We discuss whether a low-lying hybrid baryon should be defined as a three
quark - gluon bound state or as three quarks moving on an excited adiabatic
potential. We show that the latter definition becomes exact, not only for very
heavy quarks, but also for specific dynamics. We review the literature on the
signatures of hybrid baryons, with specific reference to strong hadronic
decays, electromagnetic couplings, diffractive production and production in psi
decay.Comment: Talk at ``The Physics of Excited Nucleons'' (NSTAR2000), Newport
  News, VA, 16-19 Feb. 2000. 8 pages, LaTe

Page, Philip R.

English

arXiv

The authors discuss whether a low-lying hybrid baryon should be defined as a three quark--gluon bound state or as three quarks moving on an excited adiabatic potential. We show that the latter definition becomes exact, not only for very heavy quarks, but also for specific dynamics. We review the literature on the signatures of hybrid baryons, with specific reference to strong hadronic decays, electromagnetic couplings, diffractive production and production in {psi} decay

Page, P.

UNT Digital Library

Apprwedrbrpubficrelease;~..rsmlhrr isunlimited.Title: Hybrid Baryon SignaturesAuthor(s): Philip R. Page, T- 16,Theoretical Division, Los Alamos National Lab.Submitted to: Talk at “ne Physics of Excited Nucleons” (NSTAR2000),Newport New, VA, 16-19, Feb. 2000Los AkirnosNATIONAL LABORATORYLoa AkrmoeNational Laboratcq an afiirmatk ectionkqual opportunityemployer, isoperated by the Universityof Caliinia tir the US.Department of Energy under contractW-7405-ENG-36. By acceptance of thisarticle, the puMiiher recognizes that the U.S. Governmentretains a nonexclusive,royalty-freeficenae to pubikh w repti”uce the publishedform of thiscxmfribution,or to allow others todo so,fw U.S.Government purposes.Los Aiemos National Laboratcfy requests that the publisherideniifythk articie as winlr performed under theauspicee of the U.S. Department of Energy.LoaAJamosNational Labrxatory stronglysupportsacademic freedom and a resesrche~s rtghttopublish as an institution,however, the Labo@J’y does notendorse the viewpointof a putMicationorguarantee its technicalcorrectness.Form 836 (10196jDISCLAIMERThis report was prepared as an account of work sponsoredby an agency of the United States Government. Neitherthe United States Government nor any agency thereof, norany of their employees, make any warranty, express orimplied, or assumes any legal liability or responsibility forthe accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constituteor imply its endorsement, recommendation, or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors expressed herein do notnecessarily state or reflect those of the United StatesGovernment or any agency thereof.DISCLAIMERPortions of this document may be illegibiein electronic image products. Images areproduced from the best available originaldocument.hep-ph/yymmnnnLA-UR-00-XXXXHYBRID BARYON SIGNATURESP. R. PAGEaTheoretical Division, MS-B283, Los A!amos National Laboratory,Los Aiamos, NM 875J5, USAE-mail: prp@anl.govWe discuss whether a low–lying hybrid baryon should be defined as a three’ quark—gluon bound state or as three quarks moving on an excited adiabatic potential.We show that the latter definition becomes exact, not only for very heavy quarks,but also for specific dynamics. We review the literature on the signatures of hy-brid baryons, with specific reference to strong hadronic decays, electromagneticcouplings, diffractive production and production in @ decay.1 What is a hybrid baryon?Historically a low–lying hybrid baryon was defined as a three quark – gluoncomposite. However, from the viewpoint of the Lagrangian of Quantum Chro-modymanics (QCD) this definition is non–sensical. This is because gluonsare massless, and hence there is no reason not to define a hybrid baryon, forexample, as a three quark – two gluon composite. Neither is one possibilitydistinguishable from the other, since strong interactions mix the possibilities.The place where this definition of a hybrid baryon is most useful is in large Q2deep inelastic scattering, where a Fock state expansion of a state can rigorouslybe defined, and one can at least talk about the three quark – gluon componentof such a state} However, in other situations the definition becomes perilous. Acase in point is recent work on large N= hybrid baryons, where their propertiesdepend critically on the fact that the gluon is in colour octet, and hence thethree quarks in colour octet, so that the entire state is colour singlet? The bagmodel circumvents the objections raised against this definition, since gluonsbecome massive due to their confinement inside the bag?More recently, a low–lying hybrid baryon was defined as three quarks