第二届微扰量子场论研讨会

22 Aug 2022, 08:00 → 24 Aug 2022, 18:00 Asia/Shanghai

杭州

量子场论是结合了经典场论，相对论，和量子力学的统一框架，是物理学的普适语言，在粒子物理和核物理，宇宙学，弦论，凝聚态物理，数学物理等众多领域中有广泛应用。研究量子场论的一个主要方法是微扰论。本次研讨会聚焦于量子场论中微扰论方法的发展和应用，讨论主题包括：

● 散射振幅和费曼积分的计算

● QCD和电弱辐射修正

● 量子规范场论的红外结构

● QCD因子化，重求和，和幂次修正

● 高能喷注物理

● Parton Shower和蒙特卡洛模拟

● 微扰量子场论在其它领域中的应用

会议时间：2022年8月22-24日

会议报到日期：8月21日

会议结束日期：8月24日

注册费：教师1200，学生/博士后600

会议组织方：浙江大学，上海交通大学，中国科学院高能物理研究所，山东大学

会议主办方：浙江近代物理中心，浙江大学，国家自然科学基金量子-宇宙理论物理中心

会议承办方：浙江宾馆

会议地点：杭州市西湖区浙江宾馆

注册截止日期：8月12号

Poster poster.png

Monday, 22 August

08:50 → 09:00 会议开幕致辞

09:00 → 10:25 Plenary 甲

09:00 Improved PV-Reduction Method

Improved PV-Reduction Method

Speaker: Bo Feng (北京计算科学研究中心)

R-PV-Hangzhou.pdf

09:30 Changing dimension transformation of Feynman integral

将首次介绍我们正在发展的费曼积分的“改维变换”，它结合了辅助质量微分方程和时空维数差分方程。这一变换使费曼积分计算变得极为简洁。

Speaker: Yan-Qing Ma (Peking University)

YQMa20220822.pdf

10:00 Five-loop anomalous dimensions of large charge operators in a scalar theory with $O(N)$ symmetry

We compute the complete $Q$-dependence of anomalous dimensions of traceless symmetric tensor operator $\phi^Q$ in $O(N)$ scalar theory to five-loop. The renormalization factors are extracted from $\phi^Q\rightarrow Q\phi$ form factors, and the integrand of form factors are constructed with the help of unitarity cut method. The anomalous dimensions match the known results, where the leading and subleading terms in the large $Q$ limit were obtained using a semiclassical method.

Speaker: Qingjun Jin (ITP-CAS)

10:25 → 10:55 与会者合影+茶歇

10:55 → 12:10 Plenary 甲

10:55 Introduction to symbology and bootstrap (remote)

Introduction to symbology and bootstrap.

Speaker: Song He

11:25 Graphic Expansion of Amplitudes and BG Currents

Scattering amplitudes and Berends-Giele currents can be expanded by graphs which encodes kinematic information. By the used of graphic expansions, on-shell and off-shell BCJ numerators are constructed, new amplitude relations are introduced, scattering amplitude in four dimensions can be evaluated.

Speaker: 一剑 杜 (武汉大学)

杜一剑_2022杭州v1.pdf

11:50 费曼积分函数的迭代约化

高能物理实验精度对理论预言能力提出了新的要求，而在微扰场论的理论计算中基于分部积分（IBP）关系的积分约化是最为核心的步骤之一，随着圈数与标度的增加而逐渐变得耗费更多的计算资源。我们探究了基于Baikov表示与module intersection的迭代约化方法，来充分利用IBP关系组成的矩阵所具有的稀疏性和重复结构的特性进行约化。除了基于计算代数几何来给出迭代约化关系外，我们还发现所有单圈积分和部分类型任意圈的积分可以通过统一的迭代公式进行约化。对于多圈部分，我们比较了只使用传统的约化（使用软件包FIRE6）和迭代约化公式与传统约化相结合，发现了极为显著的效率提升。

Speaker: Dr Jiaqi Chen (北京计算科学研究中心)

