arXiv subjects
Publications and source records attributed to Tord Riemann.
Phenomenologically relevant electroweak precision pseudo-observables related to Z-boson physics are discussed in the context of the strong experimental demands of future $e^+e^-$ colliders. The recent completion of two-loop Z-boson results is summarized and a prospect for the 3-loop Standard Model calculation of the Z-boson decay pseudo-observable is given.
We present Standard Model predictions for the complete set of phenomenologically relevant electroweak precision pseudo-observables related to the Z-boson: the leptonic and bottom-quark effective weak mixing angles $\sin^2θ_{\rm eff}^\ell$, $\sin^2θ_{\rm eff}^b$, the Z-boson partial decay widths $Γ_f$, where $f$ indicates any charged lepton, neutrino and quark flavor (except for the top quark), as well as the total Z decay width $Γ_Z$, the branching ratios $R_\ell$, $R_c$, $R_b$, and the hadronic cross section $σ_{\rm had}^0$. The input parameters are the masses $M_Z$, $M_H$ and $m_t$, and the couplings $α_s$, $α$. The scheme dependence due to the choice of $M_W$ or its alternative $G_μ$ as a last input parameter is also discussed. Recent substantial technical progress in the calculation of Minkowskian massive higher-order Feynman integrals allows the calculation of the complete electroweak two-loop radiative corrections to all the observables mentioned. QCD contributions are included appropriately. Results are provided in terms of simple and convenient parameterization formulae whose coefficients have been determined from the full numerical multi-loop calculation. The size of the missing electroweak three-loop or QCD higher-order corrections is estimated. We briefly comment on the prospects for their calculation. Finally, direct predictions for the $Z{\bar f}f$ vector and axial-vector form-factors are given, including a discussion of separate order-by-order contributions.
The status of numerical evaluations of Mellin-Barnes integrals is discussed, in particular, the application of the quasi-Monte Carlo integration package QMC to the efficient calculation of multi-dimensional integrals.
This document answers in simple terms many FAQs about FCC-ee, including comparisons with other colliders. It complements the FCC-ee CDR and the FCC Physics CDR by addressing many questions from non-experts and clarifying issues raised during the European Strategy symposium in Granada, with a view to informing discussions in the period between now and the final endorsement by the CERN Council in 2020 of the European Strategy Group recommendations. This document will be regularly updated as more questions appear or new information becomes available.
The future lepton colliders proposed for the High Energy and Precision Frontier set stringent demands on theory. The most ambitious, broad-reaching and demanding project is the FCC-ee. We consider here the present status and requirements on precision calculations, possible ways forward and novel methods, to match the experimental accuracies expected at the FCC-ee. We conclude that the challenge can be tackled by a distributed collaborative effort in academic institutions around the world, provided sufficient support, which is estimated to about 500 man-years over the next 20 years.
The long-standing problem of representing the general massive one-loop Feynman integral as a meromorphic function of the space-time dimension $d$ has been solved for the basis of scalar one- to four-point functions with indices one. In 2003 the solution of difference equations in the space-time dimension allowed to determine the necessary classes of special functions: self-energies need ordinary logarithms and Gauss hypergeometric functions $_2F_1$, vertices need additionally Kampé de Fériet-Appell functions $F_1$, and box integrals also Lauricella-Saran functions $F_S$. In this study, alternative recursive Mellin-Barnes representations are used for the representation of $n$-point functions in terms of $(n-1)$-point functions. The approach enabled the first derivation of explicit solutions for the Feynman integrals at arbitrary kinematics. In this article, we scetch our new representations for the general massive vertex and box Feynman integrals and derive a numerical approach for the necessary Appell functions $F_1$ and Saran functions $F_S$ at arbitrary kinematical arguments.
We introduce techniques to treat numerically Mellin-Barnes integrals in physical regions, which arise in the need of the computation of Feynman integrals for the electroweak two-loop corrections to the pseudo observables at the Z-boson resonance.
This article presents results for the last unknown two-loop contributions to the $Z$-boson partial widths and $Z$-peak cross-section. These are the so-called bosonic electroweak two-loop corrections, where "bosonic" refers to diagrams without closed fermion loops. Together with the corresponding results for the $Z$-pole asymmetries $A_l, A_b$, which have been presented earlier, this completes the theoretical description of $Z$-boson precision observables at full two-loop precision within the Standard Model. The calculation has been achieved through a combination of different methods: (a) numerical integration of Mellin-Barnes representations with contour rotations and contour shifts to improve convergence; (b) sector decomposition with numerical integration over Feynman parameters; (c) dispersion relations for sub-loop insertions. Numerical results are presented in the form of simple parameterization formulae for the total width, $Γ_{\rm Z}$, partial decay widths $Γ_{e,μ},Γ_τ,Γ_ν,Γ_{u},Γ_{c},Γ_{d,s},Γ_{b}$, branching ratios $R_l,R_c,R_b$ and the hadronic peak cross-section, $σ_{\rm had}^0$. Theoretical intrinsic uncertainties from missing higher orders are also discussed.
