Algebraic structure of lepton and quark flavor invariants andCPviolation

Abstract

Lepton and quark flavor invariants are studied, both in the Standard Model with a dimension five Majorana neutrino mass operator, and in the seesaw model. The ring of invariants in the lepton sector is highly non-trivial, with non-linear relations among the basic invariants. The invariants are classified for the Standard Model with two and three generations, and for the seesaw model with two generations, and the Hilbert series is computed. The seesaw model with three generations proved computationally too difficult for a complete solution. We give an invariant definition of the CP -violating angle θ in the electroweak sector.

Lepton mixing parameters from discrete and CP symmetries

“…Similar invariants, called I 1 and I 2 , can be defined which depend on the Majorana phases α and β [33] (see also [34][35][36]; see [11] for invariants in terms of neutrino mass matrix elements) …”

“…(35,36), only the canonical CP transformation X = X 1 leads to a direct product S 4 × CP , because only in this case AX = XA holds for all generators A (and thus all elements) of S 4 (we have used the fact that all representation matrices are real). In all the other cases AX = XA is only satisfied for one or two of the three generators S, T and U .…”

Leptonic dynamical Yukawa couplings

“…A useful earlier analysis of invariants in view of minimising flavour potentials can be found in refs. [43,44]. There are as many independent invariants as independent physical quantities; in this respect see for instance the counting in ref.…”

Baryon non-invariant couplings in Higgs effective field theory

“…The latter is a mathematical method from invariant theory to count the number of independent structures invariant under a certain symmetry group (for recent phenomenology applications see Refs. [48][49][50][51][52][53]). …”