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One-loop diagrams are usually computed as the integral over one independent momentum that can "run in the cycle". The Casimir effect , Hawking radiation and Lamb shift are examples of phenomena whose existence can be implied using one-loop Feynman diagrams, especially the well-known "triangle diagram":
Feynman parametrization is a technique for evaluating loop ... Calculation of Feynman Amplitudes, Springer, ISBN 978-3-030-80218-9 (2021). Stefan Weinzierl: Feynman ...
The Automatic Calculation project is to create the tools to make those steps as automatic (or programmed) as possible: I Feynman rules, coupling and mass generation LanHEP is an example of Feynman rules generation. Some model needs an additional step to compute, based on some parameters, the mass and coupling of new predicted particles.
The Feynman diagrams are much easier to keep track of than "old-fashioned" terms, because the old-fashioned way treats the particle and antiparticle contributions as separate. Each Feynman diagram is the sum of exponentially many old-fashioned terms, because each internal line can separately represent either a particle or an antiparticle.
For example in scalar theory in 4 dimensions, the loop integral in the calculation of one-loop renormalization of the interaction vertex has (,,) = (,,). We use the 'trick' of dimensional regularization , analytically continuing d {\displaystyle d} to d = 4 − ϵ {\displaystyle d=4-\epsilon } with ϵ {\displaystyle \epsilon } a small parameter.
The calculation is relatively straightforward [1] and the one-loop result is: =, where is the fine-structure constant. This result was first found by Julian Schwinger in 1948 [2] and is engraved on his tombstone.
However, since they can be off shell, wherever the diagram contains a closed loop, the energies and momenta of the virtual particles participating in the loop will be partly unconstrained, since a change in a quantity for one particle in the loop can be balanced by an equal and opposite change in another. Therefore, every loop in a Feynman ...
Furry's theorem allows for the simplification of a number of amplitude calculations in quantum electrodynamics. [5] In particular, since the result also holds when photons are off-shell, all Feynman diagrams which have at least one internal fermion loops with an odd number of vertices have a vanishing contribution to the amplitude and can be ignored.