The spin that runs ahead of its own motion

Magnetism is electricity seen sideways found that the electric and magnetic fields are two faces of one object that a change of frame mixes, and six numbers, one object wrote that object down. The field nobody can transform away and the one quantity a boost leaves alone found what the mixing preserves, and the beam that stops pushing itself apart applied it to a bunch of charges moving together. The turn that two pushes leave behind, in another part of the collection, found that a frame carried round a curve in velocity comes back rotated — the Thomas precession — and that the rotation halves the fine-structure splitting of hydrogen.

This essay puts those two ideas into one experiment. A particle with spin circles in a magnetic field. Its motion is turned by the field; its spin, which is a small magnet, is turned too. How the two turnings compare depends on the particle’s magnetism, on how the field looks from the particle’s moving frame, and on the rotation that frame undergoes. The comparison turns out to be independent of the particle’s speed, which is surprising, and at one special speed it becomes independent of any electric field present, which is useful: it is the principle on which the most precise measurement of the muon’s magnetism was made.

Two things that turn #

A particle of charge $e$ and mass $m$ moving through a uniform magnetic field $B$ goes round a circle, its direction of motion turning at the cyclotron rate $eB/\gamma m$. A relativistic particle turns more slowly than a slow one, because its inertia is $\gamma m$; a muon at 99.94 per cent of the speed of light, with $\gamma = 29.3$, goes round a storage ring in a field of 1.45 tesla 6.7 million times a second.

The particle’s spin is a magnetic moment, $g$ times the value the same angular momentum would have for a spinning ball of charge, and a magnetic moment in a field precesses. Dirac’s equation for the electron gave $g = 2$ exactly, and quantum electrodynamics corrects it slightly: for both the electron and the muon $g$ is larger than two by a little over a part in a thousand, and the excess, written as $a = (g-2)/2$, is called the anomalous moment. For the muon, $a = 0.00116592$.

The spin’s rate of turning, worked out from relativistic dynamics by Valentine Bargmann, Louis Michel and Valentine Telegdi in 1959, is $eB/m$ times $1/\gamma + a$. The momentum’s is $eB/m$ times $1/\gamma$. The figure plots both against the particle’s Lorentz factor. At low speed they are nearly equal: the spin turns with the motion, like a compass needle carried in a car going round a bend, and lags it only by the small anomaly. At high speed both slow down as $1/\gamma$, but the spin’s excess does not: it stays at exactly $a\,eB/m$. For the muon ring’s field that is 229 thousand turns a second, the same for a slow muon and for one at $\gamma = 29.3$.

Why the lead does not depend on speed #

That constancy is worth understanding, because it is not what naive reasoning gives. In the muon’s own rest frame the laboratory’s magnetic field is transformed: magnetism is electricity seen sideways found that a boost perpendicular to a magnetic field multiplies it by $\gamma$ and adds an electric field. The muon at rest in a field $\gamma B$ precesses at $(g/2)e\gamma B/m$, and one might expect the laboratory to see that rate — $\gamma$ times faster than a slow muon’s. It does not, because the rest frame’s clock runs slow by $\gamma$ as seen from the laboratory, and the clock that has to slow cannot be argued with. The two factors of $\gamma$ cancel, and the Larmor precession clocked in the laboratory is $(g/2)\,eB/m$ at every speed.

That alone would give the spin a rate of about $eB/m$ while the momentum turns at $eB/\gamma m$ — a lead that grows with speed. The missing piece is the Thomas precession. The muon’s rest frame is not a fixed frame: at every instant it is the frame moving with the muon, and the muon’s velocity is being turned by the field, so the rest frame is being carried round a closed curve in the space of velocities. The turn that two pushes leave behind found that such a frame comes back rotated, and the rate of that rotation is $(1 - 1/\gamma)$ times $eB/m$, backwards. The figure adds the two. The Larmor term is flat at 1.00117; the Thomas term falls from zero to $-1$ as the speed rises; their sum, the spin’s rate, is $1/\gamma + a$; and the momentum’s rate is $1/\gamma$. The difference is $a$ at every speed, because the Thomas precession removes, exactly, everything except the anomaly from the lead the Larmor term would otherwise give.

The same calculation run with $g = 2$ exactly gives a lead of zero. For a particle whose magnetism is exactly what Dirac’s equation first predicted, the spin stays locked to the direction of motion at every speed, however the field turns it. That is one of the neater facts about the value two: it is the value at which a spin in a magnetic field keeps pace with its own motion. Any measured lead is therefore a direct measurement of how far $g$ is from two, with no need to know the particle’s speed.

