Download The Fine Structure Constant and  Electron (g‐2) Factor Theory Review

4/26/2007
The Fine Structure Constant and Electron (g‐2) Factor
Nick Hutzler
Physics 135c
Spring 2007
Theory Review
• A (spin) magnetic moment μ quantifies how a particle interacts with magnetic fields
particle interacts with magnetic fields
• The electron has μ proportional to spin
μ=g
eh S
S
= gμ B
2m h
h
• The fine structure constant α characterizes the strength of the EM interaction
α=
e2
1
≈
4πε0 hc 137
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Theory Review
• The (spin) factor g is dimensionless and varies b
by particle type
ti l t
– Classical charged sphere: g = 1
– Dirac point particle: g = 2
– Neutron: g = ‐3.826 085 45(90)
– Proton: g Proton: g = 5.585
5.585 694 713(46)
• The anomalous magnetic moment is defined as a = (g‐2)/2
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Theory Review
• Why measure g?
– Probe of possible internal structure
– One of best tests of QFT/QED, new physics
– Fine structure constant α determined by g+QED
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Theory Review
• Why isn’t g = 2?
– TTree level Feynman l lF
diagram gives g = 2
– Higher order corrections from quantum fluctuations add ~.001
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Theory Review
• g and α are closely related
– a has a series expansion in α (remember that α is commonly expanded in perturbation series)
a = C1 ( απ ) + C2 ( απ ) + C3 ( απ ) + L + aμτ + ahadronic + aweak
2
3
– The Ci are calculated from QED
– The aμτ,hadronic,weak
μτ hadronic weak terms are the QED muon/tau contribution, and non‐QED hadronic and weak contributions (respectively)
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Theory Review
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History
• Earlier α measurements
– Quantum Hall effect
RK =
μ 0c
2α
– Atomic spectra
R∞ =
α 2 me c
2h
– Josephson effect
U=
h ∂φ
2e ∂t
– Recoil measurements, numerology, others
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Philosophy
• Why does α have this value?
– Anthropic Principle: small changes in the value of α
Principle: small changes in the value of α
would dramatically change the universe
– Numerology: Arthur Eddington calculated α=1/136 using numerology and gave the Eddington Number: N = 2256/ α
– Feynman: “one of the greatest damn mysteries of physics: a magic number that comes to us with no
physics: a magic number that comes to us with no understanding by man”
– James Gilson: α has a formula:
α=
cos(π / 137 ) tan(π / (137 ⋅ 29 ))
≈ 1 / 137.0359997867
137
π / (137 ⋅ 29 )
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The Experiment
• We shall discuss a recent measurement of g
and resulting calculation of α
d
lti
l l ti
f
• New Measurement of the Electron Magnetic Moment Using a One‐Electron Quantum Cyclotron
– Odom, Hanneke, D
Odom, Hanneke, D’Urso,
Urso, Gabrielse
Gabrielse
– Harvard Physics Department
– PRL 2006
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The Experiment
• Overview:
– Put an electron in cyclotron motion
– Measure jumps between lowest quantum states
– Calculate g
– Calculate α
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Penning Traps
• Static fields used to trap charged particles
charged particles
– Uniform axial B field
– Quadrupole E field
• Earnshaw’s Theorem: static electric fields alone cannot confine a
alone cannot confine a charged particle
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Penning Traps
• Four motions:
–
–
–
–
Spin (ν
Spin
(νs)
Cyclotron (νc = 2 νs/g)
Axial (νz)
Magnetron (νm)
• In the present experiment:
–
–
–
–
νs ≈ νcc’ ≈ 149 GHz
νz’ ≈ 200 MHz
νm’ ≈ 134 kHz
Prime denotes “actual”
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Particle Motion
• Spin motion is essentially free
• Cyclotron motion is damped by synchrotron radiation (~0.1 s in free space)
– However, in this trap the fields are tuned so that cyclotron motion is far from a cavity mode
– Lifetime extended to several seconds
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Energy Levels
• Measure jumps between lowest spin (ms) and cyclotron (n) levels
E (n , ms ) =
1
g
2
hν m + (n + 1 / 2 )hν c' − hδ(n + 1 / 2 + ms )
2 c s
2
• νa’ and fc’ are anomaly and spin‐up frequencies (respectively)
• νc is classical cyclotron frequency eB/2πm
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Energy Levels
• νc is obtained using the Brown Gabrielse
Brown‐Gabrielse Invariance Theorem:
(ν c )2 = (ν c' )2 + (ν z' )2 + (ν m' )2
• This allows us to calculate g/2:
l l
/2
g ν a' + ν c'
ν a' − ν 2z' / (2 f c' )
=
≅ 1+
2
νc
f c' + 3δ / 2 + ν 2z' / (2 f c' )
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What is Measured?
• Spin and cyclotron jumps induce a change
jumps induce a change in νz’
• A resonant circuit drives the axial motion at νz’
(SEO)
• Quantum Jump Quantum Jump
Spectroscopy
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What is Measured?
• Anomaly transitions are created by inducing
created by inducing spin‐flips with a perpendicular B field
• Cyclotron transitions are induced by microwaves
• Both drives are always B hd i
l
on, but tuned so that only one affects the electron
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Results
• g/2 = 1.001 159 652 180 85(76)
• That’s 76 parts per trillion
• Certainly of the most accurate measurements ever made
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Sources of Error
• Magnetic field instabilities
– After
After settling for several months, the superconducting settling for several months the superconducting
solenoid is constant to 10‐9/hours
– Maintain in 0.3K environment
– Perform measurements at night
• Fitting/statistics
– ~50 measurements/night
• Cavity radiation modes
Cavity radiation modes
• Blackbody Photon Excitation
– Cool sample to <100mK
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Fine Structure Constant
• This g‐value translates to
α‐1 = 137.035 999 710(90)(33)[0.66][0.24]
= 137.035 999 710(96)[0.70]
• The top line errors are experimental, QED respectively (second line gives total error)
• Square brackets give error in ppb
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Fine Structure Constant
• C4,C6,C8,C10 require the evaluation of 7 72 891 12 672 F
7,72,891,12 672 Feynman diagrams di
respectively
• C8 is known fairly well, C10 is still being calculated
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Important Conclusions
• QED continues to be a highly accurate d
description of the physical world
i ti
f th h i l
ld
• The electron can still be viewed as having no internal structure
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Electron Internal Structure
• Internal constituents must have m > 130 GeV
– Calculations made assume no internal structure
– Thus corrections must come from very high energy/short distance scales
– i.e. constituents must be very tightly bound
• No QED error: m No QED error: m > 600 GeV
600 GeV
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References
• New Measurement of the Electron Magnetic Moment Using a One‐Electron
Using a One
Electron Quantum Cyclotron
Quantum Cyclotron
Odom, Hanneke, D’Urso, Gabrielse. PRL 97, 030801
• New Determination of the Fine Structure Constant from the Electron g Value and QED
Gabrielse, Hanneke, Kinoshita, Hio, Odom. PRL 91 030802
• Fully Quantum Measurement of the Electron Magnetic Moment
Magnetic Moment
Brian Odom, Thesis, Harvard University, 2004
• A Finer Constant, Czarnecki, Nature 443 p. 516
• James Gilson, fine‐structure‐constant.org
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