格点量子色动力学介绍

章节主要根据Gattringer C., Lang C.B., Quantum Chromodynamics on the Lattice: An Introductory Presentation, Springer, Berlin Heidelberg, 2010.补充内容来自于报告、文章阅读等等学习笔记。从学习备注中对基础参数、研究方法等要点的记录，到各章节详细的知识点罗列，如格点QCD的基本设置、费米子行列式、手征对称性等，涵盖了量子色动力学在格点理论方面的众多关键内容，帮助学习者构建完整的知识体系。在学术研究、课程学习、知识复习等适用场景中，该思维导图都能发挥重要作用。研究人员可借助其快速回顾相关知识点，为新的研究提供思路；学生在课程学习过程中，能依据模板更好地跟上教学进度，理解复杂的理论概念；在备考复习时，它更是梳理重点、强化记忆的得力助手。是量子物理领域学习者、研究者系统梳理和深入理解相关知识的高效工具。希望对构建知识体系，发现探索兴趣有所帮助～

量子色动力学

• 学习备注
  • 学习法
    • 费曼学习法
      • 确定内容和目标
      • 想象如何讲解给新人
      • 重点回顾
      • 简化易懂
  • 误差
    • 系统误差
      • 由于仪器没有调零等因素造成的研究结果倾向性的增大或者减小
      • 具有倾向性的增大或者减小
    • 随机误差
      • 各种偶然因素所造成的观察值和真值之间的差值
      • 多次测量求平均值可以减小误差
    • 抽样误差
      • 因为抽样造成的样本统计量与总体参数之间的差异
      • 因此选择合适的抽样方法对于减少误差是必要的，如等距抽样
    • 统计误差
      • 在统计过程中，统计资料和实际情况的差别，主要包括调查误差和统计误差。 在粒子物理实验中，指模型选择时对数据处理带来的误差，计算在标准差。
    • Note: 7.1(2)(4) means the beforehand is a statistical error and the rear is a systematic error generally.
    • English definition
      • deviation
        • standard deviation square root of the variance
        • deviation, which is calculated as the difference between an individual data point and the central value of the dataset.
      • variance
        • unbiased variance
          • reason: mean is the average of the sample
      • biase
        • 从无偏估计上理解，即测量值和true value之间的差距
      • error
        • many sample which means M measurements to decide instead of one sample
        • standard error: It quantifies the variability of sample statistics across different samples from the same population.
        • In statistics, "error" can refer to different concepts depending on the context. It can represent the difference between an observed or measured value and the true or expected value. For example, in regression analysis, the error term represents the difference between the observed value and the value predicted by the regression model. "Error" can also refer to the difference between an estimate and the true population parameter(真实总体参数).
      • >standard
        • means square root or not only one data point.
  • bare parameters(裸参量)
    • bare表征无穷，实际操作中，bare还是取一个有限的值，这是格点场论中框架需要引入的，因为这和真实的物理图景（和重整化理论是相关联的，因为通过重整化，我们可以对无穷大的物理量进行处理）是相关的处理
    • 具体实例：夸克的裸质量 其次，在特定语境下，需要取一个小的裸参量以实现展开处理（参见刘川5.2对生成泛函（配分函数）的展开）
  • 点群(point group)
    • 来源：各个对称元在空间上表现为相交的一点，即原点。描述晶体结构时是相当重要的
    • 三维元素共有32个，组成分类通过纯旋转轴完成变换和其他对称方式如镜面得到 这些群元可以通过共同的矩阵形式来表现
  • 散射道(scattering channel)
    • 散射道指，对于同样的入射态，可以有不同组散射末态，每一个散射末态称之为散射道。单道散射最早研究，其实际是理想化的近似。
  • 组态(configuration)
    • 组态产生时，根据连接变量决定了胶子场，对于互不连接在一起的胶子场采用并行操作，其中并行操作的时效和规模对于准确反映低能量子色动力学具有重要作用，因此探讨组态产生的并行机制（OpenML等）、组态的存储和输出和向量在计算机上的优化对于计算机工程来说是必要的。参考：2019《格点量子色动力学组态产生和胶球测量的大规模并行及性能优化》
  • 临界温度Tc
    • 155MeV (wiki) (based on 1407.6387) 对应于禁闭的夸克转变成火球的相变温度
    • 对于QCD中的190MeV（更精确的值依赖于夸克质量等参数，约为155-160 MeV），“赝”字主要是因为这是一个Crossover，而非真正的相变。
  • 截断(cut-off)
    • 紫外截断
    • 红外截断
    • 概述： 我们并不需要将截断\Lambda在数学上趋于无穷大，需要的仅仅是保证截断\Lambda对于在相对于\Lambda低很多的能区所观察到的物理现象的影响足够小就行。 这种观念的产生，和Wilson关于重整化的思想密切相关。
  • 历史
    • Michael Creutz
      • 跟格点的关联
        • Harr Measure, SU(N)群的知识《Quark, gluon and lattice》
      • 研究领域
        • Chiral symmetry breaking 手征反常
        • nonperturbative aspects of quantum field theories
        • lattice gauge theory, 写了基础代码
        • Creutz ratio-a measure of the correlation between Wilson loops at different s.
