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

2.5. MHV rules and BCFW recursion relations 54as well as the relevant parity conjugates. This similarity to the pure QCD amplitudesholds at the MHV level, whereA (0)n (φ, 1+ , . . ., i − , . . .,j − , . . .,n + ) =〈ij〉 4∏ n−1(2.85)α=1〈α(α + 1)〉〈n1〉.The only difference being that in the pure QCD case ∑ ni=1 pµ i = 0, here ∑ ni=1 pµ i =−p µ φ. A major difference between QCD and the φ plus gluon amplitudes are theall-minus amplitudes (which are zero in QCD). The φ all-minus amplitudes have thefollowing form,A (0)n (φ, 1 − , . . .,i − , . . ., j − , . . .,n − ) =(−1) n m 4 H∏ n−1(2.86)α=1[α(α + 1)][n1].φ plus parton amplitudes have also been derived [146] and are listed in Appendix A.MHV rules have also been derived for amplitudes coupling gluons to massive(coloured) scalars [147,148]. These are relevant since eq. (2.4) shows that the onlypart of gluon loop amplitudes which are not cut-constructible are the N scalar = 0pieces. One way of implementing D dimensional unitarity is to consider the −2ǫpieces of the loop momenta as a mass µ. It has been shown [149], that one can useunitarity cuts to construct one-loop amplitudes from the MHV rules of [147,148].2.5.2 The BCFW recursion relationsThe BCFW recursion relations [140, 141] represent a remarkably simple yet deepapproach to the calculation of tree amplitudes in gauge theories. Essentially theyshow that the entire spectrum of tree amplitudes can be calculated in a theory fromknowledge of the three point vertices alone. The proof is remarkably simple andrelies only on complex analysis and the universal factorisation of tree amplitudes.There has been a huge range of applications of the recursion relations, including (butnot limited to) QCD [150–154], QED [155,156] and more exotic theories [157–160].Here we sketch the details of the proof for a pure gluonic amplitude [141].One begins by taking an on-shell amplitude A (0)nand selecting two of the gluonsp i and p j for special treatment. We wish to shift these momenta such that overall