Quantum Field Theory

4.4.4 Chiral InteractionsLet’s now look at how our interaction terms change under parity. We can look at eachof our spinor bilinears from which we built the action,P : ¯ψψ(⃗x, t) → ¯ψψ(−⃗x, t) (4.78)which is the transformation of a scalar. For the vector ¯ψγ µ ψ, we can look at thetemporal and spatial components separately,P : ¯ψγ 0 ψ(⃗x, t) → ¯ψγ 0 ψ(−⃗x, t)P : ¯ψγ i ψ(⃗x, t) → ¯ψγ 0 γ i γ 0 ψ(−⃗x, t) = − ¯ψγ i ψ(−⃗x, t) (4.79)which tells us that ¯ψγ µ ψ transforms as a vector, with the spatial part changing sign.You can also check that ¯ψS µν ψ transforms as a suitable tensor.However, now we’ve discovered the existence of γ 5 , we can form another Lorentzscalar and another Lorentz vector,¯ψγ 5 ψ and ¯ψγ 5 γ µ ψ (4.80)How do these transform under parity? We can check:P : ¯ψγ 5 ψ(⃗x, t) → ¯ψγ 0 γ 5 γ 0 ψ(−⃗x, t) = − ¯ψγ 5 ψ(−⃗x, t) (4.81){P : ¯ψγ 5 γ µ ψ(⃗x, t) → ¯ψγ − ¯ψγ 5 0 γ 5 γ µ γ 0 γ 0 ψ(−⃗x, t) µ = 0ψ(−⃗x, t) =+ ¯ψγ 5 γ i ψ(−⃗x, t) µ = iwhich means that ¯ψγ 5 ψ transforms as a pseudoscalar, while ¯ψγ 5 γ µ ψ transforms as anaxial vector. To summarize, we have the following spinor bilinears,¯ψψ : scalar¯ψγ µ ψ : vector¯ψS µν ψ : tensor¯ψγ 5 ψ : pseudoscalar¯ψγ 5 γ µ ψ : axial vector (4.82)The total number of bilinears is 1 + 4 + (4 × 3/2) + 4 + 1 = 16 which is all we couldhope for from a 4-component object.– 95 –