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February2,2008 1:55 ProceedingsTrimSize: 9inx6in trento 5 0 0 BARYON RESONANCES IN THE 1/NC EXPANSION 2 n a J RICHARDF. LEBED 4 Department of Physics & Astronomy, Arizona State University, 1 Tempe, AZ 85287-1504, USA v E-mail: [email protected] 1 2 0 The 1/Nc expansion of QCD provides a valuable semiquantitative tool to study 1 baryon scattering amplitudes and the short-lived baryon resonances embedded 0 withinthem. Ageneralizationofmethodsoriginallyappliedinchiralsolitonmodels 5 inthe1980’sprovidesthekeytoderivingarigorous1/Nc expansion. Oneobtains 0 model-independent relations among amplitudes that impose mass and width de- / h generaciesamongresonancesofvariousquantumnumbers. Phenomenologicalevi- p denceconfirmsthatpatternsofresonantdecaypredictedby1/Nc agreewithdata. - Onemayextendtheanalysistosubleadingordersin1/Nc,whereagainagreement p withdataisevident, inbothmeson-baryonscatteringandpionphotoproduction. e h : v 1. Introduction i X About 150 of the 1100 pages in the 2004 Review of Particle Properties1 r a cataloguemeasuredpropertiesofbaryons;andofthese,about100describe resonances unstable against strong decay, with lifetimes so short as to appear only as features in partial wave analyses. Such states have resisted a model-independentdescriptionfordecades. Todatethereexistsnoconvinc- ing explanation for why QCD produces any baryon resonances, much less for their peculiar observed spectroscopy, mass spacings, and decay widths. Even the unambiguous existence of numerous resonances remains open to debate, as evidenced by the infamous 1- to 4-star classification system.1 Baryonresonancesareexceptionallydifficultto study preciselybecause theyare resonancesratherthanstablestates. Forexample,treatingbaryon resonances as Hamiltonian eigenstates in quark potential models is ques- tionable,becausesuchmodelsarestrictlyspeakingvalidonlywhenvacuum qq¯pair production and annihilation is suppressed (to ensure a Hermitian Hamiltonian). It is just this mechanism, however,that provides the means bywhichbaryonresonancesoccurinscatteringfromground-statebaryons. Evenso,oneofthemostnaturaldescriptionsofexcitedbaryonsinlarge 1 February2,2008 1:55 ProceedingsTrimSize: 9inx6in trento 2 N remains an N valence quark picture. The inspiration for this choice is c c thattheground-statebaryonmultiplets (JP=1+, 3+ forN =3)neatlyfill 2 2 c asinglemultipletcompletelysymmetricundercombinedspin-flavorsymme- try [the SU(6) 56, for 3 light flavors],so that one may suppose the ground state of N quarks is also completely spin-flavor symmetric. Indeed, the c SU(6) spin-flavor symmetry for ground-state baryons is shown to become exact in the large N limit.2 Then, in analogy to the nuclear shell model, c excited states are formed by promoting a small number [O(N0)] of quarks c intoorbitallyorradiallyexcitedorbitals. Forexample,thegeneralizationof the SU(6) O(3)multiplet (70,1−)consists ofN 1quarksin the ground c × − state and one in a relative ℓ=1 state. One may then analyze observables such as masses and axial-vector couplings by constructing a Hamiltonian whosetermspossessdefinitetransformationpropertiesunderthespin-flavor symmetry and are accompanied by known powers of N . By means of the c Wigner-Eckarttheorem, one then relates observables for different states in each multiplet. This approach has been extensively studied3,4,5,6,7,8 (see Ref. 9 for a short review), but it falls short in two important respects: First, a Hamiltonian formalism is not entirely appropriate to unstable particles, since it refers to matrix elements between asymptotic external states. Indeed, a resonance is properly represented by a complex-valued poleinascatteringamplitude,itsrealandimaginarypartsindicatingmass andwidth, respectively. Moreover,a naive Hamiltoniandoes not recognize the essential nature of resonances as excitations of ground-state baryons. Second, even