Monday, 1 January 2018

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)

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 

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. 


  1. Hi Chris

    Just wanted to say hello. I came across your blog a couple of days ago by accident (serendipity of the web) and have read every post in chronological order. You write well and your story gripped me like a good thriller!

    I consider myself a permanent student (on and off) which I think partly explains my interest in your story. I've had a long interest in popular science but no academic background in science and no talent for mathematics at all (at school in the 1970s I was an arts, humanities and language student) merely a wish that I did, so science has always been a bit of a challenge. I've been quite cowardly about maths because of unpleasant memories of school maths where even O-level was a struggle.

    Anyway, throughout the Noughties, the OU provided an opportunity to revisit and take a punt at the science I missed as a youngster. I packed it in around 2010 when I accepted that I would not be affording the full "overseas student" style course fees that seem to have killed the OU's original purpose when I was exactly half way through a first degree. I was sad to have to stop but at least I was granted a couple of Certificates of Higher Education so got something formal out of the experience. These days I'm entertaining myself with a combination of science MOOCs and courses (for work).

    And now to the point of this comment... I think your blog has inspired me to have a go at maths! I did a couple of beginner courses, S151 and MU120, as part of my studies but nothing serious.I'll not do the OU again, so this will be strictly informal. Fortunately, I have the course materials for MST121 and MST221, and I will give it a go. A close friend of mine did philosphy at Cambridge and an OU degree in maths years ago and is retired now and I'm hoping to persuade him to act as my proxy tutor. I think he'll go for it, he can't resist the lure of being a teacher.

    I just wanted to thank you for your blog which has helped provide the push for me overcome the complete block/terror I have had of confronting even entry level maths. Good luck with your continuing journey and wish me luck (I'll need it!).



  2. Why thank you for your kind comments it must have taken you ages to read all the posts. I hope you do well in maths and your friend guides you. To get a good foundation then you should aim for the equivalent of M208 and MST210 and it would probably take you a couple of years to get there. Then you can go on to specialise in whatever takes your fancy be it pure or Applied
    Saylor have an archived structured maths course which you used to be able to do as part of their unofficial degree but unfortunately they stopped supporting this which is a shame anyway the link to the legacy courses is here

    If I can be of any help then my email address is

    As for me I have given up on the Open University and want to spend the next few years writing up some of the key calculations in particle physics and cosmology watch this space

  3. Best of luck with the calculations.

    I've just finished reading Peter Woit's book "Not even wrong" and I was struck by how impenetrable the modern physics debate is without an advanced mathematics background.

    Peter doesn't shirk discussing some of the technical mathematical issues as best he can (unlike most pop science books) but it seems obvious that without the actual maths, even following the controversies is very difficult. At the end of it I'm left with the memory of a load of technical terms (eg AdS/CFT correspondence, groups and representations, dualities etc) that have become familiar from repetition but about which I have zero understanding.

    Popular books for a lay audience tend to flatter to deceive in this respect. I very much suspect that Euclid was correct in saying there is no "royal road"...


  4. Yes it does seem that way but as far as I am concerned all the ideas about supergravity, superstrings etc are just speculation something with which Woit agrees with so given the limited time I have available to get a good understanding of theoretical physics and the experimental evidence for the theories one can ignore most if not all of stuff such as the AdS\CFT correspondence superstrings in 10, 11 or 26 dimensions and so forth

    To understand physics as it applies to the real world all one needs is an understanding of relativistic quantum mechanics and the standard model and General relativity the most exotic maths this involves is tensor calculus and a smattering of group theory and it's representations. Other than that a good working knowledge of mathematical methods at the second or third year of university. It is still quite an achievement to get to that stage but don't be conned into thinking that one needs to understand exotic branches of maths such as Topology, Functional Analysis, differntial forms and so forth in order to get a working knowledge of theoretical physics. I for one will ignore these branches until it can be demonstrated that new experimental results follow from them. Hopefully the notes that I will be writing up will demonstrate this

  5. Incidently Peter Woit has his own blog in which he covers developments in physics mainly attacking the pretensions of those who believe in superstrings and the Multiverse It has the same title as his book and I have put it on my bloglist

    Currently there is an interesting move by Superstring adherents such as Sean Carroll to try and shift the understanding of science away from falsifiability. Something I have no sympathy with neither does Peter Woit.

  6. I have always considered science to have a couple of key characteristics that clearly separate it from, say, religion.

    The first is that science recognises the imperfection of the human mind, prone to all sorts of biases and self-serving tendencies. It deals with this with a strong emphasis on insisting on very scrupulous/disciplined reporting of results and quamtified analysis of possible error. You're not allowed to claim more than your theory/results support.

    The second is that people are free to dream up all sorts of ideas but the final arbiter is nature - ideas have to agree with experiment.

    I don't like any move to try and redefine science as a kind of faith driven activity. Maybe a bit of faith is needed in the early days to get a tentative idea off the ground but in the end it needs to explain something that is actually out there.

    I'm not a physicist but some of the things that senior scientists have been quoted as saying do sound very self-indulgent and I've always believed that science was supposed to squash those human failings in the face of cold-hard reality. Sounds very unwelcome to me if physics is in the process of being redefined as "any bunch of fancy mathematics someone will pay me to play around with for the next 30 years irrespective if it has anything to do with actual physics".

  7. Couldn't agree more, one of the aims of this blog is to debunk many of the more extravagant claims of what passes for physics these days. My favourite philosopher David Hume summed up the situation veery well when he wrote in his enquiry

    ‘Though the chain of arguments which conduct us to it were ever so logical, there must arise a strong suspicion …that it has carried us quite beyond the reach of our faculties when it leads us to conclusions, so extraordinary, and so remote from common life and experience, We are got into fairy land.‘ [VII, 24].

    'We are got into fairy land' would make the title of a good book debunking all the extravagant claims made for say quantum physics or superstring theory.