Naturalness in the sense of 't Hooft tell us that a small parameter is a signal of a symmetry such that the parameter will be zero when the symmetry is exact. I am puzzled about how this principle is applied now that the Yukawa of the top is confirmed to be -- at low energy -- practically $y_t=1$, while all the other Yukawas are a lot smaller.

It seems that this naturalness principle claims for a symmetry that does not protect the top quark, but protects all the other quarks and leptons. Is there such symmetry in the standard model, or in the most promising BSM extensions (SUSY, GUT, etc)?

  • $\begingroup$ Footnote: if one assumes susy, this naturalness applies to a set of 84 scalars, the partners of leptons and quarks. This is a very beloved number in 11D Sugra and M-theory, as it is the number of components of the antisymmetric tensor which is the source of the membrane. Of course, it is also a common dimension for representations of Lie group, so it could have not further meaning. $\endgroup$ – user135 Sep 20 '11 at 9:37
  • $\begingroup$ I am not familiar with 't Hooft, do you have a reference? $\endgroup$ – rcollyer Sep 20 '11 at 12:44
  • $\begingroup$ I think that 't Hooft formulated it in the context of chiral symmetry breaking, but as a principle it is more general. $\endgroup$ – user135 Sep 20 '11 at 13:41

Top's Yukawa coupling of order one is technically natural, the much smaller couplings are not natural. One may find this problem discussed under the term "hierarchy of Yukawa couplings".

The previous paragraph really says that the Standard Model doesn't have any mechanism to explain the smallness of any non-top Yukawa couplings. First, let me discuss solutions that make couplings natural one by one, or subsets are natural.

In GUT-like theories, the neutrino masses are small because they're produced by the seesaw mechanism from GUT-scale right-handed neutrino Majorana masses and electroweak-scale Dirac masses which produces tiny millielectronvolt-scale Majorana masses for the well-known left-handed neutrinos.

In supersymmetric GUT-like theories, bottom and perhaps charged tau lepton may join the top quark and their masses may be technically natural because one has two Higgs doublets and if $\tan\beta$ is of order 40 – but we know from the LHC that it's almost certainly below 15 today – then the bottom quark has an $O(1)$ Yukawa coupling to the lower-vev Higgs just like the top quark has an $O(1)$ coupling to the greater-vev Higgs. Tau mass may be joined as well.

The other masses of lighter generations are even more unnatural. In string theory, one usually shows that all the generations except for the heaviest ones are strictly massless in some approximation, so the masses are generated by some subleading effects. New U(1) symmetries and/or masses produced from world sheet instantons in intersecting brane worlds are examples where low electron mass may come from.

So this is a technical industry but one must realize that these hiearchies, while obvious, are still much less severe than the lightness of the Higgs and the cosmological constant problem. The Higgs mass is $O(10^{-15})$ in Planck units; the cosmological constant is $O(10^{-123})$ in the same units. On the other hand, the electron/top is $3\times 10^{-6}$ which is less extreme than the usual hierarchy problem (low Higgs mass).


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