Simultaneous analysis of non-strange negative parity baryon Properties in large N c QCD

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1 Sr Lankan ournal of Physs, Vol. () () 7-8 Insttute of Physs - Sr Lanka Researh Artle Smultaneous analyss of non-strange negatve party baryon Propertes n large QCD Shavndra P. Premaratne, C.P. ayalath Department of Physs, Faulty of Sene, Unversty of Peradenya, Sr Lanka Abstrat on-strange l exted baryons are analyzed wthn the Large QCD framework. The Large approah s used as an effetve theory to smultaneously ft the masses and deays wth orretons up to ( / ). Values of the mxng angles for the S / and S / nuleon mass egenstates are obtaned wthn the smultaneous analyss. The observed effetve operator herarhy s onsstent wth the prevous studes onduted n ths setor and mproved results are obtaned. Keywords: Large, negatve party baryons. ITRDUCTI Quantum Chromodynams (QCD), the theory of strong nteratons, s a strongly oupled theory at low and ntermedate energes, whh does not offer a small expanson parameter. QCD has a runnng ouplng onstant, s, whh depends on the energy of the partles beng onsdered. Thus, perturbaton theory, whh had mmense suess n Quantum Eletrodynams (QED) due to the smallness of the fne struture onstant 7, fals n the low energy regme of QCD. As the naturally ourrng hadrons belong to ths energy lmt, non-perturbatve approahes were developed to study ther propertes. Varous models suh as the onsttuent quark model, the bag model, and the Skyrme model were employed to qualtatvely desrbe the baryon propertes. However, the results obtaned were dependent on the model used. At present, Latte QCD s the most wdely used non-perturbatve tehnque to study hadron propertes. Correspondng author Emal : japjayalath@gmal.om

2 S. P. Premaratne et al. /Sr Lankan ournal of Physs, Vol. (), () Hooft proposed to generalze the number of olors to an arbtrary number,, and paved the way for a method to study hadron propertes as an alternatve to methods suh as Latte QCD. The expanson parameter used n 't Hooft s model was /, and the perturbatve tehnques n terms of / ould be appled both n low and hgh energy regmes of QCD. The theory smplfed n the lmt. Wtten frst appled ths method to study baryons 7 and formulated the large ountng rules. Wtten also proved that large baryons should be omposed of valene quarks and then the baryon mass should be of ). ( Baryon spn-flavor symmetry at large was dsovered by Gervas & Sakta 8,9 n 98 and ndependently by Dashen & Manohar n 99. Ths has enabled the use of the / expanson n baryon phenomenology. In the QCD baryon setor, t was dsovered that an exat ontrated spn-flavor symmetry SU( f ) exsts for large, where f s the number of lght quark flavors 8,9,. Ths symmetry has paved the way for an expanson n / of QCD operators to be developed for baryon propertes n terms of spn-flavor SU ( f ) generators. In operator analyss, effetve operator bases are onstruted and ordered n powers of / wth the ultmate objetve of analyzng baryon observables. Hene, the / expanson s used as an effetve theory wth a base of QCD operators. In the exted baryon setor, there have not been any well-establshed modelndependent methods for analyss of baryon propertes. The onsttuent quark model s a good qualtatve theory n ths doman. The / approah has been appled to many aspets of exted baryons wth a hgh degree of suess. The symmetry for the exted baryons s ( ) SU( f ) where () orresponds to the spatal rotatons and SU ( f ) s the spn-flavor symmetry. In these resonane studes, the baryons are P lassfed nto groups as X, where X s the dmenson of the rreduble representaton n SU (), s the orbtal extaton level and P s the party of the multplet. In the urrent study, the non-strange members of the 7, mxed-symmetr baryon multplet are analyzed. It should be noted that although there are 7 states possble aordng to the group theoret analyss, a szeable porton of them are spurous states. 7, s the lowest lyng negatve-party exted baryon multplet and one of the best explored. As ths study s onerned wth the non-strange members of the multplet, only seven members are onsdered n the analyss. There are two sets of states whh have dental spn and sospn. Hene these states tend to mx, gvng the followng physal states, ' os sn sn os (') where / and / s the total spn, represent mass egenstates and the S notaton has been used wth S / and S / for the quark spn ontrbuton. ()

