The collision that wastes most of the energy

A particle accelerator is usually described by one number: the energy of its beam. It is the wrong number, and the right one is not proportional to it.

What is actually available #

Energy and momentum are both conserved, and it is the second that does the damage.

Take a beam particle of energy $E$ and momentum $p$ hitting a stationary target of mass $m$. Whatever comes out of the collision, it must carry the same total energy $E + mc^2$ and the same total momentum $p$. A collection of particles carrying that much momentum is a collection of particles that are moving, and moving costs kinetic energy that cannot be spent on anything else.

The part that is available is the invariant mass of the pair:

$$(\sqrt{s}\,c^2)^2 = (E + mc^2)^2 - p^2c^2.$$

Expand it for a fast beam, where $E \gg mc^2$, and the two terms nearly cancel: $s \approx 2mc^2E$. The square root of a near-cancellation is what makes the fixed-target line so flat.

Every four-vector has a length no observer disagrees about, and the total four-momentum of two colliding particles has one too. That length is the invariant mass, and it is the only energy actually available to make anything — the rest is bookkeeping about which frame the collision is being watched from. A collider is a machine for making that length large.

The head-on case #

Two beams of energy $E$ meeting head on have total momentum zero, by symmetry, so nothing at all is locked up in motion of the centre of mass. Every joule is available: $\sqrt{s} = 2E$.

The zero-momentum frame is where a collision is simplest, because in it nothing has to be carried away afterwards. A colliding-beam machine arranges for the laboratory to be that frame, so every joule put into the beams is a joule available to the interaction — which is the whole reason colliders replaced fixed targets, at enormous cost in difficulty.

That is the whole idea, and it was proposed by Rolf Widerøe in 1943, patented, and ignored for twenty years because nobody could make a beam dense enough to hit another beam.

What the missing energy is doing #

It has not vanished. In a fixed-target collision the centre of mass moves forward at a speed close to the beam’s, and everything produced must move with it.

Speeds do not add, so the centre-of-mass speed of a beam–target pair is not half the beam’s — it is very nearly the beam’s own. The whole system is sweeping past the laboratory at almost the speed of the incoming particle, and everything produced is dragged along with it. The energy in that bulk motion is conserved, unavailable, and most of what was put in.

So the products emerge in a narrow forward cone, at very nearly the beam’s speed, carrying nearly all of the beam’s energy as kinetic energy. That is not useless — a fixed-target experiment is the only way to make a secondary beam of anything, and every neutrino beam, every muon beam and every source of pions in the world is a fixed target — but it is unavailable for making mass.

The forward cone is narrow for a related reason. At high speed a transverse push produces far less angular deflection than the same push at low speed, so products emerging with modest transverse momentum in the centre-of-mass frame arrive in the laboratory within a degree or two of the beam line. That is why a fixed-target detector is long and thin and a collider detector is a barrel.

Why it took twenty years to use #

Widerøe’s idea is obvious once the arithmetic above has been done, and the arithmetic was not new in 1943. What stopped it was density.

A block of copper offers about $10^{29}$ nucleons per cubic metre to a beam, and every particle sent into it interacts. A beam of protons offers, at best, some $10^{15}$ per cubic metre while it is bunched and much less on average — fourteen orders of magnitude thinner. So a colliding-beam machine of the 1940s would have produced an interaction every few years, and there was nothing to be done about it with the beam intensities then available.

Three inventions closed the gap. Strong focusing, in 1952, made it possible to hold a beam in a small cross-section rather than letting it spread; storage rings, in the 1960s, let a beam be accumulated over many injections and kept circulating for hours so that each particle got millions of chances rather than one; and stochastic cooling, in the 1970s, made it possible to shrink a beam’s spread in momentum after it had been made, which is what turned antiprotons from a curiosity into a usable beam.

The first colliding-beam machine to produce physics, the Italian AdA, ran in 1964 at 250 MeV per beam — an available energy that any fixed-target machine of the day could have exceeded easily. It was built to prove that the arrangement worked at all. Everything since has been the same trade at larger and larger scale: give up rate, buy the square root back.

The cooling that looks impossible #

Of the three inventions that made colliding beams practical, the third is the one worth setting out, because on first hearing it sounds like it cannot work.

The problem it solves is that a beam of antiprotons, made by smashing protons into a target and collecting whatever comes off at roughly the right momentum, has an enormous spread in position, angle and momentum. Such a beam cannot be focused to the size a collision needs, and no arrangement of magnets improves it — a magnetic lens rearranges a beam’s phase-space distribution and cannot shrink its volume, which is Liouville’s theorem applied to a system evolving under forces.

Stochastic cooling shrinks it anyway. A pickup electrode senses the average deviation of the small number of particles passing it at that instant; the signal is amplified, sent across the ring on a chord so that it arrives before the particles do, and applied as a corrective kick. Each particle receives a kick appropriate to the average of its own sample rather than to itself, so the correction is right on average and wrong in detail — but the wrong part is random and averages down over many turns while the right part accumulates.

