EXPERIMENT: G-2 MEASUREMENT OF THE MUON

The electron, muon, tauon, and their neutrinos are fermions with spin ℏ/2. These particles (andtheir antiparticles) form the subclass of fermions known as leptons, which comes from the Greek word λεπτου meaning thin, or light in weight. The muon and tauon are like heavy electrons, except that the electron is stable, having no lighter particles into which it can decay, whereas the muon and tauon undergo radioactive decay through the weak force. Physicists have looked to see if the leptons have any structure, and at the smallest resolution they have been able to achieve, no indication of structure has been seen. It is believed that leptons occupy a physical point rather than filling some volume of space like the proton does (its size is about a femtometer).

The muon has a very long lifetime by subatomic standards, a trait which permits detailed studies of its properties. Precision measurements have been made of the muon's lifetime, mass, and magnetic moment, and at the time of this writing (2002), efforts are under way to improve the precision of the lifetime and anomalous magnetic moment measurements. In its rest frame, the muon exists for 2.19703(4) microseconds (μs), where this value represents a mean or average value of several experiments. The number in parentheses quantifies the uncertainty on the measured value (often referred to as the error), which by convention is one standard deviation, often represented by the Greek letter σ.

Many physical constants have been measured and are tabulated for easy reference (for example, see the Particle Data Group and National Institute of Standards and Technology Web sites). In the tables in these two sources is a listing in the form M ± σ, or M(σ), where M is the "best value," an average computed by the people who compiled the table, and σ is a combined standard deviation which is obtained from all the separate experiments which went into the average value. The number presented in the tables as the uncertainty on a quantity is usually the quadrature of the statistical error σ_E, and the systematic error σ_S; If these two errors are not independent quantities, then they must be added linearly, σ = σ_E + σ_S, which gives a larger total uncertainty. In precision measurements (defined below) such as the muon g-2 value, the systematic errors are as important as the statistical errors. The problem faced by the scientist is how to reasonably estimate the systematic errors while not overestimating them, which would artificially reduce the precision of the measurement.

The size of the uncertainty relative to the value of a quantity is often referred to as the precision of the measurement. A common description of the precision of a measurement is its relative error, σ/M, which for "precision measurements" can be on the order of 10^-6, parts per million (ppm), or 10^-9, parts per billion (ppb). While this level of precision is often reached in atomic physics, only a few quantities in particle physics have been measured to this level of precision. One example is the mass of the Z⁰ boson, which was measured to a precision of 20 ppm at European Laboratory for Particle Physics (CERN), using the large electron-positron collider (LEP). The lifetime of the muon given above has a precision of18.2 ppm, and the anomalous magnetic moment (the topic of this article) is currently known to a precision of 1.3 ppm.

Magnetic Moments

The magnetism associated with an elementary particle provides us with information about its structure and also about the forces that can affect it. Because of their spin, the muon, the electron, and the proton behave like tiny magnets, with the direction of the magnetic field pointing along the spin angular momentum of the particle. The classical analog of a spinning particle is sketched in Figure 1a, where an electrically charged sphere (with positive charge) is spinning about its center, just as the Earth spins about its axis. Also sketched are the magnetic field lines that emerge near the north pole and re-enter near the south pole of the magnet. For a spinning object, it is useful to define a spin angular momentum vector s←. To find the direction of the spin vector, curl your right hand fingers in the direction of rotation, and your thumb will point in the direction of the spin vector.

This spinning charge distribution will create a magnetic field, which is much like the magnetic field created by a bar magnet. Such a magnetic field is called a "dipole magnetic field," since it is set up by two magnetic poles. A useful measure of the strength of this magnetic field is a vector called the "magnetic dipole moment," or simply "magnetic moment" or "dipole moment," which is a measure of the strength

FIGURE 1

of the magnetic field. It is traditional to represent this dipole moment by the vector symbol μ⃗. (Unfortunately, physicists use the Greek letter μ to represent the magnetic dipole moment and also to represent the muon. It should be clear which is meant from the context.) For a positive charge, the magnetic dipole moment points in the same direction as the spin angular momentum vector.

