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

2.2. Quadruple cuts 39We also note that this solution only required knowledge of the two-mass easy boxmomentum structure. It is, therefore, applicable to any two-mass easy topologywith arbitrary helicity structure. However, here we see a problem with our solution,a priori we do not know whether p 1 will be in a MHV or MHV configuration. If itforms part of a MHV three point vertex we see that 〈l 2 1〉 = 0, which is not what wewanted. What should be noted is that earlier we explicitly chose ρ = 0, if however,we chose δ = 0 we would have found that,l µ = αp µ 1 + 1 2 ρ〈1|γµ |4] (2.25)Clearly the solution for α is unchanged the equation to determine ρ is given by0 = 〈1|P 123|4]〈4|P 123 |1]s 14− ρ〈4|P 123 |1]such that the second solution for l is given byρ = 〈1|P 123|4]s 14, (2.26)|l] = |1] |l〉 = |P 123|4][41](2.27)One should include the contributions from both solutions to the loop momenta whencalculating a box coefficient (although in this case one solution is always killed bythe MHV or MHV three point amplitude). For our coefficient of interest we first rewritethe integrand such that it solely depends on l 2 using momentum conservationthen substitute in our calculated result.c 4;φ|1|23|4 = 〈l 1 l 2 〉 2 [l 2l 3 ] 3 〈l 3 3〉 4 [4l 1 ] 3[l 2 1][1l 3 ] 〈l 3 2〉〈23〉〈3l 4 〉〈l 4 l 3 〉 [l 4 4][l 4 l 1 ][4|l 1 |l 2 〉 2 [l 2 |l 3 |3〉 3 〈l 3 3〉[4l 1 ]=[l 2 1][1|l 3 |2〉〈23〉〈3|l 4 |l 1 ]〈l 3 |l 4 |4]= [4|P 123|l 2 〉 2 [l 2 1]〈13〉 4〈l 2 2〉〈23〉〈34〉〈1|P 23 |4]= A (0)4 (φ, 1− , 2 + , 3 − , 4 + )[4|P 23 1P 23 |4〉= A (0)4 (φ, 1 − , 2 + , 3 − , 4 + )(s 234 s 123 − s 1234 s 23 ) (2.28)Here the coefficient is given by the tree-level amplitude multiplied by a kinematicfunction associated with the two mass box [121].