The Rosenbluth Scattering Cross Section

Hi everyone and a Happy new year

I am pleased to report that I have completed my work on the calculation of the Rosenbluth Cross section. This is a calculation of the elastic scattering cross section for an electron off a proton. It is one of the standard calculations left as an exercise for the student in many books on quantum field theory. However the actual calculation is far from straightforward and difficult to find so I have written it out in full

I have enclosed a link to the document here (hopefully there are not too many typos)

https://drive.google.com/file/d/1qA2keWm01SSdoU84jQx-HqeOZroaQYF3/view?usp=sharing

I have also enclosed a brief account of the first measurements of the cross section by Hofstadter at a proto type of what would eventually become the Standford Linear Accelerator. (SLAC). I was able to reproduce one of the early graphs from this work.

One slightly unusual feature of this work is that it does not require the use of quantum field theory and a simple derivation of the Feynman rules based on a relativistic generalisation of Fermi's Golden rule is all that is needed. The first half of the notes describes how the Feynman rules for QED can be derived. Once one has the Feynman rules then one can go onto to derive expressions for the scattering cross section or particle decay rate of whatever process you are interested in, in terms of the modulus squared of the matrix element. This stage is the same as in other allegedly more rigorous methods based on quantum field theory. It is an interesting question as to whether or not quantum field theory is as necessary to particle physics as it is sometimes made out.

As a final note the Rosenbluth cross section is expressed in terms of the electromagnetic and magnetic form factors of the proton which can only be determined from the experimental data. Once one has the form factors one can go on to derive expressions for the mean radius of the proton. This work after suitable extensions was carried out was shown to be compatible with other estimates based on the Lamb shift calculated from electron hydrogen spectroscopy. One would have thought that that would have been the end of the matter but in 2010 a series of experiments based on muon spectroscopy has come up with a radius of the proton that is about 4% smaller. Thus the proton has shrunk in size. This is shown below by this whimsical cartoon from the New York times showing the proton worried about it's shrinking waistline

There is a flurry of activity currently taking place to try and resolve the discrepancy, One collaboration (MUSE) is going to look at the scattering cross section from muon proton scattering to see if the discrepancy can be resolved. Details of which are given here 

http://www.physics.rutgers.edu/~rgilman/elasticmup/

It is of interest that calculations performed almost 70 years ago are still of relevance today. 

When I next revisit this work (don't hold your breath 😏) I shall look at the inelastic scattering of electons from protons. This led to  the somewhat surprising fact that at high energies the proton could be considered as made up of point like constituents which at the time were called partons. These were late identified as the quarks of Gell Mann (and others) and led to our current understanding of the strong interaction namely QCD. Hopefully by this time next year I will have finished describing this work. The current work provides the requisite background and I hope eventually to provide my followers with a working knowledge of the standard model of particle physics with nothing more than a smattering knowledge of relativistic quantum physics and Fermi's golden rule instead of the more arcane and somewhat mysterious knowledge of quantum field theory in it's many incarnations be it canonical formalism or the path Integral formalism.