Power counting in quantum field theory

Power counting assigns mass dimensions to fields and couplings. An interaction is relevant, marginal or irrelevant when its coupling has positive, zero or negative mass dimension, respectively.

The superficial degree of divergence $D$ is the ultraviolet power predicted by rescaling all independent loop momenta together. For a scalar graph with $L$ loops and $I$ propagators behaving as $p^{-2}$ in $d$ dimensions, $D=dL-2I$ before cancellations and subdivergences are considered.

A marginal coupling is dimensionless under classical power counting.

A relevant coupling has positive mass dimension and grows under scaling toward long distances.

An irrelevant coupling has negative mass dimension and is suppressed toward long distances.