Examples of spontaneously broken scale invariance?

Question

To me it is intuitively clear, how scale invariance of a theory can explicitely be broken by adding terms into the Lagrangian that introduce an explicit scale, such as a mass term for example. Conversely, how it can happen that scale invariance is spontaneously broken is intuitively less clear to me...

So can anybody give some enlightening examples such that I can explicitely see what kind of mechanism esxist that spontaneously break scale invariance and how this works exactly?

Comments

Just from reading the abstract (didn't read the full text), maybe this and this are examples.

Answer 1

If you take a scale invariant theory which contains some scalars then you can obtain a spontaneously broken scale invariance by giving some expectation values to the scalars. If you give some expectation values to some scalars in a gauge theory, we have the Higgs mechanism, with spontaneous breaking of the gauge symmetry. The story will be analoguous for some generic expectation values of the scalars in a scale invariant theory : at low energies, particles will obtain masses and the theory will no longer be scale invariant.

One of the simplest example of scale invariant theory is N=4 Super-Yang-Mills in four dimensions. This is a scale (in fact conformal) invariant gauge theory. There are 6 scalars in the adjoint representation of the gauge group and the expectation values of these scalars define a moduli space $M = \mathbb{R}^{6}$, called the Coulomb branch. At the origin, no scalar expectation value, unbroken gauge group, scale invariance. At a generic point, some scalar expectation values, broken gauge group by Higgs mechanism and broken scale invariance: some particles are massive.