Feynman amplitudes and Landau singularities for 1-loop graphs

A strategy for the off-shell analysis of collinear singularities in Feynman diagrams

In my PHD thesis I present a method for the off-shell singularity analysis of Feynman amplitudes based on the Speer sector decomposition of the Schwinger parametric integrals combined with the Mellin-Barnes tranform. I apply the method to one-loop corrections to Deep Inelastic Scattering.

主朗道行列式

Principal Landau Determinants

本研究为推进粒子物理计算，结合多面体几何等工具提出计算朗道奇点的新算法，定义费曼图的主朗道行列式，适配GKZ框架，完成多类图与标准模型过程的计算，实现为开源包PLD.jl。

Kinematic singularities of Feynman integrals and principal A-determinants

We consider the analytic properties of Feynman integrals from the perspective of general A-discriminants and A-hypergeometric functions introduced by Gelfand,Kapranov and Zelevinsky (GKZ). This enables us, to give a clear and mathematically rigour description of the singular locus, also known as Landau variety, via principal A-determinants. We also comprise a description of the various second type singularities. Moreover, by the Horn-Kapranov-parametrization we give a very efficient way to calculate a parametrization of Landau varieties. We furthermore present a new approach to study the sheet…

Landau Discriminants

Scattering amplitudes in quantum field theories have intricate analytic properties as functions of the energies and momenta of the scattered particles. In perturbation theory, their singularities are governed by a set of nonlinear polynomial equations, known as Landau equations, for each individual Feynman diagram. The singularity locus of the associated Feynman integral is made precise with the notion of the Landau discriminant, which characterizes when the Landau equations admit a solution. In order to compute this discriminant, we present approaches from classical elimination theory, as wel…

来自Whitney分层的朗道奇点

Landau Singularities from Whitney Stratifications

本文证明费曼积分的完整非冗余朗道奇点集可从特定映射的Whitney分层显式获得，并利用理论算法进展计算了两圈积分的例子。

A new and complete proof of the Landau condition for pinch singularities of Feynman graphs and other integrals

The Landau equations give a physically useful criterion for how singularities arise in Feynman amplitudes. Furthermore, they are fundamental to the uses of perturbative QCD, by determining the important regions of momentum space in asymptotic problems. Generalizations are also useful. We will show that in existing treatments there are significant gaps in derivations, and in some cases implicit assumptions that will be shown here to be false in important cases like the massless Feynman graphs ubiquitous in QCD applications. In this paper is given a new proof that the Landau condition is both ne…

费曼积分约简与兰道奇点

Feynman Integral Reduction and Landau Singularities

本文提出通过兰道方程解控制费曼积分约简，利用syzygy方法求解积分关系，为两环五点积分提供高效且物理透明的解法。

Leading singularities in Baikov representation and Feynman integrals with uniform transcendental weight

We provide a leading singularity analysis protocol in Baikov representation, for the searching of Feynman integrals with uniform transcendental (UT) weight. This approach is powered by the recent developments in rationalizing square roots and syzygy computations, and is particularly suitable for finding UT integrals with multiple mass scales. We demonstrate the power of our approach by determining the UT basis for a two-loop diagram with three external mass scales.

七点金字形神塔图的第一类Landau奇异性 I

Landau Singularities of the 7-Point Ziggurat I

本文计算了与七点金字形神塔图相关的四圈七点图的第一类Landau奇异性，发现其与平面N=4超杨-米尔斯理论中的七边形符号字母表子集完全一致，其余字母留待后续论文处理。

Recursive Graphical Construction for Feynman Diagrams and Their Weights in Ginzburg-Landau Theory

The free energy of the Ginzburg-Landau theory satisfies a nonlinear functional differential equation which is turned into a recursion relation. The latter is solved graphically order by order in the loop expansion to find all connected vacuum diagrams, and their corresponding weights. In this way we determine the connected vacuum diagrams and their weights up to four loops.

