In Standard model, components of a $SU(2)$ doublet (for example $u$ and $d$) have different masses. This means there is no $SU(2)$ symmetry, but I think it is okay because the $SU(2)$ symmetry is spontaneously broken.
However, before Higgs obtains non-zero vacuum expectation value, there is $SU(2)$ gauge symmetry. In this case, all fermions become massless, but Yukawa coupling constants between Higgs and components of a $SU(2)$ doublet are different (this gives different masses when symmetry is spontaneously broken). Therefore $SU(2)$ seems not to be exact symmetry even before symmetry breaking. Is it true?
If, surprisingly, it is true, I wonder whether there is any problem in gauge fixing procedures in path integral quantization, because full Lagrangian is not invariant under $SU(2)$ gauge transformation.
This question occurred to me when I studied strong CP problem. Since Yukawa coupling matrices for $(u, c, t)$ and for $(d, s, b)$ are different (in other words, mass of $u$ and $d$ are different) and mixing of $(u, c, t)$ and mixing of $(d, s, b)$ are also different, weak interaction which mediate $u$ and $d$ is CP violated.
On the other hand, QCD has no such violation, because Yukawa coupling constant to quarks with same flavor and different color are same (which corresponds to the fact that $u$_red and $u$_blue have same mass). This conditions is required by the $SU(3)$ color gauge invariance of Standard Model. I wonder why such conditions are not needed between Yukawa couplings to $u$ and $d$ in order to preserve electroweak $SU(2)$ gauge symmetry.