# What is the energy density of the zero Higgs field in SI units?

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?

• 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”? – G. Smith Oct 2 at 17:25
• @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. – peterh says reinstate Monica Oct 2 at 17:42
• Aha. I understand your question now. – G. Smith Oct 2 at 17:45

The "Mexican hat" Higgs potential is

$$V(H)=\lambda(|H|^2-v^2)^2$$

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.

• @Cham Thanks for catching that typo! – G. Smith Oct 2 at 20:07
• It feels a bit weird that the Higgs mass $m_H \approx 125~\mathrm{GeV}$ is almost exactly half the Higgs vacuum expectation value. – Cham Oct 2 at 20:16
• Agreed. I don’t think there is an accepted explanation for that. – G. Smith Oct 2 at 20:22
• @Cham: It boils down to the fact that $\lambda \approx 1/32$, but that may not be a sufficient explanation. – Michael Seifert Oct 2 at 20:38
• @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. – peterh says reinstate Monica Oct 3 at 12:29