The neutrino energy an angle fixes

The cone a decay cannot leave set a particle moving and let it break in two. In its own frame the two products fly apart back to back with fixed energies; in the laboratory the whole pattern folds forward into a cone, and the energy of either product, averaged over every direction, is spread evenly between two limits, a rectangle. The slope a spin leaves in a spectrum tilted that rectangle by giving the parent a spin. Both asked what a detector sees if it collects the products from every direction.

This essay asks the opposite question, about one direction. A detector at a fixed angle to a moving parent sees only the products that happen to go its way, and for the decay of a pion into a muon and a neutrino the energy of those products has a feature that sounds like an accident and is the whole design principle of the largest neutrino experiments of the past two decades: at a fixed angle off the beam, a wide range of parent energies gives nearly the same neutrino energy. The angle selects an energy, and the reason is a stationary point in an ordinary relativistic formula.

A fixed momentum in the rest frame #

A charged pion at rest decays, almost always, into a muon and a neutrino. With two products and no third to share the momentum, each must carry the same momentum in opposite directions, and conservation of energy fixes its size:

$$p^* = \frac{m_\pi^2 - m_\mu^2}{2m_\pi} = 29.8\ \text{MeV}/c.$$

The neutrino, with a mass far too small to matter here, carries an energy of 29.8 MeV in that frame, in a direction chosen at random. That one number is the starting point for everything below.

Now the pion is moving, with Lorentz factor γ and speed β. The neutrino’s energy in the laboratory is its rest-frame energy multiplied by the Doppler factor for its direction:

$$E_\nu = \frac{p^*}{\gamma(1 - \beta\cos\theta)} = \frac{m_\pi^2 - m_\mu^2}{2(E_\pi - p_\pi\cos\theta)},$$

where θ is its angle to the pion’s line of flight in the laboratory. A neutrino is a particle, but from the point of view of a moving source its energy follows exactly the rule a photon’s frequency does, the Doppler factor that the shift that survives at right angles followed out of the line of motion. Straight ahead, θ = 0, the factor is large, about 2γ, and the neutrino takes 0.427 of the pion’s energy. To one side, it is smaller.

The 29.8 MeV is itself an invariant in disguise. In the pion’s frame the energies of the two products are fixed by the masses alone, and the same masses fix the combination of energy and momentum that every observer agrees on — the invariant that survives a boost, the parent’s mass. A physicist who measured the muon’s and the neutrino’s energies and directions in the laboratory and combined them would recover the pion’s mass, 139.6 MeV, whatever the pion’s speed. What a boost changes is how that fixed budget is divided between energy and direction for a particular observer, and that division is all this essay is about.

Why a neutrino beam is made of pions #

A beam of neutrinos cannot be made the way a beam of protons or electrons is. Neutrinos have no charge, so no field can accelerate, steer or focus them, and they interact so rarely that no target could stop them to make them into anything else. The only way to send a beam of them somewhere is to make them in the decays of particles that can be steered, and send those particles in the right direction first.

Pions are the natural parents. A proton beam of tens of GeV striking a graphite target makes them in profusion, most of the collision’s energy going into the motion of the debris rather than into new mass, and charged pions live long enough to be steered: their mean life is 26 nanoseconds, which at γ of thirty becomes 780 nanoseconds in the laboratory, 234 metres of flight. Long enough to be focused by a magnetic horn and sent down a tunnel before decaying; short enough that a tunnel a few hundred metres long catches most of their decays. The choice of pion charge picks the neutrino or the antineutrino beam: positive pions give muon neutrinos, negative ones muon antineutrinos, and reversing the horns’ current switches between them.

Folded into a narrow beam #

The direction is transformed as well. A neutrino thrown sideways in the pion’s frame, at ninety degrees, appears in the laboratory at an angle whose tangent is 1/βγ, close to 1/γ; everything thrown into the forward half of the rest frame lands inside that angle, and most of what is thrown backwards is swept forward too. That is the sky that crowds into a cone for a moving observer, seen from the other side: a source moving fast concentrates its emission forward. A beam of pions of a few GeV, with γ of twenty or thirty, makes a beam of neutrinos a degree or two wide — narrow enough to aim across a country.

That is how accelerator neutrino beams are made. Protons strike a target and make pions of many energies; magnetic horns, pulsed with hundreds of kiloamperes, focus the pions of one charge into a beam; the pions fly down a decay tunnel hundreds of metres long and about half of them decay before they reach its end; and the neutrinos continue straight through the earth beyond, passing a near detector and then, hundreds of kilometres away, a far one. Everything but the neutrinos is stopped by the rock.

