Double Unresolved Approximations to Multiparton Scattering Amplitudes
J. M. Campbell, E. W. N. Glover
TL;DR
The paper constructs universal double-unresolved approximations to tree-level multiparton QCD amplitudes, necessary for analytic infrared subtraction at NNLO. It distinguishes colour-connected from colour-unconnected cases and derives triple-collinear and soft/collinear factorisations, including the Berends–Giele double-soft limits, with explicit expressions for e+e−→5 partons. A key consistency check is the N=1 SUSY identity for triple-collinear splitting functions, and the strong-ordered limit reduces to products of single-unresolved factors. The results lay groundwork for integrating these limits over unresolved phase space and integrating with the two-loop and one-loop contributions to compute O(αs^3) corrections to three-jet observables in electron-positron annihilation.
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 not colour connected, the approximations are simple products of the familar 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 identity. To illustrate the use of these double unresolved approximations, we have examined the singular limits of the tree level matrix elements for e+e- to 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(alpha_s^3) corrections to the three jet rate in electron positron annihilation.