3 S. P. Premaratne et al. /Sr Lankan ournal of Physs, Vol. (), () The organzaton of the paper s as follows. In seton, the tehnque used for the smultaneous analyss s desrbed. The operator bases used and the orrespondng matrx elements are gven. Seton presents the detaled results along wth graphal representatons. The dsusson of the results and the onluson are gven n seton.. AALYSIS TECHIQUE. Bakground In ths work, the / expanson s utlzed as an effetve theory and a bass of effetve operators s extrated from prevous ndependent studes of masses and nonstrange strong deays of the baryons onerned,,. A sutable subset of effetve operators s hosen based on the relatve ontrbuton and the order n /. These operators are smultaneously ft wth the avalable data to obtan a onsstent value for the mxng angles of the S / and S / nuleon mass egenstates. Data ponts wth very hgh errors are removed n favor of obtanng a more relable ft. Partal deay wdths for three unobserved hannels are predted whh are onsstent wth and mprove upon the results of prevous studes onduted. The smultaneous ft for the masses and deays s onduted wth orretons up to / ). ( The mxng angles are theoretally obtaned usng the mass matres of S / and S / nuleons. The theoretal expressons for the angles are used as nput parameters for the mxng present n strong deay ampltudes. Then the best ft for the analyss s obtaned by mnmzng the h-squared value, ( th) ( emp) ( ) dof emp ob p () where s an ndex runnng over all ob emprally known observables onsdered, p (th) s the total number of ndependent parameters n the ft, s the theoretal (emp) (emp) expresson for the observable, s the empral value and s the error assoated wth t. The above value s mnmzed usng the oeffents of the effetve operators. The degrees of freedom, dof s determned as dof () ob p. perator Bass for Mass Analyss In the studes by Carlson et al.,, nne operators are obtaned wth orretons up to ( / ). However, ther analyss reveals that the frst sx operators are the most sgnfant ontrbutors to the baryon masses. Hene, n ths study, only the frst sx operators are onsdered for the analyss.

4 S. P. Premaratne et al. /Sr Lankan ournal of Physs, Vol. (), () 7-8 The matrx elements of the operators used n the baryon mass analyss are shown n Table. The thrd and sxth rows orrespond to off-dagonal matrx elements. These operators were used to onstrut the theoretal expressons for the masses of seven baryons as, M () where are the dynamal oeffents whh have to be determned by fttng to expermental data gven by PDG. To obtan the mxng angles, frst the mass matres are formed as, ' ' ' () where / and /. Then egenvalues of above matrx are obtaned and the orrespondng egenvalues are ft wth the expermental masses. The mxng angles an then be obtaned by alulatng the egenvetors.. perator Bass for Deay Analyss In the study by Goty et al., 7 effetve operators are obtaned for strong deays up to ( / ). Aordng to the quark model, the η partle s a mxture of u, d, s quarks and u, d, s antquarks. As ths study s onerned strtly wth the nonstrange deays, any nvolvng the η partle were not nluded n the analyss. Hene, onsderng purely non-strange deays, operators are at dsposal for the analyss. All deays ontanng the emsson of a π meson, exept () D-wave deay (due to unrelablty) were used n the ft. The ndependent operators used are shown n Table. The orderng n / for the operators was performed followng Goty et al. by studyng the matrx elements and not neessarly usng the bodyness of the operators.

5 S. P. Premaratne et al. /Sr Lankan ournal of Physs, Vol. (), () 7-8 Table : Matrx elements of operators, that are lnearly ndependent for j, / ' / s () ( ) ' / / / ' / gg s tg ( ) ( ) ' / / / ( ) S S ( ) ( ) ( ) ( ) / 8( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 8 / perator verall Fator 9 ( )

6 D-Wave S-Wave S. P. Premaratne et al. /Sr Lankan ournal of Physs, Vol. (), () 7-8 Table : Bass operators for the ndependent deay analyss ame perator Bodyness rder n / B g, B, B, st,, a B,,, ts, a B, B,, gs, a,, B g,a, B, st B,, a, B,, ts, a B,,, gs, a B, B,, gs, a, B,, sg, a,,, s S, G, a, B 7 B, B,,, s S, 8 G, a [ lp, I P ] The notaton used for the operators, B [ mp, I P ], ndates the spn l P and sospn I P of the operator along wth the respetve projetons m P and I P. For smplty, the spn and sospn projetons have been omtted n some operators. The fnal set of ndependent operators has been onstruted by ouplng the orbtal transton operator, [,], wth the spn-flavor operators and alulatng the matrx elements expltly. The deay wdth for eah l, I ] hannel s gven by [ P P l P, I P lp k 8 lp M M B B P P P P P q q q q I l, I l, I l, I P () where k s the meson momentum, Λ s a sale fator hosen to be MeV, M B s the * mass of the ground state baryon resultng from the deay, M B s the mass of the exted * * [, P ] baryon, and I are the spn and sospn of the exted baryon, l P I are q

7 S. P. Premaratne et al. /Sr Lankan ournal of Physs, Vol. (), () 7-8 [, P ] dmensonless dynamal oeffents, l P I q are redued matrx elements of the [, P ] orrespondng operators and l I P q are normalzaton fators whh have been used n the analyss. The mxng of the states ours wthn the deay ampltudes. Table : Redued matrx elements of pon S-wave deays S-wave deays / / / / / /,,,, verall Fator, The redued matrx elements of S-wave deay operators are shown n Table. The orrespondng dynamal oeffents n the ft are s where,,,. The redued matrx elements of the D-wave deay operators are shown n Table. The orrespondng oeffents for the D-waves n the ft are d j where j,..., 7 (sne the eghth operator has only a small ontrbuton). The ombned h-squared value for masses and deays was onstruted usng Eq.(). The theoretal expressons for the mxng angles were alulated from the mass matres onstruted from Eq.(). These expressons were used as nput parameters for the mxng of deay ampltudes as gven n Eq.().

