Effective Field Theory for Long Strings 3 1 0 2 n M. Baker a DepartmentofPhysics,UniversityofWashington,Seattle J Box351560,SeattleWA98195,USA 8 1 E-mail:[email protected] ] h InpreviousworkweusedmagneticSU(N)gaugetheorywithadjointrepresentationHiggsscalars p to describe the long distance quark-antiquark interaction in pure Yang-Mills theory, and later - p to obtain an effectivestring theory. The empiricallydeterminedparametersof the non-Abelian e effectivetheoryyieldedZ fluxtubesresemblingthoseoftheAbelianHiggsmodelwithLandau- h N [ Ginzburg parameter equal to 1/√2, corresponding to a superconductoron the border between typeIandtypeII.However,thephysicalsignificanceofthedifferencesbetweentheAbelianand 1 v the ZN vortices was not elucidated and no principle was found to fix the value of the ’Landau- 7 Ginzburgparameter’k ofthenon-AbeliantheorydeterminingthestructureoftheZ vortices. N 6 4 Here we reexaminethis pointof view. We propose a consistency conditionon ZN vortices un- 4 derlying a confining string. This fixes the value of k . The transverse distribution of pressure . 1 p(r)intheresultingZ fluxtubesprovidesaphysicalpictureofthesevorticeswhichdifferses- N 0 sentially from that of the vortices of the Abelian Higgs model. We speculate that this general 3 1 picture is valid independentof the details of the effective magnetic gaugetheoryfromwhich it : v wasobtained. LongwavelengthfluctuationsoftheaxisoftheZ vorticesleadfromaneffective N i X fieldtheorytoaneffectivestringtheorywiththeNambu-Gotoaction.Thiseffectivestringtheory r dependsona single parameter,thestringtension s . Incontrast, the effectivefield theoryhasa a secondparameter,theintrinsicwidth1/Mofthefluxtube,andisapplicableatintermediatedis- tancesinarangebetween0.2fmand1fm,wherethecontributionoftheintrinsicwidthincreases thefluxtubewidthoverthatpredictedbyeffectivestringtheory. XthQuarkConfinementandtheHadronSpectrum 8–12October2012 TUMCampusGarching,Munich,Germany (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ EffectiveFieldTheoryforLongStrings 1. Introduction TheprincipalgoalofthistalkistoreexaminemagneticSU(N)gaugetheorywhichwe have used [1] as an effective field theoryof the long distance quark-antiquarkinteraction, andto elucidate thepropertiesofthe Z flux tubesfoundin thetheory, N In Sections 2 to 4 we write downthe Lagrangianofthe effective SU(N)gaugetheory. We obtain a relation, applicable for any configuration of the Higgs fields, between the string tension s and the transverse distribution of pressure p(r) in the resulting Z flux N ¥ tubes. In section 5, using this relation, we impose a constraint on p(r), rp(r)dr =0, 0 which we speculateis a necessaryconditionfor a fluxtubeto behaveasa string. R In sections 6 and 7 we consider SU(3), where we have found explicit classical Z flux 3 tubesolutions,andwecomparethesesolutionstothosefoundintheAbelianHiggsmodel. Weplotthepressuredistributions p(r)intheZ fluxtube,anddescribethephysicalpicture 3 thatemerges. Thepressureispositiveneartheaxisandatlargerdistancesitisnegative. It is natural to associate the boundary between the outside and inside of the string with the pointat whichthe pressure vanishes. In sections 8 and 9 we show that long wavelength fluctuations of the flux tube axis lead from an effective field theory to an effective string theory with a single parameter, s . The contribution of these fluctuations to the flux tube width [2] fixes the value of the short distancecutoff1/L oftheeffectivefieldtheoryatavaluelessthantheintrinsicwidth1/M. Thusthetheorycan resolve distancescales onthe orderof1/M. Finally,weexaminethe impactof string fluctuations onthe domainofapplicabilityofthe effective field theory. 