Abstract
We introduce the computer code Recola for the recursive generation of tree-level and one-loop amplitudes in the Standard Model. Tree-level amplitudes are constructed using off-shell currents instead of Feynman diagrams as basic building blocks. One-loop amplitudes are represented as linear combinations of tensor integrals whose coefficients are calculated similarly to the tree-level amplitudes by recursive construction of loop off-shell currents. We introduce a novel algorithm for the treatment of colour, assigning a colour structure to each off-shell current which enables us to recursively construct the colour structure of the amplitude efficiently. Recola is interfaced with a tensor-integral library and provides complete one-loop Standard Model amplitudes including rational terms and counterterms. As a first application we consider Z+2jets production at the LHC and calculate with Recola the next-to-leading-order electroweak corrections to the dominant partonic channels.
Highlights
We introduce the computer code Recola for the recursive generation of treelevel and one-loop amplitudes in the Standard Model
While rescue solutions in this context have been proposed in refs. [43, 44], it is well-known that numerical instabilities can be avoided by constructing the amplitude as a linear combination of tensor integrals, which is the natural representation in the Feynman-diagrammatic approach
While the top-quark mass is properly taken into account in closed fermion loops, finite top-quark-mass effects constrained to diagrams with external bottom quarks are neglected ((b−g) → (b−g) Z and bb → ggZ are suppressed by the bottom parton distribution functions (PDFs), gg → bbZ contributes about 1% at leading order (LO))
Summary
The tree-level recursive algorithm is inspired by the Dyson-Schwinger equations [63,64,65] and follows closely the strategy of Helac [66,67,68], using off-shell currents as basic building blocks. As a consequence of the summation in (2.2), the current w(P, {n}) depends on the particle P and on the set of generating external particles {n} but not on the particular way these particles have been combined in order to get w(P, {n}) In this respect, working with off-shell currents rather than Feynman diagrams allows to avoid recomputing identical sub-graphs contributing to different diagrams, since each current is computed just once; the summation in (2.2) reduces the number of generated objects which are passed to the step of the recursion. The initialisation part of the code builds a skeleton of the amplitude: all needed offshell currents are enumerated and, for each branch, all generating off-shell currents and the generated one are identified This part is run once for all, before giving explicit values to the momenta of the external particles, i.e. before performing the phase-space integration in a Monte Carlo program. The code has to be as efficient as possible in terms of CPU time because it must run on a large grid of points in phase-space; the computation of all non-dynamical quantities in the initialisation phase allows to avoid a repetition of those operations which can be performed once independently of the particular values of the external momenta
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