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T. Riemann

Publications and source records attributed to T. Riemann.

At least 37 records · Page 2Linked to original sources
arXiv2009

A recursive reduction of tensor Feynman integrals

We perform a recursive reduction of one-loop $n$-point rank $R$ tensor Feynman integrals [in short: $(n,R)$-integrals] for $n\leq 6$ with $R\leq n$ by representing $(n,R)$-integrals in terms of $(n,R-1)$- and $(n-1,R-1)$-integrals. We use the known representation of tensor integrals in terms of scalar integrals in higher dimension, which are then reduced by recurrence relations to integrals in generic dimension. With a systematic application of metric tensor representations in terms of chords, and by decomposing and recombining these representations, we find the recursive reduction for the tensors. The procedure represents a compact, sequential algorithm for numerical evaluations of tensor Feynman integrals appearing in next-to-leading order contributions to massless and massive three- and four- particle production at LHC and ILC, as well as at meson factories.

hep-ph↗
arXiv2009

The QCD form factor of heavy quarks at NNLO

We present an analytical calculation of the two-loop QCD corrections to the electromagnetic form factor of heavy quarks. The two-loop contributions to the form factor are reduced to linear combinations of master integrals, which are computed through higher orders in the parameter of dimensional regularization epsilon=(4-D)/2. Our result includes all terms of order epsilon at two loops and extends the previous literature. We apply the exponentiation of the heavy-quark form factor to derive new improved three-loop expansions in the high-energy limit. We also discuss the implications for predictions of massive n-parton amplitudes based on massless results in the limit, where the quark mass is small compared to all kinematical invariants.

hep-ph↗
arXiv2008

A complete reduction of one-loop tensor 5- and 6-point integrals

We perform a complete analytical reduction of general one-loop Feynman integrals with five and six external legs for tensors up to rank R=3 and 4, respectively. An elegant formalism with extensive use of signed minors is developed for the cancellation of inverse Gram determinants. The 6-point tensor functions of rank R are expressed in terms of 5-point tensor functions of rank R-1, and the latter are reduced to scalar four-, three-, and two-point functions. The resulting compact formulae allow both for a study of analytical properties and for efficient numerical programming. They are implemented in Fortran and Mathematica.

hep-ph↗
arXiv2008

On the tensor reduction of one-loop pentagons and hexagons

We perform analytical reductions of one-loop tensor integrals with 5 and 6 legs to scalar master integrals. They are based on the use of recurrence relations connecting integrals in different space-time dimensions. The reductions are expressed in a compact form in terms of signed minors, and have been implemented in a mathematica package called hexagon.m. We present several numerical examples.

hep-ph↗
arXiv2008

TERAD91: A Program package for the calculation of the cross sections of deep inelastic NC and CC scattering at HERA

We present a brief description of the recently developed Fortran code TERAD91 for semi-analytical calculations of the double differential cross sections of NC and CC deep inelastic electron-proton scattering and of some related observables. The code is mainly intended for the calculations at HERA or other ep-colliders but may be also used for similar processes like muon-proton scattering in fixed-target experiments.

hep-ph↗
arXiv2008

Automatizing the application of Mellin-Barnes representations for Feynman integrals

Feynman diagrams may be evaluated by Mellin-Barnes representations of their Feynman parameter integrals in d=4-2\eps dimensions. Recently, the Mathematica toolkit AMBRE has been developed for the automatic derivation of such representations with a loop-by-loop approach. We describe the package and exemplify its use with the \eps-expansion of the massive one-loop QED vertex function.

hep-ph↗
arXiv2008

AMBRE - a Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals

The Mathematica toolkit AMBRE derives Mellin-Barnes (MB) representations for Feynman integrals in d=4-2eps dimensions. It may be applied for tadpoles as well as for multi-leg multi-loop scalar and tensor integrals. AMBRE uses a loop-by-loop approach and aims at lowest dimensions of the final MB representations. The present version of AMBRE works fine for planar Feynman diagrams. The output may be further processed by the package MB for the determination of its singularity structure in eps. The AMBRE package contains various sample applications for Feynman integrals with up to six external particles and up to four loops.

hep-ph↗
arXiv2007

New results for 5-point functions

Bhabha scattering is one of the processes at the ILC where high precision data will be expected. The complete NNLO corrections include radiative loop corrections, with contributions from Feynman diagrams with five external legs. We take these diagrams as an example and discuss several features of the evaluation of pentagon diagrams. The tensor functions are usually reduced to simpler scalar functions. Here we study, as an alternative, the application of Mellin-Barnes representations to 5-point functions. There is no evidence for an improved numerical evaluation of their finite, physical parts. However, the approach gives interesting insights into the treatment of the IR-singularities.

hep-ph↗
arXiv2007

Pentagon diagrams of Bhabha scattering

We report on tensor reduction of five point integrals needed for the evaluation of loop-by-loop corrections to Bhabha scattering. As an example we demonstrate the calculation of the rank two tensor integral with cancellation of the spurious Gram determinant in the denominator. The reduction scheme is worked out for arbitrary five point processes.

