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

3.3. The rational pieces 99parton multiplicities. It is simple, however, to generalise the methods described inthe following sections to include increasing numbers of partons. When calculatingthe rational terms it is simplest to include the cut-completion terms with C n , wedefined the following rational termŝR n = R D n + O n − Inf A n , (3.108)merging the remaining rational terms with the cut-constructible piecesĈ n = C n + CR n . (3.109)In the above equations Inf A n , represents the pieces of the amplitude which do notvanish as z → ∞ (where z is the BCFW shift parameter). In our calculation we willfind that CR n contributes an infinite piece of this sort. In the following sections wewill analyse each of these rational contributions before putting the whole rationalpiece together.3.3.1 The cut-completion termsThe basis-set of logarithmic functions in which eq. (3.103) and eq. (3.105) are writtencontains unphysical singularities, which we remove by adding in rational pieces, theso-called cut completion terms. The new basis is given by the transformation,L 1 (s, t) = ˆL 1 (s, t),L 2 (s, t) = ˆL 1 12 (s, t) +2(s − t)(t + 1 ),s(L 3 (s, t) = ˆL 1 13 (s, t) +2(s − t) 2 t + 1 ). (3.110)sFrom the breakdown of our amplitude it is clear that only A φSnneeds to be completed.When considering the overlap terms in the next section it proves most convenientto write the cut-completion terms in the following form,CR n (φ, 1 − , . . .,m − , . . .,n + ) = Γ n[m∑n∑i=2 j=m+1( 1ρ j,i−1m1 (P (i,j−1)) + 1 ) m−1∑−s i,j−1 s i,jn∑i=2 j=m( 1ρ i,jm1 (P (i+1,j)) + 1 )s i+1,j s i,j