Experiment: The Electron Anomalous Magnetic Moment

Tests Phenomenon 93.3 and Equation (100.62). Assuming Theorem 112.9 and Proposition 112.12.

The electron magnetic moment anomaly \(a_{e}=(g-2)/2\) is the most precisely measured quantity in particle physics and the most precisely computed: the Harvard one-electron quantum cyclotron reports

a fractional uncertainty of about \(1.1\times10^{-10}\) on the anomaly itself [Fan:2023], against a QED series evaluated through tenth order [Aoyama:2019]. That 2023 measurement is the experiment this chapter is built around, and it is the year registered for the chapter in the List of Experiments; the two earlier results from the same apparatus [Odom:2006] [Hanneke:2008] are reported alongside it because the systematic budget is cumulative and the three values are not independent. This chapter is the experimental counterpart of the renormalized perturbation theory of Quantum Electrodynamics and Renormalization: it takes Dirac's \(g=2\) [Dirac:1928], the first measured departure from it [Foley:1948] [Kusch:1948], and Schwinger's one-loop \(a_{e}=\alpha/2\pi\) [Schwinger:1948], and follows the comparison down to the digit at which it currently stops.

Where it stops is the point of the chapter, and it is not where the headline figures suggest. Extracting \(a_{e}\) from theory requires an independently measured fine-structure constant, and the two best determinations — caesium recoil [Parker:2018] and rubidium recoil [Morel:2020] — disagree with each other by several standard deviations, a discrepancy far larger than either quoted uncertainty. The comparison of measurement with theory is therefore verified only to the precision of the worse-behaved input, and the chapter states that bound explicitly rather than quoting agreement to twelve digits. The same discipline is applied to the muon anomaly, where a genuine measurement–theory tension [Bennett:2006] [Abi:2021] [Aguillard:2023] is entangled with a disagreement between lattice and data-driven evaluations of the hadronic vacuum polarization [Borsanyi:2021]. The chapter follows the house experiment pattern of editorial rule 4: apparatus, procedure, observations and data, interpretation, primary references, in that order, with the historical setting and the theoretical prediction placed before the apparatus so that the reader knows what is being tested before meeting the hardware.

Historical context and the prediction under test

Dirac's $g=2$

A magnetic moment attached to an angular momentum is written

the numerical value of the Bohr magneton being the CODATA 2022 recommended one [Mohr:2025]. The dimensionless factor \(g\) is what is at issue. For a classical current loop, or for the orbital motion of a charge in an atom, \(g=1\): this follows from nothing more than the definition of the magnetic moment of a current distribution, \(\vect{\mu}=\tfrac{1}{2}\int\dd^{3}x\,\vect{x}\times\vect{j}\), applied to a charge \(-e\) of mass \(m_{e}\) moving on a closed orbit, whose angular momentum is \(\vect{L}=m_{e}\int\dd^{3}x\,\vect{x}\times \vect{v}\,\rho/(-e)\) with \(\rho\) the charge density.

Spin is not orbital motion, and the factor is not 1. Uhlenbeck and Goudsmit had to postulate \(g=2\) for the electron's intrinsic angular momentum to make the alkali doublets and the anomalous Zeeman effect come out [Uhlenbeck:1925], and Pauli inserted the same factor by hand into the nonrelativistic wave equation [Pauli:1927b]. What Dirac's relativistic equation supplies, without being asked, is exactly that factor [Dirac:1928]. The derivation is carried out in full elsewhere in this book — see Phenomenon 93.3, where squaring the minimally coupled Dirac operator produces the term \(-(q\hbar/2m_{e})\,\vect{\sigma}\cdot\vect{B}\) and the factor of two comes entirely from the algebra of \((\vect{\sigma}\cdot\vect{a})(\vect{\sigma}\cdot\vect{b})\) — and it is not repeated here.

Two features of that result are what make the present chapter possible. First, \(g=2\) is exact in the single-particle theory: it is not fitted, it contains no parameter, and it does not depend on the field strength, the electron's velocity, or anything else that could be adjusted. Second, it is a statement about a free electron, so a measurement on a free electron tests it without any intervening atomic structure. Any measured departure from 2 is therefore a measurement of something that the relativistic wave equation does not contain, and the only candidate in a theory built on it is the interaction of the electron with the quantized electromagnetic field. That is why the anomaly is diagnostic rather than merely a correction.

It is worth being explicit about the size of the effect to be found, and about a sign convention that is easy to count twice. In Equation (112.3) the orientation of the moment relative to the spin is carried by the explicit minus sign, so the \(g\) written there is a positive number, and the anomaly is defined on that magnitude:

CODATA 2022 tabulates the \(g\) factor with its sign included, \(g_{e}=-2.00231930436092(36)\), the sign recording that the moment is antiparallel to the spin for a negative charge [Mohr:2025]. Every \(g\) written in this chapter is \(\abs{g_{e}}\); inserting the tabulated negative value into Equation (112.4) would apply the same sign twice and give \(-2.0012\) instead of the anomaly. The excess over the Dirac value is therefore about one part in \(860\) — large enough to have been found in 1948 with atomic beams, small enough that measuring its own tenth significant figure is a seventy-five-year project.

The hyperfine anomaly and the Kusch–Foley measurement

The excess was not found by measuring the electron moment against an absolute standard. It was found because a set of measurements that assumed \(g_{S}=2\) failed to be self-consistent.

The atomic beam.

Kusch and Foley used the atomic-beam magnetic-resonance method that Rabi's group had developed a decade earlier [Rabi:1937] [Rabi:1938]. An oven produces a thermal beam of neutral atoms — gallium, indium and sodium in the experiments at issue — which passes in vacuum through three magnet regions in series. The first (the \(A\) magnet) has a strong transverse gradient and deflects the atoms according to the sign of their effective moment; the third (the \(B\) magnet) has a gradient of the opposite sense and refocuses onto a detector exactly those atoms whose effective moment is unchanged. Between them sits the \(C\) region: a strong, uniform field \(\vect{B}\) with a small radio-frequency field applied transverse to it. If the radio frequency matches a Zeeman transition frequency of the atom in the \(C\) field, the atom changes state, the \(B\) magnet no longer refocuses it, and the detected beam intensity drops. The observable is a dip in a beam current as a frequency is swept.

Why a ratio evades the calibration problem.

The resonance condition in the \(C\) field is

so an absolute determination of \(g_{F}\) requires knowing \(B\) to the same accuracy as \(\nu\) — and in 1948 a magnetic field could not be measured to anything like a part in \(10^{4}\), whereas a radio frequency could be counted to a part in \(10^{6}\). The way out is to measure two resonances in the same field. The ratio of their frequencies is

in which \(B\), \(h\) and \(\mu_{\mathrm{B}}\) have all cancelled. The field need only be stable for the duration of the two sweeps, not known. This is the same structural trick — measure a ratio of frequencies in a common field — that the Penning-trap experiments of Section 112.3.2 carry to a part in \(10^{13}\), and it is the single most important idea in the whole subject.

The inconsistency.

For an atom in Russell–Saunders coupling, the Landé factor of a level with quantum numbers \(L\), \(S\), \(J\) is a known function of \(g_{L}\) and \(g_{S}\),

with the orbital factor \(g_{L}=1\) up to the known reduced-mass correction.

Because the two coefficients depend on \(L\), \(S\) and \(J\), different levels weight \(g_{S}\) and \(g_{L}\) differently: the \({}^{2}P_{1/2}\) and \({}^{2}P_{3/2}\) levels of gallium and indium, and the \({}^{2}S_{1/2}\) ground state of sodium, give three independent combinations. Measuring the ratios of their \(g\) factors therefore over-determines the pair \((g_{L},g_{S})\), and the system has no solution with \(g_{S}=2g_{L}\).

Kusch and Foley found that the discrepancy is removed by allowing the electron's spin \(g\) factor to exceed twice the orbital one by about \(0.12\,\mathrm{\%}\) [Foley:1948] [Kusch:1948]. In the modern notation that is \(a_{e}\approx1.2\times10^{-3}\), and the reader should notice how coarse the measurement was by present standards and how decisive it was anyway: the quantity being tested was believed exact, so a departure of one part in \(10^{3}\), established at many times its own uncertainty, could not be absorbed into any parameter of the theory. The two 1948 papers are a matched pair — Foley and Kusch [Foley:1948] carries the interpretation and Kusch and Foley [Kusch:1948] the fuller experimental account — and they are cited together throughout this chapter for that reason.

