Effective Strange Quark/Antiquark Masses from the Chiral Soliton Models for Baryons
aa r X i v : . [ h e p - ph ] S e p Effective Strange Quark/Antiquark Masses fromthe Chiral Soliton Models for Baryons
Vladimir KopeliovichInstitute for Nuclear Research of RAS, Moscow;Moscow Institute for Physics and Ingeneering (MIPT)
Abstract
The effective strange quark and antiquark masses are estimated from the chiralsoliton approach (CSA) results for the spectrum of exotic and nonexotic baryons. Thereare problems when one tries to project results of the CSA on the simple quark models(QM): the parameter in 1 /N c expansion is so large for the case of baryon spectrumthat extrapolation to the real N c = 3 world is not possible; rigid (as well soft) rotatormodel and the bound state model coincide in the first two orders in 1 /N c , but differ inthe next orders. There is correspondence of the CSA and simple QM predictions forpentaquarks (PQ) spectra in negative S sector of the { } and { } -plets: the effectivemass of strange quark is about 135 − M eV , slightly smaller for { } . For positivestrangeness components the link between CSA and QM requires strong dependence ofthe effective ¯ s mass on particular SU (3) multiplet. The SU (3) configuration mixingis important, it pushes the spectrum towards the simplistic model (with equal massesof the strange quark and antiquark), and explanation of this nice property is lackingstill. The success of the CSA in describing many properties of baryons and lightnuclei (hypernuclei) means that predictions of pentaquark states should be consideredseriously. Existence of PQ by itself is without any doubt, although very narrow PQmay not exist. Wide, even very wide PQ should exist, therefore, searches for PQremain to be an actual task. Studies of baryon spectrum - nonstrange, strange, and with heavy flavors - remain to beone of main aims of accelerator physics. Discovery of baryon states besides well establishedoctet and decuplet, in particular, exotic baryons, could help to the progress in understandinghadrons structure. In the absence of the complete theory of strong interactions there aredifferent approaches and models; each has some advantages and certain drawbacks. Interpre-tation of hadrons spectra in terms of quark models (QM) is widely accepted, QM are ”mostsuccessful tool for the classification and interpretation” (R.Jaffe) of hadrons spectrum. TheQM are to large extent phenomenological; simplicity of QM becomes a fiction when we tryto go behind e.g. 3-quark picture for baryons, since there is no regular methods of solvingrelativistic many-body problem. The number of constituents (e.g. additional q ¯ q -pairs) is notfixed in a true relativistic theory. 1lternative approaches, in particular, the chiral soliton approach (CSA) [1, 2, 3]has certain advantages. It is based on few principles and ingrediemts incorporated in themodel lagrangian. Baryons and baryonic systems are considered on equal footing (the look”from outside”). CSA looks like a theory, but still it is a model, and some elements ofphenomenology are present necessarily within the CSA. It has been noted first in [4] and,for arbitrary baryon number, in [5] that so called exotic (i.e. containing additional quark-antiquark pairs) states appear naturally within the CSA. More definite numerical predictionsfor the mass of exotic baryon with strangeness S = +1 were