Is there a simple explanation for Schwinger's relation $g=2+\frac{\alpha}{\pi}+{\cal O}(\alpha^2)$ for the $g$-factor of the electron?

Schwinger has on his grave (it seems) the relation between the g-factor of the electron and the fine structure constant:

$$g~=~2+\frac{\alpha}{\pi}+{\cal O}(\alpha^2)$$

Did Schwinger or somebody else ever give a simple explanation for the second term of the right hand side? The 2 appears from Dirac's equation. The second term is due to the emission and absorption of a photon. Is there a simple way to see that this process leads to the expression $\frac{\alpha}{\pi}$?

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This is a loop integral which is done elegantly in Weinberg, Schwinger's way. This is as simple as it gets. –  Ron Maimon Jul 18 '12 at 18:08
Also see chapter 6 of Peskin. –  DJBunk Jul 18 '12 at 18:34

The additional correction to the magnetic moment of the electron, aptly called the 'anomalous magnetic moment,' arises from a one loop Feynman diagram calculation in quantum electrodynamics. To be specific, the Landé $g$ factor is given by,

$$g=2[1+F_2(0)]$$

where $F_2$ is a 'form factor.' The electron vertex scattering amplitude is given by,

$$\require{cancel} \int \frac{\mathrm{d}^4k}{(2\pi)^4} \, \frac{-i\eta_{\nu \rho}}{(k-p)^2}\bar{u}(p')(-ie\gamma^\nu)\frac{i(\cancel{k'}+m)}{k'^2-m^2}\gamma^\mu \frac{i(\cancel{k}+m)}{k^2-m^2}(-ie\gamma^\rho) u(p)$$

$$=2ie^2 \int \frac{\mathrm{d}^4 k}{(2\pi)^4} \, \frac{\bar{u}(p')[\cancel{k}\gamma^\mu \cancel{k}' +m^2 \gamma^\mu -2m(k+k')^\mu]}{(k-p)^2(k'^2 -m^2)(k^2-m^2)}$$

Several methods are required to compute the amplitude, including the introduction of Feynman parameters to combine the propagators, Wick rotation and Pauli-Villars regularization, which roughly corresponds to the change,

$$\frac{1}{(k-p)^2 +i\epsilon} \to \frac{1}{(k-p)^2 +i\epsilon} - \frac{1}{(k-p)^2 -\Lambda^2 +i\epsilon}$$

introducing a fictitious 'photon mass.' The final result is given by,

$$\frac{\alpha}{2\pi} \int_0^1 \mathrm{d}x \, \mathrm{d}y \, \mathrm{d}z \, \delta(x+y+z-1) \times \bar{u}(p')\left(\gamma^\mu \left[ \log \frac{z\Lambda^2}{\triangle} +\frac{1}{\triangle} ((1-x)(1-y)q^2 +(1-4z+z^2)m^2)\right]+\frac{i\sigma^{\mu \nu}q_\nu}{2m}\left[ \frac{1}{\triangle} 2m^2 z(1-z)\right] \right)$$

where new variables have been introduced for convenience. Precisely how the result is derived is available in Chapter 6 of Peskin and Schroeder. One can extract the form factor from the amplitude, and one obtains (after many manipulations),

$$F_2(q^2 =0)=\frac{\alpha}{\pi}\int_0^1 \mathrm{d}z \, \int^{1-z}_0 \mathrm{d}y \, \frac{z}{1-z} = \frac{\alpha}{2\pi}$$

The coupling constant of quantum electrodynamics is $e$, and appears in the Feynman rules due to an interaction term $e\bar{\psi}\cancel{A}\psi$, and hence its appearance in the original amplitude. The fine structure constant, $\alpha = e^2 /4\pi$ and the famous result by Schwinger is usually stated in terms of it. The correction to the $g$ factor, numerically, is

$$a_e = \frac{g-2}{2}=\frac{\alpha}{2\pi} \approx 0.0011614$$

Although a minute correction, the result was highly significant and monumental. Loop calculations are by no means trivial. For Schwinger's original paper, see Physical Review, 73, 416L (1948).

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