- Precision tests of QED
(Quantum electrodynamics **QED**), a relativistic quantum field theory of electrodynamics, is among the most stringently tested theories inphysics .Precision tests of QED consist of measurements of the electromagnetic

fine structure constant , α, in different physical systems. Checking the consistency of such measurements tests the theory.Tests of a theory are normally carried out by comparing experimental results to theoretical predictions. In QED, there is some subtlety in this comparison, because theoretical predictions require as input an extremely precise value of α, which can only be obtained from another precision QED experiment. Because of this, the comparisons between theory and experiment are usually quoted as independent determinations of α. QED is then confirmed to the extent that these measurements of α from different physical sources agree with each other.

The agreement found this way is to within ten parts in a billion (10

^{−11}). This makes QED one of the most accurate physical theories constructed thus far, afterspecial relativity , which currently is tested to 10^{−21}, [*M.P.Haugan and C.M.Will, "Testing Local Lorentz Invariance using laboratory and space technology", Advances in Space Research, Volume 9, Issue 9, (1989), pp.133–137*] the Hughes–Drever experiment: 10^{−16},Fact|date=March 2008 and the trapped atoms experiments: 3×10^{−22}.Fact|date=March 2008**Precision QED experiments**Precision tests of QED have been performed in low-energy

atomic physics experiments, high-energycollider experiments, andcondensed matter systems. The value of α is obtained in each of these experiments by fitting an experimental measurement to a theoretical expression (including higher-order radiative corrections) that includes α as a parameter. The uncertainty in the extracted value of α includes both experimental and theoretical uncertainties. This program thus requires both high-precision measurements and high-precision theoretical calculations. Unless noted otherwise, all results below are taken from [*M.E. Peskin and D.V. Schroeder, "An Introduction to Quantum Field Theory" (Westview, 1995), p. 198.*] .**Low-energy measurements****Anomalous magnetic dipole moments**The most precise measurement of α comes from the

anomalous magnetic dipole moment , or g−2 ("g minus 2"), of theelectron . [*"In Search of Alpha," New Scientist, 9 September 2006, p. 40–43.*] To make this measurement, two ingredients are needed:: 1) A precise measurement of the anomalous magnetic dipole moment, and: 2) A precise theoretical calculation of the anomalous magnetic dipole moment in terms of α.As of February 2007, the best measurement of the anomalous magnetic dipole moment of the electron was made by Gabrielse et al. [

*B. Odom, D. Hanneke, B. D'Urso, and G. Gabrielse, "New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron," Phys. Rev. Lett. 97, 030801 (2006).*] using a single electron caught in aPenning trap . The difference between the electron's cyclotron frequency and its spin precession frequency in a magnetic field is proportional to g−2. An extremely high precision measurement of the quantized energies of the cyclotron orbits, or "Landau level s", of the electron, compared to the quantized energies of the electron's two possible spin orientations, gives a value for the electron's sping-factor :: g/2 = 1.001 159 652 180 85 (76),

a precision of better than one part in a trillion. (The digits in parentheses indicate the uncertainty in the last listed digits of the measurement.)

The current state-of-the-art theoretical calculation of the anomalous magnetic dipole moment of the electron includes QED diagrams with up to four loops. Combining this with the experimental measurement of g yields the most precise value of α: [

*G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Odom, "New Determination of the Fine Structure Constant from the Electron g Value and QED," Phys. Rev. Lett. 97, 030802 (2006), Erratum, Phys. Rev. Lett. 99, 039902 (2007).*]: α

^{−1}= 137.035 999 070 (98),a precision of better than a part in a billion. This uncertainty is ten times smaller than the nearest rival method involving atom-recoil measurements.

A value of α can also be extracted from the anomalous magnetic dipole moment of the

muon . The g-factor of the muon is extracted using the same physical principle as for the electron above – namely, that the difference between the cyclotron frequency and the spin precession frequency in a magnetic field is proportional to g−2. The most precise measurement comes fromBrookhaven National Laboratory 's muon g−2 experiment, [*Pictorial overview of the Brookhaven muon g−2 experiment, [*] in which polarized muons are stored in a cyclotron and their spin orientation is measured by the direction of their decay electrons. As of February 2007, the current world average muon g-factor measurement is, [*http://www.g-2.bnl.gov/physics/index.html*] .*Muon g−2 experiment homepage, [*]*http://www.g-2.bnl.gov/*] .: g/2 = 1.001 165 920 8 (6),

a precision of better than one part in a billion. The difference between the g-factors of the muon and the electron is due to their difference in mass. Because of the muon's larger mass, contributions to the theoretical calculation of its anomalous magnetic dipole moment from

Standard Model weak interactions and from contributions involvinghadrons are important at the current level of precision, whereas these effects are not important for the electron. The muon's anomalous magnetic dipole moment is also sensitive to contributions from new physicsbeyond the Standard Model , such assupersymmetry . For this reason, the muon's anomalous magnetic moment is normally used as a probe for new physics beyond the Standard Model rather than as a test of QED. [*K. Hagiwara, A.D. Martin, Daisuke Nomura, and T. Teubner, "Improved predictions for g−2 of the muon and α*]_{QED}(M_{Z}²)", Phys.Lett. B649, 173 (2007), [*http://arxiv.org/abs/hep-ph/0611102 hep-ph/0611102*] .**Atom-recoil measurements**This is an indirect method of measuring α, based on measurements of the masses of the electron, certain atoms, and the

Rydberg constant . The Rydberg constant is known to seven parts in a trillion. The mass of the electron relative to that ofcaesium andrubidium atoms is also known with extremely high precision. If the mass of the electron can be gotten with high enough precision, then α can be found from the Rydberg constant according to:$R\_infty\; =\; frac\{alpha^2\; m\_e\; c\}\{4\; pi\; hbar\}$.

