In standard model the Higgs potential is a function of

the combination phi^dagger phi. The point to remember

is that the Higgs scalar enters only in this special combination.

phi is a two component complex flied. That is there are four real

fields. Let us call them a,b,c,d. Then the Higgs potential

is a function of

a2  + b2 + c2 + d2

That is we are allowed to change a,b,c,d as long as the

sum of squares are fixed. This is an O(4) symmetry which

is isomorphic to SU(2)xSU(2).

After VEV is given to one real component let us

say the c component, then three others (a,b,d) are

still free. That is we can still change the values

of three fields as long as the quantity a2+b2+d2

is unchanged. This is an O(3) symmetry which is

isomorphic to SU(2). That is the symmetry breaking

chain is,

SU(2)xSU(2) ---> SU(2)

The remaining SU(2) symmetry of the Higgs potential

is a custodial symmetry.

Thank you very much for this interesting explanation and specially for making this so simple and understandable.

How we can use this idea in gauge bosons case? Since it is a global symmetry, can be used this to make a particle stable?

Let us say that the Hypercharge gauge coupling is g'. Thencustodial symmetry is exact when g'=0  . That is this symmetry is only approximate. Here I have also assumed that Yukawa couplings are zero. In the presence of the Yukawa coupling splitting between top and bottom quark masses violate custodial symmetry. But let us discuss the hypothetical case that the custodial symmetry is exact. I guess that will answer your question to some extent. In the limit of g'=0, custodial symmetry forces three gauge bosons to form a degenerate triplet. When g' is non zero, this degeneracy is broken by hypercharge interactions, then we get. MW = cos thetaW MZ. That is rho= MW / (cos thetaW MZ) This relation emerges after custodial symmetry is broken by hypercharge interactions. We often have a false notion that the rho parameter is one due to the presence of custodial symmetry. But here we can see that W and Z bosons have different masses, their ratio is given by cos thetaW when rho is one.

I hope that this helps.