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(12-246)
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(12-248)
where C Land C R are real coefficients, leads to a cross section
(12-249)
for the process e(p) + v(q) ..... e(p') + v(q'). The expression for the process ev ..... ev is obtained by interchanging C L and CR. When the incident neutrino energy is much larger than the electron mass, we have
2 2 m e q' q' '" m e p' p'
E'e me 2 me E 2 - v Ev Ev
Figure 12-13 Electron-neutrino scattering to lowest order.
NONABELIAN GAUGE FIELDS
and the last term in (12-249) may be neglected. We finally obtain in the various channels
<J(vee --> vee)
= _ _e_v [4
G2 m E
sin 4 8w
+ j-(1 + 2 sin 2 8W)2]
(12-250)
These expressions are compatible with the few observed vpe and vpe events and with the value of 8w derived from neutrino-nucleon inclusive reactions sin 2 8 w "" 0.25
0.02
(12-251)
Experimental investigation of the structure of neutral currents is stilI actively underway, in particular in atomic physics.
12-6-3 Higher-Order Corrections
A renormalizable theory of weak interactions enables us to compute higher-order corrections. In fact, in the model presented so far, only leptonic processes can be considered, such as the weak contribution to the muon anomalous magnetic moment or the radiative correction to the muon decay. The weak contributions to gp - 2 are shown in Fig. 12-14. For this one-loop computation, it is safer to use a renormalizable gauge of the form (12-232). The diagrams of Fig. 12-14a, b, c give contributions of the form
2 (a)
d k O(kmp, m~) (k2 _ M2)2 (p _ k
d k O(kmp, m~) k 2 - M2 [(p - k)2 - m~J2
(12-252)
"Y--J.'" "Y7" J.lyJ.l J.lyJ.l
(a) (b) (c)
Figure 12-14 Weak corrections to the muon anomalous magnetic moment.
QUANTUM FIELD THEORY
In cases (a) and (b), M stands for the mass of the W, of the Z, or of one of the would-be Goldstone bosons cP . Assume that we have chosen a gauge where'the latter is very large, of the order of Mw such as given by Eg. (12-223) with A. finite. The one corresponding to the combination (cPo - cP6)/j2 is also large, while the physical component (cPo + cP6)/j2 has an unknown mass m</>o' We assume, however, that m</>o is much larger than mil" In the contribution (c), we have taken the muon-scalar coupling
(12-253)
into account. Since the F 2 form factor does not require any ultraviolet subtraction, we expect the previous integrals to behave as m~/M2 or m~/m~o times a possible logarithmic factor. The weak corrections to the anomalous magnetic moment are thus at most of order
(12-254)
This is precisely the order of magnitude of both the experimental and theoretical uncertainties (the latter due to the hadronic contributions):
g - 21
= (11659.22
0.09) x
10- 7
(12-255)
g - 21
= (11659.19 0.10)
10- 7
A logarithmic factor In (m~/Mij,) might, however, make the weak contributions sizable. Notice that the contribution of the Higgs boson is suppressed by a factor m~/Mij, with respect to the two others, and is negligible. In the actual computation no logarithm appears, and the weak correction is small:
g - 21
weak
2 . = Gm~2 b- sm 4 Ow h
sm Ow
1 + 4)
~2xlO-9
(12-256)
A similar calculation may be carried out for the electron. The experimental and theoretical accuracies are higher (~10- 9 ), but the weak contribution is suppressed by an extra factor m;lm~ ~ 10- 5 . Radiative corrections to muon decay may also be obtained. The result is, of course, finite when expressed in terms of renormalized quantities, and amounts to a renormalization of the Fermi coupling constant G. This correction of order ('/. might be observed in comparison with the coupling in a different weak process such as f3 decay. Unfortunately, there is no other purely leptonic decay, and a comparison with a hadronic system involves strong interaction corrections.
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