Also shown in Figure 1b is a bar magnet and its magnetic field lines. For a bar magnet, the dipole moment points from the south to the north pole as indicated by the arrow in the center of the bar. A compass needle is a familiar example of a magnetic dipole. When a dipole is placed in a magnetic field, it experiences a torque that will make it align itself with the magnetic field just as a compass aligns itself with the Earth's magnetic field, indicating which direction is north.

Any magnetic dipole will experience a torque if it is placed in a magnetic field. However, if the dipole is caused by a spin, the dipole cannot align itself with the magnetic field. The situation is like a toy top spinning about its axis in the Earth's gravitational field. Instead of falling over, the top precesses with the tip of the angular momentum vector following a circle as shown in Figure 2. A torque is produced about the contact point by gravity pulling down on the top at its center of mass. This torque causes the angular momentum to precess, since the rotational form of Newton's Second Law tells us that the angular momentum will change in the direction of the torque labeled N, which points into the page in Figure 2. The precession frequency of a top, ω_ρ, is proportional to the angular momentum, L⃗ = IS⃗ , and is given by ω_ρ = mgl /IS . The symbol ω (the lower-case Greek letter omega) is the angular frequency in radians per second. S is the spin of the top (the angular frequency of rotation about the symmetry axis), and I is the moment of inertia which depends on the distribution of mass about the line through the axis of symmetry of the top.

The magnetic moment of the muon is related to its spin through the relationship: where g is the constant of proportionality between the spin and the magnetic moment, which is called the g -factor or g -value. The symbol e is the charge of the muon, and m is its mass. For a particle with spin ℏ/2, no internal structure, and with no radiative corrections (see below), the relativistic quantum mechanics developed by Paul A. M. Dirac tells us that g is exactly two. However, if the particle has internal structure, then g is not equal to two. For example, the g -factor of the proton, which is made up of quarks and gluons, is 5.58, quite different from two. Because of its magnetic moment, a muon which is placed in a magnetic field will precess about the field, just as a toy top precesses about the (vertical) gravitational field of the Earth.

FIGURE 2

Background and History of g -2

The development of understanding the electron's g -value was at the center of the path to understanding the subatomic world. In 1921 Otto Stern proposed the famous experiment, which was later called the Stern-Gerlach experiment. In their 1924 review paper, Stern and Walther Gerlach concluded that to within 10 percent, the silver atom had a magnetic moment of one Bohr magneton. However, their interpretation was incomplete, and only with the postulate in 1925 by Samuel Goudsmit and George Uhlenbeck that the electron had an intrinsic angular momentum called spin did the full picture emerge. In 1927, motivated by the work of Stern and Gerlach and the proposal of Goudsmit and Uhlenbeck, T. E. Phipps and J. B. Taylor showed that the magnetic moment of the hydrogen atom (and thus the electron) was one Bohr magneton, in agreement with the spin hypothesis of Goudsmit and Uhlenbeck. In modern terminology, these developments told us that the g -value of the electron was two.

There were indications in several earlier published results that g might not be exactly two, but the definitive evidence came in 1947 when Polykarp Kusch and H. M. Foley obtained their results on the difference of g from two. Julian Schwinger explained this difference with one of the pioneering calculations of what is now called quantum electrodynamics (QED).

In modern language, one measures the anomalous magnetic dipole moment, sometimes called the anomaly a , where is the magnitude of the magnetic moment. It is this latter quantity which is tabulated in the tables of particle properties found at the Particle Data Group's Web site.

The g 's difference from two can be understood by examining the Feynman diagrams shown in Figure 3. The Heisenberg uncertainty principle ΔE Δt ≥ ℏ permits virtual processes that violate energy conservation to occur, as long as they happen quickly. The particles which appear out of nowhere

