REVISTA MEXICANA DE FtSICA 47 SUPLEMENTO 2. 123-127

Fingerprint of a QCD string in baryon spectraM. Kirehbaeh

Escuela de Fúica, Universidad Autónoma de ZacatecasApanado postal C-600. 98062 Zacateca.I'. Zac .. Mex/ca

Recibido ellO de enero de 2001; aceptado el 28 de febrero de 2001

DICIEMBRE 2001

Multi-spinvalued stales oflhe Rarila-Sehwinger lypc (k/2, k/2) 0 [(1/2, O) EIJ(O, 1/2)1 wilh k = 1,3, and 5 are found 10 be realized in lheex.citation spcClra of ¡he light-quark baryons. \Ve conjcclUrc that the aboye multi-rcsonance c1usters may takc thcir origin from a QCD stringdcscribcd by means of a linear actioll.

Keywords: Barionic spcctrum; QCD string

Los eSladosvalorados con multiples spines dellipo Rarila-Sehwinger (k/2, k/2) 01(1/2, O) EIJ (O, 1/2)J con k = 1,3, Y5 se encuentrande los espectros de la cxcitatiol1 de lo~ baryoncs con quarks ligeros. Conjeturamos que los multiplctcs de Lorcntz pueden lomar su origen deuna cuerda de QCD descrita por medio de una acción lineal.

Descriptores: Espectro bari6nico; cuerda de QCD

PACS: 98.80.Cq

3

Iv,) = I:Ut;!11l,), e = e, /', T. (5)i=l

The specification of particles by the mass- and spin-quantumnumbcrs is one of the most important paradigms in contem-porary field theory. The purily of this paradigm was doubtedlatest by neutrino physics experiments, which provide evi-denee [3]lhal a neutrino of a given flavor, Iv,). instead ofbeing of definite mass, 1m,), ralher exists in a linear superpo-sitian of three states of differenl masses

1. Introduction

The definition of particle states is at the very heart of contem-porary quamum theory of fields. It lakes ilS origin from Iheframe-independent invariants of the Poincaré group as wasti,,1 realized by Wigner in his work of lale thirties [1]. Onthat ground, quantum states of free particles in a Poincaré co-variant framework transform for different inertial observersaccording to

11/>') = exp [i(," P" - O"V I"v»)l1/». (1)

Here p¡, and the totally anlisymmetric tensor llw with¡t(v) = 0,1,2,3 are lhe generators of the Poincaré groupwhich satisfy the commutation relations (Poincaré algebra)

cording to

1V"II'"I1/» = -j(j + 1)01'11/»,

IV - I IVPP'p. - -'2(¡lIlPT (4)

[I¡H"Ipa] =i(gIlP1¡w-g¡wIIIQ +9p.qIIIP-gllul¡lP)' (2)

[P,.,Ip.] =i(g"vP.-g".Pp), [P",Pv] =0, (3)

where g¡W = diag(l, -1, -1, -I) is the metric tensor, and (;1'

ano 81111 are continuous paramelers. Purthermore, PI' gene-rate lranslations, Ikj = 'kjt.!, wilh k, (j, 1) = 1,2,3 are thegenerators of rotations in the kj-plane, while 101 = K¡ gener-ale the baoS! along eaeh of the I-axis. Wigner postulated thalparticle states transform as c/assicaI unitary, ano thereforeinfinite-dimensional representations of the Poincaré group.Later on, Weinberg argued 12} that quantized non-unitary2(2j + 1)-dimensional representations ean also be given par-ticle interpretation, once unitarity of the corresponding quan-lized field operalors is ensured. Weinberg's argument allowedone to consider massive $lates of pure spin. Indeed, as thePoincaré group has PP.PP. and H'p.lVP. as Casimir operators(wilh IV" slanding for the so ealled Pauli-Lubanski veelor),particles can be characterized by 11lass (m) and spin (j) ac-

Here, 1m,) denote basis states of definite mass (and henee un-determined flavor), while Un are elements of lhe 3 x 3 uni-tary mixing matrix. The latter equation shows that the neu.trino mass is undetermined beyond certain ftavor-dependentfraelional aeeuraey [4].

In lhe eonlexl of the existing baryonie speetra, we shallestablish the lhesis lhal, in addition lo slales of pure spin anddefinite mass, such as neutron, proton, etc., or, states of purespin and indefinite mass,like the neutrinos, also states of defi.nite mass and indefinite spin, i.e. multi-spin-valued $lates, doexisl. We shall present a dala-supported new interpretation ofRarita-Sehwinger fields as sueh multi-resonanee clusters.