      • 时间线
        • 1966年加利福尼亚科技大学本科毕业
  • 非微扰方法
    • 光前量子化
  • Luscher method
    • matching the finite volume and infinite volume which is the case in the scattering when calculate the cross section etc on the lattice.
  • new physics
    • >3\sigma for evidence, >5\sigma for discovery. Like the anomalous magnetic moment of the muon g-2.
  • topic about physics
    • 物理图景
    • 对称性
    • 格点的处理
    • 从哪里来
  • problem
    • get Gauss from eB[GeV^2]
      • e in SI
      • relating B in Gauss to eB in GeV^2
      • known formula from particle physics
      • use a known conversion constant
      • convert eB in GeV^2 to B in Gauss
• 第一章
  • 关键式
    • 两算符的欧几里得关联函数
  • 路径积分
    • 是和量子力学并列的处理问题的一种方式
    • 在构造时与单位1的完备关系密切相关
    • 用途:1. 传播子
    • 备注
      • 薛定谔算符，不含时 海森堡算符，含时
  • 三个空间
    • 欧几里德空间
      • 是从数学上三维空间出发的几何空间，可以表征空间位置关系，在现代数学中，欧几里德空间是推广的n维空间，距离也在原本3维的基础上进行了推广
    • 希尔伯特空间
      • 体现：量子力学中关于观察量就是在希尔伯特空间中，具体实例是线性谐振子在粒子数表象下升降算符的矩阵元
    • 福克空间
      • 将粒子数作为构造基底的根据
    • 闵氏空间
      • 这个空间与爱因斯坦的狭义相对论和广义相对论密切相关，其坐标基矢为(x, y, z, ct).
      • 四维，与量子场论和相对论密切相关，其和4维欧几里德空间的区别在于其所使用的度规是不同，而度规是反映时空弯曲的指标
  • lattice EFT
• 第二章
  • 费米子和胶子作用量
    • 两种变换
      • 规范变换
        • 旋转
        • 必要性
          • 简化问题并保证规律的不变性
            • 洛仑兹规范
          • 守恒律
            • 规范对称性在电磁学中对应于电荷守恒
          • 力的统一
            • U(1), SU(3)所包含的电磁相互作用和强相互作用
      • 洛伦兹变换
        • 最早接触是相对论参考系的变换
    • 本章核心
      • Wilson Gauge Action-格点形式的胶子作用量，并且在此引入了格子(plaquette,由连接变量组成)
      • 费米子作用量，也与link variables有关
    • 离散方案
      • staggered fermion retains a remnant of chiral symmetry.
    • 符号问题（sign problem）
      • • WV-HMC得到的是实时演化，可以直接得到输运系数而规避不适应（ill-posed）问题.
  • 场
    • 含义
      • 某一个区域，每一点都有一个数量或者向量与之对应，则称在V上定义了一个数量场或者向量场。
    • 分类
      • 梯度场
        • 由梯度给出的向量场
        • 如引力场
      • 旋度(rotation/curl)场
        • 由向量函数A的旋度所定义的向量场
        • 如磁场
      • 旋量(spinor)场
        • • \alpha, the Dirac index: 1, 2, 3, 4; c, color index, 1, 2, 3.
          • Dirac index from: 1. 洛伦兹协变性，描述粒子、反粒子以及它们的两自旋状态。
      • 矢量场
      • 数量场
  • 量子力学中的实验
    • 波粒二象性
      • 光的波动性
        • 1801年杨氏双缝干涉实验
          • 单色光照到两个狭缝上
      • 粒子的波动性
        • 物质波 电子双缝干射实验
          • 电子枪发射电子束射向双缝
• 第三章
  • 定标(scale setting)
    • 衰变常数
    • difference: 1. the coupling g in the Lagrangian 2. the bare coupling g in lattice simulation
  • 3.1 define the path integral for gluons on the lattice
    • link variable and plaquette (building block of the action)
    • integration measure: Harr measure
      • this defines the integral over all possible link variables
  • 3.2 the redundancy in the description (gauge invariance)
    • maximal tree-gauge fixing
      • save the computation time
      • the observable more transparent
      • other gauge: temporal gauge
    • Elitzur's theorem The expectation value of any non-gauge-invariant quantity (like a single link Uμ(n)) is zero.
