introduction to effective field theory

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

2、alidinalimitedenergyrange.NowstrictlyspeakingthismeansthatalmostanytheoryisanEFTexceptstringtheoriesthattrytobeatheoryofeverything(TOE).Indeedeventhequintessentialquantumeldtheory

3、quantumelectrodynamics

4、isanEFTinthatitmustbreakdownattheveryhighestenerg

5、ies.Infactwerepresentthehighenergycontributionintermsofacontertermwhich,togetherwiththedivergentcomponentfromtheloopintegration,wetintermsoftheexperimentalmassandcharge.Inordertogetabetterfeelforthisideaitisusefultoconsiderasimplerexamplefromtheregimeo

6、fordinaryquantummechanics

7、thatofRayleighscattering,orwhytheskyisblue.1.1.RayleighScatteringBeforeproceedingtoQCD,let'srstexamineeectiveeldtheoryinthesimplercon-arXiv:nucl-th/0010015v14Oct2000textofordinaryquantummechanics,inordertogetfamiliarwiththei

8、dea.Specically,weexaminethequestionofwhytheskyisblue,whoseanswercanbefoundinananal-ysisofthescatteringofphotonsfromthesunbyatomsintheatmosphere

9、Comptonscattering.[1]Firstweexaminetheproblemusingtraditionalquantummechanicsand,forsimplicity,considerelast

10、ic(Rayleigh)scatteringfromsingle-electron(hydrogen)atoms.TheappropriateHamiltonianisthen(p~eA~)2H=+e(1)2mandtheleading

11、O(e2)

12、amplitudeforComptonscatteringisgivenbytheFeynmandiagramsshowninFigure1as"e2=m1X^
<0jp~ei~qf
~rjn>^
fiAmp=

13、p^i
^f+2!i2!fmn!i+E0En!#^
<0jp~ei~qi
~rjn>^
if+(2)E0!fEnwherej0>representsthehydrogengroundstatehavingbindingenergyE0.2
Figure1.FeynmandiagramscontributingtolowenergyComptonscattering.(Notethatforsimplicitywetaketh

14、eprotontobeinnitelyheavysoitneednotbecon-sidered.)Heretheleadingcomponentisthefamiliar!-independentThomsonamplitudeandwouldappearnaivelytoleadtoanenergy-independentcross-section.However,thisisnotthecase.Indeed,asshowninahomework