Multi-loop Integrals Research Articles

(Mis-)matching type-B anomalies on the Higgs branch

Building on [1], we uncover new properties of type-B conformal anomalies for Coulomb-branch operators in continuous families of 4D mathcal{N} = 2 SCFTs. We study a large class of such anomalies on the Higgs branch, where conformal symmetry is spontaneously broken, and compare them with their counterpart in the CFT phase. In Lagrangian the- ories, the non-perturbative matching of the anomalies can be determined with a weak coupling Feynman diagram computation involving massive multi-loop banana integrals. We extract the part corresponding to the anomalies of interest. Our calculations support the general conjecture that the Coulomb-branch type-B conformal anomalies always match on the Higgs branch when the IR Coulomb-branch chiral ring is empty. In the opposite case, there are anomalies that do not match. An intriguing implication of the mismatch is the existence of a second covariantly constant metric on the conformal manifold (other than the Zamolodchikov metric), which imposes previously unknown restrictions on its holonomy group.

Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space

We introduce an algebro-geometrically motived integration-by-parts (IBP) re- duction method for multi-loop and multi-scale Feynman integrals, using a framework for massively parallel computations in computer algebra. This framework combines the com- puter algebra system Singular with the workflow management system GPI-Space, which are being developed at the TU Kaiserslautern and the Fraunhofer Institute for Industrial Mathematics (ITWM), respectively. In our approach, the IBP relations are first trimmed by modern tools from computational algebraic geometry and then solved by sparse linear algebra and our new interpolation method. Modelled in terms of Petri nets, these steps are efficiently automatized and automatically parallelized by GPI-Space. We demonstrate the potential of our method at the nontrivial example of reducing two-loop five-point non- planar double-pentagon integrals. We also use GPI-Space to convert the basis of IBP reductions, and discuss the possible simplification of master-integral coefficients in a uni- formly transcendental basis.

Vector Space of Feynman Integrals and Multivariate Intersection Numbers.

Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for the construction of multivariate intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter. We apply it to the decomposition of a few Feynman integrals at one and two loops, as first steps toward potential applications to generic multiloop integrals. The proposed method can be more generally employed for the derivation of contiguity relations for special functions admitting multifold integral representations.

Function Theory for Multiloop Feynman Integrals

Precise predictions for collider observables require the computation of higher orders in perturbation theory. This task usually involves the evaluation of complicated multiloop integrals, which typically give rise to complicated special functions. This article discusses recent progress in understanding the mathematics underlying multiloop Feynman integrals and discusses a class of functions that generalizes the logarithm and that often appears in multiloop computations. The same class of functions is an active area of research in modern mathematics, which has led to the development of new powerful tools to compute Feynman integrals. These tools are at the heart of some of the most complicated computations ever performed for a hadron collider.

Two-loop triangle integrals with 4 scales for the HZV vertex

We calculate analytically the two-loop triangle integrals entering the $\mathcal{O}(\alpha\alpha_s)$ corrections to the $HZV$ vertex with $V=Z^*,\gamma^*$ using the method of differential equations. Our result provides a prototype to study the analytic properties of multi-loop multi-scale Feynman integrals, and also allows fast numeric evaluation for phenomenological studies. We apply our results to the leptonic decay of the Higgs boson and to $ZH$ production at electron-positron colliders. Besides the top quark loop, we include also the bottom quark loop contributions, whose evaluation takes a lot of time using purely numeric methods, but is very efficient with our analytic results.

Loop-Tree Duality for Multiloop Numerical Integration.

Loop-tree duality (LTD) offers a promising avenue to numerically integrate multiloop integrals directly in momentum space. It is well established at one loop, but there have been only sparse numerical results at two loops. We provide a formal derivation for a novel multiloop LTD expression and study its threshold singularity structure. We apply our findings numerically to a diverse set of up to four-loop finite topologies with kinematics for which no contour deformation is needed. We also lay down the ground work for constructing such a deformation. Our results serve as an important stepping stone towards a generalized and efficient numerical implementation of LTD, which is applicable to the computation of virtual corrections.

A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec

The purely numerical evaluation of multi-loop integrals and amplitudes can be a viable alternative to analytic approaches, in particular in the presence of several mass scales, provided sufficient accuracy can be achieved in an acceptable amount of time. For many multi-loop integrals, the fraction of time required to perform the numerical integration is significant and it is therefore beneficial to have efficient and well-implemented numerical integration methods. With this goal in mind, we present a new stand-alone integrator based on the use of (quasi-Monte Carlo) rank-1 shifted lattice rules. For integrals with high variance we also implement a variance reduction algorithm based on fitting a smooth function to the inverse cumulative distribution function of the integrand dimension-by-dimension. Additionally, the new integrator is interfaced to pySecDec to allow the straightforward evaluation of multi-loop integrals and dimensionally regulated parameter integrals. In order to make use of recent advances in parallel computing hardware, our integrator can be used both on CPUs and CUDA compatible GPUs where available. Program summaryProgram Title: pySecDec, qmcProgram Files doi:http://dx.doi.org/10.17632/dnrkf5jxzh.2Licensing provisions: GNU General Public License v3Programming language: python, FORM, C++, CUDAExternal 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. 222 (2018) 313–326.Does the new version supersede the previous version?: YesNature of 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/GPU time.