2. Multi-spin sta tes

The massive four-dimensional representation spaee (1/2, 1/2)of the Lorentz group is lhe simpleS! example for a non-purespin state. This follows from the faet. thal Ihe only non-

124 F1NGERPRINT OF A QCD STRING IN BARYON SPECTRA

vanishing matrix elements of, say, K z in this multiplet are

(j' = O+,m = OIK,li = 1-,m = O) = i =(j' = 1-,m = OIK,li = O+,m = O). (6)

The latter are described in lerrns of totally syrnmetric trace-Iess rank-k Lorentz tensors with Dirae spinor componentsand satisfy the Dirac equation for each Lorentz index asso-ciated wilh a (1/2, 1/2) space

In coupling now lhe Dirac spinor to (k/2, k/2) from above,lhe following spin (J) and panty (P) quanlum numbers arecrealed

One of the basic qualíly tests for any model of compositebaryons is the level of accuracy reached in describing the nu-cleon and t. excitation spectra. In thal respect, the knowl-edge on the degeneracy group of baryon spectra appears asa key tool in constructing the underlying strong.interactiondynamics. To uncover it, one has first to analyze isospin byisospin how the masses ofthe resonances from the fulllistingin Ref. 12, spread wilh spin and panty. Such an analysis hasbeen perfonned in our previous work [9] where it was foundlhat Breit-Wigner masses reveal on lhe M/J plane a wellpronounced spin and parity clustering. There, it was furthershown that the quantum numbers of the resonances belongingto a particular cluster fit into Lorentz group representations of

1" 1 -" 3 -" ( 1) -"JP = "2 '"2 '"2 , ... , k + "2 (10)

In the following, we will use forthe spín-sequence in Eq. (10)the short-hand notation 021.'1' with 1 standing for the isospinof the stales considered, and a = k + l.

Now, it was initiated by Weinberg around mid sixties [2]to consider the lower-spin components of the RS fields as re-dundant, unphysical states lhat have 10 be projected out by aset of suitably chosen Lorentz covariant auxiliary conditions.In doing so, one eventually would describe a baryon of sin-gle spin-J = k + 1/2 and jixed parity as the highest-spinstate ofthe W 1L¡1L2".P", fleld and benefit from the linear differ-ential equation [Eq. (6»). Immediately, one noles thal such adescription is not necessarily compatible with the general na-tme of the boost generator. Indeed, since Lorentz and spinorindices of lhe RS fields always faclorize, the bOOSIof thisrepresentation space is most naturally obtained in boostingits bosonic part independently from the fennionic one. Theproperties of the boost to nOl preserve the quantum number ofspin, in combination with Eqs. (9) and (10), clearly illustratethal the presence of the lower-spin components in (k/2, k/2)and thereby in the associated RS cluster, guarantee in a sim.pie manner its covariance with respect to Lorentz transfor-mations. It is líule wonder, that truncated RS fields suffervarious palhologies (see Ref. 1I far details). So one has loface the unavoidable conclusion of the principal difficully lodescribe pure higher-spin states of fixed parily in terms oflinear differential equalions. NOle, that the problem arose outof the tacit assumption that nature produces baryons as purehigher-spin states. A glance at lhe baryon spectra leaches usthat actually Nalare strongly favors the excitations of mulli-spin-valued resonance clusters over that of pure higher-spinstates.

We now show that the new interprelation of the RS fieldsis supported by the data on baryonic spectra.

3. Rarita-Schwinger fields as multi.resonanceclusters

(7)

(9)

(8)

m=-l, ... ,l.

(k k) [(1 ) ( 1)]q. .- - - 0 - O EIl 0-/-11/-12""/-1,.'- 2' 2 2' 1 2 .

¡p - O" 1-" ( 1)-"- , , ...• a - 1

P ill"lU '1,1••.. :::::;1}e a '1;/- ••..'

The simplest field of the type in Eq. (7), wilh k = 1 is buillon top of the four-vector. It was considered more than twodecades earlíer by Ranta and Schwinger [6J and applied lothe descnption of a spin-3/2 partícle.

