The general Two-Higgs-Doublet Model

Authors

• Claude Duhr
  • IPPP Durham
  • claude.duhr@…
• Michel Herquet
  • NIKHEF theory group
  • mherquet@…
• Celine Degrande
  • IPPP Durham
  • celine.degrande@…

Description of the model

The two-Higgs-doublet model (2HDM) has been extensively studied for more than twenty years now, even though it has often been only considered as the scalar sector of some larger model, like the MSSM or some Little Higgs models for example. The general 2HDM considered here already displays by itself an interesting phenomenology that justifies its study like, for example, new sources of CP violation in scalar-scalar interactions, tree-level flavour changing neutral currents (FCNC's) due to non diagonal Yukawa interactions, or a light pseudoscalar state and unusual Higgs decays

The 2HDM considered here is based on two SU(2) doublets with the same hypercharge Y=+1. If one imposes only gauge invariance, the most general renormalisable scalar potential can be found in the reference quoted below. This potential has 6 real (mu_(1,2) and lambda_(1,2,3,4)) and four complex parameters (mu_3 and lambda_(5,6,7)). We assume the electromagnetic gauge symmetry is preserved, i.e. , that the vev's of two doublets are aligned in the SU(2) space in such a way that a single SU(2) gauge transformation suffices to rotate them to the neutral components.

The most general form for the Yukawa interactions contains two 3x3 complex Yukawa coupling matrices, noted Delta_i and Gamma_i, expressed in the fermion physical basis, i.e. in the basis where the fermion mass matrix are diagonal. Since the fermion mass matrix is fixed, only the Gamma_i matrices, i.e. the Yukawa couplings of the second Higgs doublet, are required. We choose as free parameters the Gamma_i matrices, while the other Yukawa couplings, the Delta_i matrices, are deduced from the matching with the observed fermion masses. Conventionally, the indices of the elements of these Yukawa matrices refer to the generations of the SU(2) doublet and singlet, respectively

The 2HDM Lagrangian implemented in FeynRules is based on the Standard Model default implementation, where the scalar potential and Yukawa interactions have been modified as explained above. An important feature of this model is the freedom to redefine the two scalar fields using arbitrary "horizontal" U(2) transformations acting on the two doublets simultaneously since this transformation leaves the gauge-covariant kinetic energy terms invariant. Since a given set of Lagrangian parameter values is only meaningful for a given basis, let us take advantage of this invariance property to select the Higgs basis where only one of the two Higgs fields acquires a non-zero vev, namely Phi1. Note that the Higgs basis is not univocally defined since a phase reparametrization of Phi2 leaves the Higgs basis condition invariant.

There are two independent minimization conditions for general 2HDM potential, one relating m_1 to lambda_1 and one relating m_3 to lambda_6. This reduces the number of free parameters in the most general 2HDM to ten (seven real parameters, three complex ones and three minimization conditions). Besides the usual three massless would-be Goldstone bosons, the physical spectrum also contains a pair of charged Higgs.

The symmetric matrix squared mass matrix for the three neutral Higgs field is diagonalized by an orthogonal matrix T which describe the relation between the physical scalar fields and the doublet neutral components. The three mixing angles defining this orthogonal matrix are externals parameters as the masses of the scalars and lambda_(2,3,7). The other parameters of the potential are internal parameters.

References

• " CP violation ", G. Branco, L. Lavoura and J. P. Silva, Clarandon Press, Oxford, 1999. Chapter 22.
• "Automatic evaluation of UV and R2 terms for beyond the Standard Model Lagrangians: a proof-of-principle", Celine Degrande, Comput.Phys.Commun. 197 (2015) 239-262.

Model files

• 2HDM.tar.gz: This archive contains all the model files. Should be expanded in the FR model directory.
• Masslessbuttb.rst: Restriction file to set all the masses of the fermions but the top and bottom quarks to zero.
• FlavorSym.rst: Restriction file to force flavor conservation.
• CPconserving.rst: Restriction file to conserve CP. It assumes that FlavorSym.rst is loaded first.
• 2HDMtbsbma.fr: add on file that introduce tan beta and sin beta - alpha for the type II restriction.
• typeIItbeta.rst: restriction for the type II. It assumes that FlavorSym.rst is loaded first.

Examples

• THDM.nb : This is an example Mathematica® notebook that loads the model and calculates Feynman rules. It also show how the NLO counterterms can be computed.
• 2HDM.nlo : The file containing the QCD R2 and UV conterterms for the 2HDM.
• 2HDMfull.nlo : The file containing the full R2 and UV conterterms for the 2HDM.
• 2HDM_NLO : UFO model at LO.
• 2HDM_NLO : UFO model with QCD corrections.
• 2HDMtII_NLO : UFO model with QCD corrections for the type II 2HDM, both the top and bottom are massive and their yukawa are renormalized like their masses, i.e. on-shell.

For the older version of this model see Old 2HDM