arXiv subjects
Publications and source records attributed to Simon Badger.
In these proceedings we report our progress in the development of the publicly available C++ library NJet for accurate calculations of high-multiplicity one-loop amplitudes. As a phenomenological application we present the first complete next-to-leading order (NLO) calculation of five jet cross section at hadron colliders.
We discuss recent progress in multi-loop integrand reduction methods. Motivated by the possibility of an automated construction of multi-loop amplitudes via generalized unitarity cuts we describe a procedure to obtain a general parameterisation of any multi-loop integrand in a renormalizable gauge theory. The method relies on computational algebraic geometry techniques such as Gröbner bases and primary decomposition of ideals. We present some results for two and three loop amplitudes obtained with the help of the Macaulay2 computer algebra system and the Mathematica package BasisDet.
We present theoretical predictions for five jet production in proton-proton collisions at next-to-leading order accuracy in QCD. Inclusive as well as differential observables are studied for collision energies of 7 and 8 TeV. In general the next-to-leading order corrections stabilize the theoretical predictions with respect to scale variations. In case of the inclusive jet cross sections, we compare with experimental data where possible and find reasonable agreement. We observe that the four-to-three and five-to-four jet ratios show better perturbative convergence than the known three-to-two ratio and are promising candidates for future alpha_s measurements. Furthermore, we present a detailed analysis of uncertainties related to parton distribution functions. The full colour virtual matrix elements used in the computation were obtained with the NJet package, a publicly available library for the evaluation of one-loop amplitudes in massless QCD.
We present a computer library for the numerical evaluation of colour-ordered n-gluon amplitudes at one-loop order in pure Yang-Mills theory. The library uses the recently developed technique of generalised unitarity. Running in double precision the library yields reliable results for up to 14 gluons with only a small fraction of events requiring a re-evaluation using extended floating point arithmetic. We believe that the library presented here provides an important contribution to future LHC phenomenology. The program may also prove useful in cross checking results obtained by other methods. In addition, the code provides a sample implementation which may serve as a starting point for further developments.
We study three and four jet production in hadronic collisions at next-to-leading order accuracy in massless QCD. We cross check results previously obtained by the BlackHat collaboration for the LHC with a centre-of-mass energy of sqrt(s)=7 TeV and present new results for the LHC operating at 8 TeV. We find large negative NLO corrections reducing the leading-order cross sections by about 40-50%. Furthermore we observe an important reduction of the scale uncertainty. In addition to the cross sections we also present results for differential distributions. The dynamical renormalization/factorization scale used in the calculation leads to a remarkably stable K-factor. The results presented here were obtained with the NJet package, a publicly available library for the evaluation of one-loop amplitudes in massless QCD.
We present a C++ library for the numerical evaluation of one-loop virtual corrections to multi-jet production in massless QCD. The pure gluon primitive amplitudes are evaluated using NGluon. A generalized unitarity reduction algorithm is used to construct arbitrary multiplicity fermion-gluon primitive amplitudes. From these basic building blocks the one-loop contribution to the squared matrix element, summed over colour and helicities, is calculated. No approximation in colour is performed. While the primitive amplitudes are given for arbitrary multiplicities we provide the squared matrix elements only for up to 7 external partons allowing the evaluation of the five jet cross section at next-to-leading order accuracy. The library has been recently successfully applied to four jet production at next-to-leading order in QCD.
We consider the maximal cut of a three-loop four point function with massless kinematics. By applying Groebner bases and primary decomposition we develop a method which extracts all ten propagator master integral coefficients for an arbitrary triple-box configuration via generalized unitarity cuts. As an example we present analytic results for the three loop triple-box contribution to gluon-gluon scattering in Yang-Mills with adjoint fermions and scalars in terms of three master integrals.
Recent advances in our understanding of tree-level QCD amplitudes in the massless limit exploiting an effective (maximal) supersymmetry have led to the complete analytic construction of tree-amplitudes with up to four external quark-anti-quark pairs. In this work we compare the numerical efficiency of evaluating these closed analytic formulae to a numerically efficient implementation of the Berends-Giele recursion. We compare calculation times for tree-amplitudes with parton numbers ranging from 4 to 25 with no, one, two and three external quark lines. We find that the exact results are generally faster in the case of MHV and NMHV amplitudes. Starting with the NNMHV amplitudes the Berends-Giele recursion becomes more efficient. In addition to the runtime we also compared the numerical accuracy. The analytic formulae are on average more accurate than the off-shell recursion relations though both are well suited for complicated phenomenological applications. In both cases we observe a reduction in the average accuracy when phase space configurations close to singular regions are evaluated. We believe that the above findings provide valuable information to select the right method for phenomenological applications.
