We continue our detour into some of the many branches of the Compton Effect, partly to show how fortunate it was that Compton was looking at the interaction of X-rays and electrons and partly to suggest how crucial it was to tease out a set of relatively simple observations to clarify the interactions of light quanta and electrons because things can get pretty complicated pretty fast with light quanta and electrons. These complications can be resolved with what are called “QED corrections” – which is to say higher-order Quantum Electro-Dynamic (QED) descriptions of interactions where the presence of fields or higher energies are represented by additional particles (including light quanta) that can be real or virtual ( virtual in the sense that they appear and disappear within an interaction). One of the nice things about the Compton Effect is that it is basically a purely QED phenomenon, that is it generally involves only light quanta (photons) and electrons (and positrons) which makes it useful for calibrating the effects of other interactions and the expected range of things like polarization in high energy interactions. Of course at really high energies (eg above 2.5 gigaelectron volts (GeV)) processes involving initially only QED objects (electrons, positrons, and light quanta) can produce quarks and those resolve themselves into mesons and baryons, but I hope to keep this blog relatively clear of quarks so we won’t go into that at the moment.
But, a digression within this digression to clarify the meaning of “QED corrections.” The assumption here – as usually illustrated and analyzed with Feynman Diagrams – is that if you allow for the impact of every possible sequence of interactions, sub-interactions and loops within an interaction, you have to put that all into some kind of order. Basically, the simplest sets of sequences are the events that contribute most to what is observed as the results of an interaction, such as these two Feynman diagrams of the Compton Effect:
But, especially at higher energies, different kinds of events and sequences can contribute to what is seen as the result of an interaction. The more complex sequences are part of an ordered perturbative series of sub-interactions that contribute less and less to the results as the complexity increases — but their contributions can be analyzed and included in shifts that can be observed. Here are a few higher order Feynman diagrams from the perturbative series that describes the Compton Effect:
As I noted in “Compton Two,” these are the “higher order” effects in the quantum electrodynamic perturbation series that involve branching and/or looping within the framework of the basic interaction.
So, to return to the main digression and to hint at some of the other more complicated things that can happen, we want to take a glance at two other aspects of the Compton Effect: higher order perturbations at higher energies and the double Compton Effect in the early universe with astrophysical plasmas.
I’ll be following three main sources:
- “Complete QED Corrections and Polarized Compton Scattering” – A. Denner and S. Dittmaier – CERN-TH/98-142 May 1998 (the source of the Feynman diagrams in this post)
- “The Double Compton Process in Astrophysical Plasmas” Andrea Ravenni and Jens Chluba – arXiv: 2005.0694v1 14 May 2020
- An Introduction to Quantum Field Theory, Michael E. Preskin and Daniel V. Schroeder, Addison Wesley, 1995
Examining the complete QED corrections to characterize polarized Compton scattering noted in the first source has two main motives – one to calibrate polarization in electron accelerators and two – to define the effects of QED corrections at high energies. In the second of the two motives, we can see why Compton was lucky to be working with the relatively low energies of X-rays hitting electrons. At the energies he was observing, QED corrections are very minor – only thousandths of a percent in terms of their effects. In extremely high energy situations, say 500 GeV (as with colliding electrons and positrons in an accelerator), there can be as much as a 2% shift in measurements due to QED corrections. Such shifts would have thrown Compton’s attempts at precise measurement into turmoil, but, when he was working, only cosmic rays (not yet even defined in those terms) would have provided such high energies to experimenters.
In higher order corrections, you can have loops in the interactions involved in the Compton Effect:
Intuitively a loop in an interaction does sound like trouble, and, in fact, if you don’t correct these loops in the corrections, integrating over all the possible momenta involved in a loop gives you infinite momenta. We won’t be looking into how correcting the corrections (or renormalization) works until this blog reaches the early 1950s – still 30 years from where we are with Compton in about 1920.
And you can get “an extra real photon” in the Double Compton (DC) Effect:
Because the DC Effect dominates interactions for electrons at higher energies and photons are even more relatively common than baryons in the early universe, the DC effect is crucial in understanding the extreme situations in the early universe. Ravenni and Chluba cite the 1952 paper of Mandl and Skyrme as the first detailed treatment of the effect ( F. Mandl and T. H. R. Skyrme, “The Theory of the Double Compton Effect,” Proceedings of the Royal Society of London, Dec 1952) It is a very cool paper and when the blog gets to the early 1950s, we will look at it again. One interesting thing about it is that Feynman diagrams are mentioned as part of the analysis but there are no Feynman diagrams in the paper.
Anyway, enough of that. We will get back to Compton in 1920 in our next post.