The RS fields are compilations of fermions of differenlspins and parities. This statement is best i1lustrated at thelevel of the group 0(4), the Wick rotated compact forro ofthe Lorentz group. For the sake of concreteness. we con-sider the coupling of, say, a positive parity Dirae fermionto an 0(4) boson composed of 0(3) states of either natu-ral (r¡ = +), or, unnalural (1/ = -) panlies. These massdegenerate 0(3) states carry aH integer internal angular mo-menta.l, with l :::::;O, ... , u - 1 and transfonn with respect tothe spaee inversion operation P as

It is obvious, that the boost does nal preserve the quantumnumber of spin and demands the (1/2,1/2) represenlationspace lO be in general covariantly inseparable into pure spin-Oand spin-l states. A more detailed analysis of the mas-sive [our-dimensional Lorentz multiplet was performed inRef. 5. There, the (1/2, 1/2) space was constructed ah initioas (1/2, O)0 (O, 1/2) direct product. The separation betweenspin-O and spin-l states was found to be possible only in thekinematica\ly restncted settings of the rest frame, on the onehand, and the helicity frame, on lhe other hand. In the mostgeneral case of a boost along a direction different bUl the axisofrotalion, lhe (1/2, 1/2) space was found to covarianlly bi-fu reate solely into a singlet and a triplel of opposite parities.without that these state carry any definite spin. [n [5] il wasfurther found that the P".4" = Ocondition, usua\ly viewedas an auxiliary condition for eliminating ane out of the fourdegrees of freedom of lhe (1/2, 1/2) space, is in fact tnvial-Iy satisfied for the tnplet part where it simply reduces to themass-she\l condition p' - m2 = O, while it can not be ful-fi\led al a\l for its singlet subspace. Thus, the (1/2, 1/2) spaceof finite-mass is in general twa-spin-valued.

The four vector is at the very heart of lhe so ca\led Ranta-Schwinger fields

Rev. Mex. Fís. 47 S2 (2001) 123-127

M. KIRCHBACH 125

FIGURE l. Rarita.Schwinger c1ustering of light-quark resonances.The full bricks stand Cor three-lo four-star resonances. the cmptybricks are onc- lOtwo-star stalcs, while the trianglcs represent statesthal are "missing" for the complClcncss oC the thrcc RS c1ustcrs.Note lhal "missing" F17 and H¡,ll nudcon cxcilalions in Fig. laappcar as four.slar resonances in the ~ spcclrum of Fig. 1b. Thc"missing" 6. excilations P3•• P33. and 033 from 63,_ are one.tolwo slar resonances in the nuclcon countcrpart 6}, _' The 6.(1600)resonance (shadowed oval) drops out of OUT RS cluster systematicsand we vicw it as an ¡ndependent hybrid stalc.

Ihe RS type. To be specilic, one finds Ihe three RS clus-ters wilh k = 1,3, and 5 in both the nucleon (N) and 21speclra. In lerms of the notations introduced aboye, all re-poned Iighl-quark baryons wilh masses below 2500 MeY(up to the 21 (1600) resonance thal is mosl probably an¡ndependen! hybrid state), have been shown in Ref. 13 tobe complelely accommodaled by the RS clusters 221.+,421 _, and 621 _, having states of highest spin-3/2-, 7/2+,ami 11/2+, re;pectively (see Fig. 1). In each one of Ihe nu-cleoo, .6., and A hyperon speclra, the natural parity c1us.ter 221.+ is always of lowest mass. We consider it lo re-side in a Fock space, :F+. built 00 lOp of a scalar va-cuum. From Eqs. (10) and (11) follows that the 221.+ clus-lers, where 1 = 1/2, 3/2, and O, always unite the lirstspin-I/2+, 1/2-, and 3/2- resonances. For the non-slrangebaryons, 221.+ is followed by Ihe unnalural parily clus-lers 421._ and 621,_. which we view 10 reside in a diffcT-ent Fock space, :F_, thal is buill on top of a pseudoscalarvacuum. Tobe specilic, one linds all the seven Ll-baryon re-sonances S31' P31, P33.D33. D35• F3s and F37 [rom 43,_ lobe squeezed within the narrow mass region from 1900 MeYlo 1950 MeY, while lhe 1 = 1/2 resonances paralleling Ihem,of which only Ihe F17 slale is still "missing" from Ihe data,are localed around 1700:!:;g MeY (see Hg. 1). Therefore,the Fl7 resonance is the only non-strange state with a massbelow 2000 MeY which is "missing" in the presenl RS clas-silication scheme. Funher paralleling baryons from the Ihirdnucleon and .ó. clusters with a ::;::G. ane finds in addition[he [OUT states H1 11' Palo P33' and D33 with rnasses abo.ve 2000 MeY to be "missing" for Ihe completeness of theocw classification scheme. The H1 11 stale is nceded 10 pa-rallel the well established H3,1l b~ryon. while the .6,-statesP311 P33' and 033 are required as partners to the (less estab-lished) PII (2100), Pd1900), and DI3(2080) nucleon reso-nances. For A hyperans, sparse data prevent a conclusive

4. Empiricallinear mass relation for themulti.resonanee clusters

(11 )

.......... &lmtr + () (4)_ pmlomiNiltd, () /JI

~!i.!~"":,.4/'"1.1 I

"

1.'_l.l:: 1.0 ••..•..................••.....

