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

2.6. The unitarity-bootstrap 62rational pieces which arise recursively and those which are defined as completionterms. Therefore to avoid double counting we must remove the overlapping residuesR D n = − ∑poles αRes z=zαĈR n (z)z− ∑poles α̂Rn (z)Res z=zα . (2.114)zSo that the final amplitude is given byA n (0) = c Γ[Ĉ n (0) + R D n + ∑poles α]ĈR n (z)Res z=zαz(2.115)This is the unitarity-bootstrap method. One first calculates C n using unitaritymethods, then after completing the L i functions one calculates the residues of ĈRassociated with the shift parameter z. Finally from the knowledge of lower pointamplitudes one calculates the rational piece R D .2.6.3 Techniques for general helicity amplitudesIn the previous section we described the unitarity-bootstrap technique in the optimumcase (where A(z) → 0 as z → ∞ and the recursive pieces contained onlysimple poles). In this section we discuss the general approach when these conditionsfail [124,165].We begin the discussion with the more serious problem associated with the appearanceof double poles. We consider a three vertex A µ (ε a , ε b ) for which a and bare external on-shell gluons and µ is the Lorentz index for the intermediate gluonwhich is going on-shell in a particular way (either 〈ab〉 or [ab] is zero). The generaltensor structure can be written as [124,166–169],) ()A µ k a · η 13(ε a , ε b ) = g 1(s ab ,ε µ ε b · k a − ε µ b(k a + k b ) · η s ε a · k b + k µ b ε a · ε bab) (k a · η kµ+g 2(s ab ,aε a · ε b − ε )a · k b ε b · k a. (2.116)(k a + k b ) · η s ab k a · k bHere the dependence on the reference vector η describes how the form factors dependon the way in which a and b go on-shell. The first tensor structure is that whichappears at tree-level,A tree,µ3 (ε a , ε b ) = 1s ab(ε µ ε b · k a − ε µ b ε a · k b + k µ b ε a · ε b). (2.117)