Some suggested changes to the 50 page EF summary.
1) VLHC: I looked at the mention of VLHC throughout the report, and I think that the examples that discuss VLHC are well chosen. I should have read more carefully before firing off my previous email about this. I would only suggest that the name "VLHC" be used everywhere the document refers to a 100 TeV collider, e.g. lines 230, 256, 807, 826. 885, 1092.
2) Lines 28-42: These refer to "phenomena of nature" not accounted for by the standard model. I think that naturalness should be mentioned after line 42, since it is one of the most important open problems in the standard model. Proposed text:
In addition, there is a major theoretical puzzle with the standard model. If the Higgs is an elementary scalar particle, as predicted by the standard model, its mass is sensitive to the masses of any heavier particle to which it couples. It appears to require a cancellation of one part in $10^{32}$ to explain why the Higgs mass is smaller than the Planck mass.
3) Lines 50-54: The above nicely ties in with this point. I do not think it is very clear to say that we have "expressed discomfort" about the Higgs. I suggest the following modification:
Second, one of the key mysteries concerns the Higgs itself. The standard model with an elementary Higgs boson is the simplest theory that describes the data, but the existence of the Higgs forces the problem of the apparent fine-tuning of elementary scalars.
4) Lines 152-196. I do not think that naturalness is a "bothersome hint" or a "slippery principle." I think it can be explained in very basic physical terms. I suggest the following:
"Naturalness" is at bottom the use of dimensional analysis to estimate unknown parameters. If a quantity such as the Higgs mass is sensitive to a physics associated with a mass $M$, then dimensional analysis suggests that the Higgs mass should be of order $M$. Of course, this does not take into account the possibility that this dependence is absent, in which case we expect to have a good reason why this sensitivity is absent, such as a symmetry or some kind of decoupling.
Decades of theoretical work in quantum field theory has shown that elementary scalar masses are generically sensitive to physics at higher scales, and only three mechanisms have been established that can avoid this sensitivity. These are supersymmetry, (SUSY), Higgs compositeness, and extra dimensions. Each of these predict a rich spectrum of new states at the scale where the new structure becomes apparent. In SUSY, these consist of the superpartners of all known particles, while in both composite and extra-dimensional models we expect towers of massive resonances. (The fact that the phenomenology is qualitatively similar is the first sign that extra-dimensional models are in fact a realization of Higgs compositeness, a fascinating and deep equivalence that was discovered in string theory and has propagated to particle phenomenology and back again to fundamental theory.)
These mechanisms allow the Higgs mass to be calculated from other more fundamental parameters, and they confirm the expectations of naturalness in the sense that the Higgs mass is indeed sensitive to the new particles associated with SUSY or compositeness. The Higgs mass therefore cannot be much smaller than the scale $M$ of new particles predicted in these models. The Higgs mass can be much smaller than $M$ only if there is an unexplained accidental cancellation, or "fine tuning."
We can see the naturalness problem even without knowing what the new fundamental physics is. If we simply assume that there is *some* new physics at a scale $M$ we can estimate the sensitivity of the Higgs mass to new physics at the scale $M$ by computing quantum loops in the standard model with a cutoff of order $M$. The parameter in the Higgs potential then receives corrections of order
Eq. (1.4) with $M$ instead of $\Lambda$
where $g_{Htt}$ is the same Yukawa coupling as in (1.2), $\alpha_w$ and $\lambda$ are the couplings of these particles, and $\theta_w$ is the weak mixing angle. Note that all terms are proportional to $M^2$, simply as a result of the fact that it is the Higgs mass squared that appears in the Lagrangian. Experience with many specific models teaches us that if there is new physics at the scale $M$, (1.4) gives a reasonable estimate of the contribution of new physics at the scale $M$ to the Higgs mass. The suppression factors in (1.4) mean that the natural expectation for the scale $M$ is that it cannot exceed the Higgs mass by about a factor of 10.
Although there is no general agreement on how to quantitatively measure the (un)naturalness of a given model, it is clear that the degree of tuning required to obtain $m_h \ll M$ grows quadratically with $M$. This means that if we increase the sensitivity to heavy particle masses by a factor of 10, we increase our probing of naturalness by a factor of 100. This provides a very strong motivation to for searches at the largest possible energies.
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Markus Luty
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Physics Department
University of California, Davis
One Shields Avenue
Davis, CA 95616
Phone: +1 530 554 1280
Skype: markus_luty