A survival that depends on the energy

The deficit of solar neutrinos was settled in 2002 by a detector that could count every flavour at once: the total flux of ⁸B neutrinos was what the standard solar model said, and only about a third of it arrived as electron neutrinos. The Sun was fine. The neutrinos were changing on the way.

“About a third” is a statement about the ⁸B neutrinos, which carry energies of several million electronvolts and are the only ones a water detector can see. The neutrinos that carry most of the Sun’s flux — the pp neutrinos, below 0.42 MeV — do something different. When a liquid scintillator finally separated them from each other, source by source, the low-energy neutrinos turned out to survive about half the time. The survival probability depends on the energy, and the dependence has a step in it.

The step is not a detail. A neutrino oscillating in empty space, with the mixing angle and mass splitting the reactors have measured, would oscillate so many times between the Sun and the Earth that only its average would survive — about 0.55 at every energy. The flat line on the left of the figure is that average, and the pp and ⁷Be neutrinos sit on it. The ⁸B neutrinos do not; they sit near 0.31, and something between the core of the Sun and the vacuum has pushed them there. What pushes them is the Sun itself, and the energy at which the push takes over — computed here from nothing but the solar density and the neutrino’s parameters — is where the step sits.

A potential that only one flavour feels #

A neutrino passing through matter is not quite passing through nothing. It scatters coherently off the particles it moves through, forward, without deflection, and the net effect is a small potential energy — a refractive index. Every flavour feels the same potential from the neutral-current scattering off protons, neutrons and electrons, and a potential that is the same for every flavour changes no relative phase and can be ignored. But an electron neutrino can also scatter off an electron through the charged current, exchanging a W boson, and the muon and tau neutrinos cannot, because there are no muons or taus in the Sun. So the electron flavour alone feels an extra potential,

$$V = \sqrt{2}\, G_F\, n_e,$$

proportional to the electron density $n_e$ and to the Fermi constant. In units that suit the Sun, it is $7.6\times10^{-14}$ electronvolts for every mole of electrons per cubic centimetre. At the centre of the Sun there are about a hundred moles per cubic centimetre, so the potential is about $8\times10^{-12}$ eV. That is an absurdly small energy — a billionth of the energy of a visible photon — and it would be irrelevant if it were not being compared with something even smaller.

What it is compared with is the energy difference that drives oscillation in the first place. Two neutrinos of the same momentum but different masses differ in energy by $\Delta m^2/2E$, and with the solar splitting $\Delta m^2 = 7.4\times10^{-5}\ \mathrm{eV}^2$ and an energy of 10 MeV that is $3.7\times10^{-12}$ eV. The two terms are the same size. Everything that follows is a consequence of that coincidence, and it is a coincidence: the density of the solar core is set by gravity and nuclear physics, the neutrino splitting by whatever gives neutrinos mass, and there is no reason the two should meet within a factor of two.

Their ratio is a single number,

$$\beta = \frac{2\sqrt{2}\, G_F\, n_e\, E}{\Delta m^2},$$

and it grows with both the density and the energy. When $\beta$ is small the neutrino barely notices the matter and oscillates as it would in vacuum. When $\beta$ is large the matter dominates, and the mixing is suppressed in a particular direction. The transition happens where $\beta$ equals $\cos 2\theta$, which for the solar mixing angle is 0.39.

Two levels that refuse to cross #

The cleanest way to see what matter does is to follow the two energy levels of the neutrino’s two-state system as the density changes, the way one would follow the energy levels of an atom as an applied field is turned up.

With no mixing at all, the two levels would be pure flavours, and the electron-flavour level would simply rise in proportion to the density, crossing the other level at one particular density. With mixing, two levels of a quantum system that are coupled cannot cross. They approach, and at the density where they would have crossed they come to a minimum separation, set by $\sin 2\theta$, and then diverge again with their identities exchanged. That density is the resonance. For a 10 MeV neutrino it is 19 moles of electrons per cubic centimetre, which the Sun reaches at 0.24 of its radius.

Now follow a neutrino. It is made in the core, at about a hundred moles per cubic centimetre — far above its resonance — as an electron neutrino, and at that density the upper level is almost pure electron flavour, so the neutrino starts almost entirely on the upper level. As it travels outward the density falls, and if it falls slowly enough, the neutrino stays on the level it started on. It passes through the resonance, where the upper level changes from mostly electron flavour to mostly the other, and emerges into vacuum on the upper level — which in vacuum is the heavier mass state, $\nu_2$. That state is only 31 per cent electron flavour, and it no longer oscillates, because it is a mass state and mass states propagate unchanged. A detector on the Earth finds an electron neutrino with probability $\sin^2\theta$, about 0.31.

