Neutrino Oscillations III

The Earth seen by neutrinos

Dmitry V. Naumov

Joint Institute for Nuclear Research, Dubna

What we know now

Vacuum: two mass waves beat; the clock ticks with $\Delta m^2L/E$.
Matter: $\nu_e$ feels the electrons, $V=\sqrt2G_Fn_e$; it matters when $2EV\sim\Delta m^2$.
Sun: slow density change converts $\nu_e$ into $\nu_2$; the Earth regenerates a little at night.

Today: the Earth

What geophysics knows, and what it does not.
Probe 1 — absorption: the Earth’s mass.
Probe 2 — oscillations: the core’s electrons.
Probe 3 — geoneutrinos: the Earth’s heat.

The Earth as geophysicists see it

What geophysics knows

PREM density against radius with the inner core, liquid outer core and mantle shaded; the electron density Y_e rho is about half of the density.

Seismic waves, the Earth’s mass and moment of inertia give the density profile (PREM).
The composition is inferred indirectly: the outer core is 5–10 % less dense than pure iron, so it contains light elements — but which ones, and how much?

PREM: Dziewonski & Anderson (1981). Density deficit: Badro, Côté & Brodholt, PNAS 111, 7542 (2014). Dashed: $Y_e\rho$ with the model compositions of the previous slides.

What neutrinos add: electrons per nucleon

\[ Y_e=\frac{\text{electrons}}{\text{nucleons}}=\sum_i w_i\,\frac{Z_i}{A_i} \]

Element	$Z/A$
H	0.992
O, Si, S, C	0.50
Mg	0.494
Fe	0.466
Ni	0.477

Seismology + neutrinos

Seismology and gravity: $\rho$.
Neutrino oscillations: $n_e\propto Y_e\,\rho$.
Together: $Y_e$, a direct handle on composition.

Core models

pure Fe: $Y_e=0.466$
Fe + 10 wt% O: $Y_e=0.469$ (+0.7 %)
Fe + 1 wt% H: $Y_e=0.471$ (+1.1 %)

Hydrogen is the outlier: one electron per nucleon. A percent-level measurement of $Y_e$ in the core would test whether the core holds hydrogen.

Derived from CIAAW standard atomic weights; mixtures by mass fraction.

Three neutrino probes of the Earth

Absorption

$E\gtrsim10$ TeV

The Earth becomes opaque: absorption counts nucleons.

→ density, mass of the core

Oscillations

$E\approx1\text{–}20$ GeV

Matter shifts the oscillation phase: forward scattering on electrons.

→ electron density, composition

Geoneutrinos

$E\approx2\text{–}3$ MeV

$\bar\nu_e$ from U and Th decays inside the Earth: counts radioactive sources.

→ radiogenic heat

Three energy ranges, three physical quantities — all through the same weak interaction. Today: one section for each.

Probe 1: absorption

The Earth as a shield

PREM Earth, CSMS cross sections; probability of no interaction along the path. Muon ranges from PDG tables of standard rock.

How opaque is the Earth?

Computed survival probability of neutrinos crossing the Earth versus zenith angle from 90 to 180 degrees and energy from 10 GeV to 100 PeV, charged-current absorption only. Near the horizon the Earth is transparent at all energies; through the core it becomes opaque above about 100 TeV. Contours at 10, 50 and 90 percent.

At 10 TeV about three quarters of the neutrinos cross the whole diameter; above ~100 TeV the core paths are dark.
The darkness depends on the column $X=\int\rho\,dl$: absorption measures mass density.

Computed: PREM, CSMS cross sections, detector 1 km deep · CC absorption only, average of $\nu$ and $\bar\nu$.

Absorption tomography: IceCube weighed the Earth

Charged-current survival probability against zenith angle at 10, 30 and 100 TeV for PREM and a homogeneous Earth of the same mass, and their relative difference.

One year of IceCube up-going muons, 0.4–20 TeV (20 145 events):

Earth’s mass $6.0^{+1.6}_{-1.3}\times10^{24}$ kg; gravity: $5.97\times10^{24}$ kg.
Core mass $2.72^{+0.97}_{-0.89}\times10^{24}$ kg; the core is denser than the mantle.

A weak-interaction measurement of the Earth’s mass, without gravity.

