An algorithm for the high-energy expansion of multi-loop diagrams to next-to-leading logarithmic accuracy

Abstract

Abstract:We present an algorithm to compute arbitrary multi-loop massive Feynman diagrams in the region where the typical energy scale √ s is much larger than the typical mass scale M , i.e. s ≫ M 2 , while various different energy and mass parameters may be present. In this region we perform an asymptotic expansion and, using sector decomposition, we extract the leading contributions resulting from ultraviolet and mass singularities, which consist of large logarithms ln(s/M 2 ) and 1/ε poles in D = 4 − 2ε dimensions. To next-toleading accuracy, at L loops all terms of the form α L ε −k ln j (s/M 2 ) with j + k = 2L and j + k = 2L − 1 are taken into account. This algorithm permits, in particular, to compute higher-order next-to-leading logarithmic electroweak corrections for processes involving various kinematical invariants of the order of hundreds of GeV and masses M W ∼ M Z ∼ M H ∼ m t of the order of the electroweak scale, in the approximation where the masses of the light fermions are neglected.

New results for loop integrals

“…2 In a loop-by-loop approach, after the first momentum integration one gets here U = 1 and a first F-function (1.8), which depends yet on one internal momentum k 1 :…”

“…For tensor Feynman integrals the expressions are a little more involved, but they have the same structure: sums of rationals in the x i combined with non-integer powers of U(x) and F(x) [2,3,4].…”

Two-loop electroweak next-to-leading logarithms for processes involving heavy quarks

“…This method is based on collinear Ward identities which permit to factorize the soft-collinear contributions from the n-fermion tree-level amplitude and isolate them in process-independent two-loop integrals. The latter are evaluated to NLL accuracy using an automatized algorithm based on the sector-decomposition technique [33] and, alternatively, the method of expansion by regions combined with MellinBarnes representations (see ref. [31] and references therein).…”

“…Global gauge invariance implies the relationÎ 2.32) between combinations of gauge and Yukawa couplings as well as the chargeconservation identity 33) which is fulfilled up to mass-suppressed terms in the high-energy limit. In (2.32), I V Φ j Φ i denotes the SU(2)× U(1) generators for the Higgs doublet, which enter the gauge couplings of the Higgs boson.…”

“…The integrals have been evaluated separately for all physical combinations of gauge-boson and fermion masses on internal and external lines. All loop integrals have been solved and cross-checked using two independent methods: an automatized algorithm based on the sector-decomposition technique [33] and the method of expansion by regions combined with Mellin-Barnes representations (see ref. [31] and references therein).…”