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In Griffith's Intro to Elementary Particles book, he wrote

there are infinitely many Feynman diagrams for any particular reaction!

Why is this true? Take for example Moller scattering that describes Coulomb repulsion between 2 electrons. The only Feynman diagram I know of is:

enter image description here

What will be the other infinite Feynman diagrams for this reaction?

TaeNyFan
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    Note that you have an $(e,e,\gamma)$ vertex and a constraint that at the "corners" of the diagram you must have two electrons entering and leaving. What you have is only the simplest "first-order" way that only uses two vertices to accomplish this. The vertices do have to come in pairs because there are no photon lines coming out of the diagram, but that does not mean that there can be only 2 of them -- there could be 4, or 6, or 8... are you assuming he means infinitely many 2-vertex diagrams? – CR Drost Mar 13 '19 at 17:00
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    Feyman diagrams are not a description of the process: they are representations of expansion terms of an infinite series and as such there are in general infinite of them (unless for some reasons at some point they all vanish). – gented Mar 13 '19 at 17:10
  • Griffiths' statement is equivalent to saying that any physical process has an infinite number of terms in the relevant perturbation expansion, nothing profound. What you have written is like the zeroth-order part of a perturbation, it may or may not be a good approximation depending on the size of the relevant coupling constant. – Tom Jun 25 '23 at 16:27

2 Answers2

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The cross section for a scattering process like Møller scattering is calculated by summing up an infinite series. Each term in this series is an integral that can be represented by a Feynman diagram. The diagram you have drawn is just the first term in the infinite series - the tree level term.

There is a nice illustration of the first few terms for Møller scattering in the Free Dictionary article on Feynman rules:

Møller scattering

After the tree level term (a) we have the one loop terms (b) to (j), then two loops then three loops and so on. The number of terms at each loop level escalates rapidly.

It is worth noting that the diagrams do not show an actual physical process. They must not be taken literally. They are just a pictorial representation of an integral called the propagator.

John Rennie
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    I think one should stress that each 1/137 entering with the loops makes the relative contribution to the crossection smaller and smaller. – anna v Mar 13 '19 at 18:17
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    Technically the propagator is only the case of diagrams with two external legs. Diagrams with more legs are scattering amplitudes or n-point correlation functions. – Javier Mar 13 '19 at 19:01
  • Correct me if I am wrong, but this picture of the Feynman diagrams is misleading. There are two tree level terms for Moller scattering https://en.wikipedia.org/wiki/Møller_scattering . Each of the diagrams (b-j) is also missing its "counterpart." Furthermore, we are often only interested in amputated diagrams, of which (g-j) aren't. – Jbag1212 Dec 03 '21 at 19:05
  • Obviously there are two tree level terms because it's a scattering, I think sometimes authors usually make a choice on which of the graphs to draw as some of them are implicitly understood. – Tom Jun 25 '23 at 16:25
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I read from the article quoted by @JohnRennie:

The illustration shows Feynman diagrams for electron-electron scattering. In each diagram, the straight lines represent space-time trajectories of noninteracting electrons, and the wavy lines represent photons, particles that transmit the electromagnetic interaction. External lines at the bottom of each diagram represent incoming particles (before the interactions), and lines at the top, outgoing particles (after the interactions). Interactions between photons and electrons occur at the vertices where photon lines meet electron lines.

Well, this is wrong. Although it's also possible to interpret Feynman diagrams as representing space-time diagrams [1] their form now in universal use is as a momentum space diagram, and a topological one. The only things a diagram is meant to show are relationships between interaction vertices and exchanged momenta.

It would be humorous to interpret the reported diagrams as space-time ones: you would have photons going from here to there following curved paths etc.

[1] Feynman diagrams

Elio Fabri
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  • Topology can be indicated by furnishing an illustrative example, in spacetime, of each set of topological connections. 2. Some lines are shown non-straight merely to reduce clutter on the diagram, but in any case since they represent an integral over worldlines you can show any worldline you like; it does not need to be straight.
    • – Andrew Steane Mar 14 '19 at 20:18