An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 6.3

Radiative Corrections: Introduction — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · qed-exotic-contributions-g-minus-2

Problem

**Exotic contributions to g−2g-2.** Any particle that couples to the electron can produce a correction to the electron-photon form factors and, in particular, a correction to g−2g-2. Because the electron g−2g-2 agrees with QED to high accuracy, these corrections allow us to constrain the properties of hypothetical new particles. (a) The unified theory of weak and electromagnetic interactions contains a scalar particle hh called the *Higgs boson*, which couples to the electron according to Hint=∫d3xλ2 h ψˉψ.H_{\text{int}} = \int d^3x \frac{\lambda}{\sqrt{2}}\, h\,\bar{\psi}\psi. Compute the contribution of a virtual Higgs boson to the electron (g−2)(g-2), in terms of λ\lambda and the mass mhm_h of the Higgs boson. (b) QED accounts extremely well for the electron's anomalous magnetic moment. If a=(g−2)/2a = (g-2)/2, ∣aexpt.−aQED∣<1×10−10.|a_{\text{expt.}} - a_{\text{QED}}| < 1 \times 10^{-10}. What limits does this place on λ\lambda and mhm_h? In the simplest version of the electroweak theory, λ=3×10−6\lambda = 3 \times 10^{-6} and mh>60m_h > 60 GeV. Show that these values are not excluded. The coupling of the Higgs boson to the muon is larger by a factor (mμ/me)(m_\mu/m_e): λ=6×10−4\lambda = 6 \times 10^{-4}. Thus, although our experimental knowledge of the muon anomalous magnetic moment is not as precise, ∣aexpt.−aQED∣<3×10−8,|a_{\text{expt.}} - a_{\text{QED}}| < 3 \times 10^{-8}, one can still obtain a stronger limit on mhm_h. Is it strong enough? (c) Some more complex versions of this theory contain a pseudoscalar particle called the *axion*, which couples to the electron according to Hint=∫d3xiλ2 a ψˉγ5ψ.H_{\text{int}} = \int d^3x \frac{i\lambda}{\sqrt{2}}\, a\,\bar{\psi}\gamma^5\psi. The axion may be as light as the electron, or lighter, and may couple more strongly than the Higgs boson. Compute the contribution of a virtual axion to the g−2g-2 of the electron, and work out the excluded values of λ\lambda and mam_a.

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