Kira - A Feynman Integral Reduction Program

In this article, we present a new implementation of the Laporta algorithm to reduce scalar multi-loop integrals---appearing in quantum field theoretic calculations---to a set of master integrals. We extend existing approaches by using an additional algorithm based on modular arithmetic to remove linearly dependent equations from the system of equations arising from integration-by-parts and Lorentz identities. Furthermore, the algebraic manipulations required in the back substitution are optimized. We describe in detail the implementation as well as the usage of the program. In addition, we show benchmarks for concrete examples and compare the performance to Reduze 2 and FIRE 5. In our benchmarks we find that Kira is highly competitive with these existing tools.Comment: 37 pages, 3 figure

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

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

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

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

Integral Reduction with Kira 2.0 and Finite Field Methods

We present the new version 2.0 of the Feynman integral reduction program Kira and describe the new features. The primary new feature is the reconstruction of the final coefficients in integration-by-parts reductions by means of finite field methods with the help of FireFly. This procedure can be parallelized on computer clusters with MPI. Furthermore, the support for user-provided systems of equations has been significantly improved. This mode provides the flexibility to integrate Kira into projects that employ specialized reduction formulas, direct reduction of amplitudes, or to problems involving linear system of equations not limited to relations among standard Feynman integrals. We show examples from state-of-the-art Feynman integral reduction problems and provide benchmarks of the new features, demonstrating significantly reduced main memory usage and improved performance w.r.t. previous versions of Kira

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]