mov-ing on the low–lying excited adiabatic potential? From the viewpoint of QCDthis can be a perfectly sensible definition. One can always evaluate the energyof a system of three fhed quarks as a function of the three quark positions,aTalk at “The Physics of Excited Nucleons” (NSTAR2000), Newport News, VA, 16-19 Feb.2000.1called the adiabatic potential. A calculation along these lines has been per-formed in flux-tube models and a first attempt has been made in lattice QCD!The three qu~ks are then allowed to move in a three-body equation, typicallya non–relativistic Schrodinger equation. Treating a three quark system via thk “two step process is called the adiabatic or Born–Oppenheimer approximation.Note that it is not appropriate to talk about case where the quarks are actuallyinfinitely heavy, because of the lack of kinetic energy. The adiabatic approxi-mation is expected to become exact if the quark masses are much greater thanthe scale of the strong interactions AQCD. The criterion for the validity ofthe adiabatic approximation is that the slow degrees of freedom (the quarks)should move much slower than the fast degrees of freedom (the gluons). It ispossible to argue that for conventional baryons the relative velocities of quarksbehave like the strong coupling constant cr,s as the quarks become heavier?Because of the asymptotic freedom of QCD, the quark relative velocities go to,zero, ensuring the validity of the adiabatic approximation. In fact, this is thebasis for the NRQCD expansion. However, since aS goes to zero only logarith-mically, one may need quarks heavier than the bottom quark for the adiabaticapproximation to be valid. Depending on the shape of the adiabatic potentialthe possibility of an NRQCD expansion for hybrid baryons~ and hence thevalidity of an adiabatic approximation, may be in jeopardy.Here we point out for the first time that for specific dynamics, the adiabaticapproximation can be exact, even for light quarks, if one redefines the adia-batic potential suitably. We call this the “redefined adiabatic approximation”,which employs a “redefined adiabatic potential”. It was noted in a flux-tubemodel that “For light quarks almost all corrections may be incorporated into aredefinition of the potentials. Mixing between [new] potentials is of the orderof 1Yo” ? We develop the following technique for obtaining the redefined poten-tial. The quark positions are still ilxed relative to each other, but the quarkand gluon positions are not defined relative to the quark positions, but relativeto the centre of mass of the quarks and the gluons, as recently pioneered: Fordynamics where the redefined adiabatic approximation is exact one can rigor-ously define a low–lying hybrid baryon as three quarks moving in the low–lyingredefined excited adiabatic potential.2 Redefined adiabatic approximationConsider the non–relativistic hamiltonian for three quarks at positions rl, rz, rsand a junction at position r4. The junction represents the gluons. The classicalhamiltonian with a simple harmonic oscillator potential is2where Al, m and k are the mass of the quarks, the mass of the junction andcoupling constant respectively. The first and second terms are the kinetic andpotential ener~ terms respectively.2.1 Exact soh.dionMaking the variable transformationsp = -$(rlz- r2) A= ~(rl +rz -2r~)U=~m~3M(rl +r2+r3-3r3) (2)one obtains the diagonal hamiltonianH = ;M(bz + i2 + ;(1 + 3$62) + ;k (/)2 + A2 + :(1 +3:)%2) (3)when working in the overall centre of mass frame. The hamiltonian describesthree independent simple harmonic oscillators so that the energies solving the(quantized) Schrodinger equation, arewhere e.g. nP = n; + n; + n;, with n;, n; and n: the degrees of excitation inthe three space directions. The vibrational frequencies are2_ kup 2=WA ——MU.2 = (1+3:)$ (5)The wave functions solving the Schrodinger equation can be denoted byk, (P) An (N A. (~) (6)where e.g. nP denotes the set {n~, n:, n; }.One can verify that the quark relative velocities - (A) *, so that theygo to zero if M >>~. The criterion for the validity of the adiabatic approxi-mation is hence satisfied for large quark masses.32.2 Adiabatic solutionFix the quark positions so that il = iz = is = O. The potential energydepends only on the differences between positions, and is hence unaffected byfixing the quark positions. The kinetic energy in Eq. 1 is $mi~, and using%/%z = –(1 + 3#)& from Eq. 2, the hamiltonian (1) isHj = ;:(1 + 3:)2&2 + ;k (p2 + A2 + ;(1 +3#)2m2) (7)Only the variable IYis dynamical. The energies (adiabatic potentials) areNow allow the quarks to move in their centre of mass frame, so thatmotion hamiltonian isHq = ~M(r; +i~ +i;) +Ej= ;M(bz +i2) + (n. + ;)W; + ;k(pz + AZ)The energies are(8)the quark(9)(lo)It is easy to see that these adiabatic approximation energies only agree withthe exact energies (4) when # >> 1, i.e. when the quarks are much heavierthan the gluons. This is in accord with one’s naive expectation for the validltyof the adiabatic approximation.2.3 Redefined adiabatic solutionWe follow the same procedure as for the adiabatic solution, with one criticalchange. The quark positions are still fixed will respect to each other, but allpositions are now defined with respect to the overall centre of mass, before weallow the quarks to move. Define the overall centre of mass aa~ = M(rl +rz +rs) +mrz 1.