Experiment: Neutrino Oscillations

Tests Phenomenon 103.50. Assuming Definition 103.46, Theorem 103.54 and Theorem 103.57.

Neutrino oscillation is the only laboratory phenomenon that unambiguously falsifies the Standard Model as originally written: it requires neutrino masses, which the massless left-handed neutrino of The Weyl Equation and Neutrinos does not have. The evidence accumulated over thirty years, beginning as an anomaly rather than a discovery — the solar electron-neutrino deficit found at Homestake [Davis:1968] [Cleveland:1998] and confirmed by Kamiokande [Hirata:1989] and the gallium detectors [Hampel:1999] [Abdurashitov:2009] — and closing with two measurements that admit no astrophysical escape: the zenith-angle dependence of atmospheric muon neutrinos in Super-Kamiokande [Fukuda:1998] and the flavour-blind neutral-current flux measured at Sudbury [Ahmad:2002], which showed the missing solar neutrinos had changed flavour rather than gone away. The Nobel Prize of 2015 recognized those two. This chapter sits after the flavour and neutrino theory of Flavour Physics and Neutrinos and The Weyl Equation and Neutrinos and supplies the parameters they use.

The prediction under test is that a neutrino changes flavour in flight. That is the single statement every instrument described below was built to confirm or to kill, and it is worth being exact about what a confirmation requires. It is not enough to find fewer neutrinos of a given flavour than expected: a deficit is what a wrong source model, a wrong cross-section or a wrong efficiency also produces. Three things, and only these three, distinguish flavour change from every mundane alternative, and the chapter is organized around them.

1. The missing flux reappears in another flavour. A detector that measures the electron-flavour rate and the flavour-summed rate in the same target, with the same exposure, can subtract one from the other; the difference is model-independent. This is the Sudbury measurement.

2. The deficit depends on \(L/E\) and on nothing else. Path length and energy enter the interference phase only in that combination. No absorption law, no decay law and no production spectrum reproduces a dependence on the ratio alone. This is the Super-Kamiokande zenith-angle measurement, and, in its sharpest form, the KamLAND spectrum.

3. The deficit is not monotone in \(L/E\) but oscillates. A normalization error can only scale a rate; only interference makes it come back. This is what a controlled terrestrial source, at a baseline chosen so that the phase is of order a radian across the detected spectrum, can show and an astrophysical source cannot.

The chapter is unusual among the experiment chapters in covering a family of instruments rather than one — radiochemical tanks, water Cherenkov detectors, heavy water, liquid scintillator, and long-baseline accelerator beams — because no single detector establishes the effect, and the argument is carried by the pattern across baselines and energies, sharpened by the matter effect of Wolfenstein, Mikheyev and Smirnov [Wolfenstein:1978] [Mikheyev:1985]. It closes on what is still unknown: the mass ordering, the \(CP\) phase, the absolute mass scale, and whether the neutrino is its own antiparticle.

Historical context and the prediction under test

Neutrinos: detection and flavour

The neutrino entered physics in 1930 as a bookkeeping device and became an object of experiment twenty-six years later. Cowan, Reines and their collaborators put a detector beside a fission reactor at Savannah River and looked for the inverse beta reaction Equation (101.3), \(\bar{\nu}_{e}+p\to n+e^{+}\), whose threshold Equation (101.4) is \(1.806\,\mathrm{MeV}\), that is \(2.894\times 10^{-13}\,\mathrm{J}\). Their target was two tanks each holding about \(200\,\mathrm{L}\) of water loaded with cadmium chloride, sandwiched between three tanks of about \(1400\,\mathrm{L}\) of liquid scintillator; the detector sat some \(12\,\mathrm{m}\) below ground level and about \(11\,\mathrm{m}\) from the reactor core. The signature was a delayed coincidence: two \(511\,\mathrm{keV}\) annihilation photons from the positron, followed some microseconds later by the several megaelectronvolts of gamma rays released when the neutron, having thermalized, is captured on cadmium. The rate with the reactor on exceeded the rate with it off by about three counts an hour [Cowan:1956] [Reines:1956], at a cross-section of order \(10^{-47}\,\mathrm{m}^{2}\) — the number derived in Equation (101.5) and recorded as Phenomenon 101.5.

Everything in this chapter descends from that apparatus. The delayed-coincidence tag on a free-proton target is still the technique of KamLAND and Daya Bay sixty years later, scaled from a few hundred litres to a kilotonne; what changed is the radiopurity of the scintillator and the photocathode coverage, not the idea.

The second ingredient is flavour. Firing a beam of neutrinos from pion decay at a ten-tonne spark chamber behind \(13.5\,\mathrm{m}\) of iron, Danby and collaborators at the Brookhaven alternating-gradient synchrotron found the charged particles produced to be muons and never electrons [Danby:1962]: thirty-four single-muon events and no electron showers, where the electron channel is kinematically the more open of the two. The null result establishes a quantum number distinguishing the two families. The third family's neutrino was observed directly at Fermilab in 2000 by the DONUT collaboration, which found the short kinked track of a tau in nuclear emulsion exposed to a beam from \(800\,\mathrm{GeV}\) protons on a tungsten target: four candidates on an expected background of \(0.34\) [Kodama:2001].

The counting is closed from the other end by the invisible width of the \(Z\) boson, which fixes the number of light neutrino species that couple to it at

[Schael:2006]; this is Phenomenon 103.22, and its derivation from the measured widths is Equation (103.15). Three flavours, conserved separately in the Standard Model as written, and therefore three quantum numbers for an oscillation experiment to violate: that is the state of knowledge against which every measurement below is read.

Mixing and oscillation

Pontecorvo proposed in 1957 that a neutrino might transform into its antiparticle by analogy with the neutral kaon, whose strangeness oscillates because the states of definite strangeness are not the states of definite mass [Pontecorvo:1957], in the paper whose English translation is [Pontecorvo:1958]. Maki, Nakagawa and Sakata gave the idea its modern form in 1962: the states produced with definite flavour are superpositions of states of definite mass, and the mismatch is a unitary matrix [Maki:1962]. That matrix is defined in Definition 103.46 and its measured magnitudes are Equation (103.49).

The formalism is derived in Flavour Physics and Neutrinos and is not rebuilt here. Three results from there are used throughout this chapter and are quoted now so that the experimental sections can refer to them without interruption. For two flavours with mixing angle \(\theta\) and squared-mass splitting \(\Delta m^{2}:=m_{2}^{2}-m_{1}^{2}\), the probability that a neutrino born as \(\nu_{\alpha}\) is found as \(\nu_{\beta}\) after a flight of length \(L\) at energy \(E\) is Equation (103.52),

Every symbol is SI: \(\Delta m^{2}c^{4}\) is an energy squared, \(\hbar cE\) is an energy squared times a length, and \(\varphi\) is a pure number. The practical form of the phase, derived as Proposition 103.52, is

and the oscillation length — the distance over which \(\varphi\) advances by \(\pi\) — is Equation (103.55),

The number \(1.26693\) is the one every experimental paper writes as “\(1.27\)” and quotes rather than derives; it is \(\left(4\hbar c\right)^{-1}\) expressed in the units the field happens to use, and nothing else.

Two consequences organize everything below. Oscillation is an interference between two mass components, so it measures a difference of squared masses and can never measure a mass; and it depends on \(L\) and \(E\) only through their ratio, so an experiment is characterized by \(L/E\) and not by either separately. A detector placed where \(\varphi\ll1\) sees nothing; one placed where \(\varphi\gg1\) sees a constant suppression by the averaged factor \(1-\tfrac{1}{2}\sin^{2}2\theta\); only a detector placed where \(\varphi\) is of order one across the detected spectrum sees the oscillation itself. Equation (114.4) is therefore the design equation of every experiment in this chapter, and Table 114.1 is the resulting map.

The last row is the reason the solar measurements are treated separately from all the others. At a phase of \(10^{7}\) radians the oscillating factor has averaged out completely long before the neutrino reaches the Earth, and a vacuum calculation would give a featureless suppression carrying no information; what the solar experiments actually measure is not vacuum oscillation at all but the matter-modified propagation of Theorem 103.57, whose signature is a dependence on energy rather than on baseline. That is taken up in Section 114.5.2.

The standard solar model prediction

The Sun burns hydrogen to helium by the pp chains, with a small contribution from the CNO cycle, and the net reaction of either is

releasing \(Q=26.73\,\mathrm{MeV}=4.283\times 10^{-12}\,\mathrm{J}\) per helium nucleus and producing exactly two electron neutrinos. The chains themselves belong to Sections 52.2.2 and 52.2.3, and the detailed model that turns them into a predicted spectrum — opacities, equation of state, composition, calibrated against the observed luminosity and radius — is that of Bahcall, Serenelli and Basu [Bahcall:2005]. Its predictions are Table 114.2.

Two features of that table decided thirty years of argument. The first is that the total is not really a prediction at all. The luminosity constraint — derived in Equation (52.36) from nothing but the observed solar constant, the energy release of Equation (114.5) and the Earth–Sun distance — fixes

independently of every uncertain input of the model, and the \(pp\) branch carries about \(94\,\mathrm{\%}\) of it. A deficit measured in the \(pp\) neutrinos therefore cannot be blamed on the model.

The second is that the \(^{8}\mathrm{B}\) flux is the least certain quantity in the table by a wide margin, because the reaction \(^{7}\mathrm{Be}+p\to{}^{8}\mathrm{B}+\gamma\) sits far out on the Gamow peak and its rate varies as a very high power of the central temperature. For three decades that was the escape route: Homestake and Kamiokande measured only the \(^{8}\mathrm{B}\) branch, and a central temperature lower by a few per cent than the model's would have removed the discrepancy without any new physics. Closing that route — by measuring a branch the model cannot get wrong, and then by measuring the flavour content directly — is what the rest of this chapter is about.

That sensitivity is now derived where it belongs, in the stellar-structure chapter: the \(^{8}\)B flux responds with the sum of the Gamow exponents of the reactions feeding and draining \(^{7}\)Be, giving \(\Phi(^{8}\mathrm{B})\propto T_{\mathrm{c}}^{\sim29}\) at fixed density and composition — self-consistent solar-model perturbation studies soften the exponent toward the customary \(T^{24}\) without changing the argument — so a central temperature a few per cent low reproduces the whole measured deficit. The derivation, and the closing of that escape route by the luminosity-pinned \(pp\) flux and by helioseismology, are carried out under Phenomenon 52.22, with the exponents tabulated in Table 52.1.

The observable and what a measurement must deliver

An oscillation experiment reports one of four observables, and it is worth naming them before any apparatus is described, because the sources of error are different in each and the tables of Section 114.4 are organized by them.

A rate. The number of interactions of a given flavour per unit time in a known target, compared with a prediction. This is the weakest observable, because it inherits the full uncertainty of the source model, the cross-section and the detection efficiency. Homestake delivered a rate, and that is why its result was arguable for thirty years.

A ratio of rates in one detector. Two channels with different flavour sensitivity, measured in the same target during the same exposure. The source normalization cancels exactly. This is Sudbury's charged-current to neutral-current ratio, and it is the strongest observable in the chapter.

A ratio of rates in two detectors. Identical detectors at two baselines from one source. Source normalization, cross-section and efficiency all cancel to the extent that the detectors are identical. This is the Daya Bay and RENO design.