Chen_Jiaqi.pptx

14:00 → 15:40 Plenary 乙

14:00 形状因子研究新进展

近些年散射振幅的研究表明，存在不同于传统费曼图的更有效的方式来理解量子场论和量子引力。其中重要的核心思想包括在壳性方法、bootstrap方法，以及一些新的对称性的应用。这些新思路可以发展应用于更一般的含有算符的形状因子等物理量，帮助研究算符分类和重整化、有效场论振幅计算、以及色动量对偶和平方关系等。在这一报告中，我们将介绍形状因子研究的一些新进展及其应用。

Speaker: Gang Yang (Institute of Theoretical Physics, CAS)

GangYang_杭州22.pdf

14:30 NLO Sector Showers and NNLO matching

We outline a new technique for the fully-differential matching of final-state parton showers to NNLO calculations, focussing here on the simplest case of leptonic collisions with two final-state jets. The strategy is facilitated by working in the antenna formalism, making use of NNLO antenna subtraction on the fixed-order side and the sector-antenna framework on the shower side. As long as the combined real-virtual and double-real corrections do not overcompensate the real-emission term in the three-jet region, negative weights can be eliminated from the matching scheme.

Speaker: Haitao Li (Shandong University)

haitao-hangzhou.pdf

14:55 Axial current renormalization and non-decoupling mass logarithms

In this talk, I will discuss the renormalization of an axial current associated with a given quark flavor, and the closely related renormalization formulae of the singlet contribution to an axial quark form factor, as well as the renormalization group equations which are used to resum the non-decoupling mass logarithms appearing in the axial quark form factors.

Speaker: Long Chen (Shandong University)

ChenLong.pdf

15:20 Transverse-momentum-dependent wave functions and Soft functions at one-loop in Large Momentum Effective Theory

In large-momentum effective theory (LaMET), the transverse-momentum-dependent (TMD) light-front wave functions and soft functions can be extracted from the simulation of a four-quark form factor and equal-time correlation functions. In this work, using expansion by regions we provide a one-loop proof of TMD factorization of the form factor. For the one-loop validation, we also present a detailed calculation of ${\cal O}(\alpha_s)$ perturbative corrections to these quantities, in which we adopt a modern technique for the calculation of TMD form factor based the integration by part and differential equation. The one-loop hard functions are then extracted. Using lattice data from Lattice Parton Collaboration on quasi-TMDWFs, we estimate the effects from the one-loop matching kernel and find that the perturbative corrections depend on the operator to define the form factor, but are less sensitive to the transverse separation. These results will be helpful to precisely extract the soft functions and TMD wave functions from the first-principle in future.

Speaker: Jun Zeng (SJTU)

JunZeng_第二届微扰量子场论研讨会-杭州.pdf

15:40 → 16:10 茶歇

16:10 → 17:40 Plenary 乙

16:10 Review on progress in scattering amplitudes and integral reduction

Review on progress in scattering amplitudes and integral reduction

Speaker: Yang Zhang (University of Science and Technology of China)

16:40 Reduction of Feynman Integrals in the Parametric Representation

We developed a systematic method to reduce Feynman integrals in the parametric representation. This method proves to be more efficient than the integration-by-part (IBP) method in the momentum space. Tensor integrals can directly be parametrized without performing tensor reductions. The integrands of parametric integrals are functions of Lorentz scalars, instead of four momenta. The complexity of a calculation is determined by the number of propagators that are present rather than the number of all the linearly independent propagators. Furthermore, the symmetries of Feynman integrals under permutations of indices are transparent in the parametric representation. This method can also be used to reduce integrals with theta functions.

Speaker: 文 陈 (浙大物理系)

Reduction_of_Feynman_Integrals_in_the_Parametric_Representation_Chen_Wen.pdf

17:00 Two-loop amplitudes for $tW$ production at Hardron colliders

We present the analytical results of the two-loop amplitudes for hadronic tW
production, focusing on the leading color and light fermion-loop contributions. The cal-
culation of the two-loop integrals is performed using the method of canonical differential
equations. The results have been expressed in terms of multiple polylogarithms and checked
by comparing the infra-red divergences with the predictions from anomalous dimensions.
Combined with the one-loop squared amplitudes we have obtained previously, we obtain
the hard function relevant for a NNLO Monte Carlo calculation.