*R= m”_‘4 = –2#(11)m+3M m+3MDefine new coordinates which are the positions of each particle with respectto the overall centre of mass. The time derivatives of these coordinates are41.r;=r4-R=-G%r;=r; =r; =–R=—2& 2m(12)The kinetic energy in terms of the new coordinates is~M(r~2+r~2+r~2)+ ~mi~2 = ;:(1 +3#)&2 (13)This kinetic energy combined with the potential in Eq. 1 isHj = ;:(1 + 3:)# + ;k (pz +A2 + ;(1 + 3:)%7 (14)We used the fact that the potential energy depends only on the differencesbetween positions. The energies (redefined adiabatic potentials) areEj = (n. + ;)U.+;k(pz+A2) (15)and the junction wave functions q%. (m). Allowing the quarks to move, thequark motion hamiltonian isIIq = *M(#+A2)+ (72. + ;)%+& +A2) (16)The energies arewhich are identical to the exact solution (4). The quark wave functions are@np (P) k.(~). In order to obtain the full wave functions of the system,we take the direct product of the quark and junction wave functions, i.e.V., (P) ~~~ (A) VJn.(~). These we identic~ to the wave functions of the exactsolution (6).In conclusion, the solution obtained via the redefinited adiabatic approxi-mation is emzct.We shall now show how the specific dynamics (1) provide a toy modelfor the quark model, in the sense that many of the features needed for thevalidity of quark model, are ezaci! in the toy model. Quark models enable thedynamics of quarks, while freezing out the dynamics of gluons. In the toymodel this corresponds to claiming that gluons are in the same wave functionfor all conventional baryons. But this is manifestly the case in the redefinedadiabatic approximation. The redefined adiabatic potential (15) is explicitly5..*.dependent on quark masses. This corresponds to quark models: the Coulombinteraction is usually postulated to have a coupling that depends on quarkmasses. On the other hand, the adiabatic potential (8) is not dependent onquark masses.The wave functions obtained from the adiabatic approximation differ fromthose in the redefined adiabatic approximation only for the gluons. When onefreezes the dynamics of gluons, as in the quark model, one does not noticethe difference between these two approximations. Even the functional’ formsof the gluon wave functions are the same: the wave functions only differ in thedimensionful scale they contain. Thus the. forms can be used interchangeably.When a process is studied that involves more than one redefined adiabaticpotential, e.g. hybrid baryon decay to a conventional baryon and meson, carehas to be taken to use the redefined adiabatic gluon wave functions insteadof the adiabatic ones. The former gluon wave functions are dependent on w~,which is itself dependent on the quark masses.3 Where can one search for hybrid baryons?We first outline the results of a recent flux-tube model calculation: Theflavour, non–relativistic spin S and Jp of the seven low–lying hybrid baryonsare (N, A) 2S+1JP = N2&+ N2; +, A4;+, A4$+, A45+z , where the firsttwo states double. The big ‘model has the same number of states. The pairN2 ~+, N2 ~+ has the same quantum numbers as in the flux-tube model? Theother five states in the bag model are A2 ~+, A2 ~+, N4 ~+, N4 ~+, N4 ~+.The state N;+ was studied in QCD sum rules!” The hyperfine interactionmoves the A states upwards and the N states downwards! Hence there arefour low–lying N hybrid baryons with a mass of 1870 + 100 MeV. This is some-what higher, but sometimes within errors, of bag models and QCD sum ruleswhich find a mass around 1.5 GeV?’10 The wave function sizes are estimatedas @ = ~ = 2.1 (conventional baryons) and 2.5 (hybrid baryons)GeV–l. Hybrid baryons are hence larger than conventional baryons.The following techniques may enable the detection of hybrid baryons:w Overabundance of state% This approach is troublesome, since not even allconventional baryons in the appropriate mass region have been discovered yet.However, hybrid baryon states are likely to be discovered before all conven-tional baryons in a relevant mass region, and hence need to be studied alongsideconventional baryons.