02 Mar 2021-Journal of High Energy Physics (Springer Science and Business Media LLC)-Vol. 2021, Iss: 3, pp 1-55

Abstract: We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers, and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.

... read more

Topics: Intersection (53%)

More

23 results found

•••

Abstract: 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 . New version program summary Program title: Kira CPC Library link to program files: https://doi.org/10.17632/v3cmsnfrnn.2 Developer's repository link: https://gitlab.com/kira-pyred/kira Licensing provisions: GNU General Public License 3 (GPL) Programming language: C++ Journal Reference of previous version: P. Maierhofer, J. Usovitsch and P. Uwer, Kira—A Feynman integral reduction program, Comput. Phys. Commun. 230 (2018) 99 [ 1705.05610 ]. Does the new version supersede the previous version?: Yes. Reasons for the new version: Implementation of new features, some of which improve the performance significantly for many problems. Summary of revisions: 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 [1,2]. This procedure can be parallelized on computer clusters with MPI . Further improvements include the expanded support for user-provided systems of equations as well as a new feature to reduce the main memory usage when generating the system of equations. Nature of problem: The reduction of Feynman integrals to a smaller set of master integrals is a central strategy for high precision calculations in theoretical particle physics, e.g. for cross sections. Furthermore, the reduction is a key ingredient in many methods to calculate the master integrals themselves. Solution method: Kira generates a system of equations employing integration-by-parts [3,4] and Lorentz-invariance identities [5] as well as symmetry relations . Linearly dependent equations are removed and master integrals are identified by solving the system over a finite field. The resulting system can be solved with two methods: Since version 1.0, Kira can solve the system algebraically with the help of Fermat [6]. New in this version is the strategy of solving the system many times over finite fields and reconstructing the master integral coefficients with the help of FireFly [1,2]. Both strategies can also be applied to arbitrary homogeneous linear systems of equations. References: [1] J. Klappert and F. Lange, Reconstructing rational functions with FireFly, Comput. Phys. Commun. 247 (2020) 106951 [ 1904.00009 ]. [2] J. Klappert, S. Y. Klein and F. Lange, Interpolation of dense and sparse rational functions and other improvements in FireFly, Comput. Phys. Commun. 264 (2021) 107968 [ 2004.01463 ]. [3] F. V. Tkachov, A theorem on analytical calculability of 4-loop renormalization group functions, Phys. Lett. B 100 (1981) 65. [4] K. G. Chetyrkin and F. V. Tkachov, Integration by parts: The algorithm to calculate β-functions in 4 loops, Nucl. Phys. B192 (1981) 159. [5] T. Gehrmann and E. Remiddi, Differential equations for two-loop four-point functions , Nucl. Phys. B580 (2000) 485 [ hep-ph/9912329 ]. [6] R. H. Lewis, Computer Algebra System Fermat , https://home.bway.net/lewis .

... read more

Topics: Integration by reduction formulae (52%), System of linear equations (50%)

45 Citations

•••

Abstract: The precision frontier in collider physics is being pushed at impressive speed, from both the experimental and the theoretical side. The aim of this review is to give an overview of recent developments in precision calculations within the Standard Model of particle physics, in particular in the Higgs sector. While the first part focuses on phenomenological results, the second part reviews some of the techniques which allowed the rapid progress in the field of precision calculations. The focus is on analytic and semi-numerical techniques for multi-loop amplitudes, however fully numerical methods as well as subtraction schemes for infrared divergent real radiation beyond NLO are also briefly described.

... read more

Topics: Higgs sector (50%)

44 Citations

•••

Abstract: We present an efficient method to shorten the analytic integration-by-parts (IBP) reduction coefficients of multi-loop Feynman integrals. For our approach, we develop an improved version of Leinartas’ multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular. Furthermore, we observe that for an integral basis with uniform transcendental (UT) weights, the denominators of IBP reduction coefficients with respect to the UT basis are either symbol letters or polynomials purely in the spacetime dimension D. With a UT basis, the partial fraction algorithm is more efficient both with respect to its performance and the size reduction. We show that in complicated examples with existence of a UT basis, the IBP reduction coefficients size can be reduced by a factor of as large as ∼ 100. We observe that our algorithm also works well for settings without a UT basis.

... read more

Topics: Partial fraction decomposition (54%), Reduction (complexity) (51%), Basis (linear algebra) (51%)

24 Citations

•••

Abstract: Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example square roots), although the final result may be expressed without algebraic extensions. In this article I present an improvement of this algorithm, which avoids algebraic extensions.

