We present a program for the numerical evaluation of form factors entering the calculation of one-loop amplitudes with up to six external legs. The program is written in Fortran95 and performs the reduction to a certain set of basis integrals numerically, using a formalism where inverse Gram determinants can be avoided. It can be used to calculate one-loop amplitudes with massless internal particles in a fast and numerically stable way.
We present a first numerical implementation of the Loop-Tree Duality (LTD) method for the direct numerical computation of multi-leg one-loop Feynman integrals. We discuss in detail the singular structure of the dual integrands and define a suitable contour deformation in the loop three-momentum space to carry out the numerical integration. Then, we apply the LTD method to the computation of ultraviolet and infrared finite integrals, and present explicit results for scalar integrals with up to five external legs (pentagons) and tensor integrals with up to six legs (hexagons). The LTD method fea…
We discuss briefly the first numerical implementation of the Loop-Tree Duality (LTD) method. We apply the LTD method in order to calculate ultraviolet and infrared finite multi-leg one-loop Feynman integrals. We attack scalar and tensor integrals with up to six legs (hexagons). The LTD method shows an excellent performance independently of the number of external legs.
We complete the generalisation of the BCFW recursion relation to the off-shell case, allowing for the computation of tree level scattering amplitudes for full High Energy Factorisation (HEF), i.e. with both incoming partons having a non-vanishing transverse momentum. We provide explicit results for color-ordered amplitudes with two off-shell legs in massless QCD up to 4 point, continuing the program begun in two previous papers. For the 4-fermion amplitudes, which are not BCFW-recursible, we perform a diagrammatic computation, so as to offer a complete set of expressions. We explicitly show an…
We present analytical results for one-loop five-point master integrals with up to three off-shell legs. The method of canonical differential equations along with the Simplified Differential Equations approach is employed. All necessary boundary terms are given in closed form, resulting to solutions in terms of Goncharov Polylogarithms of arbitrary weight. Explicit results up to weight six will be presented.
Five legs @ three loops: N=4 sYM amplitude near mass-shell
Arizona State University（亚利桑那州立大学）；Dukhov Automatics Research Institute（杜霍夫自动技术研究研究所）；Budker Institute of Nuclear Physics（布德克核物理研究所）；Joint Institute for Nuclear Research（联合核研究所）；Skobeltsyn Institute of Nuclear Physics, Moscow State University（莫斯科大学斯科别尔钦核物理研究所）；Moscow Center for Fundamental and Applied Mathematics（莫斯科基础与应用数学中心）
We present the application of a novel reduction technique for one-loop scattering amplitudes based on the combination of the integrand reduction and Laurent expansion. We describe the general features of its implementation in the computer code NINJA, and its interface to GoSam. We apply the new reduction to a series of selected processes involving massive particles, from six to eight legs.
We present the program package GoSam which is designed for the automated calculation of one-loop amplitudes for multi-particle processes in renormalisable quantum field theories. The amplitudes, which are generated in terms of Feynman diagrams, can be reduced using either D-dimensional integrand-level decomposition or tensor reduction. GoSam can be used to calculate one-loop QCD and/or electroweak corrections to Standard Model processes and offers the flexibility to link model files for theories Beyond the Standard Model. A standard interface to programs calculating real radiation is also impl…
A technique of large-charge expansion provides a novel opportunity for calculation of critical dimensions of operators $ϕ^Q$ with fixed charge $Q$. In the small-coupling regime the polynomial structure of the anomalous dimensions can be fixed from a number of direct perturbative calculations for a fixed $Q$. At the six-loop level one needs to include new diagrams that correspond to operators with five or more legs. The latter never appeared before in scalar-theory calculations. Here we show how to compute the anomalous dimension of the operator $ϕ^{Q=5}$ at the six-loop order. In combination w…
We apply a recently suggested new strategy to solve differential equations for Feynman integrals. We develop this method further by analyzing asymptotic expansions of the integrals. We argue that this allows the systematic application of the differential equations to single-scale Feynman integrals. Moreover, the information about singular limits significantly simplifies finding boundary constants for the differential equations. To illustrate these points we consider two families of three-loop integrals. The first are form-factor integrals with two external legs on the light cone. We introduce …
This article displays a proof of concept of the mixed analytical/numerical method, presented in previous publications, to compute two-loop functions with up to five massive propagators in a scalar theory having three- and four-leg vertices as the Higgs sector of the Standard Model. Several amplitudes are considered with two, three and four external legs. Some of them diverge in the UV region and we demonstrate that the method works in that case. It is shown that all these classes of amplitudes can be generated by four master topologies. Results of a numerical evaluation for some kinematics are…
This work presents a first full one-loop computation of vector boson scattering (VBS) within the non-linear effective field theory given by the bosonic sector of the usually called electroweak chiral Lagrangian (EChL). The computation is performed in the most general case of covariant $R_ξ$ gauges and is compared through all this work with the Standard Model case, whose computation in these covariant gauges is also novel and is presented also here. The calculation of the one-loop VBS amplitude is performed using the diagrammatic method by means of the one-particle-irreducible (1PI) Green funct…
We explore maximal unitarity for nonplanar two-loop integrals with up to four massive external legs. In this framework, the amplitude is reduced to a basis of master integrals whose coefficients are extracted from maximal cuts. The hepta-cut of the nonplanar double box defines a nodal algebraic curve associated with a multiply pinched genus-3 Riemann surface. All possible configurations of external masses are covered by two distinct topological pictures in which the curve decomposes into either six or eight Riemann spheres. The procedure relies on consistency equations based on vanishing of in…
We present a program to evaluate tree-level multi-gluon amplitudes with up to two of them off-shell. Furthermore, it evaluates squared amplitudes summed over colors and helicities for up to six external gluons. It employs both analytic expressions, obtained via BCFW recursion, and numerical BCFW recursion. It has been validated numerically with the help of an independent program employing numerical Dyson-Schwinger recursion.
We present a formalism for the calculation of multi-particle one-loop amplitudes, valid for an arbitrary number N of external legs, and for massive as well as massless particles. A new method for the tensor reduction is suggested which naturally isolates infrared divergences by construction. We prove that for N>4, higher dimensional integrals can be avoided. We derive many useful relations which allow for algebraic simplifications of one-loop amplitudes. We introduce a form factor representation of tensor integrals which contains no inverse Gram determinants by choosing a convenient set of bas…
We develop an iterative method for constructing four-dimensional generalized unitarity cuts in $\mathcal{N} = 2$ supersymmetric Yang-Mills (SYM) theory coupled to fundamental matter hypermultiplets ($\mathcal{N} = 2$ SQCD). For iterated two-particle cuts,specifically those involving only four-point amplitudes, this implies simple diagrammatic rules for assembling the cuts to any loop order, reminiscent of the rung rule in $\mathcal{N} = 4$ SYM. By identifying physical poles, the construction simplifies the task of extracting complete integrands. In combination with the duality between color an…
