The Basis Invariant of Generalized n-Cube Symmetries Group with Odd Degrees

The basis polynomial invariants with even degrees relatively to the symmetries group were described in cited literature. Here, the polynomial invariants with odd degrees are constructed. We give an explicit construction of all the basic polynomial invariants as algebra generators of odd degrees relatively to the symmetries group. All calculations are presented in detail.

The invariant space of multi-Higgs doublet models

In a model with more than one scalar doublet, the parameter space encloses both physical and unphysical information. Invariant theory provides a detailed description of the counting and characterization of the physical parameter space. The Hilbert series for the 3HDM is computed for the first time using partition analysis, in particular Omega calculus, giving rise to the possibility of a full description of its physical parameters. A rigorous counting of the physical parameters is given for the full class of models with N scalar doublets as well as a decomposition of the Lagrangian into irreducible representations of SU(N). For the first time we derive a basis-invariant technique for counting parameters in a Lagrangian with both basis-invariant redundancies and global symmetries.

A fully basis invariant symmetry map of the 2HDM

We derive necessary and sufficient conditions for all global symmetries of the most general two Higgs doublet model (2HDM) scalar potential entirely in terms of reparametrization independent, i.e. basis invariant, objects. This culminates in what we call a “Symmetry Map” of the parameter space of the model and the fundamental insight that there are, in general, two algebraically distinct ways of how symmetries manifest themselves on basis invariant objects: either, basis invariant objects can be non-trivially related, or, basis covariant objects can vanish. These two options have different consequences on the resulting structure of the ring of basis invariants and on the number of remaining physical parameters. Alongside, we derive for the first time necessary and sufficient conditions for CP conservation in the 2HDM entirely in terms of CP-even quantities. This study lays the methodological foundation for analogous investigations of global symmetries in all other models that have unphysical freedom of reparametrization, most notably the Standard Model flavor sector.