A rise a detector shrinks to a tilt

The chance that an electron neutrino from the Sun’s core still has its flavour when it reaches the Earth depends on its energy. Below a couple of MeV it is a little over a half; above about eight it is a third, and the difference is made by the Sun’s own electrons, which shift the neutrino’s mixing as it is born and, for the higher-energy neutrinos, carry it smoothly through a resonance on the way out. The two plateaus have both been measured. The rise between them has not: the ⁸B neutrinos, the only solar source whose spectrum covers the transition, show a survival probability that is flat within the errors from five MeV upwards, and every detector that has looked below five has seen the rise allowed and not demanded.

That has been read two ways. One is that something unconventional — a new interaction between neutrinos and matter, a light sterile state — flattens the transition. The other is that the detectors could not have seen the rise anyway. This essay works out the second reading, because it has to be excluded before the first can be believed, and the arithmetic turns out to say something quite specific about what kind of detector would see the rise and what kind cannot, however long it runs.

Why only one source can show it #

The Sun makes neutrinos in several reactions, and only one of them spans the transition in numbers that can be counted. The pp neutrinos, which carry most of the flux, stop at 0.42 MeV, far below it; the beryllium-7 and pep neutrinos are single lines at 0.86 and 1.44 MeV, each a point on the low-energy plateau. The ⁸B neutrinos are rare — about one in ten thousand — but their spectrum runs from zero to sixteen MeV and passes straight through the region where the survival probability turns over. They are made in the innermost few per cent of the Sun’s radius, the hottest part of a furnace that is, by the volume, astonishingly feeble, and they are born at the highest electron density any solar neutrino sees, which is what puts their transition at a few MeV rather than lower.

So the rise has to be read off one spectrum, from one source, by one kind of measurement at a time. There is no second solar source to check it against at the same energies, and every argument below is an argument about what a single spectral measurement can show.

What a detector records instead #

A detector does not record a neutrino’s energy. It records an electron’s. In a water or scintillator detector, the neutrino scatters off an electron in the target and the electron’s light is counted, and in a heavy-water detector the neutrino can instead be captured by a deuteron, turning it into two protons and an electron. The survival probability must be inferred from the electron’s energy spectrum, compared with the spectrum the detector would have seen if every neutrino had kept its flavour.

The opening figure makes the comparison. The survival probability (dashed) rises from 0.32 at 10 MeV to 0.44 at 3, a change of a third of its value. The scattering ratio, against the electron’s energy, rises by 0.044 over the same range — a third of the change. The capture ratio rises by 0.063, about half. And neither curve sits on the dashed one: the scattering ratio lies above it everywhere, the capture ratio below it at low energy. Each detector sees a different, smaller and displaced version of the same physics, for two reasons that the next two figures separate.

A floor and a blur #

The first reason is that scattering is not blind to the other flavours. An electron neutrino scatters off electrons through both the charged and the neutral current; a muon or tau neutrino only through the neutral current, about a sixth as often. So a neutrino that has converted is not lost to a scattering detector — it is counted at a sixth of the weight — and the ratio of counted to unconverted spectra is the survival probability plus a sixth of everything else. That raises the ratio above the survival probability and compresses its range: a change of 0.12 in survival becomes a change of about a tenth in the ratio before anything else is considered. The flavour separation that ended the solar neutrino problem used exactly this partial sensitivity, by comparing scattering with capture and with a reaction that counts every flavour equally. For the upturn it is a dilution.

The second reason is kinematic, and it is larger. A neutrino scattering off an electron gives it anything from nothing up to almost all of its energy, with a distribution that favours sharing, so an electron recorded at a given energy could have come from any neutrino above it. The figure shows which ones. A 3-MeV scattered electron is not, mostly, the product of a 3-MeV neutrino; it comes from a neutrino whose energy is more likely seven, scattering gently. The parents of low-energy electrons are spread over the whole upper half of the spectrum, where the survival probability has nearly finished rising — and so the low-energy end of the scattering spectrum, where the rise should be steepest, is mostly a record of neutrinos that have already completed it.

Capture on deuterium has no such blur. The deuteron is heavy and the reaction has a fixed energy cost, 1.44 MeV, so the electron carries the neutrino’s energy minus that cost, and each electron energy corresponds to one neutrino energy. That is why the capture ratio follows the survival curve’s shape, shifted by the 1.44 MeV and smoothed only by the detector’s resolution. It still shows only about half the rise between electron energies of 3 and 10 MeV, because a 3-MeV capture electron comes from a 4.4-MeV neutrino, already some way up the slope.