Speaker: Dr 烨凡 王

wangyefan_HangZhou.pdf

17:20 NNLO corrections to double charmonium production at B factories

Exclusive double charmonium production at electron-positron collider is among the simplest hard exclusive reactions in perturbative QCD. The double charmonium production is first observed at two B factories at the beginning of this century and has stimulated long-lasting theoretical interests. The nonrelativistic QCD (NRQCD) factorization approach, a modern effective-field-theory-based approach that explicitly exploits the nonrelativistic nature of heavy quarkonium, provides a powerful and systematic tool kit to tackle double charmonium production processes. In this talk, I will introduce the framework and discuss the NNLO predictions.

Speaker: Jia-Yue Zhang (高能所)

NNLO_Corrections_to_Double_Charmonium_Production_at_B_Factories_169_Hangzhou.pdf

18:00 → 19:30 晚宴

Tuesday, 23 August

09:00 → 10:20 Plenary 丙

09:00 Physics at Electron Ion Colliders

Overview talk on physics at Electron Ion Colliders.

Speaker: Hongxi Xing (South China Normal University)

EicC_xing.pdf

09:30 有限大动量下研究部分子横向动量依赖部分子分布函数

在报告中，汇报在有限大动量下研究部分子横向动量依赖部分子分布函数的最新结果

Speaker: 瑞林 朱 (Nanjing Normal University)

2022hangzhou-RuilinZhu.pdf

09:55 Single inclusive hadron production in forward pA collisions

With the tremendous accomplishments of RHIC and the LHC experiments and the advent of the future Electron-Ion Collider on the horizon, the quest for compelling evidence of the color glass condensate (CGC) has become one of the most aspiring goals in the high energy Quantum Chromodynamics research. Pursuing this question requires developing the precision test of the CGC formalism. By systematically implementing the threshold resummation, we significantly improve the stability of the next-to-leading-order calculation in CGC for forward rapidity hadron productions in $pp$ and $pA$ collisions, especially in the high $p_T$ region, and obtain reliable descriptions of all existing data measured at RHIC and the LHC across all $p_T$ regions. Consequently, this technique can pave the way for the precision studies of the CGC next-to-leading-order predictions by confronting them with a large amount of precise data.

Reference

[1] Y. Shi, L. Wang, S.Y. Wei, B.W. Xiao, Phys. Rev. Lett. 128, no.20, 202302 (2022).

Speaker: Shu-yi Wei (Shandong University)

threshold_resummation.pdf

10:20 → 10:45 茶歇

10:45 → 12:15 Plenary 丙

10:45 高能强子对撞实验数据在PDF和质子结构研究中的应用

在过去数十年中，人们通过DIS、fix target等实验对质子结构开展了深入研究，并通过global analysis给出了部分子分布函数（PDF）。然而，由于数据统计量小、物理反应能量相对较低等因素的限制，相关研究更多集中在中大x区域（x=0.01 ~ 0.5）的价夸克（u、d）方面。对于其它各类夸克，包括ubar、dbar和s夸克，相应的实验观测则非常有限。开展对海夸克的研究、对小x区域部分子的研究，是近年来相关领域研究的前沿。

LHC数据具有统计量大、对撞能量高的特点，其物理反应覆盖了从$10^{-5}$到$10^{-1}$的x区间的各类价、海夸克。近年来，LHC上的W、Z物理过程截面测量与当前PDF结果出现了系统性的偏差，成为了相关领域的关注重点。然而，LHC上的实验观测是各种价、海夸克在不同x区域的信息的混合结果，仅通过简单的W、Z物理截面测量是不能在具体的夸克信息和质子结构层面给出明确结论的。同时，依赖矢量玻色子物理过程，也导致PDF观测和电弱物理之间产生了不可避免的强关联性。如何寻找新的实验观测量、如何解决PDF与电弱之间的强关联性、如何解释强子对撞实验数据与DIS、fix target实验数据之间的差异，是当前研究的重点内容，其结论也将对未来PDF global analysis和质子结构研究给出方向性指导。

本次报告将围绕上述问题，介绍近期的一系列工作。

Speaker: 思奇 杨 (University of Science and Technology of China)

11:15 Recent developments in the CTEQ-TEA global analysis: CT18CS and CT18As

In this talk, we will present two recent studies from CTEQ-TEA. These two studies show the development of parton distribution global analysis with the help of input from Lattice QCD.