We investigate generic properties of one-loop amplitudes in unordered gauge theories in four dimensions. For such theories the organisation of amplitudes in manifestly crossing symmetric expressions poses restrictions on their structure and results in remarkable cancellations. We show that one-loop multi-photon amplitudes in QED with at least eight external photons are given only by scalar box integral functions. This QED `no-triangle' property is true for all helicity configurations and has similarities to the `no-triangle' property found in the case of maximal N = 8 supergravity. Results are derived both via a world-line formalism as well as using on-shell unitarity methods. We show that the simple structure of the loop amplitude originates from the extremely good BCFW scaling behaviour of the QED tree-amplitude.
We present a method for the computation of hepta-cuts of two loop scattering amplitudes. Four dimensional unitarity cuts are used to factorise the integrand onto the product of six tree-level amplitudes evaluated at complex momentum values. Using Gram matrix constraints we derive a general parameterisation of the integrand which can be computed using polynomial fitting techniques. The resulting expression is further reduced to master integrals using conventional integration by parts methods. We consider both planar and non-planar topologies for 2 to 2 scattering processes and apply the method to compute hepta-cut contributions to gluon-gluon scattering in Yang-Mills theory with adjoint fermions and scalars.
We present a numerical implementation for virtual corrections to multi-jet production at Next-to-Leading order. Using the algorithm of generalised unitarity we compute primitive amplitudes from tree-level input. These basic ingredients are then used to compute full colour and helicity summed corrections.
We present the publicly available program NGluon allowing the numerical evaluation of primitive amplitudes at one-loop order in massless QCD. The program allows the computation of one-loop amplitudes for an arbitrary number of gluons. The focus of the present article is the extension to one-loop amplitudes including an arbitrary number of massless quark pairs. We discuss in detail the algorithmic differences to the pure gluonic case and present cross checks to validate our implementation. The numerical accuracy is investigated in detail.
We present compact analytic expressions for all one-loop helicity amplitudes contributing to ttbar production at hadron colliders. Using recently developed generalised unitarity methods and a traditional Feynman based approach we produce a fast and flexible implementation.
We show some new applications of on-shell methods to calculate compact helicity amplitudes for t tbar production through gluon fusion. The rational and mass renormalisation contributions are extracted from two independent Feynman diagram based approaches.
We perform an analytic calculation of the one-loop amplitude for the W-boson mediated process 0 \to d u-bar Q Q-bar l-bar l, retaining the mass for the quark Q. The momentum of each of the massive quarks is expressed as the sum of two massless momenta and the corresponding heavy quark spinor is expressed as a sum of two massless spinors. Using a special choice for the heavy quark spinors we obtain analytic expressions for the one-loop amplitudes which are amenable to fast numerical evaluation. The full next-to-leading order (NLO) calculation of hadron+hadron \to W(\to e nu) b b-bar with massive b-quarks is included in the program MCFM. A comparison is performed with previous published work.
We construct a modified on-shell BCFW recursion relation to derive compact analytic representations of tree-level amplitudes in QED. As an application, we study the amplitudes of a fermion pair coupling to an arbitrary number of photons and give compact formulae for the NMHV and NNMHV case. We demonstrate that the new recursion relation reduces the growth in complexity with additional photons to be exponential rather than factorial.
We consider one-loop amplitudes of a Higgs boson coupled to gluons in the limit of a large top quark mass. We treat the Higgs as the real part of a complex field phi that couples to the self-dual field strengths and compute the one-loop corrections to the phi-NMHV amplitude, which contains one gluon of positive helicity whilst the remaining three have negative helicity. We use four-dimensional unitarity to construct the cut-containing contributions and a hybrid of Feynman diagram and recursive based techniques to determine the rational piece. Knowledge of the phi-NMHV contribution completes the analytic calculation of the Higgs plus four gluon amplitude. For completeness we also include expressions for the remaining helicity configurations which have been calculated elsewhere. These amplitudes are relevant for Higgs plus jet production via gluon fusion in the limit where the top quark is large compared to all other scales in the problem.
We compute the one-loop amplitude for a Higgs boson, a quark-antiquark pair and a pair of gluons of negative helicity, i.e. for the next-to-maximally helicity violating (NMHV) case, A(H, qbar-, q+, g-, g-). The calculation is performed using an effective Lagrangian which is valid in the limit of very large top quark mass. As a result of this paper all amplitudes for the transition of a Higgs boson into 4 partons are now known analytically at one-loop order.