An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 21.1

Quantization of Spontaneously Broken Gauge Theories — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · electroweak-radiative-corrections

Problem

**21.1 Weak-interaction contributions to the muon

g - 2

.** The GWS model of the weak interactions leads to two new contributions to the anomalous magnetic moments of the leptons. Because these contributions are proportional to

G_F m_\ell^2

, they are extremely small for the electron, but for the muon they might possibly be observable. Both contributions are larger than the contribution of the Higgs boson discussed in Problem 6.3. (a) Consider first the contribution to the muon electromagnetic vertex function that involves a

W

-neutrino loop diagram. In the

R_\xi

gauges, this diagram is accompanied by diagrams in which

W

propagators are replaced by propagators for Goldstone bosons. Compute the sum of these diagrams in the Feynman-'t Hooft gauge and show that, in the limit

m_W \gg m_\mu

, they contribute the following term to the anomalous magnetic moment of the muon:

a_\mu(\nu) = \frac{G_F m_\mu^2}{8\pi^2\sqrt{2}} \cdot \frac{10}{3}.

(b) Repeat the calculation of part (a) in a general

R_\xi

gauge. Show explicitly that the result of part (a) is independent of

\xi

. (c) A second new contribution is that from a

Z

-muon loop diagram and the corresponding diagram with the

Z

replaced by a Goldstone boson. Show that these diagrams contribute

a_\mu(Z) = -\frac{G_F m_\mu^2}{8\pi^2\sqrt{2}} \cdot \left(\frac{4}{3} + \frac{8}{3}\sin^2\theta_w - \frac{16}{3}\sin^4\theta_w\right).

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