Speaker: Tie-Jiun Hou (XinJiang U.)

2022.0823_PQFT_TJHou_v2.pdf

11:55 Nonperturbative Determination of Collins-Soper Kernel from Quasi Transverse-Momentum Dependent Wave Functions

In the framework of large-momentum effective theory at one-loop matching accuracy, we perform a lattice calculation of the Collins-Soper kernel which governs the rapidity evolution of transverse-momentum-dependent (TMD) distributions. We first obtain
the quasi TMD wave functions at three different meson momenta
on a lattice with valence clover quarks on a dynamical HISQ sea
and lattice spacing $a=0.12$fm from MILC, and renormalize the pertinent linear divergences using Wilson loops. Through one-loop matching to the light-cone wave functions, we determine the Collins-Soper kernel with transverse separation up to 0.6fm. We study the systematic uncertainties from operator mixing and scale dependence, as well as the impact from higher power corrections. Our results potentially allow for a determination of the soft function and other transverse-momentum dependent quantities at one-loop accuracy.

Speaker: 旻寰 楚 (Shanghai Jiao Tong University)

M.Chu_CS_kernel.pdf

14:00 → 15:30 Plenary 丁

14:00 New analyses of event shape observables and the determination of $\alpha_s$ in $e^{+}e^{-}$ annihilation

We give comprehensive analyses for event shape observables in electron-positron annihilation by using the Principle of Maximum Conformality (PMC). Conventionally the renormalization scale and theoretical uncertainties in event shape observables are often evaluated by setting the scale to the center-of-mass energy $\sqrt{s}$. The event shape distributions using this conventional scale setting are plagued by the large renormalization scale uncertainty and underestimate the experimental data. Moreover, since the renormalization scale is simply fixed to the center-of-mass energy $\sqrt{s}$, only one value of the coupling $\alpha_s$ at the single scale $\sqrt{s}$ can be extracted. In contrast, the PMC renormalization scales are determined by absorbing the nonconformal $\beta$ contributions that govern the behavior of the running coupling via the Renormalization Group Equation (RGE). The resulting PMC scales change with event shape kinematics, reflecting the virtuality of the underlying quark and gluon subprocess. The PMC scales thus yield the correct physical behavior of the scale and the PMC predictions agree with precise event shape distributions measured at the LEP experiment. More importantly, we can precisely determine the running of the QCD coupling constant $\alpha_s(Q^2)$ over a wide range of $Q^2$ in perturbative domain from event shape distributions measured at a single center-of-mass energy $\sqrt{s}$.

Speaker: 声权 王 (贵州民族大学)

event shapes_杭州.pdf

14:30 How to achieve precise prediction from a fixed-order perturbative QCD calculation ?

We suggest a novel single-scale setting approach under the principal of maximum conformality (PMC) for eliminating the conventional renormalization scheme-and-scale ambiguities following the idea of intrinsic conformality. It has been shown that the intrinsic conformality ensures the scale-invariance at each order, and vice versa. We call this newly suggested single-scale method as the PMC$_\infty$-s approach, in which an overall effective strong coupling constant, and hence an overall effective scale, is achieved by using all the known type of RG-involved non-conformal terms. Its resultant conformal series is scale-invariant and satisfies all the RG requirements. We also demonstrate that it is equivalent to the previously suggested PMC single-scale setting approach (PMCs), which further indicates that all the correctly RG-involved single-scale setting approaches are in fast the same. Finally, by using the Higgs decays to two gluons up to five-loop level as an explicit example, we show how the PMC$_\infty$-s works. The method is applicable to any perturbatively calculable physical quantities.

Speaker: Jiang Yan (重庆大学)

YanJiang.pptx

14:50 Two-loop master integrals for single-top production

The single-top production via electroweak interactions is important for studying the top quarks. One can further identify three channels in the single-top production mode, namely the t-channel, the tW associated production, and the s-channel, in the order of descending cross section at the LHC. The complete NNLO QCD corrections to single-top production involve complicated two-loop Feynman integrals with many scales at play, which makes it challenging to calculate them analytically. We present here the calculations of parts of the requisite two-loop master integrals for the single-top production at NNLO QCD. We mainly focus on the part whose differential equations can be cast into the canonical form. Complex square roots appear in the canonical differential equations. It turns out that the solutions can be expressed by multiple polylogarithms by making appropriate variable changes or simultaneously doing an expansion on one of the kinematic variables.