A shape. The dependence of the survival probability on \(L/E\) within one dataset. Nothing that scales a rate can produce a shape, so this is the observable that distinguishes oscillation from every rival hypothesis without appeal to any external prediction. Super-Kamiokande delivered it in the zenith angle, KamLAND in the reactor energy spectrum.

The chapter's argument is the accumulation of these four, in that order of strength, over thirty-five years.

Apparatus

Every instrument in this chapter solves the same two problems. A neutrino cross-section of order \(10^{-47}\,\mathrm{m}^{2}\) (Phenomenon 101.5) means that a useful counting rate demands a target of \(10^{30}\) nuclei or more, that is hundreds of tonnes; and a rate of a few events per day in such a target means that any background above a few events per day destroys the measurement. Mass and quiet are therefore the two axes along which these detectors are designed, and everything else — the choice of target nucleus, the depth, the purification chemistry, the photomultiplier coverage — follows from one or the other.

Table 114.3 collects the physical parameters. The subsections that follow give, for each class, the target, the reaction it exploits, the signal it produces and the background it must beat.

Radiochemical detectors

A radiochemical detector counts atoms, not events. The neutrino converts a stable nucleus into a radioactive one by inverse beta decay; the product is a different chemical element, so it can be separated from a target of hundreds of tonnes by ordinary chemistry and counted afterwards by its own decay. The method is uniquely insensitive to background during exposure — nothing that is not a neutrino makes the target element into the product element — and it pays for that with the total loss of every other piece of information. There is no energy, no direction, no time of arrival: only a number of atoms integrated over a run of weeks.

The two properties that matter are the threshold, fixed by the mass difference of the two nuclei, and the half-life of the product, which must be long enough to accumulate a countable population and short enough that counting does not take forever. Both detectors described here sit near the optimum: a product half-life of a few weeks against an exposure of a few weeks.

The Homestake chlorine tank

Davis's detector was a horizontal steel tank \(6.1\,\mathrm{m}\) in diameter and \(14.6\,\mathrm{m}\) long, holding \(615\,\mathrm{t}\) — about \(3.8\times 10^{5}\,\mathrm{L}\) — of tetrachloroethylene, \(\mathrm{C}_{2}\mathrm{Cl}_{4}\), an ordinary dry-cleaning fluid chosen because it is cheap, dense in chlorine and chemically inert. It was installed \(1478\,\mathrm{m}\) below the surface in the Homestake gold mine at Lead, South Dakota, an overburden of about \(4200\,\mathrm{m}\) of water equivalent [Davis:1968] [Cleveland:1998].

The reaction is

with a threshold of \(814\,\mathrm{keV}\), that is \(1.304\times 10^{-13}\,\mathrm{J}\). The threshold is what makes the detector a \(^{8}\mathrm{B}\) instrument: by Table 114.2 it is above the \(pp\) endpoint of \(420\,\mathrm{keV}\) altogether, so the branch carrying \(94\,\mathrm{\%}\) of the flux is invisible to it, and about three quarters of the predicted rate comes from \(^{8}\mathrm{B}\) with most of the remainder from \(^{7}\mathrm{Be}\).

The number of target atoms is worth computing, because every rate in Section 114.4 is quoted per target atom and the conversion to atoms per day runs through it. Tetrachloroethylene has molar mass \(165.83\,\mathrm{g}/\mathrm{mol}\), so \(615\,\mathrm{t}\) is \(3.71\times 10^{6}\) moles, containing \(4\times3.71\times 10^{6}\) moles of chlorine and hence \(8.93\times 10^{30}\) chlorine atoms; the isotopic abundance of \(^{37}\mathrm{Cl}\) is \(24.23\,\mathrm{\%}\), giving

At the measured capture rate of \(2.56\times 10^{-36}\,/\mathrm{s}\) per atom (Section 114.4.1) this is a production rate of \(5.54\times 10^{-6}\,/\mathrm{s}\), that is \(0.48\) atoms of argon per day in \(615\,\mathrm{t}\) of liquid. The saturated population, reached when production balances decay of the \(3.024\times 10^{6}\,\mathrm{s}\) half-life, is \(5.54\times 10^{-6}\,/\mathrm{s} /\left(\ln2/3.024\times 10^{6}\,\mathrm{s}\right)\approx24\) atoms. Those two dozen atoms, in \(2.2\times 10^{30}\) target nuclei, are the entire signal.

The product is counted by its own decay. \(^{37}\mathrm{Ar}\) decays by electron capture back to \(^{37}\mathrm{Cl}\) with a half-life of \(3.024\times 10^{6}\,\mathrm{s}\), that is \(35.0\) days, and the K-shell capture leaves an Auger cascade of \(2.82\,\mathrm{keV}\), that is \(4.52\times 10^{-16}\,\mathrm{J}\), deposited in a few nanoseconds within a fraction of a millimetre. A gas proportional counter of about \(0.5\,\mathrm{cm}^{3}\) internal volume — deliberately tiny, because the background scales with the volume and the signal does not — registers this as a fast-rising pulse of characteristic energy. The counters were made of low-background materials, filled with the extracted argon mixed with tritium-free methane, and operated inside a well of sodium iodide within a lead castle.

The backgrounds that had to be excluded before the deficit could be believed were of two kinds. Cosmic-ray muons surviving to \(4200\,\mathrm{m}\) of water equivalent produce \(^{37}\mathrm{Ar}\) by spallation on chlorine at an estimated \(0.047(13)\) atoms per day, some \(10\,\mathrm{\%}\) of the signal; this was measured by exposing the same target at shallower depths and extrapolating. Fast neutrons from the rock, and alpha emitters in the tank walls acting through \(^{37}\mathrm{Cl}(\alpha,n)\)-type reactions, contribute less. The counting background — a proportional counter firing when no argon atom decayed — was suppressed by rise-time discrimination, since a genuine \(2.82\,\mathrm{keV}\) Auger cascade is far more localized than a Compton electron traversing the same gas, and by requiring the decay times of a run's candidate events to be consistent with a \(3.024\times 10^{6}\,\mathrm{s}\) half-life.

The gallium detectors

The gallium experiments exist because of one number: the threshold of

is \(233\,\mathrm{keV}\), that is \(3.74\times 10^{-14}\,\mathrm{J}\), comfortably below the \(pp\) endpoint of \(420\,\mathrm{keV}\). A gallium detector therefore sees the branch whose flux is fixed by Equation (114.6) to one per cent, and a deficit there cannot be attributed to the solar model. That is the whole argument for spending a substantial fraction of the world's annual gallium production on a physics experiment.

GALLEX, and its successor GNO, operated at the Gran Sasso laboratory under about \(3300\,\mathrm{m}\) of water equivalent, with \(30.3\,\mathrm{t}\) of gallium held as about \(101\,\mathrm{t}\) of gallium trichloride in concentrated hydrochloric acid, in a tank of some \(54\,\mathrm{m}^{3}\) [Hampel:1999]. SAGE, at the Baksan Neutrino Observatory under about \(4700\,\mathrm{m}\) of water equivalent, used \(50\,\mathrm{t}\) of gallium in the metallic form — gallium melts at \(29.8\,\mathrm{^\circ\mathrm{C}}\), so the target is a liquid metal held in seven reactors of about \(7\,\mathrm{t}\) each [Abdurashitov:2009].

The target counts are, by the same arithmetic as Equation (114.8) with a molar mass of \(69.72\,\mathrm{g}/\mathrm{mol}\) and an isotopic abundance of \(39.89\,\mathrm{\%}\) for \(^{71}\mathrm{Ga}\),

At the measured rates of Table 114.4 these correspond to \(0.63\) and \(0.97\) atoms of \(^{71}\mathrm{Ge}\) per day, and to saturated populations of about ten and sixteen atoms respectively, the \(^{71}\mathrm{Ge}\) half-life being \(9.875\times 10^{5}\,\mathrm{s}\), that is \(11.43\) days.

The chemistry differs from Homestake's in one important respect: germanium tetrachloride is volatile from the acid solution, so extraction is again a gas purge, but the counting gas is germane, \(\mathrm{GeH}_{4}\), synthesized from the extracted germanium. The decay is again electron capture, with K-shell and L-shell peaks at \(10.37\,\mathrm{keV}\) and \(1.17\,\mathrm{keV}\), that is \(1.66\times 10^{-15}\,\mathrm{J}\) and \(1.88\times 10^{-16}\,\mathrm{J}\). Having two peaks is a real advantage over the single Auger line of \(^{37}\mathrm{Ar}\): a background source that mimics one rarely mimics both in the right ratio.

The gallium detectors are also the only ones in this chapter that were calibrated with an artificial neutrino source of known strength. A \(^{51}\mathrm{Cr}\) source of about \(6.3\times 10^{16}\,\mathrm{Bq}\) — \(1.7\) megacuries, produced by neutron irradiation of enriched \(^{50}\mathrm{Cr}\) in a reactor — emits electron neutrinos by electron capture at \(750\,\mathrm{keV}\), above the gallium threshold and below the chlorine one. Lowered into the tank, it turns the detector into an experiment with a known input. SAGE additionally used an \(^{37}\mathrm{Ar}\) source. What those calibrations returned is discussed in Section 114.6.4, because the answer was not quite one and the discrepancy has not gone away.

Water Cherenkov detectors

A water Cherenkov detector gives up the radiochemical method's freedom from background and buys, in exchange, everything the radiochemical method throws away: the time of each event to a few nanoseconds, its position to tens of centimetres, its energy, its direction, and — decisively for this chapter — its flavour.

The mechanism is the Cherenkov cone of Section 115.4.4. A charged particle faster than the phase velocity of light in the medium radiates on a cone of half-angle given by Equation (115.36),

which for water, \(n=1.34\), and an ultrarelativistic particle gives \(\theta_{c}=41.7^\circ\). The threshold follows from requiring \(\beta>1/n\): with \(\gamma_{\mathrm{th}}=\left(1-n^{-2}\right)^{-1/2}=1.502\), a particle of rest mass \(m\) must have total energy above \(1.502\,mc^{2}\), that is \(768\,\mathrm{keV}\) for an electron and \(159\,\mathrm{MeV}\) for a muon. The projection of the cone on the detector wall is a ring, and the number of photons per unit path length is given by Equation (115.38) — some \(340\) photons per centimetre in the visible band, of which a detector with \(40\,\mathrm{\%}\) photocathode coverage records a few per cent.

Kamiokande-II was the first instrument to detect solar neutrinos in real time. It was a cylindrical tank holding \(3\,\mathrm{kt}\) of purified water \(1000\,\mathrm{m}\) underground in the Kamioka mine, viewed by \(948\) photomultipliers of \(50\,\mathrm{cm}\) diameter giving about \(20\,\mathrm{\%}\) photocathode coverage, with a fiducial volume of \(680\,\mathrm{t}\) for the solar analysis [Hirata:1989]. It detected neutrino–electron elastic scattering,

which has no threshold of its own but is limited from below by radioactive background: the analysis threshold was \(9.3\,\mathrm{MeV}\) initially and \(7.5\,\mathrm{MeV}\) later, so that Kamiokande, like Homestake, saw only the \(^{8}\mathrm{B}\) branch. Its decisive advantage was that the recoil electron remembers the neutrino direction to within about \(25^\circ\) at these energies, so the signal appears as a peak in the distribution of event directions relative to the Sun sitting on a flat background. The same detector recorded the neutrino burst from SN 1987A [Hirata:1987].

