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Muon and electron g-2
A charged particle which has spin angular
momentum s will have also a magnetic moment m.
The ratio of the magnetic to angular moments m/s in
classical physics would be expected to be q/2m if the
charge q and mass m of the particle were distributed
in the same way. Reality is more complicated.
Deviations from this classical result are contained in
a parameter g, defined by m/s = g(e/2m).
Dirac theory predicts g=2 for
a point-like spin ½ particle.
Measured values are;
gP = 2.793...
ge = 2.002...
gm = 2.002...
The proton is not at
all like a Dirac
particle - electron
and muon are close.
Deviations of g from 2 are described by the parameter a=(g-2)/2. Within the Standard
Model non-zero a is expected from higher order corrections. The first order correction is
the same for electrons and muons and is due to this diagram 
Which gives a correction Da = a/2p = 0.001161
Beyond that things get more interesting...
Muon g-2 measurement
A muon g-2 experiment is currently running at Brookhaven and reporting updated
results every year, most recently in January 2004.
g-2 magnet
3 GeV m+ are held in a circular orbit by a vertical 1.45 T field in a 14 m diameter ring.
To extract a from wa it is necessary to know the average field in the ring. Seventeen check
mobile and 150 fixed NMR probes measure B to 0.4 ppm by reference to proton m. p=0.3×B×R
Why is (g-2) special
The Standard Model can be tested by measuring muon (g-2). The experiment is
unusually accurate compared with typical particle-physics measurements.
The subtraction of 2 ( i.e. 2.00000000000000...) is done for us by the physics, so the
measurement gets directly to the radiative corrections.
For muons in a circular orbit in a magnetic field B and zero electric field, the orbital or
cyclotron frequency is wc = (e/mc)x(B/g) and the spin precession frequency is ws =
(e/mc)x(B/g + ½(g-2)B). The experiment is sensitive to the muon polarisation at a fixed
point in the orbit*, so it measures the difference between wc and ws directly. The big
terms cancel leaving just the term containing a ( a = ½(g-2) ):
wa = wc – ws = (e/mc)aB
So all the experiment has to do is accurately measure the frequency wa and the magnetic
field B. Technology for frequency measurement is very accurate (much more than mass,
length, etc). The magnetic field is converted to a frequency with proton NMR - the value
of gP is known to 1 part in 108.
(*) The measured quantity is the counting rate of electrons with energy over some threshold, versus
time since the start of the fill. Electron energy is correlated with muon spin because the decay electron
direction is preferentially in the direction of the muon spin. If muon spin is pointing in direction of
muon motion in the lab, the electrons have higher energy because the two boosts add. If muon spin is
pointing backwards the boosts partially cancel and electron energy is low.
Magic momentum
The formulae on the previous slide were for zero
electric field. But E=0 is not practical in a real
experiment because The magnet holds the muons in position in the
horizontal direction but it does nothing to constrain
their vertical movement. A quadrupole electrostatic
field is necessary to stop them escaping up or
down.
Relativity tells us that a moving electric field is
equivalent to a magnetic field – the muons will feel
an extra magnetic field which causes a change to
their precession frequency
Dwa= e/mc . [ 1/(g2-1) – a ] . bE
There is “magic” point at g2=1+1/a or pm=3.098
GeV at which this correction cancels.
Results
The e- (or e+) count rate oscillates
at wa and dies away as the m- (or
m+) decay or escape from the ring.
It is fitted with a function which
takes account of pileup, muon
loss, betatron oscillations and wa .
The latest result for negative
muons is
am- = 11,659,214(8)(3)×10-10
Systematic
Statistical
Last year there was a similar result
for positive muons;
am+ = 11,659,204(7)(5)×10-10
Results are consistent - no CP
violation.
Theory
Many higher order corrections must be added to the basic a = 1 + a/2p to reach a Standard
Model prediction that is as accurate as the experimental value.
