# How is scale invariance broken in QCD?

It is generally believed that for the pure QCD, the classical scale invariance is broken at the quantum level (therefore anomaly rather than SSB). This breaking of scale invariance may be used to explain the quark confinement where an explicit mass scale (or mass gap for QCD) appears. Does anybody know of some references which explain or argue intuitively how this happens? Or even better, does anybody know of an argument?

• This is the so called "dimensional transmutation" phenomenon. Before QCD, it might be instructive to see how it works on 2d Gross-Neveu. A good reference is the lecture by David Gross in the IAS volumes QFT for mathematicians. I think Coleman's book "Aspects of symmetry" also has a nice discussion. – Abdelmalek Abdesselam Dec 14 '17 at 12:56

The breakdown of the scale invariance in pure Yang-Mills theories takes place due to the dependence of the running coupling constant on the renormalization mass scale. This is an anomalous breakdown, since the classical theory is invariant under scale transformations. It is manifested through the formation of a nonvanishing trace to the energy momentum tensor. Hence it is given the name "trace anomaly".

The trace anomaly can be heuristically derived from the pure QCD renormalized Lagrangian:
$$\mathcal{L} = - \frac{1}{4g^2} F_{\mu \nu}^a F^{ \mu \nu}_a$$ The trace of the energy momentum tensor can be calculated as the variation of the Lagrangian by the mass scale (logarithm of the mass parameter) $$T^{\mu}_{\mu} = \frac{\partial \mathcal{L}}{\partial \lambda} = \frac{\beta(g)}{2g^3} F_{\mu \nu}^a F^{ \mu \nu}_a$$ Where $\beta(g)= \frac{\partial g} {\partial \lambda}$ is the beta function.

Of course the above heuristic derivation does not specify the Gluon condensate upon which the trace anomaly depends.

One of the first detailed derivations of this result was given by Collins, Duncan and Joglekar

• What do you mean by "the above heuristic derivation does not precify the gluon condensate upon which the trace anomaly depends"?? Can you go through this sentence? – apt45 Dec 13 '17 at 9:36
• @apt45: I meant the following: Since pure QCD has a single dimensional parameter $\Lambda_{QCD}$, the Gluon condensate must depend on this parameter; however the above heuristic argument gives no information about this dependence. – David Bar Moshe Dec 14 '17 at 8:39