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BCCUNY-HEP/08-01 Near-integrability andconfinement forhigh-energy hadron-hadron collisions Peter Orland ∗ 1. TheNielsBohrInstitute,TheNielsBohrInternationalAcademy,Blegdamsvej17,DK-2100,CopenhagenØ,Denmark 2. TheGraduateSchoolandUniversityCenter,TheCityUniversityofNewYork,365FifthAvenue,NewYork,NY10016,U.S.A.and 3. BaruchCollege, TheCityUniversityofNewYork,17LexingtonAvenue, NewYork,NY10010, U.S.A. (Dated:January20,2008) WeinvestigateaneffectiveHamiltonianforQCDatlarges,inwhichlongitudinalgaugedegreesoffreedom are suppressed, but not eliminated. In an axial gauge the effective field theory is a set of coupled (1+1)- dimensional principal-chiral models, which are completely integrable. The confinement problem issolvable 8 inthiscontext, andwefindthelongitudinalandtransversestringtensionswithtechniquesalreadyusedfora 0 similarHamiltonianin(2+1)-dimensions.WefindsomeaposteriorijustificationfortheeffectiveHamiltonian 0 asaneikonalapproximation. Hadronsinthisapproximationconsistofpartons,whicharequarksandsoliton- 2 likeexcitationsofthesigmamodels.Diffractivehadron-hadronscatteringappearsprimarilyduetoexchangeof longitudinalfluxbetweenpartons. n a J PACSnumbers:11.10.Jj,11.10Kk,11.15.Ha,11.15.Tk,11.80.Fv,12.38.Aw,12.39.Mk,13.85.Dz 0 2 I. INTRODUCTION Astajt×esAYk)m, ruesstpseacttiisvfeylyGaaunsds’(sAlajw×Ak)a = fbacAbjAck. Physical ] h p In the last decade and a half, effective gauge-theory de- (¶ E +¶ 3E3 r )Y =0, (1.3) - scriptions for QCD at large center-of-massenergy squared s ⊥· ⊥ − p hasbeenanactiveareaofresearch[1],[2],[3], [4],[5],[6], wherer isthequarkcolor-chargedensity. Asofthiswriting, e [7], [8]. The approximation of Reference [6] was to elimi- we are not certain that this rescaled theory is definitely de- h nate some gauge-theory degrees of freedom by a longitudi- scribingQCD,butthinkitmaybeausefulphenomenological [ nal rescaling. In this paper we carefully examine the con- approachtosmall-x,large-sscattering. 2 sequences of this rescaling: x0,3 l x0,3, x⊥ x⊥, A0,3 TheHamiltonian(1.2)andtheconstraint(1.3)willberegu- 9v lno−t1eAs0t,h3e,Aco⊥m→poAn⊥en,twshoefrteheAmSU=(N→Aa)m tYa,anag=-M1i,l.l→s..fi,Nel2d−an1d,dt→hee- lwaarirzdesd,xo3nwaisllpaaltsiaolblaetmticaed[e9c]o,knetienpuionugst,ilmeaevcionngtoinnulyouths.eAtrfatnesr-- 8 transverseindices1,2aresometimescollectivelydenotedby versecoordinatesdiscrete. Wetakeanaxialgaugecondition, 3 0 ⊥. We normalize Trtatb = d ab and define ifacbtc = [ta,tb]. breakingtheprecedentofconsideringlarge-sscatteringinthe . Thecenter-of-massenergysquaredchangesass l 2s[6]. light-conegauge. Wedothisbecause,asweshowlater,(1.2) 1 − 0 Sincewewishtoconsiderhighenergies,wetake→l 1. is similar to the anisotropic (2+1)-dimensional Yang-Mills 8 If the scale factor l is small, but not zero, the≪resulting theoryinthisgauge[10],[11],[12],[13],[14].Iffurtherexpe- 0 Hamiltonian has one extremely small coupling and one ex- riencepersuadesustousealight-conelattice[15]instead,so : tremelylargecoupling.Therescaledactionis beit.Alight-cone-latticeapproachinthisspiritwasdiscussed v inReferences[16]. Thiswasthestartingpointofanargument i arX S=21g20Z d4xTr l −2F023+j(cid:229)=21F02j−j(cid:229)=21Fj23−l 2F122!,(1.1) jmtuhseatli(ifz2ya+itniog1n)t-hgderiomrueepsnc[sa1iloi7nn]g.alSucisamesdielai[nr13a(1]r,g.1[u1)m4be]ynwtasintwhaoenurietsokptnrreooswpeilncetdreegdnefooorrf- Reference[17].Weshouldmentionthatalatticegaugetheory where Fmn =¶ m An ¶ n Am i[Am ,An ]. The Hamiltonian in with differenttransverse and longitudinalcouplingshas also − − A0=0gaugeistherefore beenstudiedinReference[18]. The analogous rescaling in (2+1)-dimensions gives the H= d3x g20E2+ 1 B2 +(g′0)2E2 Hamiltonian,inA0=0gauge1, Z (cid:18) 2 ⊥ 2g20 ⊥ 2 3 g2 1 (g )2 1 H= d2x 0E2+ B2+ ′0 E2 , (1.4) + B2 , (1.2) 2 1 2g2 2 2 2(g )2 3 Z (cid:18) 0 (cid:19) ′0′ (cid:19) whereg′0=l g0,andtheelectricandmagneticfieldsareEi= whereg′0=l g0,g′0′=l −1g0 (sothatg′0g′0′=g20),theelectric id /d AiandB=(e jk¶ jAk+Aj Aj),respectively.Gauss’s andmagneticfieldsare Ei= id /d Ai andBi=e ijk(¶ jAk+ − × − 1Wenotethatthecoordinateindices1and2arereversedinReferences[10], Electronicaddress:[email