A spin lapping its own motion #

In a storage ring the lead is easy to picture. Each time the muon goes round, its spin gains $2\pi a\gamma$ on its direction of motion — 12.3 degrees at $\gamma = 29.3$. After 29.3 turns the spin has gained a full lap. The figure draws snapshots every four turns: the spin starts pointing along the motion and swings steadily round relative to it. The lap takes 4.37 microseconds. A muon at rest lives 2.2 microseconds on average, too short for a single lap; at $\gamma = 29.3$ it lives 64.4, and the spin laps its motion about fifteen times before the muon decays. Time dilation is not a correction to this experiment but the thing that makes it possible.

The speed at which the steering stops mattering #

A real storage ring cannot be a pure magnetic field. Particles that drift slightly up or down must be pushed back, and the ring does it with electric fields from plates above and below the beam. An electric field seen from a moving particle has a magnetic part — the same mixing again — and that part turns the spin too. The Bargmann–Michel–Telegdi equation gives its effect on the spin’s lead: an extra rate proportional to the electric field times $a - 1/(\gamma^2 - 1)$.

The two parts of that coefficient come from opposite effects. The anomaly $a$ is the spin’s extra response to the magnetic field the electric field becomes in the moving frame. The term $1/(\gamma^2 - 1)$ is the extra Thomas precession caused by the electric field’s own push on the particle’s direction. At low speed the Thomas term dominates and the coefficient is negative; at high speed it fades and the anomaly dominates. At one speed, $\gamma = \sqrt{1 + 1/a}$, they cancel exactly, and an electric field of any strength has no effect on the spin’s lead over the motion. For the muon that is $\gamma = 29.30$, a momentum of 3.094 GeV per $c$ — the magic momentum.

Every muon $g-2$ storage ring since the CERN experiment of the 1970s has run at the magic momentum. The ring’s focusing fields could then be electric, strong enough to keep the muons in a stable orbit, without their contributing anything to the quantity being measured. The electron’s magic momentum is 15 MeV per $c$, but electron $g-2$ is measured by other means, with single electrons held in traps, and the magic-momentum trick is a muon speciality.

Measuring the lead, not the spin #

The constant lead is not only a curiosity; it is what makes the measurement precise. The quantity of interest is $a$, the anomaly, about a thousandth of the whole $g$. An experiment that measured the spin’s precession rate and the cyclotron rate separately, and subtracted, would need each to a precision a thousand times finer than the precision wanted in $a$ — a part in ten billion, to get $a$ to a part in ten million. An experiment that measures the lead directly measures $a$ itself, and a part in ten million of the lead is a part in ten million of the answer.

That is why the whole apparatus is built around the difference. The spin and the momentum turn together, nearly, and what is recorded is only how far one has got ahead of the other. The magnetic field sets the scale, and has to be known to the precision wanted; the speed does not need to be known at all, because the lead does not depend on it. A single number, the frequency of the lead divided by the field, gives the anomaly with nothing else to measure — once the electric fields have been silenced by the magic momentum.

A spin frozen to its motion #

The same equation suggests an opposite experiment. In a ring steered only by electric fields, with no magnetic field at all, the spin’s lead over the momentum is proportional to $a - 1/(\gamma^2 - 1)$ alone, and at the magic momentum that is zero: the spin stays locked to the direction of motion, turn after turn, as though the particle’s $g$ were exactly two. This “frozen spin” is the basis of proposed searches for an electric dipole moment of the proton. A proton’s anomaly is large, $a = 1.793$, so its magic Lorentz factor is only 1.248, a momentum of 0.70 GeV per $c$. In an all-electric ring at that momentum a proton’s spin would stay aligned with its motion indefinitely, unless the proton had an electric dipole moment, which would slowly tilt the spin out of the ring’s plane in the radial electric field. Any tilt accumulating over hours would be a sign of a violation of time-reversal symmetry at a level no experiment has reached.

The same lead in a trap #

The electron’s anomaly is measured without any storage ring and without relativistic speeds. A single electron is held for months in a Penning trap — a strong magnetic field with an electric field that confines it along the axis — cooled until it occupies the lowest few quantum states of its circular motion. Its cyclotron motion is quantised into Landau levels, and its spin can point up or down along the field. The cyclotron frequency and the spin-flip frequency differ, again, by $a$ times the cyclotron frequency, and the experiment measures that difference by inducing quantum jumps between states. It is the same lead as in the muon ring, seen at $\gamma = 1$, where the Thomas precession is absent and the spin’s excess over the motion is the anomaly alone.