● Decays Except for a QCD sum rule motivated suggestion that hybridbaryons should decay to Na}l no decay calculations have been performed.However, decay of hybrid baryons to Np and Nw is a priori interesting since it6isolates states in the correct mass region, without contamination from lower–lying conventional baryons. Study of these decay modes will also yield informa-tion on photo- and electroproduction in Hall B and ‘C at Jefferson Lab, sincethe p and w couple to the photon via vector meson dominance. The process ~NT + hybrid +- NW can be studied at the D–line at Crystal Ball E913.● Diffractive TN and TN production: The detection of the hybrid meson can-didate K(1800) in diffractive TN collisions by VEg2 may indicate that hybridmesons are producted abundantly via meson–pomeron fusion. If this is thecase, one expects significant production of hybrid baryons via baryon–pomeronfusion, i.e. production in diffractive TN and nN collisions.● Production in $ decays: Naive expectations are that the gluon–rich envi-ronment of @ decays should lead to dominant production of glueballs, butalso signifant production qf hybrid mesons and baryons. The large branchingratio#3 11~(~ + p@, pm ) N 10-3 may contain hybrid baryons decaying to(P,P) w ewecially. Recently a Jp = ~+ 20 peak at mass 1834t~~ MeV was’seen in !4 * pf7q!4● Electroproduction: In the flux-tube model, i.e. the adiabatic picture ofa hybrid baryon, there is the qualitative conclusion that “ep + eX shouldproduce ordinary N*’s and hybrid baryons with comparable cross–sections” ?5However, the conclusions obtained from the three quark – gluon picture of ahybrid baryon is different. For large Q2 electroproduction, the Q2 dependenceof the amplitudes is summarized in Table 1. Since the photon has both a trans-verse and longitudinal component, the amplitude for a conventional baryon isexpected to dominate that of the hybrid baryon as Q2 becomes large? Forsmall Q2 the conclusion agrees with the large Q2 result for transverse photons,but is more dramatic for longitudinal photons: the amplitude Vanishes?GIt hasaccordingly been concluded that the (radiaUy excited) conventional baryon isdominantly electroproduced, with the hybrid baryon subdominant to the res-onances S11(1535), ~IS(1520) and A as Q2 increases~6 The Q2 dependence ofthe electroproduction of a resonance can be measured at Jefferson Lab Hall Band an energy upgraded Jefferson Lab. A hybrid baryon is expected to behavedtierent from nearby conventional baryons as a function of Q2. One needs toperform partial wave analysis at different Q2. For large Q2 cross–sections aresmall, which would make this way of distinguishing conventional from hybridbaryons challenging.AckriowledgmentsThe suggestion to study hybrid baryons with a simple harmonic oscillatorpotential was made by Nathan Isgur and Simon Capstick. The manuscript was‘7,...Table 1: Q2 dependence of amplitudes for the electroproduction of conventional or hybridbaryons with transverse or longitudinal photons}~carefully read by Noel Black. This research is supported by the Departmentof Energy under contract” W-7405-ENG-36.References1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.C.E. Carlson and N.C. Mukhopadhyay, Phgs. Rev. Lett. 67, 3745(1991).C.-K. Chow, D. Pirjol and T.-M. Yan, Phys. Rev. D 59,056002 (1999).T. Barnes and F.E. Close, Phys. Lett. B 123,89 (1983); E. Golowich, E.Haqq and G. Karl, Phys. Rev. D 28, 160 (1983); I. Duck and E. Umland,Phys. Lett. B 128, 221 (1983); C.E. Carlson and T.H. Hansson, Phys.Lett. B 128, 95 (1983); C.E. Carlson, Proc. of the 7th ht. Conf. on theStructure of Baryons (October 1995, Santa Fe, NM), p. 461, ed. B. F.Gibson et al. (World Scientific, Singapore, 1996).N. Isgur and J. Paton, Phys. Rev. D 31, 2910 (1985); Phys. Lett. B124, 247 (1983).S. Capstick and P.R. Page, Phys. Rev. D 60, 111501 (1999).C. Michael, Proc. of the 3“d Int. Conf. on Quark Confinement andHadron Spectrum (L’onjinement III), (June 1998, Newport News, VA).N. Isgur, “Hadron Spectroscopy” in Hadrons and Hadronic Matter, eds.D. Vautherin et al. (Plenum Press, 1990).A.A. Petrov, Proc. of Confinement IIlfJ. Merlin and J. Paton, J. Phys. G 11,439 (1985).L.S. Kisslinger et al., Phys. Rev. D 51,5986 (1995); Nucl. Phys. A 629,30c (1998).L.S. Kisslinger and Z.-P. Li, Phys. Lett. B 445, 271 (1999).A.M. Zaitsev (VES Collab.), Proc. of ICHEP’96 (Warsaw, 1996).Particle Data Group (C. Caso et al.), Eur. Phys. J. C 3, 1 (1998).H. Li (BES Collab.), Proc. of the 8t~ lnt. Conf. on Hadron Spectroscopy(HADRON 99), (Aug. 1999, Beijing, China); B.-S. Zou et al., hep-ph/9909204; B.-S. Zou, these proceedings.N. Isgur, Phgs. Rev. D 60, 114016 (1999).Z.-P. Li et al., Phys. Rev. D 44,2841 (1991); 46, 70 (1992).8

Hybrid baryon signatures

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Hybrid Baryon Signatures

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