protected] [11],[12],[13],[14]. ∗ 2 lawis Afterwediscusstheseconfinementmechanismsforthethe- oryin(3+1)dimensions,weattemptto justifytheeffective (¶ 1 E1+¶ 2E2 r )Y =0. (1.5) Hamiltonianasaneikonalapproximation. Thisviewwasad- · − vocatedinReference[6],butweinterpretthemeaningofthe Uponregularization,confinementofquarkscanbereadily factorl slightlydifferently. demonstrated. Thusthe hadronizationproblemis essentially Eachofthetwoincominghadronsinacollisionprocessisa solved. Thissuggeststhattheelasticpartoftheforwardam- collectionofFZparticles,joinedbylongitudinal-electricflux plitude and the soft Pomeron can eventually be understood lines. Scatteringoftwohadronscaninvolverearrangementof usingourmethods. It has been noted before that the l 0 limit of QCD is the flux lines among the FZ particles. It is also possible for → anon-AbelianphaserotationofFZparticlestooccurasthey equivalentto a set of principal-chiralSU(N) SU(N) sigma × passthrougheachother;thephasefactorisgivenbytheexact models [6], [16]. Such a sigma model has the Lagrangian L =1/(2g2)h mn Tr¶ m U†¶ n U, where U SU(N) and m = (1+1)-dimensionalexactS-matrix[19].Thesetwoprocesses 0 ∈ are notentirely distinct, ascan be seen froman examination 0,3. Thisfieldtheoryisintegrable,andtheS-matrix[19]and ofthe1/NexpansionofthisS-matrix. evencertainoff-shellinformation(forN =2)[20] isexactly Inthenextsection,weputtheHamiltonian(1.2)onaspa- known.Theintegrablenatureoflarge-sscatteringisalsoindi- tiallattice,thentransformtotheaxialgauge.InSection3,we catedbytheformofReggeizedamplitudes[21].InReference takethecontinuumlimitinthex3-directionandshowthatthis [6]itwaspointedoutthattheeffectivegluon-emissionvertex Hamiltonian is equivalent to a collection of integrable field [22] can be derived by considering longitudinal fluctuations beforetakingthetransverselimitl 0. theorieswhicharecoupledtogether. Wediscussconfinement → ofcolorandfindthetransversestringtensioninSection4. We The particles of the principal-chiral sigma model are la- findthelongitudinalstringtensioninSection5.Bycomparing beled by r=1,...,N 1 [19]. The particle with labelr has − theelectricfieldstrengthsindifferentdirections,wefindsome anantiparticlewithlabelN r. Themassspectrumis − justificationoftheeffectiveHamiltonianinSection6. InSec- sinrp 4p tion7,wedescribethenatureofhadronicstatesandbeginan mr=m1sinNp , m1=KL (g20N)−1/2e−g20N +... , (1.6) examination of diffractive hadronic scattering, focussing on N theexchangeofhadronicflux. We presentsomeconclusions whereK is a non-universalconstant, L is theultravioletcut- andmentionsomefurtherdirectionsforresearchinSection8. offandtheellipsesdenotenon-universalcorrections. WestudythephysicalstatesfortheeffectiveHamiltonian. They are very similar to those of the (2+1)-dimensional II. REGULARIZATIONANDTHEAXIALGAUGE model(1.4),(1.5),alreadyinvestigatedindetailinReferences [10], [11], [12], [13], [14]. Hadronsconsistofquarksjoined Consider a lattice of sites x, whose coordinates are xj, by electric strings. Longitudinalelectric flux consists essen- j=1,2,3,wherexj/aareintegers,andwhereaisthelattice tiallyof(1+1)-dimensionalYang-Millsstrings. Incontrast, spacing. Eachlinkisapairx, j,andjoinsthesitextox+jâ, transverseelectricfluxisbuiltoutofthemassiveparticlesof where jîsaunitvectorinthe jthdirection.Wechoosetempo- thesigmamodel.WecalltheseFaddeev-Zamolodchikov(FZ) ralgaugeA =0. Beforeanyspatialgaugefixing,thedegrees 0 particles,consistentwiththeterminologyoftheoperatoralge- offreedomareelementsofthegroupSU(N)inthefundamen- braofthe creationandannihilationoperatorsforthese parti- tal(N N)-dimensionalmatrixrepresentationU (x) SU(N) j cles. ateach×linkx, j. Inaddition,therearetheelectric-fie∈ldoper- In(2+1)dimensions,thetransverseandlongitudinalstring atorsateachlinkl (x) ,b=1,...,N2 1. Thecommutation j b tensions2 were first foundto leading orderin g′0 [10]. Later, relationsonthelatticeare − thecorrectionsofhigherordering tothelongitudinalstring ′0 tension[11],andthetransversestringtension[14]werecom- [l (x) ,l (y) ]=id d fd l (x) , j b k c xy jk bc j d puted. Stringtensionsfordifferentrepresentationsofcharges