Representations are derived for the basic scalar one-loop vertex Feynman integrals as meromorphic functions of the space-time dimension $d$ in terms of (generalized) hypergeometric functions $_2F_1$ and $F_1$. Values at asymptotic or exceptional kinematic points as well as expansions around the singular points at $d=4+2n$, $n$ non-negative integers, may be derived from the representations easily. The Feynman integrals studied here may be used as building blocks for the calculation of one-loop and higher-loop scalar and tensor amplitudes. From the recursion relation presented, higher n-point functions may be obtained in a straightforward manner.
During the last several years remarkable progress has been made in numerical calculations of dimensionally regulated multi-loop Feynman diagrams using Mellin-Barnes (MB) representations. The bottlenecks were non-planar diagrams and Minkowskian kinematics. The method has been proved to work in highly non-trivial physical application (two-loop electroweak bosonic corrections to the $Z \to b \bar{b}$ decay), and cross-checked with the sector decomposition (SD) approach. In fact, both approaches have their pros and cons. In calculation of multidimensional integrals, depending on masses and scales involved, they are complementary. A powerful top-bottom approach to the numerical integration of multidimensional MB integrals is automatized in the MB-suite AMBRE/MB/ MBtools/MBnumerics/CUBA. Key elements are a dedicated use of the Cheng-Wu theorem for non-planar topologies and of shifts and deformations of the integration contours. An alternative bottom-up approach starting with complex 1-dimensional MB-integrals, based on the exploration of steepest descent integration contours in Minkowskian kinematics, is also discussed. Short and long term prospects of the MB-method for multi-loop applications to LHC- and LC-physics are discussed.
Mellin-Barnes (MB) techniques applied to integrals emerging in particle physics perturbative calculations are summarized. New versions of AMBRE packages which construct planar and nonplanar MB representations are shortly discussed. The numerical package MBnumerics.m is presented for the first time which is able to calculate with a high precision multidimensional MB integrals in Minkowskian regions. Examples are given for massive vertex integrals which include threshold effects and several scale parameters.
The prediction of the effective electroweak mixing angle $\sin^2θ_{\rm eff}^{\rm b}$ in the Standard Model at two-loop accuracy has now been completed by the first calculation of the bosonic two-loop corrections to the $Z{\bar b}b$ vertex. Numerical predictions are presented in the form of a fitting formula as function of $M_Z, M_W, M_H, m_t$ and $Δα$, ${α_{\rm s}}$. For central input values, we obtain a relative correction of $Δκ_{\rm b}^{(α^2,\rm bos)} = -0.9855 \times 10^{-4}$, amounting to about a quarter of the fermionic corrections, and corresponding to $\sin^2θ_{\rm eff}^{\rm b} = 0.232704$. The integration of the corresponding two-loop vertex Feynman integrals with up to three dimensionless parameters in Minkowskian kinematics has been performed with two approaches: (i) Sector decomposition, implemented in the packages FIESTA 3 and SecDec 3, and (ii) Mellin-Barnes representations, implemented in AMBRE 3/MB and the new package MBnumerics.
Feynman integrals may be represented by the Mathematica packages AMBRE and MB as multiple Mellin-Barnes integrals. With the Mathematica package MBsums these Mellin-Barnes integrals are transformed into multiple sums.
A systematic study of the scalar one-loop two-, three-, and four-point Feynman integrals is performed. We consider all cases of mass assignment and external invariants and derive closed expressions in arbitrary space-time dimension in terms of higher transcendental functions. The integrals play a role as building blocks in general higher-loop or multi-leg processes. We also perform numerical checks of the calculations using AMBRE/MB and LoopTools/FF.
There are steady advances in the calculation of electroweak corrections to massive scattering problems at colliders, from the very beginning in the nineteen seventies until contemporary developments. Recent years brought a remarkable progress due to new calculational technologies. This was motivated by demands from phenomenological applications at particle accelerators: higher multiplicities of the final states, extreme kinematics, need of higher precision and thus of higher orders in perturbation theory. We describe selected contributions from the project "Massive particle production" of Sonderforschungsbereich/Transregio 9 of Deutsche Forschungsgemeinschaft.
The construction of Mellin-Barnes (MB) representations for non-planar Feynman diagrams and the summation of multiple series derived from general MB representations are discussed. A basic version of a new package AMBREv.3.0 is supplemented. The ultimate goal of this project is the automatic evaluation of MB representations for multiloop scalar and tensor Feynman integrals through infinite sums, preferably with analytic solutions. We shortly describe a strategy of further algebraic summation.
Round table discussions are in the tradition of ACAT. This year's plenary round table discussion was devoted to questions related to the use of scientific software in High Energy Physics and beyond. The 90 minutes of discussion were lively, and quite a lot of diverse opinions were spelled out. Although the discussion was, in part, controversial, the participants agreed unanimously on several basic issues in software sharing: (i) The importance of having various licensing models in academic research; (ii) The basic value of proper recognition and attribution of intellectual property, including scientific software; (iii) The user respect for the conditions of use, including licence statements, as formulated by the author. The need of a similar discussion on the issues of data sharing was emphasized and it was recommended to cover this subject at the conference round table discussion of next ACAT. In this contribution, we summarise selected topics that were covered in the introductory talks and in the following discussion.
The ZFITTER project is aimed at the computation of high-precision theoretical predictions for various observables in high-energy electron-positron annihilation and other processes. The stages of the project development are described. Accent is made on applications to the analysis of LEP data. The present status of the project and perspectives are given as well.