Calculating higher order terms for Electron Anomalous Magnetic Moment

[Adrian59]

TL;DR Summary

Does anyone know how to calculate the coefficients for higher order (sixth and beyond) terms for the electron anomalous magnetic moment?

Does anyone know how to calculate the coefficients for higher order (sixth and beyond) terms for the electron anomalous magnetic moment? I have a clear understanding of calculating the Schwinger term, but beyond that I cannot find anything in the usual QFT textbooks. The usual papers often quoted (eg Peterman, Levine, Kinoshita and Laporta) are short on detail. I know that the number of terms rises with the increasing order so that the higher order terms require computer algorithms to calculate these coefficients - often using Monte-Carlo integration methods, but these papers do not even show in broad terms how these calculations are phrased.

 

[Vanadium 50]

See https://arxiv.org/abs/hep-ph/9602417 and references therein.

 

[Adrian59]

See https://arxiv.org/abs/hep-ph/9602417 and references therein.

I do have a copy of this paper, but I can't see what the authors are doing. They clearly are using a dimensional renormalization parameter ω where the number of dimensions (n) = 4 - 2ω (note other authors use 4 - ε). ω goes to zero in the limit, so most of their integral equations I 1-18 will have diverging terms (those with ω in the denominator) or terms going to zero (those with ω in the numerator). Not much survives in these equations! There are no other variables in these equations apart from this renormalization term which has no physical interpretation.

Somehow from here we get the result for the sixth coefficient (C3) = 1.181241456, and then a result for the anomalous magnetic moment (ae) = 0.00115965222012 which does closely match the then best experimental result from Van Dyck, Schwinberg and Dehmelt from 1987 of ae = 0.0011596521884 which agree to 10 decimal places.

The setting up of the loop integrals with the Feynman terms with Dx in seems fine. Going from the coefficient to the final result is also fine, using the power series. The middle section with these bizarre integrals is just that bizarre!

 

[Haborix]

I don't think the grueling calculation is likely to be published anywhere. Have you tried to actually start working through such calculations step-by-step? Roadblocks you have along the way may be related to the key steps published in the papers you're reading (or they reveal a lack of necessary background). I'm sure the author's have much more detailed personal notes, but I'm not so sure they will share them. Nevertheless, you could always email and ask.

 

[Vanadium 50]

So you are actually asking for a different thing - you are asking how one* diagram is calculated, not a sketch of how they all are. This is not my field, but I expect you will need to study multi-loop calculations in general and not this exact set.

* Or pairs, or an otherwise small number. Often it is easier to calculate the sum of two diagrams than either one separately.

 

[Adrian59]

So you are actually asking for a different thing - you are asking how one diagram is calculated, not a sketch of how they all are. This is not my field, but I expect you will need to study multi-loop calculations in general and not this exact set.

Not exactly, One can read the one loop calculation in any serious QFT textbook. I was asking a general question, but using Laporta and Remeddi's paper as an example, as you referenced this paper. Non of the more advance papers are very transparent. Laporta and Remeddi's eighteen integrals are suppose to be the the final equations of the calculation of the 72 Feynman diagrams for the sixth order coefficient. I do not see why these equations are only a function of an inserted parameter ω which is going to be zero at the limit and has no physical interpretation. Anything remotely physical like mass or momentum appears to have been removed!

However, I think that Haborix has provided an illuminating response, see below:

I don't think the grueling calculation is likely to be published anywhere. Have you tried to actually start working through such calculations step-by-step? Roadblocks you have along the way may be related to the key steps published in the papers you're reading (or they reveal a lack of necessary background). I'm sure the author's have much more detailed personal notes, but I'm not so sure they will share them. Nevertheless, you could always email and ask.

I did try to contact one of the authors, who did help in one small way, but did not appear to want to be more helpful. It appears that much of the detail will forever be in in detailed notes of the corresponding authors but not for public consumption.

 

[Vanadium 50]

As I said, this not my field, but one loop is not two loops. When you are integrating over one variable, there is only one way to do it. If you are integrating out two, you have choices. As I understand it, the art is to set things up to a) make it easy, and b) to have nasty pieces (e.g. non-gauge invariant) pieces cancel.

I'd follow the breadcrumbs from the paper I pointed you to, or maybe look at work by some of the great calculational theorists: Dixon, Bern, Boughezal and their ilk.