Hadronic production of a Higgs boson in association with two jets at ...

1.1. The Standard Model of Particle Physics 3the conjugation ensures that the α dependence drops out). However, the derivativeterm is not invariant. This is because the derivative operation naturally acts onfields by deforming x by small amounts x → x + ǫ. However the points x and x + ǫtransform with different rotations under eq. (1.2), so no cancellation occurs. Wewish to define an object which transforms in the following wayD(x, y) → e iα(y) D(x, y)e −iα(x) (1.3)This then ensures that D(x, y)ψ(y) has the same transformation as ψ(x). Usingthis we can construct a covariant derivative which has the correct transformationproperties,n µ 1D µ ψ = lim ǫ→0 (ψ(x + ǫn) − D(x + ǫn, x)ψ(x)) (1.4)ǫwhere we have defined an arbitrary direction n µ in which the derivative acts. Wecan perform a Taylor expansion on D(x + ǫn, x)D(x + ǫn, x) = 1 − igǫn µ A µ (x) + O(ǫ 2 ). (1.5)The coefficient of the displacement ǫn µ is a new vector field A µ , which we use tobuild the covariant derivative,D µ ψ(x) = ∂ µ ψ(x) + igA µ ψ(x). (1.6)Inserting the Taylor series expansion into the transformation equation eq. (1.3) weobserve that the vector field A µ must transform in the following wayA µ (x) → A µ (x) − 1 g ∂ µα(x). (1.7)We can verify that D µ ψ(x) now behaves as we would wish,[ (D µ ψ(x) → ∂ µ + ig A µ − 1 )]g ∂ µα e iα(x) ψ(x)= e iα(x) (∂ µ + igA µ )ψ(x) (1.8)such that γ µ ψD µ ψ is now gauge invariant as required. We observe that the new termin the Lagrangian is none other than the interaction term in the QED Lagrangian.We now check that the kinetic term for the photon is also gauge invariant. This is