Schwinger's $\alpha/2\pi$

The one-loop correction to the electron–photon vertex splits, by Lorentz invariance and current conservation, into two form factors: \(F_{1}(q^{2})\), which renormalizes the charge, and \(F_{2}(q^{2})\), which multiplies the tensor structure \(\ii\sigma^{\mu\nu}q_{\nu}/ 2m_{e}c\) and therefore contributes an extra magnetic coupling. The anomaly is \(a_{e}=F_{2}(0)\), and at one loop

This chapter does not repeat that calculation: it is Theorem 100.41, proved in full in Quantum Electrodynamics and Renormalization, where the loop integral is regularized, the divergence is shown to sit entirely in \(F_{1}\), and \(F_{2}(0)\) is extracted as a convergent Feynman-parameter integral. What matters here is the epistemic shape of the result, and it is worth stating carefully because it is the reason renormalization became believable rather than merely tolerated.

The one-loop vertex is divergent. The divergence is absorbed into the definition of the electron's charge, which is not predicted by the theory and never was. What is left over, after that absorption, is a pure number times \(\alpha\) — no cutoff, no regulator, no scheme dependence, and no adjustable parameter. Schwinger's number was published in 1948, in the same year and in the same journal volume as the Kusch–Foley measurement, and it agreed with it. A theory that had been widely regarded as a formal device for organizing divergences had produced a parameter-free number that matched an experiment nobody had computed it for. Everything in the rest of this chapter is a seventy-five-year continuation of that single comparison, carried from three significant figures to ten.

The second-order term is worth one further remark. Because Equation (112.9) contains no mass, the same number is the one-loop anomaly of the muon and of the tau. Lepton-universality of the leading term is what makes the electron and the muon anomalies complementary rather than redundant: they differ only in the corrections, and the corrections are what carry information about hadrons and about anything heavier (Section 112.2.2).

What a measurement must deliver

The consequences of this structure govern everything that follows, and they are worth setting out before any hardware is described.

First, the measurement is a frequency measurement, and frequency is the SI quantity that can be realized best: the second is defined by a caesium hyperfine transition and time intervals are routinely compared to a part in \(10^{16}\), far beyond what is needed here (Measurement, SI Units, and the Theory of Errors). Nothing else in the experiment has to be that good.

Second, the difference \(\nu_{a}=\nu_{s}-\nu_{c}=a_{e}\nu_{c}\) is smaller than either frequency by three orders of magnitude, so a fractional error \(\epsilon\) in either \(\nu_{s}\) or \(\nu_{c}\) produces a fractional error of order \(\epsilon/a_{e}\approx860\,\epsilon\) in \(a_{e}\) unless it is common to both. This is the reason \(\nu_{a}\) is measured directly, as the frequency of a transition that flips the spin and lowers the cyclotron quantum number by one, rather than as a difference of two separately measured large numbers. Measured that way, the requirement on \(\nu_{c}\) is only that it be known to about \(a_{e}\) times the target precision.

Third, since \(B\) does not appear, magnetometry is replaced by a different set of systematic effects: whatever makes the trap's frequencies differ from the free-space ones. Those are the electrostatic confining potential itself, the special-relativistic mass increase, the image charges induced in the electrodes, and — dominantly — the coupling of the cyclotron motion to the modes of the conducting cavity that surrounds it. These four are the whole of the systematic problem, and Sections 112.3.2 and 112.3.3 exist to explain how each is controlled; the published line-by-line error budget is not reproduced in this book and must be read in [Fan:2023].

Fourth, and least obviously, the quantity delivered is dimensionless. No SI unit enters the final number at all, which is why \(a_{e}\) can be compared with a calculation without a single conversion factor and why its uncertainty is not limited by the realization of the kilogram, the ampere or the tesla. The uncertainty conventions used to quote it — standard uncertainty, coverage factor 1, correlations propagated explicitly — are those of Measurement, SI Units, and the Theory of Errors.

The theoretical prediction

The QED series through tenth order

Because the one-loop vertex correction of Theorem 100.41 is \(\alpha\) times a dimensionless integral, and because every additional loop adds one factor of the coupling and one dimensionless integral, the anomaly of a lepton is a power series in \(\alpha\) with pure numbers as coefficients:

The expansion parameter is written \(\alpha/\pi\) rather than \(\alpha\) because a factor \(1/\pi\) per loop is what the four-dimensional loop measure \(\dd^{4}\ell/(2\pi\hbar)^{4}\) actually supplies; with \(\alpha/\pi=2.3228\times 10^{-3}\) the terms fall by a factor of a few hundred at each order, which is why ten significant figures need only five of them. Every \(C_{n}\) is a pure number, and every term in Equation (112.13) is dimensionless in SI, as it must be for the sum to be a \(g\) factor.

The coefficients are those adopted in Quantum Electrodynamics and Renormalization, whose provenance is [Aoyama:2019]:

\(C_{1}\) is Schwinger's; \(C_{2}\) and \(C_{3}\) are known in closed form, in terms of \(\zeta(3)\), \(\ln 2\) and \(\pi^{2}\); \(C_{4}\) and \(C_{5}\) are numerical. The counting of Feynman diagrams contributing to the mass-independent part at each order is \(1\), \(7\), \(72\), \(891\) and \(12672\) [Aoyama:2012] [Aoyama:2018] [Aoyama:2019], which is the practical reason the series stops where it does: the tenth-order evaluation is a decade-long numerical project, not an algebraic one.

Two features of Equation (112.14) deserve to be flagged rather than passed over.

The tenth-order term is not negligible, and it is disputed.

With the CODATA 2022 value of \(\alpha\) the fifth-order term contributes \(C_{5}(\alpha/\pi)^{5}=4.556\times 10^{-13}\) to \(a_{e}\), which is three and a half times the experimental uncertainty \(1.3\times 10^{-13}\) of Equation (112.8). The quoted uncertainty on \(C_{5}\) itself, \(\pm0.159\), propagates to \(1.1\times 10^{-14}\), comfortably below the experiment. But that uncertainty is the error of a numerical integration, and an independent recomputation of the largest gauge-invariant subset of the tenth-order diagrams — those without closed lepton loops — does not agree with the published value within the quoted errors [Volkov:2019]. This is recorded here, as it is in Quantum Electrodynamics and Renormalization, because a single unverified numerical computation would otherwise be load-bearing in the most precisely tested prediction in physics. It does not currently change the conclusion of Section 112.7, because the \(\alpha\) input is a larger limitation than any plausible revision of \(C_{5}\); it would matter immediately if the \(\alpha\) inputs were reconciled.

The mass-dependent terms enter at fourth order.

The \(C_{n}\) of Equation (112.14) are the mass-independent coefficients: they are what remains when every internal fermion line is the same lepton as the external one. Diagrams in which a muon or tau loop is inserted into a photon propagator depend on the ratios \(m_{e}/m_{\mu}\) and \(m_{e}/m_{\tau}\) and are collected into \(a_{e}^{\mu,\tau}\). Their leading behaviour is a clean piece of physics and is derived in Section 112.2.2: a loop of mass \(M\) decouples from the anomaly of a lepton of mass \(m_{\ell}\) as \((m_{\ell}/M)^{2}\), so muon loops are suppressed in \(a_{e}\) by \((m_{e}/m_{\mu})^{2}\approx2.34\times 10^{-5}\). The full value from [Aoyama:2019] is

which is twenty times the experimental uncertainty and therefore mandatory. The logarithms \(\ln(m_{\mu}/m_{e})\) that accompany such ratios are the mass singularities analysed by Kinoshita [Kinoshita:1962]; they are finite here because the internal mass is the larger one.

