Master's Thesis in Theoretical Physics - Universiteit Utrecht

4.4 The two-Higgs doublet modelIn this section we will consider again the Higgs action with nonminimal coupling, but thistime we will consider the simplest supersymmetric extension: the two-Higgs doublet model.As the name suggests, in this model there are two Higgs doublets, H 1 and H 2 . The motivationfor the use of this model is the fact that the extra Higgs doublet naturally appears insupersymmetric models. It also introduces more couplings and four extra degrees of freedom,e.g. two extra complex fields. As we will see, of the two Higgs doublets one has a realexpectation value, whereas the other has an additional phase θ in the vacuum. Preciselythis phase provides a source of CP violation, which is an essential ingredient for baryogenesis.We will come back to this in chapter 6. For some good literature on the two-Higgsdoublet model, see for example [27].Again, considering only the Higgs sector of the action and leaving the gauge and fermioninteractions aside for now, we can write the most general matter action∫S = d 4 x −g(− ∑ω i j g µν (∂ µ H i ) † (∂ ν H j ) − V (H 1 , H 2 ) −∑)ξ i j RH † i H j .i, j=1,2i, j=1,2We have allowed for an off-diagonal nonminimal term and an off-diagonal kinetic term thatmixes the derivatives of H 1 and H 2 . The nonminimal couplings ξ i j and the constants ω i jin front of the kinetic term can in principle be complex quantities. However, if we demandour action to be real, i.e S † = S, we find that the diagonal terms ω 11 ,ω 22 and ξ 11 ,ξ 22 mustbe real and the off-diagonal terms are related as ω 12 = ω ∗ 21 and ξ 12 = ξ ∗ 21. The most generalpotential we can write for the two Higgs doublet model is (see e.g. [27]),V (H 1 , H 2 ) = λ 1 (H † 1 H 1 − v 2 1 )2 + λ 2 (H † 2 H 2 − v 2 2 )2+λ 3 [(H † 1 H 1 − v 2 1 ) + (H† 2 H 2 − v 2 2 )]2 + λ 4 [(H † 1 H 1)(H † 2 H 2) − (H † 1 H 2)(H † 2 H 1)]+λ 5 [Re(H † 1 H 2) − v 1 v 2 cosθ] 2 + λ 6 [Im(H † 1 H 2) − v 1 v 2 sinθ] 2 + λ 7 . (4.67)The couplings λ i , (i = 1,..,7) are real because of hermiticity of the action. The constant λ 7only appears such that the minimum of the potential is V = 0 and has no physical meaning.We will take λ 7 = 0 for now. v 1 and v 2 are the real expectation values of the Higgs doubletsat zero temperature, and θ is the phase difference between the doublets in the vacuum.Define the two Higgs doublets asH 1 =( φ01φ + 1), H 2 =( φ02φ + 2), (4.68)where the φ +/0 are charged and neutral complex scalar fields. We can fix a gauge such that( ) ( )φ1φ2H 1 = , H02 = , (4.69)0where φ 1 and φ 2 are complex scalar fields with expectation values 〈φ 1 〉 = v 1 and 〈φ 2 〉 = v 2 e iθ .For simplicity we take both φ 1 and φ 2 to be complex, but we remember that only theirrelative phase is important. If we now also take ω 11 = ω 22 = 1 and ω 12 = ω (making thediagonal kinetic terms canonical) we find that we can write our matter Lagrangian asL = −g[g µν ∂ µ ˜Φ ∗ Ω∂ ν ˜Φ − R ˜Φ ∗ ˜Ξ ˜Φ − V (φ 1 ,φ 2 )], (4.70)where we have defined( ) (φ11 ω˜Φ = , Ω =φ 2 ω ∗ 1) ( )ξ11 ξ 12, ˜Ξ =ξ ∗ . (4.71)12ξ 22