Master's Thesis in Theoretical Physics - Universiteit Utrecht

Implicitly we have defined two new fields φ + and φ − , and as a matter of clarity we givethese fields again(√ )1 √ ω∗φ + = 1 + |ω|2 ω φ 1 + φ 2√ )1 √ ω∗φ − = 1 − |ω|(−2 ω φ 1 + φ 2 . (4.80)We can now also invert these relations and express φ 1 and φ 2 in terms of φ + and φ − , andwe obtain√ (1 ω φ + φ −φ 1 = 2 ω ∗ − ))1 + |ω| 1 − |ω|(1 φ + φ −φ 2 = + ). (4.81)2 1 + |ω| 1 − |ω|If we now substitute these fields into the kinetic term of the Lagrangian, this becomesL der = −g [ g µν ∂ µ φ ∗ + ∂ νφ + + g µν ∂ µ φ ∗ − ∂ ]νφ −= []−g g µν ∂ µ Φ † ∂ ν Φ , (4.82)Thus we have succeeded in diagonalizing and canonically normalizing the kinetic term ofthe Lagrangian.Of course we also want to express the other parts of the Lagrangian in terms of the newfields φ + and φ − . Let us first focus on the nonminimal coupling term of the Lagrangian, L ξ .We can writewhere I have definedL ξ = −R ˜Φ † ˜Ξ ˜ΦExplicitly, the matrix terms of Ξ are= −R ˜Φ † U DD −1 DU −1 ˜ΞU DD −1 DU −1 ˜Φ= −RΦ †√ √D −1 U −1 ˜ΞU D −1 Φ= −RΦ † ΞΦ, (4.83)√√ ( )D −1 U −1 ˜ΞU D −1 ξ++ ξ +−≡ Ξ ≡. (4.84)ξ −+ ξ −−ξ ++ =ξ −− =ξ +− =ξ −+ =√ √1ω∗ ω2(1 + |ω|) (ξ 11 + ξ 22 − ξ 12ω − ξ∗ 12ω ∗ )√ √1ω∗ ω2(1 − |ω|) (ξ 11 + ξ 22 + ξ 12ω + ξ∗ 12ω ∗ )√ √1ω∗ ω2 √ 1 − |ω| (ξ 2 22 − ξ 11 − ξ 12ω + ξ∗ 12ω ∗ )√ √1ω∗ ω2 √ 1 − |ω| (ξ 2 22 − ξ 11 + ξ 12ω − ξ∗ 12). (4.85)ω∗ Now we write the complex couplings ω and ξ 12 asω = |ω|e iθ ωξ 12 = |ξ 12 |e iθ 12, (4.86)