Hadronic and electroweak contributions

The remaining two terms of Equation (112.13) are not QED. The hadronic term collects diagrams in which the virtual photon converts into quarks and gluons — hadronic vacuum polarization at leading order, and hadronic light-by-light scattering beyond it — and the electroweak term collects diagrams containing a \(W\), a \(Z\) or a Higgs boson. For the electron [Aoyama:2019] gives

Neither can be computed from first principles in perturbation theory: the hadronic one because the strong coupling is large at the relevant scale (Quantum Chromodynamics), and it is obtained instead either from measured \(e^{+}e^{-}\to\text{hadrons}\) cross sections through a dispersion relation or from lattice QCD (Remark 102.61). The electroweak one is perturbative but requires the full Standard Model.

The same statement can be checked against a case where the heavy mass is known exactly, which is a useful consistency test of the scaling argument just made. The two-loop contribution to \(a_{e}\) from a vacuum polarization loop of a heavier lepton of mass \(M\) is

which is dimensionless, as every term in Equation (112.13) must be.

With \(M=m_{\mu}\), so that \((m_{e}/m_{\mu})^{2}=2.3390\times 10^{-5}\), this gives \(5.3955\times 10^{-6}\times5.198\times 10^{-7}=2.80\times 10^{-12}\). The full mass-dependent term of Equation (112.15) is \(2.7478\times 10^{-12}\), so the leading \((m_{e}/M)^{2}\) piece accounts for it to two per cent — but note the direction: the leading piece overshoots the full value, and the residual \(2.7478\times 10^{-12}-2.8046\times 10^{-12}=-5.7\times 10^{-14}\) is negative. The same formula with \(M=m_{\tau}\) adds a tau loop of about \(+10^{-14}\), which pushes the other way, so the negative sign must come from the three-loop mass-dependent diagrams, which are the only contribution of the right sign at this size. The decoupling of a heavy state as the square of the mass ratio is therefore not an approximation invented for the hadronic estimate; it is a property of the anomaly, verified here against an independently known mass.

The fine-structure constant as an input

Equation (112.13) is a function of \(\alpha\). It predicts a number only when \(\alpha\) is supplied, and if \(\alpha\) is taken from the anomaly itself the exercise is a definition rather than a test. This subsection sets out where an independent \(\alpha\) comes from and how well it is known; Section 112.7 draws the consequence.

The best independent determinations are atom-recoil measurements, and the route is worth deriving because it shows exactly which quantity carries the uncertainty. The Rydberg constant is, by definition,

the value being CODATA 2022 [Mohr:2025]. Solving for \(\alpha^{2}\) and inserting the mass of an atom of species \(X\) in the obvious place,

Each of the three factors on the right is separately measurable. The first is Equation (112.21), known to \(1.1\times10^{-12}\) fractionally from hydrogen spectroscopy and utterly negligible here. The second, the mass of the atom in units of the electron mass, is a ratio of cyclotron frequencies in a Penning trap — the same instrument as Section 112.3.2, used for a different purpose — and is known to better than \(10^{-10}\). The third, \(h/m_{X}\), has the SI dimensions of \(\mathrm{J}\,\mathrm{s}/\mathrm{kg}\), that is \(\mathrm{m}^{2}/\mathrm{s}\), and is what the recoil experiment measures: an atom that absorbs a photon of wavevector \(k=2\pi/\lambda\) recoils with velocity

and the resulting Doppler shift, accumulated over many photon recoils in an atom interferometer, is counted as a frequency. Since Equation (112.22) gives \(\alpha\) as the square root of \(h/m_{X}\), a fractional uncertainty \(\epsilon\) on \(h/m_{X}\) becomes \(\epsilon/2\) on \(\alpha\): a determination of \(\alpha\) at \(2.0\times 10^{-10}\) requires \(h/m_{X}\) at \(4.0\times 10^{-10}\), which is the actual achievement of these experiments.

The two experiments are not variants of one technique in any way that would make a common systematic likely. Parker and collaborators [Parker:2018] used a caesium atom interferometer with Ramsey–Bordé interferometry and Bragg diffraction; Morel and collaborators [Morel:2020] used rubidium with Bloch oscillations in an accelerated optical lattice to accumulate hundreds of photon recoils. The atomic species differ, the mass ratios \(m_{X}/m_{e}\) come from different measurements, and the interferometric schemes differ. That is what makes the disagreement uncomfortable: it is not obviously attributable to one of them, and the CODATA 2022 adjustment therefore inflates the uncertainty of the recommended value \(\alpha^{-1}=137.035999177(21)\) rather than choosing between them [Mohr:2025]. Note that this recommended value is not independent of \(a_{e}\) — the adjustment uses the anomaly together with the QED series as one of its inputs — which is exactly why it cannot be used to test the series. That distinction is the subject of Section 112.7 and is the single most important statement in this chapter.

Apparatus

Free-electron precession experiments

The first measurements to use Equation (112.10) directly, on free rather than bound electrons, were made at Michigan by Crane and collaborators over the 1950s and early 1960s, culminating in the precision experiment of Wilkinson and Crane [Wilkinson:1963].

The Michigan experiment.

Electrons of about \(100\,\mathrm{keV}\), that is \(1.602\times 10^{-14}\,\mathrm{J}\), of kinetic energy were transversely polarized by Mott scattering from a thin gold foil — the spin–orbit coupling in the Coulomb field of a high-\(Z\) nucleus makes the differential cross section depend on the spin projection normal to the scattering plane, so scattering at a fixed angle selects a polarized beam. The polarized electrons were injected into a region of static, nearly uniform magnetic field of order \(0.1\,\mathrm{T}\), in which a weak axial gradient provided a shallow magnetic bottle that trapped them in a spiral orbit for a controlled dwell time. They were then ejected onto a second gold foil, and the left–right asymmetry of the Mott scattering measured the angle between the spin and the momentum at the moment of ejection.

Why the method works.

This apparatus measures the anomaly by allowing it to accumulate.

The point is the factor \(N\). A single orbit turns the spin by \(2\pi a_{e}=7.29\times 10^{-3}\) radians, far too small to measure against the analysing power of a Mott polarimeter; after \(N=10^{5}\) orbits it is \(\theta=728.6\,\mathrm{rad}\), which is \(a_{e}N=116\) full turns — a hundred and more — and the asymmetry oscillates that many times as the dwell time is swept, so the frequency of that oscillation — which is \(\nu_{a}\) — is what is fitted. The measurement has become a frequency measurement again, and the magnetic field appears only through \(\nu_{c}\), which is measured in the same apparatus. Wilkinson and Crane reached a fractional uncertainty of a few parts in \(10^{5}\) on \(a_{e}\) [Wilkinson:1963] — their \(a_{e}=1159.622(27)\times10^{-6}\) is \(2.3\times 10^{-5}\) of itself, which on the customary \(g/2\) scale is the \(2.7\times 10^{-8}\) more often quoted. That is enough to test the two-loop coefficient \(C_{2}\) of Equation (112.14), whose term is \(1.5\times 10^{-3}\) of \(a_{e}\) and therefore about sixty-five times the uncertainty, and not enough to reach the three-loop term, which is \(1.3\times 10^{-5}\) of \(a_{e}\) and sits half a standard deviation away.

What limited it.

Three things, all structural. The electrons are unconfined radially over a large volume, so \(B\) in Equation (112.11) is an average over an orbit whose extent is not precisely known; the trapping bottle needed to hold them is itself a field inhomogeneity; and the polarimetry is destructive and statistically inefficient, so each electron yields one bit and the ensemble must be re-prepared. All three are removed at once by confining a single electron in a small, well-characterized volume and reading its state nondestructively — which is the Penning trap.

The Penning trap

A Penning trap confines a charged particle using a uniform magnetic field \(\vect{B}=B\hat{\vect{z}}\) for radial confinement and a static quadrupole electrostatic potential for axial confinement. A magnetic field alone cannot confine along \(\hat{\vect{z}}\), and by Earnshaw's theorem no static electric field can confine in all three directions; the combination works because the electric field need only confine in one direction while defeating the radial anti-confinement that Laplace's equation forces on it.

The fields.

