Master's Thesis in Theoretical Physics - Universiteit Utrecht

and we can writeξ ++ =ξ −− =ξ +− =ξ −+ =where the phase θ ξ is defined as12(1 + |ω|) (ξ 11 + ξ 22 − 2|ξ 12 |cosθ ξ )12(1 − |ω|) (ξ 11 + ξ 22 + 2|ξ 12 |cosθ ξ )12 √ 1 − |ω| (ξ 2 22 − ξ 11 + 2i|ξ 12 |sinθ ξ )12 √ 1 − |ω| (ξ 2 22 − ξ 11 − 2i|ξ 12 |sinθ ξ ). (4.87)θ ξ = θ ω − θ 12 . (4.88)It is now easy to see that the new nonminimal coupling matrix Ξ is hermitean, i.e. Ξ † = Ξ.Thus we have also redefined the nonminimal term in our Lagrangian in terms of the newfields. Our result is a new nonminimal matrix Ξ that has the same properties as the original˜Ξ. Therefore in this section of the Lagrangian there is effectively no change.The final sector of the Lagrangian contains the potential term V (φ 1 ,φ 2 ) from Eq. (4.72). Wecan also rewrite this potential in terms of φ + and φ − by using Eq. (4.81). It is not hard tosee that in addition to terms φ 4 + , φ4 − and φ2 − φ2 + there will be terms φ −φ 3 + and φ +φ 3 − . Thisis a rough sketch of the modified potential, because in fact the fields are complex, althoughthe potential is real. We will not write down this potential explicitly, but instead we willwrite the Lagrangian in terms of the new fields φ + and φ − in the following way, nonminimalcouplingsL = [−g g µν ∂ µ φ ∗ + ∂ νφ + + g µν ∂ µ φ ∗ − ∂ νφ −]−Rξ ++ φ ∗ + φ + − Rξ −− φ ∗ − φ − − Rξ +− φ ∗ + φ − − Rξ −+ φ ∗ − φ + − V (φ + ,φ − ) . (4.89)If we compare this Lagrangian to the original Lagrangian in Eq. (4.70) with Ω = diag(1,1)(thus a diagonal kinetic term), we see that the Lagrangians are almost the same. By aredefinition of the fields only the potential term is changed and now contains φ − φ 3 + andφ + φ 3 − terms. Numerically we have checked that these potential terms do not qualitativelyinfluence the dynamics of inflation for typical coupling values λ i ∼ 10 −3 . Therefore, fromnow on we simply take our kinetic term to be diagonal from the start and simply work withthe fields φ 1 and φ 2 .Of course we have not taken interaction terms of the Higgs boson with the gauge bosonsor fermions into account. These have the form f φ i ψψ. If we consider Eqs. (4.81), we seethat the φ 1 fields feature a term e iθ ω. However, this constant phase can be absorbed in thefermion fields and it will not change the fermion action. The situation is different when thephase is not constant, which is the case for the phase θ between the Higgs bosons. We willcome back to this in chapter 6. For now we will focus on the Higgs sector of the Lagrangianwithout interactions. As in section 4.1 we will first derive the field equations, then try tomake approximations and see if inflation is successful. Finally we will show numericalsolutions of the field equations and actually see that inflation is a success.4.4.1 Dynamical equationsWe now derive the constraint equations for H 2 and Ḣ and the field equations as we did insection 4.1. For simplicity we take our kinetic term of the Lagrangian (4.70) to be diagonal