We discuss the theoretical bases that underpin the automation of the computations of tree-level and next-to-leading order cross sections, of their matching to parton shower simulations, and of the merging of matched samples that differ by light-parton multiplicities. We present a computer program, MadGraph5 aMC@NLO, capable of handling all these computations — parton-level fixed order, shower-matched, merged — in a unified framework whose defining features are flexibility, high level of parallelisation, and human intervention limited to input physics quantities. We demonstrate the potential of the program by presenting selected phenomenological applications relevant to the LHC and to a 1-TeV e + e − collider. While next-to-leading order results are restricted to QCD corrections to SM processes in the first public version, we show that from the user viewpoint no changes have to be expected in the case of corrections due to any given renormalisable Lagrangian, and that the implementation of these are well under way.

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The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations

SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop

Recola2: REcursive Computation of One-Loop Amplitudes 2

We discuss extensions of multi-jet matrix element and parton shower merging approaches, to also include next-to-leading order accuracy. Specifically, we generalise the so-called CKKW-L prescription and the recently developed unitarised matrix element + parton shower (UMEPS) scheme. Endowing tree-level merging methods with NLO corrections greatly enhances the perturbative accuracy of parton shower Monte Carlo programs. To generalise the CKKW-L approach, we augment the Nils-Lavesson-Leif-Lonnblad (NL3) scheme, which was previously developed for e+e−-annihilation, with a careful treatment of parton densities. This makes the application of the NL3 method to hadronic collisions possible. NL3 is further updated to use for more readily accessible next-to-leading order input calculations. We also extend the UMEPS scheme to NLO accuracy. The resulting approach, dubbed unitarised next-to-leading order + parton shower (UNLOPS) merging, does not inherit problematic unitarity-breaking features of CKKW-L, and thus allows for a theoretically more appealing definition of NLO order merging. Both schemes have been implemented in PYTHIA8. We present results for the merging of W- and Higgs-production events, where the zero- and one-jet contribution are corrected to next-to-leading order simultaneously, and higher jet multiplicities are described by tree-level matrix elements. We find that NL3 and UNLOPS yield a very similar description for W production. For Higgs production however, UNLOPS produces more stable results. The implementation of the NLO merging procedures is completely general and can be used for higher jet multiplicities and other processes, subject to the availability of programs able to correctly generate the corresponding partonic states to leading and next-to-leading order accuracy.

Merging Multi-leg NLO Matrix Elements with Parton Showers

One-loop calculations in quantum field theory: From Feynman diagrams to unitarity cuts

We present an algorithm for the numerical calculation of one-loop QCD amplitudes. The algorithm consists of subtraction terms, approximating the soft, collinear and ultraviolet divergences of one-loop amplitudes and a method to deform the integration contour for the loop integration into the complex space. The algorithm is formulated at the amplitude level and does not rely on Feynman graphs. Therefore all required ingredients can be calculated efficiently using recurrence relations. The algorithm applies to massless partons as well as to massive partons.

Numerical NLO QCD calculations

We relate an l-loop Feynman integral to a sum of phase space integrals, where the integrands are determined by the spanning trees of the original l-loop graph. Causality requires that the propagators of the trees have a modified iδ prescription, and we present a simple formula for the correct iδ prescription.

Causality and Loop-Tree Duality at Higher Loops.

We present a method to construct a suitable contour deformation in loop momentum space for multi-loop integrals. This contour deformation can be used to perform the integration for multi-loop integrals numerically. The integration can be performed directly in loop momentum space without the introduction of Feynman or Schwinger parameters. The method can be applied to finite multi-loop integrals and to divergent multi-loop integrals with suitable subtraction terms. The algorithm extends techniques from the one-loop case to the multi-loop case. Examples at two and three loops are discussed explicitly.

Direct numerical integration for multi-loop integrals

We present a method, which constructs a suitable deformation vector in loop momentum space, when the loop integration is done numerically with the help of the subtraction method. The method presented here extends previously discussed techniques from the massless case to the general case of arbitrary masses in the loop.

Direct contour deformation with arbitrary masses in the loop

I report on an implementation of an algorithm for the automated numerical calculation of spin- and colour-correlated Born matrix elements in QCD. These spin- and colour-correlated matrix elements are needed for NLO calculations in combination with the subtraction method. Both massless and massive quarks are considered. There are no restrictions on the number of external particles. As a trivial sub-case, the algorithm also applies to Born matrix elements without any correlations. These are sufficient for leading order calculations.

Stefan Weinzierl

Papers

Numerical NLO QCD calculations

Causality and Loop-Tree Duality at Higher Loops.

Direct numerical integration for multi-loop integrals

Direct contour deformation with arbitrary masses in the loop

Automated computation of spin- and colour-correlated Born matrix elements