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Figure 7-14 Lowest-order contributions to the interaction with an external field.
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348 QUANTUM FIELD THEORY
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acting on Ae. Using Gordon's identity we obtain the two equivalent expressions of the interaction hamiltonian of a slowly varying (in space and time) external field with a charged spinning particle:
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~ e fd 3X {_i lj/(x)8 ljJ(x) 2m
[1 -
_IX_ 2
(In ~ - ~ -!) oJ
Ag(x)
(7-77)
(1 + 2:) 4~ If/(X)(JvpljJ(X)F~P(X)}
In the second form the first term introduces a convective current analogous to the one pertaining to scalar particles, and its interaction with Ae is affected by infrared divergences. The second term has a simple interpretation. In the limit of a constant Fe it reduces to an effective magnetic dipole energy. Let us assume that F represents a constant magnetic field F12 = - B 3, F23 = - Bl, F 3i = - B2, (J12 = (J3, etc. It then reads
-B' -B {2: (1 + 2:)2 f
3 d x If/(x)
~ ljJ(X)J
(7-78)
When discussing the Dirac equation we found a gyromagnetic ratio 2 leading to a magnetic dipole moment e e (J -(J=2-2m 2m 2 We see that to order h quantum corrections modify this gyro magnetic ratio written in the form g = 2(1 + a), where a is called the anomaly, by a = IX/2n. This result first derived by Schwinger in 1948 has been confirmed by numerous accurate experiments, which enable us to test higher-order corrections.
The electron anomaly is mass independent to lowest order. Therefore, the same result applies as well to any elementary spin i particle such as the muon. A complete theory to higher orders requires the consideration of all charged particles interacting with the electromagnetic field. This leads to a difference in these anomalies arising from mass differences. At the present time, calculations have been pushed to third order (three loops). They read
1 IX a;h = 2 ~ - 0.328 478 445 (IX)2 + 1.183 (11) (IX) = 1159652359(282) x 10- 12 ~ ~
(Numbers in parentheses indicate uncertainties.) The latest measurements are of a similar accuracy and are likely to be improved in the near future:
a~xp =
1 159652410 (200) x 10- 12
To cope with such an improvement the theoretical program involves obtaining the (IX/n)3 term with greater accuracy and evaluating the next order [since (IX/n)4 ~ 29 x 10- 12 ] represented by as many
RADIATIVE CORRECTIONS
as 891 Feynman diagrams! Minor contributions involve the insertions of a muon-loop vacuum polarization correction in the virtual photon propagator (~2 x 10- 12 ) together with hadronic (~1.4 x 10- 12 ) or weak (~0.05 x 10- 12 ) contributions. The measurement of the electron anomaly is likely to become in the near future one of the most precise means of obtaining an accurate value of the fine structure constant ()(, by a unique combination of theoretical and experimental skills. In the case of the muon anomaly the uncertainties on hadronic corrections set a limit on the accuracy of theoretical predictions. The most recent measurement gives
a;xp = 1 165922 (9) x 10- 9
The pure electrodynamic contributions are
a~ed =
1 ()( (()()2 (()()3 :2; + 0.765 782; + 24.45 (0.06); + 128.3 (71.4) (()()4 ;
1 165851.8 (2.4) x 10- 9
where the ()(4 term is an estimate using the large ratio mp/m" and the hadronic correction is of order
d,:adr =
66.7 (9.4) x 10- 9
The theory is in reasonable agreement with experiment with a prediction
a~h = 1 165919 (10) x 10- 9
Weak interaction effects are expected to contribute at the level of a:eak ~ 2 x 10- 9 .
The first term in the effective hamiltonian (7-77) appears troublesome in view of its infrared singularity. This arose from our insistence on having an isolated pole in the corresponding Green function. However, such a state can never be separated from those including an arbitrary number of soft photons. The latter are always excited as soon as a charged particle suffers a change in velocity, no matter how small. Therefore we have to include processes involving real emission or absorption of soft photons in the external field as soon as we discuss the virtual electromagnetic effects embodied in Eq. (7-77). The fictitious photon mass disappears from the final answer to a physically correct question, being replaced by the experimental resolution.
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