Effective Hamiltonian for Scalar Theories in the G

arXiv:hep-ph/9507357v321Aug1995EffectiveHamiltonianforScalarTheoriesintheGaussianApproximationH.W.L.Naus,T.GasenzerandH.–J.PirnerInstitutf¨urTheoretischePhysikUniversit¨atHeidelberg,GermanyFebruary1,2008AbstractWeuseaGaussianwavefunctionalforthegroundstatetoreordertheHamiltonianintoafreepartwithavariationallydeterminedmassandtherest.Oncespon-taneoussymmetrybreakingistakenintoaccount,theresidualHamiltoniancan,inprinciple,betreatedperturbatively.InthisschemeweanalyzetheO(1)andO(2)scalarmodels.FortheO(2)–theorywefirstexplicitlycalculatethemasslessGoldstoneexcitationandthenshowthattheone-loopcorrectionsoftheeffectiveHamiltoniandonotgenerateamass.HD–TVP–95–1111IntroductionHamiltonianfieldtheoryhasbeenthestartingpointofmodernquantumfieldtheory,butsincethenhasbeenlessusedthanLagrangianfieldtheoryoractionorientedapproaches.TheadvantageofaHamiltonianformulation[1]istheappearanceof”wavefunctions”,whicharefunctionalsofthefieldsandallowanintuitiveunderstandingofthegroundstate.Recently,fortheanalysisofgaugetheories[2]andlight–conetheory[3]aHamiltoniantreatmenthasbeenrevived.TheconceptofaneffectiveHamiltonian,asexploitedinthispaper,hasactuallybeendevelopedrecentlyinthecontextofone–dimensionallight–conetheories[4].Thereexistsasubstantialliterature[5,6]ontheuseofvariationaltreatmentsinHamiltonianfieldtheory.OneevenfindsquantitativeestimatesoftheHiggs–mass,basedonthistechnique[7].Thevariationaltechniqueisinherentlynonperturbative,soonehopestocapturefeatureswhichgobeyondthestandardloopexpansion.Inmany–bodytheorytheGaussianwavefunctionalwithaneffectivemasscorrespondstotheself–consistentmeanfieldtheory.Infact,alsomoresophisticatedmethodsofmany–bodytheoryliketheclusterexpansion[8]ortheRPA–approximationcanbeusedtoimprovethemeanfieldresult.InthispaperwetreatO(N)–modelsforN=1,2toanalysetwomainfeatures,(i)theboundsonthemassm(1)ofthemassivescalarafterrenormalization,and(ii)themassm(2)oftheGoldstoneparticle.Intherecentliterature[7]aHiggs–massof2TeVhasbeen”predicted”fortheexperimentallygivenvacuumexpectationvalueoftheHiggsfieldintheStandardModel.WeuseavariationaltreatmentofthescalarsectorwithafinitecutoffandpresentresultsdifferentfromthepreviousestimatesinHamiltonianfieldtheory:MHiggs≤1.7TeV.InpreviousHamiltonianbasedworkontheO(2)–modelafiniteGoldstonebosonmassm(2)≃m(1)/2.06[9]isobtained,incontradictiontoGoldstone’stheorem.WewillshowthatasymmetricGaussianansatzandafurtherrediagonalizationoftheeffectiveHamiltonianexactlygivesazeromassGoldstoneexcitation.ThisresultformsthebasisforanyfurtherexplorationoftheHiggsmodelormorecomplicatedgaugetheorieslikeQCD.Possibleproblemsduetotheviolationoflocalgaugeinvariance,whichingeneralappearinapproximativetreatments,willbeavoidedbystartingwith”gaugefixedHamiltonians”fromwhichtheredundantdegreesoffreedomhavebeeneliminated[2].TherehasbeenanextensivediscussionoftheGaussianEffectivePotential(GEP)[5].Theemphasisofthatworkhasbeentocalculatetheenergyofthevacuumasafunctionofthesymmetrybreakingzeromodeofthefield.Ingeneralthemassgaporthespectrumofparticles,however,isthemoreinterestingphenomenologicalquantity.Thereforeonehastoaddresstheproblemofcalculatingtheenergyandthedispersionrelationofexcitedstates.FromourpointofviewtheGaussianwavefunctionalpresentsanefficientwayofreorderingtheHamiltonianintoquadraticandhigherpolynomialparts.Ourapproachistime–independent;recently,alsotime–dependentvariationalequationshavebeeninvestigatedinφ4fieldtheory[10].In0+1dimensions,i.e.quantummechanics,theanharmonicoscillatorisaneluci-datingexample.Itshowstheproblemswearefacinginthecaseofspontaneoussymmetrybreaking(SSB)intheO(2)–model.ThegroundstatesolutionofHH=12p2+m2x2+λx4,(1)isapproximatedbytheGaussianwavefunctionφG(x)=Ωπ12exp−12Ω(x−x0)2.(2)MinimizationoftheexpectationvalueoftheHamiltonianoperator,VG(x0,Ω)=hφG|H|φGi=14Ω+m24Ω+12m2x20+λ3Ωx20+34Ω2+x40(3)2withrespecttox0andΩgivestheequationsx0m2+6λΩ+4λx20!=0,Ω2−m2−12λx20−6λΩ=0.(4)Forthecasex06=0oneexplicitlyfindsΩ2=8λx20.(5)Equivalently,onedefinesthetrialgroundstateviacreationandannihilationoperatorsa†ΩandaΩ,aΩ|φGi=0,hφG|φGi=1,hφG|x|φGi=x0.(6)Notethatinthiswayexcitedstatesarealsoimplicitlydefined.Coordinateandmomentumarecorrespondinglydecomposedp=−isΩ2aΩ−a†Ωand˜x:=x−x0=s12ΩaΩ+a†Ω.(7)NormalorderingwithrespecttoaΩanda†Ω,whichisdenotedby::,yields:x2:=x2−12Ω,:x3:=x3−32Ω:x:,:x4:=x4−62Ω:x2:−312Ω2,:p2:=p2−Ω2.(8)ThenormalorderedHamiltonianreadsH=:12p2+m2x2+λx4+Ω4+m24Ω+3λΩx2+3λ4Ω2:.(9)Herewithoneeasilyverifieseq.(3).NowweconsiderthedifferenceHR=H−VGandusingeqs.(4)weobtainHR=:12p2+(m2+6λΩ)(x2−x20)!+λ(x4−x40):=:12p2+(Ω2−12λx20)(x2−x20)+λ(x4−x40):=:12p2+Ω2˜x2+λ˜x4+4λx0˜x3:.(10)HRcanberewrittenintermsofthecreationandannihilationoperatorsHR=H0+HI,(11)3withH0=Ωa†ΩaΩ,(12)HI=4λx012Ω32a3Ω+3a†Ωa2Ω+3a†2ΩaΩ+a†3Ω+λ12Ω2a4Ω+4a†Ωa3Ω+6a†2Ωa2Ω+4a†3ΩaΩ+a†4Ω,(13)whereweseparatedtheharmonicpartH0.TheremainingpartoftheHamiltonian,i.e.HI,willbetreatedinperturbationtheory.WewillseethattheuseofperturbationtheoryiswelljustifiedifΩislarge.Thisisobviousforx0=0,whereonlythequarticinteractionispresent.Inthe”broken”phase,howev