The electrode surfaces are shaped to produce, near the centre,

where \(V_{0}\) is the signed potential difference applied between the ring and the endcaps, \(z_{0}\) is the axial half-spacing and \(\rho_{0}\) the inner radius; \(d\) is the characteristic trap dimension, with SI units of metres. Equation (112.27) satisfies \(\nabla^{2}\Phi=0\), as it must, and the minus sign in front of \(\rho^{2}/2\) is the price: the potential that confines axially pushes the particle out radially. For a particle of charge \(-e\) the axial potential energy is \(-e\Phi=-eV_{0}z^{2}/2d^{2}\) on the axis, so trapping requires \(V_{0}<0\); the trap depth is \(\abs{V_{0}}\) and only that magnitude appears in the mode frequencies below.

The fast root \(\omega_{+}\) is the trap-modified cyclotron frequency, the slow root \(\omega_{-}\) the magnetron frequency; expanding Equation (112.28) for \(\omega_{z}\ll\omega_{c}\),

The magnetron motion is a slow \(\vect{E}\times\vect{B}\) drift of the orbit centre around the trap axis. It is a metastable mode: its energy decreases as its radius grows, so damping it would expel the electron, and it must instead be driven inwards by sideband coupling to the axial motion. In the Harvard trap the three frequencies are separated by six orders of magnitude, which is what makes them independently addressable. With \(B=5.32\,\mathrm{T}\) and \(\nu_{z}=200\,\mathrm{MHz}\),

and the anomaly frequency to be measured is \(\nu_{a}=a_{e}\nu_{c}=172.7\,\mathrm{MHz}\), close to \(\nu_{z}\). That proximity is a design choice and not a coincidence, but the dependence runs one way only: \(\nu_{a}=a_{e}eB/2\pi m_{e}\) is fixed by \(B\) alone, and \(V_{0}\) is then chosen so that \(\nu_{z}=\sqrt{e\abs{V_{0}}/m_{e}d^{2}}/2\pi\) falls near the \(\nu_{a}\) that the field has already determined. Only \(\nu_{z}\) is set by \(V_{0}\).

The invariance theorem.

A real trap is not the ideal one of Equation (112.27): its magnetic and electric axes are not exactly aligned, and the quadrupole has a small ellipticity. Both perturb \(\omega_{+}\), \(\omega_{z}\) and \(\omega_{-}\) individually, at a level far above the target precision. What rescues the measurement is that one particular combination is untouched.

Numerically the theorem matters less than it might appear, because in the ideal trap \(\omega_{+}\omega_{-}=\omega_{z}^{2}/2\) lets Equation (112.32) be rewritten as \(\omega_{c}=\omega_{+}+\omega_{z}^{2}/(2\omega_{+})\) with no approximation at all; what the general theorem buys is that the same combination survives a tilt and an ellipticity. The correction term \(\nu_{z}^{2}/(2\nu_{+})\) is \(134\,\mathrm{kHz}\) out of \(148.9\,\mathrm{GHz}\), a fractional shift of \(9\times10^{-7}\) — a million times the target precision, so it must be applied, but it depends only on \(\nu_{z}\), which is measured continuously.

The continuous Stern–Gerlach effect.

The remaining problem is reading the spin. A single electron emits no light one can collect, and a Stern–Gerlach deflection of a charged particle is defeated by the Lorentz force. Dehmelt's solution is to make the axial frequency depend on the spin state, by superimposing on the uniform field a small magnetic bottle

produced by a ring of ferromagnetic material (nickel) in the trap electrodes. In the Harvard trap \(B_{2}\approx1540\,\mathrm{T}/\mathrm{m}^{2}\).

Four hertz in two hundred megahertz is two parts in \(10^{8}\), and the axial motion can be tracked to far better than that, so a single spin flip of a single electron is a resolved, nondestructive, repeatable observation. The electron is not destroyed by the measurement and can be interrogated for months. That is what the phrase “quantum jump spectroscopy” in Section 112.4 means.

The same apparatus loads a positron as readily as an electron, and Van Dyck, Schwinberg and Dehmelt compared the two \(g\) factors in the same trap, in the same field, with the same readout [VanDyck:1987]. Because everything common to the two measurements cancels in the ratio, the bound on the difference is far sharper than the uncertainty on either value — the same structural argument as Equation (112.6). The result and its standing as a \(CPT\) test are in Section 112.5.2.

The cylindrical trap and the one-electron quantum cyclotron

The Harvard apparatus that produced Equation (112.8) is a Penning trap with three further properties, each of which removes one item that would otherwise dominate the error budget.

Cylindrical electrodes, so the cavity is calculable.

The classical Penning trap uses hyperboloidal electrodes, which generate Equation (112.27) exactly but whose enclosed volume is a cavity whose electromagnetic modes cannot be written down. The Harvard trap instead uses a stack of right circular cylinders — a ring, two endcaps and two compensation electrodes, gold-plated, of inner radius about \(4.6\,\mathrm{mm}\) and comparable total length — with the compensation electrodes tuned so that the leading anharmonic term of the potential vanishes and Equation (112.27) is recovered near the centre. The price is a slightly less perfect potential; the prize is that the enclosed volume is a right circular cylindrical cavity, whose TE and TM modes are Bessel functions with analytically known frequencies. The cavity shift of the cyclotron frequency can therefore be calculated and corrected, rather than merely bounded [Odom:2006].

That shift is not small on the scale of the measurement. The cyclotron motion is a harmonic oscillator coupled to the modes of its enclosure, and the coupling both damps it and shifts its frequency. The damping is easily estimated. In free space a cyclotron orbit radiates at the Larmor rate, giving a damping constant

where \(r_{e}\) is the classical electron radius, itself built from the same combination \(e^{2}/4\pi\varepsilon_{0}\) that appears in \(\alpha=e^{2}/(4\pi\varepsilon_{0}\hbar c)\).

With \(\omega_{c}=2\pi\times148.9\,\mathrm{GHz}\) this gives \(\gamma_{c}=11.0\,/\mathrm{s}\), a free-space lifetime of \(0.091\,\mathrm{s}\) for the first excited cyclotron state. Inside the cavity, when \(\nu_{c}\) is detuned from every mode, the density of states into which the electron can radiate is suppressed and the lifetime is lengthened by an order of magnitude. This is inhibited spontaneous emission — the counterpart of Purcell's cavity enhancement [Purcell:1946b], observed directly for a Rydberg atom between conducting plates [Hulet:1985] — and it is used here as an instrument rather than studied. Its general treatment belongs to the cavity-QED section of Quantum Optics and the Photon, which reserves the heading but does not yet carry the derivation. A cyclotron excited state that survives for seconds can be observed by watching the axial frequency, which is the whole measurement.

The frequency shift is the dispersive counterpart of the same coupling and is of order a part in \(10^{9}\) of \(\nu_{c}\), i.e. of order \(10^{-6}\) of \(a_{e}\): enormous compared with the target. Because the mode structure of a cylinder is known, the shift can be mapped by measuring the cyclotron damping rate as a function of \(B\) — the rate peaks where \(\nu_{c}\) crosses a mode — and the measurement is then made at fields chosen to sit between modes. The cavity shift and its correction remain the largest single systematic of the experiment; this book does not reproduce the published line-by-line budget, which is in [Hanneke:2008] [Fan:2023].

A dilution refrigerator, so the cyclotron motion is in its ground state.

The trap is operated at about \(100\,\mathrm{mK}\) in a dilution refrigerator, with the electrodes, the vacuum enclosure and the detection amplifier all at that temperature.

The consequence is that the electron sits in \(n=0\) and stays there: a thermally excited cyclotron quantum appears roughly once in \(10^{31}\) attempts, so any observed excitation was driven. This is what converts a classical oscillator into a two-level system that can be addressed one quantum at a time, and it is the difference between the \(100\,\mathrm{mK}\) apparatus and the \(4\,\mathrm{K}\) traps that preceded it. The same Bose–Einstein formula at \(T=4.0\,\mathrm{K}\) gives \(\bar n=1/(\ee^{7.147/4.0}-1)=0.201\), and at \(T=4.2\,\mathrm{K}\) it gives \(0.223\): about one excitation in five, so the oscillator spends a fifth of its time out of the ground state and the cyclotron frequency is smeared by the thermally fluctuating relativistic mass shift. Two tenths is not a small number when the shift per quantum is \(\delta=179\,\mathrm{Hz}\), and it is the factor \(2\times 10^{30}\) between \(0.201\) and \(9\times10^{-32}\) — the exponential \(\ee^{-7.147/T}\) read at the two temperatures — that turns the oscillator into a two-level system.