  • 3.3 observables- the interaction between color charges
    • Wilson loop
      • static potential
    • Polyakov loop
      • Polyakov correlator
  • 3.4 confinement: the force between static quarks show
    • static quark potential
      • Coulomb term B/r
      • linear term \sigma r
        • The energy of this string grows linearly with its length, r. This means it would take infinite energy to separate the quarks completely. This is color confinement.
  • 3.5 scale setting: from lattice to physical units
    • F(r_0)r_0^2 = 1.65 to r_0/a to a.
    • asymptotic freedom judging from coupling constant g.
    • Fig. 3.5
  • 3.6 generalizing this chapter: from SU(3) to other gauge groups
• 第四章
  • 大数定律和中心极限定理
    • 中心极限定理的基础是大数定律，也就是证明是需要大数定律
    • 若马尔可夫条件成立，则大数定律成立
  • 纯规范场理论的数值模拟
    • fundamental problem
      • the path integral --infinitely-dimensional in the continuum --analogy-ising model --solution-improtance sampling by Boltzmann factor
        • Ising model
          • SU(2) gauge theory & 3D ising model: same universality class reason: the same center symmetry: Z(2).
    • mechanism
      • Markov chains
        • what: it is a stochastic process where the next state U_{n+1} depends only on the current state U_n.
        • process: after an initial equilibration period, the configuration it visits are distributed according to the desired Gibbs measure.
        • how: 1. detailed balance -- ensuring the chain converges to the correct equilibrium distribution. 2. ergodicity -- able to reach any possible configuration from any other configuration. 3. based -- action Delta S = S[U'] - S[U].
    • algorithm --to achieve Markov chain
      • metropolis algorithm (oldest and most classical)
        • by a small, local change to get a candidate conf with acceptation probability min(1, exp(-Delta S)).
      • heat bath algorithm
        • ad: more efficient than Matropolis as every step is an accepted change.
        • example: SU(2), SU(2) --> SU(3)
      • over-relaxation algorithm
        • ad: Delta S = 0, acceptance is 100% and decorrelate conf quickly.
        • disad: not ergodic (solution: combined with heat bath and metropolis steps)
    • running
      • representing SU(3) matrices
      • boundary conditions
        • OBC
          • 在时间方向上\tau=0 和T 打破周期性
            • 打破a减小时能量势垒极高，拓扑测量产生统计偏差，允许拓扑荷从边界“流出”“流进”，Q可以连续变化，在热力学极限下，仍然描述真实物理，与PBC一样但autocorrelation time减小 （改善有限，TBC更佳）
        • PBC
        • TBC (空间方向)
          • • 允许k=(2pi n + \theta) / L, 满足计算散射或Form Factor时极小的动量转移
      • random numbers
      • simulation
        • initialization
          • cold start (all links=1)
          • hot start (random links)
        • equilibration
        • measurement
    • analyzing the data
      • autocorrelation
        • integrated autocorrelation time
          • then t_int = 0.9 (we take t_int<1 as the cut), it treated as uncorrelated. The reason: the mathematical error introduced by ignoring that tiny correlation is much smaller than the statistical noise itself
      • pratical process
        • 1. data blocking
        • 2. Jackknife & Bootstrap
      • challenge
        • critical slowing down
          • what: near a phase transitio, the autocorrelation time can become extremely large, making simulations prohibitively expensive.
      • example
        • the average plaquette value
          • what: average value of the real part of the trace of a 1x1 Wilson loop
          • for what: it signal phase transitions, for SU(3) it shows a smooth crossover.
        • static quark-antiquark potential
          • what: energy of a system with a static quark and antiquark separated by a distance r.
          • from what: Polyakov loops correlation functions to extract.
          • for what: a central observable for studying confinement (sigma * r to show).
• 第五章
  • fermion statistics
    • Grassmann numbers (anti-commuting)
      • 幂零性 （nilpotent）
        • 泡利不相容远离
    • integration
      • For: --> fermion determinant,
  • doubling problem
    • naive discretization
      • 1 physical (p=0) + unphysical doublers (p=pi/a).
    • cause: sin(p) = 0
  • to solve the doubling problem: Wilson fermions
    • 1. add Wilson term to the momentum space Dirac operator to distinguish between the proper pole with all p_\mu = 0.
