The angle a collider counts in logarithms

Speeds that refuse to add found that velocities along one line do not add in relativity, but that a quantity built from them does: the rapidity, $y = \tanh^{-1}(v/c)$, which simply adds when one boost follows another along the same direction. The space that speeds live in went further, showing that rapidity is a genuine distance in the curved space of velocities, and remarked in passing that particle physicists quote momenta in rapidity, or in something closely related called pseudorapidity, because differences of rapidity are unchanged by boosts along the beam. That remark carries more weight than its length suggests. Rapidity is not just a convenient bookkeeping device in collider physics; it is the variable in which the results of every hadron collider are reported, and the reasons are physical, not mathematical.

The wheel whose top cannot go twice as fast composed velocities in several directions at once round a rolling rim. The case here is simpler — everything along one axis, the beam — and that simplicity is exactly what makes rapidity powerful. For motion along a single line the composition law becomes addition, and addition turns every change of frame along that line into a sliding of the whole picture.

One spray, two descriptions #

For a particle of energy $E$ and momentum $p_z$ along the beam, the rapidity is

$$y = \tfrac12\ln\frac{E + p_z}{E - p_z},$$

which reduces to the inverse hyperbolic tangent of the velocity along the beam. A boost along the beam with rapidity $Y$ adds $Y$ to every particle’s $y$, whatever its mass, energy or direction. That one property is the whole of the case for using it.

A source that emits particles evenly in every direction is flat in the cosine of the emission angle. Seen from a frame in which the source moves along the axis at 0.8 of the speed of light, the same particles are crowded forward, the effect the brightness that is not the same for everyone found for light from a moving source; the distribution is distorted out of all recognition, with eighty-one times as many particles going straight ahead as straight back. In rapidity the two distributions are the same curve, the moving one displaced by $\tanh^{-1}0.8 = 1.099$. For massless particles the curve is $\tfrac12\operatorname{sech}^2\eta$, with a width of 1.76 units at half height: a fixed, characteristic shape that a moving source carries with it.

That is the point. A physicist who records the distribution in angle has to know the source’s velocity to interpret it. One who records it in rapidity can read off the source’s rapidity from where the curve sits and its intrinsic properties from the curve’s shape, separately.

Speeds that crowd, rapidities that do not #

Near the speed of light, velocity is a poor variable for a second reason: all the interesting differences are crowded into the last few decimal places. A proton in the Large Hadron Collider at 6.8 TeV has a Lorentz factor of about 7,250 and moves at a speed that falls short of light’s by about one part in a hundred million. A proton with a tenth of that energy falls short by one part in a million. In velocity the two differ in the seventh decimal place; in rapidity they differ by 2.3 units, the logarithm of the ratio of their energies. The same compression appears in the turns a muon makes at any energy, where a muon’s speed hardly changes over a huge range of energies while its time dilation grows enormously. Rapidity undoes the compression: at high energy it is close to $\ln(2E/mc^2)$, so equal steps of rapidity are equal ratios of energy, and a detector’s resolution, which is roughly a fixed fraction of a particle’s momentum, becomes a roughly fixed step in rapidity too.

Why the collision is moving #

At a collider with two equal beams meeting head on, it seems that the collision’s centre of mass should be at rest in the laboratory, and for electron–positron colliders it is. For protons it is not, and the reason is that a proton is not a single particle. The hard collisions that produce heavy particles happen between the proton’s constituents — quarks and gluons — and each constituent carries some fraction $x$ of its proton’s momentum, a fraction that differs from collision to collision and is spread over the whole range from nearly zero to nearly one.

If one constituent carries $x_1$ of one beam’s momentum and the other $x_2$ of the other’s, the subsystem they form moves along the beam with rapidity

$$Y = \tfrac12\ln\frac{x_1}{x_2}.$$

A collision between a constituent carrying a tenth of its proton’s momentum and one carrying a hundredth moves along the beam with rapidity 1.15. The next collision might move the other way. Every event has its own longitudinal boost, unknown in advance and usually not measured, because some of the debris escapes down the beam pipe where no detector can go.

The same formula turns the nuisance into a measurement. When the two constituents annihilate into a single particle — a Z boson, say, of mass $M$ — the particle carries the whole subsystem’s motion, so its rapidity is $\tfrac12\ln(x_1/x_2)$ and its mass fixes the product, $M^2 = x_1x_2s$, where $\sqrt{s}$ is the collision energy of the two protons. Measure the Z boson’s rapidity from its decay products and both momentum fractions follow, $x_{1,2} = (M/\sqrt s)\,e^{\pm Y}$. A Z boson produced at rapidity three, at the LHC’s 13.6 TeV, came from one constituent carrying about 13 per cent of its proton’s momentum and another carrying about three-hundredths of one per cent. Distributions of Z and W bosons in rapidity are among the main inputs to the maps of how a proton’s momentum is shared among its quarks and gluons, and they work because the rapidity of what is produced is the rapidity of the collision.