Speaker: Dr Mingming Long (University of Science and Technology of China)

longmingming-Hangzhou-0823.pdf

15:10 Three loop QCD corrections to quarkonium electroweak decays

In the framework of nonrelativistic QCD (NRQCD) factorization and thanks to AMflow method, we take finite charm mass and light-by-light effects on Upsilon decay into consideration at three loop level on the base that we confirm the existing three-loop results in literature with high numerical accuracy. In addition, we obtain a new piece of 3-loop contribution to the anomalous dimension related to the composite NRQCD bilinear of vector current that arises from the nonzero charm quark mass. Then we extend this method to quarkonium consisting of different flavor quark such as Bc and reconstruct the analytical expressions for the three-loop renormalization constant of corresponding NRQCD operator as well as its anomalous dimension. Phenomenological analysis of the above two cases are presented at the end of each sections.

Speaker: Ji-Chen Pan

Presentation 潘济陈.pdf

15:30 → 16:00 茶歇

16:00 → 17:10 Plenary 丁

16:00 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 its validity and illustrate technical details using examples with quadric integrand singularities (which the one-loop Feynman integrals belong to). Generalizations to higher-loop integrals are commented at the end.

Speaker: Ye Yuan (Z)

16:30 Form factor of Heavy-to-Light Baryonic Transitions in SCET

In the framework of the soft-collinear effective theory, we demonstrate that the leading-power heavy-to light baryonic form factors at large recoil obey the heavy quark and large energy symmetries. Symmetry breaking effects are suppressed by Λ/mb or Λ/E, where Λ is the hadronic scale, mb is the b quark mass and E ∼ mb is the energy of light baryon in the final state. At leading order, the leading power baryonic form factor ξΛ,p(E), in which two hard-collinear gluons are exchanged in the baryon constituents, can factorize into the soft and collinear matrix elements convoluted with a hard-kernel of order α_s^2.
.

Speaker: 志鹏 邢 (上海交通大学)

b-l.pdf

16:50 Discovering tauonium(S=1,L=0) in $e^+e^-\to\mu^+\mu^-$ process

Ditauonium is an exotic atom, and that is a bound state consisting of opposite-sign $\tau$ leptons by the QED interaction. We consider an impact of tauonium($1^{3}S_{1}$, $2^{3}S_{1}$ and $3^{3}S_{1}$) as an intermediates on $e^+e^-\to\mu^+\mu^-$ process. And smearing effect has a influence on signal at different experimental resolution. We estimate event distribution at $1fb^{-1}$ integrated luminosity.

Speaker: 靖航 傅 (Beihang University)

202208_FuJinghang_hangzhou.pdf

Wednesday, 24 August

09:00 → 10:20 Plenary 戊

09:00 Gluonic evanescent operators: two-loop anomalous dimensions

Evanescent operators are a special class of operators that vanish in fourdimensional spacetime but are non-zero in d dimensions. In this paper, we continue our systematic study of the evanescent operators in the pure Yang-Mills theory and focus on their two-loop renormalization. We compute the form factors of evanescent operators using the d-dimensional unitarity method. We apply both the MS scheme and the finite renormalization scheme and obtain the two-loop anomalous dimensions for the dimension-10 basis. As a consistency check, we show that the results of the two schemes coincide with each other at the Wilson-Fisher conformal fixed point. Our results show explicitly that starting from the two-loop order, evanescent operators can give rise to non-negligible physical effects in dimensional regularization.

Speaker: Rui Yu (B)

余睿.pdf

09:20 Canonical differential equations for loop integrals in four dimensions

In this talk, we introduce our work on a new type of canonical differential equation for heavy quark effective theory (HQET). This method aims at calculating the divergent part of Feynman integrals with only one region of divergence. For this special type Feynman integrals, we invent three ingredients 1) the "graded IBP operator", 2) syzygy method to control the types of divergence, 3) the "initial" algorithm for 4D canonical differential equation. We show the power of our method by calculating an example of 3-loop HQET integral family, whose 1/epsilon coefficients correspond to the angle-dependent cusp anomalous dimension.

