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

2.3. Forde’s Laurent expansion method 40The quadruple cut method reduces the extraction of the box coefficients in theone-loop basis expansion (eq. (2.5)) to an algebraic operation. For QCD amplitudesthe remaining cut-constructible pieces are the coefficients of the triangle and bubbleintegral functions. Inspired by the multiple cut techniques, new methods weredeveloped to isolate the coefficients of these integral functions. Two of these aredescribed below, Forde’s Laurent expansion method [122] and spinor integration[123].2.3 Forde’s Laurent expansion method2.3.1 The triple cut methodThe Laurent expansion method [122] allows each coefficient in eq. (2.5) to be isolatedand determined individually. Once the coefficients of the box terms have beendetermined from quadruple cuts, one applies a triple cut to the amplitude (as depictedin Fig. 2.2). For four-dimensional loop momenta, three of the componentsare fixed by the constraints. This means that the loop momenta can be expressedin terms of one free parameter t,l µ = a µ 0 t + 1 t aµ 1 + aµ 2 . (2.29)Here a i are orthogonal null four-vectors to ensure l 2 = 0, the a i are completely fixedby the kinematics of the cut however for now we are interested in the t-dependenceof the cut. The denominator of the product of three trees can contain propagatorsof the general form (l −P) 2 , which we associate with box terms. These propagatorsgo on-shell when the following equation is satisfied,(l − P) 2 = 0 =⇒ 2(a 0 · P)t + 2(a 1 · P)t+ 2(a 2 · P) − P 2 = 0. (2.30)The use of partial fractioning allows us to separate the pieces arising from poles int from the remaining terms,∫d 4 l2∏δ(l 2 i)A 1 A 2 A 3i=0