hep-ph↗
arXiv2007

Fermionic NNLO contributions to Bhabha scattering

We derive the two-loop corrections to Bhabha scattering from heavy fermions using dispersion relations. The double-box contributions are expressed by three kernel functions. Convoluting the perturbative kernels with fermionic threshold functions or with hadronic data allows to determine numerical results for small electron mass m_e, combined with arbitrary values of the fermion mass m_f in the loop, $m_e^2<<s,t,m_f^2$, or with hadronic insertions. We present numerical results for m_f = m_μ, m_τ, m_{top} at typical small- and large-angle kinematics ranging from 1 GeV to 500 GeV.

hep-ph↗
arXiv2007

Two-Loop Fermionic Corrections to Massive Bhabha Scattering

We evaluate the two-loop corrections to Bhabha scattering from fermion loops in the context of pure Quantum Electrodynamics. The differential cross section is expressed by a small number of Master Integrals with exact dependence on the fermion masses me, mf and the Mandelstam invariants s,t,u. We determine the limit of fixed scattering angle and high energy, assuming the hierarchy of scales me^2 << mf^2 << s,t,u. The numerical result is combined with the available non-fermionic contributions. As a by-product, we provide an independent check of the known electron-loop contributions.

hep-ph↗
arXiv2006

First order radiative corrections to Bhabha scattering in $d$ dimensions

The luminosity measurement at the projected International Linear $e^+e^-$ Collider ILC is planned to be performed with forward Bhabha scattering with an accuracy of the order of $10^{-4}$. A theoretical prediction of the differential cross-section has to include one-loop weak corrections, with leading higher order terms, and the complete two-loop QED corrections. Here, we present the weak part and the virtual one-loop photonic corrections. For the photonic corrections, the expansions in $ε= (4-d)/2$ are derived with inclusion of the terms of order $ε$ in order to match the two-loop accuracy. For the photonic box master integral in $d$ dimensions we compare several different methods of evaluation.

hep-ph↗
arXiv2006

The planar four-point master integrals for massive two-loop Bhabha scattering

We present the values of the complete set of planar four-point Master Integrals needed for massive Bhabha scattering in the limit of fixed angle and high energy at the two-loop level. The integrals have been calculated using direct expansions of Mellin-Barnes representations, followed by a resummation of resulting harmonic series.

hep-ph↗
arXiv2006

Differential equations and massive two-loop Bhabha scattering: the B5l2m3 case

The two-loop box contributions to massive Bhabha scattering may be reduced to two-loop box master integrals (MIs) with five, six, and seven internal lines, plus vertices and self energies. The self-energy and vertex MIs may be solved analytically by the differential equations (DE) method. This is true for only few of the box masters. Here we describe some details of the analytical determination, including constant terms in ep=(4-d)/2, of the complicated topology B5l2m3 (with 5 lines, 2 of them being massive). With the DE approach, three of the four coupled masters have been solved in terms of (generalized) standard Harmonic Polylogarithms.

hep-ph↗
arXiv2005

On the massive two-loop corrections to Bhabha scattering

We overview the general status of higher order corrections to Bhabha scattering and review recent progress in the determination of the two-loop virtual corrections. Quite recently, they were derived from combining a massless calculation and contributions with electron sub-loops. For a massive calculation, the self-energy and vertex master integrals are known, while most of the two-loop boxes are not. We demonstrate with an example that a study of systems of differential equations, combined with Mellin-Barnes representations for single masters, might open a way for their systematic calculation.

hep-ph↗
arXiv2005

Harmonic polylogarithms for massive Bhabha scattering

One- and two-dimensional harmonic polylogarithms, HPLs and GPLs, appear in calculations of multi-loop integrals. We discuss them in the context of analytical solutions for two-loop master integrals in the case of massive Bhabha scattering in QED. For the GPLs we discuss analytical representations, conformal transformations, and also their transformations corresponding to relations between master integrals in the s- and t-channel.

hep-ph↗
arXiv2005

ZFITTER: a semi-analytical program for fermion pair production in e+e- annihilation, from version 6.21 to version 6.42

ZFITTER is a Fortran program for the calculation of fermion pair production and radiative corrections at high energy e+e- colliders; it is also suitable for other applications where electroweak radiative corrections appear. ZFITTER is based on a semi-analytical approach to the calculation of radiative corrections in the Standard Model. We present a summary of new features of the ZFITTER program version 6.42 compared to version 6.21. The most important additions are: (i) some higher-order QED corrections to fermion pair production, (ii) electroweak one-loop corrections to atomic parity violation, (iii) electroweak one-loop corrections to nu-e nu-e-bar production, (iv) electroweak two-loop corrections to the W boson mass and the effective weak mixing angle.

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