Usually, in particle physics, people do not care about a constant term in scalar field potential. Rather, attentions are paid to the local profile at the minimum. But in the context of cosmology, the absolute value of the potential has a physical meaning; it is a cosmological constant and can cause the Universe to accelerate or decelerate. My impression is that the naive potential for the Higgs field has a negative value at the minima. Do people take it seriously as a negative cosmological constant? Is the dynamical change in the value of the potential at the minimum during EWSB taken into account?

  • $\begingroup$ Well, a contribution to the CC is generated spontaneously when the Higgs obtains a VEV. But it can always be cancelled by a CC is the Lagrangian. $\endgroup$ – innisfree Apr 21 at 9:09
  • $\begingroup$ @innisfree Sure, but the CC in the Lagranian is fixed once given while the Higgs potential is dynamical durning the evolution of the Universe. I want to know whether or not people seriously consider the corresponding gravitaional effects for this dynamical change. Or it is just too small that can be neglected? $\endgroup$ – Wein Eld Apr 21 at 9:27
  • $\begingroup$ This is an interesting question. My suspicion is that any such changes are too small to have an effect on cosmological evolution, provided that you fix the vacuum energy after all phase transitions, etc. complete. For instance, naively the electroweak phase transition changes the vacuum energy density by order $m_W^4$. But it takes place at temperature $T \sim m_W$ where the energy density is already $T^4 \sim m_W^4$ anyway. $\endgroup$ – knzhou Apr 22 at 5:29
  • $\begingroup$ So any effect will change the expansion rate a little around that phase transition, but will have little effect if one goes more than an $e$-fold before it. Of course, such gigantic changes in the cosmological constant occurring early in the universe do make it mysterious how the final value is so small -- but that's precisely why the cosmological constant problem is so hard. $\endgroup$ – knzhou Apr 22 at 5:30
  • $\begingroup$ @WeinEld I get it, i have added a sentence to your question, if it's OK $\endgroup$ – innisfree Apr 22 at 7:02

There are two things to consider here: the late-time accelerated expansion, often discussed within the context of the cosmological constant, $\Lambda$, and primordial inflation. Let's take the latter first.

In primordial inflation, one has a field initially evolving in the false vacuum of some potential energy function. The false vacuum must be very flat in order to get enough inflation and generate the right spectrum of density perturbations. For example,

from https://indico.cern.ch/event/180122/attachments/239069/334713/NExT_2012_Atkins.pdf

Cosmologists early on hoped that the SM Higgs, or perhaps a GUT Higgs, might have been what drove primordial inflation but the trouble was that the false vacuum wasn't flat enough. In recent years, this idea has been regained momentum when it was realized that if one coupled the Higgs to gravity (as a non-minimal coupling with the scalar curvature), then one could flatten this region and obtain successful inflation, see https://arxiv.org/abs/0710.3755.

Once the Higgs decays to the true vacuum, primordial inflation ends. If the true vacuum, however, has a nonzero vacuum energy, then it's possible that the Higgs field could contribute to the observed late-time accelerated expansion. As you note, the classical vacuum energy of the Higgs is negative, $\rho_{\rm Higgs, \,vacuum} < 0$, and this is not what we observe. Therefore, it's possible that the effective cosmological constant is something like $\Lambda_{eff} = \Lambda + \rho_{\rm Higgs,\,vacuum}$, but this subtraction would need to be fine tuned to get our tiny observed amount of cosmological constant.

(Image from: https://indico.cern.ch/event/180122/attachments/239069/334713/NExT_2012_Atkins.pdf)

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  • $\begingroup$ Thanks a lot for your answer. Although this is not strictly speaking the answer I expected but it provides useful information. What I would like to know is actually not about the cosmological constant per se but rather the discussion of the gravitational effects (at least from the change of the effective cosmological constant) during the electroweak phase transition happened in the early Universe. $\endgroup$ – Wein Eld Apr 23 at 9:20
  • $\begingroup$ As the Higgs decays, its vacuum energy drops, and so the expression I wrote above for the effective CC applies, but with $\rho_{\rm Higgs,\, vacuum}$ replaced by the time-varying vacuum energy density. Since the universe was radiation dominated during the EW transition with the vacuum making up only a tiny proportion of the energy budge, the effect of this is tiny. $\endgroup$ – bapowell Apr 23 at 12:37
  • $\begingroup$ Also, in the case of Higgs inflation, then the change in vacuum during the EW transition has a huge effect, namely it causes the inflationary epoch to transition into the radiation dominated phase. $\endgroup$ – bapowell Apr 23 at 12:54

The cosmological constant is part of the mathematical framework of General Relativity.

The Higgs field is part of the mathematical framework of quantum field theory and particle physics .

At the moment there is only effective quantization of gravity, used in cosmological models , and yes, if you google higgs field and the cosmological constant a number of publications come up, so people are examining the possible relation.

A real proof will come only when/if a unifying theory of everything embedding the standard model of particle physics and gravity is found.

In this answer I give a link to a related question, on how string theories, which include both the standard model and quantization of gravity may deal with the problem.

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  • $\begingroup$ Thanks for your answer. As I commented above, I did not mean to ask about the cosmological constant problem and not quantum gravity, but actually the gravitational effects from the change of the absolute value of the Higgs potential at the minimum. It is a classical effect, i.e. with the spacetime or graity treated as classical. $\endgroup$ – Wein Eld Apr 23 at 9:25

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