Nude Socialist (via Joseph S.) published an article called
Dark energy could be the offspring of the Higgs bosonwhich mainly discusses a June 2013 preprint by Lawrence Krauss and James Dent,
A Higgs-Saw Mechanism as a Source for Dark Energy.Funny and not terribly serious. And no, no followups to the paper appeared in the first two months.
The story quotes Frank Wilczek – without even mentioning his remotely related recent paper on Multiversality (which is somewhat more substantial). We learn that Lawrence Krauss was actually "a Higgs sceptic until the very end". What a poetic way to say "a stubborn moron".
At any rate, now Mr Krauss has apparently kindly accepted the belief that there exists a Higgs boson – something he should have learned and understood as an undergrad – so he and James Dent became convinced that they may solve all big problems of physics, too.
The first problem they have "solved" is nothing else than the cosmological constant problem – why is the observed cosmological constant (C.C.) so small in the Planck or other natural units – and they immediately self-confidently sent the solution to PRL, a prestigious journal.
The "solution" is something that every other physicist has thought about: a seesaw mechanism. Numerically, it looks like the C.C. is smaller than the fourth power of the Higgs mass by the same factor by which the fourth power of the Higgs mass is smaller than the fourth power of the Planck or GUT mass, i.e.\[
\rho_{\rm C.C.} \sim \frac{m_{\rm Higgs}^8}{m_{\rm Planck}^4}
\] On the log scale, the Higgs (electroweak) mass scale is in between the tiny mass scale calculated from the dark energy (as the fourth root of the dark energy expressed as an energy density in the \(\hbar=c=1\) units) and the huge Planck scale.
The identity above may be derived – in analogy with the neutrinos – from a pair of fields. If the matrix entries are\[
\pmatrix{ m_{\rm Planck}^2& m_{\rm Higgs}^2\\m_{\rm Higgs}^2 & 0 },
\] i.e. one entry is huge (Planckian) and the off-diagonal ones are intermediate (Higgs-like), then the eigenvalues of this squared mass matrix are of order \(m_{\rm Planck}^2\) and \(m^4_{\rm Planck} / m^2_{\rm Higgs}\), respectively. The latter formula arises because the product of these eigenvalues has to be equal to the determinant of the \(2\times 2\) matrix above which happens to be \(-m_{\rm Higgs}^4\).
Because the sum of the eigenvalues is equal to the trace \(m_{\rm Planck}^2\), one of the eigenvalues has to be close to this huge value, too. Due to the right determinant requirement above, the other eigenvalue has no choice and has to be insanely tiny to compensate the Planckian mass in the product (determinant).
Just calculate the damn eigenvalues of the matrix above if you don't know what I am saying.
At any rate, it's obvious – and has been explicitly said by many people many times – that the observed C.C. looks like one arising from such a very tiny "Higgs companion". That's all nice that we may produce new terms to the C.C. that have about the right value.
But the true difficult task is to show that except for the term we would like to emphasize, all the remaining terms cancel so that they don't spoil this nicely tiny, tuned result. In fact, this trivial point is even quoted in Nude Socialist – and attributed to Frank Wilczek although pretty much everyone else could tell them the same thing.
Needless to say, they don't have any explanation for such a coincidence (in most theories, this coincidence almost certainly doesn't occur) which means that they haven't solved the C.C. problem (in a non-anthropic way). But if the production of the seesaw-like tiny contribution is one of the steps to actually solve the C.C. problem, it's very interesting. I have thought about this possibility for quite some time and found some other possible solutions but I wouldn't be arrogant enough to send this incomplete stuff to the arXiv or even to the PRL. One must have some extra chutzpah for that.
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