In the absence of any mass scales the stress tensor trace vanishes (
It is related to the renormalization group, or the "running of coupling constants," when they are viewed at different mass scales.
However, under Weyl transformations we do not rescale the coordinates of the theory, but rather the metric and other matter fields.
In the sense of Weyl, mass (or length) are defined by the metric, and coordinates are simply scale-less book-keeping devices.
There is an extensive literature involving spontaneous breaking of Weyl symmetry in four dimensions, leading to a dynamically generate Planck mass together with inflation.
[1] [2] A conformal quantum theory is therefore one whose path integral, or partition function, is unchanged by rescaling the metric (together with other fields).
The variation of the action with respect to the background metric is proportional to the stress tensor, and therefore the variation with respect to a conformal rescaling is proportional to the trace of the stress tensor.
As a result, the trace of the stress tensor must vanish for a conformally invariant theory.
In quantum chromodynamics in the chiral limit, the classical theory has no mass scale so there is a conformal symmetry.
Beside the anomaly (believed to contribute to about 20% of the proton mass[5][6]), the rest can be attributed to the light quarks sigma terms (i.e., the fact that quark have small non-zero masses that are not associated with the trace anomaly) believed to contribute to about 17%, and the quark and gluon energies believed to contribute to about 29% and 34% of the proton mass, respectively.
Sidney Coleman and Erick Weinberg showed how spontaneous symmetry breaking of electroweak interactions involving a fundamental Higgs scalar could occur via Feynmans loops.
In fact, the Coleman–Weinberg mechanism can be traced entirely to the renormalization group running of the quartic Higgs coupling,
[8] It has been conjectured that all mass in nature is generated by trace anomalies, hence by quantum mechanics alone.
The required cancellation of the gravitational anomaly implies that the spacetime dimensionality must be equal to the critical dimension which is either 26 in the case of bosonic string theory or 10 in the case of superstring theory.