So the only descriptions that make sense across many events are ones that do not care about a boost along the beam. Transverse momentum, the component at right angles to the beam, is one: it is unchanged by any boost along the beam. Differences of rapidity are another. A pair of particles separated by two units of rapidity is separated by two units in every frame related by a boost along the beam, and a distribution in rapidity from events with different boosts is a superposition of copies of one shape, slid by different amounts, not a mixture of incompatible distortions.

What a pair of particles remembers #

The variables also make the most important single measurement at a collider — the mass of whatever decayed into the particles seen — independent of the unknown boost. Two particles whose combined energy and momentum came from the decay of a heavy parent carry that parent’s mass in their invariant mass, the quantity the invariant that survives a boost showed every observer agrees on. Written in the collider’s variables, for two particles light enough to treat as massless, it is

$$m^2c^4 = 2\,p_{T1}\,p_{T2}\,c^2\left(\cosh\Delta y - \cos\Delta\phi\right),$$

where $\Delta y$ is the difference of their rapidities and $\Delta\phi$ the difference of their directions round the beam. Every term is unchanged by a boost along the beam: transverse momenta are, differences of azimuth are, and differences of rapidity are. The Higgs boson was found in 2012 as an excess of photon pairs and of lepton quartets at one value of this mass, in events whose individual boosts along the beam were all different and all unknown. The formula is why that did not matter.

The same structure governs the spread of a decay. A particle that decays at rest throws its products in every direction, and the cone a decay cannot leave found that a moving parent confines them to a forward cone. In rapidity there is no cone, only a fixed spread — for a two-body decay into light particles, about one unit either side of the parent’s own rapidity — carried along wherever the parent goes. Physicists searching for a resonance look for a cluster of a known width in rapidity at an unknown position, which is a much easier thing to look for than a cone of unknown angle.

The angle a detector actually measures #

A detector does not measure rapidity. It measures where a particle’s track goes, and from the track’s curvature in a magnetic field its momentum; it often cannot identify the particle, so it does not know its mass, and without the mass it cannot compute its energy or its rapidity. What it can always compute is the angle $\theta$ between the track and the beam. From that, it forms the pseudorapidity,

$$\eta = -\ln\tan\frac{\theta}{2},$$

which is what the rapidity would be if the particle were massless. For a particle moving close to the speed of light, as most products of a high-energy collision are, the two are nearly equal.

The logarithm stretches the directions near the beam and compresses those near right angles. The whole range from 90° to 40° is one unit of η. The range from 15° to 9.4° is another half-unit. Between 1° and 2.7° there is again a whole unit. Detector designers think in $\eta$ for that reason: the general-purpose detectors at the Large Hadron Collider track particles to $|\eta| = 2.5$, a little under ten degrees from the beam, and measure energy with calorimeters reaching $|\eta|$ near five, within a degree. A specialised detector, LHCb, looks only forward, from $\eta$ of two to five, because the particles it studies are produced with large boosts along the beam.

The dip that mass makes #

At high momentum the difference between rapidity and pseudorapidity is negligible. At the modest momenta of most particles produced in a collision, it is not, and it leaves a mark in the data that is easily mistaken for physics.

For a particle of mass $m$ and transverse momentum $p_T$, the two variables are related by $dy/d\eta = p/E$, the particle’s speed. Far forward, every particle is fast and $\eta$ and $y$ are interchangeable. At right angles to the beam a particle moves only with its transverse speed, $p_T/m_T$ where $m_T = \sqrt{m^2 + p_T^2}$, and for a proton with half a GeV of transverse momentum that is less than half the speed of light. A distribution that is perfectly flat in rapidity therefore shows a dip near $\eta = 0$ when plotted in pseudorapidity, deepest for the heaviest particles. Measured distributions of charged particles at every hadron collider show a shallow dip of this kind in their central region. It comes from the masses of the particles being counted, not from anything the collision does there, and it is one of the first things a physicist reading such a plot has to correct for.

How much rapidity a collision has to fill #

The two beams themselves have rapidities, and those set the range a collision’s products can occupy.

