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

2.1. Unitarity 32of two tree level amplitudes to the discontinuity of a one-loop integral function. Thediscontinuity of a one-loop integral associated with a kinematic scale s = P 2 can bedetermined by replacing (or cutting) the following propagators,1l 2 + iǫ → −2πiδ(l2 ),1(l + P) 2 + iǫ → −2πiδ((l + P)2 ). (2.1)In the mid-90’s Bern, Dixon, Dunbar and Kosower used these relations to calculatea series of one-loop amplitudes [110,111]. The method involves taking a cut in acertain kinematic scale and then using the simplifying kinematics of the cut (e.g. afour dimensional on-shell loop momenta) to reduce the complexity of the coefficientsof the cut loop functions. The cut propagators are then reinstated so that the integrandreturns to being a one-loop integral. This approach successfully determinesthe coefficients of all basis integrals which contain a cut in the kinematic scale. Integralswhich do not contain a cut propagator, but can arise in the reduction process,must be dropped and recovered in the appropriate cuts.A major simplification occurs when four-dimensional tree amplitudes are usedin the cuts. This is because one can use the spinor helicity formalism (which isdescribed in Appendix A) to produce compact helicity amplitudes. However, theprice of using four-dimensional cuts is the inability to determine pieces of the amplitudewhich do not contain discontinuities in the kinematic invariants. These pieces,lacking such discontinuities, are referred to as the rational pieces [110,111]. Theirorigin, and elusiveness in four-dimensions can be thought of as follows, a generalterm in a one-loop amplitude has the following structure,(C i · I i = (c 0 + c 1 ǫ + c 2 ǫ 2 I−2+ . . .) ·ǫ + I )−1+ I 2 0 · log({s, t, . . . }) + . . . . (2.2)ǫHere C i · I i represents a basis integral I i with a coefficient C i . In general I i cancontain poles of up to second order in the dimensionally regulating parameter ǫ.The term log({s, t, . . . }) represents a generic piece of the integral which containslogarithms (these could also be di-logarithms or ln 2 ). It is these pieces which containa discontinuity in a kinematic invariant and hence enter the optical theorem.However when one uses four-dimensional trees, one loses all sensitivity to the higherorder pieces in ǫ which enter the coefficient multiplying the discontinuity. Hence,