2. Effective FieldTheory The Lagrangian Leff couples magnetic SU(N) gauge potentials, Cm to three adjoint representationscalar fieldsf . Thegaugecouplingconstantis g . i m 1 1 Leff(Cm ,f i)=2tr( Gmn Gmn + (Dm f i)2) V(f i), (2.1) −4 2 − Gmn =¶ m Cn ¶ n Cm igm[Cm ,Cn ], Dm f i=¶ m f i igm[Cm ,f i]. (2.2) − − − Thecomponentsofthe field tensorGmn definecolorelectric andmagneticfields~E and~B. 1 Ek= e Glm,Bk =Gk0. (2.3) klmn 2 TheHiggspotentialV(f )is generatedfrom oneloopgraphsofSU(N)gaugetheory:[1] i V(f )= m 2N (cid:229) 22trf 2+4Nl tr((cid:229) f 2f 2)+ 1(tr((cid:229) f 2))2+ 2 (cid:229) (trf f )2 , (2.4) i 4 i 3 i j N i N i j i ij i ij ! where the parameter m 2 has dimensions of mass squared and l is dimensionless. The SU(N)gaugesymmetry ofL reflects thatoftheoriginalSU(N)Yang-Millstheory. eff In the confining vacuum the magnetic SU(N) /Z gauge symmetry is completely bro- N ken by a Higgs condensate<f >=f =f J, wherethe three matrices J are the gener- i i0 0 i i ators of N-dimensionalirreducible representationof the three-dimensionalrotation group. TheHiggspotentialhasanabsoluteminimumatf =f : f 2= 9m 2 . i i0 0 −8(N2 1)l − 2 EffectiveFieldTheoryforLongStrings 3. Z Electric FluxTubes N Atlargedistancesrfromthefluxtubeaxisf i andCm areagaugetransformationW (q ) of thevacuumf i =f i0,Cm =0, whichwe can chooseto beAbelian; W (q )=exp(iq Y). i f i W −1(q )f i0W (q ), Cm W −1(q )¶ m W (q ). (3.1) → → g m Therequirementthatf besingle valued exp(i2p Y)is an elementofZ . i N → Asr ¥ → 1 k C~ eˆq Y, exp(igm C~ d~l) exp(2p i ), k=1,2,N 1. (3.2) → g ,r · → N − m I Assuming thegaugepotentialC~ =C(r)eˆq Y everywhereimpliesthatthe electric field 1 d(rC(r)) ~E = (cid:209) C~(~x)Y = eˆ Y. (3.3) z − × −r dr Thefiniteness oftheflux tubeenergy f =0on theflux tubeaxis. i → 4. RelationBetween String Tensionand Stress Tensor inSU(N) FluxTubes Usingthe Abelianansatz(3.3)andtheresulting classical static equation (cid:209) ~E = ~j =ig [f ,D~f ] (4.1) m i i × − to evaluate L gives the following general relation between the string tension s , the eff stress tensor component Tqq , and ~E(r=0), the color electric field on the axis of the flux tube: ¥ 2p rTqq (r)dr= 2tr(2p Yeˆ ~E(r=0)) s . (4.2) Z0 r2 − gm · − validforanyconfigurationoftheHiggsfieldsf . Thequantity 2tr(2p Yeˆ ~E(r=0))R=W, i − gm z· the work necessaryto separatea qq¯ pairlying on thez-axis bya distance R. IfTqq >0thegaugerepulsionexceedstheHiggsattractionproducedbythecirculating magnetic currents generated by the Higgs condensate, and (4.2) implies that W > s R. Thatis,whenthereisnetrepulsion,theworkW neededtoseparatetheqq¯pairadistance R inthe fixed finalvortexfield~E(r=0) isgreaterthans R,whichitself is equalto thework donein a field~E thatis beingbuiltupasthe qq¯ pairis separated. If there is compensationbetween the net attractive and repulsive contributions to the pressure p(r)=Tqq /r2 averagedoverthe widthofthe flux tube;thatis, if ¥ 2p rTqq dr= ¥ 2p rp(r)dr=0, (4.3) r2 0 0 Z Z then thestring tension s =W/R,determinedby thefield ~E(r=0) on theflux tubeaxis. 3 EffectiveFieldTheoryforLongStrings 5. Speculation onEffective FieldTheories Describing Long Strings Consider now a flux tube connecting a qq¯ pair located at z = R/2 having energy ± V (R), the heavy quark potential. The force acting on the quarks is determined by the 0 color field at the positions of the quarks and is equal to dV /dR. If the long distance 0 potentialV (R)=s Rpersiststo distancesR,thenthisfieldisfixed bythestring tensions . 