    • 2. doublers get mass \prop 1/a (decouple)
    • cost: breaks chiral symmetry
  • properties
    • hopping expansion
      • basics
        • heavy quarks, path interpretation
        • hopping parameter
        • hopping matrix (5.56): collects all nearest neighbor terms in the Dirac operator.
      • what:
        • this expansion views the quark propagator as a sum over paths
        • the quark hops from site to site, H^n represents a path of length n.
    • symmetries: preserves C, P, T
    • \gamma_5 hermiticity
      • what:
      • the Wilson action is invariant under P and C transformations
        • C is shown in (5.70) to (5.71).
      • results in: real determinant
        • essential in Monte Carlo simulations
• 第六章 (Hadron spectroscopy)
  • content
    • hadron interpolators and correlators - How do we create a specific particle (like a pion and a proton) on the lattice and measure their masses?
      • 1. hadron interpolating operator
        • built from quark and gluon fields that shares the same quantum numbers (spin, parity, flavour) as the particle I want to study
          • p: 3-quark operator with mixed symmetry
      • 2. 2pt correlation function
        • creating a particle at source m and annihilating it at sink n.
      • 3. spectral decomposition
    • quenched approximation
      • • save time
    • computing the quark propagator
      • Dirac operator D is a massive matrix
        • impossible to invert explicitly and store
      • definitions
        • point source
          • a source vector that is non-zero at only one spacetime point and one color/spin index.
        • linear system
      • solutions
        • 1. having a point source vector
        • 2. use iterative solvers to approximate the solution without ever forming the full inverse matrix
        • 3. the solution describes how a quark propagates from the source point to all other points on the lattice
    • smearing and exciting states
      • reason: real hadrons have a physical size (radius) while point sources have poor overlap with real hadrons, creating a lot of noise and excited state contamination (E_1, E_2... mixing in the E_0)
      • definition
        • smearing
          • spread the quark source out over spatial sites
        • Jacobi smearing
          • • M, a smearing operator
            • k, positive real parameter
            • N, smearing steps
      • solution
        • applying smearing to the quark source (and sink) before solving the propagator
        • then the correlation function is dominated by the ground state mass E_0 at much earlier times, improving the signal quality
    • extraction of the masses
  • 两点关联函数和强子谱的计算的步骤
    • 1.确定要研究强子的量子数，得到其正确量子数的算符集合
    • 2.构造由算符集合得到的关联函数（矩阵）
    • 3.用路径积分形式表示关联函数，借助Wick定理将费米子场缩并为夸克传播子
      • quenched approximation to update the meson correlator
      • constructs the Markov chain of gauge configurations
    • 4.在组态下计算相应的夸克传播子，拼接出目标关联函数
    • 5.进一步处理得到感兴趣的物理信息
      • 求解广义本征值
      • 根据本征值确定相应能量，即为强子谱原始数据
      • 对能量进行数值分析和外推得到最终强子谱信息
  • 介子谱的计算
    • 介子
      • 在对夸克传播子进行数值求解时涉及：点源和面源
    • 屏蔽质量和
  • 内插场算符的构造
  • Pricipal Component Analysis 主成分分析
    • what
      • 通过对原始数据的主要成分进行分析，之后用主要信息代替进行分析的方法
    • how
      • 将原始空间投影到主要元素所张成的平面上，这个过程是一个降维的过程
    • why
      • 降维的过程完成压缩无关信息和噪音 refer: https://www.cnblogs.com/pinard/p/6239403.html
  • 涂摩方法(smeared source)， 又称延展源方法(extended source)
    • 目的： 尽可能增大重叠矩阵元并降低其他矩阵元
  • 3pt correlation function
  • meson spectral function
    • classification based on the types of the meson 总宇称（表现为空间波函数）= intrinsic parity(P|particel> = eta_P |particle>) \times orbital parity ((-1)^L)
      • scalar
        • spin 0 parity +1
      • pseudo-scalar
        • spin 0 parity -1
          • e.g. \pi^0
            • 宇称变换（Parity Transformation）: 是指将所有空间坐标都反向：(x, y, z) → (-x, -y, -z)。
      • vector
        • spin 1 parity -1
        • 行为从动量在镜子中反向（宇称为-1）上类似
      • axial spectral function
        • spin 1 parity +1
    • from correlation function
      • how to get the correlation?
