Octonionic Clifford Algebra for the Internal Space of the Standard Model

Abstract

AbstractWe explore the \({\mathbb Z}_2\) graded product \(C{\ell }_{10} = C{\ell }_4 \, {\widehat{\otimes }} \, C{\ell }_6\) as a finite internal space algebra of the Standard Model of particle physics. The gamma matrices generating \(C{\ell }_{10}\) are expressed in terms of left multiplication by the imaginary octonion units and the Pauli matrices. The subgroup of Spin(10) that fixes an imaginary unit (and thus allows to write \({\mathbb O} = {\mathbb C} \oplus {\mathbb C}^3\) expressing the quark-lepton splitting) is the Pati-Salam group \(G_\textrm{PS} = Spin (4) \times Spin (6) / {\mathbb Z}_2 \subset Spin (10)\). If we identify the preserved imaginary unit with the \(C{\ell }_6\) pseudoscalar \(\omega _6 = \gamma _1 \cdots \gamma _6\), \(\omega _6^2 = -1\), then \(\mathcal{P} = \frac{1}{2} (1 - i\omega _6)\) will be the projector on the extended particle subspace, including the right-handed (sterile) neutrino. We express the generators of \(C{\ell }_4\) and \(C{\ell }_6\) in terms of fermionic oscillators \(a_{\alpha } , a_{\alpha }^* , \alpha = 1,2\) and \(b_j , b_j^* , j = 1,2,3\) describing flavour and colour, respectively. The internal space observables belong to the Jordan subalgebra of hermitian elements of the complexified Clifford algebra \({\mathbb C} \otimes C{\ell }_{10}\) which commute with the weak hypercharge \(\frac{1}{2} Y = \frac{1}{3} \sum _{j=1}^3 b_j^* b_j - \frac{1}{2} \sum _{\alpha = 1}^2 a_{\alpha }^* a_{\alpha }\). We only distinguish particles from antiparticles if they have different eigenvalues of Y. Thus the sterile neutrino and antineutrino (both with \(Y=0\)) are allowed to mix into Majorana neutrinos. Restricting \(C{\ell }_{10}\) to the particle subspace, which consists of leptons with \(Y < 0\) and quarks, allows a natural definition of the Higgs field \(\varPhi \), the scalar of Quillen’s superconnection, as an element of \(C{\ell }_4^1\), the odd part of the first factor in \(C{\ell }_{10}\). As an application we express the ratio \(\frac{m_H}{m_W}\) of the Higgs and the W-boson masses in terms of the cosine of the theoretical Weinberg angle.