Master's Thesis in Theoretical Physics - Universiteit Utrecht

In this representation the projection operators in 4 dimensions are( ) ( )1 00 0P L = , P0 0L = . (6.9)0 1Note that in principle ψ L and ψ R are 4-spinors, but because of the γ 5 matrix it is easy tosee that two components of these 4-spinors are zero. In the chiral representation we canthus write the 4-spinor ψ as two 2-spinors,( )ψLψ = . (6.10)ψ RIt is also easy to see that ( γ 5) 2 = 1 which allows us to derive the following useful relations(P L ) 2 = P L(P R ) 2 = P RP R P L = P L P R = 0. (6.11)An important property of the matrix γ 5 is that it anticommutes with all the other Diracmatrices, i.e.{γ 5 ,γ µ} = 0. (6.12)This allows us to derive the following relation( 1 − γψP L = iψ † γ 0 5)2( 1 + γ= iψ † 5)2= i ( P R ψ ) † γ0γ 0= P R ψ, (6.13)and the same relation when P L and P R are interchanged. Using this relation we can expressthe fermion action in terms of left- and righthanded fermions. For a kinetic term such asψγ α ∂ α ψ this givesψγ α ∂ α ψ = ψ(P L + P R )γ α ∂ α (P L + P R )ψ= ψ(P L )γ α ∂ α (P R )ψ + ψ(P R )γ α ∂ α (P L )ψ= P R ψγ α ∂ α (P R ψ) + P L ψγ α ∂ α (P L ψ)= ψ R γ α ∂ α ψ R + ψ L γ α ∂ α ψ L . (6.14)The terms with P L γ α P L vanish because of the anticommutation relation in Eq. (6.12) andthe fact that P L P R = 0. Thus we see that for the kinetic terms there is no mixing of theright- and lefthanded fermions.Now let us have a look at the Yukawa sector. In a very simple theory of one real scalar fieldthat couples to the fermions we have a term −f φ ¯ψψ, where f is a coupling constant. Wecan therefore identify the time dependent mass of the fermions as m = f φ. Let us expressthis mass term again in terms of the right- and lefthanded fields, which gives−f φψψ = −f φψ(P L + P R )(P L + P R )ψ= −f φ ( ψ(P L )(P L )ψ + ψ(P R )(P R )ψ )()= −f φ P R ψ(P L ψ) + P L ψ(P R ψ)= −f φ ( ψ R ψ L + ψ L ψ R). (6.15)