        • scalar
        • pseudo
        • vector
        • axial
          • axial quark number
            • defi:
              • 指的是右旋夸克和左旋夸克之差，也可表示为轴矢量流（axial current）在空间上的积分
            • 注意与价夸克不同，which是决定强子量子数的组成成分
• 第七章
  • 核心关键词
    • chiral symmetry
      • 手征磁效应简介： 当一对手征相反的Weyl点（和能级交叉相关的特殊点）出现能量差时，引入外加磁场将导致Weyl点附近产生正比于磁感应强度的拓扑电流，该现象称为手征磁效应。
      • 物理图像： 在一个强磁场中，带电粒子的运动被限制在所谓的“朗道能级”上。对于手征费米子，最低朗道能级非常特殊：右手粒子的动量只能沿磁场方向，左手粒子的动量只能沿磁场反方向。 如果存在手征不平衡（比如右手粒子比左手粒子多），那么沿着磁场方向运动的粒子就会比反方向的多，从而形成净电流
        • 意义：验证QCD真空存在拓扑结构；重离子碰撞中手征不平衡。
    • Wilson fermion action fail
      • doubling problem, breaks the chiral symmetry
    • Ginsparg-Wilson equation
      • constructed on the lattice
      • satisfy the chiral symmetry
      • solution form: overlap operator
    • spontaneously breaking of chiral symmetry
      • background: explanation of proton or neutron mass difference explanation: the spontaneous breaking (non-perturbative theory) of chiral symmetry in QCD leads to the emergence of Nambu-Goldstone bosons, such as pions, and their interactions with quarks and gluons in the QCD vacuum, thereby contributing to the mass difference between the proton and the neutron
      • related to
        • Goldstone mode
        • Goldstone bosons
        • explain why the mass of pions is light
    • topological charge
    • condition number
      • ill-conditioned example
        • treat y1 and y2 as the same points.
          • det(C)=0, invertible.
  • To know what is the chiral phase transition temperature, refer to Chiral Phase Transition Tempe in (2+1) flavour temperature To get the T, extrapolating vanishing pion mass using universal scaling analysis.
    • Universal Scaling Analysis "Universality" is the deep idea that the behavior of a system very close to a phase transition does not depend on the microscopic details of the system, but only on fundamental properties like its dimensionality and symmetries. Systems in the same "universality class".
  • 强子散射和共振态研究
    • Luscher formula:
    • • 处理两粒子系统的能级和这两个粒子在无限体积中的散射相移之间的关系
        • 三体问题
          • 一般思想：
    • GEVP（Generalized Eigenvalue Problem, 广义本征值问题），引入背景：
      • 本质上是一种变分法，思想类似于量子力学中用一组基函数来近似求解哈密顿量的本征态
  • content
    • chiral symmetry in continuum QCD what chiral symmetry is in the continuum before moving it to the lattice
      • what
        • • left
          • right
      • chiral rotation
      • chiral limit
        • • Note: in Frithjof 2212.03015, it says another case of chiral limit: infinitely heavy quarks, which is pure SU(N_c) gauge theory, Z(N_c) symmetry.
      • SSB
        • the action is symmetric under chiral rotations but the ground state is not.
        • the order parameter
        • goldstone bosons emerge from SSB.
          • poin 140MeV understood as the explicit breaking by the u and d quark masses.
    • CS and the lattice No-Go theorem- lattice action cannot be local, doubler-free, and chirally symmetric when a to 0
      • Wilson term
        • remove the doublers, but explicitly breaks CS even at m=0.
      • Nielsen-Ninomiya theorem
        • The formal proof that the continuum chiral condition fails on the lattice for local, doubler-free actions.
      • Ginsparg-Wilson (GW) equation
        • modified chiral rotation
        • allows a form of CS to exist on the lattice
    • consequences of GW equation physical and mathematical properties of any Dirac operator that satisfies the GW equation
      • GW circle
        • eigenvalues of a GW operator lie on a circle in the complex plane with r=1/a
      • Index theorerm
        • relateds the number of chiral 0 modes (n-, n+) to topo Q.
      • axial anomaly
        • what
          • the fermion determinant is not invariant under the axial vector rotations (chiral rotations):
        • The fact that the fermion measure is not invariant under U(1)_A rotations, leading to the mass of the $\eta'$ meson
        • relation
          • closely related to the chiral transition
            • U_A(1) restoration occurs whether exactly at the chiral transition temperature remains a "central topic"
      • the Banks-Casher relation
        • relates the chiral condensate to the density of near-0 eigenvalues
        • Eq. (7.77)
    • overlap operator construct an explicit, usable lattice Dirac operator that actually solves the GW equation
      • what
        • Eq. (7.78)
      • Wilson Dirac operator for the kernel (kernel operator)
        • a non-chiral operator (usually Wilson) used as the input for the overlap construction
      • sign function of the operator
        • mathematical challenge, making the operator non-ultralocal (though still "local")
        • numerical approximation
          • Polynomial approximation
• 第八章
  • content
    • fermion determinant The Grassmann integral for fermions leads to a determinant
      • key: convert the determinant into a form that can be sampled using Monte Carlo.
