1
8.5.3 Higher order corrections: Anomalous magnetic moment
1. Magnetic moment of the electron
a) Dirac equation with electron coupling to electro-magnetic field:
0)( mDiieAD
Aepp
(canonical momentum)
Ansatz for the solution as for free particle: ipx
ipx
e
e
00
0
)(
)(
meAt
i
meAt
i
Reminder:
0
0
0
00
2
Non-relativistic limit:
For this limit it makes sense to separate
interaction via charge and magnetic moment
meAmE 20,
from (2) inserted in (1): m2
0
2
2eA
mti
)()(
)(
22
1
0
0
meAt
i
eAt
i
Pauli equation.
Lower spinor component in non-relativistic limit small.
3
Beji
ji
ji
ji
222
4
1,,
0
2
22eAB
m
e
m
Aep
ti
2gwithBSm
egB
m
eg
222
with
2
1
2g
m
ee
4
Auueif
Interaction of
“spinless charge”
“Magnetic interaction”
via spin spin-flip
Auppippum
eiififf
)()(2
ipfp
A
2
i
b) Gordon decomposition for electron current:
Bm
e
2usince. Non-relativistic limit
5
2. Effect of higher order corrections
2g 2g
Auppippum
eiififf
)()(2
Auppippum
eiififf
)()()(2
12
2
1)2(
2m
ee
22
2
2
ga
g
1st order:
6
Higher order corrections to g-2
Radiative corrections g-2 are
calculated to the 4-loop level:
Feynman Graphs
O( ) 1
O( 2) 7
O( 3) 72
O( 4) 891
til O( 4) 971
Most precise QED prediction.
T. Kinoshita et al.
analytically
numerically
7
432
....9144.1....182.1...328.02
eaKinoshita 2007
2
2ga
3. Electron g-2 measurement
Experimental method: Storage of single electrons in a Penning trap
(electrical quadrupole + axial B field)
complicated electron movement (cyclotron
and magnetron precessions).
mc
eBgs
2
mc
eBC
22
Idea: bound electron:
Energy levels single electron:
Cyclotron frequency
Spin precession frequency
H. Dehmelt et al., 1987
G. Gabrielse et al., 2006
C 149 GHz 134 kHz
z 200 MHz
Leading relativistic correction
9
http://www.nobelprize.org/nobel_prizes/physics/laureates/1989/dehmelt-lecture.pdf
Excitement of axial oscillation:
Magnetron levels
(from E-field)
Axial oscillation
(E-field)
Cyclotron levels (n)
& Spin orientation
H. Dehmelt et al. 1987
)43(4188652159001.0e
a
)43(9187652159001.0e
a
4
32
....505.1
....182.1...328.02
ea
)290(133652159001.0ea
Theory
most precise value of :
)96(710999035.137)(1ea
For comparison from Quanten Hall
)270(00003036.137)(1
qH
c
cs
Bcsa
ga
Bg
2
2
)2(
Trigger RF induced transitions ( a) between
different n states or spin flips.
(change in cyclotron or spin state revealed
by axial oscillation -> feedback driven osc.)
)76(85180652159001.0ea
G. Gabrielse et al. 2006
)76(85180652159001.0ea
Phys. Rev. Lett. 97, 030801 (2006)
Phys. Rev. Lett. 97, 030802 (2006)
SEO = single electron oscillation
11
4. Experimental determination of muon g-2
m
eBC
22
Principle:
• store polarized muons in a storage ring;
revolution with cyclotron frequency c
• measure spin precession around the
magnetic dipole field relative to the
direction of cyclotron motion
EaBacm
ea
)
1
1(
2
Precession:
Difference between Lamor
and cyclotron frequency
Effect of electrical focussing
fields (relativistic effect).
GeV/c094.3
29.3 for 0
μp
First measurements:
CERN 70s
)11(911165001.0
)12(937165001.0
a
a
mc
eBgS
2
mc
eBC
22
CSa
12
ee e
e“V-A” structure of weak decay:
Use high-energy e+ from muon
decay to measure the muon
polarization
(g-2) Experiment at BNL
2 7.1 m
E=24GeV
1 / 109 protons on target
6x1013 protons / 2.5 sec
Weak charged current couples to LH
fermions (RH anti-fermions)
13
)cos(1)( 0 tAeNtN a
t
Measure electron rate:
Hz)16(59.0232292
a24 detectors
(0.7ppm)
Bcm
ea a
?
14
From a to a - How to measure the B field
<B> is determined by measuring the proton nuclear magnetic
resonance (NMR) frequency p in the magnetic field.
)1(/
~/
2
~4
2
~ a
gcm
eB
cm
ea
p
pa
p
p
a
p
p
aa
pap
paa
//
/
Frequencies can be
measured very precisely
+/ p=3.183 345 39(10)
from hyperfine splitting in muonium
W. Liu et al., Phys. Rev. Lett. 82, 711 (1999).
15
NMR trolley
17 trolley NMR probes
375 fixed NMR probes
around the ring
p /2π = 61 791 400(11) Hz (0.2ppm) ~
16
About 2.6 deviation:
• Often interpreted as sign of
new physics: SUSY
• But careful:
“Theory” has uncertainties …
… and sometimes even bugs.
• Quantum loop effects (SM or new
physics) are ~ m2 and therefore
more important for muons than
for electrons.
)7.0(10)8(21465911 10 ppma
)7.0(10)8(20365911 10 ppma
)5.0(10)6(20865911 10 ppma
17
5. Theoretical prediction of a
Beside pure QED corrections there are
weak corrections (W, Z) exchange and
„hadronic corrections“
EWHadQED aaaa
(For the electron with much lower mass
the hadronic and weak corrections are
suppressed (~m2), and can be neglected.)
Hadronic corrections
Determination of hadronic corrections
is difficult and is in addition based on
data: hot discussion amongst
theoreticians how to correctly use the
data.
Theory
Partic
le d
ata
gro
up 2
008
18
Hadronic vacuum polarization:
Hadronic corrections related to virtual intermediate
hadronic states ( , , ) – cannot be calculated.
Use the “optical theorem” to relate the loop corrections to
observable cross sections / branching ratios:
Im[ ] | hadrons |2
19
… calculations are sometimes not easy …
In 2001 Kinoshita et al. found a sign mistake in their calculation of the
light-by-light scattering amplitude:
20
Potential SUSY contribution to muon (g-2)
Potential SUSY contributions:
For muon ~40000 times larger
than in case of electrons.
SUSYEWHadQED aaaaa First sign of New Physics ??