Cutkosky's Theorem for Massive One-Loop Feynman Integrals -- Part 1

We formulate and prove Cutkosky's Theorem regarding the discontinuity of Feynman integrals in the massive one-loop case up to the involved intersection index. This is done by applying the techniques to treat singular integrals developed in \cite{app-iso}. We write one-loop integrals as an integral of a holomorphic family of holomorphic forms over a compact cycle. Then, we determine at which points simple pinches occur and explicitly compute a representative of the corresponding vanishing sphere. This also yields an algorithm to compute the Landau surface of a one-loop graph without explicitly …

Feynman Integrals of Grassmannians

We embed Feynman integrals in the subvarieties of Grassmannians through homogenization of the integrands in projective space, then obtain GKZ-systems satisfied by those scalar integrals. The Feynman integral can be written as linear combinations of the hypergeometric functions of a fundamental solution system in neighborhoods of regular singularities of the GKZ-system, whose linear combination coefficients are determined by the integral on an ordinary point or some regular singularities. Taking some Feynman diagrams as examples, we elucidate in detail how to obtain the fundamental solution sys…

基于朗道奇点的费曼积分的紧致合冲关系

Compact Syzygies for Feynman Integrals from Landau Singularities

研究基于朗道奇点构建费曼积分的紧致合冲关系，通过强调合冲与朗道奇点联系，利用该见解构建合冲解，给出单圈示例，并应用于双圈五盒族展示方法威力。

引力中的Calabi-Yau Feynman积分：表观奇异性的$\varepsilon$-因子化形式

Calabi-Yau Feynman integrals in gravity: $\varepsilon$-factorized form for apparent singularities

本文研究了一个贡献于5PM 2SF黑洞散射的四圈Calabi-Yau Feynman积分，针对维度正规化中$\varepsilon$相关的表观奇异性，提出ansatz将其转化为$\varepsilon$-因子化形式并完成验证。

Towards Analytic Structure of Feynman Parameter Integrals with Rational Curves

We propose a strategy to study the analytic structure of Feynman parameter integrals where singularities of the integrand consist of rational irreducible components. At the core of this strategy is the identification of a selected stratum of discontinuities induced from the integral, together with a geometric method for computing their singularities on the principal sheet. For integrals that yield multiple polylogarithms we expect the data collected in this strategy to be sufficient for the construction of their symbols. We motivate this analysis by the Aomoto polylogarithms, and further check…

重访 Landau 奇点：用于 Feynman 积分的计算代数几何

Landau Singularities Revisited: Computational Algebraic Geometry for Feynman Integrals

该研究利用计算代数几何提出主 Landau 行列式及相应算法，用于分类和计算 Feynman 积分的 Landau 奇点，并通过114个图和复杂2圈QCD过程验证。

朗道奇点与凸几何

Landau singularities and convex geometry

本文通过闵可夫斯基问题将朗道奇点与凸多胞形及正则细分相联系，揭示其与PL函数凸性及平面网的对偶关系。

Landau Singularities and Symbology: One- and Two-loop MHV Amplitudes in SYM Theory

We apply the Landau equations, whose solutions parameterize the locus of possible branch points, to the one- and two-loop Feynman integrals relevant to MHV amplitudes in planar $\mathcal{N}=4$ super-Yang-Mills theory. We then identify which of the Landau singularities appear in the symbols of the amplitudes, and which do not. We observe that all of the symbol entries in the two-loop MHV amplitudes are already present as Landau singularities of one-loop pentagon integrals.

Landau singularities and singularities of holonomic integrals of the Ising class

We consider families of multiple and simple integrals of the ``Ising class'' and the linear ordinary differential equations with polynomial coefficients they are solutions of. We compare the full set of singularities given by the roots of the head polynomial of these linear ODE's and the subset of singularities occurring in the integrals, with the singularities obtained from the Landau conditions. For these Ising class integrals, we show that the Landau conditions can be worked out, either to give the singularities of the corresponding linear differential equation or the singularities occurrin…

From tree- to loop-simplicity in affine Toda theories I: Landau singularities and their subleading coefficients

Various features of the even order poles appearing in the S-matrices of simply-laced affine Toda field theories are analysed in some detail. In particular, the coefficients of first- and second-order singularities appearing in the Laurent expansion of the S-matrix around a general $2N^{\rm th}$ order pole are derived in a universal way using perturbation theory at one loop. We show how to cut loop diagrams contributing to the pole into particular products of tree-level graphs that depend on the on-shell geometry of the loop; in this way, we recover the coefficients of the Laurent expansion aro…

通过机器学习揭示费曼积分中的奇点

Uncovering Singularities in Feynman Integrals via Machine Learning

本文提出基于符号回归的机器学习框架，提取多环费曼积分的完整符号字母表，揭示解析结构，为多环振幅分析提供通用工具。

基于Landau的Schubert分析

Landau-based Schubert analysis

本文提出基于Landau的Schubert分析方法，通过将几何结构与Landau奇异性联系，将Feynman积分的Landau奇异性提升为符号字母，并成功应用于平面N=4超Yang-Mills理论中振幅和形状因子的字母表研究。

Cluster patterns in Landau and Leading Singularities via the Amplituhedron

We advance the exploration of cluster-algebraic patterns in the building blocks of scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory. In particular we conjecture that, given a maximal cut of a loop amplitude, Landau singularities and poles of each Yangian invariant appearing in any representation of the corresponding Leading Singularities can be found together in a cluster. We check these adjacencies for all one-loop amplitudes up to 9 points. Along the way, we also prove that all (rational) N$^2$MHV Yangian invariants are cluster adjacent, confirming original conjectures.