The same cryogenics supply the vacuum. The trap can is sealed at room temperature and cooled; every residual gas except helium freezes onto the walls, and the pressure falls below about \(10^{-14}\,\mathrm{Pa}\). At that pressure the collision rate with background gas is such that a single electron has been held and interrogated continuously for months, which is what makes an experiment consisting of hundreds of thousands of individual quantum jumps possible at all.

A self-excited oscillator, so the axial frequency is read continuously.

The axial motion induces an image current in the endcap electrodes, which is amplified and, in the Harvard implementation, fed back to the electrode with a controlled phase and an amplitude-limiting gain. The electron's axial motion then self-oscillates at its own frequency, at a large and stable amplitude, and \(\nu_{z}\) is read out as the frequency of a strong signal rather than inferred from a weak damped ring-down. The signal-to-noise gain is what allows the \(4\,\mathrm{Hz}\) shift of Proposition 112.10 to be detected in a fraction of a second, and it is one of the refinements separating [Hanneke:2008] from [Odom:2006].

The 2023 measurement [Fan:2023] adds no new principle to this list. Its improvement by a factor of \(2.2\) over 2008 comes from better thermal isolation and control of the trap electrodes, longer averaging, a more thorough mapping of the cavity modes, and the accumulated understanding of an apparatus that had by then been operated for two decades. That is worth saying plainly: the most precise measurement in physics was improved not by a new idea but by twenty years of attention to the same one.

Procedure

Loading exactly one electron.

Electrons are produced by a field-emission point held at the trap electrode potential, or by driving a sharp tungsten tip until it emits; a burst of them is captured by pulsing the endcap potential. The resulting cloud contains an unknown number \(N\) of electrons, and the next step is to reduce it to one. The image current driven in the detection circuit by the axial motion is proportional to \(N\), so the amplitude of the axial resonance signal falls in discrete, equal steps as the trap depth is lowered and electrons are ejected one at a time. The last step before the signal vanishes identifies \(N=1\) unambiguously: this is a counting measurement, not an inference from a calibrated amplitude, and it is one of the reasons the experiment can claim to be about a single particle.

Cooling the three modes.

Each mode is brought under control by a different mechanism, and all three must be cold because each couples back to the frequencies being measured through the magnetic bottle of Equation (112.33).

• Cyclotron. Damped radiatively into the cavity at the rate of Equation (112.35), as modified by the mode structure. Since the cavity walls are at \(100\,\mathrm{mK}\), the motion thermalizes to the ground state by Proposition 112.11, in a few seconds.

• Axial. Damped resistively by the image current it drives through the detection resistor, which is also at \(100\,\mathrm{mK}\); the damping time is of order \(0.2\,\mathrm{s}\). This is also the readout channel.

• Magnetron. Cannot be damped — see the discussion after Equation (112.30) — and is instead cooled by driving the axial sideband at \(\nu_{z}+\nu_{-}\), which couples the two motions so that axial damping removes magnetron action, shrinking the magnetron orbit towards the trap axis.

Quantum jump spectroscopy.

The electron sits in the ground state \(\ket{n=0,m=-\tfrac{1}{2}}\), with \(m\) labelling the two spin projections and the level scheme as in Equation (112.37). Two drives are applied and the axial frequency is watched for the \(4\,\mathrm{Hz}\) step of Proposition 112.10:

• a microwave drive near \(\bar\nu_{+}=148.9\,\mathrm{GHz}\), injected through a waveguide, which raises the cyclotron quantum number by one at fixed \(m\);

• an anomaly drive near \(\bar\nu_{a}=172.7\,\mathrm{MHz}\), applied as an oscillating potential on a trap electrode, which drives the electron axially through the bottle field of Equation (112.33). In the electron's frame this is an oscillating transverse magnetic field at \(\bar\nu_{a}\), and it drives the simultaneous transition \(\ket{1,-\tfrac12}\to\ket{0,+\tfrac12}\) — a spin flip accompanied by the loss of one cyclotron quantum.

A spin flip is the signature, because it alone leaves the axial frequency permanently shifted while the cyclotron excitation decays back. The cyclotron line entering the denominator of Equation (112.36) is the one measured with the spin in the upper state, \(\ket{0,+\tfrac12}\to\ket{1,+\tfrac12}\); that choice is what fixes the coefficient of the relativistic correction there at \(\tfrac32\delta\) rather than \(\tfrac12\delta\), as the proof of Proposition 112.12 shows. Each attempt yields one bit: jump or no jump. The drive frequency is stepped, several hundred attempts are made at each step, and the fraction that produce a jump traces the resonance line. Because the electron is not consumed, the same electron supplies every point of every line, which removes the particle-to-particle systematics that limited Section 112.3.1.

The lineshape, which is the dominant modelling input.

The magnetic bottle that makes the readout possible also broadens the lines. The axial motion samples the bottle, so the field the cyclotron and spin motions experience is \(B+B_{2}\avg{z^{2}}\), and \(\avg{z^{2}}\) fluctuates with the axial energy. The result is a resonance whose width is set by the axial temperature and whose shape is asymmetric, with a tail on the high-frequency side, because \(\avg{z^{2}}\geq0\) can only raise the field. The line centre is therefore not the line maximum, and extracting it requires the lineshape model of [Brown:1986], in which the frequency executes a random walk driven by the axial energy's thermal fluctuations. Two limits of that model are used and their difference is taken as a systematic uncertainty. This is the single largest piece of theory entering the experimental number and the honest place to look for an unrecognized error.

Interleaving, so that field drift cancels.

The superconducting solenoid drifts. Its field is regulated and the apparatus is temperature-stabilized, but a drift of a part in \(10^{9}\) over an hour would still swamp the target. The two drives are therefore interleaved on a timescale of minutes, so that \(\bar\nu_{a}\) and \(\bar\nu_{+}\) are measured in what is effectively the same field, and the ratio of Equation (112.10) is formed from contemporaneous data. A residual linear drift is fitted and removed; the check is that the extracted \(a_{e}\) is independent of the interleaving period.

The working formula.

What is measured is not \(\nu_{s}\) and \(\nu_{c}\) but \(\bar\nu_{a}\), \(\bar\nu_{+}\) and \(\bar\nu_{z}\), and the free-space \(\nu_{c}\) has to be reconstructed from them.

The sizes are worth recording, because they show which corrections matter. At \(B=5.32\,\mathrm{T}\), \(h\nu_{c}/m_{e}c^{2}=1.21\times10^{-9}\), so \(\delta=179\,\mathrm{Hz}\) and \(\tfrac32\delta=269\,\mathrm{Hz}\), against \(\bar\nu_{+}=148.9\,\mathrm{GHz}\): a fractional correction of \(1.8\times10^{-9}\), which is \(1.6\times10^{-6}\) of \(a_{e}\) and four orders of magnitude above the target. The magnetron term \(\bar\nu_{z}^{2}/2\bar\nu_{+}=134\,\mathrm{kHz}\) is larger still. Neither is a source of uncertainty, because both are computed from measured quantities; the cavity shift is, because it is computed from a model of the cavity.

Blind analysis.

An unknown offset, drawn from a distribution wide compared with the expected uncertainty and unknown to the analysts, is added in software to the extracted \(\bar\nu_{a}\). All lineshape choices, cuts, drift models and systematic corrections are fixed with the offset in place; only then is it removed and the value revealed. This protects against the specific failure mode of a measurement whose expected answer is known to ten digits in advance, which is otherwise very hard to guard against and which the 2008 and 2023 analyses both took seriously [Hanneke:2008] [Fan:2023].

Observations and data

The measured electron anomaly

Table 112.1 is the chapter's primary data, and three features of it need comment.

Two different fractional precisions are in circulation.

The experimental papers quote parts per trillion on \(g/2\); this book quotes the fractional uncertainty on \(a_{e}\), because that is what is compared with a calculation. The two differ by the factor \((g/2)/a_{e}=(1+a_{e})/a_{e}=863\). So [Fan:2023]'s “\(0.13\) parts per trillion” is \(1.3\times10^{-13}\) absolute, which is \(1.1\times10^{-10}\) of \(a_{e}\) — about \(0.11\) parts per billion. Likewise [Hanneke:2008]'s \(0.28\) parts per trillion is \(0.24\) parts per billion of \(a_{e}\), and [Odom:2006]'s \(0.76\) parts per trillion is \(0.66\) parts per billion. Quoting the smaller number is not wrong, but a reader who compares it against the theoretical uncertainty — which is naturally expressed on \(a_{e}\) — without converting will be out by nearly three orders of magnitude.

