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

2.2. Quadruple cuts 37Minkowskii space (+ − −−) a bispinor (light-like four vector) has the form p aȧ =λ a˜λȧ. For real momenta λ and ˜λ are complex but are related to each other, ˜λ = ±λ.This means that when an invariant p i · p j = 0 both 〈ij〉 and [ij] equal zero (sincethey are conjugates of each other). If however, one drops the reality condition, suchthat λ and ˜λ are independent, one can still satisfy momenta conservation by havingeither [ij] = 0 and 〈ij〉 ≠ 0 or [ij] ≠ 0 and 〈ij〉 = 0.Choosing which spinor product should be set equal to zero is made simple byinspecting the structure of the three gluon amplitude. There are two non-zerohelicity configurations,A (0)3 (1 + , 2 + , 3 − ) = [12]3[23][31]and A (0)3 (1 + , 2 − , 3 − ) = 〈23〉3〈12〉〈31〉 . (2.9)For the (+ + −) configuration one should set 〈12〉 = 〈23〉 = 〈31〉 = 0 since theamplitude is independent of these variables and vice versa for (+ − −). This techniquewas shown in [120] to correctly reproduce one-loop amplitudes in N = 4 SYM(which contain only box terms to O(ǫ)).2.2.2 A quadruple cut exampleAs an example of the method we calculate the coefficient of a two-mass easy boxwhich appears in the amplitude A (1)4 (φ, 1 − , 2 + , 3 − , 4 + ) c 4;φ|1|23|4 , the products of fourtrees are as follows,c 4;φ|1|23|4 = A (0)3 (l − 1 , φ, l − 2 )A (0)3 (l + 2 , 1 − , l + 3 )A (0)4 (l − 3 , 2 + , 3 − , l + 4 )A (0)3 (l − 4 , 4 + , l + 1 )= −〈l 1 l 2 〉 2 [l 2l 3 ] 3 〈l 3 3〉 4 [4l 1 ] 3(2.10)[l 2 1][1l 3 ] 〈l 3 2〉〈23〉〈3l 4 〉〈l 4 l 3 〉 [l 4 4][l 4 l 1 ]Momentum conservation requires thatl 2 = l, (2.11)l 3 = l − p 1 , (2.12)l 4 = l − P 123 , (2.13)l 1 = l − P 1234 . (2.14)We choose to expand l in terms of a basis made out of the two massless legs,l µ = αp µ 1 + βpµ 4 + 1 2 (δ〈1|γµ |4] + ρ〈4|γ µ |1]). (2.15)