What is the QFT prediction for the electron $g$-factor to high precision?

Question

NIST quotes the electron $g$-factor to 15 significant figures with a fractional uncertainty of about $0.2$ parts-per-trillion:

$$ g_e = −2.002\, 319\, 304\, 362\, 56(35)$$

I know this agrees with the QFT prediction, but I can't find the theoretical prediction quoted to the same level of precision.

Aoyama, et al (2007) (arXiv link) calculate $a_e = \frac{|g_e|-2}{2}$ to high precision. I used that to calculate $g_e$ to parts-per-billion precision.

$$a_e = 1\,159\,652\,182.78 (7.72) \times 10^{-12} \implies g_e = -2.002\,319\,304\,365(15) $$

The uncertainty is dominated by uncertainty in the fine structure constant. Is there a more recent QFT calculation? Do recent measurements of $g_e$ exceed the precision of theoretical calculations?

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This question, Theoretical calculations of the electron g-factor in QED, and others like it ask about renormalization and practical concerns in doing the calculation, but do not ask for a precise numerical result.

Answer 1

The experimental results are indeed substantially more precise than the theoretical calculations. The precision of the theoretical result is limited by a couple of things. Firstly, in order to compare a prediction to the experimental number, another experimental input is needed to get the value of the fine structure constant. There are other precise measurements that may be made in QED, but comparisons are ultimately limited to the (experimental and theoretical) accuracy of the second-best-known quantity. For this reason, the measurements of the anomalous magnetic moment is often described not as a verification of the theory but rather as—assuming the theory to be correct—the most precise measurement of the fine structure constant $\alpha$.

The other limitation comes from that fact that if you want to go to really high precision in calculations of $g_{e}$, you have to go beyond just QED. For pure QED calculations, going to higher an higher orders may be tedious and involve a lot of integrals, but how to do the calculation is known. In fact, modern calculations at high orders in $\alpha$ are typically partially automated; computers are programmed to do the algebra to reduce the results from a large number of Feynman diagrams to a human-tractable calculation. However, to go to really high precision, it is necessary to include additional interactions beyond electromagnetism. For weak ($W$ and $Z$) interactions, this is not too challenging, but including strong interaction effects is really hard. For the muon anomalous magnetic moment $g_{\mu}$, the error has been dominated by effects with virtual hadrons for decades; such terms are of lesser relative importance for the electron calculation, but they are present and will make a key contribution to the theoretical error.

Answer 2

Thanks to @AccidentalFourierTransform's comment, I found what appears to be the most recent theoretical calculation from Aoyama, et al (2019). It includes 10th order QED plus strong interaction hadronic terms and weak interaction terms.

The uncertainty on the QED contribution is at the same level as the hadronic terms, but the uncertainty is still dominated by the uncertainty on the fine structure constant as determined by cesium recoil experiments. They note a $2.4\sigma$ discrepancy between their theoretically calculated $a_e$ and the best experimental result (Hanneke et al, 2008), which is the same measurement reported by NIST.

Using $a_e$ as reported by Aoyama, et al (2019), I find $$ g_e = −2.002\,319\, 304\, 363\, 21(23).$$ This uncertainty is two orders of magnitude smaller than the 2007 result quoted in the question and at the same order as the experimental result.