The three Harvard values are consistent and correlated.

The 2023 value lies \(1.4\times10^{-13}\) below the 2008 value, about half of the 2008 standard uncertainty, and \(2.6\times10^{-13}\) below the 2006 value, a third of its uncertainty. They are not independent measurements that happen to agree: the same trap, the same solenoid and much of the same systematic analysis are shared, so the correct statement is that the successive refinements did not move the number, which is a weaker but still meaningful check.

The CODATA value is not an independent measurement.

The 2022 adjustment value \(1.15965218046(18)\times10^{-3}\) [Mohr:2025] sits \(1.3\times10^{-13}\) below [Fan:2023], within the combined uncertainty. It must not be used to test the QED series, because the series is one of the inputs from which it was derived. This is the circularity that Section 112.7 is about.

Table 112.2 makes the structure of the prediction visible, and the count of figures each order buys is worth doing rather than guessing. Comparing each running sum with the measured \(1.15965218059\times10^{-3}\): one loop is off by \(1.5\times 10^{-3}\) fractionally, so it agrees to three significant figures; two loops by \(1.3\times 10^{-5}\), five figures; three loops by \(4.4\times 10^{-8}\), seven figures. Beyond that the terms themselves say where they land. The fourth-order term is \(4.8\times 10^{-8}\) of \(a_{e}\) and therefore occupies the eighth figure; the muon-loop and hadronic terms (\(2.4\times 10^{-9}\) and \(1.5\times 10^{-9}\) of \(a_{e}\)) the ninth; the fifth-order term (\(3.9\times 10^{-10}\)) the tenth, which is where the experiment's own \(1.1\times 10^{-10}\) now lives; and the electroweak term (\(2.6\times 10^{-11}\)) the eleventh, below it. Notice that \(a_{e}^{\mu,\tau}+a_{e}^{\mathrm{had}}=4.44\times 10^{-12}\) is thirty-four times the experimental uncertainty: the electron anomaly is no longer a pure QED observable, and a comparison that omitted the hadronic term would fail by an amount easily resolved.

Positrons and the CPT test

The muon anomaly

The muon anomaly is included in this chapter as a contrast, not as a digression. It is the same observable computed from the same series, measured by an entirely different technique, and its status is the opposite of the electron's: the experiment is nearly two thousand times less precise in fractional terms (\(2.1\times 10^{-7}\) against \(1.1\times 10^{-10}\), from Tables 112.1 and 112.4), the theory is far less certain, and the comparison currently does not agree. Setting the two side by side is the clearest way to show what a precision test actually consists of.

CERN and Brookhaven

A muon lives for

the width \(\Gamma_{\mu}=2.9959836\times 10^{-19}\,\mathrm{GeV} =4.800\times 10^{-29}\,\mathrm{J}\) being the PDG value [Navas:2024]. Two microseconds is not long enough to trap a muon and interrogate it for months, so the whole strategy of Section 112.3.2 is unavailable. Instead the muon is stored relativistically in a ring, and the anomaly is read out exactly as in Proposition 112.7: as an angle that accumulates between the spin and the momentum.

Storage rings at CERN.

Three generations of muon storage ring were built at CERN between 1961 and 1977. The final one stored polarized muons of both signs at a momentum of about \(3.1\,\mathrm{GeV}\)\(/c\) in a ring of \(14\,\mathrm{m}\) diameter, and reported the anomaly to about seven parts per million — together with a direct measurement of relativistic time dilation, using the same stored muons, which is written up as an experiment in its own right at Section 41.4 [Bailey:1977] [Bailey:1979]. The readout exploits the parity violation of the weak interaction (Experiment: Parity Violation): in the decay \(\mu^{+}\to e^{+}\nu_{e}\bar\nu_{\mu}\) the highest-energy positrons are emitted preferentially along the muon spin, so counting decay positrons above an energy threshold as a function of time gives a decaying exponential with a superimposed oscillation, and the oscillation frequency is \(\nu_{a}\).

The magic momentum.

A ring needs electrostatic quadrupoles for vertical focusing, and an electric field spoils Equation (112.10): a moving magnetic moment in an electric field precesses too. The Bargmann–Michel–Telegdi equation [Bargmann:1959] gives the precession of the spin relative to the momentum, for a particle of charge \(q\) and mass \(m\) moving with \(\vect{\beta}=\vect{v}/c\) perpendicular to \(\vect{B}\), as

with \(\gamma=(1-\beta^{2})^{-1/2}\) and \(a\) the anomaly of the particle in question. Both terms have SI dimensions of \(\mathrm{rad}/\mathrm{s}\): \(qB/m\) is a cyclotron frequency, and \(q\beta E/(mc)\) is the same thing with \(E/c\) playing the part of a magnetic field, which is the correct SI statement of the fact that \(\vect{E}/c\) and \(\vect{B}\) are components of one field tensor.

At that momentum, with the ring's magnetic field and radius, the kinematics is fixed and can be checked against itself. Brookhaven's E821 — and, later, Fermilab's E989, which reuses the same magnet — has a storage radius \(r=7.112\,\mathrm{m}\) and a field \(B=1.4513\,\mathrm{T}\), and the momentum a ring of that geometry selects is

which is Proposition 112.14 to five figures. The ring was built to the magic momentum, and the agreement of Equation (112.41) with Proposition 112.14 is a consistency check on the numbers quoted here rather than an independent fact.

Brookhaven's E821 measured \(a_{\mu}\) to \(0.54\) parts per million and reported a value about \(2.5\) standard deviations above the Standard Model evaluation then current [Bennett:2006]. That discrepancy is what motivated everything in Sections 112.6.2 and 112.6.3.

Fermilab E989

The E821 storage ring — a \(14.2\,\mathrm{m}\) diameter superconducting magnet, too large to disassemble — was transported intact from Brookhaven to Fermilab in 2013 and rebuilt there with a far more intense and cleaner muon beam, a completely new detector system, and roughly a threefold improvement in field uniformity. The experimental scales are worth listing, since each is derived above:

• cyclotron period \(T_{c}=2\pi r/\beta c=149\,\mathrm{ns}\);

• anomalous precession period \(1/\nu_{a}=4.37\,\mu\mathrm{s}\), from \(\nu_{a}=a_{\mu}eB/2\pi m_{\mu}=229\,\mathrm{kHz}\), so the spin turns once relative to the momentum every 29 turns of the ring;

• dilated muon lifetime \(\gamma\tau_{\mu}=29.3034\times2.197\,\mu\mathrm{s} =64.4\,\mu\mathrm{s}\), giving about 15 anomalous precession cycles per lifetime and a usable observation window of several hundred microseconds.

Two frequencies are measured. The first is \(\omega_{a}\), from the oscillation in the decay-positron time spectrum described above. The second is not \(B\) but \(\omega_{p}\), the Larmor precession frequency of a free proton in the same field, measured by pulsed nuclear magnetic resonance in water-based probes on a trolley that is driven around the storage region to map the field, cross-calibrated against a spherical-water standard. The anomaly is then extracted as

a product of one measured ratio with three externally known ones, each taken as a magnitude in the sense of Equation (112.4). Note what has happened: the field has again cancelled, and the price is that the muon anomaly depends on the proton-to-electron moment ratio, the muon-to-electron mass ratio and the electron \(g\) factor — the last of which is the subject of the rest of this chapter. The two measurements are not as independent as they look.

As at Harvard, the analysis is blind: a frequency offset unknown to the analysis teams is applied to the reference clock in hardware, and the several independent analysis groups agree their result before it is removed. The published values are \(0.46\) parts per million from the first data set [Abi:2021] and \(0.20\) parts per million from the second and third [Aguillard:2023]; they agree with each other and with Brookhaven.