Hypergeometric Feynman Integrals

In this thesis we will study Feynman integrals from the perspective of A-hypergeometric functions, a generalization of hypergeometric functions which goes back to Gelfand, Kapranov, Zelevinsky (GKZ) and their collaborators. This point of view was recently initiated by the works [74] and [150]. Inter alia, we want to provide here a concise summary of the mathematical foundations of A-hypergeometric theory in order to substantiate this viewpoint. This overview will concern aspects of polytopal geometry, multivariate discriminants as well as holonomic D-modules. As we will subsequently show, ever…

Reply to comment on "Lost in translation: topological singularities in group field theory"

In [1, arXiv:1102.1844] the author disputes the conclusion of our paper [2, arXiv:1006.0714]. He claims that the Feynman graphs of three dimensional group field theory always represent pseudo manifolds. However, [1] uses a different definition for pseudo manifolds. also, in order to apply the new definition, [1] proposes a construction which cannot be implemented in a path integral by Feynman rules. These two points invalidate the claims of [1].

七点之字形图II的朗道奇异性

Landau Singularities of the 7-Point Ziggurat II

本文通过求解朗道方程分析了九个三圈七点图的奇异性，发现Y-Δ等价性在某些解分支上失效，并识别出两个奇异性超出七边形符号字母表且非Gr(4,7)簇变量的图，进而与N=4超杨-米尔斯理论中的最大留数比较以探究其振幅抵消机制。

Landau奇点与高阶根

Landau Singularities and Higher-Order Roots

本文研究了Feynman积分中存在高阶多项式根奇点与Landau仅允许三种奇点类型之间的悖论，通过具体例子说明高阶奇点需通过高余维度运动学极限逼近，从而规避Landau论证。

任意时空维度中的Landau奇点与主导奇点

Landau and leading singularities in arbitrary space-time dimensions

本文利用多维留数理论证明任意维度单圈Feynman积分中Landau奇点与主导奇点的关系，并推广到多圈情形及正则微分方程。

正性与聚类结构在兰道分析中

Positivity and Cluster Structures in Landau Analysis

本文研究了兰道分析中正性和聚类结构的数学基础，通过定义计数Leading singularities的枚举不变量，揭示了Leading Landau奇点的递归机制，并证明了大规模Landau图在任意环数下的正性和聚类变量分解。

Landau-Ginzburg/Calabi-Yau correspondence for a complete intersection via matrix factorizations

By generalizing the Landau-Ginzburg/Calabi-Yau correspondence for hypersurfaces, we can relate a Calabi-Yau complete intersection to a hybrid Landau-Ginzburg model: a family of isolated singularities fibered over a projective line. In recent years Fan, Jarvis, and Ruan have defined quantum invariants for singularities of this type, and Clader and Clader-Ross have provided a equivalence between these invariants and Gromov-Witten invariants of complete intersections. For Calabi-Yau complete intersections of two cubics, we show that this equivalence is directly related - via Chen character - to t…

Landau Singularities from the Amplituhedron

We propose a simple geometric algorithm for determining the complete set of branch points of amplitudes in planar N=4 super-Yang-Mills theory directly from the amplituhedron, without resorting to any particular representation in terms of local Feynman integrals. This represents a step towards translating integrands directly into integrals. In particular, the algorithm provides information about the symbol alphabets of general amplitudes. We illustrate the algorithm applied to the one- and two-loop MHV amplitudes.

Lev Landau and the problem of singularities in cosmology

We consider different aspects of the problem of cosmological singularity such as the BKL oscillatory approach to singularity, the new features of the cosmological dynamics in the neighbourhood of the singularity in multidimensional and superstring cosmological models and their connections with such a modern branch of mathematics as infinite-dimensional Lie algebras. Besides, we consider some new types of cosmological singularities which were widely discussed during last decade after the discovery of the phenomenon of cosmic acceleration.