0 If condition (4.3) is met, the field ~E on the z axis near the midpoint of the flux tube is also fixed by the value of s . In this situation, it is consistent to assume the field has the same value, proportionalto s , at all points on the z axis between the qq¯ pair ; that is, the flux tube behaves like a string, consistent with the assumption that the long distance qq¯ interactionpersiststoshortdistances. Thisargumentfailsifcondition(4.3)isnotsatisfied. We assume that (4.3) must be satisfied for any effective field theory describing the confining string in SU(N)Yang-Mills theory, andwe impose this condition to constrain the parametersin L . eff 6. ClassicalStatic SU(3)Flux Tube Solutions ForSU(3)we havefoundexplicit classical static solutions[1]with l J = l , J = l , J =l , Y = 8 , (6.1) x 7 y 5 z 2 − √3 (l il ) (l +il ) f 1 = f 1(~x) 7−2 6 +f 1∗(~x) 7 2 6 , ( l il ) ( l +il ) f 2 = f 2(~x) − 52− 4 +f 2∗(~x) − 52 4 , f = f (~x)l , 3 3 2 C~ = C(r)eˆq Y, f 1(~x)=f (r)exp( iq ), f 2(~x)=f (r)exp(iq ), f 3(~x)=f 3(r). − Thecommutationrelations [Y,l il ]=l il , [Y, l il ]= ( l il ), [Y,l ]=0 (6.2) 7 6 7 6 5 4 5 4 2 − − − − − − − showthattheHiggsfieldsf f ,f carryY charge 1,1,0respectively,andthattheansatz 1 2 3 − (6.1) is consistentwith equation(4.1). We rescale the fields choosing the flux tube radius 1 as the scale of length, making M the replacementr r/M,C MC, f f f , f f f , with M=√6g f . Inserting (6.1) → → gm → 0 3 → o 3 m 0 into the effective Lagrangian (2.1) yields the energy density T and the stress tensor 00 componentTqq : 4M2 1 1d(rC) 1 1 1 df 1 df T = ( )2+ (C )2f 2+ ( )2+ ( 3)2+V(f ,f ) , (6.3) 00 3 g2 2 r dr 2 − r 2 dr 4 dr 3 m (cid:20) (cid:21) Tqq = 4M4 1(1d(rC))2+1(C 1)2f 2 1(df )2 1(df 3)2 V(f ,f ) . (6.4) r2 3 g2 2 r dr 2 − r −2 dr −4 dr − 3 m (cid:20) (cid:21) 4 EffectiveFieldTheoryforLongStrings where (f 2 1)2 (f 2 1)2 (f 2 1)(1 f 2) 25 l V(f ,f )=k 2 − +9 3 − 7 3 − − , k 2 . (6.5) 3 4 100 − 50 ≡ 9 g2 (cid:18) (cid:19) m Separatingthegaugecontribution andthe Higgscontribution to T00 andTqq gives ¥ 4M2 4M2 2p rT dr=s = s (k ) = (s (k )+s (k )), (6.6) 00 3 g2 3 g2 g h Z0 m m ¥ 2p rTqq dr = 4M2(s (k ) s (k )), (6.7) Z0 r2 3 g2m g − h where ¥ 1 1d(rC) 1 1 s (k ) 2p rdr ( )2+ (C )2f 2 , (6.8) g ≡ 0 2 r dr 2 − r Z (cid:18) (cid:19) and ¥ 1 df 1 df s (k ) 2p rdr ( )2+ ( 3)2+V(f ,f ) . (6.9) h 3 ≡ 0 2 dr 4 dr Z (cid:18) (cid:19) 7. Results forSU(3)String Tensionand Stress Tensor Figure1:Tqq /rvsr. Red,longdashed,k 2=0.5; blue,thick,k 2=0.59; green,shortdashed,k 2=0.8. Note that if f (r) has its vacuum value f =1, V(f ,f ) reduces to the Higgs potential 3 3 3 of the Abelian Higgs model with Landau Ginzburg parameter k . Furthermore, numerical solutionoftheclassicalequationsshowsthatf <1andf >1everywhere;hencetheterm 3 coupling f and f in (6.5) is attractive. This additional attraction in V(f ,f ) reduces the 3 3 energy of the Z vortex below that of the Abrikosov-Nielsen-Olesen vortex of the Abelian 3 Higgs model. The ANO vortex can then be viewed as an unstable configuration of the non-Abeliantheorythatsubsequentlydecays to the stationaryclassical solution f (r). 