lecture21 confinement (cont.) how to measure the spectrum, proof of mass gap

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1、MITOpenCourseWarehttp://ocw.mit.edu8.821StringTheoryFall2008ForinformationaboutcitingthesematerialsorourTermsofUse,visit:http://ocw.mit.edu/terms.8.821F2008Lecture21:Confinement,continuedLecturer:McGreevyJanuary30,2009•Confininggeometryandmassgap(mainlesson:massgap−→

2、smoothminimumofthewrapfactorinthedualgeometry)•BlackholemechanicsLasttimeweweretalkingaboutattemptingtofindagravitydualofadeformationto3DYMtheory(whichwegetfromtheN=4onathermalcircle1).Inthesecompactifications,the3DYMcouplingg3isoforderofthemassofthescalarsg4Ng3≈mX≈(

3、1)Rywhereg4Nisthe4Dt’Hooftcoupling.The3DtheoryisnotjustYMbutitalsohasabunchofscalarsandthisobscuresourintuitionaboutthetheory.Wecandoasimilarthingtoget4DYMtheorybyputtingND4branes(atlowenergies)onathermalcirclewitharadiusRyandagainFermionswillhaveAPBCs.TheND4theory

4、isagaugetheoryin5Dwith16superchargesanditswelldefinedatlowenergies.Theboundaryconditionsonthethermalcirclewillbreaksupersymmetryandthefermionswillget4Dmassesoforder1/Rywhilebosonswillgetmassesatoneloopoforderm2≈g42Nm2≈λ4/Ry2.AtenergiesE≈1/Ry,Xψ√the4Dand5DYMcouplings

5、arerelatedby1/g42=Ry/g52whichimpliesthatmX≈λ4/Ry.Therelationbetweenthethe4Dand5DYMcouplingscomesfromintegratingovertheextradimensioninthecompactifiedtheory.ThenicethingabouttheaboveconstructionisthatifattheUVscale(E=1/Ry),thet’Hooftcouplingλ4≪1,thenthetheoryisasympt

6、oticallyfreeandwecancalculatetheβfunctionperturbatively.TherewillbeadynamicallygenratedmassscaleΛsuchthat√e−1/λ41λ4ΛQCD≈≪≪mX≈(2)RyRyRyNoticethat1/RyplaystheroleofaUV.Itslikeputtingthe4DYMtheoryonalatticewhereRyplaystheroleofalatticespacing.Ifweprobethetheorywithene

7、rgiesoforder1/Rywewillseea5Dtheory(theUVcompletion).1RememberthatthermalcirclemeansperiodicboundaryconditionsforBosonsandantiperiodicboundaryconditionsforFermions1NoticealsothatwhenE<1/Ry,wewillhaveapureYMtheory.Thepreviousanalysiswasforλ4≪1,thegravitydualhasacurva

8、turewhichgoeslike1/λ4andwecan’ttrustthesupergravitydescription.ThegravitydualisvalidwhenΛQCD≫UV,i.e,whenλ4islarge.Atlargeλ4theQFTisn’tweaklycoupl