Phi4tools：Landau-Ginzburg-Wilson理论的费曼图汇编

Phi4tools: Compilation of Feynman diagrams for Landau-Ginzburg-Wilson theories

本文独立验证并扩展了三维四次标量场论的固定维数微扰计算至耦合常数第八阶，并通过Phi4tools软件包公开费曼图、数值及重整化群函数级数。

Causal representation of multi-loop Feynman integrands within the loop-tree duality

The numerical evaluation of multi-loop scattering amplitudes in the Feynman representation usually requires to deal with both physical (causal) and unphysical (non-causal) singularities. The loop-tree duality (LTD) offers a powerful framework to easily characterise and distinguish these two types of singularities, and then simplify analytically the underling expressions. In this paper, we work explicitly on the dual representation of multi-loop Feynman integrals generated from three parent topologies, which we refer to as Maximal, Next-to-Maximal and Next-to-Next-to-Maximal loop topologies. In…

The Hadamard Function and the Feynman Propagator in the AdS/CFT Correspondence

We construct the retarded Green function and the Hadamard function in the Lorentzian (d+1)-dimensional anti-de Sitter spacetime for the Poincaré coordinate by performing the mode integration directly. We explore the structure of singularities for the position-space Feynman propagator derived from them. The boundary scaling limits of the bulk Feynman propagator yield the bulk-boundary propagator and the boundary conformal correlation function with an extra factor.

有限费曼积分

Finite Feynman Integrals

本文提出基于朗道奇点理论构造有限费曼积分基的算法，以消除红外发散并分类渐逝积分，从而简化双圈四胶子散射振幅的计算。

朗道自举法中的新工具

New Tools in the Landau Bootstrap

介绍费曼积分解析结构的新进展，阐述两类新约束，它们在维数正规化各阶次成立，为朗道自举法提供新输入，借助费曼积分奇点和间断信息构建函数形式。

Spherical Contours, IR Divergences and the geometry of Feynman parameter integrands at one loop

Spherical contours introduced in \cite{SphericalContours} translate the concept of "discontinuity across a branch cut" to Feynman parameter space. In this paper, we further explore spherical contours and connect them to the computation of leading IR divergences of 1 loop graphs directly in Feynman parameter space. These spherical contours can be used to develop a Feynman parameter space analog of "Leading Singularities" of loop integrands which allows us to develop a method of determining Feynman parameter integrands with no reference to the momentum space loop integrand. Finally, we explore s…

格拉斯曼流形中的兰道分析

Landau Analysis in the Grassmannian

研究格拉斯曼流形中散射幅动量扭量的兰道分析，揭示其判别式与结果抗与格拉斯曼流形积中的Hurwitz和Chow形式关系，并探讨其度数、因子分解及动量区域特性。

来自在壳空间几何的序列不连续性约束

Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces

本文利用 Landau 方程的在壳空间几何与 Picard-Lefschetz 理论，提出超越 Steinmann 关系的 Feynman 积分序列不连续性约束，证明不相容 Landau 奇点对应的序列不连续性为零。

Elliptic Feynman integrals and pure functions

We propose a variant of elliptic multiple polylogarithms that have at most logarithmic singularities in all variables and satisfy a differential equation without homogeneous term. We investigate several non-trivial elliptic two-loop Feynman integrals with up to three external legs and express them in terms of our functions. We observe that in all cases they evaluate to pure combinations of elliptic multiple polylogarithms of uniform weight. This is the first time that a notion of uniform weight is observed in the context of Feynman integrals that evaluate to elliptic polylogarithms.

积分分析、主导奇点和超越多项对数之外的规范基

Integrand Analysis, Leading Singularities and Canonical Bases beyond Polylogarithms

本文探讨了超越多项对数之外的费曼积分的主导奇点与规范基的联系，通过引入复杂几何的主导奇点概念，展示了如何通过新的超越函数描述底层几何的周期，并通过例子展示其数学等价性。

递归朗道分析

Recursive Landau Analysis

本文提出一种利用幺正性原理的递归方法，可直接在运动学空间计算n点散射振幅的朗道奇点，对一大类费曼图能快速解析计算且超越现有技术，还包含标准模型中相关两圈及更高圈过程的新预测。

Notes on a singular Landau-Ginzburg family

We study some properties of a singular Landau-Ginzburg family characterized by the multi-variable superpotential $W=-X^{-1}(Y_1Y_2)^{n-1} + {1\over n} (Y_1Y_2)^n - Y_3Y_4$. We will argue that (the infra-red limit of) this theory describes the topological degrees of freedom of the $c=1$ string compactified at $n$ times the self-dual radius. We also briefly comment on the possible realization of these line singularities as singularities of Calabi-Yau manifolds.

