LOG#226. Higgs vs. lambda.

Hi, there. Short post today!

The dark energy mystery, a.k.a., the cosmological constant problem or why the observed vacuum energy is to tiny is a big problem with no consensus solution yet.

The Standard Model Higgs potential can be written as follows

    \[V_H=-m_H^2\phi^2+\lambda\phi^4+\phi_0\]

Here, m_H is the Higgs mass m_H=125GeV, \lambda=0.13 is the Higgs self-coupling from the given mass and \phi_0 is the vacuum expectation value of the Higgs field, namely

    \[\phi_0=\langle 0\vert \phi\vert 0\rangle=246GeV\]

From High School mathematics, you can easily find out the minimum of the above potential

    \[\dfrac{d V_H}{d\phi}=-2m_H^2\phi+4\lambda\phi^3=2\phi\left(-m_H^2+2\lambda\phi^2\right)=0\]

so the non null solution is given by

    \[\phi_0(min)^2=\dfrac{m_H^2}{2\lambda}\]

and the Higgs potential value at that value becomes

    \[V(\phi(min))=-\dfrac{m_H^2\phi^2}{2\lambda}+\phi_0\approx -\dfrac{m_H^2\phi^2}{4\lambda}\]

The Higgs potential provides a natural source for vacuum energy density. Using the values mentioned before we get

    \[\vert V_H(min)\vert\approx 5\cdot 10^8GeV^4\]

The value we get from Cosmology, using the LCDM model is however very tiny

    \[V(\Lambda)=\dfrac{\Lambda c^4}{8\pi G}\approx 3.1\cdot 10^{-47}GeV^4\]

thus, we have a mismatch between theory and experiment about

    \[\dfrac{\rho(Higgs)}{\rho(\Lambda)}\sim 2\cdot 10^{55}\]

Note that this is, despite the mismatch, better than the crude mismatch due to the even bigger discrepancy between Planck energy density and observed vacuum energy density, since Planck energy density reads:

    \[\rho_P=\dfrac{c^7}{G^2_N\hbar}\approx 5\cdot 10^{113}J/m^3\approx  2.2\cdot  10^{76}GeV^4\]

and thus

    \[\dfrac{\rho_P}{\rho_\Lambda}\sim 10^{122}\]

Or,as well,

    \[\dfrac{\rho_P}{\rho_\Lambda}=\dfrac{\dfrac{c^7}{G^2_N\hbar}}{\dfrac{\Lambda c^4}{8\pi G_N}}=\dfrac{8\pi}{3}\dfrac{c^3}{G_N\hbar}\dfrac{3}{\Lambda}=\dfrac{8\pi}{3}\dfrac{L_\Lambda^2}{L_P^2}\sim \dfrac{L_\Lambda^2}{L_P^2}\]

Solutions? Well, many:

  • The Higgs value is wrong, because the Higgs potential from the SM is not right, but only an approximation.
  • The Cosmological Constant is not the Higgs potential source.
  • Both values are OK, we have bad measurements only.
  • Vacuum energy is scale dependent, you can not compare them without some tricky trick.
  • Some non-perturbative effect in under the floor.
  • A new physics reason not quoted above.

I know ther are some other crude estimates from UV-cutoff given a 122-123 orders of magnitude separation. However, this is much closer. Some SUSY (supersymmetry) theories have some arguments to essentially fit some of the solutions I mentioned. But the issue is not at all clear. A very big cosmological constant would be a disaster for life. The very big vacuum energy of Quantum Field Theory grosser estimates clearly are nonsense, but we do not know yet when the calculation is wrong. Maybe a dS (de Sitter) QFT could provide a better solution using \Lambda as fundamental constant?

Note, do not confuse the Higgs self-coupling \lambda with the cosmological constant \Lambda.

What do you think? A simple minimization of the Higgs potential according the SM gives you a 55 order of magnitude split with observed vacuum energy density? Are they the same or are we lacking something more fundamental?

Off-topic news: it seems the quark-gluon plasma behaves as an ideal relativistic fluid, as the holographic hypothesis suggest, giving experimental increasing support to the bound of shear viscosity to entropy ratio predicted by those holographic models

    \[\dfrac{\eta}{s}\geq\dfrac{\hbar}{4\pi k_B}\]

or in natural units

    \[\dfrac{\eta}{s}\geq \dfrac{1}{4\pi }\]

References:

[1] Jonah E. Bernhard, J. Scott Moreland, Steffen A. Bass, «Bayesian estimation of the specific shear and bulk viscosity of quark–gluon plasma,» Nature Physics (12 Aug 2019), doi: 10.1038/s41567-019-0611-8

[2]  Kari J. Eskola, «Nearly perfect quark–gluon fuid,» Nature Physics (12 Aug 2019), doi: 10.1038/s41567-019-0643-0.

See you in another blog post!!!!

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