Muon and electron g-2

A charged particle which has spin angular momentum s will have also a magnetic moment mThe 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...

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 a = = 0.001161

Beyond that things get more interesting...

The proton is not at all like a Dirac particle - electron and muon are close.

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 + are held in a circular orbit by a vertical 1.45 T field in a 14 m diameter ring. To extract a from a it is necessary to know the average field in the ring. Seventeen mobile and 150 fixed NMR probes measure B to 0.4 ppm by reference to proton

check p=0.3×B×R

Why is (g-2) specialThe 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 c = (e/mc)x(B/) and the spin precession frequency is s = (e/mc)x(B/ + 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 c and s directly. The big terms cancel leaving just the term containing a ( a = ½(g-2) ):

a = c – s = (e/mc)aB

So all the experiment has to do is accurately measure the frequency a 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

a= e/mc . [ 1/( – a ] . E

There is “magic” point at 1+1/a or pGeV at which this correction cancels.

Results The e- (or e+) count rate oscillates at a and dies away as the (or ) decay or escape from the ring. It is fitted with a function which takes account of pileup, muon loss, betatron oscillations and a . The latest result for negative muons is

a = 11,659,214(8)(3)×10-10

Last year there was a similar result for positive muons;

a = 11,659,204(7)(5)×10-10

Results are consistent - no CP violation.

Statistical

Systematic

Theory

Many higher order corrections must be added to the basic a = 1 + 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 (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 correctionsGrey 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 s 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 ais proportional to the integral from 0 to of e+e-hadrons).(m

2/3s).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 e+e-hadrons) more accurately are in progress. This is currently the biggest uncertainty in the prediction of a.

Another possibility is to measure the energy spectrum of hadrons produced in decay. Here the experimental data is more accurate but the theory relating it to a has some approximations which are not accepted by all theorists.

Experiment versus theoryHistory of muon (g-2)

100

120

140

160

180

200

220

240

1975 1980 1985 1990 1995 2000 2005 2010

Year

a x

10^

10 -

116

5900

0

Measurement

Theory

Derived from decay

Derived from e+e-

Jump due to sign error in light-by-light scattering calculation.

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.

Theory error should improve soonish as new e+e-hadrons) results come out.

a(QED) = 1165847 ± 0.3

a(had.) = 687 ± 6

a(l-b-l) = 8 ± 4

a(weak) = 15 ± 0.2

The Future

Measurements from

CERN and

Brookhaven

SUSY limitsAt present the measured value of a is 2.7 (1.6) standard deviations higher than the Standard Model prediction where the hadronic correction is based on the e+e- ( decay) data. A real discrepancy would be a signal for physics beyond the SM and super-symmetry is one of the most popular ideas.

(*) tan is a free parameter of SUSY, thought to be in the range 4-40

There are many variations of SUSY but very roughly;

aSUSY 130×10-11 . ( 100 GeV / mSUSY )2 . tan *

If we take the theory based on the e+e- data;

at 2 confidence 53 GeVmSUSY/sqrt(tan< 136 GeV,

at 3 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.

neutralino

smuon

photon

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 (m/me)2 40000.

The QED theoretical value is know up to 4 and hadronic and electroweak corrections have negligible uncertainty.

Assuming QED is correct, ae can be interpreted as a measure of . It is consistent with and much more accurate than the value from the quantum Hall effect.

-1 from ae = 137.035 999 58(52)

-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