A
bstract
We compute a complete set of independent leading-color two-loop five-parton amplitudes in QCD. These constitute a fundamental ingredient for the next-to-next-to-leading order QCD ...corrections to three-jet production at hadron colliders. We show how to consistently consider helicity amplitudes with external fermions in dimensional regularization, allowing the application of a numerical variant of the unitarity method. Amplitudes are computed by exploiting a decomposition of the integrand into master and surface terms that is independent of the parton type. Master integral coefficients are numerically computed in either finite-field or floating-point arithmetic and combined with known analytic master integrals. We recompute leading-color two-loop four-parton amplitudes as a check of our implementation. Results are presented for all independent four- and five-parton processes including contributions with massless closed fermion loops.
A
bstract
We present the complete set of leading-color two-loop contributions required to obtain next-to-next-to-leading-order (NNLO) QCD corrections to three-jet production at hadron colliders. We ...obtain analytic expressions for a generating set of finite remainders, valid in the physical region for three-jet production. The analytic continuation of the known Euclidean-region results is determined from a small set of numerical evaluations of the amplitudes. We obtain analytic expressions that are suitable for phenomenological applications and we present a C++ library for their efficient and stable numerical evaluation.
A
bstract
We present the analytic form of all leading-color two-loop five-parton helicity amplitudes in QCD. The results are analytically reconstructed from exact numerical evaluations over finite ...fields. Combining a judicious choice of variables with a new approach to the treatment of particle states in
D
dimensions for the numerical evaluation of amplitudes, we obtain the analytic expressions with a modest computational effort. Their systematic simplification using multivariate partial-fraction decomposition leads to a particularly compact form. Our results provide all two-loop amplitudes required for the calculation of next-to-next-to-leading order QCD corrections to the production of three jets at hadron colliders in the leading-color approximation.
We present the analytic form of the planar two-loop five-gluon scattering amplitudes in QCD for a complete set of independent helicity configurations of external gluons. These include the first ...analytic results for five-point two-loop amplitudes relevant for the computation of next-to-next-to-leading-order QCD corrections at hadron colliders. The results were obtained by reconstructing analytic expressions from numerical evaluations. The complexity of the computation is reduced by exploiting physical and analytical properties of the amplitudes, employing a minimal basis of so-called pentagon functions that have recently been classified.
We present a calculation of the planar two-loop five-gluon amplitudes. The amplitudes are obtained in a variant of the generalized unitarity approach suitable for numerical computations, which we ...extend for use with finite field arithmetics. Employing a new method for the generation of unitarity-compatible integration-by-parts identities, all helicity amplitudes are reduced to a linear combination of master integrals for the first time. The approach allows us to compute exact values for the integral coefficients at rational phase-space points. All required master integrals are known analytically, and we obtain arbitrary-precision values for the amplitudes.
A
bstract
We present the leading-color two-loop QCD corrections for the scattering of four partons and a
W
boson, including its leptonic decay. The amplitudes are assembled from the planar two-loop ...helicity amplitudes for four partons and a vector boson decaying to a lepton pair, which are also used to determine the planar two-loop amplitudes for four partons and a
Z
/
γ
∗
boson with a leptonic decay. The analytic expressions are obtained by setting up a dedicated Ansatz and constraining the free parameters from numerical samples obtained within the framework of numerical unitarity. The large linear systems that must be solved to determine the analytic expressions are constructed to be in Vandermonde form. Such systems can be very efficiently solved, bypassing the bottleneck of Gaussian elimination. Our results are expressed in a basis of one-mass pentagon functions, which opens the possibility of their efficient numerical evaluation.
We present the first numerical computation of two-loop amplitudes based on the unitarity method. As a proof of principle, we compute the four-gluon process in the leading-color approximation. We ...discuss the new method, analyze its numerical properties, and apply it to reconstruct the analytic form of the amplitudes. The numerical method is universal, and can be automated to provide multiscale two-loop computations for phenomenologically relevant signatures at hadron colliders.
We describe the unitarity approach for the numerical computation of two-loop integral coefficients of scattering amplitudes. It is well known that the leading propagator singularities of an ...amplitude’s integrand are related to products of tree amplitudes. At two loops, Feynman diagrams with doubled propagators appear naturally, which lead to subleading pole contributions. In general, it is not known how these contributions can be directly expressed in terms of a product of on-shell tree amplitudes. We present a universal algorithm to extract these subleading pole terms by releasing some of the on-shell conditions. We demonstrate the new approach by numerically computing two-loop four-gluon integral coefficients.
We present the first public version of Caravel, a C++17 framework for the computation of multi-loop scattering amplitudes in quantum field theory, based on the numerical unitarity method. Caravel is ...composed of modules for the D-dimensional decomposition of integrands of scattering amplitudes into master and surface terms, the computation of tree-level amplitudes in floating point or finite-field arithmetic, the numerical computation of one- and two-loop amplitudes in QCD and Einstein gravity, and functional reconstruction tools. We provide programs that showcase Caravel's main functionalities and allow to compute selected one- and two-loop amplitudes.
Program Title:Caravel
CPC Library link to program files:https://doi.org/10.17632/rfjrxrb3rk.1
Developer's repository link:https://gitlab.com/caravel-public/caravel.git
Licensing provisions: GPLv3
Programming language: C++
External dependencies:
• Required:Python3 1, meson 2
• Optional:Doxygen 3, Eigen 4, GiNaC 5, GMP 6, Lapack 7, MPFR 8, MPI 9, PentagonLibrary 10,11, QD 12
Nature of problem: The computation of multi-loop multi-particle scattering amplitudes in quantum field theory
Solution method: The multi-loop numerical unitarity method, functional reconstruction algorithms
Additional comments including restrictions and unusual features: Current version includes tools employed in previous calculations, with the aim of showcasing details of the algorithms employed. Computations are organized by provided data files.
1http://www.python.org/2https://mesonbuild.com/3http://www.doxygen.nl/4http://eigen.tuxfamily.org/5https://ginac.de/6https://gmplib.org/7http://www.netlib.org/lapack/8https://www.mpfr.org/9https://www.open-mpi.org/10T. Gehrmann, J. M. Henn and N. A. Lo Presti, JHEP 1810 (2018) 103, arXiv:1807.09812 hep-ph11https://gitlab.com/caravel-public/pentagon-library12QD: A double-double and quad-double package for Fortran and C++, https://www.davidhbailey.com/dhbsoftware/
We present the analytic form of the two-loop four-graviton scattering amplitudes in Einstein gravity. To remove ultraviolet divergences we include counterterms quadratic and cubic in the Riemann ...curvature tensor. The two-loop numerical unitarity approach is used to deal with the challenging momentum dependence of the interactions. We exploit the algebraic properties of the integrand of the amplitude in order to reduce it to a minimal basis of Feynman integrals. Analytic expressions are obtained from numerical evaluations of the amplitude. Finally, we show that four-graviton scattering observables depend on fewer couplings than naïvely expected.
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