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

3.3. The rational pieces 101The direct recursive terms are obtained using the following formulaR D n = ∑ iA (0)L (z)R R(z) + R L (z)A (0)P 2iR (z) . (3.120)For our chosen shift (3.118), the allowed diagrams are shown in Fig. 3.12. Due toour choice of shifts the tree amplitudesA (0) (j + , ˆ1 − , −P −(1,j) ), A(0) (j + , ˆm − , −P +(m,j) )are both zero, (here j ∈ {2, 4}). Other terms that vanish are R 2 (φ, −+) which isrequired to be zero by angular momentum conservation, and R 3 (j + , ˆm − , ˆP ± ) sincethe corresponding splitting function has no rational pieces.To complete our calculation we require the one-loop gluon amplitude with onenegative helicity gluon. These are finite one-loop amplitudes and are entirely rational.The finite φ − + . . .+ amplitudes were computed for arbitrary numbers ofpositive helicity gluons in ref. [106]. As a concrete example, the three-gluon amplitudeis given by,R 3 (φ; 1 − , 2 + , 3 + ) = N P6〈12〉〈31〉[23]〈23〉 2 − 2A (0)3 (φ† ; 1 − , 2 + , 3 + ). (3.121)Similarly, the pure QCD −+. . .+ amplitudes are given to all orders in ref. [161,203].In the four gluon case, the result is,R 4 (1 − , 2 + , 3 + , 4 + ) = N P6〈2 4〉[2 4] 3[1 2] 〈2 3〉 〈3 4〉 [4 1](3.122)Finally, there the “homogenous” terms in the recursion which depend on the φ-MHV amplitude with one gluon fewer. The first few φ-MHV amplitudes are known,R 2 (φ; 1 − , 2 − ) = 2A (0) (φ, 1 − , 2 − ), (3.123)R 3 (φ; 1 − , 2 − , 3 + ) = 2A (0) (φ, 1 − , 2 − , 3 + ), (3.124)R 3 (φ; 1 − , 2 + , 3 − ) = 2A (0) (φ, 1 − , 2 + , 3 − ). (3.125)The direct rational contribution is generated by the recursion relation (3.120) andis given by,R 4 (φ, 1 − , 2 + , 3 − , 4 + ) = A (0) (φ, ˆ1 − , ˆP − 234) 1s 234R(− ˆP + 234, 2 + , ˆ3 − , 4 + )