152 research outputs found
Scattering amplitudes over finite fields and multivariate functional reconstruction
No full textSeveral problems in computer algebra can be efficiently solved by reducing them to calculations over finite fields. In this paper, we describe an algorithm for the reconstruction of multivariate polynomials and rational functions from their evaluation over finite fields. Calculations over finite fields can in turn be efficiently performed using machine-size integers in statically-typed languages. We then discuss the application of the algorithm to several techniques related to the computation of scattering amplitudes, such as the four- and six-dimensional spinor-helicity formalism, tree-level recursion relations, and multi-loop integrand reduction via generalized unitarity. The method has good efficiency and scales well with the number of variables and the complexity of the problem. As an example combining these techniques, we present the calculation of full analytic expressions for the two-loop five-point on-shell integrands of the maximal cuts of the planar penta-box and the non-planar double-pentagon topologies in Yang-Mills theory, for a complete set of independent helicity configurations
Ninja: Automated integrand reduction via Laurent expansion for one-loop amplitudes
No full textWe present the public C++ library Ninja, which implements the Integrand Reduction via Laurent Expansion method for the computation of one-loop integrals. The algorithm is suited for applications to complex one-loop processes
FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs
No full textComplex algebraic calculations can be performed by reconstructing analytic results from numerical evaluations over finite fields. We describe FiniteFlow, a framework for defining and executing numerical algorithms over finite fields and reconstructing multivariate rational functions. The framework employs computational graphs, known as dataflow graphs, to combine basic building blocks into complex algorithms. This allows to easily implement a wide range of methods over finite fields in high-level languages and computer algebra systems, without being concerned with the low-level details of the numerical implementation. This approach sidesteps the appearance of large intermediate expressions and can be massively parallelized. We present applications to the calculation of multi-loop scattering amplitudes, including the reduction via integration-by-parts identities to master integrals or special functions, the computation of differential equations for Feynman integrals, multi-loop integrand reduction, the decomposition of amplitudes into form factors, and the derivation of integrable symbols from a known alphabet. We also release a proof-of-concept C++ implementation of this framework, with a high-level interface in Mathematica
Tensor decomposition for bosonic and fermionic scattering amplitudes
No full textIn this paper, we elaborate on a method to decompose multiloop multileg scattering amplitudes into Lorentz-invariant form factors, which exploits the simplifications that arise from considering four-dimensional external states. We propose a simple and general approach that applies to both fermionic and bosonic amplitudes and allows us to identify a minimal number of physically relevant form factors, which can be related one to one to the independent helicity amplitudes. We discuss explicitly its applicability to various four- and five-point scattering amplitudes relevant for LHC physics
Physical projectors for multi-leg helicity amplitudes
No full textWe present a method for building physical projector operators for multi-leg helicity amplitudes. For any helicity configuration of the external particles, we define a physical projector which singles out the corresponding helicity amplitude. For processes with more than four external legs, these physical projectors depend on significantly fewer tensor structures and exhibit a remarkable simplicity compared with projector operators defined with traditional approaches. As an example, we present analytic formulas for a complete set of projectors for five-gluon scattering. These have been validated by reproducing known results for five-gluon amplitudes up to one-loop
Reduction to master integrals via intersection numbers and polynomial expansions
No full textIntersection numbers are rational scalar products among functions that admit
suitable integral representations, such as Feynman integrals. Using these
scalar products, the decomposition of Feynman integrals into a basis of
linearly independent master integrals is reduced to a projection. We present a
new method for computing intersection numbers that only uses rational
operations and does not require any integral transformation or change of basis.
We achieve this by systematically employing the polynomial series expansion,
namely the expansion of functions in powers of a polynomial. We also introduce
a new prescription for choosing dual integrals, de facto removing the explicit
dependence on additional analytic regulators in the computation of intersection
numbers. We describe a proof-of-concept implementation of the algorithm over
finite fields and its application to the decomposition of Feynman integrals at
one and two loops.Comment: 36 pages, 5 figures, published versio
Tensor integrand reduction via Laurent expansion
No full textWe introduce a new method for the application of one-loop integrand reduction via the Laurent expansion algorithm, as implemented in the public C++ library Ninja. We show how the coefficients of the Laurent expansion can be computed by suitable contractions of the loop numerator tensor with cut-dependent projectors, making it possible to interface Ninja to any one-loop matrix element generator that can provide the components of this tensor. We implemented this technique in the Ninja library and interfaced it to MadLoop, which is part of the public MadGraph5_aMC@NLO framework. We performed a detailed performance study, comparing against other public reduction tools, namely CutTools, Samurai, IREGI, PJFry++ and Golem95. We find that Ninja out-performs traditional integrand reduction in both speed and numerical stability, the latter being on par with that of the tensor integral reduction tool Golem95 which is however more limited and slower than Ninja. We considered many benchmark multi-scale processes of increasing complexity, involving QCD and electro-weak corrections as well as effective non-renormalizable couplings, showing that Ninja’s performance scales well with both the rank and multiplicity of the considered process
Two-loop QCD corrections to the V → qq ̄ g helicity amplitudes with axial-vector couplings
No full textWe compute the two-loop corrections to the helicity amplitudes for the coupling of a massive vector boson to a massless quark-antiquark pair and a gluon, accounting for vector and axial-vector couplings of the vector boson and distinguishing isospin non-singlet and singlet contributions. A new four-dimensional basis for the decomposition of the amplitudes into 12 invariant tensor structures is introduced. The associated form factors are then computed up to two loops in QCD using dimensional regularization. After performing renormalization and infrared subtraction, the finite parts of the renormalized non-singlet vector and axial-vector form factors are shown agree with each other, and to reproduce the previously known two-loop amplitudes. The singlet axial-vector amplitude receives a contribution from the axial anomaly from two loops onwards. This amplitude is computed for massless and massive internal quarks. Our results provide the last missing two-loop amplitudes entering the NNLO QCD corrections of vector-boson-plus-jet production at hadron colliders
Lo scambio di informazioni tra le imprese nell'epoca di Internet: considerazioni sulle regole
No full textThe article addresses the problems created in terms of competition between companies by the so-called digital revolution and the increasing use of algorithms to determine the behaviour of companies. In particular, it examines the question of the lawfulness of the exchange of information between competitors through the use of platforms, in order to verify whether the current rules laid down in Article 101 TFEU are still suitable to regulate the new forms of collusion also between the algorithms themselves. After examining both national and European case-law and practice, the author argues that the need to introduce new rules which meet the needs of the digital economy has not yet emerged, but that at the same time a mere analogical extension of the current criteria, as transposed in the Commission’s Guidelines, to innovations produced by new technologies is not at all straightforward
Local integrands for two-loop all-plus Yang-Mills amplitudes
No full textWe express the planar five- and six-gluon two-loop Yang-Mills amplitudes with all positive helicities in compact analytic form using D-dimensional local integrands that are free of spurious singularities. The integrand is fixed from on-shell tree amplitudes in six dimensions using D-dimensional generalised unitarity cuts. The resulting expressions are shown to have manifest infrared behaviour at the integrand level. We also find simple representations of the rational terms obtained after integration in 4 − 2ε dimensions
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