arXiv subjects

Tord Riemann

Publications and source records attributed to Tord Riemann.

At least 37 records · Page 2Linked to original sources

arXiv2013

Some Remarks on Non-Planar Diagrams

Two criteria for planarity of a Feynman diagram upon its propagators (momentum flows) are presented. Instructive Mathematica programs that solve the problem and examples are provided. A simple geometric argument is used to show that while one can planarize non-planar graphs by embedding them on higher-genus surfaces (in the example it is a torus), there is still a problem with defining appropriate dual variables since the corresponding faces of the graph are absorbed by torus generators.

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arXiv2012

Efficient contraction of 1-loop N-point tensor integrals

A new approach for the reduction of tensor integrals is described. The standard decomposition à la Davydychev is applied. Integrals with higher indices are then expressed in terms of scalar higher-dimensional integrals with generic indices. The approach allows to perform contractions with external momenta in a particularly efficient manner. This is due to the possibility to perform analytically the resulting sums over the indices of products of signed minors and scalar products of chords. Advantages of this approach for the calculation of cross sections are described. We are preparing the numerical package OLEC (\texttt{O}ne \texttt{L}oop \texttt{E}xternal \texttt{C}ontractions) with interfaces in Mathematica for algebraic and in C++ for numerical calculations. First numerical results are discussed.

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arXiv2012

New developments in PJFry

We report on recent progress in numerical evaluation of one loop tensor integrals. A public C++ package PJFry implementing algorithms from arXiv:1009.4436 and its extension to hexagons up to rank 6 are presented.

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arXiv2012

New results for algebraic tensor reduction of Feynman integrals

We report on some recent developments in algebraic tensor reduction of one-loop Feynman integrals. For 5-point functions, an efficient tensor reduction was worked out recently and is now available as numerical C++ package, PJFry, covering tensor ranks until five. It is free of inverse 5-point Gram determinants, and inverse small 4-point Gram determinants are treated by expansions in higher-dimensional 3-point functions. By exploiting sums over signed minors, weighted with scalar products of chords (or, equivalently, external momenta), extremely efficient expressions for tensor integrals contracted with external momenta were derived. The evaluation of 7-point functions is discussed. In the present approach one needs for the reductions a $(d+2)$-dimensional scalar 5-point function in addition to the usual scalar basis of 1- to 4-point functions in the generic dimension $d=4-2 ε$. When exploiting the four-dimensionality of the kinematics, this basis is sufficient. We indicate how the $(d+2)$-dimensional 5-point function can be evaluated.

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arXiv2010

Numerical Evaluation of Tensor Feynman Integrals in Euclidean Kinematics

For the investigation of higher order Feynman integrals, potentially with tensor structure, it is highly desirable to have numerical methods and automated tools for dedicated, but sufficiently 'simple' numerical approaches. We elaborate two algorithms for this purpose which may be applied in the Euclidean kinematical region and in d=4-2*epsilon dimensions. One method uses Mellin-Barnes representations for the Feynman parameter representation of multi-loop Feynman integrals with arbitrary tensor rank. Our Mathematica package AMBRE has been extended for that purpose, and together with the packages MB (M. Czakon) or MBresolve (A. V. Smirnov and V. A. Smirnov) one may perform automatically a numerical evaluation of planar tensor Feynman integrals. Alternatively, one may apply sector decomposition to planar and non-planar multi-loop epsilon-expanded Feynman integrals with arbitrary tensor rank. We automatized the preparations of Feynman integrals for an immediate application of the package sector_decomposition (C. Bogner and S. Weinzierl) so that one has to give only a proper definition of propagators and numerators. The efficiency of the two implementations, based on Mellin-Barnes representations and sector decompositions, is compared. The computational packages are publicly available.

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arXiv2010

News on Ambre and CSectors

Mellin-Barnes and sector decomposition methods are used to evaluate tensorial Feynman diagrams in the Euclidean kinematical region. Few software packages are shortly described and few examples demonstrate their use.