That is the matter limit on the right of the first figure. The conversion is not an oscillation at all. It is a neutrino following one energy level through a place where that level changes what it is made of — the same physics as an atom carried adiabatically through an avoided crossing, or a spin following a slowly rotating field. The electron neutrinos made in the core are not oscillating into muon and tau neutrinos on the way out; they are being turned, smoothly and irreversibly, into the mass state that is mostly muon and tau.

Which neutrinos are born above their resonance #

The resonance density depends on energy. From the definition of $\beta$ it falls as $1/E$: a more energetic neutrino resonates at a lower density. So the question of whether a given neutrino passes through a resonance on its way out is a question of whether it is born at a density above the one its own energy picks out.

The answer divides the solar spectrum in two. A pp neutrino at 0.27 MeV would need 709 moles of electrons per cubic centimetre to resonate, seven times the density at the very centre. A ⁷Be neutrino at 0.86 MeV would need 222, and a pep neutrino at 1.44 MeV would need 133. None of them is ever born at such a density, so none passes through a resonance. For them the matter is a small correction to vacuum oscillation, and they arrive with the vacuum average. A ⁸B neutrino at 10 MeV needs only 19, which it passes a quarter of the way out; at 5 MeV it needs 38, which it passes at 0.18 of the radius. Almost every ⁸B neutrino is born above its resonance and crosses it.

The boundary between the two groups, for neutrinos made at the density where most of them originate, is near 2 MeV — the vertical line in the first figure, where $\beta$ equals $\cos 2\theta$. It falls in the one stretch of the solar spectrum where there are few neutrinos: above the pep line at 1.44 MeV, below the energies where the ⁸B spectrum carries most of its flux. The two families of solar neutrinos that experiments can measure most precisely sit on opposite sides of the step. That was not arranged by anyone, and it is the reason the energy dependence could be established at all. If the transition had fallen in the middle of the ⁸B spectrum, the effect would have been a slow tilt across a single source, measured against a model of its shape. Instead it separates one set of sources from another, and each side can be measured with a different detector technology.

The Sun’s density profile enters in one more way. The electron density falls, over most of the radius, by a factor of $e$ every 0.095 of the radius, which is as close to an exponential as any real structure gets, and the resonance for any energy above about 2 MeV lies somewhere on that exponential. Where exactly does not matter much: once a neutrino has passed its resonance adiabatically, it is on the upper level and stays there. What matters is the next question — whether the passage is slow enough.

How slowly the Sun thins #

A neutrino follows its level only if the density changes slowly compared with the neutrino’s own internal clock at the resonance, where the levels are closest and the clock is slowest. The ratio of the two — the density’s scale height divided by the oscillation length at resonance — is the adiabaticity parameter $\gamma$, and the probability of jumping from one level to the other is $\exp(-\pi\gamma/2)$, the Landau–Zener formula.

For the parameters that are now measured, $\gamma$ is about 2,700 at 10 MeV and larger at lower energies. The Sun thins thousands of times too slowly to disturb a neutrino on its level, and the jump probability is zero to any precision that could be measured. The conversion is perfectly adiabatic. That is why the matter limit is so clean: every ⁸B neutrino that meets a resonance follows it, and the survival probability is fixed by the mixing angle alone.

It did not have to be so, and for a decade it was not known to be. Before 2002 several combinations of mixing angle and splitting fitted the counted rates, and one of them — a small mixing angle, $\sin^2 2\theta \approx 0.005$, with $\Delta m^2 \approx 5\times10^{-6}\ \mathrm{eV}^2$ — was the favourite for some years, because it explained the rates with a conversion that was tuned to be almost complete for some energies and absent for others. In that solution $\gamma$ is about 0.4 at 10 MeV, right at the edge, so the jump probability is 0.5 and changes quickly with energy. It predicted a strongly distorted ⁸B spectrum: the survival probability would have climbed across the range the water detectors measured.