A development of an accelerator board dedicated for multi-precision arithmetic operations and its application to Feynman loop integrals II

Evaluation of a wide variety of Feynman diagrams with multi-loop integrals and physical parameters and its comparison with high energy experiments are expected to investigate new physics beyond the Standard Model. We have been developing a direct computation method of multi-loop integrals of Feynman diagrams. One of features of our method is that we adopt the double exponential rule for numerical integrations which enables us to evaluate loop integrals with boundary singularities. Another feature is that in order to accelerate the numerical integrations with multi-precision calculations, we develop an accelerator system with Field Programmable Gate Array boards on which processing elements with dedicated logic for quadruple/hexuple/octuple precision arithmetic operations are implemented. In addition, we also develop a programming interface designed for easy use of the system. The development is continued for practical use of the system. We present the current development of our system, and the numerical results of higher-loop diagrams performed using our system.

Direct numerical computation and its application to the higher-order radiative corrections

The direct computation method(DCM) is developed to calculate the multi-loop amplitude for general masses and external momenta. The ultraviolet divergence is under control in dimensional regularization. In this paper we report on the progress of DCM to several scalar multi-loop integrals after the presentation in ACAT2016. Also the discussion is given on the application of DCM to physical 2-loop processes including numerator functions.

Rationalizing loop integration

We show that direct Feynman-parametric loop integration is possible for a large class of planar multi-loop integrals. Much of this follows from the existence of manifestly dual-conformal Feynman-parametric representations of planar loop integrals, and the fact that many of the algebraic roots associated with (e.g. Landau) leading singularities are automatically rationalized in momentum-twistor space — facilitating direct integration via partial fractioning. We describe how momentum twistors may be chosen non-redundantly to parameterize particular integrals, and how strategic choices of coordinates can be used to expose kinematic limits of interest. We illustrate the power of these ideas with many concrete cases studied through four loops and involving as many as eight particles. Detailed examples are included as supplementary material.

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 depen- dent 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. Program summaryProgram title: KiraProgram Files doi:http://dx.doi.org/10.17632/v3cmsnfrnn.1Licensing provisions: GPLv3Programming language:C++External routines/libraries used:Fermat [1], gateToFermat [2], GiNaC [3,4], yaml-cpp [5], zlib [6] and SQLite3 [7]Nature of problem: The reduction of Feynman integrals to master integrals leads in general to a system of equations which contains redundant, i.e. linearly dependent, equations. In particular, for multi-scale problems, the algebraic manipulation of these redundant equations can lead to a substantial increase in runtime and memory consumption without affecting the results.Solution method: The program identifies linearly dependent relations based on modular arithmetic with the help of an algorithm presented in Ref. [8]. Afterwards the program brings a linearly independent system of equations in a triangular form. Furthermore, the algebraic manipulations required in the back substitution are optimized.Restrictions: the CPU time and the available RAM

A systematic and efficient method to compute multi-loop master integrals

We propose a novel method to compute multi-loop master integrals by constructing and numerically solving a system of ordinary differential equations, with almost trivial boundary conditions. Thus it can be systematically applied to problems with arbitrary kinematic configurations. Numerical tests show that our method can not only achieve results with high precision, but also be much faster than the only existing systematic method sector decomposition. As a by product, we find a new strategy to compute scalar one-loop integrals without reducing them to master integrals.

Regularization with numerical extrapolation for finite and UV-divergent multi-loop integrals

We give numerical integration results for Feynman loop diagrams such as those covered by Laporta (2000) and by Baikov and Chetyrkin (2010), and which may give rise to loop integrals with UV singularities. We explore automatic adaptive integration using multivariate techniques from the ParInt package for multivariate integration, as well as iterated integration with programs from the Quadpack package, and a trapezoidal method based on a double exponential transformation. ParInt is layered over MPI (Message Passing Interface), and incorporates advanced parallel/distributed techniques including load balancing among processes that may be distributed over a cluster or a network/grid of nodes. Results are included for 2-loop vertex and box diagrams and for sets of 2-, 3- and 4-loop self-energy diagrams with or without UV terms. Numerical regularization of integrals with singular terms is achieved by linear and non-linear extrapolation methods.

Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA

The integration of differential equations of Feynman integrals can be greatly facilitated by using a canonical basis. This paper presents the Mathematica package CANONICA, which implements a recently developed algorithm to automatize the transformation to a canonical basis. This represents the first publicly available implementation suitable for differential equations depending on multiple scales. In addition to the presentation of the package, this paper extends the description of some aspects of the algorithm, including a proof of the uniqueness of canonical forms up to constant transformations. Program summaryProgram Title: CANONICAProgram Files doi:http://dx.doi.org/10.17632/fmwnmmhn77.1Licensing provisions: GNU General Public License version 3Programming language: Wolfram Mathematica, version 10 or higherNature of problem: Computation of a rational basis transformation of master integrals leading to a canonical form of the corresponding differential equation.Solution method: The transformation law is expanded in the dimensional regulator. The resulting differential equations for the expansion coefficients of the transformation are solved with a rational ansatz.