werefoundanditwasshownthatadjointsourcesarenotcon- fined[12]. Thelow-lyingmassspectrumwasstudiedinRef- [lj(x)b,Uk(y)]= d xyd jktbUj(x), (2.1) − erence[13]. allotherszero. Both transverse electric flux and longitudinalelectric flux Thelatticeversionof(1.2)isH=H +H +H ,where havesomenonzerothickness.Thethicknessoftransverseflux 0 ′ ′′ isduetocolorsmearingoftheFZparticlesinthelongitudinal direction[14]. Thethicknessoflongitudinalfluxiscausedby H =1(cid:229) g20l (x)2 (cid:229) 1 TrU(cid:3)(x) , creationanddestructionofFZparticlepairs,akintovacuum 0 a x " 2 ⊥ −j=1,24g20 j # polarization[11]. H =(g′0)2(cid:229) l (x)2, H = 1 (cid:229) TrU(cid:3)(x),(2.2) ′ 2a 3 ′′ −4(g )2a 3 x ′0′ x andwhere,asbefore,g =l g ,g =l 1g and 2InReferences [10],[11],[12],[13],[14],wecalledthetransversestring ′0 0 ′0′ − 0 tensionthe“verticalstringtension”andthelongitudinalstringtensionthe “horizontalstringtension”. Ui(cid:3)(x)=e ijkUj(x)Uk(x+jâ)Uj(x+kâ)†Uk(x)†. (2.3) 3 We need the adjoint representation of the SU(N) gauge x1,2 =0,a,2a,...,L1,2 a, andx3 =0,a,2a,...,L3. Gauss’s field Rj(x), defined by Rj(x)bctc =Uj(x)tbUj(x)†. This has lawisstillgivenby(2.5−),provided(2.6)ismodifiedto the properties Rj(x) SU(N)/ZN, Rj(x)TRj(x) = 1, and detRj(x)=1. Notice∈that D3l3(x)=d x36=L3l3(x)−d x36=0R3(x⊥,x3−a)l3(x⊥,x3−a), [Rj(x)bclj(x)c,Uk(y)]=−d xyd jkUj(x)tb. DD21ll21((xx))==ll21((xx))−RR21((xx11,−x2a,xa2,,xx33))ll21((xx11,−x2a,xa2,,xx33))., (2.7) − − − Thus l (x) generates infinitesimal SU(N) transformationson j theleftofUj(x)andRj(x)lj(x)generatesinfinitesimalSU(N) Tofixthelinksinthe3-direction,weneedtouse(2.5)and transformations on the right of Uj(x). The squares of these (2.7)tosolveforl3: operatorsarethesame x3 [lj(x)]2=[Rj(x)lj(x)]2, l3(x)= (cid:229) q(x⊥,y3) (D l )(x⊥,y3) . (2.8) − ⊥· ⊥ byvirtueoftheorthogonalityoftheadjointrepresentation. y3=0h i Colorchargeoperators,denotedbyq(x) ,satisfy b Somenon-Abeliangaugeinvarianceremains,namely [q(x) ,q(y) ]=ifdd q(x) . (2.4) b c bc xy d L3 Schro¨dingerwave functionsare complex-valuedfunctions G (x⊥)Y = (cid:229) [(D⊥·l⊥)(x)−q(x)]Y =0. (2.9) x3=0 ofallthelinkdegreesoffreedomU (x). Physicalwavefunc- j tionsY ( U )satisfyGauss’law Thisremaininggaugeinvariancemeansthat(2.8)isequivalent { } to [(D l)(x)b q(x)b] Y ( U )=0. (2.5) · − { } L3 where(withnosummationof j) l3(x)= (cid:229) q(x⊥,z3) (D l )(x⊥,z3) . (2.10) − − ⊥· ⊥ [Djlj(x)]b=lj(x)b−Rj(x−jâ)bclj(x−jâ)c. (2.6) Thenewexpz3r=esxs3+ioanhsforl3,namely(2.8)and(2.i10),allow us to completely eliminate all degrees of freedom but U . NextweimposetheaxialgaugeconditionU =1. Wetake 2,3 3 We now can write the term in the Hamiltonian (2.2) which the lattice to be open at x3 =0,L3, which means we do not dependsonthelongitudinalelectricfieldas fix any non-contractible Wilson loops. The open boundary condition means, however, that a relic of Gauss’s law must stilWlbeecihmopoosseedsp.ace to be a lattice “cylinder” of size L1 H′=−(g2′0a)2(cid:229) (cid:229)x3 (cid:229)L3 q(x⊥,y3)−(D⊥·l⊥)(x⊥,y3) L2 L3, with periodic boundary conditions in the 1- and 2×- x⊥y3=0z3=x3+ah i dire×ctions only. This means that for any function of po- q(x⊥,z3) (D l )(x⊥,z3) , sition f(x), we have f(x1+mL1,x2+nL2,x3) = f(x), for × − ⊥· ⊥ h i any m,n Z. We take components of x to have the values or ∈ H′=−(4ga′0)22(cid:229) (cid:229)L3 |y3−z3| q(x⊥,y3)−(D⊥·l⊥)(x⊥,y3) q(x⊥,z3)−(D⊥·l⊥)(x⊥,z3) , (2.11) x⊥ y3,z3=0 h ih i Intheaxialgauge,thelongitudinal-electric-field-squaredterm nature of the vacuumstate is subtle. Mandelstam, who con- H ishighlynon-localinthe3-direction. Thisnon-localityis sideredtheanalogouscontinuumHamiltonian,arguedthatthe ′ astandardfeatureofphysicalgauges. Thisfacthasasimple vacuumstate canonlyhavefinite energydensityif magnetic physical interpretation in the case of (2.11). The longitudi- condensationtakesplace[24]. Hisreasoningwasthatthisis nal electric field has been eliminated in favor of the trans- thewayinwhichthevacuumcorrelator verse degrees of freedom. Now electric flux between two charged transverse plates must be proportional to the longi- 0 [q(x⊥,y3) (D l )(x⊥,y3)] h | − ⊥· ⊥ tudinal separation of these plates. This is accounted for by [q(x⊥,z3) (D l )(x⊥,z3)] 0 , the linear factor in (2.11). On the