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Basis-independent methods for the two-Higgs-doublet model

Sacha Davidson, Howard E. Haber

TL;DR

This work develops a comprehensive, basis-independent formalism for the general two-Higgs-doublet model by exploiting $U(2)$ (and its $SU(2)$ and $U(1)_Y$ subgroups) transformations in Higgs flavor space. By constructing invariants from the Higgs potential tensors $Y$ and $Z$ together with the vev projectors $V$ and $W$, the authors express all tree-level Higgs masses and couplings in a basis-free manner and identify the physical degrees of freedom ( eleven after potential minimum conditions). They extend the formalism to basis-independent tests of discrete symmetries (notably $Z_2$) and CP-violation, deriving explicit invariant conditions and CP-violation diagnostics via invariants like $I_1$, $I_2$, and $I_3$ that reduce to simple expressions in the Higgs basis. The Yukawa sector is recast in invariant terms with $\kappa^Q$ and $\rho^Q$, clarifying when $\tan\beta$ is a physical parameter (as in type-I/II models) and how FCNCs and CP-violating couplings arise in the general type-III case. Collectively, the paper provides a robust framework to analyze 2HDMs in any basis, with clear prescriptions for connecting experimental observables to invariant quantities, and sets the stage for incorporating CP-violating potentials in future work.

Abstract

In the most general two-Higgs-doublet model (2HDM), unitary transformations between the two Higgs fields do not change the functional form of the Lagrangian. All physical observables of the model must therefore be independent of such transformations (i.e., independent of the Lagrangian basis choice for the Higgs fields). We exhibit a set of basis-independent quantities that determine all tree-level Higgs couplings and masses. Some examples of the basis-independent treatment of 2HDM discrete symmetries are presented. We also note that the ratio of the neutral Higgs field vacuum expectation values, tan(beta), is not a meaningful parameter in general, as it is basis-dependent. Implications for the more specialized 2HDMs (e.g., the Higgs sector of the MSSM and the so-called Type-I and Type-II 2HDMs) are explored.

Basis-independent methods for the two-Higgs-doublet model

TL;DR

This work develops a comprehensive, basis-independent formalism for the general two-Higgs-doublet model by exploiting (and its and subgroups) transformations in Higgs flavor space. By constructing invariants from the Higgs potential tensors and together with the vev projectors and , the authors express all tree-level Higgs masses and couplings in a basis-free manner and identify the physical degrees of freedom ( eleven after potential minimum conditions). They extend the formalism to basis-independent tests of discrete symmetries (notably ) and CP-violation, deriving explicit invariant conditions and CP-violation diagnostics via invariants like , , and that reduce to simple expressions in the Higgs basis. The Yukawa sector is recast in invariant terms with and , clarifying when is a physical parameter (as in type-I/II models) and how FCNCs and CP-violating couplings arise in the general type-III case. Collectively, the paper provides a robust framework to analyze 2HDMs in any basis, with clear prescriptions for connecting experimental observables to invariant quantities, and sets the stage for incorporating CP-violating potentials in future work.

Abstract

In the most general two-Higgs-doublet model (2HDM), unitary transformations between the two Higgs fields do not change the functional form of the Lagrangian. All physical observables of the model must therefore be independent of such transformations (i.e., independent of the Lagrangian basis choice for the Higgs fields). We exhibit a set of basis-independent quantities that determine all tree-level Higgs couplings and masses. Some examples of the basis-independent treatment of 2HDM discrete symmetries are presented. We also note that the ratio of the neutral Higgs field vacuum expectation values, tan(beta), is not a meaningful parameter in general, as it is basis-dependent. Implications for the more specialized 2HDMs (e.g., the Higgs sector of the MSSM and the so-called Type-I and Type-II 2HDMs) are explored.

Paper Structure

This paper contains 12 sections, 108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basis-independent analysis of the 2HDM
  3. Basis-Independent Description of Discrete Symmetries
  4. An exceptional region of the Higgs potential parameter space
  5. The discrete symmetry $\Phi_1\to\Phi_1$, $\Phi_2\to -\Phi_2$
  6. Explicit and spontaneous CP-violation
  7. CP-Invariant 2HDM bosonic couplings in terms of invariant parameters
  8. Yukawa couplings of the 2HDM in terms of invariant parameters
  9. Summary
  10. Basis Choices for the two-Higgs doublet model
  11. Invariants by explicit construction of eigenvectors
  12. Discovering a Discrete Symmetry

Figures (3)

  • Figure 1: Diagrammatic representation of covariant tensors. The two point vertices $V_{a{\bar{b}}}$ and $Y_{a{\bar{b}}}$ are indicated by the symbols $\hbox{\boldmath $\bf \otimes$}$ and $\hbox{\boldmath $\bf \times$}$, respectively. The four-point vertex $Z_{a{\bar{b}} c{\bar{d}}}$ is depicted by four incoming line segments (meeting at the vertex point) where the indices appear in clockwise order. Unbarred (barred) indices are represented by incoming (outgoing) directed line segments.
  • Figure 2: The one-loop bubble diagrams corresponding to (a) $Z^{(1)}_{a \bar{c}} = Z_{a \bar{b} b \bar{c}}$ and (b) $Z^{(2)}_{b \bar{c}}= Z_{a \bar{a} b \bar{c}}$. These two diagrams are distinguished, since the $Z$-vertex must be read in a clockwise fashion.
  • Figure 3: Diagrams corresponding to the potentially complex invariants of eq. (\ref{['Z6Z3Y']}).