# Implication of breakdown of scale invariance for problems with intrinsic length or time scales?

According to Wikipedia article on scale invarince, the equations for electric (and magnetic) fields : $$\nabla^2\vec{E}=\frac{1}{c^2}\frac{\partial^2\vec{E}}{\partial t^2}\hspace{0.3cm}\text{and}\hspace{0.3cm}\nabla^2\vec{B}=\frac{1}{c^2}\frac{\partial^2\vec{B}}{\partial t^2}$$ are invariant under scale transformation $\vec{r}\rightarrow \lambda\vec{r}$ and $t\rightarrow\lambda t$. This implies that, if $\vec{E}(\vec{r},t)$ (and $\vec{B}(\vec{r},t)$) is a solution of the Maxwell's equations in free space then $\vec{E}(\lambda\vec{r},\lambda t)$ (and $\vec{B}(\lambda\vec{r},\lambda t)$) having the same functional form, are also solutions ($\lambda$ is a real number).

The reason for this invariance, as I understand, is the absence of any intrinsic length scale in the problem. Whenever there is a length scale, as for fields in a conductor, existence of a solution $\vec{E}(\vec{r},t)$ doesn't necessarily guarantee the existence of a scaled-solution $\vec{E}(\lambda\vec{r},\lambda t)$ (with the same functional form) because of the breakdown of scale invariance.

Is my inference correct? Does it mean that at new scales (of time and length) newer forms of solutions can emerge?

Similarly for other quantum fields, if they are massless they would be scale invariant; an example is the scalar massless Klein Gordon field. As soon as a mass term is introduced with the introduction of an $m^2$ in the equation, it no longer is scale invariant, and the field diminishes exponentially with m as the scale.