The spectrum came out flat. The small-mixing solution had been disfavoured by the flatness of the ⁸B spectrum before the reactor experiment KamLAND confirmed the large-mixing one independently, by watching electron antineutrinos from Japanese power stations disappear over 180 kilometres. The Sun’s adiabaticity is therefore itself a measurement: the only solution that survived is the one in which the passage through the resonance is thousands of times gentler than it needs to be.

Which of the two masses is heavier #

The matter effect does something that vacuum oscillation cannot, and it is the result that most outlasts the solar neutrino problem.

In vacuum, the probability of oscillating depends on $\sin^2 2\theta$, which is the same for an angle $\theta$ and for $90^\circ - \theta$. Those two angles describe genuinely different worlds: in one, the mass state that carries most of the electron flavour is the lighter of the pair; in the other it is the heavier. No vacuum experiment can tell them apart, because vacuum oscillation depends only on the size of the mass difference, never on its sign.

Matter can, because the matter potential has a sign. It raises the electron flavour’s energy. If the electron-rich state is the lighter one, raising the electron flavour pushes it up towards the heavier state, and the two levels meet at a resonance — the case drawn in the level diagram. If the electron-rich state is already the heavier one, raising it pushes it further away, and there is no resonance at all; the matter drives the high-energy neutrinos deeper into the electron-rich state, and their survival probability would rise, to 0.65 at 10 MeV. The two worlds give the same survival for the low-energy neutrinos, where matter barely matters, and opposite answers for ⁸B.

The measured ⁸B survival is $0.317 \pm 0.016$. The first world is right: the mass state carrying most of the electron flavour is the lighter one. This is the only neutrino mass ordering that has been measured. The other one — whether the third mass state is heavier or lighter than these two — is a question that long-baseline beams and reactor experiments are still trying to answer, and the method they use is the same one: send neutrinos through enough matter, over a long enough path, that the sign of the potential makes a difference. The Sun did it first, with a hundred moles of electrons per cubic centimetre and a path of seven hundred thousand kilometres, and the answer it gave is in every calculation of neutrino physics since.

The Earth turns a few of them back #

There is a second, smaller matter effect, and it runs in the opposite direction. The neutrinos that leave the Sun are mostly in the mass state $\nu_2$, and in vacuum they stay there. At night, a detector sees the Sun through the Earth, and the Earth has electrons too — about two moles per cubic centimetre in the mantle, fifty times fewer than the solar core, and more in the iron core that the planet’s moment of inertia says has sunk to its centre. Inside the Earth the mass states are no longer the propagation states, and $\nu_2$ begins to oscillate, very slightly, back towards electron flavour.

The regeneration is small because the Earth’s density is small: its matter term is a few per cent of the vacuum term at the energies concerned, and the conversion it drives is proportional to that. For a path long compared with the oscillation length — which in the mantle is a few hundred kilometres, and every night-time path through the Earth is thousands — the gain averages to a few per cent of the flux at 10 MeV. It grows with energy, as the matter term does, and it grows as the splitting shrinks, since the matter term is measured against $\Delta m^2/2E$. A measured day–night asymmetry is therefore a measurement of the splitting made with no apparatus but the planet.

Super-Kamiokande measured it in 2014: the rate of neutrino–electron scatters was higher at night by $3.2 \pm 1.1 \pm 0.5$ per cent. That rate is diluted relative to the survival probability, because the other flavours also scatter off electrons at about a sixth of the rate, and the detector’s energy sample runs well below 10 MeV, so the asymmetry the figure computes at the neutrino level is larger than the asymmetry in the counted events by a factor of order one and a half. The measurement was the first direct evidence that the Earth’s matter affects neutrinos, at close to three standard deviations. It was also, for some years, part of a small but persistent tension: the solar data preferred a splitting nearer $5\times10^{-5}\ \mathrm{eV}^2$, while the reactor experiment measured $7.5\times10^{-5}\ \mathrm{eV}^2$. The asymmetry measured was somewhat larger than the reactor value predicted, which is the direction the figure shows a smaller splitting would go.

A rise that has not been seen #

The figure on which this whole argument rests has one feature that has never been clearly observed. Between the matter limit at high energy and the vacuum average at low energy, the survival probability has to turn up, and for the ⁸B neutrinos the turn happens between about 3 and 8 MeV.