FeynHelpers: Connecting FeynCalc to FIRE and Package-X

We present a new interface called FeynHelpers that connects FeynCalc, a Mathematica package for symbolic semi-automatic evaluation of Feynman diagrams and calculations in quantum field theory (QFT) to Package-X and FIRE. The former provides a library of analytic results for scalar 1-loop integrals with up to 4 legs, while the latter is a general-purpose tool for reduction of multi-loop scalar integrals using Integration-by-Parts (IBP) identities. Program summaryProgram Title: FeynHelpersProgram Files doi:http://dx.doi.org/10.17632/h5cfbhbbnc.1Licensing provisions: GNU Public License 3Programming language: Wolfram Mathematica 8 and higherExternal routines/libraries: FeynCalc [1,2], FeynArts [3], FeynRules [4], Package-X [5], FIRE [6]Nature of problem: FeynCalc is missing built-in capabilities to provide analytic results for scalar 1-loop integrals and to reduce multi-loop integrals using Integration-by-Parts (IBP) identities. These short-comings limit the usefulness of the package for the fully analytic evaluation of Feynman diagrams.Solution method: An easy-to-use interface implemented in Wolfram Mathematica seamlessly integrates two other Mathematica packages (Package-X and FIRE) into FeynCalc.Restrictions: The interface to FIRE currently misses the ability to recognize loop integrals that belong to the same topology, which means that each integral is processed separately. Furthermore, starting and stopping parallel kernels requires around 1.5 seconds per integral, which can be too slow, when hundreds of integrals are involved. [1]R. Mertig, M. Böhm, and A. Denner, Feyn Calc - Computer-algebraic calculation of Feynman amplitudes, Comput. Phys. Commun., 64, 345–359, (1991).[2]V. Shtabovenko, R. Mertig, and F. Orellana, New Developments in FeynCalc 9.0, Comput. Phys. Commun., 207, 432–444, (2016), arXiv:1601.01167.[3]T. Hahn, Generating Feynman Diagrams and Amplitudes with FeynArts 3, Comput. Phys. Commun., 140, 418–431, (2001), arXiv:hep-ph/0012260.[4]N. D. Christensen and C. Duhr, FeynRules - Feynman rules made easy, Comput. Phys. Commun., 180, 1614–1641, (2008), arXiv:0806.4194.[5]H. H. Patel, Package-X: A Mathematica package for the analytic calculation of one-loop integrals, Comput. Phys. Commun., 197, 276–290, (2015), arXiv:1503.01469.[6]A. V. Smirnov and V. A. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, Comput. Phys. Commun., 184, 2820–2827, (2013), arXiv:1302.5885.

Transforming differential equations of multi-loop Feynman integrals into canonical form

The method of differential equations has been proven to be a powerful tool for the computation of multi-loop Feynman integrals appearing in quantum field theory. It has been observed that in many instances a canonical basis can be chosen, which drastically simplifies the solution of the differential equation. In this paper, an algorithm is presented that computes the transformation to a canonical basis, starting from some basis that is, for instance, obtained by the usual integration-by-parts reduction techniques. The algorithm requires the existence of a rational transformation to a canonical basis, but is otherwise completely agnostic about the differential equation. In particular, it is applicable to problems involving multiple scales and allows for a rational dependence on the dimensional regulator. It is demonstrated that the algorithm is suitable for current multi-loop calculations by presenting its successful application to a number of non-trivial examples.

On the maximal cut of Feynman integrals and the solution of their differential equations

The standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try to solve them as a Laurent series in ϵ=(4−d)/2, where d are the space–time dimensions. The differential equations are, in general, coupled and can be solved using Euler's variation of constants, provided that a set of homogeneous solutions is known. Given an arbitrary differential equation of order higher than one, there exists no general method for finding its homogeneous solutions. In this paper we show that the maximal cut of the integrals under consideration provides one set of homogeneous solutions, simplifying substantially the solution of the differential equations.

Alternative method of Reduction of the Feynman Diagrams to a set of Master Integrals

We propose a new set of Master Integrals which can be used as a basis for certain multiloop calculations in massless gauge field theories. In these theories we consider three-point Feynman diagrams with arbitrary number of loops. The corresponding multiloop integrals may be decomposed in terms of this set of the Master Integrals. We construct a new reduction procedure which we apply to perform this decomposition.

Numerical multi-loop calculations: tools and applications

We report on the development of tools to calculate loop integrals and amplitudes beyond one loop. In particular, we review new features of the program SecDec which can be used for the numerical evaluation of parametric integrals like multi-loop integrals.

Generalizations of polylogarithms for Feynman integrals

In this talk, we discuss recent progress in the application of generalizations of polylogarithms in the symbolic computation of multi-loop integrals. We briefly review the Maple program MPL which supports a certain approach for the computation of Feynman integrals in terms of multiple polylogarithms. Furthermore we discuss elliptic generalizations of polylogarithms which have shown to be useful in the computation of the massive two-loop sunrise integral.