8 S. P. Premaratne et al. /Sr Lankan ournal of Physs, Vol. (), () 7-8 Table : Redued matrx elements of D-wave deays D-wave Deays / / / / / / / / / / /,,, 8 8,,,, , verall Fator , 7 7 7

9 D-Wave Deay Coeffents S-Wave Deay Coeffents Mass Coeffents (MeV) S. P. Premaratne et al. /Sr Lankan ournal of Physs, Vol. (), () 7-8. RESULTS The empral values for the baryon masses, S-wave and D-wave deays were obtaned from the Partle Data Group (PDG). In Table, the results of several fts are presented. It should be noted that the ndependent fts have been redone usng the updated empral values and do not neessarly reprodue the results of prevous studes,,. The B operators whh were omtted from prevous analyses were nluded n the present analyss sne they had a sgnfant ontrbuton to the fts., However, the operator 8 was not nluded sne ts ontrbuton was for only three of the eleven deay hannels and would have only resulted n a lesser number of degrees of freedom. Table : Ft Parameters for the Independent and Smultaneous Analyses Coeffents Mass Ft Deay Ft Smultaneous analyss 8 ± ± - ± 7 - ± 7 ± 8 ± 87 ± ± 9 8 ± 7 ± ± ± s 7.7 ± 7.9 ± s. ± 8. ± 8 s. ± 7. ± 8 s -8.8 ± 7 -. ± 7 d. ±.. ±. d -. ±. -. ±. d. ±. ± d. ±.. ±.7 d -. ±. -. ±. d -.8 ±. -.7 ±. d 7 -. ±. -. ± dof dof. ±.8. ±.9.8 ±.9. ±..9 ±.7.97 ±. The baryon masses and partal wdths for strong deays resultng from the dfferent fts are represented graphally n Fg. and Fg.. Deay hannels for whh the values are unknown or unrelable are represented wthout the empral values.

10 S. P. Premaratne et al. /Sr Lankan ournal of Physs, Vol. (), () 7-8 Fgure : Baryon masses resultng from the fts and the empral values Fgure : Partal deay wdths resultng from the fts and the empral values

11 S. P. Premaratne et al. /Sr Lankan ournal of Physs, Vol. (), () DISCUSSI AD CCLUSIS Several noteworthy aspets of the study an be dentfed. The smultaneous analyss of the masses and strong deays gve results onsstent wth the prevous studes,, and gve results well wthn the expermental errors. The expermental errors n the deays are qute hgh, resultng n hgher unertanty n the subleadng order oeffents. The ftted oeffents for S-wave and D-wave deays are separately of natural sze whh follows from the normalzed operators used n the analyss. Hene the relatve ontrbutons of the operators an be studed wthn a partular deay type but not between the S-waves and D-waves. The effetve oeffents whh arry all the dynam aspets of the strong nteraton, are very muh smlar n the ndependent and smultaneous analyses, whh onfrms the valdty of the / expanson for both masses and deays. In ths study, the mxng angles for the S /, / nuleons are obtaned as.8.9 and.97.. The value for s smlar to the one obtaned n the mass analyss and agrees well wth the values n other studes,,. The value for s a sgnfant mprovement on the value obtaned n the ndependent deay analyss. The mxng angles obtaned were also ompared wth the results from onsttuent quark model studes,7,8,9. The obtaned values for and agree well wth the values from mass analyses based on the quark model,7 and have mproved upon values from the deay analyses 8,9. The most reent study onduted n ths setor has been by de Urreta and Soola. Ther onsstent analyss nluded the study of masses, deays and the helty ampltudes. ur study dd not nlude the helty ampltude data due to ther unrelablty. Furthermore, the extenson to SU () would not have been possble wth the helty ampltudes due to the sarty of data. The results for the mxng angles obtaned n our study have mproved upon the values obtaned n the onsstent analyss. The next natural step n ths study for the future s the extenson of the smultaneous analyss to SU (). In ths setor, several mxng angles should be determned. Also the SU () symmetry s moderately broken. Hene, symmetrybreakng operators wll also be employed. The SU () analyss wll however offer a rh soure of data unlke observed n the present analyss. It s hoped the suess observed n the SU () analyss an be reprodued n the SU () analyss. When the results for the omplete exted baryon spetrum are avalable through latte smulatons, applaton of the / expanson to the results s expeted. Ths wll enable the determnaton of the quark mass dependene on the dynamal oeffents and hene the nuleon mxng angles.

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