Liouville is not violated because the system is not the one the theorem is about. The beam plus the electronics is not evolving under a Hamiltonian: a measurement has been made, information about the beam now exists outside it, and the correction is applied using that information. The phase-space volume of the beam falls and the entropy is exported into the amplifier and the resistors that dissipate its output.

That is a genuinely unusual thing for a piece of accelerator hardware to be doing — using measurement to reduce disorder, at a cost paid somewhere else — and it is the same structure as every feedback cooling scheme since, including the ones that hold single atoms and single mirrors near their quantum ground states.

What a luminosity is made of #

The rate side of the trade deserves its own expression, because it says exactly where the engineering difficulty of a collider sits.

Two bunches crossing head-on, each carrying $N$ particles, meeting $f$ times a second with $n$ bunches in the ring, and each bunch focused to a transverse size $\sigma_x$ by $\sigma_y$, give

$$\mathcal{L} = \frac{f\,n\,N^2}{4\pi\sigma_x\sigma_y}.$$

Every quantity in that is a design decision and the exponents are what matter. The particle number appears squared, because both bunches contribute, so doubling the intensity quadruples the rate. The beam size appears in the denominator, so squeezing the beams is as valuable as filling them. The bunch count and the revolution frequency enter linearly.

Putting the LHC’s numbers in — a hundred and fifteen billion protons per bunch, twenty-eight hundred bunches, eleven thousand turns a second, focused to seventeen micrometres at the interaction point — gives about $10^{38}$ per square metre per second, which is the figure quoted earlier.

The reason the squeeze is the hard part is that it fights everything else. A tighter focus needs stronger final magnets closer to the collision, which is where the detector wants to be; a larger bunch population makes the bunches repel each other during the crossing, deflecting them and blurring the focus; and more bunches mean more crossings per event, so a detector has to disentangle several collisions at once. Each of the three terms in the numerator has its own ceiling, and a collider’s design is the point where they are all pressing simultaneously.

Why fixed targets did not go away #

The essay’s arithmetic makes fixed targets look obsolete and they are not, and the reason is the one the luminosity expression makes obvious.

A collider trades rate for available energy. Any experiment whose object is a rare process at known energy therefore wants the opposite trade: as many interactions as possible at whatever energy is sufficient. Searching for a decay that happens once in $10^{10}$ times needs $10^{12}$ parent particles, and a thick target is the only way to make them.

So the modern fixed-target programme is a rare-decay and precision programme rather than an energy frontier. Kaon experiments looking for decays forbidden or heavily suppressed in the standard theory, searches for a particle that decays after travelling metres, measurements of the muon’s magnetic moment — all of them need statistics rather than reach, and all of them use a beam hitting something solid.

And every neutrino experiment in the world is a fixed-target experiment twice over: a proton beam hits a target to make pions, the pions decay in flight to make neutrinos, and the neutrinos hit a detector of several kilotonnes. Nothing about that chain could be done with two beams, because a neutrino beam cannot be stored and there is nothing to collide it with.

The two arrangements are therefore complementary rather than successive, and which one a question needs is decided by whether the answer is limited by energy or by counting.

The threshold, which is where accelerators get their size #

Run the arithmetic backwards. To make some set of products of total mass $M$, the invariant mass of what went in must be at least $M$, and the beam energy that requires depends on the arrangement.

The Bevatron case is worth having in front of a reader, because it is the clearest instance of an accelerator whose size was determined by a line of algebra rather than by what was affordable.

Making an antiproton means the reaction $p + p \to p + p + p + \bar{p}$, because charge and baryon number both have to balance and the cheapest way to conserve them is to keep the original pair and add a new one. The products therefore weigh four proton masses, 3.75 GeV — but a beam striking a stationary proton must supply a kinetic energy of 5.63 GeV to make them, three times what the new pair itself weighs. The machine was designed to 6.2 GeV to have margin, and the antiproton was found in 1955, the year after it was finished.

An arithmetic worth doing once by hand #

The square root is easy to accept and hard to feel, so it is worth putting one collision through the algebra rather than reading it off an axis.

Take the LHC’s beam energy, 6,500 GeV, and let it strike a stationary proton. The beam’s momentum is essentially its energy over c, since 6,500 GeV is seven thousand times a proton’s rest energy. Total energy: 6,500.94 GeV. Total momentum: 6,500 GeV/c. The invariant mass is the square root of the difference of the squares, and the two squares are 42,262,220 and 42,250,000 — they agree in their first four digits.

The whole of the available energy is in that fifth digit. $\sqrt{12{,}220}$ is 110.5, so 110 GeV is available out of 6,500 supplied: one part in sixty.

Turn the target round so that it is a second beam and the two squares become 169,000,000 and zero. The cancellation vanishes, and every one of the 13,000 GeV counts.

That is the entire argument, and it is worth noticing what makes it work. Nothing about protons is involved; the number 0.938 appears only in the near-cancellation, and any other target mass would give the same story with a different constant. The square root is a consequence of subtracting two large nearly equal numbers, and the two numbers are large because the beam is fast.

What a collider costs instead #

The square root is not free, and what it costs is a different quantity.

A fixed target is a solid block: $10^{29}$ nucleons per cubic metre, and every beam particle that goes through it will hit something. Two beams passing through each other are two extremely dilute gases, and almost every particle misses.

