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February 8, 2008 23:55 WSPC/INSTRUCTION FILE PRPage International JournalofModernPhysicsA c WorldScientificPublishingCompany (cid:13) 5 HYBRID AND CONVENTIONAL BARYONS IN THE FLUX-TUBE 0 0 AND QUARK MODELS 2 n a PHILIPR.PAGE J Theoretical Division,MSB283, LosAlamos NationalLaboratory, LosAlamos, NM87545, USA. 4 E-Mail: [email protected] 3 v Thestatusofconventionalbaryonflux-tubesandhybridbaryonsisreviewed.Recent 4 surprises are that a model prediction indicates that hybrid baryons are very weakly 8 produced in glue-rich Ψ decays, and an analysis of electro-production data concludes 0 that the Roper resonance is not a hybrid baryon. The baryon decay flux-tube overlap 0 has been calculated in the flux-tube model, and is discussed here. The behavior of the 1 overlapfollowsna¨ıveexpectations. 4 0 Keywords:baryon;flux-tube;hybrid;decay / h t - 1. Update on previous reviews l c Comprehensive reviews on baryon flux-tubes and hybrid baryons are available. 1,2 u n Here developments since the last review in 2002 are summarized. v: Baryon flux-tube: The consensus in quenched lattice QCD is that the baryon i potentialatsufficientdistancesisbestdescribedbythesumofaCoulomb,Y-shaped X 3 confinement, and constant term. This statement is independent of the lattice r 4 a operator, and hence physical. The difference with Abelian projected full lattice 5 5 QCDissmall. Thebaryonfluxactiondensity isdisplayedinFig.1.Thedensity clearly peaks at the three quarks and the junction. However, the detailed contour 4 lines, as well as the Y-shape, are lattice operator dependent. Abelian projected 5 full lattice QCD shows that cromo-electicflux flows in the same directions as the 5 flux-tubes, with solenoidal currents circulating around them. This supports the idea that QCD is a dual superconductor. Hybrid baryon: The first quenched lattice QCD calculation of the low-lying hy- 6 brid baryon potential has been performed. The lowest gluonic excitation energy (Fig.2)isconsistentlygreaterthan1GeV.ThisisseveralhundredMeVlargerthan 7 the flux-tube model prediction , which already gives the highest mass prediction forthelightestN1+ hybridbaryonofallthemodels1,2.Ifthelatticecalculationis 2 correct,this hybridbaryonwilllikely havea massabove2GeV, whichwouldmean that the hybrid baryon spectrum starts at a very high mass. In contrast to this, QCDsum rulepredictions for hybridbaryonmassesremainlow:Mostrecently,the Λ1+ is predicted 8 at 1.6(2) GeV. A possible reason why the lattice calculation is 2 high is that the effect of higher excited states has not been eliminated, although it 1 February 8, 2008 23:55 WSPC/INSTRUCTION FILE PRPage 2 P.R. Page 20 18 16 14 12 10 8 6 17.5 20 22.5 25 27.5 30 32.5 Fig.1. Sideviewandtop-downviewofthefluxactiondensity. l 3 4 L (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) V] (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)P e (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) G 3 (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)ρ(cid:0)(cid:1)(cid:0)(cid:1) [ (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)θ al (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)l(cid:0)(cid:1)(cid:0)(cid:1) l i nt (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) e (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) ot 2 (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) p zl Q G.S. 3 1st E.S. l 2 1 0 0.5 1 1.5 L [fm] l min 1 Fig.2. LeftisthebaryonandlowesthybridbaryonpotentialasafunctionofLmin≡l1+l2+l3. Here li is the distance between quark i and the junction equilibrium position, shown in Fig. 3. Right is a cylindrical coordinate system where the point P = (l1,l2,l3) is rewritten in terms of ρl,θl andzl.Thelinel1=l2=l3 isdenotedbyL. 6 is stated that this is not the case. Hybrid baryon production in Ψ decay,aglue-richprocess,haslongbeenthought tobepromising.2Particularly,BESatBEPCcanlookforΨ→hybridN¯.Recently, severalhighmasspeakshavebeenreported9:aJP = 1+ 2σ peakat1834+46 MeV 2 −55 seen in Ψ → pp¯η, and a 2068± 3+15 MeV peak seen in Ψ → NN¯π. A recent −40 10 model calculation finds, in contrastto na¨ıve expectations, that hybrids are very weakly produced relative to radially excited baryons in Ψ decay. This is due to a low probability of the gluon coming from the Ψ to fold in to the qqqg hybrid wavefunction, and color factors in the hybrid baryon wave function. In fact, it is 10 suggested that Ψ decay is a filter in favor of conventionalbaryons. February 8, 2008 23:55 WSPC/INSTRUCTION FILE PRPage Hybridand Conventional Baryons 3 a b C l y A A l l 3 3 y B l r r 3 A R l1 lA lB 1 A 2 A l l B 2 2 l 1 Fig.3. Coordinatesystemdescribedinthetext.Thelengthli usedinthetextequals lAi .The junctionequilibriumpositionofthefinalbaryonB isdenotedbyR. | | Hybrid baryon electro-production: A recent analysis of new Jefferson Lab data finds 11 that the amplitude for Roper resonance (N1+(1440)) production via lon- 2 gitudinalphotonsissubstantial,inconsistentwithamodelpredictionthatitshould be zero. The Roper hence does not fit well as a hybrid baryon. 