introduction to effective field theory

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1、1IntroductiontoEectiveFieldTheoryBarryR.HolsteinaaDepartmentofPhysics-LGRTUniversityofMassachusetts,Amherst,MA010031.IntroductionThebasicideabehindeectiveeldtheory(EFT)methodsistodescribenatureintermsofafullyconsistentquantumeldtheory,butonewhichisvalidinalimitede

2、nergyrange.NowstrictlyspeakingthismeansthatalmostanytheoryisanEFTexceptstringtheoriesthattrytobeatheoryofeverything(TOE).Indeedeventhequintessentialquantumeldtheory

3、quantumelectrodynamics

4、isanEFTinthatitmustbreakdownattheveryhighestenergies.Infactwerepresentthehighen

5、ergycontributionintermsofacontertermwhich,togetherwiththedivergentcomponentfromtheloopintegration,wetintermsoftheexperimentalmassandcharge.Inordertogetabetterfeelforthisideaitisusefultoconsiderasimplerexamplefromtheregimeofordinaryquantummechanics

6、thatofRayleighscatt

7、ering,orwhytheskyisblue.1.1.RayleighScatteringBeforeproceedingtoQCD,let'srstexamineeectiveeldtheoryinthesimplercon-arXiv:nucl-th/0010015v14Oct2000textofordinaryquantummechanics,inordertogetfamiliarwiththeidea.Specically,weexaminethequestionofwhytheskyisblue,whosea

8、nswercanbefoundinananal-ysisofthescatteringofphotonsfromthesunbyatomsintheatmosphere

9、Comptonscattering.[1]Firstweexaminetheproblemusingtraditionalquantummechanicsand,forsimplicity,considerelastic(Rayleigh)scatteringfromsingle-electron(hydrogen)atoms.TheappropriateHami

10、ltonianisthen(p~eA~)2H=+e(1)2mandtheleading

11、O(e2)

12、amplitudeforComptonscatteringisgivenbytheFeynmandiagramsshowninFigure1as"e2=m1X^
<0jp~ei~qf
~rjn>^
fiAmp=p^i
^f+2!i2!fmn!i+E0En!#^
<0jp~ei~qi
~rjn>^
if+(2)E0!fEnwherej0>

13、representsthehydrogengroundstatehavingbindingenergyE0.2
Figure1.FeynmandiagramscontributingtolowenergyComptonscattering.(Notethatforsimplicitywetaketheprotontobeinnitelyheavysoitneednotbecon-sidered.)Heretheleadingcomponentisthefamiliar!-independentThoms

14、onamplitudeandwouldappearnaivelytoleadtoanenergy-independentcross-section.However,thisisnotthecase.Indeed,asshowninahomework