Progress in calculating hexagon amplitudes at one–loop

The high energy experiments of the next ten years are hadron colliders operating at the multi– TeV scale. Because of the high center–of–mass energies at the Tevatron and the LHC, the final states will look very complex. One may say that we are entering the decade of multi–particle/jet physics. The expected jet rates at the LHC are huge. A leading order analysis is not adequate to make a detailed prediction for these cross sections. To address the problems with a leading order description three points can be made: 1) Large scale dependence: An N–jet cross section behaves like αs(μ) N , means scale uncertainties are amplified with the number of jets. 2) Peripheral phase space regions: Partonic cross sections are very sensitive to next–to– leading order effects, when severe cuts are applied, which is typical for background processes. 3) Jet structure: The more information about the matrix elements is known the better the internal structure of a jet can be described. It is important to note that the ability to detect New Physics at the LHC and Tevatron crucially depends on the understanding of the corresponding Standard Model backgrounds. All that is sufficient motivation to treat the generic multi– particle/jet final states at the next–to–leading order (NLO) level. Presently our computational skills for describing N–jet production at NLO end already at N=3. The pioneering work to calculate the amplitudes for two–jet processes at NLO in hadronic collisions was accomplished by Ellis and Sexton in 1986 [ 1]. The partonic amplitudes relevant for 3 jet production were provided by Bern, Dixon and Kosower [ 2] and Kunszt, Signer, Trócsányi [ 4] in 1993/94. New technology was invented for these calculations borrowed partly from string theory. Further, supersymmetry relations were exploited and helicity methods were applied at the NLO level [ 5]. Other needed ingredients were reduction formulas for Feynman parameter integrals with nontrivial numerators and representations of scalar 5–point functions (pentagon integrals) [ 6]. For QCD the step to NLO 6–point amplitudes has not been taken yet, albeit its phenomenological relevance for collider physics.