      • pseudofermions: used to represent the fermion determinant in the path integral.
      • concepts
        • weight factor
        • the pseudofermion identity
    • hybrid Monte Carlo (HMC) the det is nonlocal, changing just one gauge link requires recomputing the entire matrix.
      • logic: use Molecular Dynamics (MD) to evolve the entire gauge field simultaneously in a fictitious time
      • concepts
        • conjugate momenta: randomly generated Gaussian variables paired with gauge fields.
        • leapfrog（越级） integration: A reversible, area-preserving scheme to solve equations of motion.
        • acceptance probability
        • time unit
          • it refers to the integration time of the molecular dynamics evolution
          • one time unit = one accepted configuration
      • relation
        • Rational Hybrid Monte Carlo (RHMC)
          • standpoint: HMC only copes even flavours of fermions
          • RHMC: could handle the odd part
          • idea: approximate the fractional power using a rational approximation
          • workflow
            • HMC handles the u,d doublet
            • RHMC handles the strange quark
    • other algorithmic ideas standard HMC is designed for even numbers of mass-degenerate flavours
      • logic: use rational or polynomial approximations to linearize or "filter" the fermion force
      • concepts
        • mass preconditioning (Hasenbusch trick): splitting the determinant into a heavy and light part to reduce force fluctuations
        • even-odd preconditioning splitting the lattice to simplify matrix inversions
    • coupling-mass phase diagram find the "sweet spot" in parameter space (\beta, \kappa or m) where a to 0 while physics remains constant.
      • logic: identify lines of constant physics (mass ratios) and drive the system toward critical points.
      • concepts
        • chiral curve
          • the boundary where the pion mass vanishes.
        • lattice spacing relation
        • renormalization group scaling
• 第九章
  • the symanzik improvement program discretizing the QCD action introduces errors that are O(a) for fermions and O(a^2) for gauge fields.
    • logic: adding "improvement terms" to the lattice action to reduce the error
    • concepts
      • Symanzik improvement
        • a program to cancel lattice artifacts order-by-order in a.
      • Clover (Sheikholeslami-Wohlert) term
        • a specific d-5 operator added to the Wilson action for O(a) improvement
      • PCAC (Partially Conserved Axial Current) relation
        • to determine the improvement coefficients c_{sw} and c_A.
      • effective action
      • clover action
  • lattice actions for free fermions from RG transformations standard discretizations like Wilson action fail to maintain CS while avoiding doublers
    • logic
      • use Renormalization Group transformations to integrate out short-distance (UV) dof. "blocking continuum fields" into hypercubes.
    • concepts
      • blocking
        • integrating the continuum fields over hypercubes to construct lattice fields
      • Ginsparg-Wilson Equation
  • real space renormalization group for QCD for full QCD, the path integral in non-Gaussian and cannot be solved in closed form like free fermions
    • logic
      • 1. start from a very fine lattice (nearly continuum)
      • 2. perform discrete "block spin" transformations to a coarser lattice
      • 3. iterate this process to find a Fixed Point action where discretization effects are essentially absent.
    • concepts
      • fixed point
        • a set of action couplings that remain invariant under RG transformations at
      • saddle point analysis
        • a method to solve the RG equations at a to 0, where the path integral in dominated by the configuration that minimizes the action.
      • overlapping block spin
        • a blocking scheme where fermion fields on the fine lattice are averaged to construct gauge-invariant fields on the coarse lattice
  • mapping continuum symmetries onto the lattice determine how continuum symmetries (especially chiral symmetry) are represented on the discrete lattice
    • logic
      • 1. apply the blocking transformation to the Generating Functional of the theory
      • 2. require the lattice theory to preserve the symmetry of the continuum path integral
      • 3. derive exact lattice symmetry transformations
    • concepts
      • generating functional
      • lattice anomaly
        • the discrete counterpart to the continuum axial anomaly, emerging from the Jacobian of the lattice field transformation
      • Lüscher's symmetry
        • the specific lattice transformation generators (M, \bar M) that ensure the action is invariant while obeying the GW equation
      • anomaly
        • represent the breaking of symmetry due to the measure
• 第十章 (More about lattice fermions)
  • Staggered fermions
    • 引入： 对自由天真费米子的作用量进行交错变换（特点是将每个点的费米子场都乘以了一个平方为单位阵的适当的幂次), 进行矩阵上的考虑得到对角矩阵 (spin diago)，从而得到交错费米子。
    • What: Staggered fermions are a fermion discretization method in lattice QCD that reduces the fermion doubling problem by spreading components of the fermion field across the lattice. They preserve a partial form of chiral symmetry and are computationally efficient, making them popular for large-scale QCD simulations. However, they introduce "taste" degrees of freedom, which require careful consideration, especially in the context of taste symmetry breaking.