FIGURE 3

(physicists say the particles appear from the vacuum) are called virtual particles. In the left-hand picture of Figure 3, the muon interacts with the magnetic field by absorbing a photon (labeled γ) from the magnetic field. In the middle picture, the muon emits a virtual photon, then absorbs a photon from the magnetic field, and then reabsorbs the virtual photon. After the virtual photon is emitted, the symbol μ^* is used to remind us that the muon is "off-shell" (meaning that energy conservation must satisfy the uncertainty principle, and the muon is not a free particle). The g -value is sensitive to the ratio of charge over mass, and this virtual process changes the mass. The virtual process on the right has the effect of changing the charge distribution around the muon. The muon's magnetic dipole moment (g -value) is changed by virtual processes which change the mass, or the charge distribution surrounding the muon. The anomaly caused by the process in the middle is 0.00116140981(5). In nature, there are an infinite number of these virtual processes (called radiative corrections) involving photons and electrons, but the largest effect comes from this one process. Since the anomaly is about one thousandth of the total magnetic moment, experimentally one uses a measurement technique directly sensitive to the anomaly (g -2) rather than the total moment (g ).

The beauty of studying a process that is sensitive to virtual particles is that all particles present in nature can participate, including particles never seen before. The only requirement is that they are able to interact directly or indirectly with (physicists say "couple to") the muon. Angular momentum, charge, and other fundamental quantities (except energy for a brief time) must be conserved. The Standard Model particles that can contribute (virtually) to the anomaly at a measurable level are electrons, muons, tauons, photons, pi and Κ mesons (quarks and gluons), and the electroweak gauge bosons, the W± and the Z⁰.

One of the main motivations for measuring g precisely is its sensitivity to a wide range of new physics beyond the standard model. The g -value of the muon has traditionally served as a calibration point for new theories, since most predict a contribution to the anomaly from the constituents of the new theory. If a new theory predicts an effect on the anomaly which is ruled out by the measured value, then this theory cannot be valid without revision. Over the past decade, great interest has developed in various extensions to the standard model such as supersymmetry . Under some scenarios of this theory, the supersymmetric partners of the Wαand the Z⁰ could contribute to the anomaly at a measurable level.

In a series of three beautiful experiments at CERN, the muon anomaly was measured to an increasing precision, reaching 7.3 ppm by 1979. The final result confirmed that the muon was a lepton which obeyed the rules of quantum electrodynamics, and it also confirmed the presence of the predicted contribution from virtual pi and Κ mesons. At this level of precision, the predicted 1.3 ppm contribution of the electroweak gauge bosons was not observable. It was the desire to observe this electroweak contribution to the anomaly, along with the desire to look for effects of new physics with increased sensitivity, that motivated a new proposal to achieve an accuracy for the muon anomaly twenty times more precise than the CERN result.

The technique used in the third CERN experiment was the basis of the new Brookhaven experiment. A beam of muons with their spin pointing in the same horizontal direction is stored in a magnetic ring which forces them to go in a circle of 14 meters in diameter. The muons make up to 4,000 trips around the ring before they decay. Because their g factor is not exactly two, the muons' spin turns faster than their momentum as they go around the ring (see Figure 4). Every 29.3 turns around the ring, the spin makes one complete revolution relative to the direction the muon is moving (its momentum). Because of relativistic time dilation, the average lifetime of the muon is 64.4 microseconds, and in ten lifetimes essentially all of the muons have been lost to radioactive decay.

The frequency with which the spin turns relative to the momentum (ω_a) is written inside the circle, and this formula is the basis for the measurement. By measuring the magnetic field and the frequency with which the spin turns relative to the momentum, one can determine the anomaly. Since this frequency depends on the anomaly directly, rather than the

FIGURE 4

magnetic moment, it is called a g -2 measurement, rather than a g measurement.

The g -2 Experiment at Brookhaven National Laboratory

Based on the experience of the third CERN experiment, it was clear that one could improve the precision by a factor of twenty, provided a new, improved apparatus was built. This improvement required extending the current state-of-the art in a number of areas, if the experiment were to be successful. The physics goals were to confirm the predicted Standard Model contribution of the gauge bosons, and/or to search for contributions to the anomaly beyond the standard model. By the early 1980s the Standard Model was well established. It was becoming clear that there were deficiencies in the Standard Model (even though not all the predicted standard model particles had been discovered), and new ideas such as supersymmetry were being developed.

