# First order electroweak correction to the g-2 magnetic moment

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• dextercioby

#### dextercioby

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TL;DR Summary
Basic question for a theoretical physicist regarding perturbation theory calculations.
We know that we need to go to 5th order in perturbation theory to match 10 decimals of g-2 for electron, theory vs. experiment. But let us not assume QED is pure and independent, but it's a lower energy limit of GSW (not Green-Schwartz-Witten from superstrings) electroweak theory. Has anyone theoretically computed a 1st (2nd or whatever) order correction to the g-2 momentum of the electron? How many one loops would be required? Which decimal of g-2 only from QED would it affect?

P.S. I hope this is not a dumb question (like there is some conservation law prohibiting the 3 Ws to appear in the possible tree-level or loop FDs). While studying physics ages ago it just did not pop up in my mind.

The weak contribution is significantly smaller and contributes less than a part per billion iirc. The hadronic contribution is larger and taken into account as well.

I also believe GWS theory is typically written with Weinberg’s W in the middle although you sometimes see GSW also …

malawi_glenn and vanhees71
Yes, I know the order of publishing. So you say one per billion. Is that an SI billion (10^-12), or an American billion (10^-9) and a fraction out of what, to be precise? Of course, a proper article reference would help (also for QCD).

vanhees71
malawi_glenn, vanhees71 and dextercioby
muon g-2 has been the question of interest.

muon g-2 has been the question of interest.
OP writes:
We know that we need to go to 5th order in perturbation theory to match 10 decimals of g-2 for electron, theory vs. experiment.
Has anyone theoretically computed a 1st (2nd or whatever) order correction to the g-2 momentum of the electron?
My emphasis.

malawi_glenn
For the electron g-2 even the leading order of weak diagrams is tiny (~5% of the overall uncertainty, see the paper Orodruin linked).
For the muon the weak diagrams need to be considered: ##a=153.6 \pm 1.0 \cdot 10^{-11}## from two-loop calculations in this 2016 presentation, that's ~3 times the total theory uncertainty. It's still a well-known quantity.

vanhees71
My exact numbers below are outdated, but the starting point is that the state of the art theoretical prediction as of 2017 from the Standard Model of particle physics regarding the value of muon g-2 in units of 10-11 is:

QED 116 584 718.95 ± 0.08
HVP 6 850.6 ± 43 (part of the QCD component)
HLbL 105 ± 26 (part of the QCD component)
EW 153.6 ± 1.0

Total SM 116 591 828 ± 49

As a practical matter, the QED and EW parts of the calculation could be much less precise than they are now and still make now material impact on the overall SM calculation of muon g-2, because these errors are dwarfed by the QCD uncertainties. The QED part could be about 100 times less precise and still make little difference in the overall uncertainty (since components of the uncertainty are added in quadrature to the extent that they are uncorrelated). And, the weak force part could be about 10 times less precise.

Which decimal of g-2 only from QED would it affect?

It isn't a QED only calculation although the QCD and weak force effects for electron g-2 are very small and the errors are equally tiny, so the potential uncertainty from the other parts of the calculation aren't that important.

Electron g-2 is reviewed, for example, here and here. In this case, most of the uncertainty comes from the experimental determination of the electromagnetic force coupling constant since the weak force and QCD contributions are much smaller relative to the total result than in the muon and tau cases (basically and non-rigorously because there is less mass-energy available in the system studied to leverage into virtual particles with non-negligible impacts on the overall result).

In step with the progress of measurement, the theory of ae, expressed as a power series in α, has been pushed to the fifth power of α. Including small contributions from hadronic effects and weak interaction effect and using the best non-QED value of α: α-1 = 137.035999049(90), one finds ae (theory) = 1159652181.72 (77) ×10-12. The uncertainty is about 0.66 ppb, where 1 ppb =10-9 . The intrinsic uncertainty of theory itself is less than 0.1 ppb . The overall uncertainty comes mostly from the uncertainty of non-QED α mentioned above, which is about 0.66 ppb .

(Source).

Since the predicted value and the experimental value match (in part, because QCD contributions with great uncertainty are so much less important), we know that we don't need the corrections from BSM theories yet, and any BSM theory that has an effect before the 10th significant digit is very likely wrong.

I haven't, in a very quick and dirty search, been able to locate papers quantifying electron g-2 value modifications expected from various beyond the Standard Model modifications of QED or the Standard Model more generally. I'm sure that they are out there for some particular theories although not necessarily the specific one mentioned in the opening question.

Last edited:
vanhees71