a Hamiltonian constructed to respect the instability of the resonances would not necessarily give states in the simple quark-shell baryonmultiplets as its eigenstates. Just asin the nuclearshell model, the possibility of configuration mixing suggeststhat the true eigenstates might consist of mixtures of states with 1, 2, or several excited quarks. In contrast to quark potential models, chiral soliton models naturally accommodatebaryonresonancesas excitationsresulting fromscatteringof mesonsoffground-statebaryons. Suchmodelsareconsistentwiththelarge N limit because the solitons are heavy, semiclassical objects compared to c the mesons. As has been known for many years,10 a number of predictions followingfromthe Skyrme andother chiralsolitonmodels areindependent of the details of the soliton structure, and may be interpreted as group- theoretical,model-independentlargeN results. Indeed, the equivalenceof c group-theoreticalresultsforground-statebaryonsintheSkyrmeandquark models in the large N limit was demonstrated11 long ago. Compared to c quark models, chiral soliton models tend to fall short in providing detailed February2,2008 1:55 ProceedingsTrimSize: 9inx6in trento 3 spectroscopy and decay parameters for baryon resonances, particularly at higher energies. It is therefore gratifying that large N provides a point of c reference where both pictures share common ground. In the remainder of this talk I discuss how the chiral soliton picture (no specific model) may be used to study baryon resonances as well as the full scattering amplitudes in which they appear, and also its relation to the quark picture (again, no specific model). It summarizes a series of papers written in collaboration with Tom Cohen (and more recently our students),12,13,14,15,16,17,18 and updates an earlier version19 of this talk. 2. Amplitude Relations In the mid-1980’s a series of papers20,21,22,23,24 uncovereda number of linear relations between meson-baryonscattering amplitudes in chiral soliton models. The fundamentally group-theoretical nature of these results, as was pointed out, suggested consistency with the large N limit. c Standard N counting25 shows that ground-state baryons have masses c of O(N1), but meson-baryonscattering amplitudes are O(N0). Therefore, c c the characteristicresonantenergyofexcitationabovethegroundstateand resonance widths are both generically expected to be O(N0). To say that c twobaryonresonancesaredegeneratetoleadingorderin1/N thusactually c means equal masses at both the O(N1) and O(N0) levels. c c A prototype of these linear relations was first derived in Ref. 22. For a ground-state(N or ∆) baryonof spin = isospin R scattering with a meson (indicatedbythesuperscript)ofrelativeorbitalangularmomentumL(and primes for analogous final-state quantum numbers) through a combined channel of isospin I and spin J, the full scattering amplitudes S may be expanded in terms of a smaller set of “reduced” scattering amplitudes s: SLπL′RR′IJ = (−1)R′−R [R][R′] [K](cid:26)RK′ LI′ J1(cid:27)(cid:26)KR LI J1(cid:27)sπKL′L ,(1) p XK Sη = δ δ(LRJ)sη , (2) LRJ KL K XK where[X] 2X+1,andδ(j j j )indicatestheangularmomentumaddition 1 2 3 ≡ triangle rule. Both are consequences of a more general formula26 involving 9j symbols that holds for mesons of arbitrary spin and isospin, which for brevity we do not reproduce here. The basic feature inherited from chiral soliton models is the quantum number K (grand spin) with K I+J, ≡ conservedby the underlying hedgehogconfiguration,whichbreaksI andJ February2,2008 1:55 ProceedingsTrimSize: 9inx6in trento 4 separately. The physical baryon state is a linear combination of K eigenstates that is an eigenstate of both I and J but no longer K. K is thus a good (albeit hidden) quantum number that labels the reduced amplitudes s. The dynamical content of relations such as Eqs. (1)–(2) lies in the s amplitudes,whichareindependent foreachvalueofK allowedbyδ(IJK). Infact,K conservationturnsouttobe equivalenttothe largeN limit. c The proof12 begins with the observation that the leading-order (in 1/N ) c t-channel exchanges have I =J ,27 which in turn is proved using large N t t c consistency conditions28—essentially, unitarity order-by-order in 1/N in c meson-baryon scattering processes. However, (s-channel) K conservation wasfound—yearsearlier—tobe equivalenttothe(t-channel)I =J rule,24 t t due to the famous Biedenharn-Elliott sum rule,29 an SU(2) identity. ThesignificanceofEqs.