Editing Feynman diagram (section)

==== Loop order ====
A forest diagram is one where all the internal lines have momentum that is completely determined by the external lines and the condition that the incoming and outgoing momentum are equal at each vertex. The contribution of these diagrams is a product of propagators, without any integration. A tree diagram is a connected forest diagram.

An example of a tree diagram is the one where each of four external lines end on an {{mvar|X}}. Another is when three external lines end on an {{mvar|X}}, and the remaining half-line joins up with another {{mvar|X}}, and the remaining half-lines of this {{mvar|X}} run off to external lines. These are all also forest diagrams (as every tree is a forest); an example of a forest that is not a tree is when eight external lines end on two {{mvar|X}}s.

It is easy to verify that in all these cases, the momenta on all the internal lines is determined by the external momenta and the condition of momentum conservation in each vertex.

A diagram that is not a forest diagram is called a ''loop'' diagram, and an example is one where two lines of an {{mvar|X}} are joined to external lines, while the remaining two lines are joined to each other. The two lines joined to each other can have any momentum at all, since they both enter and leave the same vertex. A more complicated example is one where two {{mvar|X}}s are joined to each other by matching the legs one to the other. This diagram has no external lines at all.

The reason loop diagrams are called loop diagrams is because the number of {{mvar|k}}-integrals that are left undetermined by momentum conservation is equal to the number of independent closed loops in the diagram, where independent loops are counted as in [[homology theory]]. The homology is real-valued (actually {{math|'''R'''<sup>''d''</sup>}} valued), the value associated with each line is the momentum. The boundary operator takes each line to the sum of the end-vertices with a positive sign at the head and a negative sign at the tail. The condition that the momentum is conserved is exactly the condition that the boundary of the {{mvar|k}}-valued weighted graph is zero.

A set of valid {{mvar|k}}-values can be arbitrarily redefined whenever there is a closed loop. A closed loop is a cyclical path of adjacent vertices that never revisits the same vertex. Such a cycle can be thought of as the boundary of a hypothetical 2-cell. The {{mvar|k}}-labellings of a graph that conserve momentum (i.e. which has zero boundary) up to redefinitions of {{mvar|k}} (i.e. up to boundaries of 2-cells) define the first homology of a graph. The number of independent momenta that are not determined is then equal to the number of independent homology loops. For many graphs, this is equal to the number of loops as counted in the most intuitive way.