The flat spectrum a fast pion must paint

The energy above which faster arrives later used the relation between energy and momentum in a proton ring. The same protons, striking a target or a cloud of gas at the end of their acceleration, produce pions in great numbers, and about a third of those are neutral. A neutral pion lives for less than a hundred-million-billionth of a second and then, almost always, decays to two photons. In its own rest frame the rule could not be simpler: momentum conservation makes the photons leave back to back, and energy conservation gives each exactly half the pion’s rest energy, or 67.5 MeV.

That is a sharp line and it would be easy to recognise. Almost no neutral pion, however, is at rest in the laboratory, or in space. What does a sharp line turn into when its source flies past at high speed, with each photon’s direction chosen at random? One’s first guess is a smeared bump, broader for a faster pion. The right answer is a flat rectangle with sharp edges. And when a whole population of such rectangles is added up, the sharp line comes back in a form that has nothing to do with its original cause, and it gives gamma-ray astronomers one of their few unambiguous signatures.

A decay with no preferred energy #

Work in the pion’s frame, with the photons along some axis at an angle $\theta^*$ from the direction in which the pion is going to be seen moving. In its rest frame the decay has no way to prefer any direction, because a pion has no spin, so all directions are equally likely. A uniform distribution over the sphere is uniform in $\cos\theta^*$. That small geometric fact, which the sky that crowds into a cone used for aberration, is the whole of the result.

Boost to the laboratory. A photon’s energy transforms by the Doppler factor, linear in the cosine,

$$E = \gamma\frac{m}{2}\left(1 + \beta\cos\theta^*\right),$$

where units have $c = 1$. Energy is a linear function of a uniformly distributed variable, and so it is uniformly distributed itself. There can be no peak.

The edges are at $\tfrac12 m\gamma(1 \pm \beta)$: the photon sent directly forward has the greatest energy and the photon sent directly backward has the least. A pion of 400 MeV has these at 388.3 and 11.7 MeV. So the two photons of a single decay land at positions in this range that add up to the pion’s energy, one high and one low, or both in the middle. Their distribution is flat, regardless of pion energy. Sampling confirms this to within statistical noise: every one of the forty bins here lies within 8 per cent of the mean.

The flatness is a very special feature of two-body decays to massless particles, and it is worth noting how narrowly it applies. For a charged pion decaying to a muon and a neutrino, the neutrino’s energy has the same linear relation, and the same box follows, which the neutrino energy an angle fixes turned around, to get an almost monochromatic beam at a small fixed angle. For a three-body decay, the energy distribution in the rest frame is already spread, as the plane in which three bodies are flat discusses, and the shape in flight is not a box. For a two-body decay where the decay has a preferred direction, because its parent is polarised, the top of the box tilts. A muon’s decay, for example, shows a tilted spectrum, and the turns a muon makes at any energy depends on it. A pion has no spin to do that with.

Every box is centred on the same energy #

The two edges of a box hold another surprise. They are related by a reciprocal Doppler factor, $\gamma(1 + \beta)$ and $\gamma(1 - \beta)$, whose product is

$$\gamma^2(1 - \beta^2) = 1.$$

Therefore the edges multiply to a constant, independent of the pion’s speed:

$$E_+E_- = \left(\frac{m}{2}\right)^2.$$

On a logarithmic axis, the geometric mean of the edges is the midpoint. So every box is exactly centred on $\log(m/2)$, whatever the pion’s energy. A faster pion gives a wider box, and as a box must hold two photons per decay, it is also lower, with height $2/(\gamma\beta m)$. A very fast pion’s box reaches from a tiny fraction of an MeV to nearly the pion’s own energy, and it is very low. Yet it still contains 67.5 MeV. A faster pion does not move away from that energy in any sense on this plot: it just spreads out from there. This fact, which follows simply from the Doppler factor and its reciprocal, is what makes the next result so robust.

The logarithmic axis is natural here for a physical reason. Rapidity, which the angle a collider counts in logarithms uses as its proper coordinate, is additive under boosts, and a Doppler factor is the exponential of a rapidity. A boost slides a log-energy coordinate by a fixed amount. The forward and backward photons slide by equal amounts in opposite directions, starting from the same place, which is why they are symmetric on this axis about the point they began at. Nothing else is needed.