The prediction is a rise of about 0.12 between 10 and 3 MeV. The measured spectra are flat. SNO, Super-Kamiokande and Borexino have each reported the ⁸B survival probability as a function of energy, and each is consistent with no slope at all; the upturn is allowed by the data at one or two standard deviations, and not seen. The reason is partly a question of where the rise is. The figure shades the energies below the thresholds of the water detectors: 5 MeV, above which most of their data were taken, and 3.5 MeV, the lowest analysis threshold reached. Half of the predicted rise happens below 5 MeV, and the steepest part of it happens exactly where radioactive backgrounds in the detectors overwhelm the signal. The one feature of the curve that would demonstrate the transition inside a single source sits where the measurements are thinnest.

It matters because the transition region is where anything unconventional would show. Non-standard interactions between neutrinos and matter, a light sterile neutrino, or a density profile different from the model’s would all shift or reshape the upturn while leaving the two plateaus alone. The plateaus are measured precisely; the thing between them is not, and every proposal that the flat spectrum means something new is a proposal about the shape of that rise.

What the picture takes for granted #

Every curve in these figures was computed from five numbers — the mixing angle, the splitting, the small third mixing angle, the Fermi constant and the electron density — and the density is the least certain of them. It comes from the standard solar model, and the model’s electron density in the core is not independently measured. The sound speed, which helioseismology does measure, constrains the density and composition together; the heavy elements that a revised spectroscopy removed from the Sun’s surface change the interior at the per-cent level, and the resonance positions move with them. At the precision of the plateaus that barely matters. At the precision needed to see the upturn, it would.

The production density is also a simplification. Each source is made over a range of radii — the ⁸B neutrinos in the innermost twentieth of the Sun, where the temperature is highest, the pp neutrinos over a region several times wider — and a proper calculation averages the survival probability over each source’s production profile. The figures use a single density for each curve, 90 moles per cubic centimetre for the survival curve and 93 for ⁸B. The averaging smooths the step slightly and moves no plateau.

Two further effects are left out entirely. The figures treat the conversion as the only thing happening to the neutrinos, which is right for the Sun but wrong for a supernova, where the neutrino density is so high that neutrinos scatter off each other and the flavour evolution becomes collective and non-linear. And they assume three neutrinos with standard interactions. Both assumptions are the ones the upturn would test.

A measurement of the particle made with a star #

The history runs in an unusual direction. For thirty years the solar neutrino problem was treated as a problem about the Sun, and the proposed solutions were changes to the solar interior — a cooler core, a mixed core, a rapidly rotating core. The fusion rates were re-measured, the opacities recomputed, the model’s convection adjusted. Helioseismology then fixed the interior so tightly that none of those escapes remained, and the problem became a problem about the neutrino. The step in the survival probability is the form in which that reversal is finished: a detailed measurement of a property of an elementary particle, obtained by using a star as the apparatus.

The Sun is a good apparatus for three reasons that have nothing to do with each other. Its core density puts the matter potential within a factor of two of the vacuum term for the neutrinos it happens to produce. Its density falls smoothly and slowly enough to make the conversion perfectly adiabatic. And its neutrino spectrum has a gap in exactly the place the transition falls. Any one of these could have been otherwise. A star twice as dense in the core would have moved the transition down onto the ⁷Be line; a steeper density profile would have made the conversion partial and energy-dependent in a way that is hard to disentangle; a spectrum without a gap would have smeared the step across a source.

The mass ordering it fixed is the one that shows the depth of the result. Before the Sun was used this way, there was no conceivable measurement that could say which of two neutrino mass states was the heavier; now that fact is an input to the analysis of every long-baseline experiment, and it came out of a hundred moles of electrons per cubic centimetre.

Still open: whether the rise is where it should be #

The two plateaus are measured to a few per cent, the day–night regeneration to three standard deviations, and the mass ordering of the first two states without ambiguity. What is not measured is the transition itself. A detector that could see ⁸B neutrinos down to 2 or 3 MeV with low background — a large liquid scintillator deep underground, or a water detector loaded to lower its threshold — would trace the upturn directly, and the same detector would see the CNO neutrinos whose flux measures the metals in the solar core. The rise will either appear where the curve puts it, closing the solar chapter of neutrino physics, or it will not, and the flatness will become the first observation of an interaction nobody has written down. The same question can be asked of other stars only at much greater cost: the next time a nearby star collapses into a neutron star, its neutrinos will pass through densities a billion times higher than the Sun’s, and the level crossings that turned ⁸B neutrinos smoothly into $\nu_2$ will happen in a medium where the neutrinos themselves are part of the matter.