Is the Higgs Field Potential Still Symmetric Post-Electroweak Symmetry Breaking?

[ChrisVer]

Hi, a fast question... I was having somekind of a discussion, and I actually hit a dead end to the way I could explain my statement.
I believe that after the Higgs Mechanism and SSB, the resulting potential of the Higgs Field should no longer be symmetric to rotations : I am saying this because by the time we choose a vev for the Higgs Field (doing the gauge fixing), we are at one chosen point on the $(\phi_1,\phi_2)$-plane and there should be no longer any degenerate ground state [the massless GBs have been gauged out]. To me, this sounds like that the ground state of the mexican hat is slightly changed, deeper for the choice of our gauge. However, wouldn't that mean the breaking of the residual U(1)? Where am I wrong?

 

[nikkkom]

Hi, a fast question... I was having somekind of a discussion, and I actually hit a dead end to the way I could explain my statement.
I believe that after the Higgs Mechanism and SSB, the resulting potential of the Higgs Field should no longer be symmetric to rotations : I am saying this because by the time we choose a vev for the Higgs Field (doing the gauge fixing), we are at one chosen point on the $(\phi_1,\phi_2)$-plane

Here's how I understand it.

Unlike Mexican Hat used in visual analogy, it's not a plane - it has 4 dimensions, not 2. (Follows from Higgs field having four components.)

"4-dimensional Mexican Hat potential" has a minimum which is a 3-sphere (whereas 2-dimensional one has a minimum which is a circle). While Higgs field value in this minimum, it can move in three directions without having to climb a potential hill - those three directions correspond to three massless "higgses" which are "eaten" by W+- and Z. The fourth direction, "perpendicular" to 3-sphere, is the direction which requires climbing a hill. That's the component which we observe as massive Higgs boson.