The new experiment had a goal of 0.35 ppm relative error, which would be adequate to observe the effect of virtual W and Z⁰ gauge bosons. This design goal meant that the new experiment might be sensitive to new particles such as the supersymmetric particles (if they exist), and if no effect were found, it would at least restrict what their properties might be. The new experiment would also serve to further restrict muon or W boson substructure. Evidence for any of these effects outside of the standard model would represent a major new discovery.

The proposal was submitted to Brookhaven National Laboratory and was approved by the Laboratory in 1986. A major new feature of the Brookhaven

experiment was to form a beam of muons external to the experiment, bring this beam into a storage ring, and then give it a kick to move the beam into a stable orbit within the ring. The heart of the new experiment is a unique storage ring magnet, 14 meters in diameter and weighing 700 tons, which is the world's largest superconducting magnet. This magnet has been shimmed to an average uniformity of 1 ppm over the region where the muon beam is stored. It now appears that the systematic errors associated with the magnetic field will be about 0.3 ppm, which should permit the collaboration to reach close to their design sensitivity.

In February 2001 a new result was reported by the collaboration, a_μ = 0.0011659202(14)(6), where the first error in parentheses is statistical and the second is systematic. This result, when averaged with previous measurements, was two and one-half standard deviations larger than the value expected from the Standard Model, and statistically there was less than a 2 percent chance that this result was compatible with the Standard Model prediction. This result generated a great deal of excitement among those who believe that supersymmetry is the correct theory to extend the Standard Model, since the magnitude of the discrepancy is easily accommodated by this theory.

Unfortunately, a small piece of the Standard Model theoretical value called the hadronic light-by-light contribution that had been assigned a negative sign turned out to be positive. This mistake was discovered by Marc Knecht and Andreas Nyffeler at the University of Marseille and was soon confirmed by the authors who had originally obtained the negative sign. This incident demonstrates the critical interplay between theory and experiment in the progress of physics. When the correct sign is used, the discrepancy with the theory is reduced to 1.6 standard deviations, which implies an 11 percent chance of agreeing with the Standard Model.

The collaboration has collected seven times as much data as were reported on in 2001, which should reduce the statistical error by the square root of seven, and the systematic error will also be reduced. When these additional data are analyzed, they will determine if there is a meaningful discrepancy with the Standard Model or not.

Organization of the Experiment

With less than 100 collaborators worldwide, the g -2 collaboration is a modest-sized collaboration by the standards of particle physics. The collaborators come from Boston University; Brookhaven National Laboratory; Budker Institute of Nuclear Physics, Novosibirsk, Russia; Cornell University; Fairfield University; Rijksuniversiteit Groningen, the Netherlands; University of Heidelberg, Germany; University of Illinois; KEK Japan; University of Minnesota; Tokyo Institute of Technology; and Yale University. The experiment is organized along the lines of almost all large particle experiments, with cospokespersons and a formal management structure. The word spokesperson has different meanings. In high-energy physics, the spokespersons are the leaders of the collaboration, but the duties vary greatly from collaboration to collaboration. In g -2, the cospokespersons are Vernon W. Hughes (Yale) and B. Lee Roberts (Boston). Gerry Bunce (Brookhaven) is the project manager, and William M. Morse (Brookhaven) is resident spokesperson.

The principal governing body of the experiment is an executive committee (EC) that consists of the cospokespersons plus representatives from each of the institutions. The chairpersons of the EC have been drawn from the senior members of the collaboration. Over the past twelve years there have been five different chairpersons (B. Lee Roberts, David Hertzog–Illinois, Priscilla Cushman–Minnesota, Klaus Jungmann–Heidelberg), with the 2002 chairperson being James Miller (Boston).

The first paper was signed by over 100 authors from fourteen institutions. The current collaboration list, links to papers, and pictures of the experiment are available on the Muon (g -2) Collaboration's Web site. Since the scale of the project was large, teams were set up to carry out the many tasks, and their responsibilities and the names of the team leaders are also given on the Web site.

Many graduate students and postdoctoral research associates have made major contributions to the experiment, and they are also listed on the Web site along with their institutions. The g-2 experiment has served as an important training ground for graduate and undergraduate students.

See also:Lepton; Muon, Discovery of