An electroweak theory of leptons in six dimensions

We present a theory of electroweak interactions in the lepton sector extended to six dimensions. This extension involves adding two additional time-like dimensions to the standard four-dimensional theory, resulting in a six-dimensional spacetime corresponding to the Clifford algebra Cℓ(3,3). A matrix representation of Cℓ(3,3) is used to construct Lorentz invariant quantities from its elements. We demonstrate that three specific elements of Cℓ(3,3) are sufficient to describe the eight lepton states of a single generation. Using these elements, we define a representation of SU(2)×U(1), leading to an electroweak theory that incorporates the chiral nature of the interactions. Our results show that components of the gauge bosons in the extra time-like dimensions form the graded superalgebra $\mathrm{su}\left(4\right|4)$, which generates quadratic and quartic terms for the extra-dimensional components of the W bosons. These terms facilitate a vacuum expectation value and spontaneous symmetry breaking, providing a novel interpretation of the Higgs potential. We predict a tree-level Higgs mass from this potential to be half the vacuum expectation value, or $m_H=123.11$ GeV. Furthermore, the model predicts a Weinberg angle of ${\theta }_W={30}^{\circ }$. This approach contrasts with previous models that utilized internal gauge symmetries and superalgebras like $\mathrm{su}\left(2\right|1)$ or $\mathrm{su}\left(2\right|2)$, by deriving the superalgebra $\mathrm{su}\left(4\right|4)$ directly from the spacetime geometry of Cℓ(3,3).