On-the-fly reduction of open loops

Building on the open-loop algorithm we introduce a new method for the automated construction of one-loop amplitudes and their reduction to scalar integrals. The key idea is that the factorisation of one-loop integrands in a product of loop segments makes it possible to perform various operations on-the-fly while constructing the integrand. Reducing the integrand on-the-fly, after each segment multiplication, the construction of loop diagrams and their reduction are unified in a single numerical recursion. In this way we entirely avoid objects with high tensor rank, thereby reducing the complexity of the calculations in a drastic way. Thanks to the on-the-fly approach, which is applied also to helicity summation and for the merging of different diagrams, the speed of the original open-loop algorithm can be further augmented in a very significant way. Moreover, addressing spurious singularities of the employed reduction identities by means of simple expansions in rank-two Gram determinants, we achieve a remarkably high level of numerical stability. These features of the new algorithm, which will be made publicly available in a forthcoming release of the OpenLoops program, are particularly attractive for NLO multi-leg and NNLO real-virtual calculations.Comment: v2 as accepted by EPJ C: extended discussion of the triangle reduction and its numerical stability in section 5.4.2; speed benchmarks for 2->5 processes included in section 6.2.1; ref. adde

A new method for one-loop amplitude generation and reduction in OpenLoops

We describe a new method for the automated construction of one-loop amplitudes based on the open-loop algorithm, where various operations are performed on-the-fly while constructing the integrand. In particular, an on-the-fly reduction interleaved with the construction steps of the amplitude keeps the maximum tensor rank in the loop momentum at two throughout the algorithm, thus drastically reducing the complexity of the calculation. The full reduction to scalar integrals is unified with the amplitude construction in a single recursion within the OpenLoops framework. This approach strongly exploits the factorisation of one-loop integrands in a product of loop segments. The on-the-fly approach, which is also applied to helicity summation and the merging of different diagrams, increases the speed of the original open-loop algorithm in a very significant way. A remarkably high level of numerical stability is achieved by exploiting freedoms in reduction identities and through simple expansions in rank-two Gram determinants. These features are particularly attractive for NLO multi-leg and NNLO real-virtual calculations. The new algorithm will be made public in a forthcoming release of the OpenLoops program.Comment: Contribution to the proceedings of the 13th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology), 25-29 September, 2017, St. Gilgen, Austri

NNLO QCD×\timesEW corrections to Z production in the qqˉq\bar{q} channel

We present the first results for the ${\cal O}(\alpha\alpha_s)$ corrections to the total partonic cross section of the process $q\bar q\to Z+X$, with the complete set of contributions, that include photonic and massive weak gauge boson effects. The results are relevant for the precise determination of the hadronic $Z$ boson production cross section. Virtual and real corrections are calculated analytically using the reduction to the master integrals and their evaluation through differential equations. Real corrections are dealt with using the reverse-unitarity method. They require the evaluation of a new set of two-loop master integrals, with up to three internal massive lines. In particular, three of them are expressed in terms of elliptic integrals. We verify the absence, at this perturbative order, of initial state mass singularities proportional to a weak massive virtual correction to the quark-gluon splitting.Comment: 6 pages, 1 figur

On-the-fly reduction of open loops

We describe new developments in the OpenLoops framework based on the recently introduced on-the-fly method. The on-the-fly approach exploits the factorisation of one-loop diagrams into segments in order to perform various operations, such as helicity summation, diagram merging and the reduction of Feynman integrands in between the recursion steps for the amplitude construction. This method significantly reduces the complexity of scattering amplitude calculations for multi-particle processes, leading to a major increase in CPU efficiency and numerical stability. The unification of the reduction to scalar integrals with the amplitude construction in a single algorithm, allows to identify problematic kinematical configurations and cure numerical instabilities in single recursion steps. A simple permutation trick in combination with a one-parameter expansion for a single topology, which is now implemented to any order, eliminate rank-two Gram determinant instabilities altogether. Due to this any-order expansion, the numerical accuracy of the algorithm can be determined with a rescaling test. The on-the-fly algorithm is fully implemented for double and quadruple precision, which allows for true quadruple precision benchmarks with up to 32 correct digits as well as a powerful rescue system for unstable points. We present first speed and stability results for these new features. The on-the-fly algorithm is part of the forthcoming release of OpenLoops 2.Comment: Contribution to the Proceedings of Loops and Legs in Quantum Field Theory (LL2018), 29 April 2018 - 04 May 2018, St. Goar, German

Mixed QCD-electroweak corrections to on-shell Z production at the LHC

We present the first complete calculation of mixed QCD-electroweak corrections to the production of on-shell $Z$ bosons in hadron collisions and their decays to massless charged leptons. Our computation is fully differential with respect to final state QCD partons and resolved photons, allowing us to compute any infra-red safe observable pertinent to the $pp \to Z \to l^+ l^-$ process in the approximation that the $Z$ boson is on shell. Although mixed QCD-electroweak corrections are small, at about the per mill level, we observe that the interplay between QCD-QED and QCD-weak contributions is subtle and observable-dependent. It is therefore not possible to avoid computing one or the other if ${\cal O}(\alpha_{EW} \alpha_s)$ precision is desired.Comment: 6 pages, 2 figure

Mixed QCD-electroweak corrections to on-shell Z production at the LHC

We present the first complete calculation of mixed QCD-electroweak corrections to the production of on-shell Zbosons in hadron collisions and their decays to massless charged leptons. Our computation is fully differential with respect to final state QCD partons and resolved photons, allowing us to compute any infra-red safe observable pertinent to the pp →Z→l+l−process in the approximation that the Zboson is on shell. Although mixed QCD-electroweak corrections are small, at about the per mill level, we observe that the interplay between QCD-QED and QCD-weak contributions is subtle and observable-dependent. It is therefore not possible to avoid computing one or the other if O(αEWαs)precision is desired

Five-Parton Scattering in QCD at Two Loops

We compute all helicity amplitudes for the scattering of five partons in two-loop QCD in all the relevant flavor configurations, retaining all contributing color structures. We employ tensor projection to obtain helicity amplitudes in the 't Hooft-Veltman scheme starting from a set of primitive amplitudes. Our analytic results are expressed in terms of massless pentagon functions, and are easy to evaluate numerically. These amplitudes provide important input to investigations of collinear-factorization breaking and to studies of the multi-Regge kinematics regime.Comment: fixed typos in Eqs.(8,9) and Table 1; updated numerical results in Table 2; analytic results unchanged; extended examples in the repository at https://zenodo.org/records/1022768