# How can Higgs Field contribute to the vacuum energy density if its vev is zero?

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?

• It's not clear here what exactly you are talking about. Could you elaborate a bit more? – ACuriousMind Feb 1 '17 at 16:32
• have a look www2.ph.ed.ac.uk/~playfer/PPlect17.pdf the HIggs vev is 246 Gev – anna v Feb 2 '17 at 13:17
• yeah thanks anna v. This means the original field with no particle interpretation is still physically relevant since it is the one contributing? Meaning it is not just a component of the stress energy tensor that contributes to the vacuum but also the vev of all scalars? – bonez001 Feb 2 '17 at 13:23
• @bonez001 the fields in field theory are the fields of the particles in the elementary particle table en.wikipedia.org/wiki/… . The creation and annihilation operators generate the particles. The vacuum expectation values of all particle-fields except the Higgs-field are zero. In the standard model the only scalar field is the Higgs field. You have to distinguish between fields and the particles which are riding on the fields with creation and annihilation operators.. – anna v Feb 3 '17 at 15:29
• I really don't get it if the fields are linear in annihilation and creation operators then the expectation value of all fields must be zero. – bonez001 Feb 5 '17 at 1:04

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.

• "Higgs mechanism for fermions" means how the mechanism gives the fermion's mass? I think I get it now. If we redefine the Higg's field, there will always be a constant term because of the redefinition. And we can interpret it as the fields expectation value. – bonez001 Feb 5 '17 at 1:06

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.