The peak no pion spectrum can move #

Take any population of pions, with energies distributed in any way. Its total photon spectrum is the sum of these boxes, each weighted by the number of pions at that energy. To reach a photon energy $E$, a pion must have enough energy that its box contains $E$. So at a given $E$ one must add up all the boxes with $E_-\leq E\leq E_+$. Solving for the lowest pion energy that can do this:

$$E_{\pi,\min} = E + \frac{m^2}{4E}.$$

$$\frac{dN}{dE} = 2\int_{E_{\pi,\min}}^{\infty}\frac{f(E_\pi)}{\sqrt{E_\pi^2 - m^2}}\,dE_\pi.$$

The dependence on $E$ is only through the lower limit. It is largest when the lower limit is smallest, since $f$ is positive, and $E + m^2/4E$ is smallest at exactly $E = m/2$. So the spectrum peaks at 67.5 MeV for absolutely any $f$. And since $E + m^2/4E$ does not change when $E$ is replaced by $m^2/4E$, the spectrum is exactly symmetric on a logarithmic axis around that point.

The peak has nothing to do with a pion being at rest. Most of the contributing pions are moving at highly relativistic speeds, and there may be no slow ones at all. The peak arises because every box covers that energy, and slower boxes have a greater height. Numerical integration for three pion spectra, $E^{-2}$, $E^{-2.7}$ and $E^{-4}$, gives peaks at the same place, with widths that differ, and the check at a point and its mirror image, at 30 MeV and at its counterpart, agrees to within half a per cent. A steeper spectrum gives a narrower peak because the slow pions contribute more of the total. Their position does not change.

Floyd Stecker worked this out in 1971, in the context of gamma rays expected from cosmic rays striking interstellar gas. The feature is known as the pion bump. On astronomers’ usual plots, which multiply the spectrum by a power of the energy, it appears as a characteristic sharp rise below a few hundred MeV.

What a peak at 67.5 MeV is worth #

Cosmic rays reach the Earth with energies up to at least $10^{20}$ eV, mostly as protons, and since they are charged, their tracks have been scrambled by the galaxy’s magnetic field, so that a cosmic ray’s direction does not point back to its source. For about a century that has left open the question of where they come from. Supernova remnants have long been the leading candidate, because the expanding shells have the energy to supply them, and a shock has a mechanism to accelerate them. Fast electrons were already seen by their radio and X-ray emission in those remnants, but electrons are only about one per cent of cosmic rays, and evidence was needed for protons.

The protons would give themselves away by colliding with gas near the remnant and producing neutral pions, whose photons travel straight. Their spectrum, then, would show the bump. The trouble was that electrons can also produce gamma rays, by inverse Compton scattering and bremsstrahlung, and in many remnants there is no way to tell from the high-energy spectrum alone which source is responsible. What electrons do not have is any reason to produce a peak at 67.5 MeV: their spectra simply continue down in energy without that feature. A rise of the right shape at low energies would therefore be unique to pions.

The Fermi Gamma-ray Space Telescope looked for this at about 100 MeV, in the two remnants IC 443 and W44, and in 2013 its team reported exactly that characteristic rise; the AGILE satellite had reported something similar in W44 shortly before. It is the most direct evidence so far that supernova remnants do accelerate protons. Whether they reach the very highest energies is another question, and the bump does not address that, since it comes from the low end. The same shape also appears in the diffuse gamma-ray glow from the galactic plane, as well as, strikingly, in Earth’s own atmosphere: cosmic rays colliding with the upper air produce a pion-bump gamma-ray glow, seen by Fermi from its orbit.

What every pair has in common #

A single decay has one further rule, exact and unaffected by any boost. The invariant mass of the two photons is the pion’s mass, because that is what the invariant that survives a boost means. Two massless particles with energies $E_1, E_2$ at an opening angle $\theta$ have

$$m^2 = 2E_1E_2(1 - \cos\theta).$$

The figure shows it working over a large range of energies, with each decay generated in the rest frame, boosted to the laboratory and checked against the formula. The pairs vary widely in energy and angle, and they all lie on one line. This is how a neutral pion is identified in practice: two photons are measured in a detector, and the invariant mass of the pair is computed. In a very high-background environment, like a particle collision, the result is a small pion peak on a smooth background of accidental pairs. The mass weighed without one of its pieces treats the harder case where one decay product escapes detection. The Higgs boson’s discovery, in 2012, was in part a peak in the same plot at 125 GeV.

A subtle but significant consequence follows. The formula contains the opening angle as an equal partner of the energies, so the mass measurement depends on how well that angle is known. A detector needs fine position resolution to go with its energy resolution, and that is where an energetic pion becomes troublesome.

Two photons that stop looking like two #

Most decays give an opening angle close to the smallest possible value, which occurs when the two photons carry equal energy. That is $\sin(\theta_{\min}/2) = 1/\gamma$. The distribution piles up there for the same reason as the flat box: it comes from a Jacobian, with the angle having a stationary minimum as a function of the decay’s orientation. At 1 GeV the minimum is about 16°, and two separate photon showers are easy to resolve. At 50 GeV it is about 0.31°. Two showers a third of a degree apart, at a metre, are separated by half a centimetre, and in a calorimeter with cells a few centimetres across they merge into what looks like one photon.

