One-loop Feynman diagram
Encyclopedia
In physics
, a one-loop Feynman diagram is a connected
Feynman diagram
with only one cycle
(unicyclic). Such a diagram can be obtained from a connected tree diagram
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 cycles
, while the word loop
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 semiclassical contributions.
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":
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
:
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a one-loop Feynman diagram is a connected
Connected component (graph theory)
In graph theory, a connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices. For example, the graph shown in the illustration on the right has three connected components...
Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
with only one cycle
Cycle (graph theory)
In graph theory, the term cycle may refer to a closed path. If repeated vertices are allowed, it is more often called a closed walk. If the path is a simple path, with no repeated vertices or edges other than the starting and ending vertices, it may also be called a simple cycle, circuit, circle,...
(unicyclic). Such a diagram can be obtained from a connected tree diagram
Tree diagram
The term tree diagram refers to a specific type of diagram that has a unique network topology. It can be seen as a specific type of network diagram, which in turn can be seen as a special kind of cluster diagram.-Applications:...
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 cycles
Cycle (graph theory)
In graph theory, the term cycle may refer to a closed path. If repeated vertices are allowed, it is more often called a closed walk. If the path is a simple path, with no repeated vertices or edges other than the starting and ending vertices, it may also be called a simple cycle, circuit, circle,...
, while the word loop
Loop (graph theory)
In graph theory, a loop is an edge that connects a vertex to itself. A simple graph contains no loops....
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 semiclassical contributions.
One-loop diagrams are usually computed as the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
over one independent momentum that can "run in the cycle". The Casimir effect
Casimir effect
In quantum field theory, the Casimir effect and the Casimir–Polder force are physical forces arising from a quantized field. The typical example is of two uncharged metallic plates in a vacuum, like capacitors placed a few micrometers apart, without any external electromagnetic field...
, Hawking radiation
Hawking radiation
Hawking radiation is a thermal radiation with a black body spectrum predicted to be emitted by black holes due to quantum effects. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974, and sometimes also after the physicist Jacob Bekenstein...
and Lamb shift 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 zero-mass particles in the cycle of the diagram (infrared divergence
Infrared divergence
In physics, an infrared divergence or infrared catastrophe is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very small energy approaching zero, or, equivalently, because of physical phenomena at very long distances.The infrared ...
) or due to insufficient falloff of the integrand for high momenta (ultraviolet divergence
Ultraviolet divergence
In physics, an ultraviolet divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very high energy , or, equivalently, because of physical phenomena at very short distances. An infinite answer to a question that should have a...
). 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 programRenormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
.
The one-loop corrections lead to the following effective action
Effective action
In quantum field theory, the effective action is a modified expression for the action, which takes into account quantum-mechanical corrections, in the following sense:...
: