Historical research guide · 0663 · 2013
The Cosmological Constant Problem: Why it's hard to get Dark Energy from Micro-physics
C. P. Burgess
In one minute
Cliff Burgess wrote these Les Houches lectures to state, as precisely as anyone has, why the vacuum’s energy is the hardest open question in physics. His argument runs in three moves. First, Lorentz invariance decides how a vacuum energy must gravitate: pressure equal to minus the energy density, exactly the behaviour dark energy shows. Second, the famous headline number — theory beating observation by 10¹²² — is, he says, a straw man, because the cutoff it depends on cancels out of any real prediction. Third, the honest version is worse in a more interesting way: once everything is written in measured masses, the electron alone contributes about 10³⁰ times too much vacuum energy, and the Z boson about 10⁵⁴. That is the gap to explain. Burgess sets three tests any solution must pass, backs two supersymmetric large extra dimensions as the most promising route, and closes that the problem is a message of hope: new physics must be close.
Connection
Chapter 2 rests on the vacuum having a real and enormous energy density, and this is the clearest published statement of what that costs us: the mismatch is not an artefact of a cutoff, it is there in the electron. Chapter 13 needs exactly this, because any unified picture in which zero-point energy and dark energy are one quantity has to say what happens to the other fifty orders of magnitude — and Burgess supplies the criteria to judge an answer by, plus a route that gets tested by benchtop gravity at micron distances rather than by argument.
From the source
Lorentz invariance settles how a vacuum energy would gravitate. The only stress-energy tensor consistent with an invariant vacuum is proportional to the inverse metric, conservation of that tensor then forces the energy density to be a constant, and comparing with a perfect fluid gives a pressure equal to minus the energy density — an equation-of-state parameter of exactly minus one, which is what the dark-energy observations show.Section 1.1, Eqs. 1.1 to 1.3
The famous headline mismatch — cut the mode sum off at the Planck mass and the predicted vacuum energy exceeds the observed value by a factor of about 10¹²² — is described by Burgess as something of a straw man. The cutoff is an artificial scale introduced to regulate integrals, and in every other part of quantum field theory that dependence is absorbed by renormalisation of the parameters of the Lagrangian; tracking ultraviolet divergences is therefore misleading when asking whether a theory is natural.Section 1.3, Eqs. 1.21 to 1.23; Section 3, point 3
The sharper version of the problem survives renormalisation and is stated in measured quantities. Each particle species contributes a term of order its mass to the fourth power divided by sixteen pi squared, independent of any cutoff. The electron alone overshoots the observed dark-energy density by about 10³⁰, and the Z boson by about 10⁵⁴ — so the honest size of the discrepancy is of order 10⁵⁰, and it is carried by the particle we understand best.Section 1.3, Eqs. 1.24 and 1.25
Technical naturalness is the criterion Burgess proposes, and it asks two separate questions of any small parameter: why it is small in the underlying microscopic theory, and why it stays small as the higher-energy modes are integrated out to give the effective theory at the scale where it is measured. For the cosmological constant, matching the sub-eV value requires the constant in the effective theory above the electron mass and the electron’s own quantum correction to cancel to better than thirty decimal places, and every other well-understood hierarchy of scale between cosmology and particle physics does have a technically natural explanation.Section 1.4, Eq. 1.28; Section 3, point 1
Three minimal criteria are set for any candidate solution: it must work beyond the classical approximation, since there is no problem at all classically; it must apply at scales above an electronvolt, since that is where the quantum problem starts; and it must do no harm to the physics already established at those scales. The rule of thumb Burgess gives is to ask whether a proposal can include the electron, and his assessment of the field in 2013 is that none of the popular models satisfies all three.Section 1.5; Section 3, point 4
His own preferred direction is two supersymmetric large extra dimensions, because extra dimensions break the direct link between a four-dimensional Lorentz-invariant vacuum energy and a large four-dimensional curvature, and supersymmetry in the bulk combines scale invariance with supersymmetry in a way that evades Weinberg’s no-go result. What to watch: the picture is falsifiable at the bench. It requires Newton’s inverse-square law to cross over to an inverse fourth power at distances tied to the dark-energy density — of order a micron, with the longest Kaluza-Klein wavelength near 0.9 micron for extra dimensions of about 1.3 micron — no superpartners at the LHC, and energy loss into the extra dimensions if the higher-dimensional Planck scale is near 10 TeV. Burgess’s closing verdict is that the cosmological constant problem is a message of hope, not despair: by being large, the Universe is telling us that new physics is just around the corner.Section 2.2, Opportunities and worries; Section 3, point 5
The way in
https://arxiv.org/abs/1309.4133How to cite it
C. P. Burgess (2013) The Cosmological Constant Problem: Why it's hard to get Dark Energy from Micro-physics arXiv:1309.4133