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

2.2. Quadruple cuts 35Here the LHS of eq. (2.6) represents the application of the four on-shell constraintsδ(l 2 i ) to the LHS of eq. (2.5), shown schematically in Fig. 2.1, whilst the RHSrepresents the application of the four cuts to the basis integral. In four-dimensionsthe loop momenta has the same number of components as there are cut propagatorsand as such is frozen by the cuts. Since the integral over the cut phase space merelycontributes a Jacobian, which is identical on both sides of the equation, we canrelate the coefficient of a box (with a given topology P) to the solution of the fourproducts of treesA (0)1 (l 1 , l 2 )A (0)2 (l 2 , l 3 )A (0)3 (l 3 , l 4 )A (0)4 (l 4 , l 1 ) = c 4,{P } , (2.7)where the loop momenta are solved via the on-shell constraints.There is a subtlety, however, regarding the application of four cuts to a one-loopamplitude. There are six classes of box integral which can be classified in terms ofthe number of legs bunched at each corner. Two or more legs at a specific cornerresults in a “mass” since the squared sum of corner momenta is no longer zero. Forfour external (massless) particles it is thus possible to have a zero-mass box. As thenumber of external particles increase so can the number of masses, one can drawa one-mass box, two two-mass boxes 1 , a three-mass and a four-mass box. For allbut the the four-mass box, after the application of the four-cut, one finds a cornercontaining a three-point vertex A (0)3 (1λ 1, 2 λ 2, 3 λ 3). Since this amplitude is Lorentzinvariant it must depend only on squares of the individual momenta p 2 i, or invariantproducts between them p i · p j . For massless momenta p 2 i = 0 so the amplitudecan only depend on invariant products between momenta. However, conservation ofmomenta forces each product to be zero,(p i + p j ) 2 = p 2 k = 0 =⇒ p i · p j = 0, (2.8)which implies A (0)3 (1 λ 1, 2 λ 2, 3 λ 3) = 0. For real momenta this appears to be a majorflaw in the quadruple cut approach. The crucial observation in [120] is that ifone switches to complex momenta this obstacle can be overcome. Specifically, in1 The integral in which the massive corners are opposite is called the easy configuration. Theother configuration in which the masses are adjacent is referred to as the hard configuration