I just returned from the first lecture delivered by Lisa Randall that I attended - her course is about the physics beyond the Standard Model.

Today, she focused on technicolor, i.e. hypothetical strong gauge dynamics behind the electroweak symmetry breaking. Her final verdict was, of course, that the explicit theories (especially ETC, extended technicolor) suffer from very serious problems - especially from the flavor-changing neutral currents that are inevitable if the theory is made able to give quarks (and leptons) realistic masses (and mixings): those two things - one of them necessary, the other undesirable - are simply generated with roughly the same magnitude. ETC is also problematic because "too much new stuff near a TeV" typically destroys the agreement with precision electroweak measurements. And getting all the things right amounts to a huge number of assumptions about ETC which makes it less convincing.

Nevertheless she reinforced my desire to give a natural, "exponential" explanation of the hierarchy problem.

The QCD scale is much smaller than the Planck scale, and we know why. The QCD (or the GUT) coupling at the Planck scale (or the GUT scale) is naturally a number of order one (although slightly smaller than one, imagine 1/25 for the fine structure constant). The RG running is responsible for dimensional transmutation - the dimensionless coupling is converted to the information about the scale where this coupling becomes of order one. And because the dimensionless coupling depends on the scale just logarithmically, you must go to exponentially small scales compared to the GUT scale to see the coupling approach one - and this is why you get the QCD scale so small. This is what I call a satisfactory qualitative explanation.

It's just very natural to imagine a similar explanation for the gap between the electroweak scale (such as the Higgs' vev or its mass) and the Planck scale (or another high scale). Technicolor theories are just very natural in this respect because they try to mimic QCD almost exactly.

But are we really sure that we cannot make this stuff happen without the extra fermions and gauge theory? For a fixed value of the quartic coupling, the Higgs mass is a (decreasing) function of the Higgs vev. Cannot there be some symmetry that guarantees that the Higgs mass must be (up to a factor) equal to the Higgs vev at the relevant (electroweak) scale? Perhaps some residual Z_2 symmetry of a duality? If it were so, they would both to be comparable to scale where the quartic coupling is of order one (which is meant to be around a TeV), and the scale where this coupling approaches one could be governed by similar logic as the QCD scale.

Moreover, have we/they explored all possible loopholes of the argument that the beta function for spin 0 and spin 1/2 theories must be positive, as Gross and Coleman (?) proved? Is not there some new coupling, analogous to the Green-Schwarz mechanism or something like that, that could lead to a negative beta function?

People have obviously tried a lot of various things, and me reassure you that I still believe that the SUSY is the most justified explanation of the hierarchy problem. But we must be prepared for the situation that SUSY at low energies does not exist. The anthropic people tend to say recently that the Landscape predicts a high supersymmetry breaking scale. Moreover, even though SUSY makes the Higgs mass stable against quantum corrections, it still does not really explain where the large ratio between the electroweak scale and the Planck scale comes from, am I wrong? An exponential mechanism like one mentioned above may be desirable even in the context of SUSY phenomenology.

ReplyDeleteNevertheless she reinforced my desire to give a natural, "exponential" explanation of the hierarchy problem.Hmm let me try one... the Weinberg angle must run until reaching the smallest possible value for [the square matrix element of] Z0 decay, ie for the object \Gamma(Z0)/M_Z^3. That sounds very physical to me :-)