G. Heinrich

Numerical evaluation of multi-loop integrals by sector decomposition

Abstract In a recent paper [Nucl. Phys. B 585 (2000) 741] we have presented an automated subtraction method for divergent multi-loop/leg integrals in dimensional regularisation which allows for their numerical evaluation, and applied it to diagrams with massless internal lines. Here we show how to extend this algorithm to Feynman diagrams with massive propagators and arbitrary propagator powers. As applications, we present numerical results for the master 2-loop 4-point topologies with massive internal lines occurring in Bhabha scattering at two loops, and for the master integrals of planar and non-planar massless double box graphs with two off-shell legs. We also evaluate numerically some two-point functions up to 5 loops relevant for beta-function calculations, and a 3-loop 4-point function, the massless on-shell planar triple box. Whereas the 4-point functions are evaluated in non-physical kinematic regions, the results for the propagator functions are valid for arbitrary kinematics.

golem95: A numerical program to calculate one-loop tensor integrals with up to six external legs

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.

Four-particle phase space integrals in massless QCD

The inclusive four-particle phase space integral of any 1→4 matrix element in massless QCD contains divergences due to the soft and collinear emission of up to two particles in the final state. We show that any term appearing in this phase space integral can be expressed as linear combination of only four master integrals. These four master integrals are all computed in dimensional regularisation up to their fourth order terms, relevant to next-to-next-to-leading order jet calculations, both in an analytic form and purely numerically. New analytical and numerical techniques are developed in this context. We introduce the tripole parametrisation of the four-parton phase space. Furthermore, we exploit unitarity relations between multi-parton phase space integrals and multi-loop integrals. For the numerical calculation, the iterated sector decomposition of loop integrals is extended to phase space integrals. The results in this paper lead to infrared subtraction terms needed for the double real radiation contributions to jet physics in e+e− annihilation at the next-to-next-to-leading order.

A Proposal for a standard interface between Monte Carlo tools and one-loop programs

Many highly developed Monte Carlo tools for the evaluation of cross sections based on tree matrix elements exist and are used by experimental collaborations in high energy physics. As the evaluation of one-loop matrix elements has recently been undergoing enormous progress, the combination of one-loop matrix elements with existing Monte Carlo tools is on the horizon. This would lead to phenomenological predictions at the next-to-leading order level. This note summarises the discussion of the next-to-leading order multi-leg (NLM) working group on this issue which has been taking place during the workshop on Physics at TeV Colliders at Les Houches, France, in June 2009. The result is a proposal for a standard interface between Monte Carlo tools and one-loop matrix element programs. Dedicated to the memory of, and in tribute to, Thomas Binoth, who led the effort to develop this proposal for Les Houches 2009. Thomas led the discussions, set up the subgroups, collected the contributions, and wrote and edited this paper. He made a promise that the paper would be on the arXiv the first week of January, and we are faithfully fulfilling his promise. In his honour, we would like to call this the Binoth Les Houches Accord.

Determination of the strong coupling constant using matched NNLO+NLLA predictions for hadronic event shapes in e+e− annihilations

We present a determination of the strong coupling constant from a fit of QCD predictions for six event-shape variables, calculated at next-to-next-to-leading order (NNLO) and matched to resummation in the next-to-leading-logarithmic approximation (NLLA). These event shapes have been measured in e+e? annihilations at LEP, where the data we use have been collected by the ALEPH detector at centre-of-mass energies between 91 and 206 GeV. Compared to purely fixed order NNLO fits, we observe that the central fit values are hardly affected, but the systematic uncertainty is larger because the NLLA part re-introduces relatively large uncertainties from scale variations. By combining the results for six event-shape variables and eight centre-of-mass energies, we find ?s(MZ) = 0.1224???0.0009?(stat)???0.0009?(exp)???0.0012?(had)???0.0035?(theo), which improves previously published measurements at NLO+NLLA. We also carry out a detailed investigation of hadronisation corrections, using a large set of Monte Carlo generator predictions.

Tensorial reconstruction at the integrand level

We present a new approach to the reduction of one-loop amplitudes obtained by reconstructing the tensorial expression of the scattering amplitudes. The reconstruction is performed at the integrand level by means of a sampling in the integration momentum. There are several interesting applications of this novel method within existing techniques for the reduction of one-loop multi-leg amplitudes: to deal with numerically unstable points, such as in the vicinity of a vanishing Gram determinant; to allow for a sampling of the numerator function based on real values of the integration momentum; to optimize the numerical reduction in the case of long expressions for the numerator functions.

A numerical evaluation of the scalar hexagon integral in the physical region

Abstract We derive an analytic expression for the scalar one-loop pentagon and hexagon functions which is convenient for subsequent numerical integration. These functions are of relevance in the computation of next-to-leading order radiative corrections to multi-particle cross sections. The hexagon integral is represented in terms of n -dimensional triangle functions and ( n +2)-dimensional box functions. If infrared poles are present this representation naturally splits into a finite and a pole part. For a fast numerical integration of the finite part we propose simple one- and two-dimensional integral representations. We set up an iterative numerical integration method to calculate these integrals directly in an efficient way. The method is illustrated by explicit results for pentagon and hexagon functions with some generic physical kinematics.

A numerical method for NNLO calculations

Abstract A method to isolate the poles of dimensionally regulated multi-loop integrals and to calculate the pole coefficients numerically is extended to be applicable to phase space integrals as well.

NNLO moments of event shapes in e+e− annihilation

We compute the next-to-next-to-leading order (NNLO) QCD corrections to the first five moments of six event shape variables related to three-particle final states in electron-positron annihilation; the thrust, the heavy jet mass, the C-parameter, the wide and total jet broadenings and the three-to-two-jet transition parameter in the Durham algorithm Y3. The NNLO corrections to the first moment are moderate for all event shapes, while the renormalisation scale dependence of the theoretical prediction is substantially reduced compared to the previously existing NLO results. From a comparison with data from JADE and OPAL, we observe that the energy dependence of the moments of the wide jet broadening and Y3 can be largely explained without any non-perturbative power corrections, while the other observables exhibit a clear need for power-like contributions at low centre-of-mass energy.

Loop induced interference effects in Higgs Boson plus two jet production at the LHC.

We calculate the order (α2αs3) interference effect between the gluon fusion and weak boson fusion processes allowed at the one-loop level in Higgs boson plus 2 jet production at the LHC. The corresponding one-loop amplitudes, which have not been considered in the literature so far, are evaluated analytically using dimensional regularisation and the necessary master integrals with massive propagators are reported. It is discussed in detail how various mechanisms conspire to make this contribution numerically negligible for experimental studies at the LHC.