made somewhat later in [6]and (quite definite!) in [7]. Results obtained within CSA for the spectrum of baryons withdifferent values of strangeness mimic some features of baryons spectrum within quark modelsdue to Gell-Mann - Okubo relations.In the next section basic features and properties of the CSA are described and theGell-Mann — Okubo relations for the spectrum of baryons within the rigid rotator model(RRM) of skyrmions quantization are presented at arbitrary number of colors N c . Section3 describes the bound state model (BSM) results for the baryon spectrum, the differenebetween the RRM and the BSM results is fixed, and the way to remove it is discussed. Firstterms of the 1 /N c expansion for the effective strange quark/antiquark masses are presntedhere. Section 4 contains numerical estimates of the effetive strange quark/antiquark masseswithin simple quark model where the quark/antiquark masses make additive contribution tothe baryon mass. Final section summarizes the main problems and conclusions. Arbitrary SU (2) skyrmion is described by three profiles { f, α, β } ( x, y, z ) which parametrizethe unit vector on the 3-sphere S , and the baryon number of the configuration is the degreeof the map of the 3-dimensional space R to the 3-sphere S . Masses, binding energies ofclassical configurations, moments of inertia Θ I , Θ J and some other characteristics of chiralsolitons contain implicitly information about interaction between baryons. Minimizationof the static energy (mass) functional M class provides three profiles and allows to calculatemoments of inertia, etc. The details can be found in [3, 8, 9, 10].The observed spectrum of states is obtained by means of the quantization procedureand depends on the baryon quantum numbers and static characteristics of the skyrmion,moments of inertia Θ, Σ-term (Γ), etc. In SU (2) case, the rigid rotator model (RRM) ismost effective and successfull in describing the properties of nucleons, ∆-isobar [3], of lightnuclei [11] and also ”symmetry energy” of nuclei with A < ∼
20 [12].In the SU (3) case the mass formula takes place, also for RRM [13] M ( p, q, Y, I, J ) = M cl + K ( p, q, J )2Θ K + J ( J + 1)2Θ π + δM ( Y, I ) , (1)where these terms scale as functions of the number of colors N c as ∼ N c , ∼ , ∼ N − c and ∼ , correspondingly; it is in fact expansion in powers of 1 /N c . The quantity K ( p, q, J ) = C ( SU − J ( J + 1) − N c B /
12 (1 a )contains difference of the second order Casimir operators of the SU (3) and SU (2) groups, C ( SU
3) = ( p + q + pq ) / p + q , where p, q are the numbers of upper and lower indeies2n the spinor describing the SU (3) multiplet, C ( SU
2) = J ( J + 1), J being the spin of thebaryon. It is worth to mention here that formula (1) takes place only for the particularcase when the inident SU (2) skyrmion is located in the ( u, d ) SU (2) subgroup, i.e. it isnonstrange. Other possibilities for the starting skyrmion configuration have been consideredas well [14]. Remarkable property of Eq. (1) is that the total splitting of the whole SU (3)multiplet is ∼ N c .Mass splittings δM are due to the term in the lagrangian L M ≃ − ˜ m K Γ s ν , (2) ν is the angle of rotation into strange direction, ˜ m K = F K m K /F π − m π includes the SU (3)-symmetry violation in flavor decay constants, the sigma - term Γ ∼ Gev − , moments ofinertia Θ π ∼ (5 − Gev − , Θ K ∼ (2 − Gev − , see [8, 9] and references here. All momentsof inertia Θ ∼ N c . Strange, or kaonic inertia Θ K contains important contribution due toflavor symmetry breaking in meson decay constants, F K /F π ≃ . K = 18 Z (1 − c f ) " F K + 1 e f ′ + 2 s f r ! d r. (3)This expression is valid for skyrmions originally located in the ( u, d ) SU (2) subgroup of SU (3). ”Strangeness contents” of baryons C S = < s ν / > B (4)can be calculated exactly with the help of the wave functions in SU (3) configuration space,for arbitrary number of colors N c [8, 9]. Some examples of the C S values at arbitrary numberof colors N c are: C S (” N ”) = 2( N c + 4)( N c + 3)( N c + 7) , C S (”Ξ”) = 4 N c + 7 , C S (”Θ”) = 3 N c + 9 , (5)Approximately at large N c C S ≃ | S | N c . (6)The Gell-Mann - Okubo formula takes place in the form [9] C S = − A ( p, q ) Y − B ( p, q ) h Y / − ~I i + C ( p, q ) , (7) A ( p, q ) , B ( p, q ) , C ( p, q ) depend on particular SU (3) multiplet. For the ”octet”, [ p, q ] =[1 , ( N c − / A (”8”) = N c + 2( N c + 3)( N c + 7) , B (”8”) = 2( N c + 3)( N c + 7) , C (”8”) = 3( N c + 7) . (8)If we try to make expansion in 1 /N c , then the parameter is ∼ /N c . For ”decuplet” ([ p, q ] =[3 , ( N c − / p, q ] = [0 , ( N c + 3) / ∼ /N c and becomes worse for greater multiplets, ” { } ”-plet, ” { } ”-plet, etc. Apparently,3or the realistic world with N C = 3 the 1 /N c expansion does not work and some propertiesof baryon spectrum which take place at large N c may be not correct at the realistic value N c = 3. Any chain of states connected by relation I = C ′ ± Y / m effS ∼ ˜ m K Γ[ A ( p, q ) ∓ ( C ′ + 1 / B ( p, q )] , (9) C ′ = 1 for decuplet (antidecuplet). This is valid if the flavor symmetry breaking is included inthe lowest order of perturbation theory. At large N c , m effS ∼ ˜ m K Γ /N c , too much, ∼ . GeV if extrapolated to N c = 3.If we make expansion in the RRM, we obtain for the ”octet” of baryons δM N ≃ m K Γ N c − N c ! , δM Λ ≃ ¯ m K Γ N c − N c ! δM Σ ≃ ¯ m K Γ N c − N c ! , δM Ξ ≃ ˜ m K Γ N c − N c ! , (10)For the ”decuplet” of baryons we have after such expansion δM ∆ ≃ m K Γ N c − N c ! , ... δM Ω ≃ ¯ m K Γ N c − N c ! , (11)equidistantly for all components. For positive strangeness components of exotic multipletswe obtain from our previous results [9, 10] δM Θ ,J =1 / ≃ ˜ m K Γ N c − N c ! , δM Θ ,J =3 / ≃ ¯ m K Γ N c − N c ! , It is instructive to note here that physics implications of the large N c extrapolation are ambiguous. Inparticular, the electric charge of quarks is not fixed. If one takes the charge of the u ( c, t ) quark equal to 2 / d ( s, b ) quark equal to − /
3, then the hypercharge of a baryon consisting of N c quarksis integer only if N c / Y = N c B/ S + ..., see, e.g. [15] and [8]. The charges of the ”proton” and neutron are Q p = ( N c + 3) / , Q n = ( N c − / N c , Q u ( c,t ) = (1 + 1 /N c ) / , Q d ( s,b ) = ( − /N c ) / . These expressions follow from the Gell-Mann - Nishijima relations, Q = I + Y / Y = B + S + C + t + b ,the latter equality is generalization of the original relation for the hypercharge, Y = B + S , with flavors C (charm), b (beauty) ant t (truth) included. B is the baryon number, B q = 1 /N c for quarks in the QCD with N c colors. Evidently, the charge of any baryon is an integer number in this case [16]. Experimental check ofthe quark properties in the hypothetical (gedanken) large