Super-Kamiokande is the same idea built seventeen times larger. The tank is \(39\,\mathrm{m}\) in diameter and \(42\,\mathrm{m}\) high and holds \(50\,\mathrm{kt}\) of water; the inner detector is viewed by \(11146\) photomultipliers of \(50\,\mathrm{cm}\) diameter for a photocathode coverage of \(40\,\mathrm{\%}\), and is optically separated from an outer shell instrumented with \(1885\) smaller tubes that serves as an anticoincidence veto against entering cosmic-ray muons. The fiducial volume for atmospheric neutrinos is \(22.5\,\mathrm{kt}\). It sits under \(1000\,\mathrm{m}\) of rock, about \(2700\,\mathrm{m}\) of water equivalent, in the Mozumi mine at Kamioka, where the residual cosmic-ray muon rate through the detector is a few per second — a rate that would swamp everything were the outer veto not there [Fukuda:1998]. The water is continuously recirculated through filtration, reverse osmosis, ion exchange and ultraviolet sterilization, reaching a resistivity near \(1.8\times 10^{5}\,\mathrm{\Omega}\,\mathrm{m}\) and a light attenuation length beyond \(70\,\mathrm{m}\); radon-reduced air is kept above the water surface, because \(^{222}\mathrm{Rn}\) dissolved in the water is the limiting background at low energy.

The property that makes Super-Kamiokande an oscillation instrument rather than merely a large one is flavour tagging by ring shape. A charged-current interaction of a \(\nu_{\mu}\) produces a muon, which is heavy, loses energy slowly and travels in a straight line: its Cherenkov ring has a sharp outer edge. A charged-current interaction of a \(\nu_{e}\) produces an electron, which scatters and radiates, initiating an electromagnetic shower whose many low-energy tracks smear the ring into a fuzzy disc. The two are separated event-by-event with a misidentification probability of a few per cent. Nothing else in this chapter measures the two flavours of a single natural source side by side in one detector, and that is precisely why the atmospheric measurement is decisive.

Heavy water and the neutral-current channel

The Sudbury Neutrino Observatory was designed around one property of the deuteron: it is the lightest nucleus that a neutrino can break up by the neutral current, so a heavy-water target gives access, in one detector, to a reaction that counts electron neutrinos and a reaction that counts all three flavours equally.

The apparatus was \(1\,\mathrm{kt}\) of heavy water of \(99.92\,\mathrm{\%}\) isotopic purity — on loan from the Canadian strategic reserve, the loan being the reason the experiment was possible at all — held in a spherical acrylic vessel \(12\,\mathrm{m}\) in diameter with a wall about \(5\,\mathrm{cm}\) thick. The vessel was surrounded by \(7\,\mathrm{kt}\) of ultrapure ordinary water acting as shielding against radioactivity from the rock and the photomultipliers, and viewed by \(9456\) photomultipliers of \(20\,\mathrm{cm}\) diameter mounted on a geodesic sphere of \(17.8\,\mathrm{m}\) diameter. The whole assembly sat in a cavity \(2092\,\mathrm{m}\) below the surface in the Creighton nickel mine near Sudbury, Ontario, an overburden of about \(6010\,\mathrm{m}\) of water equivalent — the greatest of any detector in this chapter, and necessary because a cosmic-ray muon passing through the heavy water would produce spallation neutrons indistinguishable from the neutral-current signal. At that depth the muon rate through the detector is about three per hour [Ahmad:2001] [Ahmad:2002].

Three reactions were used.

The charged-current reaction has a threshold of \(1.442\,\mathrm{MeV}\) (\(2.31\times 10^{-13}\,\mathrm{J}\)) and, because only the electron flavour has a charged-current partner light enough to be produced at solar energies, counts \(\nu_{e}\) alone. The neutral-current reaction is the breakup of the deuteron, so its threshold is the deuteron binding energy, \(2.224\,\mathrm{MeV}\) or \(3.56\times 10^{-13}\,\mathrm{J}\); the \(Z\) couples identically to all three flavours, so its rate is proportional to the total active flux with no flavour weighting whatever. Elastic scattering is sensitive to all three but not equally: \(\nu_{e}\) can scatter through both the charged and the neutral current while \(\nu_{\mu}\) and \(\nu_{\tau}\) have only the neutral one, so the effective flux it measures is

The weight is a ratio of cross-sections and is worth deriving, since it is used quantitatively twice below.

The experimental difficulty is entirely in the neutral-current channel, whose only observable is a free neutron. Three successive configurations were run, deliberately chosen so that the systematic errors of the neutron measurement would be unrelated.

Phase I, pure heavy water. The neutron captures on a deuteron, \(n+d\to{}^{3}\mathrm{H}+\gamma\), emitting a single photon of \(6.25\,\mathrm{MeV}\) (\(1.00\times 10^{-12}\,\mathrm{J}\)) which Compton-scatters an electron above the Cherenkov threshold. The capture cross-section of deuterium is small, so the overall neutron detection efficiency was only about \(14\,\mathrm{\%}\), and the resulting Cherenkov light is similar in amount and in isotropy to a charged-current event: the separation between the two channels rests on a statistical fit to the distributions of energy, radius and direction.

Phase II, salt. Two tonnes of sodium chloride were dissolved in the heavy water. Neutrons then capture predominantly on \(^{35}\mathrm{Cl}\), which has a far larger capture cross-section and de-excites through a cascade of photons totalling \(8.6\,\mathrm{MeV}\) (\(1.38\times 10^{-12}\,\mathrm{J}\)). Both the efficiency and the discrimination improve: efficiency rises to about \(40\,\mathrm{\%}\), and a multi-photon cascade produces a markedly more isotropic pattern of hit tubes than the single electron of a charged-current event, so the two channels separate on event topology rather than on energy alone.

Phase III, discrete counters. The salt was removed and an array of \(^{3}\mathrm{He}\) proportional counters — some forty vertical strings, about \(400\,\mathrm{m}\) of counter in total — was deployed in the heavy water. A neutron captured on \(^{3}\mathrm{He}\) gives \(n+{}^{3}\mathrm{He}\to p+{}^{3}\mathrm{H}\) with \(764\,\mathrm{keV}\) released, detected as an ionization pulse in the counter and not in the photomultipliers at all. The neutral-current measurement is then completely decoupled from the Cherenkov measurement of the charged current, so the statistical anticorrelation between the two channels that limited the first two phases disappears.

Liquid scintillator detectors

An organic liquid scintillator emits some ten thousand photons per megaelectronvolt of deposited energy, against the few hundred of a Cherenkov radiator, and it emits them isotropically and with no velocity threshold. The trade is exact: the scintillator sees far lower energies, and it loses all directional information. For a reactor experiment, where the source direction is known and the energies are a few megaelectronvolts, that is the right trade.

The reaction is the one Reines and Cowan used, inverse beta decay Equation (101.3) on the free protons of the hydrocarbon, with the threshold of Equation (101.4), \(1.806\,\mathrm{MeV}\). The signature is the delayed coincidence described in Section 101.1.3: a prompt pulse from the positron carrying \(E_{\bar{\nu}}-0.78\,\mathrm{MeV}\) plus its two annihilation photons, followed by a delayed pulse when the neutron, having thermalized, is captured. That the reaction is charged-current on a proton means it detects \(\bar{\nu}_{e}\) and nothing else, which is what makes a reactor experiment a pure disappearance measurement.

KamLAND placed \(1\,\mathrm{kt}\) of liquid scintillator — about \(80\,\mathrm{\%}\) dodecane and \(20\,\mathrm{\%}\) pseudocumene by volume, with \(1.5\,\mathrm{g}/\mathrm{L}\) of the fluor PPO — inside a balloon of nylon and ethylene-vinyl-alcohol copolymer \(13\,\mathrm{m}\) in diameter, itself suspended in \(1800\,\mathrm{t}\) of non-scintillating mineral oil inside a steel sphere \(18\,\mathrm{m}\) in diameter carrying \(1879\) photomultipliers, for a photocathode coverage of about \(34\,\mathrm{\%}\). A water Cherenkov outer detector of \(3.2\,\mathrm{kt}\) vetoes cosmic-ray muons. It occupies the cavern that held Kamiokande, under the same \(2700\,\mathrm{m}\) of water equivalent [Eguchi:2003]. Neutrons capture on the free protons of the scintillator itself, giving a \(2.22\,\mathrm{MeV}\) photon (\(3.56\times 10^{-13}\,\mathrm{J}\)) after a mean capture time near \(200\,\mu\mathrm{s}\).

What makes KamLAND a solar-parameter experiment rather than a reactor experiment is its position. Japan's fifty-odd power reactors, some \(70\,\mathrm{GW}\) of thermal power in total, are distributed at distances from \(140\,\mathrm{km}\) to several hundred kilometres, with a flux-weighted mean baseline near \(180\,\mathrm{km}\). A thermal power \(P\) produces antineutrinos at a rate of about

since fission releases about \(200\,\mathrm{MeV}\) and the resulting fragments beta-decay about six times each on the way to stability, so the Japanese reactor complex is a source of order \(1.3\times 10^{22}\,/\mathrm{s}\). At \(180\,\mathrm{km}\) that is a flux of order \(10^{10}\,/\mathrm{m}^{2}/\mathrm{s}\) at the detector — and, above the prompt-energy threshold actually analysed, about three hundred detected events per year per kilotonne of scintillator. Only part of the kilotonne is usable: the analysis is restricted to a fiducial volume of \(408\,\mathrm{t}\), well inside the balloon, so that the radioactivity of the balloon film and of the surrounding mineral oil cannot reach it. That is why the first result rested on \(54\) events.

Daya Bay and RENO use the same reaction at a hundredth of the baseline and with a completely different logic. Daya Bay observes six reactors of \(2.9\,\mathrm{GW}\) thermal each, \(17.4\,\mathrm{GW}\) in total, from three underground halls: two near halls at flux-weighted baselines of about \(470\,\mathrm{m}\) and \(576\,\mathrm{m}\) under \(250\,\mathrm{m}\) and \(265\,\mathrm{m}\) of water equivalent, and a far hall at about \(1648\,\mathrm{m}\) under \(860\,\mathrm{m}\) [An:2012]. Each hall holds one or more functionally identical antineutrino detector modules. A module is a nest of three volumes: an inner acrylic vessel \(3\,\mathrm{m}\) in diameter holding \(20\,\mathrm{t}\) of liquid scintillator doped with \(0.1\,\mathrm{\%}\) gadolinium by mass; an outer acrylic vessel holding a further \(20\,\mathrm{t}\) of undoped scintillator, which catches photons escaping from the inner volume; and \(40\,\mathrm{t}\) of mineral oil, optically transparent and non-scintillating, separating both from the \(192\) photomultipliers of \(20\,\mathrm{cm}\) diameter that view the assembly. The whole module is immersed in a pool of purified water \(2.5\,\mathrm{m}\) deep instrumented as a Cherenkov muon veto, itself covered by resistive plate chambers.