~ 1.1/. .,;.;.,;,;,,,.,,"',.;.;....... 60._ ~

:;: 1,6 40.- r1.4 +.:':.'''' 10.+

I.l ~

",-~~~===-=!¡+ rrJ+j+r7+:r-,+,- 2J1t

The reported mass averages of the resonances fram the RSmulliplels wilh k = I,:l, and 5 are well described by meansof the following simple empirical recursive relation

M•. - M. = mi [:, - (7\2]I [(7')2_I_a2_1]

+21112 2 2'

(a) (b)

FIGURE 2. a) Clustering in the A hypcron spcctrum and b) 0(4) ro-talional bands of nucleon (N) and (~) cxcilalions (right). Notalionsas in Fig. l.

analyses. Even so, see Fig. 2, Ihe RS paIlem is alreadyapparenl in the reponed spectrum. The (approximate) de-generacy group of baryon srectra as already suggested inReL 13 and funher juslified here, is, Iherefore, found tobe SU(2) 1 00(1, 3)¡" i.e .• isospin0space-time symmetry.

Within OUT scheme, Ihe inter-c1usler spacing of 200lo 300 MeY is larger by a faclor of 3 to 6 as compared tothe mass spread within the c1usters. For example, ¡he 21.+'23.+,41._, and 43,_ c1usters carry the maximal mass split-ting of 50 to 70 MeY. The above consideralions establish thal

1. Observed exciled lighl-fiavor baryons preferably existas multi-resonance c1usters that are described in termsof the Ihree RS mulliplets 22/.+' 42/.- and 62/._'

2. The above RS c1usters exhaust all lhe resonances ob-served in the" N scaIlering channel lup to Ihe 21(1600)statel. They constitute an almost accomplished excita-tion mode in its own rights. as only 5 resonances are"missing" for the completeness of this structure.

3. As long as the inlemal parily of the c1uslers changesfram natural for the first one, to unnatural, for the sub.sequent two, the question arises whether this changcsignals a phase Iransilion for baryons. In ReL 14we further showcd how presence or absence of theindependent cluster excitation sequence 32/,_,32/,+'

and 52/.+' could probe Ihe scale of the chiral phasetransition for baryons. In this way. the "missing"state search program of Ihe CLAS collaboralionat JLAB [11 J obtains an additional motivalion, thatis conceptually differenl from the SU(6)SF 00(3)Lclassification schcme.

(b)

H

~::-~'::::::1~U •~I ... --.;,;._..•"••..••..•

~ I •.....

IJU N

r rrrS+5"'7+¡'-ru+ 2J'It(a)

Rev. Mex. Fú. 4752 (2001) 123-127

126 HNGERPRINT OF A QCD STRING IN BARYON SPECTRA

5. QCD string wilh linear aclion

where. again. a = k + 1. The two mass parameters take lhevalues mi = 600 MeY. ano 11l'l = 70 MeV, respectively.The tirsl lerm on lhe RHS in Eq. (12) is Ihe lypical differencebetween lhe energies of 1wo single particle stales of princi-pal quantum numbers a. and al, respeclively. occupied by aparticle wilh mass T1l moving in a Coulomb-like potential ofstrength (le Wilhmi = (1~m/2. The term

in Eq. (12) is Ihe generalizalion of lhe thece-dimensio-nal j(j + 1) rule (wilh j = k/2) lo four Euclidean dimen-sioos and descrihcs a generalizcd 0(4) rotatimlOl batid (seeFig.2).

The parameter 11m2 = 2.82 fm corresponds lo lhe 1110-

menL of ¡nenia ~1= 2/5J\I R2 of sorne "effective" rigid-body resonance wilh mass ¡\f = 1085 MeY and a ra-dius R = 1.13 fm. Nulc Ihal while Ihe splilling belween lheCoulol11b~like states dccreases with increasing a. lhe diffe-reoce between the energies ol' the rotational states increaseslinearly with (J so that the net effect is a slightly increasingspacing belween lhe baryoll c1usler Icvels.