N c world is not possible, of course. By this reasonwe could guess that charges of the quarks are Q u ( c,t ) = (1 + α ) / , Q d ( s,b ) = ( − α ) / , baryon consist of 1 /α quarks, α being arbitrary. It would be in analogy with consideration of D -dimensionalspace- time with noninteger D . M Θ ,J =5 / ≃ ˜ m K Γ N c − N c ! , (12)These results are summarized in Table 1. Within the bound state model (BSM) antikaon field is bound by the SU (2) skyrmion [17, 18].The mass formula takes place M = M cl + ω S + ω ¯ S + | S | ω S + ∆ M HF S (13)where strangeness and antistrangeness excitation energies ω S = N c ( µ − / K , ω ¯ S = N c ( µ + 1) / K , (14) µ = q m K /M ≃ m K M , M = N c / (16ΓΘ K ) ∼ N c , µ ∼ N c . (15)The hyperfine splitting correction depending on hyperfine splitting constants c and¯ c , and ”strange isospin” I S = | S | / M HF S ( S, I, J ) = J ( J + 1)2Θ π + ( c S − J ( J + 1) − I ( I + 1)] + (¯ c S − c S ) I S ( I S + 1)2Θ π (16)and is small at large N c , ∼ /N c , and for heavy flavors (see [18, 8, 10] where the hyper-fine splitting constants c S , c ¯ S are presented). For anti-flavor (positive strangeness) certainchanges should be done: ω S → ω ¯ S and c S → c ¯ S in the last term. The baryon states inthe BSM are labeled by their strangeness (flavor in general case), isospin and spin, but donot belong apriori to a definite SU (3) multiplet [ p, q ], and can be some mixture of different SU (3) multiplets.In this way we obtain for the ”octet” [8] δM N = 2 ˜ m K Γ N c , ... δM Ξ = ˜ m K Γ N c − N c ! , (17)and for ”decuplet” δM ∆ ≃ m K Γ N c , ... δM Ω ≃ ¯ m K Γ N c − N c ! , (18)Total splitting of the ”octet” and ”decuplet”∆ tot (”8” , BSM ) = ˜ m K Γ N c − N c ! , ∆ tot (”10” , BSM ) = ˜ m K Γ N c − N c ! . (19)In the BSM the mass splittings are bigger than in the RRM.For exotic S = +1 Θ- hyperons we obtain in BSM [8, 9, 10] δM BSM Θ ,J =1 / = ¯ m K Γ N c − N c ! , δM BSM Θ ,J =3 / = ¯ m K Γ N c − N c ! , M BSM Θ ,J =5 / = ¯ m K Γ N c − N c ! (20)and again considerable difference from the RRM results presented in Eq. (12) takes place.From the comparison of these results with previous section we conclude that the RRMused for prediction of pentaquarks in [4, 6, 19] is different from the BSM model, used in thepaper [20] to disavow the Θ + . { } { } { } { } { } m RRMs − /N c − /N c − − /N c − / N c m BSMs − /N c − /N c − − − m RRM ¯ s − − − /N c − /N c − /N c m BSM ¯ s − − − /N c − /N c − /N c Table 1.
First terms of the /N c expansion for the effective strange quarkmass (the upper two lines) and the antiquark mass (the lower two lines)within different SU (3) multiplets, in units ¯ m K Γ /N c . Empty spaces are leftin the cases of theoretical uncertainty. The assumption concerning strangequarks/antiquarks sea should be kept in mind, see explanation in the text. The mass m s for the ”octet” is defined as half of the splitting between nucleon ( Y =1 , I = 1 /
2) and Ξ-hyperon ( Y = − , I = 1 / m s is defined as 1 / Y = 1 , I = 3 / Y = − , I = 0. For highermultiplets the strange quark masses are obtained from the negative strangeness sectors, asdifferences of masses of states with ( Y, I ) = (0 ,
2) and ( − , / − , /
2) and ( − ,
1) forthe ” { } ”-plet, and states with ( Y, I ) = (1 , /
2) and (0 , ,
