Generalised Proper Time as a Unifying Basis for Models with Two Right-Handed Neutrinos

Abstract

Models with two right-handed neutrinos are able to accommodate solar and atmospheric neutrino oscillation observations as well as a mechanism for the baryon asymmetry of the universe. While economical in terms of the required new states beyond the Standard Model, given that there are three generations of the other leptons and quarks this raises the question concerning why only two right-handed neutrino states should exist. Here we develop from first principles a fundamental unification scheme based upon a direct generalisation and analysis of a simple proper time interval with a structure beyond that of local 4-dimensional spacetime and further augmenting that of models with extra spatial dimensions. This theory leads to properties of matter fields that resemble the Standard Model, with an intrinsic left-right asymmetry which is particularly marked for the neutrino sector. It will be shown how the theory can provide a foundation for the natural incorporation of two right-handed neutrinos and may in principle underlie firm predictions both in the neutrino sector and for other new physics beyond the Standard Model. While connecting with contemporary and future experiments the origins of the theory are motivated in a similar spirit as for the earliest unified field theories.

Octonions in Particle Physics through Structures of Generalised Proper Time

“…In particular the inroads into the properties of the Standard Model obtained through generalised proper time go much further and in a much more direct manner than can be achieved for models with extra spatial dimensions (as alluded to in the opening of section 2), that is by models effectively based on equation 4 as a restricted case of equations 5 and 7 with n > 4 but with p = 2 only. In fact through a similarly direct analysis the properties obtained for matter fields in the case of extra spatial dimensions bear no resemblance to the Standard Model (as discussed for [30] equation 8-9 and figure 1). Moreover the inroads into the Standard Model achieved for the present theory continue with a correspondence identified with some of the further esoteric features discussed following table 1, as we review here.…”

“…vector space: symmetry: The initial stage is to write equation 3 in the form of equation 7 and to arrange the four {v a } components in a 2 × 2 Hermitian complex matrix with v 4 ≡ h ∈ h 2 C in a standard way such that L 2 (v 4 ) SL(2,C) = det(h) = 1 as a matrix determinant, where SL(2, C) is the double cover of the Lorentz group SO + (1, 3) ( [30] equations 15 and 18). This structure can then be directly subsumed within an augmented 3 × 3 Hermitian complex matrix form L 3 (v 9 ) SL(3,C) = det(v 9 ) = 1 for the 9-component vector v 9 ∈ h 3 C with an SL(3, C) symmetry ( [30] equations [16][17][18][19][20].…”

“…The octonions enter at the next stage via the augmentation C → O from the SL(3, C) level to the 27-dimensional form L 3 (v 27 ) E 6 ≡SL(3,O) = det(v 27 ) = 1, where v 27 ∈ h 3 O is a 3×3 Hermitian octonion matrix (and hence an element of the exceptional Jordan algebra), since such a determinant can still be defined for the octonion case ( [30] equation [30][31]. In constructing the symmetry transformations upon this cubic norm through an octonion composition care is needed for the non-associativity property, as described explicitly in ([9] chapter 4, [10] section 6, as reviewed in [2] chapter 6).…”

“…The space F (h 3 O) can be decomposed as two sets of h 3 O subcomponents and two further real R subcomponents. While the quartic norm q(v 56 ) is not itself a determinant the transformations of the E 6 ⊂ E 7 subgroup embedding are straightforward to describe in acting upon one h 3 O subspace as reviewed above for equation 8 and the other h 3 O subspace as the complex conjugate representation (as reviewed for [30] equation 34), while the E 6 ⊂ E 7 action is trivial on the two real subcomponents. The full set of E 7 transformations is described in ([2] equations 9.29-9.32 and the accompanying references).…”

“…The four components of v 4 ∈ TM 4 projected out of the 56-dimensional vector v 56 ∈ F (h 3 O), as subsumed from the components of v 4 ≡ h ∈ h 2 C for the original 4-dimensional case in table 2 and as intimately associated with the symmetry breaking for the higher-dimensional forms of proper time, are associated with a non-standard Higgs in this theory as indicated in table 3 and as discussed in ( [36] after figure 4). The manner in which the Higgs structure in this theory accounts for the 'origin of mass' consistently with the mutual relation between energy-momentum and spacetime curvature as conceived in the framework of general relativity is described for ( [30] equations [22][23][24]. The spectrum of individual particle masses is attributed to Yukawa coupling factors deriving from the vacuum values of scalar invariant components (which might also have some connection with a 'dark sector') as listed in the bottom line of table 3 and as proposed in ([30] subsection 4.2).…”