I am thinking about all the possible contributions to the vacuum energy. One of them comes from the minimum of the potential of scalar field which is called the the classical contribution. For the case of the Higgs field (assuming a Mexican Hat potential), most sources say that it gives a nonzero contribution which for me is surprising since I think you will eventually redefine the Higgs field that gives particle interpretation and to be consistent with perturbation theory. This mean the new field has zero vaccum expectation value (vev). Am I missing something? Is the physically relevant field the original one that also gives the vev?
The nonzero vev of the Higgs field is part of the cosmological constant problem, along with the QCD condensates and the zero-point energies that should also contribute.
And no one seems to have any good ideas except the anthropic solution, which says that it's all cancelled out by other contributions (e.g. string-theoretic branes and fluxes), in a messy random way, because if it didn't cancel, there couldn't be atoms or galaxies or life.
So we are at some random place in the string landscape of vacua, where the various vacuum energy contributions happen to cancel out, and there's no deeper explanation.
It's a logically possible explanation but certainly not proven, so one can look for other answers. And maybe it's conceivable that for some reason, the "new field" that results from the redefinition, is the real baseline that one should use in calculating the Higgs contribution to the vacuum energy.
But there's two problems with this. First, the Higgs mechanism for fermions seems to require the existence of a genuinely nonzero energy density. Second, this doesn't account for all the other contributions that should be creating a huge vacuum energy.
You can't just handwave away the VEV, because the $U(1)$ symmetry is spontaneously broken in its choosing of a complex VEV's phase before you try to translate it away. There will still be a circle of potential-minimizing field amplitudes, which gives a nontrivial vacuum contribution. To think of it another way, the EOM is nonlinear so the usual decomposition into ladder operators can't be used to prove the field averages to zero in vacuo.