  • different inputs for the pion mass
    • 目的
      • chiral extrapolation to physical pion mass
      • tuning lattice parameters
      • studying finite volume effects
        • Finite volume effects become more pronounced for smaller pion masses because the pion is the lightest particle, and its long-range interactions are more affected by the limited size of the simulation box.
  • domain wall fermions
    • another version of chiral lattice QCD
  • twisted mass fermions
    • a mass term provides a useful IR regulator and utilized to obtain O(a) improvement of the lattice formulation
  • heavy quarks on the lattice quarks like charm and bottom have masses larger than the typical inverse lattice spacing 1/a. This leads to massive discretization errors, making them impossible to simulate with the standard relativistic actions
    • logic
      • use effective field theories to expand the action in powers of 1/m_q or the velocity v, which "integrates out" the heavy mass scale
    • concepts
      • NRQCD (Non-Relativistic QCD)
        • an expansion in powers of v, suitable for heavy-heavy systems like bottomonium
      • HQET (Heavy Quark Effective Theory)
      • static limit
        • the m_q to \infinity wehre the heavy quark becomes a fixed color source
  • TMD, GPD
    • • 纵向动量分数 x：沿着质子平均动量方向，表示部分子携带的纵向动量比例。 二维横动量 k_T：部分子在横向动量空间中的分布，反映“动量侧”的二维结构。 二维横向位置 b_T: 部分子在质子内部横向空间的分布，反映“位置侧”的二维结构。
    • relation: LaMET is the method to get these on lattice simulation.
• 第十一章
  • low-energy parameters How can we relate lattice field operators to the observed particle states?
    • logic
      • 1. use Ward-Takahashi identieies to relate expectation values of transformed field variables
      • 2. ensure the partition function remains invariant under symmetry transformations
      • 3. derive relations like the GMOR (names) relation to calculate the quark condensate and pion decay constant. Eq. (11.38)
    • concepts
      • interpolating field operator
        • an operator (e.g. pseudoscalar P^a) that couples to all states sharing specific quantum numbers J^{PC}.
      • PCAC (Partially Conserved Axial Current)
        • a hypothesis relating the divergence of the axial vector current to the pion field, defining the pion decay constant F_{pi}.
      • Ward-Takahashi identities
        • identities expressing the invariance of the partition function under infinitesimal symmetry transformations of the fermion fields
      • GMOR relation
        • a foundational relation for the chiral limit
  • renormalization To match the lattice results to experimental results or continuum schemes like \bar{MS}, we must account for scale-dependent devergences and renormalization factors
    • logic
      • 1. employ nonperturbative methods like the Rome-Southampton (RI/MOM) method to determine renormalization constants
      • 2. these constants bridge the gap between lattice calculations and continuum perturbation theory
    • concepts
      • renormalization constant
        • the proportionality factor between an interacting lattice matrix element and its free (tree-level) counterpart
        • denotes the five types: S, V, A, T, and P.
      • RI/MOM scheme
        • a regularization-independent scheme implemented in both lattice simulations and continuum perturbation theory to allow for scheme conversion
      • Landau gauge fixing
        • a common gauge choice for computing gauge-variant renormalization constants, typically implemented via iterative minimization of a functional of the link variables
  • hadronic decays and scattering finite size effects exist in lattice. Masses show exponentially small volume dependence, 2-particle interactions (like scattering) show power-law suppression, making it difficult to extract scattering amplitudes directly.