(1)–(2)liesinthefactthatthereexistmorefull observable scattering amplitudes S than reduced amplitudes s. Therefore, one obtains a number of linear relations among the measured amplitudes holding at leading [O(N0)] order. In particular, a resonant pole appearing c in one of the physical amplitudes must appear in at least one reduced amplitude; but this same reduced amplitude contributes to a number of other physicalamplitudes,implyingadegeneracybetweenthemassesandwidths of resonances in several channels.12 For example, we apply Eqs. (1)–(2) to negative-paritya I=1, J= 1 and 3 states (called N , N ) in Table 1. 2 2 2 1/2 3/2 Noting that neither the orbitalangularmomenta L,L′ nor the mesons π,η that comprise the asymptotic states can affect the compound state except by limiting availabletotal quantum numbers (I, J, K), one concludes that a resonance in the SπNN channel (K = 1) implies a degenerate pole in 11 DπNN, because the latter contains a K=1 amplitude. One thus obtains 13 towers of degenerate negative-parity resonance multiplets labeled by K: N , ∆ , (K=0: sη), 1/2 3/2 ··· 0 N , ∆ , N , ∆ , ∆ , (K=1: sπ , sπ ), 1/2 1/2 3/2 3/2 5/2 ··· 100 122 ∆ , N , ∆ , N , ∆ , ∆ , (K=2: sπ , sη). (3) 1/2 3/2 3/2 5/2 5/2 7/2 ··· 222 2 Itisnowfruitfultoconsiderthequark-shellpicturelargeN analogueof c the firstexcitednegative-paritymultiplet [the (70,1−)]. JustasforN =3, c there are two N and two N states. If one computes the masses to 1/2 3/2 O(N0) for the entire multiplet in which these states appear, one finds only c three distinct eigenvalues,6,12,30 which are labeled m , m , and m and 0 1 2 listed in Table 1. Upon examining an analogous table containing all the aParityenters byrestrictingallowedvaluesofL,L′.13 February2,2008 1:55 ProceedingsTrimSize: 9inx6in trento 5 Table1. ApplicationofEqs.(1)–(2) tosamplenegative-paritychannels. State QuarkModelMass PartialWave, K-Amplitudes N1/2 m0,m1 S1π1NN = sπ100 D1π1∆∆ = sπ122 S1η1NN = sη0 N3/2 m1,m2 D1π3NN = 21 sπ122+sπ222 D1π3N∆ = 12(cid:0)sπ122−sπ222(cid:1) S1π3∆∆ = sπ1(cid:0)00 (cid:1) D1π3∆∆ = 21 sπ122+sπ222 D1η3NN = sη2(cid:0) (cid:1) states in this multiplet,12 one quickly concludes that exactly the required resonant poles are obtained if each K amplitude, K=0,1,2, contains precisely one pole, which is located at the value m . The lowest quark-shell K multipletofnegative-parityexcitedbaryonsisfoundtobecompatible with, i.e., consistofa complete setof,multiplets classifiedby K. Butthe quark- shellmassesarereal Hamiltonianeigenvalues,andthereforepresentaresult less general than that obtained from the K amplitude analysis. Onecanprove13 this compatibility for allnonstrangebaryonmultiplets intheSU(6) O(3)shellpicture.b Itisimportanttonotethatcompatibility × does not imply SU(6) is an exact symmetry at large N for resonances as c it is for ground states.2 Instead, it says that SU(6) O(3) multiplets are × reducible multipletsatlargeN . Intheexamplegivenabove,m eachlie c 0,1,2 only O(N0) abovethe groundstate, but are separatedby O(N0) intervals. c c We emphasize that large N by itself does not mandate the existence c of any resonances at all; rather, it merely tells us that if even one exists, it must be a member of a well-defined multiplet. Although the soliton and quark pictures both have well-defined large N limits, compatibility is a c remarkable feature that combines them in a particularly elegant fashion. 