Hadronic vacuum polarization: lattice versus data-driven

By Phenomenon 112.4 the hadronic contribution to \(a_{\mu}\) is some \(4.3\times 10^{4}\) times that to \(a_{e}\), which puts it at about \(7\times10^{-8}\) — three hundred times the experimental uncertainty of Table 112.4. It is therefore the dominant theoretical uncertainty, and since it cannot be computed perturbatively (Quantum Chromodynamics) it must be obtained some other way.

The dispersive route.

The photon self-energy is an analytic function of \(s=(pc)^{2}\) with a cut along the positive real axis, and the discontinuity across the cut is fixed by the optical theorem to be the total cross section for the photon to make hadrons — which is measured, in \(e^{+}e^{-}\) annihilation, as the ratio \(R(s)\) of Phenomenon 102.15. A dispersion relation then converts the measured cross section into the contribution to the anomaly:

with \(s_{\mathrm{th}}=4m_{\pi}^{2}c^{4}\) the two-pion threshold, \(R\) and \(K\) both dimensionless, and \(s\) carrying SI dimensions of \(\mathrm{J}^{2}\). The asymptotic form of the kernel is where the \(m_{\ell}^{2}\) scaling of Equation (112.17) comes from, and the \(1/s\) weighting is why the integral is dominated by the lowest resonances: about \(73\,\mathrm{\%}\) of the muon integral comes from the \(\rho\) resonance region, that is from \(e^{+}e^{-}\to\pi^{+}\pi^{-}\) below \(1\,\mathrm{GeV}\), equivalently \(1.602\times 10^{-10}\,\mathrm{J}\). The factor \(1/3\) in the asymptotic kernel is the same one that carried the derivation of Equation (112.18), where the known two-loop coefficient \(1/45\) fixes it independently.

The two evaluations disagree.

Carrying out Equation (112.43) over the compiled world data gives the leading-order hadronic term \(a_{\mu}^{\mathrm{HVP,LO}}=693.1(4.0)\times10^{-10}\) [Aoyama:2020], which is the input to the data-driven prediction in Table 112.4. Computing the same quantity directly from the Euclidean correlator on the lattice, without any cross-section data, gives \(707.5(5.5)\times10^{-10}\) [Borsanyi:2021]. The difference, \(14.4(6.8)\times10^{-10}\), is about \(2.1\) standard deviations between two calculations of one number, and it is nearly the whole of the measurement–prediction gap. Substituting the lattice value moves the prediction from the \(1.16591810(43)\times10^{-3}\) of Table 112.4 to about \(1.16591954\times10^{-3}\), and against the Fermilab measurement \(1.16592055(24)\times10^{-3}\) of that same table [Aguillard:2023] the difference falls from \(2.45\times 10^{-9}\), which is \(5.0\) combined standard deviations, to \(1.01\times 10^{-9}\), which is \(1.6\). No world average is used here: both numbers are rows of Table 112.4, so the arithmetic can be checked against the table.

And the input data are themselves in conflict.

The CMD-3 experiment at Novosibirsk measured the \(e^{+}e^{-}\to\pi^{+}\pi^{-}\) cross section from threshold to \(1.2\,\mathrm{GeV}\), that is \(1.923\times 10^{-10}\,\mathrm{J}\), and obtained a result several per cent above every previous measurement in the dominant \(\rho\) region [Ignatov:2024]. Used in Equation (112.43) it would raise the data-driven \(a_{\mu}^{\mathrm{HVP,LO}}\) towards the lattice value and remove much of the discrepancy — but it is inconsistent with the earlier data sets, so it cannot simply be averaged in. The honest description of the present situation is therefore a chain of three disagreements: between two theoretical evaluations of the hadronic term, between two sets of hadronic cross-section measurements, and between the resulting prediction and the muon measurement. Only the last of these would be evidence of anything new, and it cannot be assessed until the first two are settled.

The subleading hadronic light-by-light term, of order \(9\times10^{-10}\), carries an uncertainty of about \(2\times10^{-10}\) [Aoyama:2020]: smaller than the vacuum polarization discrepancy, but not negligible, and it is not constrained by any dispersion relation as directly.

Status

Three things would settle the question, and none of them is a new theory.

1. Independent lattice computations of the same quantity. The lattice result [Borsanyi:2021] is one collaboration's. The sharpest way to test it is not to recompute the whole integral but to compare restricted “window” observables — the correlator integrated over a bounded range of Euclidean time — which are cleaner on the lattice and can also be extracted from the cross-section data. Several groups now do this, and it localizes the disagreement in Euclidean time rather than merely confirming it.

2. A resolution of the two-pion cross-section conflict. Either the CMD-3 measurement [Ignatov:2024] or the earlier body of data is wrong, and the discrepancy is large enough that it should be resolvable by further measurements with different systematics.

3. The complete Fermilab data set. The measurement in Table 112.4 uses part of the collected data; the final analysis roughly halves the experimental uncertainty again. This sharpens the comparison but cannot resolve it, because the limitation is theoretical.

A fourth route deserves mention because it is orthogonal to all three: measuring the hadronic contribution to the running of \(\alpha\) directly in elastic muon–electron scattering, which would give \(a_{\mu}^{\mathrm{HVP,LO}}\) from a single spacelike measurement with systematics unrelated to either the annihilation data or the lattice. A proposal of this kind exists at CERN; no primary reference for it is available in this book's bibliography, and the claim is therefore made without a citation rather than with an invented one.

What must not be said is that the muon anomaly shows physics beyond the Standard Model. Under this book's first editorial rule the tension is reported, not interpreted: the measurements agree with one another, the theoretical evaluations do not agree with one another, and the difference between the two theoretical evaluations is comparable to the difference between theory and experiment. The open problems chapter What We Observe but Do Not Understand collects the questions this book regards as unresolved; the muon anomaly is not among the entries currently written there, and on the reading given here it does not yet qualify — it is an unresolved calculation, not an unexplained observation.

Interpretation

The question this section answers is: how many digits are actually verified? The headline numbers invite the answer “twelve”, and that answer is wrong. The comparison has to be run in both directions before it can be stated honestly.

Direction one: theory plus $\alpha$ against the measured $a_{e}$

Evaluate Equation (112.13), with the coefficients of Equation (112.14) and the non-QED terms of Equations (112.15) and (112.16), at each of the independently measured values of \(\alpha\) in Table 112.3, and subtract the measured \(a_{e}\) of Equation (112.8).

Table 112.5 is the chapter's central result, and it should be read slowly. The same measurement, the same series and the same non-QED corrections give a discrepancy of \(3.9\) standard deviations with one input for \(\alpha\) and \(2.1\) with the other. Neither is a comfortable agreement; they are on opposite sides of the measurement; and the difference between the two rows is entirely due to the disagreement between two atom-recoil experiments, which is a problem in atomic physics and has nothing whatever to do with QED.

Direction two: the measurement plus theory against $\alpha$

The same three ingredients can be arranged differently. Take the QED series as correct, insert the measured \(a_{e}\), and solve Equation (112.13) for \(\alpha\). Inverting the series numerically with the coefficients of Equation (112.14) at the measured value of Equation (112.8) gives

the uncertainty being the experimental \(1.3\times10^{-13}\) on \(a_{e}\) propagated back through \(\delta\alpha=2\pi\,\delta a_{e}\). That is a fractional uncertainty of \(1.1\times10^{-10}\), and it is worth checking that figure against Table 112.3 rather than repeating the claim usually attached to it. The three fractional uncertainties on \(\alpha^{-1}\) are \(1.5\times 10^{-8}/137.036= 1.09\times 10^{-10}\) here, \(2.7\times 10^{-8}/137.036=1.97\times 10^{-10}\) for the caesium recoil [Parker:2018], and \(1.1\times 10^{-8}/137.036=8.03\times 10^{-11}\) for the rubidium one [Morel:2020]. So the anomaly route is \(1.8\) times sharper than caesium and \(1.4\) times blunter than rubidium: it is competitive with the best recoil measurement, not better than it, and Table 112.3 says so already in its \(\delta a_{e}\) column, where the rubidium entry \(9.32\times 10^{-14}\) sits below the experimental \(1.3\times 10^{-13}\) that this paragraph propagates.