other hand, the vacuum × − ⊥· ⊥ | i expectation value of H must be proportional to the spacial canfalloffsufficientlyquicklyin y3 z3 . ′ volume L1L2L3. The non-locality of (2.11) means that the What of the remainder of the H| am−ilto|nian H +H after 0 ′′ 4 gaugefixing?SettingU1(x)=1,wefind respectively,wherem =0,3.TheoperatorJ0L(x⊥,x3,j)pro- ducesanSU(N)rotationattheedgeoftheband(x ,j)atx . ⊥ ⊥ H0=(cid:229) H0(x⊥,2)+H0(x⊥,3) , (2.12) TheoperatorJ0R(x⊥,x3,j)producesanSU(N)rotationatthe x⊥h i otheredgeoftheband(x⊥,j)atx⊥+jâ. where H H0(x⊥,j)= (cid:229)L3 g20lj(x)2 (cid:10)H(cid:10)(cid:10)H(cid:10)(cid:10)HH(cid:10)HH Ocohfnathnthegeegdoa,uthsgieenrcfihe−ealindtLxd(cid:229)33.d=−,e0taxph32e=egn01l20do2asnaogRnietluyTdroinUnajlt(hmxe)aUtgrajn(nexst⊥vice,xrtse3er+mcoaHm)†′p′.,o(in2se.u1nn3ts-) x63 (cid:10)(cid:10)(cid:30)x2 H(cid:10)(cid:10)HH(cid:10)(cid:10)HHH(cid:10)(cid:10)(cid:10)(cid:10)HHHH(cid:10)(cid:10)(cid:10)HHHH(cid:10)(cid:10)(cid:10)(cid:10)HHHH(cid:10)(cid:10)(cid:10)(cid:10)HHHHH(cid:10)(cid:10)H(cid:10)(cid:10)HHHH(cid:10)(cid:10)(cid:10)H(cid:10)(cid:10)HHH(cid:10)(cid:10)H(cid:10)(cid:10)(cid:10)(cid:10)HH(cid:10)H(cid:10)HH(cid:10)(cid:10)(cid:10)H(cid:10)HH(cid:10)(cid:10) NowthattheHamiltonianhasbeenrecastinanaxialgauge, (cid:10) wemay,atleastinprinciple,takethethermodynamiclimit,in (cid:10) H whichL1,L2 andL3 allbecomeinfinity. H H Hj x1 III. COUPLED(1+1)-DIMENSIONALFIELDTHEORIES FIG.1: Spacewithx discreteandx3 continuousresemblesthein- ⊥ Next let us examine each of the terms of the Hamilto- teriorofawinetransitbox. nian more closely. We will take a continuumlimit in the 3- direction. Thus only the transverse coordinateswill be latti- cized. Theresultingstructureresemblesatransitboxforbot- tlesofwineasshowninFig. 1. EachtermH0(x⊥,j),defined WewriteH0as in(2.13), isalattice (1+1)principal-chiralnonlinearsigma model, with couplingconstant g . We will regard H as the Hasaminitletroancitaionnosf. tNheotuicnepethrtautrtbheedc0toheefofircyieanntds torfeaetacHh′o0afndtheHs′e′ H0=x(cid:229) ,jZ dx321g20[J0L(x⊥,x3,j)2+J3L(x⊥,x3,j)2].(3.1) terms is small, by virtue of l 1. Thus one coupling, g ⊥ 0 ≪ issmall(totakethecontinuumlimit),anothercouplingg is ′0 muchsmallerstillandthethirdcouplingg iscomparatively The connection between (2.12), (2.13) and (3.1) is made ′0′ extremelylarge.Weshallsaymoreaboutthesizesofthecou- through the Heisenberg equation of motion for Uj. This plingsinthenextsection. gives lj(x) ≈ ag−02J0L(x⊥,x3,j), for small a. Similarly, priWnceipwali-lclhaisrsaulmsiegmthaemlaottdiceelssapsacnineagristhsemiratlhlearnmdodtryenaatmthiec Risnj(oxw)ljJ(x)L≈R.ag−02J0R(x⊥,x3,j). Thetransverseelectricfield 0 andcontinuumlimits. Eachsigmamodellivesinaband,that isaatwo-dimensionalstripoflengthL3 ¥ andwidtha,in The residualgauge-invarianceconditionandthelongitudi- the j-3 plane, where j=1,2 (we referre→dto bandsas layers nal-electric-field-squaredtermarewrittenbythesamesubsti- inthepaperson(2+1)-dimensionalgaugetheories. Inthree tution into (2.9) and (2.11), respectively. Residual gaugein- spatial dimensions, a different name seems appropriate). In varianceisnowtheconditiononphysicalstatesY our wine-transit-boxanalogy, thereare fourbandssurround- ing each wine bottle, and two wine bottles adjacent to each abtanxd⊥.+Wjêadbeyno(txe⊥t,hje).baTnhdebleetfwt-heeanndthedelainndearitgxh⊥t-haannddtehdecliunre- Z dx3j=(cid:229) 1,2hJ0L(x⊥,x3,j)−J0R(x⊥−jâ,x3,j) rentsofsigmamodelintheband(x⊥,j)are r (x ,x3) Y =0, (3.2) ⊥ − JmL(x⊥,x3,j)b = iTrtb¶ m Uj(x⊥,x3)Uj(x⊥,x3)†, i JmR(x⊥,x3,j)b = iTrtbUj(x⊥,x3)†¶ m Uj(x⊥,x3), andthelongitudinal-electric-field-squaredtermis 5 H′= − 4(gg4′0a)22(cid:229) (cid:229) dx3 dy3|x3−y3| J0L(x⊥,x3,j)−J0R(x⊥−jâ,x3,j)−r (x⊥,x3) 0 x⊥ j=1,2Z Z h i × J0L(x⊥,y3,j)−J0R(x⊥−jâ,y3,j)−r (x⊥,y3) (3.3) h i wherer (x ,x3)isalinearchargedensity,satisfyingthealge- We can regard a line in the 3-direction as a choice of x . ⊥ ⊥ bra Consider now the four bands which meet at this line x , ⊥ namely(x ,1), (x ,2), (x 1â,1)and(x 2â,2),shown ⊥ ⊥ ⊥ ⊥ [r (x⊥,x3)b,r (y⊥,y3)c] = id x⊥y⊥d (x3−y3) inFig. 