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

TL;DR

This paper develops a basis-invariant framework for the scalar sector of the 2HDM, deriving six-loop RGEs for invariants without Feynman diagrams, using invariant theory and syzygies to understand the model's physical parameters.

Contribution

It introduces a basis-invariant polynomial ring for 2HDM scalars and computes six-loop RGEs for these invariants using invariant theory techniques.

Findings

Constructed a basis-invariant polynomial ring for 2HDM scalars.

Derived six-loop renormalization group equations for invariants.

Applied invariant theory and syzygies to analyze relations among invariants.

Abstract

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.