Two-loop amplitudes with nested sums: Fermionic contributions toe+e−→qq¯

Abstract

We present the calculation of the n f -contributions to the two-loop amplitude for e + e − → qqg and give results for the full one-loop amplitude to order ε 2 in the dimensional regularization parameter. Our results agree with those recently obtained by Garland et al.. The calculation makes extensive use of an efficient method based on nested sums to calculate two-loop integrals with arbitrary powers of the propagators. The use of nested sums leads in a natural way to multiple polylogarithms with simple arguments, which allow a straightforward analytic continuation.

Jet algorithms in electron–positron annihilation: perturbative higher order predictions

“…In addition, the two-jet rate can be deducted at N 3 LO from the knowledge of the total hadronic cross section at order α 3 s and the numbers above. The numerical Monte Carlo program relies heavily on research carried out in the past years related to differential NNLO calculations: Integration techniques for two-loop amplitudes [18][19][20][21][22][23][24][25], the calculation of the relevant tree-, one-and two-loop-amplitudes [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40], routines for the numerical evaluation of polylogarithms [41][42][43], methods to handle infrared singularities and experience from the NNLO calculations of e + e − → 2 jets and other processes [54,[68][69][70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85]…”

Next-to-Next-to-Leading Order Corrections to Three-Jet Observables in Electron-Positron Annihilation

“…These singlet contributions are expected to be numerically small [16,17,18] and neglected in the present calculation. The computation of the NNLO coefficient C O requires the knowledge of the amplitudes for the three-parton final state e + e − →qqg up to two-loops [18,19], the amplitudes of the four-parton final states e + e − →qqgg and e + e − →qqqq up to one-loop [20,21,22,23] and the fiveparton final states e + e − →qqggg and e + e − →qqqqg at tree level [24,25]. Taken separately, the three-, four-and five-parton contributions are all individually infrared divergent.…”

Numerical evaluation of multiple polylogarithms

“…Whereas we expect harmonic polylogarithms to be sufficient to express the results for 2 → 2 scattering processes in massless quantum field theories, amplitudes with more external particles and/or massive particles involve additional scales, which naturally leads to multiple polylogarithms. Since many recent results [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] of higher order calculations are expressed in terms of multiple polylogarithms, a numerical evaluation routine for multiple polylogarithms is of immediate use for perturbative calculations in particle physics. Algorithms for the numerical evaluation of multiple polylogarithms are the subject of this article.…”