当前位置：文档之家 › Non-perturbative O(a) improvement of the vector current

Non-perturbative O(a) improvement of the vector current

  • 格式：pdf
  • 大小：97.16 KB
  • 文档页数：3

a

r

Xiv:

h

e

p

-lat/9709088v1 23 Sep 19971CERN97-256

DESY97-173

Non-perturbativeO(a)improvementofthevectorcurrent∗

M.GuagnelliaandR.Sommerb

aCERN,TheoryDivision,1211Geneva23,Switzerland

bDESY-IfH,Platanenallee6,D-15738Zeuthen,Germany

Wediscussnon–perturbativeimprovementofthevectorcurrent,usingtheSchr¨odingerFunctionalformalism.ByconsideringasuitableWardidentity,wecomputetheimprovementcoefficientwhichgivestheO(a)mixingofthetensorcurrentwiththevectorcurrent.

1.INTRODUCTION

ItiswellknownthatinWilson’slatticeQCD

discretizationerrorsreceivecontributionsthat

arelinearinthelatticespacing.Themostobvi-

ousrecipetoreducethem—namelytogotothe

continuumlimitbyperformingnumericalsimula-

tionsatsmallerlatticespacings—representsstill

ahardtask,eveninthequenchedapproximation.

FollowingSymanzik[1],latticeartifactscanbe

removedorderbyorderinabyaddingappro-

priatehigher–dimensionaloperatorstotheaction

andtothefieldswhosecorrelationfunctionsare

ofinterest.Ifonerestrictsoneselfbyrequiring

improvementonlyforon–shellquantities[2],such

asparticlemassesandmatrixelementsbetween

physicalstates,thestructureoftheimprovedac-

tionforQCDandoftheimprovedcurrentsis

rathersimple.Thegeneraltheorywasdeveloped

indetailinref.[3].Inthefollowingwewilluse

thesamenotationwithoutfurtherreference.

Inthequenchedapproximation,theimproved

actionwasdeterminednon–perturbativelyfor

couplings0≤g0≤1[4].

Therenormalizedandimprovedaxial–vector

andvectorcurrentsareparametrizedas

(AR)aµ=ZA(1+bAamq)(AI)aµ,

(VR)aµ=ZV(1+bVamq)(VI)aµ,

with

(AI)aµ=Aaµ+acA˜∂µPa,˜∂µ=1

∗TalkgivenbyM.GuagnelliattheInternationalSym-posiumonLatticeFieldTheory,21−27Juli1997,Edin-burgh,Scotlandareknownto1–looporderofperturbationthe-

ory[5,6].Moreover,thenormalizations(ZAand

ZV)andtheimprovementcoefficientscAandbVhavebeendeterminednon–perturbativelyinthe

quenchedapproximationforg0≤1[4,7].Asa

nextstep,wedeterminecV,thuscompletingthe

knowledgeoftheimprovedvectorcurrent.The

computationofotherimprovementcoefficientsis

discussedin[9].

KnowledgeofcVis,forexample,requiredfor

thecomputationofvectormesondecayconstants.

Therelativecontributionofa˜∂νTµνtothesede-

cayconstantscanbe,atg20=1,aslargeas

0.3×cV[8].Althoughtheperturbativeesti-

mate[6],

cV=−0.01225(1)×g20CF+O(g40),CF=4/3,(1)

suggestsaneffectoflessthan1%onthedecay

constants,ourpreliminarynon–perturbativere-

sultsdeterminecVtobemuchlargerinmagni-

tudeforg20≃1.

2.THESTRATEGY

ChiralWardidentitiesrelatecorrelationfunc-

tionsofaxial–vectorandvectorcurrents.Inthe

O(a)improvedtheoryandforzeroquarkmass

theseidentitiescanbewritteninaformwhichis

validuptoerrortermsthatarequadraticina.

SincetheO(a)–improvedaxial–vectorcurrentas

wellasZVandZAareknown,onecanuseapar-

ticularWardidentityinwhichtheonlyunknown

iscV.

OurstartingpointistheWardidentity(inthe

continuum):

󰀂

∂Rdσµ(x)ǫabc󰀂Aaµ(x)Abν(y)Qc󰀃−(2)2

2m󰀂

Rd4xǫabc󰀂Pa(x)Abν(y)Qc󰀃=2i󰀂Vcν(y)Qc󰀃,

wherethespace–timeregionRwithboundary∂R

containsthepointyandQcisasourcelocated

outsideR.

WethenimposeSchr¨odingerFunctionalbound-aryconditions[10],chooseν=kandspecifythe

source,

Qck=a6󰀁

y,z¯ζ(y)γkτc

9󰀂Vak(x)Qak󰀃,(5)

kT(x0)=−1

18󰀁

xǫabc󰀂(AI)a0(x)(AI)bk(y)Qck󰀃,(7)

eq.(4)mayberewritten,fort1

ZV[kV(x0)+acV˜∂0kT(x0)]=

Z2A[kIAA(t2,x0)−kIAA(t1,x0)]+O(a2).(8)

Atzeroquarkmassm,thisequationcanbesolved

forcV.However,duetoapeculiarityoftheFigure1.DependenceofYVonthequarkmass.

quenchedapproximation,i.e.theappearanceof

zero–modesinthequarkpropagator,itisnotal-

wayspossibletosimulatedirectlyatzeroquark

mass[4].Fortheparticularboundaryconditions

andlatticesizeemployed,here,thisphenomenon

isrelevantwhenβ=6/g20≤6.4.Inthisre-

gionwehavetoperformanextrapolationtozero

quarkmass.However,inthecourseofthenumer-

icalsimulationsitturnedoutthatcV,implicitly

definedbyeq.(8)alsoforfinitem,isasteepfunc-

tionofmandcouldnotbereliablyextrapolated

tozeroquarkmasswhenwewerenotabletosim-

ulateatverysmallmasses.Thereasonforthis

strongdependenceisrelatedtothefactthatin

theWardidentity(4)themasstermwasleftout:

asaconsequence,eq.(8)isnotevenvalidinthe

continuumlimit.