    • logic
      • 1. use Lüscher's formula to relate the energy spectrum of 2 particles in a finite, periodic box to their elastic scattering phase shifts
      • 2. by varying the spatial volume L, one can map out the momentum-dependent phase shift and identify resonances like the rho-meson
    • finite size effects
      • lattice size squeezes the virtual polarization cloud of a particle within a finite box
    • scattering phase shift
      • a momentum-dependent shft acquired in the interaction region that characterizes the scattering behavior of 2 particles
    • level-crossing avoidance
      • a phenomenon in the finite-volume energy spectrum that signals the presence of a resonating phase shift
  • matrix elements probing the internal structure of a hadron (charge, spin, and momentum distribution) requires calculating off-forward matrix elements, which is computationally expensive because each propagator calculation requires an inversion of the Dirac operator
    • logic
      • implement the sequential source method to reduce the number of required inversions by computing a "sequential propagator" for a fixed sink momentum.
      • for: this allows for the efficient calculation of 3-point correlators and the extraction of form factors from plateau values
    • concepts
      • form factor
        • a function (like F^\pi(Q^2) ) that describes how a hadron couples to a probe, such as a photon, at a given momentum transfer.
      • sequential source method
        • a technique where a propagator is fixed at the sink to allow multiple operator insertions to be calculated with a single additional inversion
      • operator product expansion (OPE)
        • a framework that relates light-cone matrix elements to matrix elements of local operators, facilitating the study of parton distributions
      • effective weak Hamiltonian
        • a sum of local operators multiplied by Wilson coefficients used to separate short-distance (weak) physics from long-distance (QCD) effects
• 第十二章 （temperature and chemical potential）
  • temperature introduction
    • Matsubara frequencies
      • boson fields:
      • fermion fields:
  • confinement / deconfinement
    • Polyakov loop's average: <P> = 0, confinement <P> != 0, deconfinement
      • 1. related to the Center Symmetry breaking 2. but the proving of CS breaking is unresolved 3. only high T limit, V could be perturbative calculated and minima at z\in Z(N).
      • based:
        • confinement, F to infinity, CS deconfinement, F finite, no CS
      • related to the plaquette, which could be also signal the confinement/deconfinement.
      • dynamical quarks explicitly break the Center Symmetry (fermion action is not invariant). --> then 1st order transition becomes a smooth crossover.
    • full QCD (gluons + dynamical quarks)
      • Chiral Symmetry restoration: != 0, low T, spontaneously broken to 0, high T, symmetry restored
      • Is T_c of deconfinement crossover equal to the CS crossover T_{ch}? Evidence suggests they are very close (150-190 MeV) and deep connected.
  • chemical potential introduction - simulate systems with a net density of quarks (more quarks than anti-quarks)
    • introduce
    • concepts
      • \hat{N}_q is the quark number operator, \mu is the quark chemical potential
      • sign problem: lattice implementation of \mu causes the fermion determinant to a complex number. Then the Boltzmann weight now becomes complex.
      • • temporal component of the conserved vector current
        • to show that the naive way by introducing \mu linearly would cause energy density diverge when a to 0.
        • then the correct way is by (12.37) considering the forward f(a\mu) and backward hopping factors f(a\mu)^{-1}.
      • imaginary chemical potential
        • when \mu is purely imaginary, then the determinant is real.
        • Using the imaginary is the foundation for most modern techniques.
    • Monte Carlo techniques for \mu >0
      • reweighting
        • generating configuration at \mu=0, then re-weight
        • limitation: The SNR decays exponentially with volume and \mu. It works for very small \mu and small lattices.
      • taylor expansion
        • physical quantities are even functions of \mu/T. then the expansion coefficients can be calculated at \mu=0. applied for \mu/T < 1.
      • analytic continuation from imaginary \mu
        • 1. perform at several values of imaginary chemical potential 2. fit the results into a function 3. analytically continue the function to real chemical potential.
        • limitation: notoriously unstable numerical problem which depends heavily on the quality of the imaginary \mu and the choice of the fitting function.
        • 解析延拓
          • 把实数解析延拓成复数
            • Dyson-Schwinger 方程(用在FRG的解析延拓中)
      • canonical partition function
        • • at imaginary \mu, this becomes a Fourier series.
          • limitation: The Fourier integral is highly oscillatory for large n, then the estimation is statistically challenging.
    • QCD phase diagram
      • • 定义：在热力学和物理学中，临界终点（Critical End Point, CEP）通常指代不同相平衡曲线终止的特殊状态。
      • T_pc 对化学势的依赖 Tab. 3 [2002.02821]
      • T_pc 0 \mu_B [1812.08235]
• local & nonlocal distinction: topological
  • local operator: defined at a specific spacetime point x
    • example: electric current-
  • nonlocal: on a sub-manifold (a line, a surface, or the whole space volume)
    • example: charge
• 拓扑
• 费米子行列式中包含裸夸克质量信息，因为行列式可以展开成关于大夸克质量从而取quenched近似