Both reasons were present in every solar detector that has measured ⁸B. The model behind these figures — the standard tree-level scattering cross-section, a capture cross-section scaled to reproduce the heavy-water detector’s count, an allowed beta-decay spectrum for ⁸B tuned to its measured mean energy, and a water detector’s resolution — reproduces that detector’s 1,967 capture events in its first 306 days to within 7 per cent, so the dilution is not an artefact of a careless model.

Counting is not the problem #

A rise of 0.044 in a scattering ratio is a tilt of about ten per cent across a spectrum, and a tilt that size is easy to see with enough events.

The comparison is between the predicted spectrum and the best spectrum a flat survival probability can give, with the flat value and the overall normalisation both free — the question being not whether the survival probability is below one, which is known, but whether it changes with energy. Counting alone, a scattering detector reaches three standard deviations in sixty kiloton-years above 3.5 MeV. The largest water detector has a fiducial mass of twenty-two and a half kilotons and has been taking solar data for over two decades, an exposure of several hundred kiloton-years. If counting were the limit, the upturn would have been seen many times over, even allowing for the efficiency losses this idealised count leaves out.

The capture detector is more striking still: two kiloton-years, because its ratio carries half the rise rather than a third and its electrons point to their parents’ energies. The heavy-water detector that made the flavour measurement had a kiloton of target and ran for about three years. By counting alone it had what it needed.

So the reason the rise has not been seen is not that too few neutrinos have been counted. It is something that grows with the count rather than shrinking — a systematic uncertainty — or something that removes the counted events from use.

A tilt that looks like a miscalibration #

The most obvious such uncertainty is the energy scale: the conversion from the light an electron produces to the energy it had. A detector calibrates it with a small electron accelerator lowered into the tank and with radioactive sources, and knows it to a few tenths of a per cent. As with the comparison stars in a photometric measurement, the calibration is not a preliminary to the measurement but part of it, and its error enters every bin with a shape of its own. A few tenths of a per cent sounds negligible until it is compared with what the rise does to the spectrum.

The rise, compared with a flat survival probability with the same total, is a spectrum a few per cent high at the low end and a few per cent low at the high end, crossing over near 6 MeV. A miscalibrated energy scale moves every electron to a slightly wrong energy, and because the ⁸B spectrum falls steeply towards its endpoint, that shift makes the high-energy end of the spectrum appear higher or lower by many times the miscalibration. An energy scale too low by half a per cent depresses the counts at 12 MeV by nearly five per cent, which is about what the rise does there.

The two do not look the same everywhere, which is the saving grace. The miscalibration leaves the low end of the spectrum almost alone, where the rise is largest. A fit that allows the energy scale to float within its calibrated uncertainty can still separate the two, but it pays for the separation: at five hundred kiloton-years, the significance falls from 8.7 standard deviations to 4.5 when a half-per-cent scale uncertainty is admitted. The high-energy half of the evidence has been absorbed by the calibration; what is left is the low-energy half.

Where the evidence sits #

The evidence the upturn offers a scattering detector is not spread evenly over the spectrum. It is concentrated in two places: below about 4.5 MeV, where the counted spectrum is a few per cent above a flat survival probability’s, and between about 7 and 12 MeV, where it is a few per cent below. Between them, near 5 to 6 MeV, the two predictions cross and the spectrum says nothing. The high-energy half is the half an energy-scale uncertainty absorbs. The low-energy half sits where water detectors are worst.

Below about five MeV, a water detector’s counts are dominated not by solar neutrinos but by radioactivity: radon dissolved in the water and emanating from the detector’s walls, whose daughters emit electrons and gamma rays of a few MeV, and the gamma rays of thallium and bismuth in the photomultipliers and the rock. These events are not neutrinos and do not point away from the Sun, and they are removed statistically, by fitting the angular distribution of events in each energy bin: solar neutrino recoils point away from the Sun and background does not. That works, but every background event adds its own fluctuation to the subtraction, and a bin with ten background events for each solar one carries the statistical weight of a bin with a tenth of the solar events.

A background, not a threshold #

The background is drawn here with a simple shape — equal to the solar signal at some energy and rising tenfold every 1.15 MeV below it, a shape chosen to resemble the steep radon-dominated background in water rather than to reproduce any one detector’s — and the energy at which it takes over is varied. The result is the essay’s central point. With no background, lowering the threshold from 3.5 MeV to 2.5 raises the significance from 4.5 to 5.8 standard deviations. With the background taking over anywhere above about 5 MeV, the two thresholds give the same answer: the extra counts below 3.5 MeV are background, and they buy nothing. What moves the significance is not the threshold but the energy at which the background takes over, and at six MeV a detector with five hundred kiloton-years is at 2.4 standard deviations, before the efficiency losses and the further systematics this model leaves out.