2. Therearefivecolo−rchargesatthelin−ex⊥,onefrom fdr (x ,x3) . (3.4) eachbandandonefromthecolorsource dx3r (x⊥,x3)onthe × bc ⊥ d line. Thenontrivialconstraint(3.2) meansthat sum of these R ThelasttermintheHamiltonianisthelongitudinal-magnetic- fivecolorchargesiszero. field-squaredterm H H = 1 (cid:229) dx3ReTrU(cid:3)(x ,x3), (3.5) (cid:10)(cid:10) HH ′′ −4(g′0′)2a2 x⊥Z ⊥ (cid:10)(cid:10) HHH (cid:10) (cid:10)(cid:10) whereU(cid:3)(x⊥,x3)isgivenby(2.3),asbefore. (cid:10) (cid:10)(cid:10)(cid:10)HH (cid:10) (cid:10) (cid:10)(cid:10) HHYHH(cid:10) H H(cid:10)(cid:10)(cid:10)(cid:29) (cid:10) H H (cid:10) (cid:10)(cid:10) x3 H H(cid:10) (cid:10) 6 (cid:10) (cid:10)(cid:10) HH x2 HH (cid:10)(cid:10)(cid:10)(cid:30) H (cid:10) HHj HH(cid:10) (cid:10)(cid:30) HH (cid:10)(cid:10) H (cid:10) H(cid:10)(cid:10) (cid:10) H H (cid:10) (cid:10) (cid:10) x3 (cid:10) HH H 6 Hj x1 x2 (cid:10)(cid:30) (cid:10) (cid:10) FIG. 3: The Wilson loop U(cid:3)(x ,x3), directed counter-clockwise H ⊥ (cid:10) aroundfourbands. H H H H (cid:10)(cid:10) H Hj H x1 (cid:10) The interaction H′, as discussed before, is simply the E12 (cid:10) termoftheHamiltonianinaxialgauge. FInally, the interaction H is a discrete version of the in- ′′ FIG.2:Thefourbandsmeetingatthelineoffixedx . tegral of the square of the longitudinal magnetic flux. The ⊥ quantityTrU(cid:3)(x ,x3)istheWilsonlooparoundfourbands(a ⊥ winebottle standsin the middleofthese fourbands),shown ThetermsintheHamiltonian(3.1),(3.3),(3.5)andthecon- inFig. 3. straint(3.2) may seem a bitcomplicated, butthey each have a straightforwardgeometricalinterpretation. A sigma model lives in each band (x ,j), whose Hamiltonian is H (x ,j). ⊥ 0 ⊥ TheexcitationsofH aretheFZparticles,whichbehavelike IV. TRANSVERSECONFINEMENT 0 solitons, though they are not quantized versions of classical solutions. Withouttheinteractionterms,theseparticlesmove Nextweexplainhowtransverseconfinementworksforour onlyin the 3-direction,scattering with a nontrivialS-matrix. effectivegaugetheory. The mechanismdoesnotrely onour Thescatteringisintegrable,whichimpliesthatthereisnopar- takingthe continuumlimit ofthe coordinatex3; it holdsjust ticle creationordestruction. The elementaryr=1FZ parti- aswellonthethree-dimensionallattice. Thekeyingredients clesareadjointgluon-likeparticles(seeequation(1.6)).They arethemassgapofH0 andresidualgaugeinvariance(2.9)or can be thought of a color dipoles with a right fundamental (3.2). Wewillshowthatourexplanationmakessensefor color charge (anti-charge)at x and a left fundamentalanti- ⊥ ochuatrogfeth(cehsaerrge=)a1t“xd⊥if+frajˆcat.ivAellgoluthoenrs”F.Zparticlescanbebuilt (g′0)2= (gg′0′40)2 ≪ g10e−4p /(g20N). (4.1) 6 OurargumentsareadaptedfromReference[10]. Letusfirstconsidertheextremecaseofl =0.Inthiscase, g =0 and g =¥ , so that H =H . Let us place a quark ′0 ′0′ 0 at u ,u3 and an antiquark at v ,v3. We now ask what the ⊥ ⊥ ground-state energy is, with these sources introduced. This | energyisaneigenvalueofH . SinceH isthesumofnonlin- 0 0 | | earsigmamodels,wetrytoputasmanyaspossibleofthese q¯ | | | | |t sigmamodelsintheirgroundstates. Inotherwords,wewant | as many bands of our wine transit box as possible to be un- occupiedbyFZ particles. Considerthe effectoftheresidual x2 | conditionon states, (3.2). This tells us that we cannotmake 6 | allfourofthebandsmeetingatu inacolor-singletstate,due ⊥ | || | to the presence of the quark at u . Now we want the sigma ⊥ | modelineachofthesebands(u ,1),(u ,2),(u 1â,1)and ⊥ ⊥ ⊥ t (u 2â,2)tobeinaneigenstate(sinceweseek−thelowest- q ⊥ − energyeigenstateofH ). Nowatleastoneofthesefoursigma 0 models cannot be in a color-singlet state. The Hohenberg- -x1 Mermin-Wagnertheoremtellsusthatthereisnospontaneous symmetrybreakingforthe vacuumof a (1+1)-dimensional FIG. 4: The transverse projection of a transverse string, built out field theory. Hence thevacuummustbe a singlet. Anynon- ofFZparticles(representedbylargebullets),joiningaquarktoan singlet eigenstate must therefore have energy of at least the antiquark(representedbysmallbullets).Longitudinalfluxlinesjoin massgap,m . Soatleastoneofthefoursigmamodelsmust theparticlestogether.Thesefluxlinesareparalleltothelineofsight, 1 containanFZparticle,say(u ,1). Ifthereisnocolorsource hencearenot visible. Theneighboring