Gadolinium is the essential ingredient. Its thermal-neutron capture cross-section is some five orders of magnitude larger than hydrogen's, so at \(0.1\,\mathrm{\%}\) doping the great majority of neutrons capture on gadolinium rather than on protons; the capture releases a gamma cascade totalling about \(8\,\mathrm{MeV}\) (\(1.28\times 10^{-12}\,\mathrm{J}\)) after a mean time near \(28\,\mu\mathrm{s}\). A delayed pulse at \(8\,\mathrm{MeV}\) is far above natural radioactivity, whereas the \(2.22\,\mathrm{MeV}\) of capture on hydrogen sits in the middle of it; and the shorter capture time narrows the coincidence window, cutting accidental backgrounds by a further order of magnitude. RENO is built on the same plan with two identical detectors of \(16\,\mathrm{t}\) of gadolinium-loaded scintillator at about \(294\,\mathrm{m}\) and \(1383\,\mathrm{m}\) from the centre of a six-reactor array of \(16.5\,\mathrm{GW}\) thermal [Ahn:2012].

Borexino is the extreme case of the same technology, optimized not for reactor antineutrinos but for the lowest-energy solar neutrinos. It holds \(278\,\mathrm{t}\) of pseudocumene with \(1.5\,\mathrm{g}/\mathrm{L}\) of PPO in a nylon vessel of \(8.5\,\mathrm{m}\) diameter, inside a stainless steel sphere of \(13.7\,\mathrm{m}\) carrying \(2212\) photomultipliers, inside \(2.1\,\mathrm{kt}\) of water serving as shield and muon veto, at Gran Sasso under about \(3800\,\mathrm{m}\) of water equivalent [Bellini:2014]. It detects elastic scattering Equation (114.12) on electrons, which has no threshold, so the limit is set entirely by radiopurity. The achieved contamination is the lowest ever reached in a bulk material: below \(10^{-19}\,\mathrm{g}/\mathrm{g}\) for \(^{238}\mathrm{U}\) and below \(6\times 10^{-19}\,\mathrm{g}/\mathrm{g}\) for \(^{232}\mathrm{Th}\), with an isotopic ratio \(^{14}\mathrm{C}/^{12}\mathrm{C}\) near \(3\times 10^{-18}\). The last of these is irreducible — \(^{14}\mathrm{C}\) is chemically identical to \(^{12}\mathrm{C}\) and cannot be purified away — and its beta spectrum, with endpoint \(156\,\mathrm{keV}\), is what sets the lower edge of the \(pp\) measurement.

Accelerator beams

An accelerator gives what no natural source can: a beam whose flavour, energy spectrum, direction, time structure and intensity are all under the experimenter's control, so that the same detector can be compared with itself with the beam on and off, and a near detector can measure the unoscillated flux directly.

The production chain is the same one that makes atmospheric neutrinos (Section 115.7.2), run deliberately. Protons of tens of gigaelectronvolts strike a thin target of graphite or beryllium; the secondary pions and kaons are focused by magnetic horns — pulsed conductors carrying currents of order \(200\,\mathrm{kA}\) whose toroidal field sign-selects and collimates one charge — and are allowed to decay in an evacuated or helium-filled tunnel a few hundred metres long. Charged pion decay is two-body, \(\pi^{+}\to\mu^{+}\nu_{\mu}\), so the beam is almost pure \(\nu_{\mu}\); the residual \(\nu_{e}\) contamination, from three-body kaon decay and from muon decay in the tunnel, is at the level of \(1\,\mathrm{\%}\) and is the irreducible background of every appearance measurement. A beam dump and hundreds of metres of rock absorb the surviving hadrons and muons.

K2K sent a beam from the \(12\,\mathrm{GeV}\) proton synchrotron at KEK, with a mean neutrino energy of \(1.3\,\mathrm{GeV}\), over \(250\,\mathrm{km}\) to Super-Kamiokande, using the same detector that had made the atmospheric measurement [Ahn:2006].

MINOS used the NuMI beam from Fermilab's \(120\,\mathrm{GeV}\) main injector over \(735\,\mathrm{km}\) to a detector in the Soudan mine: \(5.4\,\mathrm{kt}\) of alternating steel and plastic scintillator planes, magnetized to about \(1.3\,\mathrm{T}\) so that the muon charge, and hence whether the parent was a neutrino or an antineutrino, can be read from the track curvature. A functionally similar near detector of \(1\,\mathrm{kt}\) sat \(1\,\mathrm{km}\) from the target [Michael:2006].

T2K and NOvA introduced the off-axis technique, which is the main advance in beam design of the last two decades. Because pion decay is two-body, the neutrino energy in the pion rest frame is fixed by the two masses alone,

and the Lorentz transformation to the laboratory at a fixed angle \(\theta\) to the pion direction gives

where the second form follows from \(\gamma_{\pi}=E_{\pi}/m_{\pi}c^{2}\) and \(\gamma_{\pi}\beta_{\pi}=p_{\pi}c/m_{\pi}c^{2}\), so that \(\gamma_{\pi}\left(1-\beta_{\pi}\cos\theta\right) =\left(E_{\pi}-p_{\pi}c\cos\theta\right)/m_{\pi}c^{2}\). On axis (\(\theta=0\)) and for \(E_{\pi}\gg m_{\pi}c^{2}\) the denominator is \(m_{\pi}^{2}c^{4}/E_{\pi}\) and the result is \(E_{\nu}\approx0.427\,E_{\pi}\): the neutrino spectrum simply copies the pion spectrum, which is broad. Off axis it does not. At fixed \(\theta\) the right-hand side rises with \(E_{\pi}\), turns over at \(E_{\pi}\approx m_{\pi}c^{2}/\theta\) and falls again, so across a wide band of parent energies \(E_{\nu}\) is almost independent of \(E_{\pi}\). A detector placed slightly off the beam axis therefore sees a narrow band of energies rather than the broad spectrum on axis, which is exactly what an oscillation measurement wants: all the flux is concentrated where \(\varphi\) is near \(\pi/2\), and the high-energy tail that would otherwise feed the \(\nu_{e}\) background is suppressed. T2K sends a beam from the \(30\,\mathrm{GeV}\) J-PARC main ring \(295\,\mathrm{km}\) to Super-Kamiokande at \(2.5\,^\circ\) (\(43.6\,\mathrm{mrad}\)) off axis, peaking at \(0.6\,\mathrm{GeV}\) [Abe:2011]; Equation (114.21) at that angle gives \(0.61\,\mathrm{GeV}\) at \(E_{\pi}=2\,\mathrm{GeV}\), \(0.68\,\mathrm{GeV}\) at the stationary point \(E_{\pi}=3.2\,\mathrm{GeV}\) and \(0.62\,\mathrm{GeV}\) at \(E_{\pi}=5\,\mathrm{GeV}\). NOvA sends the NuMI beam \(810\,\mathrm{km}\) to a \(14\,\mathrm{kt}\) segmented liquid scintillator detector at Ash River, \(14.6\,\mathrm{mrad}\) off axis, peaking near \(2\,\mathrm{GeV}\) [Acero:2022]; there the same expression gives \(1.84\,\mathrm{GeV}\), \(2.04\,\mathrm{GeV}\) and \(1.85\,\mathrm{GeV}\) at \(E_{\pi}=6\,\mathrm{GeV}\), \(9.6\,\mathrm{GeV}\) and \(15\,\mathrm{GeV}\).

OPERA solved the hardest problem of the field: detecting the appearance of \(\nu_{\tau}\), whose charged-current interaction produces a tau that decays after travelling of order a millimetre. Nothing with electronic readout resolves that. OPERA used \(1.25\,\mathrm{kt}\) of target in the form of about \(150000\) bricks, each a stack of fifty-seven lead plates \(1\,\mathrm{mm}\) thick interleaved with nuclear emulsion films — lead to supply the mass, emulsion to supply a position resolution of under a micrometre. Electronic trackers identified which brick had been hit; the brick was then extracted, developed and scanned automatically. The CNGS beam from CERN, of mean energy \(17\,\mathrm{GeV}\), travelled \(730\,\mathrm{km}\) through the Earth to Gran Sasso [Agafonova:2015].

What all of them have in common

Three requirements recur in every one of these instruments, and it is worth stating them once.

Depth. The rate of cosmic-ray muons falls roughly as the inverse cube of the water-equivalent overburden over the range covered by Table 114.3. The depth an experiment needs is set by its signal rate and by whether muons produce a background that mimics its signal. Homestake, with a signal of \(0.5\) atoms a day and a muon-induced spallation background in the same channel, needed \(4200\,\mathrm{m}\) of water equivalent; SNO, whose neutral-current signature is a free neutron and for which every muon-induced spallation neutron is an irreducible fake, needed \(6010\,\mathrm{m}\); Daya Bay, whose delayed-coincidence tag rejects almost everything and whose signal rate is hundreds of events a day, manages with \(250\,\mathrm{m}\).

Radiopurity. Every material within sight of the target contributes. The dominant chains are \(^{238}\mathrm{U}\) and \(^{232}\mathrm{Th}\), whose daughters emit betas and gammas up to a few megaelectronvolts, and \(^{40}\mathrm{K}\). Above about \(5\,\mathrm{MeV}\) natural radioactivity essentially stops, which is why every Cherenkov measurement in this chapter has an analysis threshold in the range \(5\text{–}9\,\mathrm{MeV}\) and why reaching below that — Borexino's achievement — requires purity at the level of \(10^{-19}\,\mathrm{g}/\mathrm{g}\).

Calibration with a known source. A rate measurement is only as good as the efficiency it is divided by. The instruments in this chapter are calibrated with deployed radioactive sources of known strength: gamma sources of a few megaelectronvolts to fix the energy scale, neutron sources such as \(^{252}\mathrm{Cf}\) or \(\mathrm{Am}\)–\(\mathrm{Be}\) to fix the neutron detection efficiency that the SNO and reactor measurements rest on, and, in the gallium case, an artificial neutrino source that calibrates the whole apparatus end to end.

Procedure

The radiochemical run cycle

A radiochemical run is a single measurement lasting weeks, and it proceeds in five steps that have not changed since 1968.

Reset. The tank is purged, removing all product atoms accumulated so far, and a measured quantity of a stable isotopic carrier is added: about \(0.1\,\mathrm{cm}^{3}\) at standard temperature and pressure of \(^{36}\mathrm{Ar}\) or \(^{38}\mathrm{Ar}\) for Homestake, a comparable trace of stable germanium for the gallium detectors. The carrier is the key to the whole procedure: it is chemically identical to the product, it is present in a quantity that can be measured by ordinary means, and whatever fraction of it is recovered at the end is the fraction of the signal atoms recovered too. Chemical efficiency is thereby measured in every run rather than estimated once.

Exposure. The tank stands undisturbed for a period comparable with the product half-life — some \(3\times 10^{6}\text{–}9\times 10^{6}\,\mathrm{s}\) at Homestake, that is between one and three months, and about \(1.8\times 10^{6}\,\mathrm{s}\) for the gallium runs. The number of product atoms present at time \(t\) after a reset obeys

with \(R\) the production rate and \(\lambda=\ln2/t_{1/2}\) the decay constant. Exposing for one half-life recovers half the saturated population; exposing for three recovers seven eighths. There is no gain beyond about two half-lives, which is why the exposure time is what it is.

Extraction. A large volume of carrier gas is circulated through the target. At Homestake, several hundred thousand litres of helium were bubbled through the \(615\,\mathrm{t}\) of perchloroethylene, sweeping the dissolved argon — a noble gas, chemically bound to nothing — into a charcoal trap at liquid-nitrogen temperature. At GALLEX, nitrogen was swept through the acid solution, carrying volatile \(\mathrm{GeCl}_{4}\) with it. The extraction is repeated and the recovered carrier measured, giving efficiencies of about \(95\,\mathrm{\%}\).

