Scale in mathematics vs in quantum physics

[RockyMarciano]

In general, one should distinguish symmetry of general laws from symmetry of special solutions.

Take, for instance, one free photon with 4-momentum $k$. It is not a Lorentz invariant object (and hence not Poincare invariant object) because in another Lorenz frame it has momentum $k'\neq k$. Nevertheless, this photon is described by Lorentz invariant laws. But one photon with one particular momentum is a particular solution of these laws, and it is not a Lorentz invariant solution. In free QED, which is the theory of free photons and electrons, the only Lorentz invariant solution is the vacuum.

Of course, free particles are not very interesting from a physical point of view, but the example above serves to understand the difference between symmetry of laws and symmetry of solutions. Equipped with this conceptual understanding, now we can try to understand something less trivial.

Now consider interacting QED. Take, for instance, scattering of two electrons with given initial momenta. In calculation of the scattering amplitude one encounters loop diagrams which appear divergent. To make them convergent one takes some cut-off, and this violates scale invariance and Poincare invariance. Physically, the effective (i.e. renormalized) coupling constant, which initially was a true constant, now depends on energy. Energy is not scale and Lorentz invariant, so naively one would say that scale and Poincare invariance are violated. But one should recall that it is obtained during a study of a special case: scattering of two electrons with given initial momenta. So this "violation" is merely the absence of symmetry in the special solution. The general laws are still Poincare and scale invariant.

Thanks for replying.
I'm aquainted with the distinction between symmetry in the laws vs the solutions. But note that I'm not referring to solutions in this thread, the theme in this thread is about comparing the symmetries in the laws of renormalized qft that reflect the absence of physical invariance of scale with the scale invariant symmetries of the mathematical model (gauge groups acting on Minkowski space) employed to embody those laws. And asking without succes if from this could be deduced an obstruction to the mathematical soundness of such model and if not why not.
So my question is more formal than about the physics behind.