Landau equations and asymptotic operation

The pinched/non-pinched classification of intersections of causal singularities of propagators in Minkowski space is reconsidered in the context of the theory of asymptotic operation as a first step towards extension of the latter to non-Euclidean asymptotic regimes. A highly visual distribution-theoretic technique of singular wave fronts is tailored to the needs of the theory of Feynman diagrams. Besides a simple derivation of the usual Landau equations in the case of the conventional singularities, the technique naturally extends to other types of singularities e.g. due to linear denominator…

Infrared singularities in Landau gauge Yang-Mills theory

We present a more detailed picture of the infrared regime of Landau gauge Yang-Mills theory. This is done within a novel framework that allows one to take into account the influence of finite scales within an infrared power counting analysis. We find that there are two qualitatively different infrared fixed points of the full system of Dyson-Schwinger equations. The first extends the known scaling solution, where the ghost dynamics is dominant and gluon propagation is strongly suppressed. It features in addition to the strong divergences of gluonic vertex functions in the previously considered…

运动学、簇代数与费曼积分

Kinematics, cluster algebras and Feynman integrals

本文研究了四维 conformal 费曼积分的平面运动学的簇代数，发现其作为 G(4,n) 簇代数的子代数，并通过八点三环积分示例展示簇代数对费曼积分奇点的编码作用。

朗道的利维坦

Landau's Leviathans

提出一种新方法，通过有限域计算费曼积分欧拉示性数下降点直接确定朗道奇点，适用于多圈多尺度积分，发现超出已知复杂度的新奇点。

Three-Dimensional Gorenstein Singularities and SU(3) Modular Invariants

Using N=2 Landau-Ginzburg theories, we examine the recent conjectures relating the SU(3) WZW modular invariants, finite subgroups of SU(3) and Gorenstein singularities. All isolated three-dimensional Gorenstein singularities do not appear to be related to any known Landau-Ginzburg theories, but we present some curious observations which suggest that the SU(3)_n/SU(2)xU(1) Kazama-Suzuki model may be related to a deformed geometry of C^3/Z_{n+3}xZ_{n+3}. The toric resolution diagrams of those particular singularities are also seen to be classifying the diagonal modular invariants of the SU(3)_n …

RG流中的奇点

Singularities in the RG flow

本文研究了RG流中因红外奇点导致微扰不相关算子变为相关的现象，以四费米子相互作用为例，揭示了无限多个模式对RG流的影响。

BPS and non-BPS states in a supersymmetric Landau-Ginzburg theory

We analyze the spectrum of the N=(2,2) supersymmetric Landau-Ginzburg theory in two dimensions with superpotential W=X^{n+2}-lambda X^2. We find the full BPS spectrum of this theory by exploiting the direct connection between the UV and IR limits of the theory. The computation utilizes results from the Picard-Lefschetz theory of singularities and its extension to boundary singularities. The additional fact that this theory is integrable requires that the BPS states do not close under scattering. This observation fixes the masses of non-BPS states as well.

Mirror symmetry for exceptional unimodular singularities

In this paper, we prove the mirror symmetry conjecture between the Saito-Givental theory of exceptional unimodular singularities on Landau-Ginzburg B-side and the Fan-Jarvis-Ruan-Witten theory of their mirror partners on Landau-Ginzburg A-side. On the B-side, we compute the genus-zero generating function from a perturbative formula of primitive forms introduced by the first three authors recently. This computation matches the orbifold-Grothendieck-Riemann-Roch and WDVV calculations in FJRW theory on the A-side. The coincidence of the full data at all genera is established by reconstruction tec…

椭圆领头奇点与正则被积函数

Elliptic leading singularities and canonical integrands

本文将零亏格下由整数领头奇点构造 dlog 被积函数的方法推广到椭圆一亏格情形，通过选取避免导数的第二类代数一形式，得到具有特殊微分方程且 ε 逐阶解为纯函数的 Feynman 积分。

ADE Singularities and Coset Models

We consider the compactification of the IIA string to (1+1) dimensions on non-compact 4-folds that are ALE fibrations. Supersymmetry requires that the compactification include 4-form fluxes, and a particular class of these models has been argued by Gukov, Vafa and Witten to give rise to a set of perturbed superconformal coset models that also have a Landau-Ginzburg description. We examine all these ADE models in detail, including the exceptional cosets. We identify which perturbations are induced by the deformation of the singularity, and compute the Landau-Ginzburg potentials exactly. We also…