2. Baryon decay flux-tube overlap When a baryon decays via OZI allowed quark-antiquark pair creation to a baryon and a meson, it has not hitherto been considered what happens to the flux-tubes. There is certainly an effect, as the pair is created from gluons. Here we report on the first preliminary calculation. Such a calculation exists for meson decay to two mesons. 12,13 The structure of the decay can be investigated by fixing the initial quarks in the baryon, and evaluating the overlap of the initial flux configuration with the final baryon and mesons, for a fixed quark-antiquark pair created from the initial flux-tube: the “flux-tube overlap”. This quantity enters in models of baryon decay where it is traditionally assumed to be a constant. This is equivalent to taking the Y-shaped flux structure to have no effect on decay. In such a picture the main effect of confinement on the decay is that it specifies the decay operator for quark-antiquark pair creation, e.g. the 3P0 decay operator 13. This operator is typically taken to act locally with no knowledge of the global Y-shaped structure. Here the concern is not with the local characterof decay (the decay operator), but with imposing constraints from Y-shaped confinement (the flux-tube overlap).The baryon decay flux overlap N3A−2 3 NiA NC +1 γ(l ,y)≡ d3r( d2yiA)δ2(y3A −(y−lA− (r−lA))) i Z j NC+1 3 NA+1 3 NXC=1 iY=1jY=1 3 ×ψA(r,yiA) ψB∗(rB,yiB) ψC∗(yC) (1) j j j February 8, 2008 23:55 WSPC/INSTRUCTION FILE PRPage 4 P.R. Page b a −2 0.25 0.2 1] − 0.15 V γ 0.1 [Ge30 y 0.05 0 −2 −2 0 0 2 y1 [GeV−1] 22 y3 [GeV−1] , −2 y1 [G0eV−1] 2 Fig.4. (a)Sideviewand(b)top-downviewofthebaryondecayflux-tubeoverlapγinGeV.Itis invariantundery3 y3 (thereisnothingspecialaboutwhetherpaircreationisaboveorbelow →− theplaneofthethreequarks),buttheinvarianceunder y1 y1 onlyensuesbecause l1=l2. →− measures the overlap of the flux-tube wavefunction of initial baryon A, with those of final baryonB and meson C, assuming that pair creation takes place at position y. The overlap is evaluated analytically in the flux-tube model of Isgur and Pa- ton7,14 aswasdone forthe mesoncase 12,13.Inthis model the flux is represented by oscillating beads, and for the baryon the flux-tubes meet at a junction. The co- ordinates of the quarksand junction are displayedin Fig. 3(b). The position of the junction of the initial baryon, measured from its equilibrium position, is denoted by r. In Eq. 1 the position of bead j associated with quark i is denoted by yi. j For the calculation shown, the constituent quark masses we taken to be 330 MeV, the string tension that of mesons (0.18 GeV2 ≈ 1 GeV/fm), 7,14 and all the bead masses equal. Bead masses are fixed by the condition that their sum equals the potential energyin the flux-tubes when the system is in equilibrium. The flux-tube overlap γ depends on the six variables y and l , shown in Fig. 3(a). There is also i dependence on the number of beads associated with each flux-tube in the initial baryon(NA).Ifpaircreationcantakeplaceonmorethanonebead,γ isthesumof i the possible overlaps.(Hence the sum overthe number of beads on the final meson NC inEq.1.)Withoutlossofgeneralitypaircreationassociatedwiththeflux-tube of the third quark is studied. The components of y perpendicular to the vector lA 3 are y1 (in the three-quark plane) and y3 (perpendicular to the three-quark plane). The component of y along lA3 is y2. When any variables are fixed in the Figures, the following choices are made, unless indicated otherwise: y1 = y3 = 0, y2 = 1.25 GeV−1 =0.25 fm, l =2.5 GeV−1 =0.5 fm and NA =3. i i Thedependenceofγonthepaircreationpositionyisasfollows.Thedependence on y1 and y3 is shown in Fig. 4. A clear Gaussian fall-off with flattened top with a diameter ∼0.4 fm think can be seen. This follows closely what would be expected for a thick flux-tube which falls off away from lA as predicted by the flux-tube 3 12 model (see Fig. 2 of Ref. ). The Gaussian fall-off