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Multi-loop Scattering Amplitudes via Algebraic Geometry

Applicant Zhang Yang
Number 161341
Funding scheme Ambizione
Research institution
Institution of higher education Institution abroad - IACH
Main discipline Particle Physics
Start/End 01.09.2015 - 31.08.2018
Approved amount 377'520.00
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All Disciplines (2)

Discipline
Particle Physics
Mathematics

Keywords (3)

high energy physics; algebraic geometry; quantum field theory

Lay Summary (German)

Lead
Mein Forschungsprojekt „Multi-Loop-Streuamplituden über algebraische Geometrie“ ist an der Grenze zur Teilchenphysik und seiner Verbindung zur Mathematik. Ich benutze ein neues mathematisches Werkzeug, rechnerische algebraische Geometrie, um die Streuung der fundamentalen Teilchen zu berechnen.
Lay summary

Um neue Elementarteilchen und neue Physik-Gesetze zu finden, wurde der Large Hadron Collider (LHC) in der Nähe von Genf in der Schweiz gebaut. Am LHC werden Protonen auf extrem hohe Geschwindigkeit (etwa 99,9999991 % der Lichtgeschwindigkeit) beschleunigt und kollidieren dann, um andere Teilchen wie Quarks, Higgs-Bosonen oder mögliche neue Teilchen zu erzeugen. Die meisten der Streuprozesse erzeugen bekannte Teilchen, und wir nennen diese „den Hintergrund“. Um Anzeichen von neuen Teilchen zu finden, müssen wir den Hintergrund mit der „Feynman-Diagramm“-Methode in hoher Präzision berechnen.

Allerdings ist traditionell sehr schwierig, Feynman-Diagramme zu berechnen, weil es zu viele Integrale gibt. Die aktuelle Rechenleistung entspricht nicht den Anforderungen des LHC. 

Ich werde dieses Problem durch „rechnerische algebraische Geometrie“ lösen. Algebraische Geometrie ist ein Teilgebiet der Mathematik, das Studienobjekte aus Kurven zu abstrahierten Varianten überführt. Es löst komplizierte nichtlineare Beschränkungen, deshalb werde ich es einsetzen, um (1) die Anzahl der Feynman-Diagramme zu reduzieren und (2) jedes der Feynman-Diagramme zu vereinfachen. Das Ziel ist es, ein hocheffizientes Streuungsberechnungsverfahren zu entwerfen. Nach dem Abschluss dieses Projekts erwarte ich, dass unsere Fähigkeit zur Berechnung den LHC-Anforderungen entsprechen wird.

 

Ich erwarte, dass dieses Projekt die Forschung in der Mathematik stärken wird. Es bietet viele interessante Beispiele für algebraische Geometrie. Die für dieses Projekt entwickelten Algorithmen werden hilfreich für andere Wissenschaftszweige wie beispielsweise die statistische Physik,die Kryptographie und die Robotik sein. 

 

Direct link to Lay Summary Last update: 16.09.2015

Responsible applicant and co-applicants

Employees

Publications

Publication
Differential equations for loop integrals without squared propagators
Larsen Kasper, Bosma Jorrit, Zhang Yang (2018), Differential equations for loop integrals without squared propagators, in Loops and Legs in Quantum Field Theory, St. Goar, GermanyLoops and Legs in Quantum Field Theory, St. Goar, Germany.
Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections
Böhm Janko, Georgoudis Alessandro, Larsen Kasper J., Schönemann Hans, Zhang Yang (2018), Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections, in Journal of High Energy Physics, 2018(9), 24-24.
Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals
Böhm Janko, Georgoudis Alessandro, Larsen Kasper J., Schulze Mathias, Zhang Yang (2018), Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals, in Physical Review D, 98(2), 025023-025023.
Differential equations for loop integrals in Baikov representation
Bosma Jorrit, Larsen Kasper J., Zhang Yang (2018), Differential equations for loop integrals in Baikov representation, in Physical Review D, 97(10), 105014-105014.
Maximal cuts in arbitrary dimension
Jorrit Bosma, Mads Sogaard, Yang Zhang (2017), Maximal cuts in arbitrary dimension, in Journal of High Energy Physics, 051.
Scattering amplitudes via computational algebtraic geometry
Zhang Yang (2016), Scattering amplitudes via computational algebtraic geometry, in Computeralgebra-Rundbrief, 58, 9-13.
The Polynomial Form of the Scattering Equations is an H-Basis
Bosma Jorrit, Søgaard Mads, Zhang Yang (2016), The Polynomial Form of the Scattering Equations is an H-Basis, in Phys. Rev., D94(4), 041701-041701.
Integration-by-parts reductions from unitarity cuts and algebraic geometry
Larsen Kasper J., Zhang Yang (2015), Integration-by-parts reductions from unitarity cuts and algebraic geometry, in Phys. Rev., D93(4), 041701-041701.
Scattering Equations and Global Duality of Residues
Søgaard Mads, Zhang Yang (2015), Scattering Equations and Global Duality of Residues, in Phys. Rev., D93(10), 105009-105009.
Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves
Georgoudis Alessandro, Zhang Yang (2015), Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves, in JHEP, 12, 086-086.
Azurite: An algebraic geometry based package for finding bases of loop integrals
Alessandro Georgoudis, Kasper J. Larsen, and Yang Zhang, Azurite: An algebraic geometry based package for finding bases of loop integrals, in Computer Physics Communications, 1.
Integration-by-parts reductions from the viewpoint of computational algebraic geometry
Larsen Kasper, Zhang Yang, Integration-by-parts reductions from the viewpoint of computational algebraic geometry, in PoS, LL2016, 29.