It is also not a test of anything. Equation (112.44) assumes the QED series in order to produce \(\alpha\); using it to check the QED series would be circular, and the same is true of the CODATA recommended value, which is an adjustment containing Equation (112.44) as one of its inputs. This is the distinction the chapter has been building towards, and it can now be stated flatly:

• If \(\alpha\) is taken from the anomaly itself, the exercise is a definition. It produces the best available value of \(\alpha\) and tests nothing.

• If \(\alpha\) is taken from an independent measurement — and atom recoil is the only method currently competitive — the exercise is a test, and its sharpness is bounded by Phenomenon 112.15.

Notice, incidentally, that Equation (112.44) lies between the caesium and rubidium values and is closer to the rubidium one, differing from it by \(4.0\times10^{-8}\) and from the caesium one by \(1.2\times10^{-7}\). That is a statement about which recoil experiment is more likely to be in error if QED is right; it is not evidence about QED.

What the agreement bounds

Whatever else is true, the electron anomaly agrees with a parameter-free calculation to about \(10^{-12}\) in absolute terms, and that constrains anything not in the calculation.

Electron substructure.

If the electron were a bound state of constituents held together at a scale \(\Lambda\), the anomaly would acquire a contribution of order \((m_{e}c^{2}/\Lambda)^{2}\) from the compositeness. Requiring it to be smaller than the \(\delta a_{e}=1.4\times10^{-12}\) of Equation (112.25),

using the CODATA electron rest energy [Mohr:2025]. The electron is pointlike down to a distance \(\hbar c/\Lambda\approx4.6\times 10^{-19}\,\mathrm{m}\), less than a thousandth of the proton charge radius \(8.4075(64)\times 10^{-16}\,\mathrm{m}\) [Mohr:2025].

New particles coupling at loop level.

The weaker but more generic bound applies to a state that couples to the electron with roughly electromagnetic strength and therefore contributes only at one loop. Equation (112.20) then gives

using \(\delta a_{e}=1.4\times10^{-12}\) again. The muon's sensitivity in the same sense is read off Table 112.4 as the combined uncertainty of the comparison there, \(\delta a_{\mu}=\sqrt{(2.4\times 10^{-10})^{2}+(4.3\times 10^{-10})^{2}} =4.9\times 10^{-10}\), the two entries being the Fermilab measurement and the data-driven Standard Model evaluation. The same expression with the muon's mass then gives \(Mc^{2}\gtrsim230\,\mathrm{GeV}=3.69\times 10^{-8}\,\mathrm{J}\). The muon reaches about eleven times higher while being some \(350\) times less sensitive in \(\delta a\), two and a half orders of magnitude, because the reach grows only as \(\delta a^{-1/2}\): the factor \(350\) costs \(\sqrt{350}\approx19\) in mass, and the mass ratio \(206.7683\) more than repays it. That is Remark 112.5 made numerical. The two anomalies are not competitors: the electron's is the test of QED, the muon's is the search.

What it does not bound.

The agreement says nothing about physics that does not couple to the electron's magnetic moment, and nothing about the parts of the Standard Model that do not enter at this order — the electroweak contribution to \(a_{e}\) is \(3\times10^{-14}\), below the experimental uncertainty, so the electron anomaly does not test the electroweak sector of Electroweak Unification and the Higgs Boson at all. It is a test of quantum electrodynamics specifically, and it is the sharpest one there is.

A closing accounting

Set out plainly, the state of the comparison is this. The measurement is secure to \(1.3\times10^{-13}\) and has been improved twice by the same group without moving. The QED series is computed to fifth order, contributes at the eleventh figure, and its fifth-order coefficient is disputed by an independent recomputation [Volkov:2019] at a level that does not currently matter but would if the input improved. The hadronic contribution is thirteen times the experimental uncertainty and is taken from the muon programme's dispersive machinery, so the electron anomaly is no longer purely a QED observable. And the fine-structure constant, the one ingredient that must come from elsewhere, is known from two experiments that disagree by five and a half standard deviations.

The correct summary sentence is therefore not “QED has been verified to twelve significant figures”. It is: the measured and calculated electron anomalies agree to about nine significant figures, which is the precision of the best independent fine-structure constant, and the remaining three figures of experimental precision are waiting for atomic physics to resolve a discrepancy of its own. That is a less impressive sentence and a true one.

Primary references

The chapter rests on four groups of sources, and they are of different kinds.

The prediction.

Dirac's 1928 paper [Dirac:1928] is where \(g=2\) appears, as a consequence rather than an assumption; the derivation reproduced in this book is at Phenomenon 93.3. Schwinger's 1948 letter [Schwinger:1948] is the one-loop anomaly, a page long, and it is the founding calculation of renormalized quantum field theory as a predictive rather than a formal enterprise; the full derivation is Theorem 100.41. For the modern series the standing reference is Aoyama, Kinoshita and Nio's review [Aoyama:2019], which supersedes the same group's earlier tenth-order papers [Aoyama:2012] [Aoyama:2018] and is the source of every coefficient in Equation (112.14) and of the mass-dependent, hadronic and electroweak terms. It should be read together with Volkov's independent five-loop computation [Volkov:2019], which does not agree with it; citing the review alone would misrepresent the state of the calculation, which is why both appear wherever the fifth-order coefficient is used. Kinoshita's analysis of mass singularities [Kinoshita:1962] is background for the logarithms in the mass-dependent terms.

The measurement.

Foley and Kusch [Foley:1948] and Kusch and Foley [Kusch:1948] are the discovery pair; they are cited together throughout because the interpretation and the experimental detail are split between them. The method they used is Rabi's [Rabi:1937] [Rabi:1938]. Wilkinson and Crane [Wilkinson:1963] is the best of the free-precession experiments and the bridge to the trap era. For the Penning trap itself, Brown and Gabrielse's review [Brown:1986] is the standard reference and the source of the invariance theorem, the lineshape model and the relativistic level scheme of Equation (112.37); it is a review rather than a primary paper, and is cited as such. The Harvard sequence is Odom et al. [Odom:2006], Hanneke et al. [Hanneke:2008] and Fan et al. [Fan:2023]; the last is the source of Equation (112.8) and the experiment this chapter is registered under. Van Dyck, Schwinberg and Dehmelt [VanDyck:1987] is the electron–positron comparison of Section 112.5.2. The cavity-QED effect exploited in Section 112.3.3 traces to Purcell [Purcell:1946b] and was observed in its inhibitory form by Hulet, Hilfer and Kleppner [Hulet:1985].

The input.

Parker et al. [Parker:2018] and Morel et al. [Morel:2020] are the two atom-recoil determinations of \(\alpha\) whose disagreement bounds this chapter's conclusion. They are primary papers in atomic physics, not in particle physics, and the reader who wants to judge which is more likely to be right must read them directly; this book takes no position beyond reporting the disagreement and its consequence.

The data sets.

Two machine-readable sources carry every constant quoted here. CODATA 2022 [Mohr:2025] supplies the Bohr magneton, the electron and muon masses, the Rydberg constant, the recommended \(\alpha\), and the recommended electron and muon anomalies; the values used are those in the repository copy of the NIST listing, and every number in Tables 112.1, 112.3 and 112.4 attributed to it has been checked against that file. The PDG 2024 compilation [Navas:2024] supplies the muon rest energy and the muon width used in Equation (112.38). Where a value appears in both, CODATA is used for constants of the electron and PDG for properties of the muon as a particle, and the two are consistent within their stated uncertainties.

The muon programme.

Bailey et al. [Bailey:1977] [Bailey:1979] report the CERN storage ring, the second of these being the final CERN account of both the anomaly and the time-dilation measurement written up at Section 41.4. Bennett et al. [Bennett:2006] is the Brookhaven E821 final report, and Abi et al. [Abi:2021] and Aguillard et al. [Aguillard:2023] the first two Fermilab results. The Standard Model evaluation is the Muon \(g-2\) Theory Initiative's white paper [Aoyama:2020]; the competing lattice evaluation of the hadronic vacuum polarization is Borsanyi et al. [Borsanyi:2021]; and the cross-section measurement that unsettles the dispersive input is CMD-3 [Ignatov:2024]. The spin-precession formula behind the magic momentum is Bargmann, Michel and Telegdi [Bargmann:1959].