2) and ( − , / { } ”-plet.To define the masses of the strange antiquarks we assumed first that the strangequarks sea within exotic multiplets is the same as in the ”octet” and ”decuplet”, i.e. C RRMS ( sea ) = C RRMS (”8” , sea ) ≃ C RRMS (”10” , sea ) ≃ N c (cid:18) − N c (cid:19) , (21)and similar for the BSM: C BSMS ( sea ) = C BSMS (”8” , sea ) ≃ C BSMS (”10” , sea ) ≃ N c , (22)Strangeness contents of the nucleon and delta-isobar coincide in the leading and next-to-leading orders of the 1 /N c expansion. Then, the strange antiquark mass equals m ¯ s ( B ) = ¯ m K Γ [ C S ( B ) − C S ( sea )] , (23)where B is the exotic, strangeness S = +1 baryon. These assumptions lead to the resultspresented in Table 1.It follows from Table 1 that the addition of the term to the BSM result, possible dueto normal ordering ambiguity present in the BSM (I.Klebanov, VBK, 2005, unpublished)∆ M BSM = − m K Γ N c (2 + | S | ) (24)6rings results of the RRM and BSM in agreement - for nonexotic and exotic states. Thisprocedure looks not quite satisfactorily: if we believe to the RRM, there is no need to considerthe BSM and to bring it in correspondence with the RRM. Anyway, the RRM and the BSMin its accepted form are different models .The SU (3) configuration mixing of exotic baryon multiplets has been studied in [19],but was ignored in most of previous papers devoted to exotic baryons. For antidecupletmixing with nonexotic components of the octet is important, it decreases slightly the totalsplitting, and pushes N ∗ and Σ ∗ toward higher energy. Apparent contradiction with simplestassumption of equality of masses of strange quarks and antiquarks m ( s ) = m (¯ s ) takes place.For decuplet configuration mixing with components of the { } -plet increases totalsplitting of the decuplet considerably, but approximate equidistancy in the position of thedecuplet components still remains. The configuration mixing should be included within theQM as well: states with different numbers of q ¯ q pairs can mix, and this is complicated notresolved problem, see next section.The rotation-vibration approach (RVA) by H.Weigel and H.Walliser [21] unifies theRRM and BSM in some way, Θ + has been confirmed with somewhat higher energy andconsiderable width (Γ Θ ∼ M eV ). It is possible to make comparison of the CSA results with expectations from simple quarkmodel in pentaquark approximation, or N c + 2 approximation for arbitrary N c (projection ofCSA on QM). The masses m s , m ¯ s and m ( s ¯ s ) come into play. The following diquark-diquark-antiquark wave functions are considered usually. For antidecuplet | , Y = 2 , I = 0 , I = 0 > (Θ +0 ) ∼ ¯ s [ du ][ du ]; | , , I = 1 / , I = 1 / > ∼ − ¯ d [ du ][ du ] + ¯ s [ su ][ du ] + ¯ s [ du ][ su ]; | , , I = 1 , I = 1 > ∼ − ¯ d [ su ][ du ] + ¯ d [ du ][ su ] + ¯ s [ su ][ su ]; | , − , I = 3 / , I = 3 / > ∼ Ξ ∗ / ∼ − ¯ d [ su ][ su ] (25) . [ q q ] means antitriplet in color, singlet in spin, antisymmetric in flavor combination (an-titriplet) considered in [22], see also [23], and called ”good” diquark, according to [24]. Thesewave functions can be obtained by action of the U -spin operator, U d = s, U ¯ s = − ¯ d , andshould be normalized properly. For each of the isomultiplets the states with other possible3-d projection of the isospin I can be obtained from the wave function of the highest statewith I = I by action of the lowering operator I − . Obviously, the weight of the s ¯ s pair is0 , /
