Coming from the bottom up, we can use the renormalization group equations to calculate if there are any fixed points and if yes, where they lie.
Fixed points correspond to scale invariant theories, i.e. theories where the beta functions vanish.
In solid-state physics the position of the fixed points are related to external parameters that can be controlled, like, for example, temperature and pressure. If the we tune these parameters to special values, namely the fixed point, we can observe the scale invariant behaviour.
What is controlling the positions of the fixed points in particle physics?
In other words, if we come from the top down, i.e. when we start with a scale invariant theory, how can the theory move away from the fixed point, if the theory is scale invariant? What is controlling the perturbation of the theory away from the fixed point and therefore the non-scale invariant behaviour?
As far as I know, in in particles physics one usually uses the normal RGEs to search for fixed points. However, there are no "external" control parameters and the behaviour is completely determined by the scale $\mu$. Thus the behaviour is quite different as in solid state physics. How can a theory by scale invariant at some scale $\mu_C$ and non-scale invariant at some lower scale?