Speaker: Ma Rourou (U)

2022_Hangzhou_Rourou.pdf

09:40 Automatic computation of Feynman integrals containing linear propagators via auxiliary mass flow

We proposed a recipe to systematically calculate Feynman integrals containing linear propagators using the auxiliary mass flow method. The key of the recipe is to introduce a quadratic term for each linear propagator and then using differential equations to get rid of their effects. As an application, we calculated all master integrals of vacuum integrals containing a gauge link up to four loops, and we checked the results by nontrivial dimensional recurrence relations.

Speaker: Zhi-Feng Liu (school of physics Peking Uniersity)

220824-report3.pdf

10:00 自举方法在形状因子计算中的应用和最大超越度原理

在过去二十年里，现代散射振幅理论发展了许多新的方法，在计算和理解量子场论中与散射过程相关的可观测量方面也得到了许多新的结果，它们为许多高能物理现象提供了更为精确的理论预言与解释。而近些年，起源于上个世纪S-matrix program思想的自举(bootstrap)方法，在计算场论中的物理量方面有一些重要的进展。要计算量子场论中的物理量，往往需要大量极其繁琐的中间步骤，而最终结果有时是极其简单的。自举方法的主要思想，便是通过合适地假设具有一般形式的结果，再利用一些已知的物理约束，直接限定得到最终结果，从而省去不必要的中间过程。除此之外，不同理论往往满足一些普适的物理约束，这能用于解释不同理论结果之间的唯一性，例如最大超越度原理。最大超越度原理是指，在最大超对称杨-米尔斯理论(${\cal N}=4$ SYM)和量子色动力学理论(QCD)中，对于一类物理量，如反常量纲、形状因子的圈图修正等，其最大超越度部分是相同的。本次报告将介绍利用基于主积分(master integral)的自举方法计算量子场论中的两圈四点形状因子的相关进展，我们使用利用了红外发散、共线因子化、非物理极点消除、幺正性等物理约束，得到了形状因子的圈图修正中最大超越度部分的结果。由于这些物理约束具有一定意义上的一般性，因此它们也可以用于讨论最大超越度原理。

Speaker: 圆宏 郭 (中国科学院理论物理研究所)

Bootstrap_Guo.pdf

10:20 → 10:40 茶歇

10:40 → 12:00 Plenary 戊

10:40 A new method of Feynman integral reduction

In this talk, I will introduce the Block-triangular-form Feynman integral reduction method and its application to N3LO QCD corrections of top quark pair production at future e+e– colliders.

Speaker: 鑫 关 (北京大学)

2022杭州研讨会.pdf

11:00 色-动量对偶研究进展

“色-动量对偶”是近年来散射振幅研究的一个重要发现，满足“色-动量对偶”的振幅可以通过平方关系直接得到引力理论的振幅，因此对于引力理论的研究也有很大的意义。树图水平的“色-动量因子对偶”已经被证明是存在的，但是在圈图层次这依然还是一个假设。在超对称场论中，“色-动量因子对偶”有着许多高圈的构造例子，但是对于纯杨米尔斯理论这样的构造还十分的有限，并且相对于超对称场论要困难许多。本次报告将简要介绍纯杨米尔斯场论中形状因子和振幅的色动量对偶构造的一些研究进展。

Speaker: Dr 泽宇 李 (理论物理研究所)

色-动量对偶研究进展（杭州）.pdf

11:20 The symbology of Feynman integrals from twistor geometries I, II

In this talk, we describe a method producing the symbol letters of planar Feynman integrals evaluating to multiple-polylogarithms (MPL) from geometries in momentum twistor space, generated from “Schubert problems”. We establish the method in the case of dual conformally invariant (DCI) integrals in the first part of the talk, and extend the method to non-DCI integrals (eg. those appearing in QCD) in the latter. We present many one- and two-loop examples, as well as preliminary attempts to extend the method to non-planar and beyond-MPL cases. Our construction provides new insight into how the symbol letters organize into unions of A-type cluster algebras, and is expected to be useful for bootstrapping individual Feynman integrals.

Speakers: Mr 一朝 唐 (ITP) , 清霖 杨 (ITP)

twistor_geometry_YQL_TYC.pdf