cross section as a function of scattering angle, $\chi$, for three
representative neutrino energies, $E_{\nu}=$ 0.5, 1.5 and 5.0 MeV, integrated
over the allowed kinematic range, $0<T<T_{\mathrm{max}}$. Figure 10 shows the
angular distribution of electron recoils induced by neutrinos from the Earth’s
crust, the Earth’s mantle, the Sun and worldwide nuclear power reactors,
calculated at Gran Sasso, with angular smearing according to equation (1) and
electron recoil energy $T>$ 250 keV.

### Sensitivity Analysis Method

For each contribution to the total geo-$\overline{\nu}_{e}$ flux and for a
given exposure, we run a set of 1000 pseudo-experiments, treating a single
contribution as signal and all other sources, including other
geo-$\overline{\nu}_{e}$s, as background. Template distributions are
constructed by simulating a large sample ($\geq$ 500M each) of
geo-$\overline{\nu}_{\ell}$, solar-$\nu_{\ell}$ and
reactor-$\overline{\nu}_{\ell}$ events, with event times distributed randomly
over the year, according to the predicted incident angular distributions shown
in Fig. 3, the energy spectra from Fig. 8, Table 2 and equation (2), and the
scattering cross section and kinematics presented in the previous section. The
number of events in each pseudo-experiment is fluctuated according to the
statistical error on the predicted event rates from Tables 4 and 5 and scaled
to the total exposure of the pseudo-experiment in tonne-years.

We assume a track reconstruction efficiency of 100% for electron recoil
energies $T>$ 200 keV. This is a reasonable assumption given the performance
of past detectors [29], provided there are sufficient measurement points along
the track. Reconstructed angles are smeared with a Gaussian distribution
according to the angular resolution parameterization given in equation (1).
This is perhaps an overly conservative scenario since the as-built detector
would likely have finer point resolution in the readout plane, relative to
that of the MUNU detector.

Figure 11: Solar angular separation of electron recoils induced by neutrinos.
Angular separation from the Sun ($\theta_{\mathrm{sun}}$) of electron recoils
induced by solar neutrinos (blue striped), geo-neutrinos (green hatched) and
reactor antineutrinos (orange weave), calculated at Gran Sasso (Italy).
Pseudo-experimental data (black squares) are also shown, with their
statistical errors. Distributions are normalized to an exposure of 11 1
ktonne-year or 11 10 ktonne-years, including the effect of neutrino
oscillation, with an applied electron energy threshold of $T>$ 11 250 keV or
11 800 keV. Pseudo-experimental data is fit using the functional form from
equation (9), with $\chi^{2}/N_{\mathrm{dof}}=$ 11 6.64/6 and 11 2.16/6.
Reconstructed angles have been smeared with a Gaussian distribution according
to the angular resolution parameterization given in equation (1).

We define $\theta_{\mathrm{sun}}$ as the angular separation between the
electron recoil direction and the Sun-to-Earth vector at a given event time.
This variable is a powerful discriminant between solar-$\nu_{\ell}$ and
geo-$\overline{\nu}_{\ell}$ events since the former mostly point parallel to
the Sun (cos $\theta_{\mathrm{sun}}=1$), while the latter are distributed
evenly from $-1\leq\mbox{cos\,}\theta_{\mathrm{sun}}\leq 1$. For each of the
studies described here, a cut on cos $\theta_{\mathrm{sun}}$ is applied in
order to reject background events associated to solar $\nu_{e}$s. Figure 11
shows the simulated $\theta_{\mathrm{sun}}$ distribution for two pseudo-
experiments at Gran Sasso, scaled to an exposure of (a) 1 ktonne-year and (b)
10 ktonne-years, for electron recoils with energies $>250$ keV and $>800$ keV,
respectively. Together, the two plots illustrate the energy dependence of the
angular spread of recoils relative to the incident neutrino direction. The
pseudo-experimental data in this figure has been fit with a von-Mises
distribution, summed with a flat pedestal, to describe the solar-$\nu_{\ell}$
plus reactor- and geo-$\overline{\nu}_{\ell}$ background contributions,
respectively:

$\displaystyle
f(\mathrm{cos\,}\theta)=A\frac{e^{\kappa\mathrm{cos\,}(\theta-\mu)}}{2\pi
I_{0}(\kappa)}+C,$ (9)

where $I_{0}(\kappa)$ is the modified Bessel function of order 0 and $A$, $C$,
$\mu$ and $\kappa$ are parameters of the fit. The parameters $\mu$ and
$1/\kappa$ are analogous to the mean ($\mu$) and variance ($\sigma^{2}$) of a
Gaussian distribution. The mean $\chi^{2}/N_{\mathrm{dof}}$ for 1000 pseudo-
experiments, each scaled to an exposure of 1 ktonne-year, is 8.83/6. The
results of the fit at a given energy threshold are used to determine the
precise cut value on $\theta_{\mathrm{sun}}$, following an optimization of the
calculated sensitivity. The cut values used and their calculated acceptance of
signal and background contributions are discussed in the main text for each of
the geo-$\overline{\nu}_{e}$ contributions under study.

For each set of pseudo-experiments, we assess two statistics: (i) the level of
the signal geo-$\overline{\nu}_{e}$ flux that can be excluded at 95% (90%) CL,
under the hypothesis that the data contain only background events; and (ii)
the level of the signal geo-$\overline{\nu}_{e}$ flux that excludes the null
hypothesis at 95% (90%) CL, assuming that the data contain both background and
geo-$\overline{\nu}_{e}$ signal. Confidence intervals are calculated using a
profile likelihood analysis, taking into account systematic uncertainties on
the measured signal and predicted background distributions, discussed in the
main text. To assess (i), we calculate the 95% (90%) confidence interval upper
limit on the predicted geo-$\overline{\nu}_{e}$ signal, assuming that the data
contain only background events, and determine the exposure needed to achieve
an upper limit equal to the predicted signal flux. To assess (ii), we
calculate the 95% (90%) confidence interval, assuming that the data contain
both signal and background events, and determine the exposure at which 95%
(90%) of pseudo-experiments have a non-zero lower limit. Results have been
cross-checked with the frequentist CL ($\mathrm{CL_{\mathrm{s}}}$,
$\mathrm{CL_{\mathrm{b}}}$) computation Junk (1999), with the profile
likelihood method chosen here performing marginally better.

###### Acknowledgements.

J.M. thanks Mark Chen, Enectali Figueroa-Feliciano and Nikolai Tolich for
critical discussions at various stages of the study. M.L. thanks Cécile
Lapoire for helpful consistency checks of the analysis. M.L. acknowledges
support from the Marie Skłodowska-Curie Fellowship program, under grant 665919
(EU, H2020-MSCA-COFUND-2014), Ministerio de Economia, Industria y
Competitividad (MINECO), Agencia Estatal de Investigación (AEI) and Fondo
Europeo de Desarrollo Regional (FEDER), under grants FPA2014-77347-C2-2 and
SEV-2012-0234. S.D. appreciates past support from the Cooperative Studies of
the Earth’s Deep Interior (CSEDI) and the Cooperative Institute for Dynamic
Earth Research (CIDER) programs funded by the U.S. National Science
Foundation. J.M. acknowledges support from the MIT Pappalardo Fellowship
program, the U.S. Department of Energy (contract number DE-FG02-05ER41360),
the European Research Council Project ERC StG 279980, the UK Science &
Technology Facilities Council (STFC) grant ST/K002570/1 and the Leverhulme
Trust grant number ECF-20130496.

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