Counting. The purified product is loaded into a proportional counter of about \(0.5\,\mathrm{cm}^{3}\) together with a counting gas, and the counter is observed for several months — long enough that the decay curve itself can be fitted. Each pulse is recorded with its energy and its rise time.

Analysis. A candidate event must satisfy three independent conditions: its energy must match the K- or L-capture peak, its rise time must be that of a localized Auger cascade rather than of an extended ionizing track, and — the decisive one — the times of the run's candidate events must be distributed as an exponential of the known half-life on a flat background. A maximum-likelihood fit to the time distribution separates the product decays from the constant counter background, and it is this fit, not a raw count, that yields the number quoted. In practice, of order twenty candidate pulses per run.

The result of one run is a production rate in atoms per day, converted to a capture rate per target atom by dividing by the target count of Equation (114.8) or Equation (114.10). The published number is the average over more than a hundred such runs.

Reconstructing a Cherenkov event

A water Cherenkov detector records, for every photomultiplier, the arrival time of the first photon and the integrated charge. Everything else is inference, and the inference proceeds in a fixed order.

Vertex. Cherenkov light from a point of origin arrives at the tubes at times \(t_{i}=t_{0}+n\abs{\vect{x}_{i}-\vect{x}_{0}}/c\), with \(n\) the refractive index. Minimizing the spread of the implied \(t_{0}\) over all hit tubes locates the vertex. With a timing resolution of a few nanoseconds per tube and several thousand tubes hit, the vertex resolution is a few tens of centimetres. This is what defines the fiducial volume: events reconstructed within about \(2\,\mathrm{m}\) of the wall are discarded, because the wall and the tubes themselves are the dominant source of radioactivity.

Direction and ring. With the vertex fixed, the pattern of hit tubes is fitted to a cone of half-angle \(\theta_{c}\) from Equation (114.11). The cone axis is the particle direction; for a multi-\(\mathrm{GeV}\) muon the angular resolution is a degree or two.

Flavour. The fitted ring is tested for sharpness. The figure of merit is essentially the goodness of fit of a sharp-edged cone against that of a diffuse disc, and the classification is calibrated on beams of electrons and muons of known momentum injected into the detector, and cross-checked on the sample of cosmic-ray muons that stop in the detector and decay, which supplies a muon and, microseconds later, an electron from the same vertex.

Energy. The total collected charge, corrected for water attenuation, tube-by-tube gain and geometrical acceptance, is proportional to the track length and hence to the energy. The proportionality constant is fixed by deployed sources and by the end-point of the electron spectrum from stopping-muon decay, which is a known \(52.8\,\mathrm{MeV}\).

Zenith angle to baseline. The step that makes the atmospheric measurement possible is the last one. The reconstructed direction of the outgoing lepton approximates the direction of the incoming neutrino, to within the scattering angle of the interaction — of order \(15^\circ\) to \(20^\circ\) at \(1\,\mathrm{GeV}\) and better at higher energy. The zenith angle \(\theta_{z}\) of the neutrino then fixes the distance it travelled from its production point at an altitude \(h\approx15\,\mathrm{km}\),

which runs from \(15\,\mathrm{km}\) directly overhead (\(\cos\theta_{z}=1\)) through \(437\,\mathrm{km}\) at the horizon (\(\theta_{z}=90^\circ\), where the expression reduces to \(\sqrt{2R_{\oplus}h+h^{2}}\)) to \(12800\,\mathrm{km}\) directly below. A single detector therefore scans three orders of magnitude in \(L\) at fixed \(E\), with the same target, the same efficiency and the same analysis — the cleanest \(L/E\) scan available anywhere in physics, and it costs nothing but the angular resolution.

Separating the three Sudbury channels

SNO's three reactions all end as Cherenkov light in the same detector — Equations (114.13), (114.14) and (114.15) — and separating them is the entire experimental problem. The separation is statistical, not event by event, and rests on three distributions.

Energy. The charged-current electron carries essentially the full neutrino energy less \(1.442\,\mathrm{MeV}\), so its spectrum follows the \(^{8}\mathrm{B}\) spectrum. The neutral-current signal is monoenergetic in the sense that every neutron capture gives the same photon energy — \(6.25\,\mathrm{MeV}\) on deuterium, \(8.6\,\mathrm{MeV}\) on chlorine — so it appears as a bump.

Radial distribution. Charged-current and elastic events occur uniformly in the heavy water, so their density in the cube of the radius is flat out to the acrylic vessel. Neutrons diffuse before capturing, so the neutral-current events are depleted near the vessel wall.

Direction. Elastic scattering is strongly forward-peaked, so its events cluster at \(\cos\theta_{\odot}\to1\) with respect to the direction from the Sun. Charged-current events on deuterium have a mild backward asymmetry, roughly \(1-\tfrac{1}{3}\cos\theta_{\odot}\). Neutron captures are isotropic, having lost all memory of the incident direction during thermalization.

An extended maximum-likelihood fit to the joint distribution in energy, radius and direction returns the three normalizations at once. The three are anticorrelated — the fit can trade charged-current events against neutral-current events to some extent — and it is that anticorrelation, not the statistics, which dominated the uncertainty in the first phase. Adding salt sharpens the separation by adding a fourth variable, the isotropy of the hit pattern, which distinguishes a multi-photon cascade from a single electron; and the \(^{3}\mathrm{He}\) counters of the third phase remove the correlation entirely by moving the neutral-current measurement into a separate instrument.

The neutron detection efficiency, which multiplies the whole neutral-current result, is measured by deploying a calibrated \(^{252}\mathrm{Cf}\) neutron source at many positions in the vessel and mapping the response.

The delayed coincidence and the reactor flux model

A reactor experiment selects an antineutrino by requiring two pulses in sequence.

The prompt pulse is the positron: its kinetic energy plus the two \(511\,\mathrm{keV}\) annihilation photons, so that the visible energy is \(E_{\bar{\nu}}-0.78\,\mathrm{MeV}\). The prompt spectrum is therefore a shifted copy of the antineutrino spectrum, which is what makes a reactor experiment a spectral measurement and not merely a counting one.

The delayed pulse is the neutron capture: \(2.22\,\mathrm{MeV}\) on hydrogen after about \(200\,\mu\mathrm{s}\), or about \(8\,\mathrm{MeV}\) on gadolinium after about \(28\,\mu\mathrm{s}\).

The coincidence conditions are a time window a few capture times wide, a spatial separation below a metre or so, and energy windows on both pulses. Their product rejects everything uncorrelated. The residual backgrounds are of three kinds, and all three are measured rather than modelled: accidental coincidences of two uncorrelated radioactive decays, whose rate is obtained by opening an off-time window; correlated events from fast neutrons produced by cosmic-ray muons, which give a proton recoil followed by the same capture and are measured from the muon-tagged sample; and the \(\left(\alpha,n\right)\) reactions of \(^{210}\mathrm{Po}\) on \(^{13}\mathrm{C}\) in the scintillator, which are estimated from the measured polonium activity.

The flux prediction is the one place where a reactor experiment depends on something it does not measure. The antineutrino emission of a reactor core is computed from four ingredients: the thermal power, known to a few tenths of a per cent from the coolant flow and temperature rise; the fission fractions of \(^{235}\mathrm{U}\), \(^{238}\mathrm{U}\), \(^{239}\mathrm{Pu}\) and \(^{241}\mathrm{Pu}\), which drift over a fuel cycle and are tracked by the reactor operator's own core simulation; the energy released per fission of each isotope; and the antineutrino spectrum per fission of each. The last is the weak link, carrying an uncertainty of a few per cent in normalization and having twice been revised. For KamLAND, where the observable is the survival probability itself, that uncertainty enters the result directly and is the second-largest systematic. For Daya Bay it very nearly cancels, and the next subsection is why.

The near-to-far ratio

The design of Daya Bay and RENO is the reason \(\theta_{13}\) could be measured at the per-cent level within months of turning on, and the argument is short enough to give in full.

The proposition converts a hard absolute measurement into an easy relative one, and it explains every feature of the apparatus. The detector modules are built to the same drawings, filled from the same batch of scintillator so that the proton density is common, and their masses measured to better than a part in a thousand; at Daya Bay two modules were placed in the same near hall specifically so that their measured rates could be compared, giving a direct experimental bound of a few tenths of a per cent on the residual difference in efficiency. The far hall is placed where Equation (114.4) makes the phase near maximal, the near halls where it is small, so that the numerator of Equation (114.25) is suppressed and the denominator is not.

The one thing the ratio does not cancel is the relative contribution of the six reactors to each hall, since the halls are at different distances from each core; that is why the baselines quoted in Section 114.2.6 are flux-weighted, and why the reactors' individual thermal powers must still be tracked.

Calibration, blindness and the treatment of systematics

Two procedural points deserve recording because they are what makes the numbers in Section 114.4 usable.

Calibration is end-to-end where possible. The gallium experiments are the only ones that could be calibrated with a neutrino source of known activity, and they were: a \(^{51}\mathrm{Cr}\) source of \(6.3\times 10^{16}\,\mathrm{Bq}\) lowered into the target tests the capture cross-section, the extraction chemistry, the counter efficiency and the analysis in a single measurement. Every other instrument is calibrated in pieces — an optical calibration for the light transport, a gamma source for the energy scale, a neutron source for the capture efficiency — and the composition of those pieces is itself a systematic.

The analyses were blind. SNO, KamLAND, Daya Bay and T2K each fixed their selection cuts, their efficiency corrections and their background estimates on a subset of the data, or on data with an unknown offset applied, before looking at the number the experiment was built to measure. This matters more here than in most fields because the expected answer was known in advance: an experiment looking for a factor of three that everyone expects to find will find it if the cuts are chosen after the fact. The published SNO result was unblinded once, and the value quoted is the value that appeared.

Observations and data

The solar neutrino deficit

Table 114.4 gives the radiochemical results. Every entry is a rate per target atom, in SI, together with the prediction of the standard solar model of Table 114.2 and the ratio of the two.

The Homestake number, \(2.56(23)\times 10^{-36}\) captures per target atom per second [Cleveland:1998], is the average over one hundred and eight extractions between 1970 and 1994. Multiplied by the \(2.16\times 10^{30}\) target atoms of Equation (114.8) it is a production rate of

against a cosmic-ray-induced background in the same channel of \(0.047(13)\) atoms per day and a predicted solar rate near \(1.5\) atoms per day. Twenty-five years of running, in a target of six hundred tonnes, to establish a shortfall of one atom of argon per day.

The gallium results [Hampel:1999] [Abdurashitov:2009] are the ones that closed the astrophysical escape route, and the reason is arithmetic rather than statistical. Their threshold of \(233\,\mathrm{keV}\) admits the \(pp\) neutrinos, whose flux is fixed by Equation (114.6) to about one per cent from the observed solar luminosity alone. Of the predicted \(1.26\times 10^{-34}\) captures per second per atom, some \(55\,\mathrm{\%}\) comes from the \(pp\) branch; even setting the entire \(^{8}\mathrm{B}\) and \(^{7}\mathrm{Be}\) contribution to zero and moving the whole luminosity-constrained flux into \(pp\) — which no solar model allows, but which brackets the astrophysical uncertainty — leaves a predicted rate of \(7.4\times 10^{-35}\) per second per atom, against a measured \(6.5\times 10^{-35}\) to \(6.9\times 10^{-35}\). The deficit is reduced by that extreme assumption but not removed, and the assumption itself is contradicted by the direct measurements of the two branches it sets to zero.

