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 ??