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

3.2. The cut-constructible parts 89As expected we see that G factors onto the tree-level amplitude multiplying ahelicity blind function. The denominators in the above equations have exactlythe form we expect from our four-cut calculations. Spinor products of the form〈lj〉 ∝ (l − p j ) 2 /[lj] are linked to Feynman propagators and can be associated withbox diagrams. Indeed the residues of these propagators correspond to setting afurther propagator on-shell and as such correspond to a four-cut. At first glancewe observe three spinor products associated with inserting additional propagators(l − p j ) 2 , (l − p j+1 ) 2 and (l − p i+1 ) 2 . Of these the first two correspond to two-masseasy boxes and the third corresponds to a two-mass hard topology. We observed inthe previous section that there are no such contributions to the φ-MHV amplitude,implying that somehow this residue must not contribute to a box-coefficient. Uponcloser inspection we see that there is indeed no residue associated with this termsince the non-vanishing three-point vertex in the triangle requires that |l〉 ∝ |i〉 andas such when the solution for the loop-momenta is inserted there is a cancellationbetween 〈i(i + 1)〉 in the denominator and numerator.To determine F and S one merely has to insert the parameterisation for the loopmomentum in terms of eq. (3.62) and take the t 0 coefficient in a series expansionaround t = ∞. We find,F 2,1m (1, i, j) = A (0)n(〈j(j − 1)〉 〈im〉〈i1〉 2 〈m|P j;i |i]+ 〈im〉2 〈i1〉〈1|P j;i |i]〈1m〉 2 〈ij〉〈i(j − 1)〉 〈ij〉〈i(j − 1)〉− 〈im〉2 〈i1〉 2 〈j|P j;i |i]〈ij〉 2 〈i(j − 1)〉)− 〈im〉2 〈i1〉 2 〈(j − 1)|P j;i |i]. (3.70)〈ij〉〈i(j − 1)〉 2After using the Schouten identity to simplify the above formula we findF 2,1m (1, i, j) = A (0)n (−bij 1m + b i(j−1)1m )〈i|P j;i |i], (3.71)where b ij1m is defined as in eq. (3.46). The calculation for S is identical to that ofF (although here the intermediate formulae are more complicated so we quote onlythe final result)S 2,1m (1, i, j) = A (0)n ((bij 1m )2 − (b i(j−1)1m )2 )〈i|P j;i |i]. (3.72)With these solutions in hand we are now able to calculate the coefficient of anyone- or two-mass triangle appearing in the φ-MHV amplitude. All that remains is