with flattened top continues to February 8, 2008 23:55 WSPC/INSTRUCTION FILE PRPage Hybridand Conventional Baryons 5 a 0.4 0.4 θ = 0 θl = π/3 , 5π/3 θl = 2π/3 , 4π/3 0.3 θl = π 0.3 l V] γ [Ge 0.2 γ [GeV]0.2 0.1 0.1 0 0 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 y2 [GeV−1] , y2 / l3A Fig. 5. The variation of γ with li shown in (b) is achieved by redefining li in terms of ρl,θl and zl defined in Fig. 2, and plotting the variation as a function of θl. The calculations are for ρl=ρmax/2(ρmax=zl/√2)andzl=√3 2.5GeV−1. × be observedas y2 is varied. Fig. 5(a) shows the dependence ony2. The asymmetric shape is because pair creation can take place beyond the position of the third quark (y2 = l3 = 2.5 GeV−1). Vanishing pair creation at y = 0 is an artifact of the small-oscillations approximation 7,12,13,14 made in this calculation, and is not a prediction of the flux-tube model. The dependence of γ on quark position l i variationsisshowninFigs.5(b)and6.Theextremesofthevariationareconsistently (a)smallestγ whenl3 <l1 =l2 (redlineθl =0),and(b)largestγ whenl3 >l1 =l2 (bluelineθl =π).Theseextremesareeasilyunderstoodintuitively:Whenl3issmall / large, there is less / more opportunity for pair creation associated with the third quark.Fig.5(b)showsthatpaircreationbecomesmoreconfinedtotheregioninside the third quark (0 ≤ y2 ≤ l3) as l3 increases. There is also a dependence on the number of beads NA. Calculations were also performed for NA = 4. The number i i of beads determine the inter-bead spacing, a regularization parameter, which is unphysical. The qualitative features of the dependence of γ on y and l discussed i aboveare independent ofNA, andhence physical.They arealso withoutexception i what one would have expected before the calculation was performed. The overall magnitude of decay is strongly dependent on NA, and hence unphysical. This is i as expected, since the magnitude of decay is normalized to phenomenology in pair creation models. 3. Conclusions Baryon: At sufficient distances the potential is Y-shaped with a clear junction. The decayflux-tube overlaphasbeenevaluatedinthe flux-tube model,andfollows na¨ıve expectations. Y-shaped baryon confinement can henceforth be incorporated February 8, 2008 23:55 WSPC/INSTRUCTION FILE PRPage 6 P.R. Page a b θ = 0 00..34 θθθθθlllll ===== 245ππ/πππ3///333 00..34 θθθθllll ==== 02ππ/π3/ 3, 5, 4π/π3/3 l V] V] e e G G γ [0.2 γ [0.2 0.1 0.1 0 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 y1 [GeV−1] , y3 [GeV−1] Fig.6. Conventions asinFig.5(b),except that y2=l3/2. in models of decay. Hybrid baryon: The only known way to study them rigorously is via excited adiabatic potentials. 2 The low–lying state in models is N1+ with 2 2 mass 1.5 – 1.8 GeV. If recent lattice QCD calculations are correct, this mass is most likely above 2 GeV. A recent model calculation shows that glue-rich Ψ decay veryweaklyproduceshybridbaryons.TheRoperresonancedoesnotappeartobea hybridbaryonfromarecentanalysisofelectro-productionvialongitudinalphotons. Acknowledgments Collaboration with N.F. Black (University of Tennessee, Knoxville) on the baryon decay flux-tube overlap,and hospitality at MENU2004, are acknowledged. References 1. T. Barnes, contribution at the COSY Workshop on Baryon Excitations (May 2000, Ju¨lich, Germany), nucl-th/0009011. 2. P.R. Page, Proc. of the 9th International Conference on the Structure of Baryons (BARYONS 2002) (March 2002, Newport News, VA),nucl-th/0204031. 3. T.T. Takahashi, H. Matsufuru, Y. Nemoto and H. Suganuma, Phys. Rev. Lett. 86, 18 (2001); Phys. Rev. 65, 114509 (2002); C. Alexandrou, P. de Forcrand and O. Jahn, Nucl. Phys. (Proc. Suppl.)119, 667 (2003). 4. F. Okiharuand R.M. Woloshyn, Nucl. Phys. (Proc. Suppl.) 129, 745 (2004). 5. V.G. Bornyakov et al. (DIKCollab.), hep-lat/0401026. 6. T.T. Takahashi and H. Suganuma,Phys. Rev. Lett. 90, 182001 (2003). 7. S. Capstick and P.R.Page, Phys. Rev. C66, 065204 (2002). 8. L.S. Kisslinger, Phys. Rev. D69, 054015 (2004). 9. H.Lietal.(BESCollab.),Phys.Lett.B510,75(2001);M.Ablikimetal.(BESCollab.), hep-ex/0405030. 10. R.G. Ping, H.C. Chiang and B.S. Zou, nucl-th/0408007. 11. I.G. Aznauryan et al.,nucl-th/0407021; V.D. Burkert,these proceedings. 12. N.Dowrick, J. Paton and S.J. Perantonis, J. Phys., 423 G13 (1987). 13. R.Kokoski and N. Isgur, Phys. Rev. D35, 907 (1987). February 8, 2008 23:55 WSPC/INSTRUCTION FILE PRPage Hybridand Conventional Baryons 7 14. N. Isgurand J. Paton, Phys. Rev. D31, 2910 (1985).

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