Restricting the ℤ2-graded tensor product of Clifford algebras   <math> <mi>C</mi> <msub> <mi>ℓ</mi> <mn>4</mn> </msub> <mover> <mo>⊗</mo> <mo>̂</mo> </mover> <mi>C</mi> <msub> <mi>ℓ</mi> <mn>6</mn> </msub> </math>  $$ C{\mathrm{\ell}}_4\hat{\otimes}C{\mathrm{\ell}}_6 $$  to the particle subspace allows a natural definition of the Higgs field Φ, the scalar part of Quillen’s superconnection, as an element of   <math> <mi>C</mi> <msubsup> <mi>ℓ</mi> <mn>4</mn> <mn>1</mn> </msubsup> </math>  $$ C{\mathrm{\ell}}_4^1 $$ . We emphasize the role of the exactly conserved weak hypercharge Y, promoted here to a superselection rule for both observables and gauge transformations. This yields a change of the definition of the particle subspace adopted in recent work with Michel Dubois-Violette [13]; here we exclude the zero eigensubspace of Y consisting of the sterile (anti)neutrinos which are allowed to mix. One thus modifies the Lie superalgebra generated by the Higgs field. Equating the normalizations of Φ in the lepton and the quark subalgebras we obtain a relation between the masses of the W boson and the Higgs that fits the experimental values within one percent accuracy.

France

Bulgaria

Ivan Todorov

Superselection of the weak hypercharge and the algebra of the Standard Model

The notion of superconnection devised by Quillen in 1985 and used in gauge-Higgs field theory in the 1990's is applied to the spin factors (finite-dimensional euclidean Jordan algebras) recently considered as representing the finite quantum geometry of one generation of fermions in the Standard Model of particle physics.

Michel Dubois-Violette

Superconnection in the spin factor approach to particle physics

We continue the study undertaken in [13] of the relevance of the exceptional Jordan algebra <math><msubsup><mrow><mi>J</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>8</mn></mrow></msubsup></math> of hermitian <math><mn>3</mn><mo>×</mo><mn>3</mn></math> octonionic matrices for the description of the internal space of the fundamental fermions of the Standard Model with 3 generations. By using the suggestion of [30] (properly justified here) that the Jordan algebra <math><msubsup><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>8</mn></mrow></msubsup></math> of hermitian <math><mn>2</mn><mo>×</mo><mn>2</mn></math> octonionic matrices is relevant for the description of the internal space of the fundamental fermions of one generation, we show that, based on the same principles and the same framework as in [13], there is a way to describe the internal space of the 3 generations which avoids the introduction of new fundamental fermions and where there is no problem with respect to the electroweak symmetry.

Exceptional quantum geometry and particle physics II

The paper gives a historical survey of the causal position space renormalization with a special attention to the role of Raymond Stora in the development of this subject. Renormalization is reduced to subtracting the pole term in analytically regularized primitively divergent Feynman amplitudes. The identification of residues with “quantum periods” and their relation to recent developments in number theory are emphasized. We demonstrate the possibility of integration over internal vertices (that requires control over the infrared behavior) in the case of the massless φ4 theory and display the dilation and the conformal anomaly.