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

2.6. The unitarity-bootstrap 63The second tensor structure vanishes when a and b have opposite helicities,(ε + a · ε − b − ε+ a · k b ε − b · k )a= − [aq b]〈bq a 〉k a · k b 〈aq a 〉[bq b ] + [ab]〈bq a〉〈ba〉[aq b ]〈aq a 〉[bq b ]〈ab〉[ba] = 0. (2.118)The definitions of the polarisation vectors in terms of their reference vectors q i aregiven in Appendix A. Since the tensor structure vanishes at tree-level it should alsovanish for all loops (because the loops only change the form factor g 2 ). Thereforewhen the two gluons a and b have opposite helicity only the tree-like tensor structureenters the game and we understand the factorisation for complex momenta. However,when the two momenta have the same helicity the tensor structure no longervanishes,(ε + a · ε+ b − ε+ a · k b ε + b · k )ak a · k b= [ab]〈q bq a 〉〈aq a 〉〈bq b 〉 − [ab]〈bq a〉[ba]〉〈aq b 〉〈aq a 〉〈bq b 〉〈ab〉[ba]()[ab]= −〈ab〉〈q a q b 〉 + 〈bq a 〉〈aq b 〉〈ab〉〈aq a 〉〈bq b 〉= − [ab]〈ab〉 . (2.119)This tensor structure still vanishes if we approach the on-shell limit such that [ab] →0 (or 〈ab〉 → 0 for the two negative case). In these cases the tree-level also piece alsovanishes. If this tensor structure survives in the on-shell limit then it can producedouble poles in 〈ab〉 (when multiplied by the 1/s ab ). These double poles producesubleading single poles whose complex momenta behaviour is not fully understood,and, at the moment, there is no systematic way to include them into the recursionrelations.Therefore when using the recursion relations at one-loop we have to ensure thatthere are no shifts which contain a three point vertex with two external gluons ofthe same helicity. However, as mentioned above if the tree-level amplitude for aparticular shift vanishes these diagrams do not contribute, e.g. if we shift |i〉 →|i〉 + z|k〉 and i is a negative gluon found in a tree amplitude with j and ̂P ij thenwe find,̂P 2ij = 0 =⇒ 0 = P 2ij + z〈kj〉[ji] =⇒ z = −〈ij〉〈kj〉 . (2.120)