3. Phenomenology ConfrontingtheseformallargeN resultswithexperimentposestwosignif- c icant challenges, both of which originate from neglecting O(1/N ) correc- c tions. First,thelowestmultipletofnonstrangenegative-paritystatescovers quiteasmallmassrange(only1535–1700MeV),whileO(1/N )masssplit- c bStudiestoextendtheseresultstoflavorSU(3)areunderway17;whilethegrouptheory ismorecomplicated, itremainstractable. February2,2008 1:55 ProceedingsTrimSize: 9inx6in trento 6 tings can genericallybe as largeas O(100 MeV). Any claims that twosuch statesaredegeneratewhiletwoothersarenotmustbecarefullyscrutinized. Second, the number of states in eachmultiplet increaseswith N , meaning c that a number of large N states are spurious in N =3 phenomenology. c c For example, for N 7 the analogue of the 70 contains three ∆ states, c 3/2 ≥ but only one [∆(1700)] when N = 3. As N is tuned down from large c c valuestoward3,thespuriousstatesmustdecouplethroughthe appearance of factors such as (1 3/N ), which in turn requires one to understand c − simultaneously leading and subleading terms in the 1/N expansion. c Nevertheless, it is possible to obtain testable predictions for the decay channels,evenusing justthe leading-orderresults. For example,note from Table 1 that the K =0(1) N resonance couples only to η(π). Indeed, 1/2 the N(1535) resonance decays to ηN 30–55% of the time despite lying barely above that threshold, while the N(1650) decays to ηN only 3–10% of the time despite having much more comparable phase space to πN and ηN. This pattern clearly suggests that the π-phobic N(1535) should be identified with K=0 and the η-phobic N(1650) with K=1, the first fully field theory-based explanation for these phenomenological facts. 4. Configuration Mixing As mentioned above, one does not expect quark-shell baryon states with a fixed number of excited quarks to be eigenstates of the full QCD Hamilto- nian. Rather, configuration mixing likely clouds the situation. Consider, forexample,theexpectationthatbaryonresonanceshavegenericallybroad [O(N0)] widths. One may ask whether some states escape this restriction c and turn out to be narrow in the large N limit. Indeed, some of the first c work5 on excited baryons combined large N consistency conditions and a c quarkdescriptionofexcitedbaryonstatestopredictthatbaryonsinthe70- analoguehavewidthsofO(1/N ),whilestatesinanexcitednegative-parity c spin-flavor symmetric multiplet (56′) have O(N0) widths. c In fact there arise, even in the quark-shell picture, operators induc- ing configuration mixing between these multiplets.14 The spin-orbit and spin-flavortensor operators(respectively ℓs and ℓ(2)gG in the notationof c Refs. 6,7,30), which appear at O(N0) and are responsible for splitting the c eigenvalues m , m , and m , give nonvanishing transition matrix elements 0 1 2 between the 70 and 56′. Since states in the latter multiplet are broad, configuration mixing forces at least some states in the former multiplet to be broad as well. One concludes that the possible existence of any excited February2,2008 1:55 ProceedingsTrimSize: 9inx6in trento 7 baryon state narrow in the large N limit requires a fortuitous absence of c significant configuration mixing. 5. Pentaquarks Thepossibleexistenceofanarrowisosinglet,strangeness+1(andtherefore exotic)baryonstateΘ+(1540),claimedtobe observedbynumerousexper- imental groups (but not seen by several others), remains an issue of great dispute. Although the jury remains out on this important question, one may nevertheless use the large N method described above to determine c the quantum numbers ofits degeneratepartners.15 For example, if one imposes the theoreticalprejudice J =1, then there must also be pentaquark Θ 2 stateswithI=1,J=1,3 andI=2,J=3,5,withmassesandwidthsequal 2 2 2 2 that of the Θ+, up to O(1/N ) corrections. c The large N analogue of the “pentaquark” actually carries the quan- c