Monday, January 30, 2006

Yukawa couplings in the heterotic MSSM

Start with the unique known heterotic background whose visible spectrum matches the pure MSSM - namely the heterotic MSSM. What are the Yukawa couplings and the fermion masses? Braun, He, and Ovrut compute the answer (in the zeroth approximation) tonight:

The resulting textures, affected by the selection rules implied by the Calabi-Yau geometry, make one generation of quarks and leptons naturally light. The word "light" means that the cubic terms contribute zero and the actual masses of the light family arise from higher-order and non-perturbative effects. That means that the cubic approximation is not enough for you to compute the mass of the electron.

The nonzero Yukawa couplings connect the "generation 1" with "generation 2", or "generation 1" with "generation 3", using the appropriate Higgs in each case (which is different for up and down quarks and/or leptons). The "generation i" are just some basis states, not the mass eigenstates. You can see that there are 8 couplings that are complex a priori. With two massive generations, you can assume, without a loss of generality, that the couplings are real. If you don't calculate the numerical value of the couplings, you will have no prediction for the quark and lepton masses: there are 4 quarks and 4 leptons in the two heavier generations, precisely parameterized by the 8 Yukawa couplings.

On the other hand, you may follow and surpass the authors and try to analyze the Yukawa couplings expressed as the integrals in the equation 22. If you're lucky enough, you could predict a non-trivial relation between the masses of the two heavier fermionic families. (The large masses of the right-handed neutrinos will, however, complicate the analysis in the neutrino sector.) The reason why it is complicated is that it is expected that with the non-trivial bundles turned on, the Yukawa couplings won't be simple integer-valued intersection numbers that are constant over the moduli space. Instead, they will be general functions of the moduli. The values of the stabilized moduli are therefore needed to predict the quark and lepton masses.

If you analyze what one can predict without much bigger effort than the already difficult calculation of Braun, He, and Ovrut that may be comprehensible roughly to 12 people in the world, it seems that one can exactly predict the lightness of a single generation relatively to the other two - which seems to be a correct prediction or more precisely a postdiction.


  1. it's all GREAT except it's been shown recently by mathematicians and physicist that the hidden sector bundle in this model is NOT 'STABLE'.

    there comes a chance for those who believes in keeping this alive.

  2. Lubos:

    I protest that you invoke my name in your dispute with Peter. I have no part in this since it is a waste of time to dig out the technical details of this paper. After some very tedious and complicated derivations of symbols and formalisms, which makes one's brain numb, you find at the end, if your brain hasn't become numb yet, that after all, the authors have done NO CALCULATION. Not a single numerical value was given, used or calculated. The only thing worth meantioning is that they figured out some quarks are lighter than others, which is true but nonsense: Of couse unless all quarks are the same, some must necessarily be lighter than others.

    These sentence from the paper sums it all up: "Of course, to explicitly compute the quark/lepton masses one needs, in addition, the K¨ahler potential, which determines the correct normalization of the fields."

    They have not calculated the particle masses.


  3. It seems that the "fast comments" section on this piece could be more interesting than usual and I almost feel tempted to continue here.

    Now I think that quantoken name is well invoked here if as an alert about how easy is to derive "physical results" from random especulation. Or remember our (Hans and me) best example, that mass of the electron equals to mass of W times the vacuum polarization part of the giromagnetic ratio of muon (regretly while this result is exact with the current measure of W, the counterpart, to calculate mass of muon from mass of Z and giromagnetic ratio of electron, gives a bad 105.7 MeV). I find it hard to put more faith in the string derived zero order results than in these random observations.