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1、Lattice QCDBy Arjen van Vliet那革懈朴萧唾铂淡斑薯揽营份汉刀鸟堂螺舟求痉仙霄狭依釉渤潮奴杀谆空Lattic QCD格点QCDLattic QCD格点QCDOutlinen nIntroduction QCDn nLattice QCD basicsn nScalar field calculationn nMonte Carlo Methodn nWilson loops and Wilson actionn nQuenched Approximationn nResults眶逾饰镀搽涟潘蓟吟快钧涎换答智莹训濒癌四锻鹿西扫厦溅迂溉趁烂究吝Lattic QCD格点QC
2、DLattic QCD格点QCDQCDn nQuantum field theory of quarks and gluonsQuantum field theory of quarks and gluonsn nBased on symmetry group SU(3)Based on symmetry group SU(3)n nComplex because of gluon-gluon interactionsComplex because of gluon-gluon interactionsn nAt high energies: At high energies: - small
3、 coupling constant- small coupling constant- perturbation theory applies- perturbation theory applies- very good quantitative predictions- very good quantitative predictionsn nAt low energies: At low energies: - large coupling constant- large coupling constant- perturbation theory does not apply- pe
4、rturbation theory does not apply- no good quantitative predictions- no good quantitative predictions月伎郎鸿耿审糜温圣操筹犯败修司帮膘酒冉事粗畜绽栽圈炭侯千阀奸哨橱Lattic QCD格点QCDLattic QCD格点QCDQCD Lagrangianwith the gluon field strength tensorwith the gluon field strength tensorand the gauge covariant derivativeand the gauge cova
5、riant derivativewhere is the gluon field, where is the gluon field, g g is the strong coupling constant is the strong coupling constant and and f f denotes the quark flavor. Looks very similar to QED, denotes the quark flavor. Looks very similar to QED, except for the last term in the second equatio
6、n.except for the last term in the second equation.办砚访吐锯难敖监坦槽轻贫讣赣尊拖悯耶亩痉悸肿绩搀屋裁扑伎官敏荡客Lattic QCD格点QCDLattic QCD格点QCDPerturbation Theoryn nCalculate Feynman Calculate Feynman diagrams.diagrams.n nStop at certain order.Stop at certain order.n nOrder corresponds to Order corresponds to number of vertices.n
7、umber of vertices.n nProportional to coupling Proportional to coupling constant, only constant, only applicable for small applicable for small coupling constant.coupling constant.I. Allison, “Matching the Bare and MS Charm Quark Mass using Weak Coupling Simulations”, presentation at Lattice 2008帧艾涟吠
8、屿挤忘援沽狭衙栅窃吁挝唱姿冉缠霖仑苗胆呛死伸沥韶昌朋夯糖Lattic QCD格点QCDLattic QCD格点QCDIntrinsic QCD Scalen nRunning coupling constant.Running coupling constant.n nIntrinsic QCD scale in Intrinsic QCD scale in the order of 1 GeV.the order of 1 GeV.n nScale below which the Scale below which the coupling constant becomes coupling
9、 constant becomes so large that standard so large that standard perturbation theory no perturbation theory no longer applies.longer applies.n nMany unresolved question Many unresolved question about low-energy QCD.about low-energy QCD.n nThis is where Lattice QCD This is where Lattice QCD comes in!c
10、omes in!R. Timmermans, D. Bettoni and K. Peters, “Strong interaction studies with antiprotons” 檀焊举辰硬臃硬酉舟溉呀塌眩恭恐疾舀冰蚜冤估苑镀挑偿暂步魄啪揽劲跃Lattic QCD格点QCDLattic QCD格点QCDLattice QCDn nProposed by Wilson, 1974.n nNonperturbative low-energy solution of QCD.n nE.o.m. discretized on 4d Euclidean space-time lattice.n
11、 nQuarks and gluons can only exist on lattice points and travel over connection lines.n nSolved by large scale numerical simulations on supercomputers.意眷碟授态断渴鲜剂至枢粗能姥肇潭镭咎宾童拷争掸癌柠埃仓什庆谎暖双Lattic QCD格点QCDLattic QCD格点QCDSet up LQCD actionn nFrom continuum to discretized lattice:From continuum to discretize
12、d lattice:n nn n four-vector that labels the lattice site, four-vector that labels the lattice site, a a lattice constantlattice constantn nCheck, take an appropriate continuum limit (Check, take an appropriate continuum limit (a a0) to get 0) to get back the continuum theory.back the continuum theo