The directional and real-time solar measurements

Kamiokande-II was the first detector to see solar neutrinos in real time and the first to see that they come from the Sun. Its signal is the elastic scattering Equation (114.12), whose recoil electron retains the neutrino direction well enough that the distribution of events in \(\cos\theta_{\odot}\), the cosine of the angle between the reconstructed electron direction and the direction away from the Sun, shows a clear forward peak on a flat background [Hirata:1989]. The measured \(^{8}\mathrm{B}\) flux was

for an analysis threshold of \(7.5\,\mathrm{MeV}\) in recoil electron energy. Against the \(^{8}\mathrm{B}\) entry of Table 114.2, \(5.69(91)\times 10^{10}\,/\mathrm{m}^{2}/\mathrm{s}\), that is a ratio of \(0.48(12)\). The ratio quoted in the source itself is \(0.46(15)\) [Hirata:1989], taken against the solar model current in 1989; the two are not the same comparison, and this chapter keeps them apart for the reason given in Section 114.8.

Two things follow that the radiochemical experiments could not establish. The neutrinos are solar, not terrestrial or cosmic — the angular distribution says so directly. And the deficit is about a factor of two rather than the factor of three seen in chlorine, at the same \(^{8}\mathrm{B}\) branch. That difference is not an inconsistency: elastic scattering retains sensitivity to the other flavours through the neutral current, with the weight of Equation (114.16), so a detector that measures \(\Phi_{e}+0.154\,\Phi_{\mu\tau}\) must report a smaller deficit than one measuring \(\Phi_{e}\) alone — if and only if \(\Phi_{\mu\tau}\) is not zero. The comparison of the two numbers was in fact the first quantitative hint of flavour change, well before SNO measured it in a single detector; but it required comparing two experiments with different targets, thresholds and systematics, and no one was prepared to draw the conclusion from it.

Atmospheric zenith-angle dependence

Super-Kamiokande's atmospheric measurement is the first in this chapter that needs no external prediction at all.

Two observables carry it. The first is the double ratio

in which the absolute atmospheric flux — uncertain by some \(20\,\mathrm{\%}\) — cancels, leaving only the flavour ratio, which by Equation (115.61) is fixed at two by the decay chain \(\pi\to\mu\nu_{\mu}\), \(\mu\to e\nu_{e}\bar{\nu}_{\mu}\) and is reliable to a few per cent. The second is the up-down asymmetry

with \(U\) and \(D\) the numbers of events with \(-1<\cos\theta_{z}<-0.2\) and \(0.2<\cos\theta_{z}<1\) respectively, in which the flux normalization, the cross-section and the detector efficiency all cancel, because the same detector counts both samples. The atmospheric flux is up-down symmetric above a few gigaelectronvolts to better than one per cent — the geomagnetic field distorts it at lower energies, which is why the asymmetry is quoted for the multi-\(\mathrm{GeV}\) sample.

Sudbury and the flavour-blind flux

The internal consistency of Table 114.6 is worth one more line, because it is a check the experiment supplies on itself. Given \(\phi_{e}\) and \(\phi_{\mu\tau}\) from the first and third rows, Equation (114.16) predicts an elastic-scattering flux of \(1.76+0.154\times3.41=2.29\), in units of \(10^{10}\,/\mathrm{m}^{2}/\mathrm{s}\), against the measured \(2.39(27)\). Three channels, two unknowns, one consistency condition satisfied.

KamLAND and reactor antineutrinos

The first KamLAND exposure was \(162\,\mathrm{t}\,\mathrm{yr}\) — \(145.1\) days of live time in a fiducial mass of \(408\,\mathrm{t}\) inside the kilotonne target, \(145.1\) days being \(0.397\) of a year and \(0.397\times408\,\mathrm{t}=162\,\mathrm{t}\,\mathrm{yr}\) — and it yielded \(54\) antineutrino candidates above a prompt-energy threshold of \(2.6\,\mathrm{MeV}\), against \(86.8(56)\) expected in the absence of oscillation from the reactor bookkeeping of Section 114.3.4, on an estimated background of \(1.0(10)\) events [Eguchi:2003]. The ratio is

the first uncertainty statistical, the second systematic. A ratio of one is excluded at better than \(99.95\,\mathrm{\%}\).

The third mixing angle

Accelerator beams: disappearance and appearance

The accelerator programme does three things the natural sources cannot: it confirms the atmospheric parameters with a source whose spectrum and composition are known independently, it observes the appearance of a flavour rather than the disappearance of one, and it provides the only access to the \(CP\) phase.

K2K's deficit corresponds to \(\abs{\Delta m^{2}_{32}}c^{4}\) near \(2.8\times 10^{-3}\,\mathrm{eV}^{2}\) at maximal mixing, and excludes the no-oscillation hypothesis at about four standard deviations [Ahn:2006]; MINOS, with ten times the statistics and a magnetized detector, gives \(\abs{\Delta m^{2}_{32}}c^{4}=2.74\times 10^{-3}\,\mathrm{eV}^{2}\) with an uncertainty of about \(15\,\mathrm{\%}\) and \(\sin^{2}2\theta_{23}>0.87\) [Michael:2006]. Both agree with the atmospheric determination from a completely different source, which is the point of doing them.

The appearance measurements are the harder and the more informative. T2K's six electron-neutrino candidates on an expected background of \(1.5(3)\) were a two-and-a-half standard deviation indication in 2011 [Abe:2011], published months before the reactor measurements of Table 114.7 and consistent with them: \(\nu_{\mu}\to\nu_{e}\) appearance requires \(\theta_{13}\neq0\) by the same three-flavour expression that gives Equation (114.40), so the two are measurements of the same parameter in different channels. OPERA's five tau candidates on a background of \(0.25\) close the three-flavour picture from the other side [Agafonova:2015]: the flavour that the atmospheric and accelerator disappearance measurements infer must be there, because the muon neutrinos go somewhere, is observed directly.

The global fit

All of the above, together with the solar and reactor spectra in full, is combined into a fit for six parameters — two independent squared-mass splittings, three mixing angles and one phase. Table 114.9 is the result. It repeats Table 103.3 deliberately, because an experiment chapter must state the numbers its measurements produce, and because the third column here is different: it records which measurement dominates each parameter.

Two properties of that table are physics rather than bookkeeping. The two splittings differ by a factor of about thirty-three, which is why the solar and atmospheric sectors decouple and why each could be measured before the other was understood. And two of the three angles are large — \(\sin^{2}2\theta_{23}=0.99\) and \(\sin^{2}2\theta_{12}=0.85\) — in flat contradiction to the near-diagonal quark mixing matrix of Section 103.4.2, whose largest off-diagonal magnitude is about \(0.22\). Two mixing matrices generated by the same mechanism in the same theory, one nearly the identity and one nearly democratic: that is a fact with no explanation. It belongs to the flavour puzzle of Section 133.1, which catalogues the Yukawa couplings the Standard Model accommodates without predicting, though the contrast between the two matrices is not itemized there and is recorded here.

Interpretation

What the data force

The interpretation of Section 114.4 is a chain of four steps, and it is worth setting it out as a chain, because each link excludes something the previous one did not.

Step one: the deficit is real. Three radiochemical detectors with different thresholds and different chemistry, one water Cherenkov detector with direction, and one heavy-water detector all find fewer solar electron neutrinos than the model predicts (Tables 114.4 and 114.6). The gallium threshold reaches a branch whose flux is fixed by Equation (114.6) independently of the model, so the deficit is not an artefact of the astrophysics.

Step two: the missing neutrinos are still there. Sudbury's neutral-current channel measures the flavour-summed flux and finds it in agreement with the model, while the charged-current channel finds a third of it (Equation (114.35)). The subtraction is model-free. Whatever happened to the electron neutrinos, they were not absorbed and they did not decay into something invisible.

Step three: the change depends on \(L/E\) and oscillates. Super-Kamiokande's zenith-angle distribution is a scan in \(L\) at fixed \(E\) over three orders of magnitude with one detector (Equation (114.23)), and the depletion follows \(\sin^{2}\) of Equation (114.3). KamLAND's prompt-energy spectrum is a scan in \(E\) at fixed \(L\), and the survival probability returns rather than falling monotonically (Equation (114.38)). Absorption, decay and decoherence are all monotone in \(L/E\); interference is not.

Step four: therefore the neutrino has mass. By Equation (114.2) the phase is proportional to \(\Delta m^{2}c^{4}\), so a non-zero oscillation amplitude requires \(m_{2}^{2}\neq m_{1}^{2}\) and hence at least one non-zero mass. The left-handed neutrino of The Weyl Equation and Neutrinos has none, and the Standard Model as originally written is therefore false. That is the only place where a laboratory measurement contradicts it.

Three technical points belong here rather than in the observational sections.

Plane waves. The derivation of Equation (114.2) gives the two mass components the same momentum and the same flight time, both of which are approximations. The conditions under which they are good — production and detection regions small compared with the oscillation length, and wave packets still overlapping on arrival — are stated and evaluated in Remark 103.51, and the tightest case in this chapter is KamLAND, examined in Remark 114.7.

Three flavours. The two-flavour formula is used throughout the observational sections because the two splittings differ by a factor of thirty-three and each experiment sits in a window where one dominates. The exact three-flavour expression is Theorem 103.54, and Equation (114.39) shows explicitly how it reduces at a short reactor baseline, with the neglected term quantified at \(1.6\times 10^{-3}\). Wherever the two-flavour statement is used here, the correction is of order \(\cos^{4}\theta_{13}=0.957\) or smaller.

What oscillation does not measure. The probabilities depend on \(\Delta m^{2}\), on the mixing angles and on one phase. They do not depend on the absolute masses, on whether the mass term is of Dirac or of Majorana type (Proposition 103.47), or on the mechanism that generates the masses. Everything in Table 114.9 is silent on all three.

Matter effects: the MSW mechanism

The solar measurements are not measurements of vacuum oscillation, and treating them as such gives the wrong answer by a factor of two. The theory is Theorem 103.57; what belongs here is the evidence for it.

The mechanism in one sentence: matter contains electrons and contains no muons or taus, so \(\nu_{e}\) alone can forward-scatter coherently through the charged current and acquires the extra potential energy Equation (103.57), \(V_{e}=\sqrt{2}G_{F}n_{e}\), which at the centre of the Sun is \(1.22\times 10^{-30}\,\mathrm{J}\), that is \(7.6\times 10^{-12}\,\mathrm{eV}\). Because the shift is flavour-dependent it changes the mixing angle, and by Equation (103.61) the effective mixing becomes maximal at the energy Equation (103.62), \(E_{\mathrm{res}}=1.9\,\mathrm{MeV}\). That single number splits the solar neutrino spectrum in two.

Table 114.10 is the test, and it is worth saying exactly what makes it one. A survival probability that is constant in energy is what vacuum oscillation over an averaged phase gives; the measured values are not constant, and they fall with energy in the direction and by the amount the matter effect predicts. Two energy decades, one curve, no adjustable parameter. The intermediate points — \(pep\) at \(1.44\,\mathrm{MeV}\), sitting almost exactly at the resonance — are where the transition happens, and they are also where the measurement is weakest, which is an honest statement about what has and has not been established: the two asymptotes are solid, the shape of the transition between them is not yet resolved.

