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.Comment: Presented by K. Bielas at the International Conference of Theoretical Physics "Matter To The Deepest", Ustron 201

Numerical integration of massive two-loop Mellin-Barnes integrals in Minkowskian regions

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.Comment: Proceedings for 13th DESY Workshop on Elementary Particle Physics: Loops and Legs in Quantum Field Theory (LL2016), final PoS versio

Complete electroweak two-loop corrections to Z boson production and decay

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, $\Gamma_{\rm Z}$, partial decay widths $\Gamma_{e,\mu},\Gamma_{\tau},\Gamma_{\nu},\Gamma_{u},\Gamma_{c},\Gamma_{d,s},\Gamma_{b}$, branching ratios $R_l,R_c,R_b$ and the hadronic peak cross-section, $\sigma_{\rm had}^0$. Theoretical intrinsic uncertainties from missing higher orders are also discussed.Comment: 10 page

Optimizing the Mellin-Barnes approach to numerical multiloop calculations

The status of numerical evaluations of Mellin–Barnes integrals is discussed, in particular the application of the quasi-Monte Carlo integration method to the efficient calculation of multidimensional integrals

New prospects for the numerical calculation of Mellin-Barnes integrals in Minkowskian kinematics

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.Comment: Presented at the Epiphany Cracow conference 2017, refs adde

Non-planar Feynman diagrams and Mellin-Barnes representations with AMBRE 3.0

We introduce the Mellin-Barnes representation of general Feynman integrals and discuss their evaluation. The Mathematica package AMBRE has been recently extended in order to cover consistently non-planar Feynman integrals with two loops. Prospects for the near future are outlined. This write-up is an introduction to new results which have also been presented elsewhere

The two-loop electroweak bosonic corrections to sin⁡2θeffb\sin^2\theta_{\rm eff}^{\rm b}

The prediction of the effective electroweak mixing angle $\sin^2\theta_{\rm eff}^{\rm b}$ 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 $\Delta{\alpha}$, ${\alpha_{\rm s}}$. For central input values, we obtain a relative correction of $\Delta\kappa_{\rm b}^{(\alpha^2,\rm bos)} = -0.9855 \times 10^{-4}$, amounting to about a quarter of the fermionic corrections, and corresponding to $\sin^2\theta_{\rm eff}^{\rm b} = 0.232704$. 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.Comment: 14 pp; v2: some explanations and Tab.2 added, version published in PL

Mellin-Barnes Integrals: A Primer on Particle Physics Applications

We discuss the Mellin-Barnes representation of complex multidimensional integrals. Experiments frontiered by the High-Luminosity Large Hadron Collider at CERN and future collider projects demand the development of computational methods to achieve the theoretical precision required by experimental setups. In this regard, performing higher-order calculations in perturbative quantum field theory is of paramount importance. The Mellin-Barnes integrals technique has been successfully applied to the analytic and numerical analysis of integrals connected with virtual and real higher-order perturbative corrections to particle scattering. Easy-to-follow examples with the supplemental online material introduce the reader to the construction and the analytic, approximate, and numeric solution of Mellin-Barnes integrals in Euclidean and Minkowskian kinematic regimes. It also includes an overview of the state-of-the-art software packages for manipulating and evaluating Mellin-Barnes integrals. These lecture notes are for advanced students and young researchers to master the theoretical background needed to perform perturbative quantum field theory calculations.Comment: This is a preprint of the following work: Ievgen Dubovyk, Janusz Gluza and Gabor Somogyi, Mellin-Barnes Integrals: A Primer on Particle Physics Applications, 2022, Springer reproduced with permission of Springer Nature Switzerland AG. 280 page

Non-planar Feynman integrals, Mellin-Barnes representations, multiple sums

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.Comment: Contribution to the proceedings of the Loops and Legs 2014 conferenc

30 years, some 700 integrals, and 1 dessert or: electroweak two-loop corrections to the Z ̄bb vertex

The one-loop corrections to the weak mixing angle sin2 qb eff, derived from the Z¯bb vertex, are known since 1985. It took another 30 years to calculate the complete electroweak two-loop corrections to sin2 qb eff. The main obstacle was the calculation of the O(700) bosonic two-loop vertex integrals with up to three mass scales, at s = M2 Z. We did not perform the usual integral reduction and master evaluation, but chose a completely numerical approach, using two different calculational chains. One method relies on publicly available sector decomposition implementations. Further, we derived Mellin-Barnes (MB) representations, exploring the publicly available MB suite. We had to supplement the MB suite by two new packages: AMBRE 3, a Mathematica program, for the efficient treatment of non-planar integrals and MBnumerics for advanced numerics in the Minkowskian space-time. Our preliminary result for LL2016, the “dessert”, for the electroweak bosonic two-loop contributions to sin2 qb eff is: Dsin2 qb(a2;bos) eff = sin2 qW Dk(a2;bos) b , with Dk(a2;bos) b = 1:0276 104. This contribution is about a quarter of the corresponding fermionic corrections and of about the same magnitude as several of the known higher-order QCD corrections. The sin2 qb eff is now predicited in the Standard Model with a relative error of 10-4 [1]