3. anomalous magnetic moment - physikalisches institutmenzemer/pp_ss2012/magmo… · 1 8.5.3 higher...
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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
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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.
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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
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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
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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:
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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
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432
....9144.1....182.1...328.02
eaKinoshita 2007
2
2ga
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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
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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
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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
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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
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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)
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)cos(1)( 0 tAeNtN a
t
Measure electron rate:
Hz)16(59.0232292
a24 detectors
(0.7ppm)
Bcm
ea a
?
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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).
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NMR trolley
17 trolley NMR probes
375 fixed NMR probes
around the ring
p /2π = 61 791 400(11) Hz (0.2ppm) ~
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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
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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
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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
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… calculations are sometimes not easy …
In 2001 Kinoshita et al. found a sign mistake in their calculation of the
light-by-light scattering amplitude:
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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 ??