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

2.3. Forde’s Laurent expansion method 442.3.2 A triple cut exampleAs an example of the triple cut method we describe the calculation of C 3;φ|12|34 ,the coefficient of a three-mass triangle which appears in the φ-NMV amplitudeA (1)4 (φ, 1 + , 2 − , 3 − , 4 − ). The product of the three amplitudes has the following form,C 3;φ|12|34 = A (0)2 (φ, l − 0 , l− 2 )A(0) 4 (l + 2 , 4− , 3 − , l + 1 )A(0) 4 (l − 1 , 2− , 1 + , l + 0 ) (2.42)= −〈l 0 l 2 〉 2 〈34〉 3 〈l 1 2〉 3〈l 2 4〉〈3l 1 〉〈l 1 l 2 〉 〈12〉〈1l 0 〉〈l 0 l 1 〉(2.43)We now insert the definitions for l i in terms of K ♭µ1t dependent function,and K♭µ 2 , generating the following∆〈34〉 3 (t〈K ♭C 3;φ|12|34 = −12〉 + 〈K22〉α ♭ 11 ) 3〈12〉(t〈K1 ♭1〉 + 〈K♭ 2 1〉α 01)(t〈K1 ♭3〉 + 〈K♭ 2 3〉α 11)(t〈K1 ♭4〉 + 〈K♭ 2 4〉α 21)(2.44)where∆ =(α 01 − α 21 ) 2(α 11 − α 21 )(α 01 − α 11 )(2.45)and the definitions for α ij can be found in [122]. The triangle coefficient is foundby taking the t 0 coefficient in a series expansion of eq. (2.44) around t = ∞.∑C 3;φ|12|34 (φ, 1 + , 2 − , 3 − , 4 − ) =γ=γ ± (p φ ,p 1 +p 2 )m 4 φ−〈K♭ 1 2〉3 〈34〉 32γ(γ + m 2 φ )〈K♭ 11〉〈K13〉〈K ♭ 14〉〈12〉 , ♭(2.46)2.3.3 The double cut methodThe Laurent expansion method also includes double cuts [122]. Two cut propagatorsimplies that two parameters are needed to encapsulate the remaining degrees offreedom of the loop momenta,l µ = y2t aµ 0 + y t aµ 1 + ya µ 2 + ta µ 3 + a µ 4. (2.47)The general approach is the same as the triple cut. One wishes to find a parameterisationof the loop momenta which cleanly separates triangle box and bubblecontributions such that we can extract the bubble contribution. Although more