Abstract
We analyze the running at one-loop of the gauge couplings in the spectral Pati-Salam model that was derived in the framework of noncommutative geometry. There are a few different scenario's for the scalar particle content which are determined by the precise form of the Dirac operator for the finite noncommutative space. We consider these different scenarios and establish for all of them unification of the Pati-Salam gauge couplings. The boundary conditions are set by the usual RG flow for the Standard Model couplings at an intermediate mass scale at which the Pati-Salam symmetry is broken.
Highlights
Where F4, F2, F0 are coefficients related to the function f and ak are Seeley deWitt coefficients, expressed in terms of the curvature of M and the gauge and scalar fields
There is a hierarchy of three energy scales: SM, an intermediate mass scale mR where symmetry breaking occurs and which is related to the neutrino Majorana masses (1011 − 1013GeV), and the GUT scale Λ
We have analyzed the running of the Pati-Salam gauge couplings for the spectral model, considering different scalar field contents corresponding to the assumptions made on the finite Dirac operator
Summary
And as explained in the introduction, the above algebra comes out as a solution of the two-sided Heisenberg quantization relation between the Dirac operator D and the two maps from the four spin-manifold and the two four spheres S4×S4 [10, 11] This removes the arbitrary symplectic constraint and replaces it with a relation that quantizes the four-volume in terms of two quanta of geometry and have far reaching consequences on the structure of space-time. All primed indices A correspond to the Hilbert space of conjugate spinors It is acted on by both the left algebra M2 (H) and the right algebra M4 (C). We distinguish three cases: 1) Left-right symmetric Pati-Salam model with fundamental Higgs fields ΣbaJI , HaIbJ. At the mass scale mR the Pati-Salam symmetry is broken to that of the Standard Model, and we take it to be the same scale that is present in the see-saw mechanism It should be of the order 1011 − 1013GeV. We discuss the three models, in order of complexity
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