3; 1 / , and 0 in these 4 states of the antidecuplet.For { } -plet we have | , Y = 2 , I = 1 , I = 1 > (Θ ++1 ) ∼ ¯ s ( uu )[ du ] + ¯ s ( du )( uu ); | , , I = 3 / , I = 3 / > ∼ − ¯ d ( uu )[ du ] + ¯ s ( uu )[ su ]; | , , I = 2 , I = 2 > ∼ − ¯ d ( uu )[ su ]; | , − , I = 3 / , I = 3 / > ∼ − ¯ d ( su )[ su ];7 , − , I = 1 , I = 1 > ∼ − ¯ d ( ss )[ su ] (26)The weight of the s ¯ s pair is 0 , / , , q q ) is triplet in spin, symmetric inflavor (i.e. 6-plet in flavor) diquark configuration, so called ”bad” diquark (in color it is alsoantitriplet, similar to ”good” diquark).For the { } -plet we have, also for the components with maximal isospin | , Y = 2 , I = 2 , I = 2 > (Θ +++2 ) ∼ ¯ s ( uu )( uu ); | , Y = 1 , I = 5 / , I = 5 / > ∼ ¯ d ( uu )( uu ); | , Y = 0 , I = 2 , I = 2 > ∼ ¯ d ( su )( uu ) + ¯ d ( uu )( su ); | , Y = − , I = 3 / , I = 3 / > ∼ d ( su )( su ) + ¯ d ( uu )( ss ) + ¯ d ( ss )( uu ); | , Y = − , I = 1 , I = 1 > ∼ ¯ d ( su )( ss ) + ¯ d ( ss )( su ); | , Y = − , I = 1 / , I = 1 / > ∼ ¯ d ( ss )( ss ) . (27)There are no s ¯ s pairs in the QM wave functions of the 35-plet components. Table 2, givenalso previously in [8, 10], summarizes these results, in comparison with the rigid rotatormodel calculations. The upper lines (for each of the SU (3) multiplets) in the table show thecontribution of masses of the strange quark m s , antiquark m ¯ s and the quark-antiquark pair m s ¯ s into the mass of the quantized state in the simple quark model where the quark massmakes additive contribution to the mass of the baryon, according to wave functions given ibEq. (25)-(27). E.g., for the antidecuplet it is m ¯ s , 2 m s ¯ s / m s + m s ¯ s / m s (the numberof colors N c = 3). Here we keep different masses for the strange quark and antiquark, and m ¯ ss may be different from the sum m ¯ s + m s . However, without configuration mixing simplerelations take place for the mass of the strange quark-antiquark pair m s ¯ s : m s ¯ s | { } = m s | { } + m ¯ s | { } , (28)and m s ¯ s | ( { } = m s | { } + m ¯ s | { } . (29)These equalities are in fact the consequences of the Gell-Mann — Okubo relations for themasses of the SU (3) multiplet components.The expressions (25) − (27) for the wave functions will be more complicated if thenumber of colors N c >
3, and the N c + 2 approximation should be investigated insteadof the pentaquark approximation. The contribution of the m s ¯ s to the masses of stateswith hypercharge Y = 1, or zero strangeness will be ( N c + 1) / ( N c + 3) for ”antidecuplet”,( N c − / ( N c + 1) for the ” { } ”-plet, and ( N c − / ( N c −
1) for ” { } ”-plet, which obviouslygo over to 2 /
3, 1 / s − quark/antiquarkmasses, since in the differences of masses of states which belong to the same SU (3) multiplet8he rotational energies cancel, according to previously given in [8, 10] expressions, as wellas the strange quarks sea contributions. From the total splitting of antidecuplet we obtain,before the SU (3) configurations mixing[2 m s − m ¯ s ] = ¯ m K Γ / ≃ M ev. (30)Configuration mixing decreases this quantity to 247
M eV . | , , > | , , > | , , > | , − , >m ¯ s + ... m s ¯ s / ... m s + m s ¯ s / ... m s + ...
564 655 745 836600 722 825 847 | , , > | , , > | , , > | , − , > | , − , >m ¯ s + ... m s ¯ s / ... m s + ... m s + ... m s + ...
733 753 772 889 1005749 887 779 911 1048 | , , > | , , > | , , > | , − , > | , − , > | , − , >m ¯ s + ... ... m s + ... m s + ... m s + ... m s + ... Table 2.
