Is the electroweak $SU(2)$ gauge symmetry an exact symmetry in Standard Model before spontaneous symmetry breaking? 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.


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*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.
 A: Yes, and the gauge symmetry is still exact even after "spontaneous symmetry breaking" (SSB).  The name SSB really isn't appropriate in this context, even though it's widely used, even by people who know better. (I'm guilty of this, too.) A better way to say it is that the model is "in the Higgs phase." The gauge symmetry is still exact, even though it may be hidden. This is also emphasized in another post.
More generally, gauge symmetries are always exact. This is emphasized in
Witten (2017) “Symmetry and Emergence” (http://arxiv.org/abs/1710.01791), who goes even farther and suggests that the only exact symmetries are gauge symmetries. 
To put the "Higgs phase" in context, note that a quantum field model can have a rich phase diagram as a function of the model's parameters. Typical phases include the "Coulomb phase" (like in QED), the "Higgs phase" (like in the electroweak sector), and the "confinement phase" (like in QCD). Even a simple model can exhibit all three of these phases for different values of the coefficients in the Lagrangian. This is analogous to the phase diagram of a thermodynamic system as a function of things like temperature and pressure. 
To drive home the point that the gauge symmetry is never really broken, note that in some models we can go continuously from the Higgs phase to the confinement phase  by varying the model's parameters in the right way. Then the Higgs and confinement phases are not really separate phases, much like the liquid and vapor phases of water are not really separate phases (we can go continuously from one to the other without crossing the boiling-line). Here are some references about this:


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*Bhanot and Creutz (1980), "Phase diagram of Z(N) and U(1) gauge theories in three dimensions," Physical Review D 21, pages 2892-2902 (http://inspirehep.net/record/152445?ln=en)

*Creutz (1980), "Phase diagrams for coupled spin-gauge systems," Physical Review D 21, page 56 (https://quark.phy.bnl.gov/~creutz/mypubs/pub036.pdf)

*Fradkin and Shenker (1979), "Phase diagrams of lattice gauge theories
with Higgs fields," Physical Review D 19, pages 3682-3697 (https://journals.aps.org/prd/abstract/10.1103/PhysRevD.19.3682)

*Montvay and Munster (1994), Quantum Fields on a Lattice, page 334
Page 334 in the last book says this about the phase structure of a particular SU(2) Higgs model:

At strong gauge coupling... the distinction between 'confinement' and the 'Higgs mechanism' loses its meaning... 

We would not say that the gauge symmetry is broken in the confinement phase,  and it's not really broken in the Higgs phase, either — whether or not the Higgs and confinement phases are connected.
