Higgs boson has a mass, and a vacuum expectation value, both are in electronvolts.

An uncommon feature of the Higgs field is that to its zero energy density belongs a non-zero field value.

Is it possible, probably with QFT, to calculate its energy density (per unit volume)? In layman terms, how would it work?

  • $\begingroup$ As you said, it has zero energy density, so why are you asking how to calculate the energy density? And, as you said, the field has a nonzero VEV, so why does your title refer to the “zero Higgs field”? $\endgroup$ – G. Smith Oct 2 '19 at 17:25
  • $\begingroup$ @G.Smith If I understood that maxican hat thing well, I understand the origin in that potential diagram. Maybe I formulated something badly? Effectively, I am asking for, how high is the origin in the mexican hat potential. In SI units. Feel free to edit the post, if it makes it better. $\endgroup$ – peterh - Reinstate Monica Oct 2 '19 at 17:42
  • $\begingroup$ Aha. I understand your question now. $\endgroup$ – G. Smith Oct 2 '19 at 17:45

The "Mexican hat" Higgs potential is


where $v$ is the Higgs vacuum expectation value, 246 GeV, and the dimensionless Higgs coupling $\lambda$ is about 0.0323.

To get this value for $\lambda$, look at the quadratic "mass term" when expanding the Higgs field around its VEV, $H=v+h$. It is $4\lambda v^2 h^2$, so the mass of the Higgs is $m_H^2=8\lambda v^2$ and thus $\lambda=m_H^2/8v^2$. Using 125 GeV as the measured Higgs mass, one gets $\lambda\approx 125^2/(8\times 246^2)$.

When the Higgs field $H$ is zero, the potential is $\lambda v^4$, which is about $1.18\times 10^8\text{ GeV}^4$. The conversion for energy density between natural units and SI units is

$$1\text{ GeV}^4\approx 2.08\times10^{37}\text{ J/m}^3$$

so the zero-Higgs-field energy density is about $2.45\times 10^{45}\text{ J/m}^3$. The corresponds to a mass density of $2.72\times 10^{28}\text{ kg/m}^3$. This is about 10 orders of magnitude higher than density at the center of a neutron star, but about 68 orders of magnitude lower than the Planck density.

Note that this energy density is not the Higgs energy density now, when $H=v$. That energy density is, as you said, zero.

  • $\begingroup$ @Cham Thanks for catching that typo! $\endgroup$ – G. Smith Oct 2 '19 at 20:07
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    $\begingroup$ It feels a bit weird that the Higgs mass $m_H \approx 125~\mathrm{GeV}$ is almost exactly half the Higgs vacuum expectation value. $\endgroup$ – Cham Oct 2 '19 at 20:16
  • $\begingroup$ Agreed. I don’t think there is an accepted explanation for that. $\endgroup$ – G. Smith Oct 2 '19 at 20:22
  • $\begingroup$ @Cham: It boils down to the fact that $\lambda \approx 1/32$, but that may not be a sufficient explanation. $\endgroup$ – Michael Seifert Oct 2 '19 at 20:38
  • $\begingroup$ @G.Smith Thanks! Wiki says, neutron star central density is only about $10^{17} \frac{\rm{kg}}{\rm m^3}$. Thus, the difference is only 9 orders of magnitude. $\endgroup$ – peterh - Reinstate Monica Oct 3 '19 at 12:29

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