Eur. Phys. J. C (2009) 61: 247-278
https://doi.org/10.1140/epjc/s10052-009-1013-3

Regular Article - Theoretical Physics

Mixing angles of quarks and leptons in quantum field theory

¹ LPTHE tour 24-25, 5 ème étage, UPMC Univ Paris 06, BP 126, 4 place Jussieu, 75252, Paris Cedex 05, France
² Unité Mixte de Recherche UMR 7589, CNRS/UPMC Univ Paris 06, Paris, France
³ SSC RF ITEP, lab. 180, Bolshaya Cheremushkinskaya Ul. 25, 117218, Moscow, Russia

^* e-mail: machet@lpthe.jussieu.fr

Received: 7 November 2008
Revised: 26 January 2009
Published online: 3 April 2009

Abstract

Arguments coming from Quantum Field Theory are supplemented with a 1-loop perturbative calculation to settle the non-unitarity of mixing matrices linking renormalized mass eigenstates to bare flavor states for non-degenerate coupled fermions. We simultaneously diagonalize the kinetic and mass terms and counterterms in the renormalized Lagrangian. SU(2) _L gauge invariance constrains the mixing matrix in charged currents of renormalized mass states, for example the Cabibbo matrix, to stay unitary. Leaving aside CP violation, we observe that the mixing angles exhibit, within experimental uncertainty, a very simple breaking pattern of SU(2) _f horizontal symmetry linked to the algebra of weak neutral currents, the origin of which presumably lies beyond the Standard Model. It concerns on the one hand the three quark mixing angles; on the other hand a neutrino-like pattern in which θ ₂₃ is maximal and tan (2θ ₁₂)=2. The Cabibbo angle fulfills the condition tan (2θ _c )=1/2 and θ ₁₂ for neutrinos satisfies accordingly the “quark–lepton complementarity condition” θ _c +θ ₁₂=π/4. θ ₁₃=±5.7⋅10⁻³ are the only values obtained for the third neutrino mixing angle that lie within present experimental bounds. Flavor symmetries, their breaking by a non-degenerate mass spectrum, and their entanglement with the gauge symmetry, are scrutinized; the special role of flavor rotations as a very mildly broken symmetry of the Standard Model is outlined.