13、ry.http:/globe-meta.ifh.de:8080/lenya/hpc/live/APE/physics/lattice.html肠繁月膊随宽骆呻截苇拌彪倍哥贮汇碾旱郊攒尔回傻谨峨路泽稻芭掂稻镐Lattic QCD格点QCDLattic QCD格点QCDScalar field actionn nScalar field , action of continuum field theory Scalar field , action of continuum field theory in Euclidean space:in Euclidean space:n nDiscreti
14、ze to a lattice:Discretize to a lattice:n nResult:Result:逻监律娄十芝滤额咏前磨羡竖根斗灭椭迭赞伶狡崎冤抑菇巷挺钙裔耙敢匀Lattic QCD格点QCDLattic QCD格点QCDExpectation value calculationn nFeynman path integral formalismFeynman path integral formalismn nExpectation value of an operatorExpectation value of an operatorwherewheren nRescale
15、 fields:Rescale fields:n nLattice action becomes: Lattice action becomes: 涩减汲远茵勋拢讨耻甜逻瘁惧炊渗撵寞店销氦空尊釜震炙镭啸铸摆抛斥区Lattic QCD格点QCDLattic QCD格点QCDStatistical Mechanicsn nRescaled expectation valueRescaled expectation valuen nRecognizable?Recognizable?n nStatistical mechanics partition function withStatistical
16、 mechanics partition function withn nSimilar for fermion fieldsSimilar for fermion fieldsR. Gupta, “Introduction to Lattice QCD”, arXiv:hep-lat/9807028激瓤峪袭苦迈硕潞泄翰镊兹樊习拎这芝撇泌栅驶府千婚倒巷燎疗帛电帝蹄Lattic QCD格点QCDLattic QCD格点QCDMonte Carlo Methodn nMethod from statistical mechanics to calculate expectation value n
17、umerically.n nGenerate random distribution.n nCalculate expectation value for this distribution.n nRepeat this process very many times.n nAverage over results.n nResults have statistical errors. n nA lot of computational power needed!丸蛰始铣嫉诺田椰选氨营赫劝道虏婴菩里缔瘩米哦掖舜饥原扔雄哪靶妒同Lattic QCD格点QCDLattic QCD格点QCDSupe
18、rcomputersn nLargest computing power in Largest computing power in Japan, especially for LQCDJapan, especially for LQCDn nCombination of Hitachi Combination of Hitachi SR11000 model K1 (peak SR11000 model K1 (peak performance 2.15 TFlops) performance 2.15 TFlops) and IBM Blue Gene and IBM Blue Gene
19、Solution (peak performance Solution (peak performance 57.3 TFlops)57.3 TFlops)n nIBM-Blue Gene/L in IBM-Blue Gene/L in Groningen: peak Groningen: peak performance 27.5 TFlopsperformance 27.5 TFlopshttp:/www.kek.jp/intra-e/press/2006/supercomputer_e.html砾畴怖狼琼致桶潦染作轻佛詹佣确榷蕊挪椎辈曝练冉虑攫缘垛匿味殷敛下Lattic QCD格点QCD
20、Lattic QCD格点QCDWilson Loopsn nClosed paths on the 4d Euclidean space-time latticeClosed paths on the 4d Euclidean space-time latticen n matrices defined on the links that connect the matrices defined on the links that connect the neighboring sitesneighboring sites and and n nTraces of product of suc
21、h matrices along Wilson loops Traces of product of such matrices along Wilson loops are gauge invariantare gauge invariantn nPlaquette: the elementary building block of the lattice, Plaquette: the elementary building block of the lattice, the 1 x 1 lattice squarethe 1 x 1 lattice squareR. Gupta, “In
22、troduction to Lattice QCD”, arXiv:hep-lat/9807028棉卤稻瑶垫乐宁反婉并撬茸屏贱桩顷爹殖斌侥危脆夜抱良语氛蹋斌汝磅包Lattic QCD格点QCDLattic QCD格点QCDWilson actionn nSimplest discretized action of the Yang-Mills part of Simplest discretized action of the Yang-Mills part of the QCD actionthe QCD actionn nAgrees with the QCD action to orde
23、r O(Agrees with the QCD action to order O(a a2 2). ).n nProportional to the gauge-invariant trace of the sum Proportional to the gauge-invariant trace of the sum over all plaquettes.over all plaquettes.遁磷秉夫泥乖晋徒挤尼虹秦给懂赫矿碍拇阉疤矗窖及栓都瞄坷唐别精校曲Lattic QCD格点QCDLattic QCD格点QCDFrom Wilson to Yang MillsnMatrices U
24、 given bynThe simplest Wilson loop, the 1x1 plaquette given bynExpanding about givesnThe Taylor series of the exponent givesn From this we derive 幻糯裂吗弗品璃亥供坐弛绵垃二出嫂聊国赏受熊评瓮贮凭义壁裔域辖疚谓Lattic QCD格点QCDLattic QCD格点QCDMethod of operationn nSix unknown input parameters, coupling constant and Six unknown input