tumnumbersofN +2quarks,consistingof(N +1)/2spin-singlet,isosinglet c c ud pairs and an s¯quark. The quark operator picture, for example, shows the partner states we predict to belong to SU(3) multiplets 27 (I=1) and 35 (I=2).31 However,the existence of partners does not depend upon any particular picture for the resonance or any assumptions regarding configu- rationmixing. Since the generic width for such baryonresonancesremains O(N0), the surprisingly small reported width (<10 MeV) does not appear c tobeexplicablebylargeN considerationsalone,butmaybeaconvergence c of small phase space and a small nonexotic-exotic-pioncoupling. 6. 1/Nc Corrections All the results exhibited thus far hold at the leading nontrivial order (N0) c inthe1/N expansion. WesawinSec.3that1/N correctionsareessential c c notonly toexplainthe sizesofeffects apparentinthe data,butinthe very enumerationofphysicalstates. Clearly,ifthis analysisis tocarryrealphe- nomenological weight, one must demonstrate a clear path to characterize 1/N corrections to the scattering amplitudes. Fortunately, such a gener- c alization is possible: As discussed in Sec. 2, the constraints on scattering amplitudesobtainedfromthelargeN limitareequivalenttothet-channel c requirement I =J . In fact, Refs. 27 showed not only that the large N t t c limit imposes this constraint, but also that exchanges with I J =n are t t | − | suppressed by a relative factor 1/Nn. c Thisresultpermitsonetoobtainrelationsforthescatteringamplitudes February2,2008 1:55 ProceedingsTrimSize: 9inx6in trento 8 incorporating all effects up to and including O(1/N ): c 1 R′ I L′ R′ J SLL′RR′IsJs = (cid:20)R 1 I =s (cid:21)(cid:20)R L J =s (cid:21)stJLL′ XJ t J t J 1 R′ I L′ R′ J 1 s s st(+) −Nc (cid:20)R 1 I = (cid:21)(cid:20)R L J = +1(cid:21) JLL′ XJ t J t J 1 1 R′ Is L′ R′ Js st(−) +O( 1 ),(4) −Nc XJ (cid:20)R 1 It=J (cid:21)(cid:20)R L Jt=J−1(cid:21) JLL′ Nc2 One obtains this expression by first rewriting s-channel expressions such as Eqs. (1)–(2) in terms of t-channel amplitudes. The 6j symbols in this casecontainI andJ asarguments(whichforthe leadingtermareequal). t t Onethenintroduces16newO(1/N )-suppressedamplitudesst(±),forwhich c J I = 1. The square-bracketed 6j symbols in Eq. (4) differ from the t t − ± usual ones only through normalization factors, and in particular obey the same triangle rules. Relationsbetweenobservableamplitudesthatincorporatethelargerset st, st(+), and st(−) are expected to be a factor of N =3 better than those c merely including the leading O(N0) results. Indeed, this is dramatically c evident in πN π∆, where sufficient numbers of amplitudes are measured → (Fig. 1). For example, (c) and (d) in the first four insets give the imag- inary and real parts, respectively, of partial wave data for SD ( ) and 31 ◦ (1/√5)DS ((cid:3)),whichareequaluptoO(1/N )corrections;in(c)and(d) 13 c of the secondfour insets, the points againareSD data, while ♦ repre- 31 ◦ sent √2DS , and by Eq. (4) these are equal up to O(1/N2) corrections. − 33 c 7. Pion Photoproduction Meson-baryon scattering is not the only process that can be considered in the soliton-inspired picture. As long as one knows the isospin and spin quantum numbers of the field coupling to the baryon along with the corresponding N power suppressionof eachcoupling, one may carry out pre- c cisely the same sort of analysis as described above. The processes we have in mind are those involving real or virtual photons (photoproduction,18 electroproduction, real or virtual Compton scattering). One minor complication is that the electromagnetic interaction breaks isospin, in that the photon is a mixture of isoscalar (I = 0) and isovector (I=1) sources. The former is suppressed by a factor 1/N com- c February2,2008 1:55 ProceedingsTrimSize: 9inx6in trento 9 1.2 0.1 1 (a) Im T Im T 0 0.8 0.6 −0.1 0.4 −0.2 0.2 0 −0.3 (c) −0.2 −0.4 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1.2 