A maximum at one over gamma #

At a fixed laboratory angle the neutrino energy depends on the pion’s energy in two competing ways. A faster pion carries more energy to share, which raises the neutrino’s. But a faster pion also folds its decay further forward, so a neutrino emerging at the fixed angle θ must have been thrown further backward in the pion’s frame, where the Doppler factor is smaller. For small angles the result is

$$E_\nu \approx \frac{0.427\,E_\pi}{1 + \gamma^2\theta^2},$$

which rises in proportion to the pion’s energy while γθ is small and falls as $1/E_\pi$ once γθ is large. In between it has a maximum, at γθ = 1, and its height is

$$E_\nu^{\max} = \frac{p^*}{\sin\theta} \approx \frac{p^*}{\theta},$$

the rest-frame momentum divided by the angle. At 2.5 degrees that is 0.68 GeV, from a pion of 3.2 GeV.

A maximum is where a function is flat, and flatness is the point. Around γθ = 1 the neutrino’s energy barely changes as the pion’s does: every pion between two and ten GeV — a factor of five in energy, most of a real beam — sends neutrinos at 2.5 degrees with energies between 0.61 and 0.68 GeV. The sideways neutrinos are the rest-frame ninety-degree neutrinos of pions whose γ matches the angle, and the stationary point gathers pions of many energies into a narrow band of neutrino energies, a Jacobian peak of the same kind that piles up the edges of the rectangle a decay makes when its products are collected from every direction.

One beam, two spectra #

The figure computes it directly. A beam of pions with a broad spread of energies is decayed, each pion isotropically in its own frame, and the neutrinos that leave within a tenth of a degree of a chosen direction are kept. Along the axis the spectrum copies the pions’: broad, a GeV and a half wide at half maximum, with a long tail to high energy. At 2.5 degrees off axis the same pions give a peak a tenth of a GeV wide, and essentially nothing above the kinematic edge.

The off-axis detector receives fewer neutrinos in total, since the beam is densest along its axis. What it receives is concentrated where it is wanted, and what it gives up is mostly neutrinos at energies that would only add background.

The energy the oscillation wants #

The reason an experiment wants a particular energy is neutrino oscillation. A neutrino made as a muon neutrino is a superposition of states of slightly different mass, which travel with slightly different phases, and after a distance L the chance it is still a muon neutrino is

$$P = 1 - \sin^2 2\theta_{23}\,\sin^2\!\left(\frac{1.27\,\Delta m^2 L}{E}\right),$$

with L in kilometres, E in GeV and the mass-squared difference Δm² in eV². Over the 295 kilometres from the J-PARC accelerator at Tokai to the Super-Kamiokande detector under a mountain at Kamioka, with Δm² of 2.5 × 10⁻³ eV², the oscillation reaches its first maximum, where the muon neutrinos have almost entirely turned into other flavours, at 0.59 GeV.

That is where the experiment learns most: the depth of the dip measures the mixing, and its position measures Δm². It is also near that energy that a small fraction of the muon neutrinos turn into electron neutrinos, the appearance channel that tests whether neutrinos and antineutrinos oscillate differently, a violation of the symmetry between matter and antimatter. Neutrinos far above the dip carry little information about it and make backgrounds that look like signal. The T2K experiment, running since 2010, points its beam 2.5 degrees away from Super-Kamiokande for exactly this reason, so that its neutrino spectrum peaks at the dip. A detector near the target, on the same off-axis line, measures the beam before it has oscillated.

Counting a few neutrinos a day #

The detector that receives the beam at Kamioka is fifty thousand tonnes of ultrapure water in a steel tank a kilometre under a mountain, its walls lined with eleven thousand photomultiplier tubes. A neutrino that interacts in the water makes a muon or an electron moving faster than light does in water, and that charged particle radiates the cone of blue light that the glow of a fast charge traced, which arrives on the tank’s wall as a ring. A muon, heavy and slowly scattered, leaves a sharp ring; an electron, which scatters and showers, leaves a fuzzy one. The difference tells a muon neutrino from an electron neutrino, which is the measurement the oscillation requires.

Even aimed at the detector with all the beam’s intensity, the neutrinos interact so rarely that the experiment records a handful of beam events on a good day, and its results are built from a few hundred events accumulated over years. That scarcity is why the shape of the spectrum matters so much. Every neutrino that arrives at an energy where the oscillation leaves no trace is an event spent on nothing, or worse, a background that mimics the electron-neutrino appearance the experiment is looking for. The off-axis beam gives up raw numbers to make the few events it gets count.

Choosing an energy by choosing an angle #

A longer baseline puts the first oscillation maximum at a higher energy, in proportion to the distance, and the off-axis method follows: the peak energy goes inversely as the angle, so a longer baseline wants a smaller angle. The NOvA experiment in the United States, whose beam from Fermilab near Chicago travels 810 kilometres to a detector at Ash River in Minnesota, sits 14.6 milliradians off its beam’s axis, where the peak lies near two GeV. The next generation of experiments plans detectors that can be moved across the beam, measuring the flux at many angles so that a combination of off-axis spectra reconstructs the response to any energy.

