PERTURBATIVE AND NON-PERTURBATIVE QCD JET CROSS SECTIONS

Two loop matrix elements

Babis Anastasiou, Lee Garland, Nigel Glover, Thanos Koukoutsakis, Carlo Oleari, Maria Tejeda Yeomans

In the past twelve months there has been enormous progress in the field of two loop integrals for massless 2 -> 2 scattering. These are the loop integrals necessary for next-to-next-to-leading order corrections for gg -> gg, qg -> qg and qQ -> qQ as well as Bhabha scattering.

First, some of the basic two loop scalar integrals for massless two-to-two scattering have been evaluated. Scalar integrals are those with no loop momenta in the numerator and usually one power of each propagator in the denominator. Smirnov has provided analytic expressions for the two loop planar box, and Tausk has done likewise for the non-planar box graph. We have studied [1], [2] other related (but simpler) scalar integrals. In [2] we also discuss how tensor integrals - those with loop momenta in the numerator are related to scalar integrals with additional powers of the propagators in higher dimensions.

Second, the master integrals connected with the tensor integrals have been determined. The question is how many master integrals do you need to know in order to specify the tensor integrals. For both the planar and cross box graphs, there are two. Smirnov and Veretin have found the second master integral for the planar two loop box and, together with our German collaborators, we have evaluated [3] the extra master integral needed for the cross box tensor integrals.

[1] C. Anastasiou, E.W.N. Glover and C. Oleari, Application of the negative dimension approach to massless scalar box integrals, Nucl. Phys. B565 (2000) 445, hep-ph/9907523.

[2] C. Anastasiou, E.W.N. Glover and C. Oleari, The two-loop scalar and tensor pentabox graph with light-like legs, to be published in Nucl. Phys., hep-ph/9912251.

[3] C. Anastasiou, T. Gehrmann, C. Oleari, E. Remiddi and J.B. Tausk The tensor reduction and master integrals of the two-loop massless crossed box with lightlike legs, to be published in Nucl. Phys. hep-ph/0003261.