On the scalar sector of 2HDM: ring of basis invariants, syzygies, and six-loop renormalization-group equations

We consider a generating set of reparametrization invariants that can be constructed from the couplings and masses entering the scalar potential of the general Two-Higgs-Doublet Model (2HDM). Being independent of higgs-basis rotations, they generate a polynomial ring of basis invariants that represent the physical content of the model. Ignoring for the moment gauge and Yukawa interactions, we derive six-loop renormalization group equations (RGE) for all the invariants entering the set. We do not compute a single Feynman diagram but rely heavily on the general RGE results for scalar theories. We use linear algebra together with techniques from Invariant Theory. The latter not only allow one to compute the number of linearly independent invariants entering beta functions at a certain loop order (via Hilbert series) but also provide a convenient tool for dealing with polynomial relations (so-called syzygies) between invariants from the generating set.