Abstract
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.
Highlights
In high-energy physics symmetries play a fundamental role in model building of both theory and phenomenology multi-scalar theories
Invariant theory provides a detailed description of the counting and characterization of the physical parameter space
We studied in detail the group structure of the matrices in the scalar potential of multiHiggs doublet models
Summary
In high-energy physics symmetries play a fundamental role in model building of both theory and phenomenology multi-scalar theories. The Hilbert series of the 2HDM was first obtained in [15] and later studied in the context of CP violation in [16] With this technique, the complete roadmap to the basis-invariant description of the 2HDM built on invariant theory was achieved in [17]. It provides a framework for a full group-theoretical perspective of the parameter space As complete as it may be, invariant theory relies heavily on the computation of a formal quantity, known as the Hilbert series. With the knowledge on how the vector space of a Lagrangian decomposes in irreducible representations of any group, we derive a technique that counts the number of parameters in a Lagrangian with both basis-invariant redundancies and global symmetries. This method does not require knowledge of invariant theory, only of the group structure of the symmetry group G
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