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Geometric Models: E8, SO(10)

  1. Dec 6, 2007 #1


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    Geometric Models: E8, SO(10),
    Which ansatzs will prove to be right by CERN?
    1. Ali H. Chamseddine and Alain Connes , SO(10)
    …. the spectral action associated with this noncommutative space unifies gravitation with the Standard Model at the unification scale.
    … Therefore the bare action we obtained and associated with the spectrum of the standard model is consistent within ten percent provided the cutoff scale is taken to be _ ∼ 1015 Gev at which the action becomes geometrical.

    2. A. Garrett Lisi, E8
    … A non-compact real form of the E8 Lie algebra has G2 and F4 subalgebras which
    break down to strong su(3), electroweak su(2) x u(1), gravitational so(3,1), the frame-Higgs, and three generations of fermions related by triality.

    Everyone is trying to get away from the Planck Scale.

    If the geometry is expressed at the QCD why would the cut off scale need to go to the unification scale?
  2. jcsd
  3. Dec 6, 2007 #2
    If you're going to talk about being "proven right by CERN" though, then of course we should keep in mind that both of these models are already in conflict with experiment by way of proton decay. As GUTs both SO(10) and E8 predict proton decay, right? But as far as any experiment that has been done so far can tell, proton decay does not happen...
  4. Dec 6, 2007 #3
    On what time scale do the theories predict decay? I seem to recall the decay time is long enough to go to the big crunch...
  5. Dec 6, 2007 #4


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    SO(10) does not address the family problem; E8 does not solve it minimally.
  6. Dec 7, 2007 #5


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    People have been trying to make SO(10) Models work for quite a while. As a result I was able to find quite a bit of info.
    I like this explanation of “family”.
    The discussion above indicates that for an adequate baryon asymmetry to survive, a few -violating Higgs bosons must be significantly lighter than the rest. In general, Higgs-boson masses are determined by minimization of the effective Higgs potential and diagonalization of the resulting mass matrices. Some models exhibit approximate symmetries, broken on scales , which lead to approximately degenerate sets of Higgs bosons corresponding to irreducible representations of the approximate symmetry. Models with large approximate symmetry groups may therefore be unable to account for the observed baryon asymmetry. In the absence of approximate symmetries, one must usually resort to a purely statistical treatment, taking the boson masses as random variables. In the simplest case, one takes the boson masses to be independent random variables, distributed over a definite, say, unit, interval. With 5 particles, the average spacing of the lowest two masses is then ; the spacing decreases to for 10 particles, to with 20, and to with 50. With more than 20 particles, the probability that the lowest two particles are separated by more than 10% in mass falls below 0.1. An alternative and perhaps more realistic approach takes the elements of the boson-mass matrices to be independent Gaussian random variables (with, say, zero mean and unit valance). [Note, however, that the Clebsch-Gordan coefficients on which the mass matrix elements depend are not in fact Gaussian distributed at least in the case of SO(3).[28]] The distribution of eigenvalues for such matrices becomes semicircular when the dimension n of the matrix exceeds about 5.[29] Monte Carlo simulation shows that the mean spacing between the lowest masses, normalized by the total range of masses, falls roughly linearly with n, taking on a value for . For , the probability for a spacing larger than 10% between the lowest masses again falls below 0.1.

    Calculation of Cosmological Baryon Asymmetry in Grand Unified Gauge Models (1982)
    7. SO(10) Models
    Alain Connes papers seem to be trying to get get people interested in his ansatzs that proposes to solve the “plethora of Higgs fields or the infinite tower of states.”
    His approach keeps the “Higgs” within the QCD range but he still wants to go to the unification scale.