First there are the higher order QED corrections, now known up to 5th order in (a/p), their
coefficients are approximately ½, 0.765, 24.1, 126, 930. The theoretical uncertainty from these
corrections is negligible.
Then there are electroweak corrections, such as the left-hand diagram above. They are known
to second order and their theoretical uncertainty is negligible. One of them involves the Higgs
but it is very small, because the Higgs-muon coupling is small, because the muon is light.
Finally there are corrections which involve quark loops, such as the two shown above right.
Hadronic corrections
Grey area represents something not calculable by perturbative QCD. Initial production of
q-qbar is easy (QED). But as soon as created the quarks start to emit gluons, gluons emit
more gluons, gluons split to q-qbar, etc. All low energy  high as  non-perturbative. The
situation appears hopeless. But theorists have a trick up their sleeve:
The grey blob is this diagram is closely related to the grey area in this diagram, and this cross
section has been measured. In fact the correction to am is proportional to the integral from 0 to
 of s(e+e-hadrons).(mm2/3s).ds . Where the 1/s term means that in practice the upper
limit is irrelevant and it is the low end of the energy range that matters, around 0.3 to 3 GeV.
Old experimental data has been dug out from the time when e+e- collisions at a few GeV was
frontier physics. New experiments to re-measure s(e+e-hadrons) more accurately are in
progress. This is currently the biggest uncertainty in the prediction of am.
Another possibility is to measure the energy spectrum of
hadrons produced in t decay. Here the experimental data is
more accurate but the theory relating it to am has some
approximations which are not accepted by all theorists.
Experiment versus theory
Measurements from
CERN and
History of muon (g-2)
Brookhaven
240
a x 10^10 - 11659000
220
200
180
Measurement
Theory
160
Derived from t decay
140
Derived from e+e-
120
100
1975
1980
1985
1990
1995
2000
2005
2010
Year
The Future
The Brookhaven experiment that
made the recent measurements has
now finished. There will probably be
another improvement sometime maybe not soon - gap of 25 years last
time.
Jump due to sign
error in light-by-light
scattering calculation.
Theory error should improve soonish as new
s(e+e-hadrons) results come out.
Dam(QED) = 1165847 ± 0.3
Dam(had.) = 687 ± 6
Dam(l-b-l) = 8 ± 4
Dam(weak) = 15 ± 0.2
SUSY limits
At present the measured value of am is 2.7 (1.6) standard deviations higher than
the Standard Model prediction where the hadronic correction is based on the
e+e- (t decay) data. A real discrepancy would be a signal for physics beyond the
SM and super-symmetry is one of the most popular ideas.
There are many variations of SUSY but very roughly;
neutralino
DamSUSY  130×10-11 . ( 100 GeV / mSUSY )2 . tanb *
(*) tanb is a free parameter of SUSY,
thought to be in the range 4-40
smuon
If we take the theory based on the e+e- data;
at 2s confidence 53 GeV < mSUSY/sqrt(tanb) < 136 GeV,
photon
at 3 s confidence the upper limit goes to .
In supersymmetry every SM particle has a partner with the same quantum numbers
except for spin which differs by half. E.g. the muon has a spin 0 partner called a scalar
muon or smuon. The photon has a spin ½ partner called a photino, or more generally if
the neutral SUSY particles are mixed states they are called neutralinos.
Electron g-2
The electron g-2 is even more precisely measured than the muon value ( 350 times more ).
However, it is much less sensitive to new physics.   The lower mass of the electron
means that there is less energy around to borrow for the creation of new particles. The effect
of massive particles is suppressed by a factor of (mm/me)2  40000.
The QED theoretical value is know up to a4 and hadronic and electroweak corrections have
negligible uncertainty.
Assuming QED is correct, ae can be interpreted as a measure of a . It is consistent with and
much more accurate than the value from the quantum Hall effect.
a-1 from ae =
137.035 999 58(52)
a-1 from QH =
137.036 003 00(270)
Electron g-2 is a brilliant test of QED but does not test other aspects of the standard model