Particle physics

Two is the magic number

In quantum theory, the magnetic moment of the muon should be twice the value calculated classically, although in fact their ratio is not exactly two. Theory and experiment disagree on quite how far from two it is.

There are essentially three ways of exploring the strange phenomena that might occur at very high energies or, equivalently, at very high temperatures: build a powerful accelerator to produce new particles (as was done for the discoveries of the W and Z particles and the top quark); search in the present-day Universe for any ‘fingerprints’ of exotic particles that might have existed fleetingly in the first moments after the Big Bang; or make precise measurements of those particle properties that might be influenced by high-energy quantum processes. In this last category, one such measurement is that of the anomalous magnetic moment of the muon. Its latest value, reported by G. W. Bennett and colleagues1, does not match the theoretical prediction as well as might be expected — and might be a hint of something as yet unknown.

The muon is, as far as we know, a truly elementary particle, similar in almost all respects to the electron except that it is some 200 times heavier. Like the electron, the muon has negative charge, and has a positively charged antiparticle. Both the electron and muon have one half-unit of the quantum property known as spin. Any spinning charge generates a magnetic field aligned along its spin axis, so the muon and the electron have a magnetic moment, which can be calculated semi-classically from the particle's spin, charge and mass. But when the first measurements were made of the magnetic moment of the electron in the 1920s, its magnitude was found to be twice the expected value. However, one of the triumphs of Dirac's formulation of a quantum theory consistent with special relativity was that it predicted that the electron's magnetic moment would be exactly twice the semi-classical value. The ratio of the two is usually known as g.

In fact, life is a little more complicated. A particle's properties can be modified by other quantum effects — such as the emission and absorption of particles on short timescales, allowed by Heisenberg's uncertainty principle and illustrated in the ‘loop diagram’ of Fig. 1a. The magnetic moment of the electron or muon is slightly modified in this way and deviates from the expected value of two, so that (g−2)/2 is not exactly zero. This quantity is known as the anomalous magnetic moment and can be calculated in principle. Any significant difference between the calculated value of the anomalous magnetic moment of either the electron or muon and the corresponding measured value is a clear indication that there must be something else, perhaps some as yet unknown particles, that should be included in the calculation (Fig. 1a).

Figure 1: Loop corrections and the anomalous magnetic moment of the muon.

a, The uncertainty principle allows momentary fluctuations such as the emission of a photon by a muon, provided that the photon is reabsorbed by the muon some short time later. Similarly, the photon itself might fluctuate into two other particles (electrons, quarks or perhaps something unknown) that then recombine. Theorists must try to estimate the effects of all possible loops on a particle's properties. b, This magnetic storage ring at Brookhaven National Laboratory, New York, is fed with muons. As the particles circle the ring, they gradually decay, releasing electrons that are detected by one of the 24 detectors (blue) lining the inside of the ring. c, The number of electrons detected varies as the muon spin direction precesses and gradually decreases as the muon population inside the ring becomes depleted. Starting from the top, the precession can be traced, in 100-µs segments, over a period that is equivalent to about eight times the average muon lifetime. Such precision gives an accurate measurement of the anomalous magnetic moment of the muon. (The data shown are for positively charged muons2.)

Because the uncertainty principle allows very energetic fluctuations, provided that the timescale is suitably short, much higher energy scales can be probed through measuring the anomalous magnetic moment than can be reached directly. The effective energy scale explored is governed by two factors: the precision with which the anomalous magnetic moment can be measured, and the accuracy of the theoretical calculations. Both of these present enormous technical challenges.

On the experimental side, Bennett et al.1 — the Muon (g−2) Collaboration, at Brookhaven National Laboratory in New York — have measured the anomalous magnetic moment of the negative muon (aµ) with an astonishing accuracy of less than one part in a million: aµ=0.0011659214±0.0000000008. Six months ago the same group published their result2 for positive muons with the same level of accuracy and in very good agreement with previous measurements. Although this is a tricky experiment to perform, the principle behind it is fairly simple. Muons are collected inside a storage ring (Fig. 1b), constrained to move in a circular orbit by a ring of magnets. Initially, the direction of the muon spins is aligned with their velocity. But as they travel round the ring, the direction of the muon spins changes a little faster than the direction of their motion. The resulting misalignment (which grows by about 12° for each complete orbit) is directly proportional to the deviation of the g-factor from two. In fact, the muons only live a short while, decaying into an electron and two other particles called neutrinos. The direction in which the electron emerges is correlated with the direction of the decaying muon's spin, enabling measurement of the precession, and hence of aµ (Fig. 1c).

The challenge for the theorists is equally great. There are three sets of diagrams to be incorporated in the calculation. The largest effects come from electrodynamics (processes involving photons, such as Fig. 1a); calculations have been done with up to five ‘loops’ in the process, to ensure that the theoretical uncertainties are smaller than the experimental errors. Weak processes (involving W and Z particles) are also significant, and calculations with up to two loops have been performed. These are computational challenges. However, the greatest uncertainty arises from loops involving other, strongly interacting particles. The strength of their interactions is such that standard computational techniques (exemplified by Fig. 1a) do not work. As more computer power becomes available, the calculation of these effects will eventually be possible. But, until then, they can only be estimated by using measurements of other processes with a similar structure.

There have been heroic efforts to work out the contribution of pions, which are made from a quark and an anti-quark and interact strongly. One method uses data from collisions of electrons and anti-electrons (see, for example, ref. 3); another uses data from the decays of τ particles4 (an even heavier relation of the electron) into pions. The two theoretical calculations are not in precise agreement — they differ beyond the eighth decimal place — for reasons that are not yet understood. Nevertheless, there are indications of a discrepancy between the measurements and the theoretical expectations (at the level of 2.7 and 1.4 standard deviations, respectively). If this discrepancy is confirmed by further measurements and more refined calculations, then there is clear evidence that a new ingredient is needed in the recipe.

What might this new ingredient be? There are, of course, many theories (and many theorists) trying to answer that question. One of the most popular ideas is that there is a new species of very massive particle, called supersymmetric particles — one consequence of their existence being a change in the anomalous magnetic moment of only a few parts in a billion. There are many other reasons why theorists would welcome such an addition to the standard model of particle physics. And there are compelling reasons to expect that, if supersymmetric particles exist, they will be found at the Large Hadron Collider, now being built at CERN in Switzerland.


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Peach, K. Two is the magic number. Nature 427, 688–689 (2004).

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