    Since E8 and SO(10) are geometric models, then some of the argument used by one approach would also apply to the other approach.
    Ali H. Chamseddine and Alain Connes
    27 June 2007
    The purpose of this letter is to remove the arbitrariness of the ad hoc choice of the algebra and its representation in the noncommutative approach to the Standard Model, which was begging for a conceptual explanation. We assume as before that space-time is the product of a four-dimensional manifold by a finite noncommmutative space F. The spectral action is the pure gravitational action for the product space. To remove the above arbitrariness, we classify the irreducibe geometries F consistent with imposing reality and chiral conditions on spinors, to avoid the fermion doubling problem, which amounts to have total dimension 10 (in the K-theoretic sense). It gives, almost
    uniquely, the Standard Model with all its details, predicting the number of fermions per generation to be 16, their representations and the Higgs breaking mechanism, with very little input. The geometrical model is valid at the unification scale, and has relations connecting the gauge couplings
    to each other and to the Higgs coupling. This gives a prediction of the Higgs mass of around 170 GeV and a mass relation connecting the sum of the square of the masses of the fermions to the W mass square, which enables us to predict the top quark mass compatible with the measured experimental value. We thus manage to have the advantages of both SO(10) and Kaluza-Klein unification, without paying the price of plethora of Higgs fields or the infinite tower of states.

    It was shown in [4] that the fermions lie in the desired representations, and that the spectral action associated with this noncommutative space unifies gravitation with the Standard Model at the unification scale.

    (4) ftp://ftp.alainconnes.org/standard.pdf
    Conceptual Explanation for the Algebra in the Noncommutative Approach to the
    Standard Model
    Ali H. Chamseddine and Alain Conne
    21 Nov 2007

    The freedom in the choice of the gauge group and the fermionic representations have led to many attempts to unify all the gauge interactions in one group, and the fermions in one irreducible representation. The most notable among the unification schemes are models based on the SO(10) gauge group and groups containing it such as E6, E7 and E8. The most attractive feature of SO(10) is that all the fermions in one family fit into the 16 spinor representation and the above delicate hyper-charge assignments result naturally after the breakdown of symmetry. However, what is gained in the simplicity of the spinor representation and the unification of the three gauge coupling constants into one SO(10) gauge coupling is lost in the complexity of the Higgs sector.
    To break the SO(10) symmetry into SU(3)c × U(1)em one needs to employ many Higgs fields in representations such as 10, 120, 126 [1]. The arbitrariness in the Higgs sector reduces the predictivity of all these models and introduces many arbitrary parameters, in addition to the
    unobserved proton decay.

    The noncommutative geometric approach [2] to the unification of all fundamental interactions, including gravity, is based on the three ansatz [3], [4]:
    • Space-time is the product of an ordinary Riemannian manifold M by a finite noncommutative space F.
    • The K-theoretic dimension (defined below) of F is 6 modulo 8.
    • The physical action functional is given by the spectral action at unification scale.
    The empirical data taken as input are:
    • There are 16 chiral fermions in each of three generations.
    • The photon is massless.
    • There are Majorana mass terms for the neutrinos.
    Furthermore one makes the following “ad hoc” choice
    • The algebra of the finite space is taken to be C ⊕ H ⊕M3 (C) where H is the algebra of quaternions andM3 (C) is the algebra of complex 3×3 matrices.
    One of the main purposes of this letter is to show how this algebra arises.
    …. the spectral action associated with this noncommutative space unifies gravitation with the Standard Model at the unification scale.

    We conclude that our approach predicts a unique fermionic representation of dimension 16, with gauge couplings unification. These properties are only shared with the SO(10) grand unified theory.
    The geometrical model is valid at the unification scale, and relates the gauge coupling constants to each other and to the Higgs coupling. When these relations are taken as boundary conditions valid at the unification scale in the renormalization group (RG) equations, one gets a predic-
    tion of the Higgs mass to be around 170 } 10 GeV, the error being due to our ignorance of the physics at unification scale.
    We note that general studies of the Higgs sector in the standard model [8] show that when the Higgs and top quark masses are comparable, as in our case, then the Higgs mass will be stable under the renormalization group equations, up to the Planck scale.
    Since the Higgs fields have not been detected both approach (E8 and SO(10) ) have got to make the assumption that something similar exist to make their model work.
    I also gather from the papers that what ‘Higgs” could be doing is problematic.

    If the geometry is expressed at the QCD why would the cut off scale need to go to the unification scale? Do we really need all those extra particles? Do we need to go to the higher energy scale of the unification?

  7. Dec 7, 2007 #6


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    "DRESS FOR THE BEGGER" and "Conceptual Explanation for the Algebra in the Noncommutative Approach to the Standard Model" are the same article. It's just that the editor wanted a formal title for publication.
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