lecture18 wilson loops (cont.)

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1、MITOpenCourseWarehttp://ocw.mit.edu8.821StringTheoryFall2008ForinformationaboutcitingthesematerialsorourTermsofUse,visit:http://ocw.mit.edu/terms.8.821F2008Lecture18:WilsonLoopsLecturer:McGreevyDecemebr28,20081MinimumSurfacesTheexpectationvalueofWilsonloopoperato

2、rs
W[C]intheCFTcanbecomputedbycomputingthestringtheorypartitionfunctiononAdSwithastringworldsheetendingontheloopC.Thatis,
W[C]CFT=Zstringtheory[stringendingonC]Furtherwesawthat,allargeλandlargeN,theworldsheetpathintegralisdominatedbythesaddlepoint.Inthesaddlepo

3、intapproximation,thestringtheorypartitionfunctionisgivenby,����Area(Σ)Zstringtheory∼exp−extremum
.∂Σ=CαHence,thecomputationof
W[C]intheCFTcanbemappedtofindinganminimumsurfaceinAdS.Theareaofstringworldsheet(classical)isgivenby,����L4Area[X]=dσdτdet∂αXA∂βXBgAB(X)=d

4、σdτdet∂αXA∂βXBδABz4L2HerewehaveusedthefactthatgAB=z2δABforEuclideanAdS.Noticethat,����L2√ln
W=extremumArea(Σ)=λextremumArea(Σ)α
∂Σ=C∂Σ=CL=1L=1Further,theareaofworldsheetisinfinite.ThisdivergencearisesfromthefactthattheworldsheetgoesallthewaytotheboundaryofAdS.Let

5、uszoomintoaregionclosetotheboundaryandintroduceacut-offatz=whichisequivalenttotheUVcut-offofthefieldtheory.WecanthinkofthedirectionalongthecurveCastheτcoordinateoftheworldsheetnearz=andσextendsalongtheradialdirectionnearz=(pleaseseeFig.1).Theareaevaluateduptothec

6、ut-offhasthefollowingbehavior,���1length[C]A=dzdτdet∂αXA∂βXBδAB∼+...z2CItcanbeseenthatAdivergesas−1.LetuscomparethisresultwithWilsonloopcalculationinthefieldtheory.Inthefieldtheory,theWilsonloopisthetrajectoryofaheavyquark.Quarkscouple1Figure1:(a)Worldsheetendi

7、ngoncurveC(b)Worldsheetneartheboundary.S=mlength[C]_wlmm__qqFigure2:DiagramcontributingtoMassRenormalizationofheavyquarktogluonsandtheycanemitandreabsorbgluons.Anexampleofsuchaprocessistheleadingone-loopcorrectiontothequarkpropagator(showninFig.2).Thisdiagramcorr

8、espondstothemassreno�rmalization(δmq)oftheheavyquarkatzeroquarkmomentum.Thisimplieswemustaddδmqdstotheworldlineactionofthequark.So,the−1divergenceintheWilsonl