Anomalous magnetic dipole moment
Encyclopedia
In quantum electrodynamics
, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics
, expressed by Feynman diagram
s with loops, to the magnetic moment
of that particle. (The "magnetic moment
", also called "magnetic dipole moment", is a measure of the strength of a magnetic source.)
The "Dirac" magnetic moment
, corresponding to tree-level Feynman diagrams (which can be thought of as the classical result), can be calculated from the Dirac equation
. It is usually expressed in terms of the g-factor; the Dirac equation predicts
. For particles such as the electron
, this classical result differs from the observed value by a small fraction of a percent. The difference is the anomalous magnetic moment, denoted
and defined as
The one-loop contribution to the anomalous magnetic moment—corresponding to the first and largest quantum mechanical correction—of the electron is found by calculating the vertex function
shown in the diagram on the right. The calculation is relatively straightforward and the one-loop result is:
where
is the fine structure constant.
This result was first found by Schwinger
in 1948. As of 2009, the coefficients of the QED formula for the anomalous magnetic moment of the electron have been calculated through order
, and are known analytically up to 
.
The QED prediction agrees with the experimentally measured value to more than 10 significant figures, making the magnetic moment of the electron the most accurately verified prediction in the history of physics
. (See precision tests of QED
for details.)
The current experimental value and uncertainty is:
According to this value, a is known to an accuracy of around 1 part in 1 billion (109). This required measuring g to an accuracy of around 1 part in 1 trillion (1012)).
is calculated in a similar way; its measurement provides a precision test of the Standard Model
. The prediction for the value of the muon anomalous magnetic moment includes three parts: αμSM = αμQED + αμEW + αμhad. The first two components represent the photon and lepton loops, and the W boson and Z boson loops, respectively, and can be calculated precisely from first principles. The third term represents hadron loops, and cannot be calculated accurately from theory alone. It is estimated from experimental measurements of the ratio of hadronic to muonic cross sections (R) in e+e- collisions. As of November 2006, the measurement disagrees with the Standard Model by 3.4 standard deviation
s, suggesting beyond the Standard Model
physics may be having an effect (or theoretical/experimental errors not completely under control).
The E821 experiment at Brookhaven National Laboratory
(BNL) studied the precession of muon and anti-muon in a constant external magnetic field as they circulated in a confining storage ring. The E821 Experiment reported the following average value (from the July 2009 review by Particle Data Group)
where the first errors are statistical and the second systematic.
, which is made up of charged quark
s, and the neutron
, which has a magnetic moment even though it is electrically neutral.
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, expressed by Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s with loops, to the magnetic moment
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
of that particle. (The "magnetic moment
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
", also called "magnetic dipole moment", is a measure of the strength of a magnetic source.)
The "Dirac" magnetic moment
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
, corresponding to tree-level Feynman diagrams (which can be thought of as the classical result), can be calculated from the Dirac equation
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
. It is usually expressed in terms of the g-factor; the Dirac equation predicts
. For particles such as the electronElectron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
, this classical result differs from the observed value by a small fraction of a percent. The difference is the anomalous magnetic moment, denoted
and defined asAnomalous magnetic moment of the electron
Vertex function
In quantum electrodynamics, the vertex function describes the coupling between a photon and an electron beyond the leading order of perturbation theory...
shown in the diagram on the right. The calculation is relatively straightforward and the one-loop result is:
where
is the fine structure constant.This result was first found by Schwinger
Julian Schwinger
Julian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...
in 1948. As of 2009, the coefficients of the QED formula for the anomalous magnetic moment of the electron have been calculated through order
, and are known analytically up to .The QED prediction agrees with the experimentally measured value to more than 10 significant figures, making the magnetic moment of the electron the most accurately verified prediction in the history of physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. (See precision tests of QED
Precision tests of QED
Quantum electrodynamics , a relativistic quantum field theory of electrodynamics, is among the most stringently tested theories in physics....
for details.)
The current experimental value and uncertainty is:
According to this value, a is known to an accuracy of around 1 part in 1 billion (109). This required measuring g to an accuracy of around 1 part in 1 trillion (1012)).
Anomalous magnetic moment of the muon
The anomalous magnetic moment of the muonMuon
The muon |mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...
is calculated in a similar way; its measurement provides a precision test of the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
. The prediction for the value of the muon anomalous magnetic moment includes three parts: αμSM = αμQED + αμEW + αμhad. The first two components represent the photon and lepton loops, and the W boson and Z boson loops, respectively, and can be calculated precisely from first principles. The third term represents hadron loops, and cannot be calculated accurately from theory alone. It is estimated from experimental measurements of the ratio of hadronic to muonic cross sections (R) in e+e- collisions. As of November 2006, the measurement disagrees with the Standard Model by 3.4 standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
s, suggesting beyond the Standard Model
Beyond the Standard Model
Physics beyond the Standard Model refers to the theoretical developments needed to explain the deficiencies of the Standard Model, such as the origin of mass, the strong CP problem, neutrino oscillations, matter–antimatter asymmetry, and the nature of dark matter and dark energy...
physics may be having an effect (or theoretical/experimental errors not completely under control).
The E821 experiment at Brookhaven National Laboratory
Brookhaven National Laboratory
Brookhaven National Laboratory , is a United States national laboratory located in Upton, New York on Long Island, and was formally established in 1947 at the site of Camp Upton, a former U.S. Army base...
(BNL) studied the precession of muon and anti-muon in a constant external magnetic field as they circulated in a confining storage ring. The E821 Experiment reported the following average value (from the July 2009 review by Particle Data Group)
where the first errors are statistical and the second systematic.
Anomalous magnetic moment of composite particles
Composite particles often have a huge anomalous magnetic moment. This is true for the protonProton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
, which is made up of charged quark
Quark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
s, and the neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
, which has a magnetic moment even though it is electrically neutral.