Figure computed: PREM vs homogeneous Earth of the same mass, CC absorption only. Measurement: 68 % credible intervals · Donini, Palomares-Ruiz & Salvado, Nature Physics 15, 37 (2019)

Probe 2: oscillations

Paths through the Earth

Left: Earth cross-section with mantle, outer and inner core and four paths from below to a detector at the top. Right: electron density along each path; steeper paths enter the core and see a jump to more than twice the mantle value.

The arrival direction $\cos\theta_z$ fixes the path: $L=2R_\oplus|\cos\theta_z|$.
Paths with $\cos\theta_z<-0.84$ cross the core; below $-0.98$ also the inner core.

A neutrino crossing the Earth

Direction, cos θ −0.95 Energy 4.0 GeV Born as

path length—

Normal ordering, NuFIT 6.1; PREM Earth.

Press play: the flavor changes fastest where the density jumps. Try 4 GeV through the core, then 6 GeV through the mantle only.

The oscillogram of the Earth

Probability of muon to electron neutrino conversion through the PREM Earth as a function of energy from 1 to 20 GeV and of the cosine of the zenith angle. Neutrinos show a large mantle resonance near 6 GeV and structured core features below cos theta of -0.84; antineutrinos show almost nothing.

Mantle resonance near 6 GeV: $P(\nu_\mu\to\nu_e)$ grows from a few % in vacuum to almost 50 %.
Core-crossing paths: mantle and core layers interfere — sharp features at 2–5 GeV.
Antineutrinos (normal ordering): no resonance. The asymmetry reveals the mass ordering.

Computed: three flavors, NuFIT 6.1 best fit (normal ordering), PREM with $Y_e^{\rm mantle}=0.497$, $Y_e^{\rm core}=0.466$; detector at the surface. Core–mantle enhancement: Akhmedov, Nucl. Phys. B 538, 25 (1999); Petcov, Phys. Lett. B 434, 321 (1998).

Explore a path

Direction, cos θ −0.95

path length—

deepest layer—

Core Yₑ 0.466

Channel Mass ordering antineutrinos

Move the direction from the horizon to the nadir. Change the core $Y_e$: which energies and directions react? Switch to antineutrinos and to the inverted ordering.

Build your own Earth

Core Yₑ 0.466 Core density × 1.00 no core (rock instead)

Channel

status—

largest |ΔP|—

Left: the oscillogram of your Earth. Right: how it differs from PREM with an iron core. Remove the core; add hydrogen; make the core 5 % lighter.

How large is the composition signal?

$Left: change of the muon to electron conversion probability when the core electron fraction rises by one percent, as a map over energy and direction below the core boundary; changes reach about plus or minus 0.03 on sharp features. Right: the probability at cos theta = -0.95 for the two core compositions.$

$+1\,\%$ in the core $Y_e$ changes $P$ by up to $\pm0.03$, only for core-crossing paths of 2–10 GeV; resolution smears much of this fine pattern.

Computed with the same model; only the core $Y_e$ changed; no detector response or flux.

What it takes

If smearing leaves a 1 % change in the event rate, seeing it at 3σ needs $3/\sqrt N\approx0.01$, i.e. $N\approx10^5$ events in that window: megaton-scale detectors over many years.

Studies for real detectors

After 20 years KM3NeT/ORCA and DUNE reach $Z/A$ of the outer core to 3–4 %, Hyper-Kamiokande to 2.5 %; 1 wt% of hydrogen reaches only 1σ after 50 years of combined data.

Why so hard

Few atmospheric neutrinos cross the core: $\cos\theta_z<-0.84$ is 16 % of the lower hemisphere.
The detector must measure both energy and direction of GeV neutrinos.
Flux and cross-section uncertainties also enter.

A geophysical measurement with particle-physics detectors: the effect is percent-level, the exposure must be megatons × decades.

Rott, Taketa & Bose, Sci. Rep. 5, 15225 (2015) · Maderer et al., Front. Earth Sci. 11, 1008396 (2023)

Probe 3: geoneutrinos

Antineutrinos from inside the Earth

$\beta$ decays in the $^{238}$U and $^{232}$Th chains emit $\bar\nu_e$. Only four decays are energetic enough to be seen through $\bar\nu_e+p\to e^++n$ (threshold 1.806 MeV):

Chain	Visible decays	Highest $\bar\nu$ energy
$^{238}$U	$^{234\rm m}$Pa, $^{214}$Bi	3.27 MeV
$^{232}$Th	$^{228}$Ac, $^{212}$Bi	2.25 MeV
$^{40}$K	—	1.31 MeV: invisible

Heat per kilogram

U: 98.5 μW/kg
Th: 26.3 μW/kg
K: 0.0033 μW/kg

Chondritic ratio Th/U = 3.9.

