T. Zirke

SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop

Abstract SecDec is a program which can be used for the factorization of dimensionally regulated poles from parametric integrals, in particular multi-loop integrals, and the subsequent numerical evaluation of the finite coefficients. Here we present version 3.0 of the program, which has major improvements compared to version 2: it is faster, contains new decomposition strategies, an improved user interface and various other new features which extend the range of applicability. Program summary Program title: SecDec 3.0 Catalogue identifier: AEIR_v3_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEIR_v3_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 123828 No. of bytes in distributed program, including test data, etc.: 1651026 Distribution format: tar.gz Programming language: Wolfram Mathematica, perl, Fortran/C++. Computer: From a single PC to a cluster, depending on the problem. Operating system: Unix, Linux. RAM: Depending on the complexity of the problem Classification: 4.4, 5, 11.1. Catalogue identifier of previous version: AEIR_v2_1 Journal reference of previous version: Comput. Phys. Comm. 184(2013)2552 Does the new version supersede the previous version?: Yes Nature of problem: Extraction of ultraviolet and infrared singularities from parametric integrals appearing in higher order perturbative calculations in gauge theories. Numerical integration in the presence of integrable singularities (e.g. kinematic thresholds). Solution method: Algebraic extraction of singularities within dimensional regularization using iterated sector decomposition. This leads to a Laurent series in the dimensional regularization parameter, where the coefficients are finite integrals over the unit-hypercube. Those integrals are evaluated numerically by Monte Carlo integration. The integrable singularities are handled by choosing a suitable integration contour in the complex plane, in an automated way. Reasons for new version: • Improved user interface. • Additional new decomposition strategies. • Usage on a cluster is much improved. • Speed-up in numerical evaluation times. • Various new features (please see below). Summary of revisions: • Implementation of two new decompositions strategies based on a geometric algorithm. • Scans over large ranges of parameters are facilitated. • Linear propagators can be treated. • Propagators with negative indices are possible. • Interface to reduction programs like Reduze, Fire, LiteRed facilitated. • Option to use numerical integrator from Mathematica. • Using CQUAD for 1-dimensional integrals to improve speed of numerical evaluations. • Option to include epsilon-dependent dummy functions. Restrictions: Depending on the complexity of the problem, limited by memory and CPU time. Running time: Between a few seconds and several hours, depending on the complexity of the problem.

Higgs Boson Pair Production in Gluon Fusion at Next-to-Leading Order with Full Top-Quark Mass Dependence.

We present the calculation of the cross section and invariant mass distribution for Higgs boson pair production in gluon fusion at next-to-leading order (NLO) in QCD. Top-quark masses are fully taken into account throughout the calculation. The virtual two-loop amplitude has been generated using an extension of the program GoSam supplemented with an interface to Reduze for the integral reduction. The occurring integrals have been calculated numerically using the program SecDec. Our results, including the full top-quark mass dependence for the first time, allow us to assess the validity of various approximations proposed in the literature, which we also recalculate. We find substantial deviations between the NLO result and the different approximations, which emphasizes the importance of including the full top-quark mass dependence at NLO.

Full top quark mass dependence in Higgs boson pair production at NLO

A bstractWe study the effects of the exact top quark mass-dependent two-loop corrections to Higgs boson pair production by gluon fusion at the LHC and at a 100 TeV hadron collider. We perform a detailed comparison of the full next-to-leading order result to various approximations at the level of differential distributions and also analyse non-standard Higgs self-coupling scenarios. We find that the different next-to-leading order approximations differ from the full result by up to 50 percent in relevant differential distributions. This clearly stresses the importance of the full NLO result.We study the effects of the exact top quark mass-dependent two-loop corrections to Higgs boson pair production by gluon fusion at the LHC and at a 100 TeV hadron collider. We perform a detailed comparison of the full next-to-leading order result to various approximations at the level of differential distributions and also analyse non-standard Higgs self-coupling scenarios. We find that the different next-toleading order approximations differ from the full result by up to 50 percent in relevant differential distributions. This clearly stresses the importance of the full NLO result.

Top-quark mediated effects in hadronic Higgs-Strahlung

Novel contributions to the total inclusive cross section for Higgs-Strahlung in the Standard Model at hadron colliders are evaluated. Although formally of order

vh@nnlo - Higgs Strahlung at hadron colliders

\alpha_{s}^{2}

Gluon-induced Higgs-strahlung at next-to-leading order QCD

, they have not been taken into account in previous NNLO predictions. The terms under consideration are induced by Higgs radiation off top-quark loops and thus proportional to the top-quark Yukawa coupling. At the Tevatron, their effects to WH production are below 1% in the relevant Higgs mass range, while for ZH production, we find corrections between about 1% and 2%. At the LHC, the contribution of the newly evaluated terms to the cross section is typically of the order of 1%–3%. Based on these results, we provide updated predictions for the total inclusive Higgs-Strahlung cross section at the Tevatron and the LHC.