FZparticlesandsourcesin ⊥ in the line u +1â, then by the constraint(4.1), at least one thisfigurearenotadjacent,ingeneral,sincetheymaybelocatedat oftheothert⊥hreesigmamodelsinbandsadjacenttothisline, differentvaluesofx3. namely(u +1â,1),(u +1â,2)and(u +1â 2â,2),must ⊥ ⊥ ⊥ − also be excited, by the same reasoning. Continuing in the Eachofthecorrelationfunctionsinthisproductdecaysexpo- this way, there must be a connected two-dimensional union nentiallyas ofbandsinwhichallthesigmamodelsareexcited,terminat- ing at u⊥. For the energy to be finite, this two-dimensional 0 U1(x⊥,x3)U1(x⊥+1ˆM1,x3)† 0 e−m1M1 , unionofbandsmustalsoterminateatv ,wheretheantiquark h | | i∼ ⊥ ispresent. where 0 is the vacuum of the sigma model. Therefore the | i The physical picture of transverse confinement is the fol- Wilsonloopbehavesas lowing. Imagine we look at the quark-antiquarkpair from a A exp s M1M3. (4.3) great distance along the 3-direction. We see a string of FZ ∼ − ⊥ particlesinthetransverseplane,joiningthequarktotheanti- Loops oriented in the x2-x3-plane behave exactly the same quark,asinFig. 4. Thepotentialisthesumofthemassesof way. alltheseFZparticles. Asithappens,thepotentialisnotrota- Let us now suppose we increase l a bit, so that it is no tion invariant, even in the transverseplane. If the quarkand longer zero, but so that (4.1) is satisfied. Then the coeffi- antiquarkareseparatedalongthe1-or2-directions,however, cients of H and H are small compared to m /a. We have ′ ′′ 1 the potential between these sources is linear with transverse three mass scales in the problem, namely m , (g )2/a and stringtension 1 ′0 1/[(g )2a]. If the first of these mass scales is much greater ′0′ s = m1 . (4.2) thantheothertwo,transverseconfinementwillstillhold. To ⊥ a determine the correction to the potential, as g′0 is increased andg is decreasedwill requirethe applicationof exactma- Thelackofrotationalinvariancearoundthex3-axisseemsun- ′0′ trixelements;currentformfactorsforH [20]andfieldform realistic, butwewillargueattheendofthissectionthatthis ′ factors for H [23]. The inclusion of these terms will mean invariancecomesaboutwhenH andH areincluded. ′′ ′ ′′ that FZ particles interactbetween adjacentbands. There are WecanreadilyseethatcertainspacelikeWilsonloopshave alsobeprocesseswherebyFZparticlescanbecreatedorde- an area law. Consider a rectangular loop in the x1-x3-plane. stroyed,duetotheseinteractions.Theseeffectsmeanthatthe ByvirtueofthegaugeconditionU =1,thisloopbreaksapart 3 theoryisnolongerintegrable,thoughitisnearlyso,provided intoaproductofsigma-modelcorrelationfunctions.Suppose (4.1)holds. the dimensions of the loop are M1 and M3 in the 1- and 3- The correction to the transverse string tension for (2+ directions, respectively. Thenthe Wilson loopisroughly(in 1)-dimensional SU(2) gauge theory comes from the lower- thatwearebeingsloppywithcontractionsofgroupindices) dimensional analogue of H (there is no term analogous to ′ y3+M3 H′′)[14]. Thiscorrectionisduetofluctuationsofpositionsof A (cid:213) 0 U (x ,x3)U (x +1ˆM1,x3)† 0 . FZparticles,boundbylongitudinalstrings. Thelongitudinal- 1 ⊥ 1 ⊥ ∼ x3=y3 h | | i string potential(which we discuss in the nextsection) is not 7 linearatshortdistances,butquadratic;thisisduetofactthat cle,boostthemtothelabframe,andconsiderthewavefunc- thecolorofanFZparticleissmearedinthelongitudinaldirec- tionofthesecondparticleinthepresenceofthosefields(af- tion. TheactualdistributionofcolorisapproximatelyGaus- terwards one can improve the result by iteratively imposing sian, which can be foundusing a form factor for the current unitarity).Wepointoutinthissectionthattherescalingofthe operatorofthesigmamodel[20]. couplingconstantin(1.1),(1.2)and(2.2)produces“boosted” We expect that corrections coming from the inclusion of electricfields. H makethepotentialinvariantwithrespecttorotationsabout Thevalueofg mustbechosensothattheratioofthelongi- ′′ ′0 the3-axis. ThisisbecauseH actingonstringstatesdeforms tudinalstringtensiontothetransversestringtensionissmall. ′′ contourofthestringinthetransverseplane. Perturbationthe- Thisratio,from(4.2)and(5.1)is ory in this term should therefore cause the string