A first test of the framed standard model against experiment

The framed standard model (FSM) is obtained from the standard model by incorporating, as field variables, the frame vectors (vielbeins) in internal symmetry space. It gives the standard Higgs boson and 3 generations of quarks and leptons as immediate consequences. It gives moreover a fermion mass matrix of the form: m = mTαα†, where α is a vector in generation space independent of the fermion species and rotating with changing scale, which has already been shown to lead, generically, to up–down mixing, neutrino oscillations and mass hierarchy. In this paper, pushing the FSM further, one first derives to 1-loop order the RGE for the rotation of α, and then applies it to fit mass and mixing data as a first test of the model. With 7 real adjustable parameters, 18 measured quantities are fitted, most (12) to within experimental error or to better than 0.5 percent, and the rest (6) not far off. (A summary of this fit can be found in Table 2 of this paper.) Two notable features, both generic to FSM, not just specific to the fit, are: (i) that a theta-angle of order unity in the instanton term in QCD would translate via rotation into a Kobayashi–Maskawa phase in the CKM matrix of about the observed magnitude (J ~ 10-5), (ii) that it would come out correctly that mu < md, despite the fact that mt ≫ mb, mc ≫ ms. Of the 18 quantities fitted, 12 are deemed independent in the usual formulation of the standard model. In fact, the fit gives a total of 17 independent parameters of the standard model, but 5 of these have not been measured by experiment.

NONCOMMUTATIVE FERMION MASS MATRIX AND GRAVITY

The first part is an introductory description of a small cross-section of the literature on algebraic methods in nonperturbative quantum gravity with a specific focus on viewing algebra as a laboratory in which to deepen understanding of the nature of geometry. This helps to set the context for the second part, in which we describe a new algebraic characterization of the Dirac operator in noncommutative geometry and then use it in a calculation on the form of the fermion mass matrix. Assimilating and building on the various ideas described in the first part, the final part consists of an outline of a speculative perspective on (noncommutative) quantum spectral gravity. This is the second of a pair of papers so far on this project.

DEVELOPING THE FRAMED STANDARD MODEL

The framed standard model (FSM) suggested earlier, which incorporates the Higgs field and three fermion generations as part of the framed gauge theory (FGT) structure, is here developed further to show that it gives both quarks and leptons hierarchical masses and mixing matrices akin to what is experimentally observed. Among its many distinguishing features which lead to the above results are (i) the vacuum is degenerate under a global su(3) symmetry which plays the role of fermion generations, (ii) the fermion mass matrix is "universal," rank-one and rotates (changes its orientation in generation space) with changing scale μ, (iii) the metric in generation space is scale-dependent too, and in general nonflat, (iv) the theta-angle term in the quantum chromodynamics (QCD) action of topological origin gets transformed into the CP-violating phase of the Cabibbo–Kobayashi–Maskawa (CKM) matrix for quarks, thus offering at the same time a solution to the strong CP problem.

A SOLUTION TO THE STRONG CP PROBLEM TRANSFORMING THE THETA ANGLE TO THE KM CP-VIOLATING PHASE

It is shown that in the scheme with a rotating fermion mass matrix (i.e. one with a scale-dependent orientation in generation space) suggested earlier for explaining fermion mixing and mass hierarchy, the theta angle term in the QCD action of topological origin can be eliminated by chiral transformations, while giving still nonzero masses to all quarks. Instead, the effects of such transformations get transmitted by the rotation to the CKM matrix as the KM phase giving, for θ of order unity, a Jarlskog invariant typically of order 10-5, as experimentally observed. Strong and weak CP violations appear then as just two facets of the same phenomenon.

CATEGORIFIED NONCOMMUTATIVE MANIFOLDS

We construct a noncommutative geometry with generalised 'tangent bundle' from Fell bundle C*-categories (E) beginning by replacing pair groupoid objects (points) with objects in E. This provides a categorification of a certain class of real spectral triples where the Dirac operator D is constructed from morphisms in a category. Applications for physics include quantization via the tangent groupoid and new constraints on D finite (the fermion mass matrix).

NEW ANGLE ON THE STRONG CP AND CHIRAL SYMMETRY PROBLEMS FROM A ROTATING MASS MATRIX

It is shown that when the mass matrix changes in orientation (i.e. rotates) in generation space for a changing energy scale, the masses of the lower generations are not given just by its eigenvalues. In particular, these masses need not be zero even when the eigenvalues are zero. In that case, the strong CP problem can be avoided by removing the unwanted θ term by a chiral transformation not in contradiction with the nonvanishing quark masses experimentally observed. Similarly, a rotating mass matrix may shed new light on the problem of chiral symmetry breaking. That the fermion mass matrix may so rotate with the scale has been suggested before as a possible explanation for up–down fermion mixing and fermion mass hierarchy, giving results in good agreement with experiment.