{"title":"Principal Landau determinants","authors":"Claudia Fevola , Sebastian Mizera , Simon Telen","doi":"10.1016/j.cpc.2024.109278","DOIUrl":null,"url":null,"abstract":"<div><p>We reformulate the Landau analysis of Feynman integrals with the aim of advancing the state of the art in modern particle-physics computations. We contribute new algorithms for computing Landau singularities, using tools from polyhedral geometry and symbolic/numerical elimination. Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we define the principal Landau determinant of a Feynman diagram. We illustrate with a number of examples that this algebraic formalism allows to compute many components of the Landau singular locus. We adapt the GKZ framework by carefully specializing Euler integrals to Feynman integrals. For instance, ultraviolet and infrared singularities are detected as irreducible components of an incidence variety, which project dominantly to the kinematic space. We compute principal Landau determinants for the infinite families of one-loop and banana diagrams with different mass configurations, and for a range of cutting-edge Standard Model processes. Our algorithms build on the <span>Julia</span> package <span>Landau.jl</span> and are implemented in the new open-source package <span>PLD.jl</span> available at <span>https://mathrepo.mis.mpg.de/PLD/</span><svg><path></path></svg>.</p></div><div><h3>Program summary</h3><p><em>Program title:</em> <span>PLD.jl</span></p><p><em>CPC Library link to program files:</em> <span>https://doi.org/10.17632/7h5644mm4n.1</span><svg><path></path></svg></p><p><em>Developer's repository link:</em> <span>https://mathrepo.mis.mpg.de/PLD/</span><svg><path></path></svg></p><p><em>Licensing provisions:</em> Creative Commons by 4.0</p><p><em>Programming language:</em> <span>Julia</span></p><p><em>Supplementary material:</em> The repository includes the source code with documentation (PLD_code.zip), a jupyter notebook tutorial providing installation and usage instructions (PLD_notebook.zip), a database containing the output of our algorithm on 114 examples of Feynman integrals (PLD_database.zip).</p><p><em>Nature of problem:</em> A fundamental challenge in scattering amplitude is to determine the values of complexified kinematic invariants for which an amplitude can develop singularities. Bjorken, Landau, and Nakanishi wrote a system of polynomial constraints, nowadays known as the Landau equations. This project aims to rigorously revisit the Landau analysis of the singularity locus of Feynman integrals with a practical view towards explicit computations.</p><p><em>Solution method:</em> We define the principal Landau determinant (PLD), which is a variety inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ). We conjecture that it provides a subset of the singularity locus, and we implement effective algorithms to compute its defining equation explicitly.</p><p><em>References:</em> OSCAR <span>[1]</span>, HomotopyContinuation.jl <span>[2]</span>, Landau.jl <span>[3]</span></p></div>","PeriodicalId":285,"journal":{"name":"Computer Physics Communications","volume":null,"pages":null},"PeriodicalIF":7.2000,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computer Physics Communications","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0010465524002017","RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
We reformulate the Landau analysis of Feynman integrals with the aim of advancing the state of the art in modern particle-physics computations. We contribute new algorithms for computing Landau singularities, using tools from polyhedral geometry and symbolic/numerical elimination. Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we define the principal Landau determinant of a Feynman diagram. We illustrate with a number of examples that this algebraic formalism allows to compute many components of the Landau singular locus. We adapt the GKZ framework by carefully specializing Euler integrals to Feynman integrals. For instance, ultraviolet and infrared singularities are detected as irreducible components of an incidence variety, which project dominantly to the kinematic space. We compute principal Landau determinants for the infinite families of one-loop and banana diagrams with different mass configurations, and for a range of cutting-edge Standard Model processes. Our algorithms build on the Julia package Landau.jl and are implemented in the new open-source package PLD.jl available at https://mathrepo.mis.mpg.de/PLD/.
Program summary
Program title:PLD.jl
CPC Library link to program files:https://doi.org/10.17632/7h5644mm4n.1
Supplementary material: The repository includes the source code with documentation (PLD_code.zip), a jupyter notebook tutorial providing installation and usage instructions (PLD_notebook.zip), a database containing the output of our algorithm on 114 examples of Feynman integrals (PLD_database.zip).
Nature of problem: A fundamental challenge in scattering amplitude is to determine the values of complexified kinematic invariants for which an amplitude can develop singularities. Bjorken, Landau, and Nakanishi wrote a system of polynomial constraints, nowadays known as the Landau equations. This project aims to rigorously revisit the Landau analysis of the singularity locus of Feynman integrals with a practical view towards explicit computations.
Solution method: We define the principal Landau determinant (PLD), which is a variety inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ). We conjecture that it provides a subset of the singularity locus, and we implement effective algorithms to compute its defining equation explicitly.
References: OSCAR [1], HomotopyContinuation.jl [2], Landau.jl [3]
期刊介绍：
The focus of CPC is on contemporary computational methods and techniques and their implementation, the effectiveness of which will normally be evidenced by the author(s) within the context of a substantive problem in physics. Within this setting CPC publishes two types of paper.
Computer Programs in Physics (CPiP)
These papers describe significant computer programs to be archived in the CPC Program Library which is held in the Mendeley Data repository. The submitted software must be covered by an approved open source licence. Papers and associated computer programs that address a problem of contemporary interest in physics that cannot be solved by current software are particularly encouraged.
Computational Physics Papers (CP)
These are research papers in, but are not limited to, the following themes across computational physics and related disciplines.
mathematical and numerical methods and algorithms;
computational models including those associated with the design, control and analysis of experiments; and
algebraic computation.
Each will normally include software implementation and performance details. The software implementation should, ideally, be available via GitHub, Zenodo or an institutional repository.In addition, research papers on the impact of advanced computer architecture and special purpose computers on computing in the physical sciences and software topics related to, and of importance in, the physical sciences may be considered.
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