How far something gets before it interacts depends on the density of what it is passing through, and a beam is not dense. That is what a collider pays instead: the available energy is maximal and the collision rate is minute, so the machine is judged on luminosity rather than on energy alone, and most of the engineering is about squeezing bunches rather than accelerating them.

The quantity that measures the trade has a name and a unit that says what it is. Luminosity is the number of particles crossing per unit area per unit time, so multiplying it by a cross-section — an area — gives a rate. The LHC runs at about $10^{38}$ per square metre per second; a fixed-target experiment with a thin target reaches that easily and with a thick one exceeds it by orders of magnitude. Everything about a collider that is difficult — the vacuum, the magnet alignment, the bunch structure, the beam lifetime — is difficulty about that one number.

So the trade is available energy against event rate, and the history of the subject is a sequence of decisions about which was scarcer. The Bevatron, the AGS and the SPS in its fixed-target years produced enormous rates at modest available energy; the ISR, the SPS collider, the Tevatron and the LHC produce very small rates at energies fixed targets could never reach.

The same invariant-mass arithmetic works at the other end of the scale. A photon bouncing off an electron shifts its wavelength by an amount that follows from conserving the total four-momentum and nothing else — the identical calculation, at energies twelve orders of magnitude lower, with the same quantity conserved.

What it costs #

The target has been a free proton. A real fixed target is a nucleus, whose nucleons are moving inside it with a few hundred MeV of momentum — so a fixed-target collision has a spread of available energies rather than one value, and the spread is what the phrase Fermi motion names.

Everything above ignores what the beam loses on the way in. A beam traversing a thick target loses energy continuously by ionisation, so the collisions that happen deep inside it are at lower energy than the nominal one. Thin targets avoid this and give up rate.

And the products have been treated as a total mass. In practice a threshold calculation has to know what the final state is, and the cheapest final state consistent with every conservation law is not always the obvious one — the antiproton case is easy because baryon number is simple, and a great many others are not.

Where the model stops #

Asymmetric colliders break the symmetry deliberately. Two beams of different energies give a centre of mass that moves, which is usually a waste and is sometimes exactly what is wanted: at the B factories the products are short-lived and a moving centre of mass stretches their laboratory lifetime enough to measure where they decayed. What looks like throwing energy away is buying a length.

Synchrotron radiation is the real limit on a circular electron machine. A charge going round a circle radiates, at a rate going as the fourth power of the energy over the mass, and LEP at 104.5 GeV per beam was losing about three per cent of each beam’s energy per turn. That is why the next electron collider will be linear and why the LHC is protons.

The available energy is not the same as the useful energy. Two protons colliding at 13 TeV are two bags of quarks and gluons, and what actually collides is one constituent from each, carrying a fraction of its parent’s momentum drawn from a distribution. The invariant mass of the pair that really interacts is therefore typically a small fraction of the 13 TeV, and the machine’s headline energy is a ceiling reached only in the rarest events. That is why raising the energy of a hadron collider by a factor of two raises its reach for a heavy particle by much more than a factor of two: it moves the whole distribution, and the tail moves fastest.

And a collider’s reach is not its energy. What can be found depends on the rate as much as the energy, and a process a thousand times too rare to see is out of reach at any energy. The two axes of the first figure are one half of the problem.

A charge in a magnetic field goes round a circle of radius $mv/qB$, so the curvature of a track measures a momentum. That is how all of this becomes data: the invariant mass of a set of products is reconstructed from the curvatures of their tracks, and a resonance is a bump in the distribution of that reconstructed number.

What the pictures cannot show #

The first figure plots two straight lines on logarithmic axes and its content is a difference of slopes, which is the least dramatic possible presentation of the largest fact in accelerator physics. Drawn linearly, the fixed-target curve would be indistinguishable from the horizontal axis over almost the whole range.

Nor can any of these figures show the collision. What is drawn is arithmetic about the initial state: energies, momenta, an invariant. What actually happens between two protons at 13 TeV involves their constituents, only a fraction of each proton’s momentum participates, and the effective available energy in the hard interaction is drawn from a distribution rather than being the number on the axis. The figure gives a ceiling and the events sit well below it.

Where this ladder goes next #

Two rungs stand on relativistic-dynamics. The first found that force and acceleration stop being parallel, and that Newton’s second law survives only as $\mathbf{F} = d\mathbf{p}/dt$. This one uses nothing but the conservation of that momentum and of energy, and finds that the two together make most of a beam’s energy unspendable.

The habit worth carrying away is about which quantity a design should be stated in. When two conserved quantities constrain an outcome together, the useful variable is usually the invariant they build, not either one alone. The energy of a beam is the number in the brochure and the invariant mass is the number that decides what can be made; quoting the first is quoting the input, and the whole art is knowing what fraction of an input survives into the thing being paid for.

What is left on this ladder is what happens to a body under a force applied for a long time. A constant proper acceleration produces a hyperbola with a light ray as its asymptote, and there are events a steadily accelerating traveller can never receive — which turns a statement about dynamics into a statement about which parts of spacetime an observer has access to.