To get the mass of the electron, this method actually measures the mass of an

^{86}Rb atom by measuring the recoil speed of the atom after it emits a photon of known wavelength in an atomic transition. Combining this with the ratio of electron to^{86}Rb atom, the result for α is, [*Pierre Cladé, Estefania de Mirandes, Malo Cadoret, Saïda Guellati-Khélifa, Catherine Schwob, François Nez, Lucile Julien, and François Biraben, "Determination of the Fine Structure Constant Based on Bloch Oscillations of Ultracold Atoms in a Vertical Optical Lattice," Phys. Rev. Lett. 96, 033001 (2006).*]: α

^{−1}= 137.035 998 78 (91).Because this measurement is the next-most-precise after the measurement of α from the electron's anomalous magnetic dipole moment described above, their comparison provides the most stringent test of QED, which is passed with flying colors: the value of α obtained here is within one standard deviation of that found from the electron's anomalous magnetic dipole moment, an agreement to within ten parts in a billion.

**Neutron Compton wavelength**This method of measuring α is very similar in principle to the atom-recoil method. In this case, the accurately known mass ratio of the electron to the

neutron is used. The neutron mass is measured with high precision through a very precise measurement of itsCompton wavelength . This is then combined with the value of the Rydberg constant to extract α. The result is,: α

^{−1}= 137.036 010 1 (5 4).**Hyperfine splitting**Hyperfine splitting is a splitting in the energy levels of anatom caused by the interaction between themagnetic moment of the nucleus and the combined spin and orbital magnetic moment of the electron. The hyperfine splitting inhydrogen , measured using Ramsey's hydrogenmaser , is the most precisely known quantity in physics. Unfortunately, the influence of theproton 's internal structure limits how precisely the splitting can be predicted theoretically. This leads to the extracted value of α being dominated by theoretical uncertainty:: α

^{−1}= 137.036 0 (3).The hyperfine splitting in

muonium , an "atom" consisting of an electron and an antimuon, provides a more precise measurement of α because the muon has no internal structure:: α

^{−1}= 137.035 994 (18).**Lamb shift**The

Lamb shift is a small difference in the energies of the 2 S_{1/2}and 2 P_{1/2}energy levels of hydrogen, which arises from a one-loop effect in quantum electrodynamics. The Lamb shift is proportional to α^{5}and its measurement yields the extracted value:: α

^{−1}= 137.036 8 (7).**Positronium**Positronium is an "atom" consisting of an electron and apositron . Whereas the calculation of the energy levels of ordinary hydrogen is contaminated by theoretical uncertainties from the proton's internal structure, the particles that make up positronium have no internal structure so precise theoretical calculations can be performed. The measurement of the splitting between the 2^{3}S_{1}and the 1^{3}S_{1}energy levels of positronium yields: α

^{−1}= 137.034 (16).Measurements of α can also be extracted from the positronium decay rate. Positronium decays through the annihilation of the electron and the positron into two or more

gamma-ray photons. The decay rate of the singlet ("para-positronium")^{1}S_{0}state yields: α

^{−1}= 137.00 (6),and the decay rate of the triplet ("ortho-positronium")

^{3}S_{1}state yields: α

^{−1}= 136.971 (6).This last result is the only serious discrepancy among the numbers given here, but there is some evidence that uncalculated higher-order quantum corrections give a large correction to the value quoted here.

**High-energy QED processes**The cross sections of higher-order QED reactions at high-energy electron-positron colliders provide a determination of α. In order to compare the extracted value of α with the low-energy results, higher-order QED effects including the running of α due to

vacuum polarization must be taken into account. These experiments typically achieve only percent-level accuracy, but their results are consistent with the precise measurements available at lower energies.The cross sections for $e^+e^-\; o\; e^+e^-e^+e^-$ yields

: α

^{−1}= 136.5 (2.7),and the cross section for$e^+e^-\; o\; e^+e^-\; mu\; ^+mu\; ^-$ yields

: α

^{−1}= 139.9 (1.2).**Condensed matter systems**The

quantum Hall effect and the ACJosephson effect are exotic quantum interference phenomena in condensed matter systems. These two effects provide a standardelectrical resistance and a standardfrequency , respectively, which are believedFact|date=February 2007 to measure the charge of the electron with corrections that are strictly zero for macroscopic systems.The quantum Hall effect yields

: α

^{−1}= 137.035 997 9 (3 2),and the AC Josephson effect yields

: α

^{−1}= 137.035 977 0 (7 7).**References****External links*** [

*http://pdg.lbl.gov/ Particle Data Group (PDG)*]

* [*http://pdg.lbl.gov/2007/reviews/g-2_s004219.pdf PDG Review of the Muon Anomalous Magnetic Moment as of July 2007*]

* [*http://pdg.lbl.gov/2007/listings/s003.pdf PDG 2007 Listing of particle properties for electron*]

* [*http://pdg.lbl.gov/2007/listings/s004.pdf PDG 2007 Listing of particle properties for muon*]

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