to fluctu- ateenoughtomakearotation-invariantpotential(thisiswhat s L = (g′0)2 = l 2g20 happens in Hamiltonian-strong-couplingperturbation theory s m a m a 1 1 [25]). ⊥ Wecanturnthisexpressionaroundtofind m a s V. LONGITUDINALCONFINEMENT l 2= g12 s L . (6.1) 0 ⊥ Next we will show how longitudinal confinement takes Supposenowweconsidertwoincominghadrons,travelingin place. Again, the reasoning is essentially that of Reference the3-direction,withvelocity v. Iftheelectricandmagnetic ± [10]. fieldsofahadroninitsrestframeareE andB ,respectively, k′ k′ Letusnextconsideraquarkandantiquarkseparatedinthe theninthelabframe 3-direction only, with coordinates u ,u3 and u ,v3, respec- ⊥ ⊥ E vB E vB tthiveelsyo.urIcfegs,′0s=inc0e, tlhoenrgeituisdijnuastl ealeccotnrisctaflnutxpoctoesntstianlobeentweregeyn. E1= √1′1± v22′, E2= √2′1∓ v21′, E3=E3′. (6.2) − − Ifweassumeg =0and(4.1)instead,thenelectricfluxdoes ′06 If we assume the magnetic fields inside the hadron are ran- costsomeenergy. Furthermorethisfluxwillbeconcentrated dom,thentheaverageoftheratioofthelongitudinalelectric alongthelineu . Thereasonforthisconcentrationisthatif ⊥ fieldtothetransverseelectricfieldis√1 v2. We substitute apnliyesfltuhxatleoanveeosfththiselfionuer,breasniddsua(luga,1u)g,e(iunv,a2ri)a,n(cue(4.11ˆ)a,i1m)-, thisfors L/s in(6.1)toobtain − ⊥ ⊥ ⊥ ⊥ (u 2â,2)cannotbeinasingletstate. Henceatlea−stoneof m a ⊥− l 2= 1 1 v2. (6.3) thesefourbandsisexcitedwithanenergyatleastthesigma- g2 − 0 modelgapm1. Themassgaptendstopreventfluxspreading p transversely. Note that (6.3) means that large velocity v implies small l . We can now estimate the longitudinalstring tension. It is Wehavetherebyinterpretedtherescalingofthecouplingcon- simplythestringtensionofaYang-Millstheoryinonespace stantsin(1.1),(1.2)and(2.2)astheresultofhadronsmoving andonetimedimension,withcouplingg′0/a: at high velocities. To make such an interpretation sensible, however,thevelocityin thetransformedcoordinates(thatis, s = (g′0)2C , (5.1) the coordinates defined after the rescaling) should be small. L a2 N Otherwise, the expression (6.1) cannotbe used. This means thattoconsiderlarge-sprocesses,l shouldbetakensmall,but whereCN is the Casimir of SU(N). Notice thatthe physical velocitiesinournewcoordinatesshouldbenonrelativistic. mechanismoflongitudinalconfinementisadualMeissneref- Ourjustificationoftherescalingasaneikonalapproxima- fect, though no assumptions of magnetic condensation have tion is rather heuristic. It seems strange, because it contra- beenmade. dicts the fact that under a simple rescaling, the longitudinal Notethatifwesimplytookl =0[6],wewouldhavetrans- componentofvelocitydoesnotchange. We are arguingthat verseconfinement,butnolongitudinalconfinement.Theterm there really is an anomalous transformation of the velocity. H′ isessentialforthelongitudinalstringtension. Beforerescalingthevelocityisclosetozero,butafterwards, Correctionsofhigherorderin(g′0)to(5.1)comefromvir- itisgivenby(6.3).Itseemsworthmakingtheargumentmore tualpairsofFZparticles.Theanalogouscalculationin(2+1) rigorous. This could conceivably be done with anisotropic dimensionshasalreadybeendone[11]. renormalization-groupmethods[16],[17],[13],[14]. VI. THEEFFECTIVEHAMILTONIANASANEIKONAL VII. HADRONICSTATESANDDIFFRACTIVE APPROXIMATIONFORQCD SCATTERING We nowarguethatthe effectiveHamiltonianis aneikonal The structure of hadrons for our effective action is rather approximationforQCD.Aconventionalapproachtothisap- similar to that of the strong-couplingpicture [9], despite the proximation[26]istotakethefieldsfromoneincomingparti- factthatonlyonecouplingg ,isstrong,allothersbeingweak ′0′ 8 (the same is true in the (2+1)-dimensional case where all couplingsareweak).Hadronsarebuiltoutofstringsconnect- ing quarks. Strings terminate at quarks and N of them can meet in junctions. Thus we have a string-parton picture, in whichthepartonsarequarksandFZparticles. Xj Xj Xj jr jr jr jr jr jr Xj Xj Xj x2 eqq q rj X⊗j 6 jr Xj q rjA jr rj rjeqq X⊗j rj Xj rj x2 Xj rj Xj 6 Xj Bj Xj Xj Xj Xj q - x1 Xj Xj jr ⊗ Xj FIG.6:Anexampleoflongitudinalfluxexchange(LFE)isresonance jr betweentheconfigurationsshownhere,resembling(A)inFigure5. eqrq