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what are Transition Distribution Amplitudes respectively Deeply Virtual Compton Scatterings ? I'm so unsure about the terms I can't even distinguish (in case they are not really related to each other). I found a lecture which claims to be an introduction ( https://www.ectstar.eu/sites/www.ectstar.eu/files/talks/Dieter_Mueller.pdf ) but no, it's not. At least not for me. Can someone please explain how to understand compton scattering in particle physics respectively how to use in Feynman graphs or just in general?

This reads better:

Deeply Virtual Compton Scattering (DVCS) is the simplest hard exclusive process related to Generalized Parton Distributions (GPDs): the scattering of an electron off a proton through the exchange of a photon of virtuality Q2, accompanied by the re-emission of a real photon. http://irfu.cea.fr/en/Phocea/Vie_des_labos/Ast/ast_visu.php?id_ast=4192

But I guess there are too much topics involved I just don't know. Isn't Compton $ \gamma \to e^- $ ? So here it is $ e^- \to p $ ?

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May be seeing the Feynman diagram will help you :

DVCS

Searching for Feynman diagrams for Deeply Virtual Compton Scattering (DVCS), this is the clearer I found , the one on the left.

The concept is used in calculating cross sections for electron proton scattering. Protons are composed by quarks and a sea of gluons and antiquarks and to get useful crossection (electromagnetic) one has to use approximations for all those charges moving around it. So a mathematical model is made up , which one would have to study to get the real form, where the two charged entities ( electron and approximation) interact through a virtual photon. This is analogous to real photon scattering off and electron , except it is the nucleus construct it scatters off. The output is a gamma and an electron and a photon with different four vectors.

The other two diagrams show how the same topology can be calculated through different mathematics, with the photon coming from the electron and a simple virtual scatter with the proton.

DVCS to be true to D needs a lot of $Q^2$ transferred from the electron to the process. The C comes by thinking of the virtual gamma as the real one in Compton scattering. You would have to read the model to understand the mathematics of it.

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