One-loop Feynman diagram

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In physics, a one-loop Sander-Feynman diagram is a connected Feynman diagram with only one cycle (unicyclic). Such a diagram can be obtained from a tree diagram by adding an edge connecting two internal vertices (of course, this is only possible in general if we ignore the vertex matching rules).

Diagrams with loops (i.e. cycles, which is not to be confused with a loop in graph theory which is an edge connecting a vertex with itself) correspond to the effects of quantum physics. Because one-loop diagrams only contain one cycle, they express the next-to-classical contributions called the semiclassical contributions.

One-loop diagrams are usually computed as the integral over one independent momentum that can "run in the cycle". The Casimir effect and Hawking radiation are examples of phenomena whose existence can be proved using one-loop Feynman diagrams.

The evaluation of one-loop Feynman diagrams usually leads to divergent expressions, which are either due to zero-mass particles in the cycle of the diagram (infrared divergence) or due to insufficient falloff of the integrand for high momenta (ultraviolet divergence). The former are usually dealt with by assigning the zero mass particles a small mass λ, evaluating the corresponding expression and then taking the limit , the latter are dealt with in the renormalization program.

The one-loop corrections lead to the following effective action: