Suppose it is understood how the Spontaneous Symmetry Breaking work mathematically to give masses to Fermions through Yukawa Interactions and to Gauge Bosons via Higgs Mechanism.
In this case, my question: how is a theory phenomenological description at energy scales $E \ll v$ different from the description at $E \gg v$ (where $v$ is the VEV responsible for SSB)?
To motivate the doubt above, suppose I have a breaking pattern $SU(3) \times SU(2) \to SU(2)' \times U(1)$ and I want to calculate the running of $g_{SU(3)}$. To do that, I should match both theories at a single scale. This matching is not analogous to what one does when integrating out heavy degrees of freedom suplying local interactions to a low energy effective model -- rather, it is about ensuring the lagrangians above and below that 'symmetry threshold' describe the same theory, numerically.
For example, if we have that the breaking pattern is such that $SU(2)' \subset SU(3)$, in that scale we must have $g_{SU(2)'}=g_{SU(3)}$, and if we have something like
$$ Y =\sum_\alpha p_\alpha T^\alpha, $$
where $Y$ is the generator of $U(1)$ and $T^\alpha$ represent collectively the 11 generators of $SU(3) \times SU(2)$, then the matching condition is
$$ \frac{1}{g_Y^2} = \sum_\alpha \frac{p_\alpha^2}{g_\alpha^2}, $$
where $g_\alpha$ is the coupling constant of the group corresponding to $T_\alpha$. This is what happens in the SM.
So, stating my question again: what is the use of this construction? At scales $E \gg v$, where one could say 'the symmetry is still unbroken', are the mass eigenstates of vector bosons still the degrees of freedom which propagate?