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arXiv2010

Some variations of the reduction of one-loop Feynman tensor integrals

We present a new algorithm for the reduction of one-loop \emph{tensor} Feynman integrals with $n\leq 4$ external legs to \emph{scalar} Feynman integrals $I_n^D$ with $n=3,4$ legs in $D$ dimensions, where $D=d+2l$ with integer $l \geq 0$ and generic dimension $d=4-2ε$, thus avoiding the appearance of inverse Gram determinants $()_4$. As long as $()_4\neq 0$, the integrals $I_{3,4}^D$ with $D>d$ may be further expressed by the usual dimensionally regularized scalar functions $I_{2,3,4}^d$. The integrals $I_{4}^D$ are known at $()_4 \equiv 0$, so that we may extend the numerics to small, non-vanishing $()_4$ by applying a dimensional recurrence relation. A numerical example is worked out. Together with a recursive reduction of 6- and 5-point functions, derived earlier, the calculational scheme allows a stabilized reduction of $n$-point functions with $n\leq 6$ at arbitrary phase space points. The algorithm is worked out explicitely for tensors of rank $R\leq n$.

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arXiv2010

A recursive approach to the reduction of tensor Feynman integrals

We describe a new, convenient, recursive tensor integral reduction scheme for one-loop $n$-point Feynman integrals. The reduction is based on the algebraic Davydychev-Tarasov formalism where the tensors are represented by scalars with shifted dimensions and indices, and then expressed by conventional scalars with generalized recurrence relations. The scheme is worked out explicitly for up to $n=6$ external legs and for tensor ranks $R\leq n$. The tensors are represented by scalar one- to four-point functions in $d$ dimensions. For the evaluation of them, the Fortran code for the tensor reductions has to be linked with a package like QCDloop or LoopTools/FF. Typical numerical results are presented.

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arXiv2009

New results for loop integrals: AMBRE, CSectors, hexagon

We report on the three Mathematica packages hexagon, CSectors, AMBRE. They are useful for the evaluation of one- and two-loop Feynman integrals with a dependence on several kinematical scales. These integrals are typically needed for LHC and ILC applications, but also for higher order corrections at meson factories. hexagon is a new package for the tensor reduction of one-loop 5-point and 6-point functions with rank R=3 and R=4, respectively; AMBRE is a tool for derivations of Mellin-Barnes representations; CSectors is an interface for the package sector_decomposition and allows a convenient, direct evaluation of tensor Feynman integrals.

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arXiv2008

Virtual Hadronic and Heavy-Fermion O(alpha^2) Corrections to Bhabha Scattering

Effects of vacuum polarization by hadronic and heavy-fermion insertions were the last unknown two-loop QED corrections to high-energy Bhabha scattering and have been first announced in \cite{Actis:2007fs}. Here we describe the corrections in detail and explore their numerical influence. The hadronic contributions to the virtual O(alpha^2) QED corrections to the Bhabha-scattering cross-section are evaluated using dispersion relations and computing the convolution of hadronic data with perturbatively calculated kernel functions. The technique of dispersion integrals is also employed to derive the virtual O(alpha^2) corrections generated by muon-, tau- and top-quark loops in the small electron-mass limit for arbitrary values of the internal-fermion masses. At a meson factory with 1 GeV center-of-mass energy the complete effect of hadronic and heavy-fermion corrections amounts to less than 0.5 per mille and reaches, at 10 GeV, up to about 2 per mille. At the Z resonance it amounts to 2.3 per mille at 3 degrees; overall, hadronic corrections are less than 4 per mille. For ILC energies (500 GeV or above), the combined effect of hadrons and heavy-fermions becomes 6 per mille at 3 degrees; hadrons contribute less than 20 per mille in the whole angular region.

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arXiv2008

Model independent QED corrections to the process $ep \longrightarrow eX$

We give an exhaustive presentation of the semi-analytical approach to the model independent leptonic QED corrections to deep inelastic neutral current lepton-nucleon scattering. These corrections include photonic bremsstrahlung from and vertex corrections to the lepton current of the order ${\cal O}(α)$ with soft photon exponentiation. % A common treatment of these radiative corrections in several variables -- leptonic, hadronic, mixed, Jaquet-Blondel variables -- has been developed and double differential cross-sections are calculated. In all sets of variables we use some structure functions, which depend on the hadronic variables and which do not have to be defined in the quark parton model. The remaining numerical integrations are twofold (for leptonic variables) or onefold (for all other variables). For the case of hadronic variables, all phase space integrals have been performed analytically. Numerical results are presented for a large kinematical range, covering fixed target as well as collider experiments at HERA or LEP$\otimes$LHC, with a special emphasis on HERA physics.