Unit of signal

1 TNU = 1 event per $10^{32}$ free target protons per year.

Fiorentini, Lissia & Mantovani, Phys. Rep. 453, 117 (2007) · Dye, Rev. Geophys. 50, RG3007 (2012), Table 3 · Q-values: NNDC NuDat 3 · Borexino, PRD 101, 012009 (2020)

Geoneutrinos: the Earth’s radioactive heat

Schematic Earth with crust and mantle sources of geoneutrinos reaching a detector.

Crust enlarged; paths illustrative.

$^{238}$U and $^{232}$Th chains emit $\bar\nu_e$ up to 3.27 and 2.25 MeV; detected by $\bar\nu_e+p\to e^++n$ above 1.806 MeV.
The Earth loses $47\pm2$ TW of heat. How much is radiogenic, how much primordial?
Radiogenic U+Th heat: Borexino $38.2^{+13.6}_{-12.7}$ TW; KamLAND $15.4^{+8.3}_{-7.9}$ TW. KamLAND sees geoneutrinos at 8.5σ.
Geoneutrinos oscillate too: averaged over many oscillation lengths, $P_{ee}\approx0.55$.

Heat flow: Davies & Davies, Solid Earth 1, 5 (2010) · Borexino, PRD 101, 012009 (2020) · KamLAND, GRL 49, e2022GL099566 (2022), Th/U free · $P_{ee}=c_{13}^4(1-\tfrac12\sin^22\theta_{12})+s_{13}^4$ with NuFIT 6.1.

How much of the Earth’s heat is radiogenic?

Horizontal bars of power in terawatts: Bulk Silicate Earth models give 11, 20 and 33 TW of total radiogenic heat; the continental crust alone 6.8 TW; KamLAND measures 15.4 TW and Borexino 38.2 TW from uranium and thorium with large errors; the total surface heat flow is 47 TW.

Earth models disagree by a factor of 3: 11–33 TW of radiogenic heat out of 47 TW. The rest is primordial heat.
The crust is sampled by rocks; geoneutrinos see the U and Th of the mantle directly.

Šrámek et al., EPSL 361, 356 (2013), Table 2 · Huang et al., G³ 14, 2003 (2013) · Davies & Davies (2010) · Borexino 2020 · KamLAND 2022 (Th/U free). Models and crust: U+Th+K; measurements: U+Th only.

JUNO: geoneutrinos by the hundred

≈ 400 per year

expected geoneutrino events in JUNO

More events per year than KamLAND and Borexino have collected in total.
About 70 % of the signal comes from the crust, 44 % from the crust within a few hundred kilometres: local geology must be mapped.
The goal: subtract the crust and measure the mantle.

JUNO Yellow Book, J. Phys. G 43, 030401 (2016), Sec. 8: $408\pm60$ events per year; bulk crust 28.2 of 39.7 TNU. Prediction of 2015.

Three lectures in one slide

What neutrinos can tell about the Earth

Probe	Energy	Physics	Measures	Status
Absorption	> 10 TeV	weak interaction with nucleons	density, mass of the core	first measurement (IceCube)
Oscillations	1–20 GeV	refraction on electrons	electrons per nucleon: composition	3–4 % on $Z/A$ in ~20 years
Geoneutrinos	2–3 MeV	U and Th decays	radiogenic heat	observed; mantle at 2.3σ (Borexino)

Vacuum: a neutrino is two interfering waves; the clock ticks with $\Delta m^2L/E$ — sensitive to $10^{-21}$ of the energy.
Matter: $\nu_e$ feels electrons; resonance and adiabatic following explain the Sun and reveal the mass ordering.
The Earth: the same physics becomes a tool — to weigh the planet, probe its core and measure its heat.

Chain	Visible decays	Highest \(\bar\nu\) energy
\(^{238}\)U	\(^{234\rm m}\)Pa, \(^{214}\)Bi	3.27 MeV
\(^{232}\)Th	\(^{228}\)Ac, \(^{212}\)Bi	2.25 MeV
\(^{40}\)K	—	1.31 MeV: invisible