Theory of Anomalous Magnetic Dipole Moments of the Electron
The anomalous magnetic dipole moment of the electron, the so-called electron \(g-2\) , provides us with a high-precision test of quantum electrodynamics (QED), which is the relativistic and quantum-mechanical generalization of electromagnetism, and helps
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Theory of Anomalous Magnetic Dipole Moments of the Electron Masashi Hayakawa
Abstract The anomalous magnetic dipole moment of the electron, the so-called electron g − 2, provides us with a high-precision test of quantum electrodynamics (QED), which is the relativistic and quantum-mechanical generalization of electromagnetism, and helps to determine the value of the fine structure constant α, one of the fundamental physical constants. This article intends to give a pedagogical introduction to the theory of g − 2, in particular, the computation of high-order quantum electrodynamics in g − 2.
2.1 Introduction A single electron is known to become magnetized due to its intrinsic charge and spin. Its magnetic dipole moment is given by µ = −ge
e s, 2m e c
(2.1)
where m e and s denote the mass and spin of an electron, respectively. The constant ge represents the strength of the magnetic dipole moment in units of the Bohr magneton, and is called the g-factor of the electron. At the zeroth order of perturbation, QED predicts that gψ is equal to 2 for every massive particle ψ with spin 21 . The quantum correction in general shifts gψ from 2, depending on the particle species ψ. It is thus convenient to focus on this shift, called the ‘anomalous magnetic dipole moment’, by introducing a symbol
M. Hayakawa (B) Department of Physics, Nagoya University, 464-8602 Nagoya, Japan e-mail: [email protected] W. Quint and M. Vogel (eds.), Fundamental Physics in Particle Traps, Springer Tracts in Modern Physics 256, DOI: 10.1007/978-3-642-45201-7_2, © Springer-Verlag Berlin Heidelberg 2014
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aψ ≡
gψ − 2 . 2
(2.2)
We shall call this quantity g − 2 of ψ hereafter. An important point to notice is the fact that the electron g −2, ae , can be measured very accurately. The most accurately measured value of the electron g − 2 ae (HV08) = 1 159 652 180.73 (28) × 10−12 ,
(2.3)
where the numerals in the parenthesis represent the uncertainty in the final few digits, was obtained by the Harvard group using a Penning trap with cylindrical cavity [1, 2]. The uncertainty (2.3) is smaller than the one obtained by Washington university group in 1987 [3] by a factor 15. See the previous chapter by Gabrielse et al. for a detailed explanation about how such a large reduction of uncertainty has been achieved. Even if g −2 is measured very accurately, one may wonder what physical implication it possesses. Worthy of note is the fact that aψ is a predictable quantity in so far as the theory is renormalizable in the framework of quantum field theory. (Section 2.2 introduces the notion and the foundation of the quantum field theory for the readers not specialized in particle physics.) We are thus inclined to question the validity of a renormalizable model of elementary particles by asking the compatibility of its theoretical prediction aψ (th) with the experimentally measured value aψ (exp). The standard model of elementary particles has endured most of tests for these forty years. It is a renormalizable theory and
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