This has large practical implications. In collider experiments, neutral pions from jets are far more common than genuine high-energy single photons, and a merged pion mimics a real photon. Finding the Higgs boson’s decay to two photons required separating the true photon pair from a far larger number of such events. The ATLAS experiment has a first layer of its calorimeter with strips only about 4 mm wide, finely segmented precisely to tell one shower from two, and CMS uses a crystal calorimeter with a similar goal, based on the shapes of the showers.

Cosmic gamma rays have the same issue in reverse. A gamma-ray telescope looking down at the atmosphere sees this glow, and it can resolve a single photon from a single decay at most at one energy. It simply measures the energy of the photon that arrives; the other goes elsewhere. That is one reason why the spectrum and not the pair is what astronomers analyse.

The same box in other decays #

A box of this kind is a general feature of a two-body decay, and it appears whenever a detector measures the energy of one daughter and not the angle. If a parent of mass $M$ decays into two daughters, each daughter has a fixed energy $E^*$ in the rest frame, and in flight its energy runs linearly from $\gamma(E^* - \beta p^*)$ to $\gamma(E^* + \beta p^*)$. Only massless daughters give edges with the convenient reciprocal property, but the flatness holds for all. In a two-body decay of a charged pion, the muon’s energy in the laboratory is a box, and a particle physicist looking at muon energy spectra can read off the pion’s energy from the two sharp edges.

Particle physicists also use the edges in the opposite direction. A heavy particle decaying in a detector produces a box whose upper edge depends on its mass and its boost, and the sharp edge of a spectrum is much easier to fit than the location of a smeared peak. The search for the top quark’s decay products, and later the measurement of its mass, used edges in reconstructed quantities that have this same origin. The mass weighed without one of its pieces has an edge at half the W boson’s mass, in a variable built specifically to preserve it.

A further feature of the box is what it says about thermal Doppler broadening, and the shift that survives at right angles gives the second-order piece of that. A hot gas of emitters at low speed broadens a line by a roughly Gaussian shape, not a box, because the speeds are distributed and low. A box needs all sources at one speed, and a true top-hat profile is a sign of a monoenergetic source: a shell of gas expanding at one speed produces exactly that, and supernova remnants have shown flat-topped emission lines for this very reason.

What the sampling assumes #

The decays here are ideal two-body decays, with an isotropic rest-frame distribution. A neutral pion has no spin, so this is exact. Around one decay in eighty goes instead to an electron–positron pair and a photon, called a Dalitz decay, which has a distinct shape, and a real measurement must allow for it. The population sums are for simple power-law pion spectra, chosen to demonstrate the independence of the peak’s location on the spectrum. The peak’s location is independent of $f$; its width and the height of its sides are not. Real cosmic-ray collisions have kinematic thresholds that remove low-energy pions in a way that depends on the proton spectrum, and helium in the gas and in the cosmic rays changes the normalisation.

An observer also multiplies the spectrum by the detector’s response and a power of the energy for display, so the feature that is seen on a published plot is a rise and not a peak. And none of the arguments about the bump proves that any particular remnant’s emission is mainly from pions, for an electron contribution can add a smooth component underneath. What Fermi could do was show that the data required the rise and that the electron models could not fit without it.

Still open: where the highest-energy cosmic rays come from #

The pion bump identifies cosmic protons at a site but says little about their highest energies. The cosmic-ray spectrum extends smoothly to a feature near $3 \times 10^{15}$ eV called the knee, and galactic sources are generally assumed to be able to accelerate protons to that energy. To find such sources, known as PeVatrons, requires looking for photons at about a tenth of the proton’s energy, near several hundred TeV, and their pions’ boxes are enormously wide. The LHAASO array in Tibet has reported a dozen or so sources emitting photons above 100 TeV, and in some cases electrons seem insufficient to explain them. Whether most of them are hadronic, whether supernova remnants are involved at all at these energies, and which source class supplies the knee, are questions that remain open.

An independent check is possible with neutrinos, because a charged pion made in the same collision decays to neutrinos, with their own box. The IceCube detector reported in 2023 that it had seen neutrinos from the galactic plane at about the level expected from cosmic rays interacting with interstellar gas, and that is a direct proof of pion production there, by the very process that yields the bump.

An isotropic two-body decay is flat in energy, because the Doppler factor is linear in the cosine. The edges of that box multiply to $(m/2)^2$, so every box at every pion energy contains 67.5 MeV, and any sum of boxes must peak there. No pion needs to be slow for the peak to exist. It says only that neutral pions were there, and it is why a rise near 100 MeV can show where cosmic-ray protons are.