Longitudinalfluxlinesarepresent,ingeneral,inthesecondandthird diagrams.TheparticlesshowncanbeeitherquarksorFZparticles. -x1 VIII. CONCLUSIONS FIG.5:Longitudinallyapproachingbaryons.Largecirclesrepresent FZparticlesandsmallcirclesrepresentquarks.Thesymbol means ⊙ thattheconstituentparticle(FZparticleorquark)iscomingoutofthe Inthispaper,we haveconsideredthe anisotropiceffective pageandthesymbol meansthatitisgoingintothepage. Strings theory of Reference [6], which is a longitudinally-rescaled ⊗ of FZ particles can intersect at lines of fixed x⊥ (A) or overlap at Yang-Mills theory, with rescaling parameter l . We have transversebands(B). shownthatthistheoryconfinesforsufficientlysmalll anda naturalstring-partonpictureemerges.Thepartonsarevalence The transverse projection of a state of two baryons ap- quarksandthesoliton-likeFZparticlesoftheprincipal-chiral proachingeach other longitudinallyis shownin Fig. 5. The non-linear sigma model. The interpretation of this effective hadronscancollideatcoincidentlinesor bands. Letussup- actionasaneikonalapproximationisrathersubtle,sinceelec- pose we neglect H +H . In the case of overlapping sites tricfieldsare muchstrongerin thetransversethanthe longi- ′ ′′ (butwithoutoverlappingat adjacentlinks), longitudinalflux tudinal direction, even for slow-moving hadrons. We have exchange(LFE) atlinesoffixedx canhappen,butnothing argued that this interpretation is correct, provided colliding ⊥ else occurs. TheLFE amplitudecan be thoughtofas due to hadrons are moving anomalously slowly in the rescaled co- resonancebetweendifferentfluxarrangementsandisoforder ordinates. Animportantprocessinhadronicscatteringinthe 1/N,asweexplaininthenextparagraph. forwarddirectionistheexchangeoflongitudinalflux,evenif TheLFEprocessisanalogoustothecaseofmesonscatter- FZparticlescollide. ingin(1+1)dimensions. WecanthinkofthetwoFZparti- Thereareessentiallytwoareaswhichcometomindforfur- clesadjacenttothelinex (inonehadron)asbeingsimilarto therinvestigation. ⊥ two sources of fundamentalcolor in that line (recall that FZ We would like to improve our understanding of confine- particles are dipoles, and each of them has one fundamental mentfortheanisotropicgaugetheoryconsideredhere.Thisis sourceadjacenttotheline).ThusLFEisessentiallysimilarto notastrong-couplinggaugetheory,butratherahybridgauge flux exchangein (1+1)-dimensionalQCD. This non-planar theorywhereonecoupling,namelyg , isstrongandtherest ′0′ processisoforder1/N[27]. AnexampleofLFEisshownin areweak.Ademonstrationthatconfinementstillholdsifg is ′0′ Fig. 6. weakwouldberealprogressontheQCD confinementprob- We notethatworldsheetoflongitudinalelectricfluxlines lem. We havebeeninvestigatingexpansionsin (g ) 2 using ′0′ − during LFE has the same topology as that of Pomeron ex- fieldformfactorsofthesigmamodel[23],butasyethaveno changeinstringmodels. simpleargumentthatconfinementholdsforsmallg . Inany ′0′ In the case of two hadrons intersecting the same band, case, it seems possible to generalize our calculations of the i.e. withcollidingFZparticles,theS-matrixoftheprincipal- corrections to the longitudinal [11] and string tensions [14], chiralsigmamodelcomesintoplay.Wenotethatthis(1+1)- andtothemassspectrum[13]to(3+1)dimensions,withg′0′ dimensional S-matrix is unity in the large-N limit, N ¥ , large,butnotinfinite. g2N fixed. Thus in this limit, no interaction occurs. I→n the Theotherproblemofimportanceisthe forwardscattering 0 1/NexpansionofthisS-matrix,however,LFEoccurs[19]. amplitude of hadron-hadronscattering. We hope that its so- 9 lutionwillyieldaquantitativeunderstandingofthePomeron. Acknowledgments ThesolutionwillrequiresomecontrolofLFEprocesses. We believethatthisproblemistractable. WehavearguedinSec- tion6 thatin thenew coordinates,momentashouldbetaken as small as possible; in this way the electric fields are lon- We thankGregoryKorchemskyforabrief,butveryinfor- gitudinally boosted as they should be. Thus, LFE processes mative discussion, Jamal Jalilian-Marian for inspiration and canbestudiedinanonrelativisticcontext. We hopetomake PoulHenrikDamgaardforencouragement. progressonthisproblemsoon. 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