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arXiv2008

Leptonic QED Corrections to the process $ep \longrightarrow eX$ in Jaquet-Blondel variables

For the study of deep inelastic scattering at HERA, the use of Jaquet-Blondel variables is advantageous in several respects. We calculate the complete leptonic ${\cal O}(α)$ QED corrections for the reaction $ep \to eX$ in these variables. All but one phase space integrations are performed analytically. After exponentiation of soft photon corrections, an accuracy of at least 1% is matched. Numerical results are presented and compared to estimates based on the leading logarithmic approximation. Equation (17) is corrected in accordance with equation (7.44) of DESY 94--115. The misprint did not influence the numerics.

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arXiv2008

Virtual Hadronic Corrections to Massive Bhabha Scattering

Virtual hadronic contributions to the Bhabha process at the NNLO level are discussed. They are substantial for predictions with per mil accuracy. The studies of heavy fermion and hadron corrections complete the calculation of Bhabha virtual effects at the NNLO level.

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arXiv2008

Virtual Hadronic and Leptonic Contributions to Bhabha Scattering

Using dispersion relations, we derive the complete virtual QED contributions to Bhabha scattering due to vacuum polarization effects in photon propagation. We apply our result to hadronic corrections and to heavy lepton and top quark loop insertions. We give the first complete estimate of their net numerical effects for both small and large angle scattering at typical beam energies of meson factories, LEP, and the ILC. The effects turn out to be smaller, in most cases, than those corresponding to electron loop insertions, but stay, with amounts of typically one per mille, of relevance for precision experiments. Hadronic corrections themselves are typically about 2-3 times larger than those of intermediate muon pairs (the largest heavy leptonic terms).

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arXiv2008

A new treatment of mixed virtual and real IR-singularities

We discuss the determination of the infrared singularities of massive one-loop 5-point functions with Mellin-Barnes (MB) representations. Massless internal lines may lead to poles in the $\eps$ expansion of the Feynman diagram, while unresolved massless final state particles give endpoint singularities of the phase space integrals. MB integrals are an elegant tool for their common treatment. An evaluation by taking residues leads to inverse binomial sums.

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arXiv2007

Planar two-loop master integrals for massive Bhabha scattering: N_f=1 and N_f=2

Recent developments in the computation of two-loop master integrals for massive Bhabha scattering are briefly reviewed. We apply a method based on expansions of exact Mellin-Barnes representations and evaluate all planar four-point master integrals in the approximation of small electron mass at fixed scattering angle for the one-flavor case. The same technique is employed to derive and evaluate also all two-loop masters generated by additional fermion flavors. The approximation is sufficient for the determination of QED two-loop corrections for Bhabha scattering in the kinematics planned to be used for the luminosity determination at the ILC.

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arXiv2006

Factorizable electroweak O(alpha) corrections for top quark pair production and decay at a linear e+e- collider

We calculate the standard model predictions for top quark pair production and decay into six fermions at a linear e+ e- collider. We include the factorizable electroweak O(alpha) corrections in the pole approximation and QED corrections due to the initial state radiation in the structure function approach. The effects of the radiative corrections on the predictions are illustrated by showing numerical results for two selected six-fermion reactions e+ e- -> b nu_mu mu+ bar{b} mu- bar{nu_mu} and e+ e- -> b nu_mu mu+ bar{b} d bar{u}.

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arXiv2005

Towards high precision predictions for top quark pair production and decay at a linear collider

We report on the progress in work on improving precision of the standard model theoretical predictions for the top quark pair production and decay into six fermions at a linear collider. Two programs have been combined into a single Monte Carlo program: 'eett6f', a MC program for e+e- -> 6f, and 'topfit', a program for electroweak radiative corrections to e+e- -> t\bar t. The MC program is described and preliminary numerical results are shown.

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