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Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019)最新文献

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On the Application of Intersection Theory to Feynman Integrals: the univariate case 交点理论在费曼积分中的应用:单变量情况
Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019) Pub Date : 2021-02-02 DOI: 10.22323/1.383.0017
H. Frellesvig, L. Mattiazzi
{"title":"On the Application of Intersection Theory to Feynman Integrals: the univariate case","authors":"H. Frellesvig, L. Mattiazzi","doi":"10.22323/1.383.0017","DOIUrl":"https://doi.org/10.22323/1.383.0017","url":null,"abstract":"This document is a contribution to the proceedings of the MathemAmplitudes 2019 conference held in December 2019 in Padova, Italy. A key step in modern high energy physics scattering amplitudes computation is to express the latter in terms of a minimal set of Feynman integrals using linear relations. In this work we present an innovative approach based on intersection theory, in order to achieve this decomposition. This allows for the direct computation of the reduction, projecting integrals appearing in the scattering amplitudes onto an integral basis in the same fashion as vectors may be projected onto a vector basis. Specifically, we will derive and discuss few identities between maximally cut Feynman integrals, showing their direct decomposition. This contribution will focus on the univariate part of the story, with the multivariate generalisation being discussed in a different contribution by Gasparotto and Mandal.","PeriodicalId":173323,"journal":{"name":"Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019)","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129730182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
Applications of intersection numbers in physics 交点数在物理学中的应用
Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019) Pub Date : 2020-11-05 DOI: 10.22323/1.383.0021
S. Weinzierl
{"title":"Applications of intersection numbers in physics","authors":"S. Weinzierl","doi":"10.22323/1.383.0021","DOIUrl":"https://doi.org/10.22323/1.383.0021","url":null,"abstract":"In this review I discuss intersection numbers of twisted cocycles and their relation to physics. After defining what these intersection number are, I will first discuss a method for computing them. This is followed by three examples where intersection numbers appear in physics. These examples are: tree-level scattering amplitudes within the the CHY-formalism, reduction of Feynman integrals to master integrals and correlation functions on the lattice.","PeriodicalId":173323,"journal":{"name":"Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019)","volume":"283 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122106856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
Module Intersection for the Integration-by-Parts Reduction of Multi-Loop Feynman Integrals 多环Feynman积分分部积分化简的模交
Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019) Pub Date : 2020-10-14 DOI: 10.22323/1.383.0004
Dominik Bendle, J. Boehm, W. Decker, A. Georgoudis, F. Pfreundt, M. Rahn, Y. Zhang
{"title":"Module Intersection for the Integration-by-Parts Reduction of Multi-Loop Feynman Integrals","authors":"Dominik Bendle, J. Boehm, W. Decker, A. Georgoudis, F. Pfreundt, M. Rahn, Y. Zhang","doi":"10.22323/1.383.0004","DOIUrl":"https://doi.org/10.22323/1.383.0004","url":null,"abstract":"In this manuscript, which is to appear in the proceedings of the conference \"MathemAmplitude 2019\" in Padova, Italy, we provide an overview of the module intersection method for the the integration-by-parts (IBP) reduction of multi-loop Feynman integrals. The module intersection method, based on computational algebraic geometry, is a highly efficient way of getting IBP relations without double propagator or with a bound on the highest propagator degree. In this manner, trimmed IBP systems which are much shorter than the traditional ones can be obtained. We apply the modern, Petri net based, workflow management system GPI-Space in combination with the computer algebra system Singular to solve the trimmed IBP system via interpolation and efficient parallelization. We show, in particular, how to use the new plugin feature of GPI-Space to manage a global state of the computation and to efficiently handle mutable data. Moreover, a Mathematica interface to generate IBPs with restricted propagator degree, which is based on module intersection, is presented in this review.","PeriodicalId":173323,"journal":{"name":"Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019)","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124854451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 11
Computing cohomology intersection numbers of GKZ hypergeometric systems GKZ超几何系统的上同交数计算
Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019) Pub Date : 2020-08-07 DOI: 10.22323/1.383.0013
Saiei-Jaeyeong Matsubara-Heo
{"title":"Computing cohomology intersection numbers of GKZ hypergeometric systems","authors":"Saiei-Jaeyeong Matsubara-Heo","doi":"10.22323/1.383.0013","DOIUrl":"https://doi.org/10.22323/1.383.0013","url":null,"abstract":"In this review article, we report on some recent advances on the computational aspects of cohomology intersection numbers of GKZ systems developed in cite{GM}, cite{MH}, cite{MT} and cite{MT2}. We also discuss the relation between intersection theory and evaluation of an integral of a product of powers of absolute values of polynomials.","PeriodicalId":173323,"journal":{"name":"Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019)","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126694038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
Building blocks of closed and open string amplitudes 闭弦和开弦振幅的组成部分
Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019) Pub Date : 2020-07-17 DOI: 10.22323/1.383.0022
P. Vanhove, Federico Zerbini
{"title":"Building blocks of closed and open string amplitudes","authors":"P. Vanhove, Federico Zerbini","doi":"10.22323/1.383.0022","DOIUrl":"https://doi.org/10.22323/1.383.0022","url":null,"abstract":"In this text we review various relations between building blocks of closed and open string amplitudes at tree-level and genus one. We explain that KLT relations between tree-level closed and open string amplitudes follow from the holomorphic factorisation of conformal correlation functions on conformal blocks. We give a simple hands-on evaluation of the $alpha'$-expansion of tree-level closed string amplitudes displaying the special single-valued nature of the coefficients. We show that the same techniques can be used also at genus-one, where we give a new proof of the single-valued nature of the coefficients of 2-point closed string amplitudes. We conclude by giving an overview of some open problems.","PeriodicalId":173323,"journal":{"name":"Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019)","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131553449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
A double integral of dlog forms which is not polylogarithmic 不是多对数的对数形式的二重积分
Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019) Pub Date : 2020-06-16 DOI: 10.22323/1.383.0005
F. Brown, C. Duhr
{"title":"A double integral of dlog forms which is not polylogarithmic","authors":"F. Brown, C. Duhr","doi":"10.22323/1.383.0005","DOIUrl":"https://doi.org/10.22323/1.383.0005","url":null,"abstract":"Feynman integrals are central to all calculations in perturbative Quantum Field Theory. They often give rise to iterated integrals of dlog-forms with algebraic arguments, which in many cases can be evaluated in terms of multiple polylogarithms. This has led to certain folklore beliefs in the community stating that all such integrals evaluate to polylogarithms. Here we discuss a concrete example of a double iterated integral of two dlog-forms that evaluates to a period of a cusp form. The motivic versions of these integrals are shown to be algebraically independent from all multiple polylogarithms evaluated at algebraic arguments. From a mathematical perspective, we study a mixed elliptic Hodge structure arising from a simple geometric configuration in $mathbb{P}^2$, consisting of a modular plane elliptic curve and a set of lines which meet it at torsion points, which may provide an interesting worked example from the point of view of periods, extensions of motives, and L-functions.","PeriodicalId":173323,"journal":{"name":"Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019)","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115761945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 27
Status of Intersection Theory and Feynman Integrals 交理论与费曼积分的研究现状
Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019) Pub Date : 2020-02-24 DOI: 10.22323/1.383.0016
Sebastian Mizera
{"title":"Status of Intersection Theory and Feynman Integrals","authors":"Sebastian Mizera","doi":"10.22323/1.383.0016","DOIUrl":"https://doi.org/10.22323/1.383.0016","url":null,"abstract":"We give a pedagogical review of the recently-introduced notion of a \"scalar product\" between Feynman integrals and how it helps us understand the analytic structure of the perturbative S-matrix. (This article is a contribution to the proceedings of the workshop \"MathemAmplitudes 2019: Intersection Theory and Feynman Integrals\" held in Padova, Italy on 18-20 December 2019.)","PeriodicalId":173323,"journal":{"name":"Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019)","volume":"225 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131645714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 36
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