Conformal anomaly

A conformal anomaly, scale anomaly, trace anomaly or Weyl anomaly is an anomaly, i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory.

In quantum field theory when we set Planck constant

ℏ

{\displaystyle \hbar }

to zero we have only Feynman tree diagrams, which is a "classical" theory (equivalent to the Fredholm theory of a classical field theory).

One-loop (N-loop) Feynman diagrams are proportional to

ℏ

{\displaystyle \hbar }

(

ℏ

N

{\displaystyle \hbar ^{N}}

).

If a current is conserved classically (

ℏ

=

0

{\displaystyle \hbar =0}

) but develops a divergence at loop level in quantum field theory (

∝

ℏ

{\displaystyle \propto \hbar }

), we say there is an anomaly. A famous example is the axial current anomaly where massless fermions will have a classically conserved axial current, but which develops a nonzero divergence in the presence of gauge fields.

A scale invariant theory, one in which there are no mass scales, will have a conserved Noether current called the "scale current." This is derived by performing scale transformations on the coordinates of space-time. The divergence of the scale current is then the trace of the stress tensor. In the absence of any mass scales the stress tensor trace vanishes (

ℏ

=

0

{\displaystyle \hbar =0}

), hence the current is "classically conserved" and the theory is classically scale invariant. However, at loop level the scale current can develop a nonzero divergence.