Double unresolved approximations to multiparton scattering amplitudes

Abstract

We present approximations to tree-level multiparton scattering amplitudes which are appropriate when two partons are unresolved. These approximations are required for the analytic isolation of infrared singularities of n + 2 parton scattering processes contributing to the next-to-next-to-leading order corrections to n jet cross sections. In each case the colour ordered matrix elements factorise and yield a function containing the singular factors multiplying the n-parton amplitudes. When the unresolved particles are colour unconnected, the approximations are simple products of the familiar eikonal and Altarelli-Parisi splitting functions used to describe single unresolved emission. However, when the unresolved particles are colour connected the factorisation is more complicated and we introduce new and general functions to describe the triple collinear and soft/collinear limits in addition to the known double soft gluon limits of Berends and Giele. As expected the triple collinear splitting functions obey an N = 1 SUSY identify. To illustrate the use of these double unresolved approximations, we have examined the singular limits of the tree-level matrix elements for e + e − → 5 partons when only three partons are resolved. When integrated over the unresolved regions of phase space, these expressions will be of use in evaluating the O(α s 3) corrections to the three-jet rate in electron-positron annihilation.