Two further consequences of Theorem 103.57 are the subject of current work, and Remark 103.60 names them both.

Day–night asymmetry. Solar neutrinos arriving at night have crossed the Earth, whose electron density partially regenerates the \(\nu_{e}\) component, so the night-time rate should exceed the daytime rate by a few per cent with no free parameter beyond those already measured. The predicted asymmetry for the \(^{8}\mathrm{B}\) flux is a few per cent and the measurements sit at about the level of two standard deviations: an indication, not an observation, and this treatise records it as one. No source for the day–night measurement is cited in this treatise; the size of the effect is stated here without one, and nothing later in the chapter rests on it.

The sign of the splitting. This is the deepest consequence. Vacuum oscillation depends on \(\Delta m^{2}\) only through \(\sin^{2}\varphi\), which is even in \(\Delta m^{2}\) and therefore blind to its sign; the matter term is not, because \(V_{e}>0\) for matter and \(V_{e}<0\) for antimatter, so the resonance condition Equation (103.61) can be satisfied for neutrinos or for antineutrinos but not both. The solar data therefore determine the sign of \(\Delta m^{2}_{21}\) — \(\nu_{2}\) is heavier than \(\nu_{1}\) — which no vacuum measurement could. The same argument applied to \(\Delta m^{2}_{32}\) with the Earth's density in place of the Sun's is the route to the mass ordering, and it is why the long-baseline experiments of Table 114.8 matter beyond their statistics.

The PMNS matrix and the measured parameters

Collecting Table 114.9 into the mixing matrix Definition 103.46 gives magnitudes Equation (103.49): every entry of the first two rows is of order one half, and only \(\abs{U_{e3}}=\sin\theta_{13}=0.148\) is small. This is Phenomenon 103.48.

Three observations about the parameter set, each of which is a statement about the evidence rather than about the theory.

The measurements are over-determined and they agree. The solar splitting is fixed by KamLAND's terrestrial reactor spectrum and, independently, by the energy dependence of the solar survival probability in Table 114.10; the atmospheric splitting by Super-Kamiokande's zenith distribution and, independently, by MINOS and NOvA with a controlled beam; \(\theta_{13}\) by two reactor experiments at different sites and, independently, by T2K's appearance channel. In each case the sources are astrophysical against terrestrial, or disappearance against appearance, with unrelated systematics. That is the strongest form of confirmation an experimental result can have.

The parameter count is the honest statement of ignorance. Three neutrino masses, three mixing angles and one phase — seven new numbers, and nine if the neutrino is of Majorana type — are added to the Standard Model by the results of this chapter, and not one of them is predicted by anything. The free-parameter bookkeeping is in the appendix on the fundamental parameters of physics, which sets out how the count changes according to the mass mechanism assumed.

The contrast with quark mixing is unexplained. The quark mixing matrix of Section 103.4.2 is nearly the identity, its largest off-diagonal magnitude being \(0.22\) and its smallest \(0.0037\); the lepton mixing matrix is nearly democratic. Both arise, in the Standard Model extended by neutrino masses, as the same kind of misalignment between two bases. There is at present no evidence-based account of why the two should look so different, and it belongs among the mass and mixing puzzles of Section 133.1.

What oscillation cannot say

Modern repetitions and precision

Borexino and the solar fusion chains

Borexino did what no earlier solar detector could: it measured the individual branches of the pp chain in real time, one at a time, at energies below a megaelectronvolt.

The measurement is a spectral decomposition. Elastic scattering Equation (114.12) has no threshold, so a monoenergetic neutrino line produces a recoil-electron spectrum with a sharp Compton-like edge at \(T_{\max}=2E^{2}/\left(m_{e}c^{2}+2E\right)\) — \(665\,\mathrm{keV}\) for the \(862\,\mathrm{keV}\) \(^{7}\mathrm{Be}\) line, \(261\,\mathrm{keV}\) for the \(420\,\mathrm{keV}\) \(pp\) endpoint. The observed spectrum is fitted as a sum of those shapes plus the known beta and alpha spectra of the residual contaminants, and the branch fluxes come out of the fit.

Three results follow from Table 114.11.

The \(pp\) flux of \(6.6(7)\times 10^{14}\,/\mathrm{m}^{2}/\mathrm{s}\) [Bellini:2014] agrees with the luminosity constraint Equation (114.6) and with the model entry of Table 114.2. This is a direct check that the Sun is powered by the reaction Equation (114.5) at the rate its luminosity requires — and, because neutrinos leave the core immediately while photons take of order \(10^{5}\) years to diffuse out, that the solar output has been steady over that interval. This is the content of Phenomenon 52.23.

The energy dependence of the survival probability across two decades, Table 114.10, is the direct confirmation of the matter effect discussed in Section 114.5.2.

The CNO detection [Agostini:2020a] establishes that the cycle operates in the Sun at all, supplying about one per cent of its power. Its flux depends on the abundance of carbon, nitrogen and oxygen in the solar core, so it is in principle an independent handle on the solar metallicity, on which the spectroscopic determination and the helioseismic one — the latter belonging to Section 52.6 — have disagreed for two decades. At the present uncertainty of about \(40\,\mathrm{\%}\) the measurement does not yet discriminate between them, and saying so is part of reporting it.

Supernova and astrophysical neutrinos

SN 1987A. On 23 February 1987 the collapse of a blue supergiant in the Large Magellanic Cloud, at about \(5\times 10^{4}\,\mathrm{pc}\), produced the only neutrino burst ever detected from a stellar collapse. Kamiokande-II recorded eleven events within \(13\,\mathrm{s}\) with energies between about \(7\,\mathrm{MeV}\) and \(36\,\mathrm{MeV}\) [Hirata:1987], and the Irvine–Michigan–Brookhaven detector recorded eight in \(6\,\mathrm{s}\) [Bionta:1987]. Nineteen events, and they remain the entire observational database of core-collapse neutrino emission.

The bound they give on the neutrino mass is kinematic and follows from the arrival-time spread. A neutrino of mass \(m\) and energy \(E\) travels at a group velocity \(v/c=\left[1-m^{2}c^{4}/E^{2}\right]^{1/2} \approx1-m^{2}c^{4}/2E^{2}\), so two neutrinos emitted together at energies \(E_{1}\) and \(E_{2}\) arrive separated by

with \(D\) the distance. Requiring the observed spread to be no larger than the plausible emission duration bounds \(mc^{2}\) below \(17\,\mathrm{eV}\), the value obtained in Equation (115.67) — weak by comparison with the laboratory limit of Table 114.12, but obtained over a baseline \(10^{15}\) times longer than any terrestrial one, and hence a bound of a completely different kind. The burst itself is Phenomenon 115.48.

Neutrinos in ice. A cubic kilometre of Antarctic ice instrumented with photomultipliers detects the same atmospheric neutrinos as Super-Kamiokande, at energies from tens of gigaelectronvolts upwards, and its densely instrumented core has been used to measure the atmospheric oscillation parameters in that range [Aartsen:2018a] — an independent measurement in a medium, a geometry and an energy range unrelated to the water Cherenkov detectors of Section 114.2.4. The same instrument established a diffuse astrophysical neutrino flux of much higher energy [Aartsen:2013] and associated one event with a flaring blazar [Aartsen:2018]. Those results belong to Section 115.7.4 and are named here only because the astrophysical flux arrives with a flavour composition that is itself an oscillation measurement over cosmological baselines: whatever the production ratio, the phases average completely and the observed composition should be near one to one to one.

Absolute mass: kinematics, cosmology, double beta decay

Three experiments bound three different combinations of the three masses, defined in Definition 103.61. They must be reported separately, and Table 114.12 does so.

The kinematic measurement. KATRIN measures the shape of the tritium beta spectrum within a few electronvolts of its \(18.6\,\mathrm{keV}\) endpoint. A non-zero neutrino mass removes the last \(m_{\nu}c^{2}\) of the spectrum and distorts what remains, by Proposition 101.20; the observable is \(m_{\beta}^{2}\), an incoherent sum \(\sum_{i}\abs{U_{ei}}^{2}m_{i}^{2}\), and the measurement is of that squared quantity, so the fitted value may come out negative without anything being wrong. The current result is \(m_{\beta}^{2}c^{4}=0.26(34)\,\mathrm{eV}^{2}\), which is consistent with zero and yields the limit of Table 114.12 [Aker:2022]. This is Phenomenon 101.21 and Phenomenon 103.62.

The cosmological bound. Massive neutrinos are relativistic when they decouple and non-relativistic today; they free-stream out of small-scale density perturbations and suppress the growth of structure below a characteristic scale. Fitting the microwave background together with the distribution of galaxies bounds \(\sum_{i}m_{i}c^{2}<0.12\,\mathrm{eV}\) [Aghanim:2020]. It is the tightest bound available and the most model-dependent one: it assumes the standard cosmological model, three species with standard thermal history, and a particular treatment of the nonlinear scales. Compare it with the minimum from Equation (114.42), \(\sum_{i}m_{i}c^{2}\geq0.059\,\mathrm{eV}\) in the normal ordering, and the window is a factor of two. This is Phenomenon 103.68.

Neutrinoless double beta decay. If the neutrino is its own antiparticle then a nucleus may undergo \(2\beta\) decay emitting two electrons and no neutrinos, violating lepton number by two units. It has not been observed. KamLAND-Zen, using enriched \(^{136}\mathrm{Xe}\) dissolved in the KamLAND scintillator, bounds the half-life above \(2.3\times 10^{26}\,\mathrm{yr}\) and the effective Majorana mass in the range of Table 114.12 [Abe:2023]. The range, rather than a number, is the nuclear matrix element: converting a half-life limit into a mass limit requires a many-body calculation whose results differ by a factor of two to three between methods (Remark 103.65, and the nuclear structure of Nuclear Forces and Nuclear Structure). This is Phenomenon 103.64.

The three are logically independent, and a treatise of evidence must not collapse them. A positive double beta signal would establish lepton number violation and hence a Majorana mass term whatever the mechanism; a positive KATRIN signal would fix the scale model-independently; a cosmological detection would fix the sum within one cosmology. None has happened.

Anomalies and sterile searches

A treatise of evidence must state which of its anomalies have not gone away. Four are outstanding, and none of them is established.

LSND. A stopped-pion source at Los Alamos produced \(\bar{\nu}_{\mu}\) of a few tens of megaelectronvolts, and a liquid-scintillator detector \(30\,\mathrm{m}\) away found an excess of \(\bar{\nu}_{e}\)-like events: \(87.9(232)\) events above background, an apparent transition probability of \(2.64(81)\times 10^{-3}\), at about \(3.8\) standard deviations [Aguilar:2001]. At that \(L/E\) the required splitting would be \(\Delta m^{2}c^{4}\) of order \(1\,\mathrm{eV}^{2}\) — two orders of magnitude above \(\abs{\Delta m^{2}_{32}}c^{4}\), and therefore a fourth mass state. By Equation (114.1) a fourth state cannot couple to the \(Z\), so it would have to be sterile.

MiniBooNE. A different beam, a different baseline and a different detector technology found an excess of low-energy electron-like events consistent in size with LSND [AguilarArevalo:2018]. Whether the excess is electrons or photons — which a Cherenkov detector cannot always distinguish, since a converted photon makes an electromagnetic shower too — was the central question, and what follows is the measurement that tests it.

