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

2.4. Spinor integration 46Here we have used l α, ˙α = tλ α˜λ ˙α . The integration over t is actually frozen by thesecond delta function,δ((l − P) 2 ) = δ(P 2 − t〈λ|P |λ]) =⇒ t = P 2〈λ|P |λ] . (2.50)A generic double cut integrand has the following form,∫P 2c 2;P = 〈λ, dλ〉[˜λ, d˜λ]〈λ|P |λ] 2g(λ, ˜λ), (2.51)where g(λ, ˜λ) arises from the product of the tree amplitudes. The main crux of theidea behind spinor integration is that in the above equation one can systematicallyperform one of the integrations (either in λ or ˜λ), whilst the remaining contourintegration can be done via the Residue Theorem. Two different methods havebeen proposed in order to perform this first integration. In the following sectionwe will discuss performing the integration by the application of Stokes’ Theorem[132]. Firstly we describe the method proposed in the original paper [123] via theholomorphic anomaly.We begin by considering the integration of a simpler function g(λ) which dependson λ only, i.e. it has the general form,∏ ki=1g(λ) =〈λ, A i〉∏ kj=1 〈λ, B j〉 . (2.52)The following identity follows from the Schouten identity,[λ dλ]〈λ|P |˜λ] = ∂ ( )[˜λη]2 −d˜λċ , (2.53)∂˜λċ 〈λ|P |˜λ]〈λ|P |η]and holds for all values of λ, apart from those where the denominator vanishes alongthe integration contour (where ˜λ = λ) at this point,−d˜λċ ∂∂˜λċ1= 2πδ(〈λ, χ〉) (2.54)〈λχ〉the definition of δ is such that it freezes the integration in λ,∫〈λdλ〉δ(〈λ, χ〉)B(λ) = −iB(χ). (2.55)With the knowledge of the holomorphic anomaly (2.54) in hand, we can re-writeeq. (2.53) for use with our function g(λ),( )[λ dλ] ∂ [˜λη]g(λ)= −d˜λċ〈λ|P |˜λ] 2g(λ) ∂˜λċ 〈λ|P |λ]〈λ|P |η]