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We review the recent progress on the numerical implementation of the Loop-Tree Duality Method (LTDM) for the calculation of scattering amplitudes. A central point is the analysis of the singularities of the integrand. In the framework of the LTDM some of these singularities cancel out. The ones left over are dealt with by contour deformation. We present details on how to achieve this as well as first results., Comment: 4 pages, Proceedings

We review the recent developments of the loop-tree duality method, focussing our discussion on analysing the singular behaviour of the loop integrand of the dual representation of one-loop integrals and scattering amplitudes. We show that within the loop-tree duality method there is a partial cancellation of singularities at the integrand level among the different components of the corresponding dual representation. The remaining threshold and infrared singularities are restricted to a finite region of the loop momentum space, which is of the size of the external momenta and can be mapped to the phase-space of real corrections to cancel the soft and collinear divergences., Comment: 8 pages 3 figures, Proceedings of Loops and Legs in Quantum Field Theory, 27 April 2014 - 02 May 2014, Weimar, Germany

We analyse the singular behaviour of one-loop integrals and scattering amplitudes in the framework of the loop--tree duality approach. We show that there is a partial cancellation of singularities at the loop integrand level among the different components of the corresponding dual representation that can be interpreted in terms of causality. The remaining threshold and infrared singularities are restricted to a finite region of the loop momentum space, which is of the size of the external momenta and can be mapped to the phase-space of real corrections to cancel the soft and collinear divergences., Comment: 12 pages, 2 figures. Final version to appear in JHEP

We develop the Tree-Loop Duality Relation for two- and three-loop integrals with multiple identical propagators (multiple poles). This is the extension of the Duality Relation for single poles and multiloop integrals derived in previous publications. We prove a generalization of the formula for single poles to multiple poles and we develop a strategy for dealing with higher-order pole integrals by reducing them to single pole integrals using Integration By Parts., Comment: 21 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1007.0194

Calculation of amplitudes in perturbative quantum field theory involve large loop integrals. The complexity of those integrals, in combination with the large number of Feynman diagrams, make the calculations very difficult. Reduction methods proved to be very helpful, lowering the number of integrals that need to be actually calculated. Especially, the reduction at the integrand level technique, improves the speed and set-up of these calculations. In this article we demonstrate, by counting the numbers of tensor structures and independent coefficients, how to write such relations at the integrand level for one- and two-loop amplitudes. We clarify their connection to the so-called spurious terms at one loop and discuss their structure in the two-loop case. This method is also applicable to higher loops, and the results obtained apply to both planar and non-planar diagrams.

The analytical package written in FORM presented in this paper allows the computation of the complete set of Feynman Rules producing the Rational terms of kind R2 contributing to the virtual part of NLO amplitudes in the Standard Model of the Electroweak interactions. Building block topologies filled by means of generic scalars, vectors and fermions, allowing to build these Feynman Rules in terms of specific elementary particles, are explicitly given in the Rxi gauge class, together with the automatic dressing procedure to obtain the Feynman Rules from them. The results in more specific gauges, like the 't Hooft Feynman one, follow as particular cases, in both the HV and the FDH dimensional regularization schemes. As a check on our formulas, the gauge independence of the total Rational contribution (R1 + R2) to renormalized S-matrix elements is verified by considering the specific example of the H --> gamma-gamma decay process at 1-loop. This package can be of interest for people aiming at a better understanding of the nature of the Rational terms. It is organized in a modular way, allowing a further use of some its files even in different contexts. Furthermore, it can be considered as a first seed in the effort towards a complete automation of the process of the analytical calculation of the R2 effective vertices, given the Lagrangian of a generic gauge theory of particle interactions.

We present the complete set of Feynman rules producing the rational terms of kind R_2 needed to perform any 1-loop calculation in the Electroweak Standard Model. Our formulae are given both in the R_xi gauge and in the Unitary gauge, therefore completing the results in the 't Hooft-Feynman gauge already presented in a previous publication. As a consistency check, we verified, in the case of the process H -> gamma gamma and in a few other physical cases, the independence of the total Rational Part R_1+R_2 on the chosen gauge. In addition, we explicitly checked the equivalence of the limits xi -> infinity after or before the loop momentum integration in the definition of the Unitary gauge at 1-loop., Comment: 16 pages,3 figures

Achieving a precise description of multi-parton final states is crucial for many analyses at LHC. In this contribution we review the main features of the HELAC-NLO system for NLO QCD calculations. As a case study, NLO QCD corrections for tt + 2 jet production at LHC are illustrated and discussed., Comment: 7 pages, 4 figures. Presented at 10th DESY Workshop on Elementary Particle Theory: Loops and Legs in Quantum Field Theory, Worlitz, Germany, April 25-30, 2010

This report summarizes the activities of the SM and NLO Multileg Working Group of the Workshop "Physics at TeV Colliders", Les Houches, France 8-26 June, 2009., Comment: 169 pages, Report of the SM and NLO Multileg Working Group for the Workshop "Physics at TeV Colliders", Les Houches, France 8-26 June, 2009

We present the complete set of Feynman rules producing the rational terms of kind R_2 needed to perform any 1-loop calculation in the Electroweak Standard Model. Our results are given both in the 't Hooft-Veltman and in the Four Dimensional Helicity regularization schemes. We also verified, by using both the 't Hooft-Feynman gauge and the Background Field Method, a huge set of Ward identities -up to 4-points- for the complete rational part of the Electroweak amplitudes. This provides a stringent check of our results and, as a by-product, an explicit test of the gauge invariance of the Four Dimensional Helicity regularization scheme in the complete Standard Model at 1-loop. The formulae presented in this paper provide the last missing piece for completely automatizing, in the framework of the OPP method, the 1-loop calculations in the SU(3) X SU(2) X U(1) Standard Model., Comment: Many thanks to Huasheng Shao for having recomputed, independently of us, all of the ${\rm R_2}$ effective vertices. Thanks to his help and by comparing with an independent computation we performed in a general $R_\xi$ gauge, we could fix, in the present version, the following formulae: the vertex $A l \bar l$ in Eq. (3.6), the vertex $Z \phi^+ \phi^-$ in Eq. (3.8), Eqs (3.16), (3.17) and (3.18)

We point out that the compact Feynman rules for Majorana fermions proposed by Denner et al. are in fact a convention for the complex phases of (anti)spinors, valid for both Majorana and Dirac fermions. We establish the relation of this phase convention with that common in the use of spinor techniques., Comment: 5 pages, comment and reference added

The one-loop effective potential is a powerful tool in studying the electroweak symmetry breaking of supersymmetric theories, whose precise calculation may have important phenomenological consequences. In this work, we are correctly treating the contribution of the Higgs sector to the effective potential and refine the radiative corrections to the Higgs mixing parameter $\mu$, which is known to affect greatly the supersymmetric spectrum. Working at the average stop scale to minimize the effect of the stop sector, we find additional corrections which can play a dominant role in the Focus Point region of the parameter space of the MSSM. The comparison of our results with those of the literature is discussed. We also discuss the gauge dependence of the effective potential and its effect on the $\mu$ parameter in analyses where this is determined from the 1-loop minimization conditions of the effective potential., Comment: 13 pages LateX, 3 eps figures. Revisions to text, comments and relevant references added