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Abstract

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.