Higgs Strahlung at the Large Hadron Collider in the 2-Higgs-doublet model

A numerical program for the evaluation of the inclusive cross section for associated Higgs production with a massive weak gauge boson at hadron colliders is described, σ(pp/pp→HV), V∈{W,Z}. The calculation is performed in the framework of the Standard Model and includes next-to-next-to-leading order QCD as well as next-to-leading order electro-weak effects.

pySecDec: a toolbox for the numerical evaluation of multi-scale integrals

A bstractGluon-induced contributions to the associated production of a Higgs and a Z boson are calculated with NLO accuracy in QCD. They constitute a significant contribution to the cross section for this process. The perturbative correction factor (K-factor) is calculated in the limit of infinite top-quark and vanishing bottom-quark masses. The qualitative similarity of the results to the well-known ones for the gluon-fusion process gg → H allows to conclude that rescaling the LO prediction by this K-factor leads to a reliable NLO result and realistic error estimate due to missing higher-order perturbative effects. We consider the total inclusive cross section as well as a scenario with a boosted Higgs boson, where the Higgs boson’s transverse momentum is restricted to values pT,H> 200 GeV. In both cases, we find large correction factors K ≈ 2 in most of the parameter space.

Soft gluon resummation for gluon-induced Higgs Strahlung

A bstractWe present a calculation of all relevant contributions to associated production of a Higgs boson with a weak gauge boson in the 2-Higgs-doublet model (2HDM) at the LHC, pp → ϕ, with ϕ ∈ {h, H0, A} and V ∈ {W, Z}. While for the W ϕ mode, this mostly amounts to a simple rescaling of the Standard Model (SM) cross section, the Zϕ cross section depends on several 2HDM parameters. The ratio σWϕ/σZϕ, for which we present the currently most complete SM prediction, therefore appears to be a sensitive probe of possible New Physics effects. We study its numerical dependence on the top and bottom Yukawa couplings, including their sign. Furthermore, we consider the Wϕ/Zϕ ratio in exemplary 2HDM scenarios and briefly address the effects in the boosted regime. Analogous studies for other 2HDM scenarios will become possible with an upcoming version of the program vh@nnlo which incorporates the 2HDM effects.

Numerical multi-loop calculations: tools and applications

Abstract We present py SecDec , a new version of the program SecDec , which performs the factorization of dimensionally regulated poles in parametric integrals, and the subsequent numerical evaluation of the finite coefficients. The algebraic part of the program is now written in the form of python modules, which allow a very flexible usage. The optimization of the C++ code, generated using FORM , is improved, leading to a faster numerical convergence. The new version also creates a library of the integrand functions, such that it can be linked to user-specific codes for the evaluation of matrix elements in a way similar to analytic integral libraries. Program summary Program Title: pySecDec Program Files doi: http://dx.doi.org/10.17632/3y8bbz9c9v.1 Licensing provisions: GNU Public License v3 Programming language: python, FORM, C++ External routines/libraries: catch [1], gsl [2], numpy [3], sympy [4], Nauty [5], Cuba [6], FORM [7], Normaliz [8]. The program can also be used in a mode which does not require Normaliz. Journal reference of previous version: Comput. Phys. Commun. 196 (2015) 470–491. Nature of the problem: Extraction of ultraviolet and infrared singularities from parametric integrals appearing in higher order perturbative calculations in quantum field theory. Numerical integration in the presence of integrable singularities (e.g. kinematic thresholds). Solution method: Algebraic extraction of singularities within dimensional regularization using iterated sector decomposition. This leads to a Laurent series in the dimensional regularization parameter ϵ (and optionally other regulators), where the coefficients are finite integrals over the unit-hypercube. Those integrals are evaluated numerically by Monte Carlo integration. The integrable singularities are handled by choosing a suitable integration contour in the complex plane, in an automated way. The parameter integrals forming the coefficients of the Laurent series in the regulator(s) are provided in the form of libraries which can be linked to the calculation of (multi-) loop amplitudes. Restrictions: Depending on the complexity of the problem, limited by memory and CPU time. References: [1] https://github.com/philsquared/Catch/ . [2] http://www.gnu.org/software/gsl/ . [3] http://www.numpy.org/ . [4] http://www.sympy.org/ . [5] http://pallini.di.uniroma1.it/ . [6] T. Hahn, “CUBA: A Library for multidimensional numerical integration,” Comput. Phys. Commun. 168 (2005) 78 [hep-ph/0404043], http://www.feynarts.de/cuba/ . [7] J. Kuipers, T. Ueda and J. A. M. Vermaseren, “Code Optimization in FORM,” Comput. Phys. Commun. 189 (2015) 1 [arXiv:1310.7007], http://www.nikhef.nl/ form/ . [8] W. Bruns, B. Ichim, B. and T. Romer, C. Soger, “Normaliz. Algorithms for rational cones and affine monoids.” http://www.math.uos.de/normaliz/ .