That is the arithmetic of the null result. Two decades of a large water detector have the counting statistics to see the upturn many times over; an energy-scale uncertainty of half a per cent halves the evidence, and a background that dominates below five or six MeV halves it again. The flat spectrum is what a detector with those properties sees whether the rise is there or not, and the detectors’ own published limits are consistent with this: the upturn is allowed by the data at one or two standard deviations, and a flat survival probability is allowed too.

The deuterium detector is held back by the same thing in a different way. Its capture events carry more of the rise and need a fraction of the exposure, but the heavy-water detector’s background at low energy was dominated by the same radioactive chains, and below about 3.5 MeV it could not separate capture events from those of the neutral-current reaction it was also counting. Its capture-only spectrum is the best measurement of the transition that exists, and it was cut off where the rise was still steepening.

What would see it #

A detector that sees the rise needs three things in a particular order of difficulty. It needs to be clean at low energy, so that the background takes over below about three MeV rather than five or six; that is the dominant requirement, and it is a matter of material purity and shielding rather than size. It needs an energy scale known to a few tenths of a per cent, which existing detectors already have. And only then does it need exposure, which for a clean scattering detector is tens of kiloton-years and for a capture detector a few.

Liquid scintillator meets the first requirement better than water. It produces many times more light per MeV, so its resolution is better, and it can be purified to radioactive levels many orders of magnitude below water’s. The detector that measured the CNO neutrinos did so by reaching those levels in three hundred tons; a scintillator of twenty kilotons, now operating for reactor neutrinos, will see ⁸B scattering down to about three MeV if its backgrounds there are as low as its design. A large argon detector built for accelerator neutrinos would see ⁸B by capture on argon, which, like deuterium, preserves most of the neutrino’s energy — but it brings its own low-energy background from argon-39 and from neutrons captured in the argon, which is to say the same problem in a new material.

A test of the method, made by the instrument #

The argument in this essay is about detectors, but it is the same argument that applies whenever a small shape is sought in a spectrum that has a steep slope, a dominant background at one end and a calibration uncertainty at the other. A gap in the masses of compact objects poses the same question with orbits instead of electrons: whether the counting can test the feature, and whether systematic effects can fill it. In both, the counting arithmetic is the easy part, and the conclusion depends on whether the instrument’s own errors have a shape that resembles the feature or not.

Here they partly do. The energy-scale error resembles the high-energy half of the upturn’s signature, and the background swamps the low-energy half. The model’s survival curve, computed from the Sun’s density profile and measured from its oscillations to a fraction of a per cent, is not in question; what is in question is whether the instruments could see it, and the answer is that the ones built so far could not.

The same arithmetic for a dying star #

The next nearby supernova will test the same instruments on a different problem. A collapsing core releases nearly all of its energy as neutrinos of every flavour in about ten seconds, at energies of ten to twenty MeV, and the flavour conversions inside the star — through the same resonances, at densities a billion times the Sun’s — leave their mark on the spectrum each detector records. A water detector would count thousands of events, mostly by a capture reaction on protons that preserves the energy, rather than by scattering, so the blur that hides the solar rise would matter less. The background would hardly matter at all, because ten seconds of a supernova outshines years of radon. What would matter is the same thing as here: whether the detector’s view of the spectrum, rather than the physics, decides what a flat or a curved spectrum means.

What the model leaves out #

The ⁸B spectrum is an approximation tuned to the measured mean energy; the true spectrum has a slightly different shape set by the broad excited state of beryllium-8 the decay feeds, and that shape is itself an uncertainty the detectors fit. The detector efficiency is one, there are no event-selection cuts, and the energy resolution is a single water-like curve. The background has one parametric shape with no uncertainty of its own, while a real background’s normalisation is fitted and its uncertainty adds to the error. The energy-scale prior is the only systematic; the resolution’s own uncertainty and the ⁸B spectral-shape uncertainty both add more. And the day–night effect, in which neutrinos crossing the Earth at night partly regain their flavour, is not included — it adds a few per cent at all energies and has been measured separately.

None of these makes the rise easier to see. Each one adds a systematic, and the conclusion that a flat spectrum is what these detectors expect to see whether the rise is there or not becomes stronger, not weaker.

Still open: whether the rise is where the curve puts it #

The question the flat spectrum posed — whether something unconventional flattens the transition — is not answered by showing that the detectors could not have seen it. It is only postponed. The Sun’s ⁸B flux is already a thermometer for its core, and the shape of its spectrum will be the test of the neutrino physics: a large, clean scintillator detector with a three-MeV threshold and a few years of data would see the predicted rise at more than three standard deviations or not see it at all. If it sees the rise where the curve puts it, the solar chapter of neutrino oscillation closes on its last predicted feature. If it sees a spectrum still flat at three MeV, the arithmetic above will have removed the instrumental explanations, and the flatness will be the first clean measurement of an interaction the standard picture does not contain.