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The \(g-2\) Experiments

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The Anomalous Magnetic Moment of the Muon

Part of the book series: Springer Tracts in Modern Physics ((STMP,volume 274))

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Abstract

The main concepts and early results from the CERN muon storage ring experiment have been summarized in Bailey et al. (Nuovo Cim A, 9:369, 1972, [1]). There are a number of excellent reviews on this subject and I am following in parts the ones of Combley, Farley and Picasso (Phys Rep, 14C:1, 1974, [2], Quantum Electrodynamics, World Scientific, Singapore, p 479, 1990, [3]) and of Vernon Hughes (The anomalous magnetic moment of the muon, 2001, [4]).

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Notes

  1. 1.

    Formulas like (6.1) presented in this first overview will be derived below.

  2. 2.

    L is a matrix operator acting on four–vectors. The \(\cdot \) operation at the right of the spacial submatrix means forming a scalar product with the spatial part of the vector on which L acts.

  3. 3.

    Note that \(\mathrm{d}\, \gamma =\gamma ^3 \,\mathbf {v} \cdot \mathrm{d}\mathbf {v}/c^2\) and the equation of motion implies

    $$\begin{aligned} \mathbf {v} \cdot \frac{\mathrm{d}\, (\gamma m\mathbf {v})}{\mathrm{d}t}=m\gamma ^3 \mathbf {v} \cdot \frac{\mathrm{d}\mathbf {v}}{\mathrm{d}t}=e \,\mathbf {v} \cdot \mathbf {E}\,, \end{aligned}$$

    as \(\mathbf {v} \cdot (\mathbf {v} \times \mathbf {B})\equiv 0\). This has been used in obtaining (6.35).

  4. 4.

    Magnetic focusing using an inhomogeneous field \(B_z=B_0\,(r_0/r)^n\), which by Maxwell’s equation \(\nabla \times \mathbf {B}=0\) implies \(B_r\simeq -n/r_0\,B_0\,z\) for \(r \simeq r_0\), leads to identical betatron oscillation equations (6.23) as electrostatic focusing.

  5. 5.

    The pitch frequency here should not to be confused with the proton precession frequency \(\omega _p\) appearing in (6.7).

  6. 6.

    Remembering the normalization: the magnetic and electric dipole moments are given by \(\mu =\frac{g}{2}\,\frac{e\hbar }{2mc}\) and \(d=\frac{\eta }{2}\,\frac{e\hbar }{2mc}\), respectively.

  7. 7.

    Only the recently established phenomenon of neutrino oscillations proves that lepton number in fact is not a perfect quantum number. This requires that neutrinos must have tiny masses and this requires that right–handed neutrinos (\(\nu _R\)’s) must exist. In fact, the smallness of the neutrino masses explains the strong suppression of lepton number violating effects.

  8. 8.

    At Brookhaven the 24 GeV proton beam extracted from the AGS with \(60\times 10^{12}\) protons per AGS cycle of 2.5 s impinges on a Nickel target of one interaction length and produces amongst other debris–particles a large number of low energy pions. The pions are momentum selected and then decay in a straight section where about one third of the pions decay into muons. The latter are momentum selected once more before they are injected into the \(g-2\) storage ring.

  9. 9.

    Note that the original electron phase space element \(\mathrm{d}V_e\equiv \frac{\mathrm{d}^3p_1}{E_e}\) is L–invariant such that with \( \mathrm{d}^3 p_1=-p_e^2 \mathrm{d}p_e\, \mathrm{d}\cos \theta \, \mathrm{d}\varphi \), after integrating over the azimuthal angle \(\varphi \), giving a factor \(2\pi \), and using \(p_e \mathrm{d}p_e= E_e \mathrm{d}E_e\) we infer that \(\mathrm{d}V_e \rightarrow 2\pi \,\sqrt{E_e^2-m_e^2} \mathrm{d}E_e\, \mathrm{d}\cos \theta \) is independent of the frame. While in the rest frame \(u_0p_1=-\mathbf {P}_\mu \mathbf {p}_1=-P_\mu p_e \cos \theta \) in the laboratory frame \(u_0=\left( 1,\frac{\mathbf {p}_0}{E_0-m}\right) \,\frac{\mathbf {p}_0\mathbf {P}_\mu }{m}\) and thus \(u_0p_1=\cos \theta _{\mu }\,\frac{p_\mu }{m}\,\left( E_e-\frac{p_\mu p_{1x}}{E_0-m}\right) \).

  10. 10.

    With \(Q^2{=}m_\mu ^2+m_e^2-2p_0p_1\), \(Qp_0{=}m_\mu ^2-p_0p_1\), \(Qp_1{=}p_0p_1-m_e^2\), and \(p_0p_1{=}m_\mu E_e\), \(E_e=xW,\,m_\mu =2W\) the curly bracket of (6.56) reads \(\left\{ \cdots \right\} =8W^4\,x\,\left[ (3-2x)+P_\mu \cos \theta \left( 2x-1\right) \right] +O(m_e^2/m_\mu ^2)\).

  11. 11.

    Note that, this is not what one gets by writing (6.56) in terms of laboratory system variables. It is rather a matter of how the geometrical acceptance of the decay positrons/electrons is affected by boosting the system.

  12. 12.

    This value is replacing \(\lambda =3.18334539(10)\) used in [24].

  13. 13.

    The gyromagnetic ratios of the bound electron and muon differ from the free ones by the binding corrections [38]

    $$\begin{aligned} g_J= g_e\,\left( 1-\frac{\alpha ^2}{3}+\frac{\alpha ^2}{2}\frac{m_e}{m_\mu } +\frac{\alpha ^3}{4\pi } \right) ~~,~~~ g_\mu '= g_\mu \,\left( 1-\frac{\alpha ^2}{3}+\frac{\alpha ^2}{2}\frac{m_e}{m_\mu } \right) \;. \end{aligned}$$

    .

  14. 14.

    What I mean is that, as in Sect. 6.3, one solves the Dirac equation in en external field (the first of the QED field Eq. (3.1) with zero radiation field \(A_\mu (x)\equiv 0\)) rather than the coupled QED field equations.

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  1. Institut für Physik, Humboldt-Universität zu Berlin, Berlin, Germany

    Friedrich Jegerlehner

  2. Deutsches Elektronen-Synchrotron (DESY), Zeuthen, Germany

    Friedrich Jegerlehner

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Jegerlehner, F. (2017). The \(g-2\) Experiments. In: The Anomalous Magnetic Moment of the Muon. Springer Tracts in Modern Physics, vol 274. Springer, Cham. https://doi.org/10.1007/978-3-319-63577-4_6

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