3 Condition (4.3) along with (6.7) yield s (k )= s (k ), which determines the physical g h valueofk ;k 2 0.6. Thestring tensions (k 2 0.6) 3.1, approximatelyequaltoits value ≈ ≈ ≈ in the Abelian Higgs model at k 2 =1/2. Figure 1 shows Tqq /r evaluated at the classical 5 EffectiveFieldTheoryforLongStrings solution as a function of r for three values of k 2. For k 2 0.6, where condition (4.3) is ≈ satisfied,Tqq =0atr r 1.7/M; thereisrepulsionatr r andattractionatr>r . Itis ≡ ∗∼ ≤ ∗ ∗ naturaltoidentifyr asa boundary,separatingthe insideofthe fluxtubefrom its exterior. ∗ In contrast, in the Abelian Higgs modelk = 1 is a BPS state [3], and condition (4.3) √2 is satisfied exactly because the componentsTqq andTrr ofthe stress tensor vanish for all r [4], andthustheprofile ofTqq (r) doesnotrevealtheboundaryof theflux tube. We speculate that the difference between the properties of the stress tensor inside (positive netpressure) andoutside the flux tube (negative netpressure) is a fundamental physical property of flux tubes giving rise to a confining string. The difference between the Abelian and non-Abelian theories is caused by the additional attractive interaction between the scalar particles, which breaks the supersymmetry [5] giving rise to the BPS AbelianHiggsvortex, andstabilizes thenon-Abelianflux tube. Indeed,aswe haveseen, the additional attraction in the Higgs potential of the non-Abelian theory is approximately balancedatk 2 0.6 bythe additionalrepulsionassociated with the fact thatk 2>1/2. ≈ 8. From Effective FieldTheory to Effective String Theory The Higgs fields f vanish on the axis L of the static flux tube. Long wavelength fluctuations of the axis L of a flux tube connecting a quark-antiquark pair sweep out a spacetimesurfacex˜m (z )onwhichf vanishes. TheWilsonloopW(G )ofYang-Millstheory is the path integral over all field configurations for which the Higgs fields vanish on some surface x˜m (z ) whoseboundaryis the loopG . W(G )= DCm Df exp(iS(Cm ,f ), S(Cm ,f )= dxLeff(Cm ,f ). (8.1) Z Z We transformW(G ) to apath integraloverthevortex sheetsx˜m (z ) in two stages: 1. Wefixthelocationx˜m (z )ofthevortexandintegrateoverfieldconfigurationsCm (x), f (x) for which f (x)x=x˜(z ) =0. The integration (8.1) over these configurations Seff(x˜), | → the action ofthe effective string theory. 2. We then integrate over all surfaces x˜m (z ). This integration putsW(G ) into the form of apartition functionofan effective string theory: [6] W(G )= Dx˜m ...exp[iS (x˜m )]. (8.2) eff Z Thepathintegral(8.2)goesoverthe two transverse fluctuationsoftheworld sheetx˜m (z ). m ThefieldmodescontributingtoS [x˜ ]havemasses>M. Fluctuationsofwavelength eff > 1/M are string fluctuations accounted for by (8.2). We can then replace integrations (8.1) over field configurationsCm ,f by the classical configuration minimizing S(Cm ,f ) for fixed position xm (z ) ofthe vortex. exp(iS (x˜m (z ))) exp(iSclass(x˜m (z ))). eff ≈ Whencondition(4.3)issatisfied,thelinearpotentialpersistswhenastraightfluxtube isshortened. Likewise,bendingthefluxtubeslightlygivesachangeinenergyproportional 6 EffectiveFieldTheoryforLongStrings to the change D R in length: D E =s D R. The action of the effective of the effective field theorybecomesthe Nambu-Gotoactionproportionalto the areaofthe vortexsheet. S (x˜m )=s d2x g(x ) S (x˜m ). (8.3) eff NG − ≡ Z p 9. HeavyQuarkPotentials andFlux Tube Shape To obtain the heavy quark potential V (R) and transverse energy profiles between 0 static quarks separated by distance R we couple the vector potentialC~ to a Dirac string, writing 2p ~E = (cid:209) C~ d (x)d (y)(q (z+R/2) q (z R/2))eˆ Y (9.1) z − × − g − − m in the Lagrangian L , and solving the resulting static equations [7]. We compared the eff results [8] with lattice data for heavy quark potentials [9], and found that g 3.91; i. e., m ≈ M 1.9√s . Furthermore,thesecalculationswereconsistentwithSU(2)latticesimulations ≈ [10]fortransverseenergyprofilesforarangeofinterquarkspacings0.25/√s R<2/√s . ≤ Theabovecalculationsdidnotexplicitly includethe contributionofstring fluctuations. However,stringfluctuationsrenormalizetheintrinsicwidthandthereforetheyaretosome extent accounted for in the empirically determined value of M. For distances larger than 1/√s , string fluctuations become dominant, leading to the logarithmic increase of the ∼ meansquarewidth oftheflux tubeatits midpoint[11]; d 2 R w2(R/2)= − log . (9.2) 2ps r 0 Recent lattice simulations of (2+1)dSU(2) Yang-Mills theory [2] extending to dis- tances R=36/√s gave excellent agreement with the predictions of effective string the- ory for distances R>1.5/√s , and determined the value of r =0.364/√s . (Interpreting 0 1/r =L asthecutoffoftheeffectivefieldtheorygivesL 2.75√s 1.41M.) However,for 0 ≈ ≈ distances1.5/√s >R>0.2/√s thelattice simulationsofw2(R/2)lie abovetheprediction (9.2),indicatingthattheintrinsicwidthofthefluxtubemustbetakenintoaccountatthese qq¯ separations. It is in this intermediate range, shown schematically in Figure 2, that we can test the physical picture oftheconfining string given bythe effective field theory. Figure2: SchematicshowingapproximatedomainsofapplicabilityofEffectiveFieldTheory(EFT)(solid blueline)andEffectiveStringTheory(EST)(reddashedline). 7 EffectiveFieldTheoryforLongStrings With use of analytic regularization [12], string fluctuations do not renormalize the string tension s , andhence its physical interpretationas the energyperunit lengthof the classical flux tube is preserved. The leading large distance correction to the heavy quark potential is the Lüscher term p (d 2)/24R [13], which can be regarded as a renormal- − − ization of the intrinsic width at the intermediate distances shown as the region of overlap in Figure2. 10. Summary and Future Work We have presented a physical picture of the Z flux tubes giving rise to a confining N string. In this picture net positive pressure in the interior of the Z vortices balances N net negative pressure outside. (Perhaps at the deconfinement temperature the flux tube ’bursts’! ) Wespeculatethatthisgeneraldescriptionisvalid,independentofthedetailsof the effective magneticgaugetheoryfrom whichit wasobtained. Comparisonwithlatticesimulationsatintermediatedistanceswouldtestourhypothe- sisthatthereisaneffectivefieldtheoryunderlyingtheconfiningstring. SinceM 3 times ∼ T , the SU(3) deconfinementtemperature, the theory should be applicable for a range of C temperatures in the deconfined phase, where it was used [14] in a preliminary study of spatialWilson loopsandwhereweexpect somemanifestationofthe Higgsfield. Acknowledgment: I would like to thank the organizersfor the opportunityto participate in this very stim- ulating conference. References [1] M.Baker,J.S.BallandF.Zachariasen,Phys.Rev.D41,2612(1990). [2] F.Gliozzi,M.PepeandO.-JWeise,Phys.Rev.Lett.104,232001(2010). [3] E.B.Bogomolny,Sov.J.Nucl.Phys.24,449(1976)[Yad.Fiz.24,861(1976)];M.K.Prasad andC.M.Sommerfield,Phys.Rev.Lett.35,760(1975). [4] H.J.deVegaandF.A.Schaposnik,Phys.Rev.D14,1100(1976). [5] P.Fayet,IlNuovoCimento31A,626(1976). [6] M.BakerandR.Steinke,Phys.Rev.D63,094013(2001);D65,114042(2002). [7] M.Baker,J.S.BallandF.Zachariasen,Phys.Rev.D44,3328(1991). [8] M.Baker,J.S.BallandF.Zachariasen,Phys.Rev.D56,4400(1997). [9] G.S.Bali,K.SchillingandA.Wachter.Phys.Rev.D562566(1997). [10] A.M.Green,C.MichaelandP.S.Spencer,Phys.Rev.D55,1216(1997). [11] M.Lüscher,G.MünsterandP.Weisz,Nucl.Phys.B180,1(1981). [12] K.DietzandT.Filk,Phys.Rev.D27,2944(1982). [13] M.Lüscher,Nucl.PhysB180,317(1981). [14] M.Baker,Phys.Rev.D78,014009,(2008). 8