The requirement of Lorentz covariancc of Eq. (14) restricts Wto be a Lorentz multiplet construct and imposes conditionsonto the form of the rll matrices. To be specific. these matri-

One of lhe frontier problcms of contemporary theoreti-cal physics is (O understand how the properties of low-energy hadrun physics relale lo QCD -lhe gauge lheoryof strong inleraction-. Because of lhe non-Abelian charac-ler of lhe color gauge gruup SU(3)e. Ihe associaled gaugebosons, the gluons, posscss various self-interaClions andbnng highly non-linear lerms inlO Ihe QCD Lagrangian.This circumstance introduces serious difficulties in makingdirecI QCD prediclions, and oblaining solulions lhal de-scribe the hadronic degrees of freedom. As a possible ansatzfor such a solution, a string in four space-time dimensionswas considered [12, 13J. A particular lype of lhis solution isknown under the name "gonihedric" string, or simply. QCDstring.Q The gonihedric string has been defined as a modelof random surfaces with an action which is proportional tothe linear size of the surface [13]. In the continuum limit. itsequation describes the propagation of a state that is a multi-fermion system as it sprcads over a space conlour. When thesurfacc shrinks to a single world line. thl: action bccomcs pro-purtional 10 Ihe lenglh of Ihe palh and Ihe melhod resemblesFeynman's palh integral for a point-like Dirac l'ennion. Thepropagalion of a multi-fennion sta te thal contains a spin-l/2fermiol1 in the point-particlc limil can be described by mean sof a Dirac-type equíltion for an intinitc dimensional wavefunclion

u"-I k(k )-- =2- - + 12 2 2 '

(if,,D" - ",)IJI = O.

(12)

( 13)

ces have to satisfy Majorana's commutation relations

[f,,'!,p] = y",fp - Y.pf,. (14)In Ref. 18 various Solulions of Eq. (14) al Ihe malrix elemenllevel were suggesled. In Ihe following we shall focus onone of them. that de serves our special altention as it appliesdirectly 10 low-energy hadron physics. It is lhe so called "ge-neralized solulion" of Ref. 18. lhal predicIs a mass speclrumcharaclenzed by a (2u - 1)-fold degeneracy according lo

[Afj;~)'J2=",2(1_ 4~2) [j~~:r,u=2,4,6, ... , (15)

wilh jIu) ranging from j~~! = 1/2 lu ji::; = u - 1/2.Though a similar degeneracy is known from the hydrogenatom, where a acts as the principal quantum number of theCoulomb problem, Ihe quadralic mass Eq. (16) should nolbe confused wilh lhe Salmer-series, where Ihe energy enlersIinearly.

The dilTerence between the squared masses of the spin-c1uslers fullowing from Eq. (16) can be cast inlo lhe furm

[Mj;;'~~,]2 _ [Mj:;,;,.]2 = 4m2 [j;::: + 1]

",2 ['(U)J2[ 1 1]+4 J..... u' - (u + 2)' . (16)

Now Ihe queslion arises. how Ihe Iheury of lhe gonihe-dric QCD slring relales lo hadron speclroscopy.

6. Signalure for QCD string in baryonexcilations

Comparison of Eqs. (12) and (11)-( 17) reveals a slriking si-milarilY belween lhe speclrum of Ihe gonihedric QCD stringand the reported spectra of the non-strange baryons. Thissimilarity concerns

i) The (2u - I)-fold degeneracy oflhe stales.ii) The occurrence of even values of a.iii) The _l/u' pattern in Ihe mass distribulion.

Wir/zill rhis COllle:a, Qne is ratller rempted 10 cOlljeclUre tharlhe obsen'ed hydrogell./ike degelleracy j,1 baryofl spee/mmay take j{S roots in lhe gOflilzedric QCD str;lI~.