It is worth noticing what the method does not need. The accelerator, the target, the magnetic horns and the decay tunnel are the same as for an on-axis beam. The energy is set by geometry alone: by the line from the target to the detector and the direction the beam is pointed, decided when the beamline was dug into the ground. T2K’s beamline at Tokai was built to point 2.5 degrees away from the line to Kamioka, so that its far detector would sit at that angle for the life of the experiment.

A decay that hands over a spin as well #

The pion has no spin, and the neutrino from it always spins against its direction of motion, a fact the slope a spin leaves in a spectrum used to explain how pion decays make polarised muons. The two products leave back to back with no orbital angular momentum to spare, so the muon’s spin must cancel the neutrino’s: in the pion’s frame the muon too spins against its motion, completely. The neutrino’s handedness is fixed by the weak interaction; it is one of the facts that makes the weak interaction violate the symmetry between left and right.

For a neutrino beam that handedness is a second constraint the experiment can use. Antineutrinos, made from negative pions, spin the other way, interact with matter differently and less often, and the difference between how neutrinos and antineutrinos oscillate is the effect the experiments are built to measure. Running the beam in both modes, with the horns’ current reversed, gives two off-axis spectra peaked at the same energy, because the kinematics depend on p* and θ and not on the sign of anything.

The off-axis idea itself was first proposed in the mid-1990s for an experiment at Brookhaven that was never built. It was adopted by T2K in Japan and NOvA in the United States; T2K reported in 2020 a first hint that neutrinos and antineutrinos do not oscillate in exactly the same way, which NOvA’s data neither confirm nor rule out.

The same formula elsewhere #

The maximum at γθ = 1 is not special to neutrinos. Any two-body decay of a fast parent into a light particle has it, and so does any source whose emission is concentrated by its motion into a cone of half-angle 1/γ. A photon from a decaying neutral pion seen at a fixed angle has the same peaked energy; the light of a fast electron passing through an undulator, which the magnet period that comes out as an X-ray followed, has an energy that falls with angle as 1/(1 + γ²θ²) by the same Doppler factor. In astrophysics the radiation from a jet moving at Lorentz factor γ is brightest and bluest for observers within 1/γ of its direction, and an observer slightly off the jet’s axis sees emission dominated by the parts of a decelerating flow whose 1/γ has grown to match the viewing angle.

The particle physics case is the cleanest because the rest-frame energy is a single number fixed by two masses. A detector at a fixed angle reads, in effect, the Doppler factor’s stationary value, and that value depends on p* and θ and on nothing about the beam.

Where the idealised beam stops #

The figures make three simplifications that the real beams do not. They take every pion to fly exactly along the beam’s axis, where in a real beam the horns leave the pions with a spread of directions of several milliradians, so the angle between a pion’s path and the detector varies from pion to pion; that smears the peak and moves it down, which is why T2K’s measured spectrum peaks near 0.6 GeV rather than at the 0.68 GeV kinematic edge drawn here. They ignore where along the decay tunnel each pion decays, which changes its angle to a distant detector slightly. And they ignore other parents: kaons decay to muons and neutrinos with a rest-frame momentum of 236 MeV/c and give a harder off-axis spectrum, and muons from the pion decays themselves decay to give electron neutrinos, a background to the appearance measurement. The real flux is predicted by simulating the whole beamline and is constrained by measurements of how pions are produced in a replica target.

The kinematic statement survives all of that: at a fixed angle, the energy from a two-body decay has a maximum at γθ = 1 of height p*/sin θ, and a broad parent spectrum is compressed into a narrow band below it.

Still open: how well the flux can be predicted #

The precision of the oscillation measurements now depends as much on knowing the neutrino beam as on counting events. The number of neutrinos in each energy bin at the far detector, and how they interact with the nuclei in the detector, have uncertainties of several per cent, and the measurements that are most sought — whether neutrinos and antineutrinos oscillate differently, which of the neutrino masses is heaviest — are effects of a few per cent. Combining near detectors at several off-axis angles, measuring hadron production for the exact target used, and modelling neutrino–nucleus interactions from first principles are all in progress, and how far the systematic uncertainties can be pushed down is open.

The habit worth carrying away is to look for the flat place in a relation before averaging over it. A pion’s neutrino carries p = 29.8 MeV/c in its rest frame, and at a fixed angle θ in the laboratory its energy, 0.427E_π/(1 + γ²θ²), has a maximum p/θ at γθ = 1 — 0.68 GeV at 2.5° — around which pions from 2 to 10 GeV all give neutrinos within a tenth of a GeV.** An experiment that needs one energy chooses it by where it puts its detector.