350
11. Loops and Renormalization II: QED
or, equivalently,
α
[(g − 2)/2] Schwinger =
≈ 0.0011614.
(11.83)
2π
Note that since κ is a dimensionless quantity, it cannot depend on the mass m
of the internal fermion in (11.66). Contributions from two-loop (and higher)
diagrams can involve different leptons in internal lines, and hence can depend
on lepton mass ratios.
The prediction (11.83) may be compared with the experimental values
which are, for the electron (Hanneke et al. 2008)
a e,expt ≡ [(g e − 2)/2] expt = 115 965 218 0.73 (0.28) × 10
−12 [0.24 ppb] (11.84)
and for the muon (Bennett et al. 2006)
a μ,expt ≡ [(g μ − 2)/2] expt = 116 592 080 (63) × 10
−11 [0.54 ppm], (11.85)
where the bracketed figures are the quoted uncertainties (statistical and systematic combined in quadrature). Of course, in Schwinger’s day the experimental accuracy was far different, but there was a real discrepancy (Kusch
and Foley 1947) with the Dirac value (a = 0). Schwinger’s one-loop calculation provided a fundamental early confirmation of QED, and was the start
of a long confrontation between theory and experiment which still continues.
The interested reader is referred to the extensive review by Jegerlehner and
Nyffeler (2009), upon which we shall draw in the following.
The extraordinarily precise values in (11.84) and (11.85) represent the
result of ever more sophisticated and imaginative experimentation. The measurement of a e,expt is some 2250 times more accurate than that of a μ,exp . Yet
the latter is capable of probing the Standard Model more deeply, for an interesting reason. Consider expanding the vacuum polarization formula (11.18)
in powers of m/Λ, having done the momentum integrals as in (10.51) and
removed the ln Λ divergence by the subtraction (11.33). The resulting expression will be finite as Λ → ∞, but for finite Λ it will contain Λ-dependent
terms, the first being of order (m
2 /Λ
2 ). This suggests that the contribution
of a ‘beyond QED physics’ scale to a μ,theory (modelled crudely by our cut-off)
would be enhanced by a factor (m μ /m e )
2
≈ 43 000 relative to its contribu1
tion to a e,theory . This outweighs by a factor of 19 the greater experimental
accuracy in a e,exp .
This is both good news and bad news. We may distinguish three distinct
contributions to ‘beyond QED physics’ in a e,theory and a μ,theory : (i) SM weak
interactions; (ii) SM strong (or hadronic) interactions; (iii) beyond the SM
physics. Representative diagrams contributing to (i) and (ii) are shown in
figure 11.10 (a) and (b) respectively. Sensitivity of a e,theory to effects under (i)
is welcome, since they are calculable, and in principle may provide precision
1 The sensitivity would be even greater for aτ of course, but the very short lifetime of
the τ precludes an accurate measurement of its magnetic moment, at present.
11. Loops and Renormalization II: QED
or, equivalently,
α
[(g − 2)/2] Schwinger =
≈ 0.0011614.
(11.83)
2π
Note that since κ is a dimensionless quantity, it cannot depend on the mass m
of the internal fermion in (11.66). Contributions from two-loop (and higher)
diagrams can involve different leptons in internal lines, and hence can depend
on lepton mass ratios.
The prediction (11.83) may be compared with the experimental values
which are, for the electron (Hanneke et al. 2008)
a e,expt ≡ [(g e − 2)/2] expt = 115 965 218 0.73 (0.28) × 10
−12 [0.24 ppb] (11.84)
and for the muon (Bennett et al. 2006)
a μ,expt ≡ [(g μ − 2)/2] expt = 116 592 080 (63) × 10
−11 [0.54 ppm], (11.85)
where the bracketed figures are the quoted uncertainties (statistical and systematic combined in quadrature). Of course, in Schwinger’s day the experimental accuracy was far different, but there was a real discrepancy (Kusch
and Foley 1947) with the Dirac value (a = 0). Schwinger’s one-loop calculation provided a fundamental early confirmation of QED, and was the start
of a long confrontation between theory and experiment which still continues.
The interested reader is referred to the extensive review by Jegerlehner and
Nyffeler (2009), upon which we shall draw in the following.
The extraordinarily precise values in (11.84) and (11.85) represent the
result of ever more sophisticated and imaginative experimentation. The measurement of a e,expt is some 2250 times more accurate than that of a μ,exp . Yet
the latter is capable of probing the Standard Model more deeply, for an interesting reason. Consider expanding the vacuum polarization formula (11.18)
in powers of m/Λ, having done the momentum integrals as in (10.51) and
removed the ln Λ divergence by the subtraction (11.33). The resulting expression will be finite as Λ → ∞, but for finite Λ it will contain Λ-dependent
terms, the first being of order (m
2 /Λ
2 ). This suggests that the contribution
of a ‘beyond QED physics’ scale to a μ,theory (modelled crudely by our cut-off)
would be enhanced by a factor (m μ /m e )
2
≈ 43 000 relative to its contribu1
tion to a e,theory . This outweighs by a factor of 19 the greater experimental
accuracy in a e,exp .
This is both good news and bad news. We may distinguish three distinct
contributions to ‘beyond QED physics’ in a e,theory and a μ,theory : (i) SM weak
interactions; (ii) SM strong (or hadronic) interactions; (iii) beyond the SM
physics. Representative diagrams contributing to (i) and (ii) are shown in
figure 11.10 (a) and (b) respectively. Sensitivity of a e,theory to effects under (i)
is welcome, since they are calculable, and in principle may provide precision
1 The sensitivity would be even greater for aτ of course, but the very short lifetime of
the τ precludes an accurate measurement of its magnetic moment, at present.