g-2) collaboration [1] announced a new meaurement of the muon
g-factor, disagreeing with theory by 2.6 standard deviations. Either some
term in the theory has been wrongly calculated or this could be the first
sign of physics beyond the standard model. To see why this matters let us
look back at the electron. The magnetic moment of the electron is g times the
natural unit which is its angular momentum The muon is 206 times heavier than the electron so its magnetic moment is 206 times smaller, but the virtual particles in the quantum soup can be more massive. As a result the anomalous moment is 40,000 times more sensitive to undiscovered particles and new physics at short distances,
where How can one measure B
the orbit angular frequency is
where
If
It would be convenient to calibrate the magnetic field by measuring muon precession at rest, combining equations (3) and (4) to give
Then ( µ
where µ_{µ}/µ_{p}_{µ} and µp are the magnetic moments of the muon and
proton. The value of l is known to 26 ppb from
the hyperfine structure of muonium [3]. In summary, one injects polarised muons into a
magnetic field calibrated by proton NMR and measures the rate at which the
spin turns relative to the momentum; this determines am rather than
µs givng hundreds of cycles to
measure. In 1956 ( In 1957 parity was violated and the muons from p
m + n decay were found
to be longitudinally polarised. In a footnote to their first paper on muon
precession Garwin, Lederman and Weinrich [4] invoked the (
In 1960 muons from the CERN cyclotron were focused
onto a beryllium block inside a long bending magnet, fig.1, so that they
lost energy and were trapped in the field [5, 6]. A transverse gradient
(high field at the bottom of fig.1, lower field at the top) made the
orbits walk to the right, 2 cm per turn near the beryllium falling to 0.4
cm per turn in the middle of the magnet. When the muon reached the end it
encountered a much steeper gradient, stepped along at 11 cm per turn,
emerged from the field and was stopped in a field free absorber. After an
average of 2.2 to 0.4%.
At the time the result was a surprise, it confirmed QED up to a mass scale
of 1 GeV (while the pundits including R.P. Feynman had been anticipating a
discrepancy) and it gave no sign of a new field postulated to explain the
muon mass. Now the muon was accepted as a heavy electron and nothing more,
and this was confirmed by experiments at high energy, muon pair production
by photons and the trident process (muon pairs produced by high energy
muons), which showed that the muon obeys Fermi-Dirac statistics.a_{µ}
In 1962 the CERN proton synchrotron (PS) was running
and high energy muons were available. Could we use them for ( When the muon decays some energy is lost to
neutrinos, so the decay electron is bent to a smaller radius, emerges on
the inside of the ring and hits a counter. By selecting large pulses from
the lead/scintillator sandwich one can select high energy electrons and
these must come from forward decay in the muon rest frame. Selecting
energy in the laboratory selects angle in the moving frame. Therefore, as
the muon spin rotates, the number of detected electrons is modulated. One
sees an exponential decay with the dilated muon lifetime (27
To ensure vertical focusing, essential for storing
muons for many turns, the field of the ring magnet had a radial gradient
of 54 ppm/mm. So to determine the mean field and calculate
^{3}-terms in the
QED expansion, connected theoretically with the scattering of light by
light which has never been observed. They found a surprisingly large
coefficient of 18.6, which brought the theory into agreement with the
data.
In 1969 the main obstacle to further improvement was
the radial magnetic gradient needed for vertical focusing. It was
considered impossible to know the radius of the muons to better than 1 mm.
So in the third CERN experiment the magnet had a uniform magnetic field
with vertical focusing provided by electric quadrupoles. Positive plates
above and below the orbit repelled the It is the vertical component of the electric field
that does the focusing, but inevitably it is associated with a horizontal
electric field that varies with radius. In general a radial electric field
affects the spin motion, so it is still necessary to know the muon radius.
Nothing is gained ñ unless one operates at a carefully chosen energy! If
however a) = 29.3 a radial electric field
does not change the (g-2) frequency; then one can use a uniform
magnetic field with vertical focusing provided by electric quadrupoles;
and the (g-2) frequency will be the same all over the aperture. One
does not need to know where the muons are! This so-called ìmagicî energy
is 3.096 GeV, easily accessible with current accelerators.
In fact a correction is required for muons which are not exactly centred in the aperture; their momentum is not exactly magic and there is some radial electric field. On average this correction (calculated from the radial distribution of muons obtained from the orbit frequency data) is about 0.5 ppm. The third CERN experiment [8] used a
14 m diameter storage ring operating at the magic energy. To gain beam
intensity and reduce background, pions were injected (instead of protons).
The magnetic field was stabilised at 40 points by feedback from NMR
probes, and was very reproducible. The measurement to 7.3 ppm agreed with
theory, and confirmed the existence of virtual hadrons in the quantum soup
around the muon, calculated to contribute 58 ppm to the value of
^{+}p^{ñ},
r, f, and w resonances and all sorts of higher states. The
equations which govern the strong interactions (quantum chromodynamics,
QCD) are presumed known, but no one can solve them for the strongly
coupled low energy states which contribute most to
. How the effect is estimated is discussed
below.
a_{µ} Also present in the soup are the intermediate bosons
of the weak interaction,
To see this effect was a primary objective of the new
more accurate measurement of muon ( To measure the magnetic field a trolley, carrying NMR probes and a computer, was made to move round the ring inside the vacuum chamber and measure the field along the track of the muons at any moment. A single coaxial cable carried DC power to the electronics and a 62 MHz reference frequency and served to pull the trolley round the ring. It also carried output signals from the computer giving the results of the measurements. During the muon storage runs the trolley was withdrawn into a special garage, (also under vacuum) and the field was monitored by 350 fixed NMR probes deployed above and below the vacuum chamber. The average field calculated from these fixed probes tracked with the average meaured by the trolley to within ±0.2 ppm.
^{-9}
with an error of 1.34 ppm. Combining with the previous measurements gives
the best experimental value
= 1,165,920.3 x 10a_{µ}^{-9}
with an error of 1.27 ppm compared to the theoretical prediction
= 1,165,916.0 x 10a_{µ}^{-9}
with an error of 0.57 ppm [10]. The discrepancy of (3.7 ± 1.4) ppm
warrants some discussion.
It could be a statistical fluke but that is unlikely. It could be a ìharbinger for new physicsî [10]. In particular the theory of Supersymmetry [11] (which has many adjustable parameters) predicts a new contribution to am and can easily accommodate our value. In this theory every known boson is matched by a supersymmetric fermion, and vice versa, so many new particles would be there awaiting discovery as soon we can reach sufficient energy; and the theory gets rid of some mathematical infinities that plague the standard model. but new experimental data could still change
the value. On the other hand there are more complex hadronic processes
involving several virtual photons and these contribute to
a_{µ}a at the level of 1 ppm and in this area no one is
quite sure of the calculation._{µ}
Copyright EPS and EDP Sciences, 2001 | |||||||||||||||||||||||||||||||