We derive general tree-level recursion relations for amplitudes which include massive propagating particles. As an illustration, we apply these recursion relations to scattering amplitudes of gluons coupled to massive scalars. We provide new results for all amplitudes with a pair of scalars and n ≤ 4 gluons. These amplitudes can be used as building blocks in the computation of one-loop 6-gluon amplitudes using unitarity based methods. * On Leave from Princeton University At one loop, the MHV rules have been successfully applied to supersymmetric amplitudes [25][26][27][28][29][30][31]. New classes of amplitudes have been derived with the new methods in Refs. [5,[32][33][34][35][36][37][38][39][40][41][42]. All of these results give complete amplitudes only in supersymmetric theories which are cut-constructible in 4 dimensions [6,7]. In non-supersymmetric gauge theories, the new methods apply only to the 4D cut-constructible parts of the amplitudes.The second motivation of this paper is to assemble together the pieces necessary for the complete calculation of one-loop amplitudes in non-supersymmetric gauge theories.

We use tree-level perturbation theory to show how non-supersymmetric oneloop scattering amplitudes for a Higgs boson plus an arbitrary number of partons can be constructed, in the limit of a heavy top quark, from a generalization of the scalar graph approach of Cachazo, Svrček and Witten. The Higgs boson couples to gluons through a top quark loop which generates, for large m t , a dimension-5 operator H tr G µν G µν . This effective interaction leads to amplitudes which cannot be described by the standard MHV rules; for example, amplitudes where all of the gluons have positive helicity. We split the effective interaction into the sum of two terms, one holomorphic (selfdual) and one antiholomorphic (anti-selfdual). The holomorphic interactions give a new set of MHV vertices -identical in form to those of pure gauge theory, except for momentum conservationthat can be combined with pure gauge theory MHV vertices to produce a tower of amplitudes with more than two negative helicities. Similarly, the anti-holomorphic interactions give anti-MHV vertices that can be combined with pure gauge theory anti-MHV vertices to produce a tower of amplitudes with more than two positive helicities. A Higgs boson amplitude is the sum of one MHV-tower amplitude and one anti-MHV-tower amplitude. We present all MHV-tower amplitudes with up to four negative-helicity gluons and any number of positive-helicity gluons (NNMHV). These rules reproduce all of the available analytic formulae for Higgs + n-gluon scattering (n ≤ 5) at tree level, in some cases yielding considerably shorter expressions.

We present a method for the direct extraction of rational contributions to one-loop scattering amplitudes, missed by standard four-dimensional unitarity techniques. We use generalised unitarity in D = 4 − 2ǫ dimensions to write the loop amplitudes in terms of products of massive tree amplitudes. We find that the rational terms in 4 − 2ǫ dimensions can be determined from quadruple, triple and double cuts without the need for independent pentagon contributions using a massive integral basis. The additional massdependent integral coefficients may then be extracted from the large mass limit which can be performed analytically or numerically. We check the method by computing the rational parts of all gluon helicity amplitudes with up to six external legs. We also present a simple application to amplitudes with external massless fermions.

We compute the planar part of the two-loop five gluon amplitude with all helicities positive. To perform the calculation we develop a D-dimensional generalized unitarity procedure allowing us to reconstruct the amplitude by cutting into products of six-dimensional trees. We find a compact form for the integrand which only requires topologies with six or more propagators. We perform cross checks of the universal infra-red structure using numerical integration techniques.