Ultraviolet divergence

An ultraviolet divergence comes from arbitrarily large loop momentum, equivalently from singular short-distance behavior. Locality makes its regulator dependence cancellable by local counterterms in a renormalizable theory.

An amplitude is ultraviolet finite if its regulated high-loop momentum contribution has a finite regulator-removal limit after whatever lower-order subtractions are specified. A zero superficial degree of divergence does not force divergence: gauge invariance can cancel the candidate local term. Conversely, superficial convergence alone does not exclude divergent subgraphs. Ultraviolet finiteness makes no claim about infrared divergences or physical threshold singularities.

An ultraviolet divergence appears as a pole of a dimensionally continued Feynman integral. In $d=4-2ϵ$, a logarithmically divergent one-loop integral typically has a $1/ϵ$ term. Dimensional regularization can also produce infrared divergences with the same formal regulator, so the origin must be identified before interpreting a pole as an ultraviolet counterterm.

An ultraviolet subdivergence arises when loop momenta internal to a proper subgraph tend to infinity while the momenta entering that subgraph are held fixed. It can occur even if simultaneous scaling of all momenta of the full Feynman diagram is convergent. Recursive local counterterms subtract these subgraph divergences before the remaining overall divergence is analyzed. A tadpole diagram inserted into an otherwise convergent six-point phi-fourth theory Feynman diagram is a simple massive example.