22 research outputs found
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 -boson partial widths and -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 -pole asymmetries , which have been presented earlier,
this completes the theoretical description of -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, , partial decay widths
,
branching ratios and the hadronic peak cross-section,
. 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 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
The prediction of the effective electroweak mixing angle in the Standard Model at two-loop accuracy has now been completed
by the first calculation of the bosonic two-loop corrections to the vertex. Numerical predictions are presented in the form of a fitting
formula as function of and , . For central input values, we obtain a relative correction of
, amounting
to about a quarter of the fermionic corrections, and corresponding to
. 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]