Collaboration

Group / person Country
Types of collaboration
CEA Saclay France (Europe)
- in-depth/constructive exchanges on approaches, methods or results
SLAC United States of America (North America)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Uppsala University Sweden (Europe)
- in-depth/constructive exchanges on approaches, methods or results
University of Kaiserslautern Germany (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
CERN Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
Loopfest 2018 Talk given at a conference Integration-by-parts reduction via algebraic geometry methods 17.07.2018 Lansing, Michigan, United States of America Zhang Yang;
Loopfest XVI Talk given at a conference AZURITE: a package to determine master integrals via computational algebraic geometry 31.05.2017 Argonne National Lab, USA, United States of America Zhang Yang;
Scattering Amplitudes and Beyond Talk given at a conference Integral reduction and differnetial equations in Baikov representation 08.05.2017 Santa Barbara, USA, United States of America Zhang Yang;
Amplitudes 2016 International Conference Talk given at a conference Integration-By-Parts Reduction from Unitarity, an Algebraic Geometry Story 04.07.2016 Stockholm, Sweden Zhang Yang;
LHC Run II and the Precision Frontier Talk given at a conference Integral reduction via Unitarity and Algebraic Geometry 23.05.2016 Santa Barbara, CA, United States of America Zhang Yang;
MHV@30: Amplitudes and Modern Applications Talk given at a conference Integral Reduction via Tangent Algebra of Affine Varieties 16.03.2016 Fermilab, United States of America Zhang Yang;
Amplitudes in Asia 2015 Talk given at a conference Two-loop integral reduction via unitarity and algebraic geometry 02.11.2015 Taipei, Taiwan Zhang Yang;


Awards

Title Year
Central Committee of the Chinese Communist Party, 1000 Youth Talents Program 2018
Collaborative Research Center SFB Fellowship, Hamburg University, 2017 2017
Physics review D, highlighted article by the editor http://journals.aps.org/prd/issues/94/4 2016

Abstract

I work with the support of Ambizione grant from Swiss national science foundation, on the research project “Multi-loop Scattering Amplitudes via Algebraic Geometry”, which aims at developing new highly efficient methods for calculating the particle scattering processes in high energy physics, especially for Large Hadron Collider (LHC). The key of my idea is to apply the new mathematical method, computational algebraic geometry.The goal of high energy physics is to find new particles and new physical laws in the high energy regime, which is reached experimentally by particle scattering processes on colliders. Scattering processes are quantitatively characterized by scattering amplitudes in quantum physics. On colliders, possible new physics signals are hidden in the dominating Standard Model (SM) background, hence to extract signals of new physics, we have to theoretically calculate the SM scattering amplitudes to high precision.Traditionally, scattering amplitudes are calculated by Feynman diagrams. However, the method becomes obscure in the multi-loop orders. Some SM scattering amplitudes, crucial for LHC Run II, are untouchable from Feynman diagram approach.In my viewpoint, the difficulty of Feynman diagram approach originates from the large number of complex variables for high-loop scattering processes. I believe that the powerful mathematical method, algebraic geometry, which is the ideal tool for multivariate complex analysis, will lead to highly efficient scattering amplitude calculation methods. In this research project, I will apply the mathematical tools of algebraic geometry, like multivariate complex analysis, Gröbner basis, Riemann-Roch theorem, differential Galois theory, etc, to study two-loop and three-loop amplitudes thoroughly, and to evaluate several important multi-loop amplitudes for LHC Run II. I am also going to publish multi-loop amplitude public codes, to automate the amplitude calculation via algebraic geometry.
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