A beam particle’s rapidity is $\cosh^{-1}(E/mc^2)$, close to $\ln(2E/mc^2)$, so it grows only logarithmically with the beam’s energy. The Large Hadron Collider’s protons, at 6.8 TeV each, sit at $\pm9.58$; the whole span between them is about nineteen units, of which a central detector covers five. A beam striking a fixed target spans the range from the target’s rapidity, zero, to the beam’s — and the centre of mass sits at exactly half the beam’s rapidity, because the two colliding particles have equal masses. In rapidity the fixed-target experiment and the collider are the same span seen from different places, which is a cleaner way of putting the point the collision that wastes most of the energy made with invariant masses: a fixed target spends most of its beam’s energy on moving the centre of mass, and in rapidity that motion is just the offset of the span.

HERA, which collided 27.5 GeV electrons with 920 GeV protons in Hamburg until 2007, shows the other use. Its beams were deliberately unequal, so the collisions moved strongly along the proton’s direction, by about 1.76 units, and its detectors were built asymmetrically to match. The asymmetric electron–positron colliders built to study B mesons used the same trick on purpose: by colliding beams of unequal energy, they made the B mesons move along the beam fast enough to travel a measurable fraction of a millimetre before decaying, so that their lifetimes, and the differences between matter and antimatter in their decays, could be read off distances.

A flat plateau, seen in angle #

High-energy collisions produce most of their particles with small transverse momenta, spread over rapidity in a broad, nearly flat plateau. Richard Feynman suggested in 1969 that the plateau should be flat and its height independent of energy, and although the height turned out to grow slowly with energy, the plateau’s flatness in the central region is one of the most robust features of hadron collisions. James Bjorken built a model of heavy-ion collisions on it in 1983, assuming that the hot matter formed in the collision looks the same in every frame moving along the beam — boost invariance — and his estimate of the energy density reached in such collisions, read off the plateau’s height, is still the first number quoted for each new collider.

In angle, the same plateau is anything but flat. Per degree it piles up towards the beam as $1/\sin\theta$; per unit solid angle, as $1/\sin^2\theta$. A spray that is perfectly even in rapidity looks, to anyone measuring in angle, like a jet of debris thrown mostly forward and backward along the beam. That is why the detector elements closest to the beam receive the highest radiation doses at a hadron collider, and why their forward parts are rebuilt most often.

The variable also makes absences visible. In some collisions no particles at all are produced across several units of rapidity — a rapidity gap — and such gaps are the signature of an exchange between the two protons that carries no colour charge. In angle the same gap is a cone near the beam with nothing in it, easy to overlook; in rapidity it is a hole in a flat plateau, and it was studied at HERA and at the Tevatron as a window on the strong force at long range.

A line, massless products and no transverse motion #

Every figure here treats motion along the beam alone, as relativity’s composition law does when it reduces to adding rapidities; a boost in any other direction does not simply shift rapidity, and the Wigner rotations of the space that speeds live in reappear. The spray in the first figure is of massless particles from a point-like source with no transverse motion; massive products from a real moving source have a broader rapidity distribution that depends on their mass and temperature. The dip figure takes a single transverse momentum, while real particles have a spread of them, which smears the dip; the beam figure ignores the beams’ own small transverse spread. The plateau is an idealisation of measured distributions, which are flat only in the central few units and fall away towards the beam rapidities. The domain is products of high-energy collisions moving along or near the beam axis, with rapidities between the beams’.

Still open: what spreads correlations across the plateau #

The plateau was long taken to be the sum of many independent clusters of particles, each local in rapidity. In 2010, measurements of proton–proton collisions with very many products found that pairs of particles were correlated in their directions round the beam even when separated by several units of rapidity — a long, narrow “ridge” in the correlation plot. Something had correlated particles that, in a boost-invariant picture, should know nothing about each other. Similar ridges had been seen in collisions of heavy nuclei and interpreted as the collective flow of a droplet of quark–gluon plasma, but in proton–proton collisions no one expected a droplet. Whether the ridge in small systems comes from a tiny fluid, from correlations already present in the colliding protons’ gluon fields, or from both, is argued in a large and unfinished literature. The tool for asking the question is the one drawn here: a variable in which a correlation’s reach along the beam is a frame-independent number.

The underlying statement is short. Along one axis boosts add rapidities, $y = \tfrac12\ln\frac{E+p_z}{E-p_z}$, so a collision moving along the beam — and at a hadron collider every one does, with rapidity $\tfrac12\ln(x_1/x_2)$ — shifts its products’ rapidity distribution without changing its shape; detectors measure $\eta = -\ln\tan(\theta/2)$ instead, equal to $y$ for massless particles and dipping at $\eta = 0$ to the transverse speed for massive ones — 0.47 for 0.5 GeV/c protons — and a flat plateau in $\eta$ is 132 times denser per steradian at 5° than at 90°. Collider physics counts angles in logarithms because, along the beam, that is how relativity counts speeds.