MicroBooNE. A liquid-argon time projection chamber of \(85\,\mathrm{t}\) in the same beam images the interaction vertex with millimetre resolution and separates a single electron from a converted photon by the ionization density at the start of the track. It finds no excess in the electron-neutrino channel across several final-state topologies [Abratenko:2022]. That does not by itself close the question — it constrains the electron interpretation, not every interpretation of the MiniBooNE excess — but it removes the simplest one.

The gallium anomaly. The \(^{51}\mathrm{Cr}\) and \(^{37}\mathrm{Ar}\) source calibrations of Section 114.3.6, which exist to verify the detectors, returned rates below expectation: the ratio of measured to predicted is about \(0.87(5)\) when the four exposures are combined [Hampel:1999] [Abdurashitov:2009]. A deficit at a baseline of a few metres would require a splitting \(\Delta m^{2}c^{4}\) of order \(1\,\mathrm{eV}^{2}\), which is the LSND scale; alternatively the predicted cross-section for Equation (114.9) into the excited states of \(^{71}\mathrm{Ge}\) is wrong. The discrepancy is small enough not to affect any conclusion of Section 114.4.1 — it would move the gallium ratio from \(0.55\) to about \(0.63\), still far below one — and it has not been resolved.

Reactor rate anomaly. Measured reactor antineutrino rates at short baselines sit a few per cent below the predicted flux. Part of this was resolved by recalculating the fission spectra, and part remains; a spectral bump near \(5\,\mathrm{MeV}\) in the measured prompt spectrum, seen by all the modern experiments and absent from every prediction, is a defect of the flux model and not of the neutrinos, since it appears identically at near and far detectors and therefore cancels in Equation (114.25). No source for the rate anomaly or for the spectral bump is cited in this treatise; both are stated here without one, and the near–far cancellation is the reason neither affects any oscillation result quoted above.

No sterile neutrino is established. The anomalies do not form a consistent picture with one another: the parameter regions preferred by LSND, by the reactor rates and by the gallium sources overlap only partially, and the disappearance searches that a sterile state would also require have found nothing. This chapter records them as open.

Open questions

The mass ordering

The sign of \(\Delta m^{2}_{32}\) is not measured. Normal ordering means \(m_{1}<m_{2}<m_{3}\), the close pair at the bottom; inverted means \(m_{3}<m_{1}<m_{2}\), the close pair at the top. Global fits prefer the normal ordering at a level between two and three standard deviations that depends on which datasets are combined [Esteban:2020] [Navas:2024], which is a preference and not a measurement.

Two routes will settle it, and they are independent.

The matter effect in a long-baseline beam. By the sign argument of Section 114.5.2, the Earth's matter enhances \(\nu_{\mu}\to\nu_{e}\) and suppresses \(\bar{\nu}_{\mu}\to\bar{\nu}_{e}\) if the ordering is normal, and does the reverse if it is inverted. The effect grows with baseline and with energy, which is why NOvA's \(810\,\mathrm{km}\) is more sensitive to it than T2K's \(295\,\mathrm{km}\), and why a longer baseline still would settle the question outright. The difficulty is that the same comparison is also the measurement of \(\delta_{CP}\), and the two enter with a partial degeneracy.

The vacuum interference pattern at a medium baseline. A reactor experiment at some \(50\,\mathrm{km}\), where the solar and atmospheric phases are comparable, sees the fast atmospheric oscillation modulating the slow solar one; the phase of the fast modulation relative to the slow envelope differs between the two orderings. This requires no matter effect and no beam, only an energy resolution of about three per cent — which is the hard part, since it is a factor of two better than any large scintillator has achieved.

The answer matters beyond bookkeeping. In the inverted ordering the effective Majorana mass of Table 114.12 has a lower bound of about \(15\,\mathrm{meV}/c^{2}\), within reach of the next generation of double beta decay experiments, so a null result there would exclude the inverted ordering for a Majorana neutrino; in the normal ordering the effective mass can vanish by cancellation and no null result excludes anything. The cosmological sum likewise has different minima in the two orderings (Equation (103.75)).

The $CP$ phase

The phase \(\delta_{CP}\) is the one parameter of Table 114.9 that is constrained rather than measured. It is observable only through a difference between \(\nu_{\mu}\to\nu_{e}\) and \(\bar{\nu}_{\mu}\to\bar{\nu}_{e}\), because by Theorem 103.54 the \(CP\)-odd term is proportional to the leptonic Jarlskog invariant

which vanishes if any angle vanishes. That is why Phenomenon 114.8 reorganized the field: before 2012 it was possible that \(\theta_{13}=0\) and that leptonic \(CP\) violation was therefore unobservable in principle.

T2K's data disfavour \(CP\) conservation at about the \(95\,\mathrm{\%}\) level and prefer a value near \(-\pi/2\) [Abe:2020]; NOvA's are compatible with \(CP\) conservation [Acero:2022]. Combining them gives a preference, not a measurement, and this treatise records it as such. Three degeneracies stand in the way and all three must be broken together: with the mass ordering, through the matter effect just described; with the octant of \(\theta_{23}\), since \(\abs{U_{\mu3}}^{2}=\sin^{2}\theta_{23}\) enters the appearance channel unsquared while disappearance sees only \(\sin^{2}2\theta_{23}\), which is symmetric about \(\pi/4\); and with the overall normalization of the beam, which is what the near detector is for.

Leptogenesis is often named as the motivation. It is worth stating plainly, as Remark 103.72 does, that a measured \(\delta_{CP}\) would not establish it: the phases entering the decays of the hypothetical heavy states are not the phases of Definition 103.46, and the connection is a model assumption and not a theorem.

Dirac or Majorana

Whether the neutrino is its own antiparticle is not answerable by any experiment in this chapter. Oscillation cannot answer it, by Proposition 103.47: the extra phases of a Majorana mass matrix cancel from every oscillation probability. The kinematic and cosmological observables cannot, being phase-independent. The only practical discriminator is a lepton-number-violating process, and the searches [Abe:2023] have found nothing — with the interpretation of any half-life limit further limited by the nuclear matrix elements of Remark 103.65 and Nuclear Forces and Nuclear Structure.

The neutrino sector is therefore the one place where the Standard Model is known to be incomplete from laboratory data alone. Every other experimental anomaly in this book is either a measurement that agrees with the theory to remarkable precision, or an astrophysical observation whose interpretation requires assumptions the laboratory cannot check. Here the laboratory itself says the theory is wrong, and says almost nothing about what replaces it: not the mass scale, not the ordering, not the mechanism, not whether lepton number is conserved. Of that list Section 133.1 records the absolute mass scale and the Dirac–Majorana question; the ordering and the mechanism are recorded here and nowhere else in this book.

Primary references

The literature of this chapter divides into five groups, and the division matters because the groups differ in what they can be relied on for.

The theoretical origin. Pontecorvo's paper [Pontecorvo:1957] — the Russian original, whose English translation [Pontecorvo:1958] carries a later year and is the same work, so the bibliography lists it twice for one paper — proposes oscillation by analogy with the neutral kaon and is about neutrino–antineutrino transitions, not flavour transitions; the flavour-mixing formulation that this chapter actually tests is that of Maki, Nakagawa and Sakata [Maki:1962]. It is a common misattribution to credit the second idea to the first papers, and the distinction is worth preserving. The matter effect is Wolfenstein's [Wolfenstein:1978], with the resonance and its consequences for solar neutrinos due to Mikheyev and Smirnov [Mikheyev:1985]; note that the latter, though universally dated 1985, appeared in a 1986 issue, and the bibliography entry records the journal's own date. Giunti and Kim [Giunti:2007] is the standard monograph and the place to go for the wave-packet treatment that Remark 103.51 summarizes.

The instrument papers. Davis, Harmer and Hoffman [Davis:1968] describes the chlorine detector and reports the first upper limit; the twenty-five-year result is Cleveland and collaborators [Cleveland:1998], which remains the authority for the Homestake apparatus, its extraction efficiencies and its backgrounds. For the gallium detectors, [Hampel:1999] covers GALLEX and [Abdurashitov:2009] the third and final SAGE dataset, each with its own source calibrations. Hirata and collaborators [Hirata:1989] is the Kamiokande-II solar measurement and [Hirata:1987] the supernova burst from the same detector. The Super-Kamiokande atmospheric result [Fukuda:1998] is the paper that made the case, and it should be read for its treatment of the up-down asymmetry rather than for its best-fit parameters, which are superseded; the later \(L/E\) analysis of the same atmospheric sample is [Ashie:2004]. Sudbury published the charged-current rate first [Ahmad:2001] and the neutral-current comparison a year later [Ahmad:2002]; only the second contains the flavour-blind measurement, and citing the first for it is an error that appears in the secondary literature. The salt phase, with its enhanced neutral-current sensitivity, is [Ahmed:2004], and the combined analysis of all three phases is [Aharmim:2013].

The reactor papers. KamLAND's first result [Eguchi:2003] established the deficit on fifty-four events; the demonstration that the survival probability oscillates as a function of \(L/E\) rather than falling monotonically came with later exposures, and is the spectral distortion of [Araki:2005], which is the source for the oscillatory dependence stated in Phenomenon 114.6. The final reactor analysis, taken across the shutdown of the Japanese reactors, is [Gando:2013]. Daya Bay [An:2012] and RENO [Ahn:2012] are the \(\theta_{13}\) measurements; the first is the more detailed on the near-to-far cancellation.

The accelerator papers. K2K [Ahn:2006] and MINOS [Michael:2006] are the disappearance confirmations; T2K [Abe:2011] is the first appearance indication and [Abe:2020] its \(\delta_{CP}\) constraint; NOvA [Acero:2022] is the current precision measurement of \(\abs{\Delta m^{2}_{32}}\) and \(\theta_{23}\); OPERA [Agafonova:2015] is the direct \(\nu_{\tau}\) appearance. The detection of the tau neutrino itself, which these presuppose, is [Kodama:2001], and the two-neutrino experiment on which the whole notion of lepton flavour rests is [Danby:1962]. The original detection of the neutrino is [Cowan:1956] [Reines:1956], and the counting of the light species is [Schael:2006].

The compilations and the model. The oscillation parameters of Table 114.9 are the Particle Data Group's [Navas:2024] and the global fit of Esteban and collaborators [Esteban:2020]; the two use overlapping data and are not independent, and where they differ this chapter follows the former. The dataset shipped with this treatise for [Navas:2024] carries the particle masses used in the kinematic derivations; the constants are CODATA 2022 [Mohr:2025]. The standard solar model is Bahcall, Serenelli and Basu [Bahcall:2005], whose BS05(OP) fluxes are Table 114.2; the much earlier prediction against which the original deficit was measured is [Bahcall:1968], and quoting the modern fluxes as though they were the ones Davis was compared with in 1968 misrepresents the history. Borexino's pp-chain and CNO results are [Bellini:2014] [Agostini:2018] [Agostini:2020a]. The absolute mass bounds are KATRIN [Aker:2022], Planck [Aghanim:2020] and KamLAND-Zen [Abe:2023]. The anomalies are LSND [Aguilar:2001] and MicroBooNE [Abratenko:2022], and the astrophysical neutrino flux is [Aartsen:2013] [Aartsen:2018].