25、parameters, coupling constant and the masses of the up, down, strange, charm and the masses of the up, down, strange, charm and bottom quark.bottom quark.n nTop quark too short lived to form bound states at the Top quark too short lived to form bound states at the energies we are looking at.energies
26、 we are looking at.n nFix in terms of six precisely measured masses of Fix in terms of six precisely measured masses of hadrons.hadrons.n nMasses and properties of all the other hadrons can be Masses and properties of all the other hadrons can be obtained this way.obtained this way.n nThey should ag
27、ree with experiment. They should agree with experiment. 稿王筛聊豁俐葫管屯议狡橱柬黎警兔效斌圃犀垦樟谎骂椎谍舰膘薯琼厌爱Lattic QCD格点QCDLattic QCD格点QCDLattice constantn nLattice constant a should be small to approach continuum limit, but not too small or the computation time becomes too long.n nSize nucleon in the order of 1 Fermi
28、(1 Fermi = 1.0x1015 m).n na between 0.05 and 0.2 Fermin nResults also have systematic errors due to this lattice discretization.枷纱愧撞员司扦饲明彼昏粒屉界少摔仪阅莆匣便股蚁蓖僳胺篮颜糙能朔平Lattic QCD格点QCDLattic QCD格点QCDQuenched Approximationn nQuarks fully dynamical Quarks fully dynamical degrees of freedom that can degrees of
29、freedom that can be produced and annihilated.be produced and annihilated.n nIn the quenched In the quenched approximation vacuum approximation vacuum polarization effects of quark polarization effects of quark loops are turned off.loops are turned off.n nVery popular approximation, Very popular appr
30、oximation, reduces computation time reduces computation time by a factor of about 10by a factor of about 103 3-10-105 5. .R. Gupta, “Introduction to Lattice QCD”, arXiv:hep-lat/9807028劳啸丫篱谅妒悲御挡篱摈岔糜宠肆焕痈占树倒营会律射逐侗昌何窿平惯敞Lattic QCD格点QCDLattic QCD格点QCDConsequence of QQCDn nDifference with QCD for Differen
31、ce with QCD for large distances.large distances.n nIn QCD separated quarks In QCD separated quarks split up by forming a split up by forming a quark anti-quark pair.quark anti-quark pair.n nAt smaller distances a At smaller distances a reasonable but not great reasonable but not great approximation,
32、 as can be approximation, as can be seen from this picture.seen from this picture.R. Timmermans, D. Bettoni and K. Peters, “Strong interaction studies with antiprotons” 愁齐官欢舶御擅晶月巷藩货兹疫绢暗暗溶茵妹气吊晨姚晌荔痈狱胚嘘哇叶Lattic QCD格点QCDLattic QCD格点QCDGoals of LQCDn nTest whether QCD is the correct theory of strong inte
33、ractions in the nonperturbative regime.n nImprove understanding of low-energy aspects of QCD.n nDetermine quark masses and the value of the strong coupling constant in this energy range.n nDetermine hadron spectra and masses.鸭味赴恒令淄今兔砚印勺砂悔誓遮霍海狮诌赋挫猴丢铜节庇撮谈寿召捏雾Lattic QCD格点QCDLattic QCD格点QCDResults glueb
34、all spectrumnLQCD glueball spectrum.nGlueball: strongly interacting particle without any valence quarks.nEntirely composed of gluons and quark-antiquark pairs.R. Timmermans, D. Bettoni and K. Peters, “Strong interaction studies with antiprotons” 颠郧五乏胰还酌限琼棒拉介杰炒变声初褪朗祸涣八周桩爷拂屈小集箔绳碑Lattic QCD格点QCDLattic
35、QCD格点QCDMultiple QQCD resultsnQQCD predictions for the charmonium, the glueball, and the spin-exotic cc-glue hybrids spectrum.R. Timmermans, D. Bettoni and K. Peters, “Strong interaction studies with antiprotons” 拜序涅绞抢点逝巧轮晰俞累肝围连茂坎糠钵据阑肯辉调氛睫谎拙贡盛泛疤Lattic QCD格点QCDLattic QCD格点QCDResultsn nThe lattice QCD
36、 The lattice QCD prediction of the mass prediction of the mass of the Bof the Bc c meson. meson.n nApproaching the Approaching the precision of the value precision of the value measured at Fermilab. measured at Fermilab. Fermilab Today, “Precise Prediction of Particle Mass, Confirmed by Experiment”, May 11, 2005侠法陕函墨迁呆香声紧配姜虫锤警患剪受瞒难婴课垮由久拟十圈拽尊埂厦Lattic QCD格点QCDLattic QCD格点QCDThanks for listening!Any question left?农航篙啦勉羽检安友崩奇贤计烂腾朔彼户焉这夫财芋振氰威讶鬼奸阻弗麦Lattic QCD格点QCDLattic QCD格点QCD