0.3 1 (b) Re T 0.2 Re T 0.8 0.1 0.6 0 0.4 0.2 −0.1 0 −0.2 −0.2 −0.3 (d) −0.4 −0.4 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 W (GeV) W (GeV) 1.2 0.1 1 (a) Im T Im T 0 0.8 0.6 −0.1 0.4 −0.2 0.2 0 −0.3 (c) −0.2 −0.4 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1.2 0.3 1 (b) Re T 0.2 Re T 0.8 0.1 0.6 0 0.4 0.2 −0.1 0 −0.2 −0.2 −0.3 (d) −0.4 −0.4 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 W (GeV) W (GeV) Figure1. RealandimaginarypartsofπN →π∆scatteringamplitudes. Thefirstfour insetsgivetwoparticularpartialwavesequaltoleadingorder[henceindicatingthesize ofO(1/Nc)corrections]. Thesecondfourinsetsgivetwoparticularlinearcombinations ofthesamedatagoodtoO(1/N2). c paredto the latter since baryoncouplings carryingboth a spin index (coupling to the photon polarization vector) and an isospin index are larger than those carrying just a spin index by a factor N .32 c Moreover,electromagneticprocessesaretypicallyparametrizedinterms of multipole amplitudes, which combine the intrinsic photon spin with its relativeorbitalangularmomentum;infact,thisisveryconvenient,because then the photon can be treated effectively as a spinless field whose effec- tive orbital angular momentum is the order of the multipole. Note that this makes processes with virtual photons just as simple as those with real photons,eventhoughtheformercancarrynotonlyspin-1butspin-0ampli- February2,2008 1:55 ProceedingsTrimSize: 9inx6in trento 10 tudes as well. With these caveatsin mind, carryingoutananalysis ofpion photoproductionamplitudes, including 1/N corrections(leading plus sub- c leadingI=1amplitudesandleadingI=0amplitudes),isstraightforward.18 For example, a relationship receiving only O(1/N2) corrections reads c m,p(π+)n m,n(π−)p L+1 m,p(π+)n m,n(π−)p M =M M M , (5) L,L,− L,L,− −(cid:18) L (cid:19)h L,L,+ − L,L,+ i where the superscript m means magnetic multipoles, N(πa)N′ means the process Nγ N′πa, and the subscripts L,L, mean that an electromag- → ± netic multipole oforderLcreatesapionintheLth partialwave,withtotal J=L 1. Including just the first term on the right-handside (r.h.s.) gives ±2 a relation valid up to O(1/N ) corrections, and the quality of both this c relation and its extension to next-to-leading order may be assessed. A sample result appears in Fig. 2, where the left-hand side (l.h.s.) is a solid line, the O(1/N ) result is dotted, and the O(1/N2) is dashed. While c c the agreementat first glance may not seem impressive,some very hearten- ingfeaturesmaybediscerned. First,theagreementintheregionbelowthe appearance of resonances is quite good, and indeed improves at O(1/N2). c Second, unlike the solid line [containing D (1520)], the dotted line gives 13 no hint of a resonance but the dashed line does [D (1675)]; and the fact 15 that their positions do not precisely match should not alarm us, as one expects them to differ by an amountof O(Λ /N ) 100 MeV. One may QCD c ∼ in fact use the helicity amplitudes compiled1 for these two resonances and relate them directly to the amplitudes appearing in Eq. (5). In order to obtain dimensionless and scale-independent results, one divides the linear combination of helicity amplitudes corresponding to Eq. (5) by the same expressionwithallsignsmadepositive. The O(1/N )andO(1/N2)combi- c c nationsgive18 0.38 0.06and 0.13 0.06,respectively,showingthatthe − ± − ± 1/N expansion works beautifully—even better than one might expect. c 8. Conclusions: The Way Forward There now exist reliable and convincing calculational techniques using the 1/N expansion of QCD that handle not only long-lived ground-state c baryons,butalsounstablebaryonresonancesandthescatteringamplitudes inwhichtheyappear. Theapproach,originallynotedinchiralsolitonmod- elsbuteventuallyshowntobeatrueconsequenceoflargeN QCD,isfound c tohavephenomenologicalconsequences[suchasthelargeη branchingfrac- tion of the N(1535)] that compare favorably with real data.