One-loop Feynman diagram

In <a href="/facts/Physics/a7aH2y08">physics</a>, a one-loop Feynman diagram is a <a href="/facts/Connected_component_(graph_theory)/5owVSXtY">connected</a> <a href="/facts/Feynman_diagram/BRFNvDdr">Feynman diagram</a> with only one <a href="/facts/Cycle_(graph_theory)/Is44edrv">cycle</a> (<a href="/facts/Unicyclic_graph/Fm5fK5BH">unicyclic</a>). Such a diagram can be obtained from a connected <a href="/facts/Feynman_diagram/BRFNvDdr">tree diagram</a> by taking two external lines of the same type and joining them together into an edge.
Diagrams with loops (in graph theory, these kinds of loops are called <a href="/facts/Cycle_(graph_theory)/Is44edrv">cycles</a>, while the word <a href="/facts/Loop_(graph_theory)/jCuevIad">loop</a> is an edge connecting a vertex with itself) correspond to the quantum corrections to the classical field theory. Because one-loop diagrams only contain one cycle, they express the next-to-classical contributions called the <a href="/facts/Semiclassical_physics/8v1CG3RP">semiclassical</a> contributions.
One-loop diagrams are usually computed as the <a href="/facts/Integral/V7lyV997">integral</a> over one independent momentum that can "run in the cycle". The <a href="/facts/Casimir_effect/xRhpDH6T">Casimir effect</a>, <a href="/facts/Hawking_radiation/J6R7W2wL">Hawking radiation</a> and <a href="/facts/Lamb_shift/p4rqJV2j">Lamb shift</a> are examples of phenomena whose existence can be implied using one-loop Feynman diagrams, especially the well-known "triangle diagram":

The evaluation of one-loop Feynman diagrams usually leads to divergent expressions, which are either due to:

<ul><li>zero-mass particles in the cycle of the diagram (<a href="/facts/Infrared_divergence/x9ErWwNk">infrared divergence</a>) or</li>
<li>insufficient falloff of the integrand for high momenta (<a href="/facts/Ultraviolet_divergence/pfVuulfl">ultraviolet divergence</a>).</li></ul>
Infrared divergences are usually dealt with by assigning the zero mass particles a small mass λ, evaluating the corresponding expression and then taking the limit 
 
 
 
 λ
 →
 0
 
 
 {\displaystyle \lambda \to 0}
 
. Ultraviolet divergences are dealt with by <a href="/facts/Renormalization/B4PdTB4o">renormalization</a>.

One-loop Feynman diagram open-in-new

One-loop Feynman diagram