This statement is further supported by the realistic masssplitting of lhe baryoll spin-c1uslers concluded frum lhe QCDslring lheory on lhe grounds of Eq. (17). To lix Ihe unknownparameler m2 there, one may tit the differcncc between thesquared masses of the 2nd and 1st c1usters to data. From theempiricalobservation

(Ar,.:,)2 - (M~('j,)2= 0.61 GeY",J J

one finds ",2 = 0.061 Gey2. In using Ihis value lo calculale(Ar;~i,n.,)2 _ (hr;~il"•.•)2 we find

J J

(M~;;)"'f - (Af~:;,;,)2 = l.l GeV2•J J

This numher is close lo the empirical one, where oneencounters

Rev. Mex. Fís. 4752 (2001) 123-127

M. KIRCHBACH 127

TABLE 1. String-like baryon excitations (in MeV). The experimen-tal values of the averaged masses of the multi.fermion clusters aredcnoted by Af;xP. while i\!;~:;)g.and J\f:s, stand in tum for thetheoretical predictions of the quadratic striog, and the linear RSmass formuJac. The difference il", = Afo+2 - Al" as calculatcdfrom the QCD slriog model (.ó.~h;"g). lhe RS schcrnc (.ó.:S) andas obscrved experimentally (L\~"P).For other "alarioos, (sce tcxt).

a",n /vrlring Af!:S AJ:."P Ó"Ir;ng L\RS L\':xp., j("') v u '" q v

Moreover, O1lso the linear mass difference Arlring - ArUin"J(<,,+2) j(<T)

can be calculated and compared to data. This is done inTable 1 whieh also eontains the predielions following fromEq, (12), The undertaken eomparison shows thal lhe QCDslring rnodel leads to an approximate equidistant spacing be-tween the spin-clusters. At the first glance, on the experimen-tal side, the equidistant mass spacing does not seem as wellpronounced as on the theoretical side. Note, however, thatmost of (he resonances participating the (J = 6 spin-clusterare not well established and the experimental estimate of theaveraged mass here may not be realistic. A reliable know-ledge of the precise locations of the resonances in this massregion is urgently desirable and a challenge to experimen-tal resonanee programs. Finally, lhe absolule values of lhec1usler masses following from the QCD slring theory can befound also in Table 1, where one finds them strongly under-eslimated though with inereasing mass, the effeet nolieeablydecreases. '

3708641360

149816892102

488496

322350

191413

Lorenlz boost and should nol be projecled OUl.These fieldsare realized in the spectra of the light-flavor baryons. wherethe landscape of the ex cita tion s is well structured alongthe RS c1assifiealion seheme, ralher than populaled by ran-domly distributed resonances. The (approximate) degene-raey symmelry luroed out lo be SU(2)1 (8) 0(1,3)/" Withinthis eonlext, lhe 221,+' 421,_' and 62/,_ RS c1uslers ob-served so far in the "N seallering ehannel by lhe LAMPFat LANL, constitute an almost accomplished excitation modewith only five states "missing". The dynamical origin forthe RS clustering is still lacking a unique explanation and itremains a challenge for future research. The factorization ofisospin from lhe spaee-lime symmelry in SU(2)1 (8) 0(1,3)/,is strongly supported by QCD, where the isodoublel lighlquarks and the isosinglet heavy-flavor quarks are the es-tablished isospin degrees of freedom, On lhe olher hand,any QCD solution has necessarily to be a Lorentz covariantobjeel. In Ref. 14 lhe bosonie parts of the Lorenlz c1uslers,sueh as (k/2, k/2) from lhe RS field in Eq, (17), were eonsid-ered as independent fundamental bosonic degree of freedomof baryon slrueture, and lhe term "hyperquark" was eoinedfor them. There, lhe elustering was modeled after a 0(4)invarianl quark-hyperquark corcelalion, From a slightly dif-ferent QCD perspective, the clusters can also be viewed asstring solutions associated with a linear action [14J. On thewhole, our view is that a 'itructured baryon spectrum thatshares comrnon flavor and relativistic syrnmetries with QCDis more likely to be linked via an appropriale effeelive theoryto first principies of strong-interaction dynarnics lhan a spec-trumwith states distributed al randorn.

7. Surnrnary and Outlook

We eonclude lhal lhe lower-spin eomponenlS of the Rarita-Sehwinger fields are foreed upon by lhe properties of lhe

Q The presentation of this section follows Savvidy's work [14].

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2, S, Weinberg,Phys, Rev. 133 (1964) 8318,

3. The Super-Kamiokande collaboration, Phys. Rev. Let!. 82(1999) 2644; LSND eollaboration, ibid, 81 (1998) 1774,

4. G.Z. Adunas, E. Rodríguez-Milla. and D.V. Ahluwalia. Phys.Lell. B 485 (2000) 215,

5, D.Y.Ahluwalia and M, Kirchbach, hep-phlOlOlOO9,

6, W.Rarita and J, Schwinger, Phys, Rev, 60 (1941) 61.

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Acknowledgrnents

Work supported by CONACyT Mexieo,

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Rev, Mex, Fís, 4752 (2001) 123-t27