Itisthereforenaturaltorepeatthecalculation,

keepingthemassterm.Eq.(4)isthenmodified

byanadditionaltermandhascorrectionsoforder

O(am)insteadofO(m).Wesolvetheresulting

equationformallyforcVanddenotethesolution

atfinitemassbyYV,whichcoincideswithcVas

m→0.Asexpected,thislimitisnowreached

verysmoothly(seefigure1).

Inanon–perturbativecalculation,cVdepends

onthechoicesmadeforthevariouskinemat-

icalparameterssuchasT/Lortheparticular

Schr¨odingerFunctionalbackgroundfield.Differ-

entchoicesleadtoavariationofcVitselfthatis

O(a).Thisisunavoidableandonemustmakea

合集下载

ResearchObjectives

ResearchObjectives

ResearchObjectives

TheMILCCollaborationisengagedinabroadresearchprograminQuantumChromodynamics

(QCD).Thisresearchaddressesfundamentalquestionsinhighenergyandnuclearphysics,and

isdirectlyrelatedtomajorexperimentalprogramsinthesefields.Itincludesstudiesofthemass

spectrumofstronglyinteractingparticles,theweakinteractionsoftheseparticles,andthebehavior

ofstronglyinteractingmatterunderextremeconditions.

TheStandardModelofHighEnergyPhysicsencompassesourcurrentknowledgeofthefunda-

mentalinteractionsofsubatomicphysics.Itconsistsoftwoquantumfieldtheories:theWeinberg-

Salaamtheoryofelectromagneticandweakinteractions,andQCD,thetheoryofthestronginterac-

tions.TheStandardModelhasbeenenormouslysuccessfulinexplainingawealthofdataproduced

inacceleratorandcosmicrayexperimentsoverthepastthirtyyears;however,ourknowledgeof

itisincompletebecauseithasbeendifficulttoextractmanyofthemostinterestingpredictions

Recently

Recently

Recently,topological insulator has attracted great interests due to its peculiar band structures and

electronic properties.The two-dimensional quantum spin Hall systems and three-dimensional TIs

including Bi2Se3,Bi2Te3,and Bi1-xSbx have theoretically proposed and experimentally

observed.They have a bulk excitation gap and gapless edge states for 2D TIs and 2D surface at

boundaries:1D edge states for 2D TIs and 2D surface states for 3DTIs.The surface states of 3D TIs

with an odd number of Dirac cones are robust against(weak and nonmagnetic)disorder scattering

and many-body interactions.The Bi2Se3 material has been predicted and verified to have a bulk

gap of 0.3 eV and a single Dirac come of surface states.

The 3D TIs are expected to show several unique properties when the time reversal symmetry is

broken.The latter can be realizes directly by a ferromagnetic insulating(FI) layer attached to the

Bi2Se3未考虑vdw的错误汇总

Bi2Se3未考虑vdw的错误汇总

在没有考虑vdw作用之前，算Bi2Se3材料soc中出现的错误汇总

VASP自旋轨道耦合计算错误汇总

静态计算时，报错：

VERY BAD NEWS! Internal内部 error in subroutine子程序 IBZKPT:

Reciprocal倒数的 lattice and k-lattice belong to different class of lattices. Often results are still useful... 48

INCAR参数设置：

对策：根据所用集群，修改INCAR中NPAR。将NPAR=4变成NPAR=1，已解决！

错误：sub space matrix类错误

报错：静态和能带计算中出现警告： WARNING: Sub-Space-Matrix is not hermitian共轭 in DAV

结构优化出现错误：

WARNING: Sub-Space-Matrix is not hermitian in DAV 4 -4.681828688433112E-002

对策：通过将默认AMIX=0.4，修改成AMIX=0.2（或0.3），问题得以解决。

以下是类似的错误：

WARNING: Sub-Space-Matrix is not hermitian in rmm -3.00000000000000

RMM: 22 -0.167633596124E+02 -0.57393E+00 -0.44312E-01 1326 0.221E+00BRMIX:

very serious problems the old and the new charge density differ old charge density: 28.00003 new 28.06093 0.111E+00

错误：

Physics Letters B 389 ( 1996) 112- 118 PHYSICS LETTERS B

Physics Letters B 389 ( 1996) 112- 118 PHYSICS LETTERS B

ELSEVIER 5 December 19%

Physics Letters B 389 ( 1996) 112- 118 PHYSICS LETTERS B

A Monte Carlo study of old, new and tadpole improved actions

Margarita Garcia PCrez, Jeroen Snippe, Pierre van Baal

Instituut-Lorentz for Theoretical Physics, University of Leiden, PO. Box 9506, NL-2300 RA Leiden, The Netherlands

Received 9 August 1996 Editor: PV. Landshoff

Abstract

Scaling of mass ratios in intermediate volumes, obtained with improved W(2) lattice actions is tested against analytic results for the Wilson and continuum action. A new improved action is introduced by adding a 2 x 2 plaquette to the Symanzik action. Completing a square leads to a covariant propagator that simplifies perturbative calculations. Data is presented on lattices of size 43 x 128, with lattice spacings of approximately 0.02 and 0.12 fermi. For the latter case no further improvement as compared to the tree-level action was observed when including the Lepage-Mackenzie tadpole correction to the one-loop improved Liischer-Weisz Symanzik action.