The most recent such measurement, in 2023, determined the electron’s magnetic moment to about one part in ten trillion, the most precise measurement of any property of an elementary particle. Combined with the theory of quantum electrodynamics, it gives the fine-structure constant, and its agreement with the value from atom-recoil measurements is one of the sharpest tests of that theory.

Reading a spin that nobody can see #

The muon’s spin cannot be observed directly. What makes the measurement possible is the way the muon decays: into a positron and two neutrinos, with the positron thrown out preferentially along the direction of the muon’s spin, a consequence of the weak interaction’s violation of mirror symmetry. In the laboratory, the most energetic positrons are those thrown forward along the muon’s motion, so the number of high-energy positrons reaching detectors round the ring is largest when the spin points along the motion and smallest when it points back.

The count therefore falls exponentially with the muons’ dilated lifetime and oscillates as the spin laps the motion. That oscillation is the famous “wiggle plot”. Its frequency divided by the magnetic field is $a\,e/m$, so a measurement of the frequency and of the field — the field measured with probes that read the precession of protons in water, calibrated in turn against the proton’s own magnetic moment — gives the anomaly. The exponential is time dilation, measured incidentally to a part in ten thousand; the oscillation is $g - 2$.

What the measurement has found #

The anomaly of the muon is computed in the Standard Model of particle physics as a sum of contributions: the dominant one from quantum electrodynamics, known to five loops; a small one from the weak interaction; and a troublesome one from the strong interaction, in which the muon’s field briefly creates and reabsorbs quarks. That last part cannot be computed by expanding in a small coupling, and for two decades it was obtained from measurements of electron–positron collisions producing hadrons.

Measurements at Brookhaven and then at Fermilab, running at the magic momentum in the same ring, reached a precision of 127 parts per billion by 2025. For several years the measured value disagreed with the prediction built on collision data by more than four standard deviations, a discrepancy widely discussed as a possible sign of new particles. Meanwhile calculations of the strong-interaction contribution made on a lattice of spacetime points, with supercomputers, gave a larger value that brought the prediction into line with the measurement, and newer collision data moved towards the lattice results. The prediction adopted in 2025 agrees with the measured anomaly. What remains open is why the older collision data and the lattice disagreed, which is a question about the strong interaction rather than about the muon.

Where the equation stops #

The Bargmann–Michel–Telegdi equation treats the spin as a classical vector carried by a point particle moving in slowly varying fields. For a particle in a storage ring that is an excellent approximation; the quantum corrections to it are far below any measured effect. The magic-momentum cancellation is exact only for particles moving exactly perpendicular to the field at exactly the magic momentum, and a real beam has a spread of momenta and small vertical oscillations; the residual effects of the electric field on particles slightly off the magic momentum, and of vertical motion — the “pitch” correction — are among the corrections the experiments compute and apply, at the level of a few tenths of a part per million. An electric dipole moment, if the muon had one, would add a precession that tilts the spin out of the ring’s plane, and the experiments search for that too.

What the pictures cannot show #

The figures draw the spin as an arrow in the plane of the ring, which is its expectation value; a single muon’s spin is not an arrow but a quantum state, and the wiggle appears only in the average over millions of muons. The ring figure spreads its snapshots round the circle for legibility, when in fact the muon returns to the same point every turn and the spin is simply further round each time. And none of the figures shows the magnetic field’s uniformity, which is where much of the experimental difficulty lies: the field of the 14-metre ring had to be known, averaged over the muons’ paths, to tens of parts per billion.

Still open: what the strong interaction contributes #

The muon’s anomaly is now measured more precisely than it can be predicted, and the limiting uncertainty in the prediction is the contribution of virtual quarks. The lattice calculations and the data-driven estimates, after moving towards each other, still disagree in their details, and new electron–positron data from several colliders disagree among themselves. A separate experiment in Japan plans to measure the muon’s anomaly with a completely different method — ultra-cold muons in a compact ring with no electric focusing at all, so that the magic momentum is not needed — and would test whether anything about the magic-momentum method has hidden a systematic error. Whether the muon’s magnetism hides anything beyond the Standard Model is, for now, a question about how well the strong interaction can be computed.

The habit worth carrying away is to follow every factor of $\gamma$ to the end before trusting an estimate of a relativistic effect. A moving particle sees a magnetic field γ times stronger and clocks it γ times slower, and its rest frame is rotated backwards by the Thomas precession, so the spin’s lead over the momentum is a·eB/m at every speed — and at $\gamma = \sqrt{1 + 1/a}$ the electric fields that steer the particle stop adding to it. A g of exactly two would lock spin to motion; the muon’s lead of one part in a thousand, lapping its motion every 29 turns, is how the difference is measured.