... read more

Topics: Algebraic number (59%), Vector space (54%), Algebraic geometry (54%) ... read more

21 Citations

•••

Abstract: Canonical Feynman integrals are of great interest in the study of scattering amplitudes at the multi-loop level. We propose to construct dlog-form integrals of the hypergeometric type, treat them as a representation of Feynman integrals, and project them into master integrals using intersection theory. This provides a constructive way to build canonical master integrals whose differential equations can be solved easily. We use our method to investigate both the maximally cut integrals and the uncut ones at one and two loops, and demonstrate its applicability in problems with multiple scales.

... read more

Topics: Intersection theory (50%), Representation (mathematics) (50%)

19 Citations

More

96 results found

••

Abstract: The following statement is proved: the counterterm for an arbitrary 4-loop Feynman diagram in an arbitrary model is calculable within the minimal subtraction scheme in terms of rational numbers and the Riemann ζ-function in a finite number of steps via a systematic “algebraic” procedure involving neither integration of elementary, special, or any other functions, nor expansions in and summation of infinite series of any kind. The number of steps is a rapidly increasing function of the complexity of the diagram. To demonstrate further possibilities offered by the technique we compute the 5-loop diagram contributing to the anomalous field dimension γ 2 ( g ) in the ϕ 4 model, that defied, heretofore, analytical calculation by other methods.

... read more

Topics: Algebraic number (56%), Field (mathematics) (55%), Rational number (54%) ... read more

1,639 Citations

•••

Abstract: We present a technique which utilizes unitarity and collinear limits to construct ansatze for one-loop amplitudes in gauge theory. As an example, we obtain the one-loop contribution to amplitudes for n -gluon scattering in N = 4 supersymmetric Yang-Mills theory with the helicity configuration of the Parke-Taylor tree amplitudes. We prove that our N = 4 ansatz is correct using general properties of the relevant one-loop n -point integrals. We also give the “splitting amplitudes” which govern the collinear behavior of one-loop helicity amplitudes in gauge theories.

... read more

Topics: MHV amplitudes (72%), Supersymmetric gauge theory (59%), Hamiltonian lattice gauge theory (58%) ... read more

1,203 Citations

•••

Abstract: We identify a large class of one-loop amplitudes for massless particles that can be constructed via unitarity from tree amplitudes, without any ambiguities. One-loop amplitudes for massless supersymmetric gauge theories fall into this class; in addition, many non-supersymmetric amplitudes can be rearranged to take advantage of the result. As applications, we construct the one-loop amplitudes for n -gluon scattering in N = 1 supersymmetric theories with the helicity configuration of the Parke-Taylor tree amplitudes, and for six-gluon scattering in N = 4 super-Yang-Mills theory for all helicity configurations.

... read more

Topics: MHV amplitudes (75%), Supersymmetric gauge theory (57%), Amplituhedron (56%) ... read more

1,041 Citations

•••

Abstract: We present a technique which utilizes unitarity and collinear limits to construct ansatze for one-loop amplitudes in gauge theory. As an example, we obtain the one-loop contribution to amplitudes for $n$ gluon scattering in $N=4$ supersymmetric Yang-Mills theory with the helicity configuration of the Parke-Taylor tree amplitudes. We prove that our $N=4$ ansatz is correct using general properties of the relevant one-loop $n$-point integrals. We also give the ``splitting amplitudes'' which govern the collinear behavior of one-loop helicity amplitudes in gauge theories.

... read more

Topics: Unitarity (57%), Gauge theory (55%), Helicity (51%) ... read more

880 Citations

•••

Abstract: One-loop amplitudes of gluons in N = 4 gauge theory can be written as linear combinations of known scalar box integrals with coefficients that are rational functions. In this paper we show how to use generalized unitarity to basically read off the coefficients. The generalized unitarity cuts we use are quadruple cuts. These can be directly applied to the computation of four-mass scalar integral coefficients, and we explicitly present results in next-to-next-to-MHV amplitudes. For scalar box functions with at least one massless external leg we show that by doing the computation in signature ( − − + + ) the coefficients can also be obtained from quadruple cuts, which are not useful in Minkowski signature. As examples, we reproduce the coefficients of some one-, two-, and three-mass scalar box integrals of the seven-gluon next-to-MHV amplitude, and we compute several classes of three-mass and two-mass-hard coefficients of next-to-MHV amplitudes to all multiplicities.

... read more

Topics: Unitarity (57%), Scalar (mathematics) (56%), MHV amplitudes (55%) ... read more

856 Citations