Dear Colleagues,

Sally has a very nice analysis of the difference between the naturalness sections of the long report
(section 1.2.2) proposed by me and by Markus. The current draft was agreed upon between Chip
and me before we sent it to you, but I will take responsibility for its attitude.  Sally's reply to my
email yesterday is pasted in below.  I sent Markus' version yesterday, and it appears again below.

Anyone who wants to weigh in on this -- especially to object to what is in the current draft -- should write
back by Monday morning if possible.  My attitude is that if I have an honest difference of opinion with one
of the conveners, I should win, but if I have an honest difference of opinion with most of the conveners,
their (your) opinion should win.  So, let us all know your opinion by replying to snowmass-ef.

I do think it is important to say that it is more likely to find the first new particles at 1 TeV than at 5 TeV.  
Otherwise, why is LHC so highly motivated?

Thanks,

Michael

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The naturalness sections that Michael and Marcus wrote reflect honest
differences of scientific opinion.  Michael is trying to quantify naturalness
and Marcus is arguing that this isn't really well defined.  From what Marcus
wrote, the reader would infer that 5 TeV is just as likely as 1 TeV for new
particles so we should look at as high an energy as possible.  From what Michael
wrote, you would take home that 1 TeV new particles are much more likely
than 5 TeV.

I subscribe to Marcus's view, but as long as the naturalness section which
Michael wrote refrains from saying that there must be particles at 1 TeV,
I'm ok.

Sally

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 Michael E. Peskin                           [log in to unmask]
 HEP Theory Group, MS 81                       -------
 SLAC National Accelerator Lab.        phone: 1-(650)-926-3250
 2575 Sand Hill Road                       fax:     1-(650)-926-2525
 Menlo Park, CA 94025 USA              www.slac.stanford.edu/~mpeskin/
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________________________________________

from Markus:


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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