The strange quark (antiquark) masses contributions to the massesof baryons according to the simple wave functions in the pentaquark approx-imation, N c = 3 (first lines for each of exotic SU (3) multiplets of baryons).Numerical values, in M eV , are given for the mass differences of the baryonstate and the nucleon, second and 3-d lines for each of multiplets, within theCSA. 2-d lines - without configuration mixing, and 3-d line - configurationmixing included. Θ K = 2 . GeV − , Θ π = 5 . GeV − and Γ = 1 . GeV − [19]. In the simplistic model ( m ¯ s = m s = m ¯ ss /
2) we would obtain ∆( { } ) = m s , incontradiction with numerical value from the CSA. Remarkably, that configuration mixingdecreases the splitting of antidecuplet (247 M ev instead of 272
M ev ), thus pushing the resultof the CSA toward the simplistic QM. If we believe that the strange quark mass within theantidecuplet is in usually accepted interval, ∼ − M ev , then the strange antiquarkmass should be unusually small, even negative.From splittings within { } -plet we obtain, before mixing[ m s − m ¯ s ] = ¯ m K Γ /
56 = 39
M ev, (31)Configuration mixing decreases this to 30
M ev . So, strange antiquark is lighter than strangequark, and again the configuration mixing pushes the effective strange antiquark mass to-wards the simplistic model. From the negative strangeness sector of the { } -plet we get[ m s ] = 3 ¯ m K Γ / ≃ M ev, (32)before mixing and [ m s ] ≃ M ev after the configuration mixing, in reasonable agreementwith accepted value of the strange quark mass. The mass of the cryptoexotic component ofthe 27-plet S = 0 , I = 3 / } { } { } { } { } m RRMs ( no mix ) 1 /
12 1 / − / m RRM ¯ s ( no mix ) − − − /
56 13 / m s − m ¯ s ) RRM ( no mix ) − − / / − / Table 3.
The values of the strange quark/antiquark masses in the RRMwithout configuration mixing (no mix), in units ¯ m K Γ in the realistic case N c = 3 . The strangeness contents C S of the states used for this evaluationare taken from [9, 10]. From the { } -plet the difference of energies of states with S = +1 and S = 0 givesthe mass of the strange antiquark[ m ¯ s ] = 13 ¯ m K Γ / ≃ M ev. (33)Configuration mixing deceases this quantity down to ≃ M ev . From the negative strangenesssector of the 35-plet we obtain m s | = 5 ¯ m K Γ / ≃ M ev, (34)and the configuration mixing increases this value up to ∼ M ev , again in reasonableagreement with commonly accepted value.Our results for the strange quark, antiquark masses and the combination 2 m s − m ¯ s are presented in Table 3 (in units of m K Γ) and in Table 4 (numerically, in
M tV ). Strongdependence of the s -antiquark mass on the multiplet takes place, in qualitative agreementwith the 1 /N c expansion, presented in Table 1. It is a challenge to theory to understand, isit an artefact of CSA, or has physical meaning.For the octet the quark mass m s is the half of the total mass splitting, for decupletit is 1 / m s is defined by splittings in thenegative strangeness sector, components with largest isospin. { } { } { } { } { } m RRMs ( no mix ) 181 91 −
117 113 m RRMs ( mix ) 196 134 −
135 130 m RRM ¯ s ( no mix ) − − −
78 295 m RRM ¯ s ( mix ) − − −
105 270(2 m s − m ¯ s ) RRM ( no mix ) − −
272 156 − m s − m ¯ s ) RRM ( mix ) − −
247 165 − Table 4.
Numerical values of the effective strange quark/antiquark masses(in
M eV ) in the RRM without configuration mixing (no mix), and with con-figuration mixing (mix). The results of previous papers [8, 9, 10] are usedhere. Θ K = 2 . GeV − , Θ π = 5 . GeV − and Γ = 1 . GeV − [19]. Strong dependence of the strange antiquark mass and the combination 2 m s − m ¯ s onthe SU (3) multiplet becomes somewhat softer after inclusion of the configuration mixing,according to Table 4. It should be kept in mind, however, that the above presented wavefunctions for the pentaquarks, Eq. (25 −
27) do not hold after configuration mixing, and thedefinition of the quark masses itself becomes more shaky in this case.10
Problems and conclusions
We conclude with following remarks and statements. The parameter in 1 /N c expansion islarge for the case of the baryon spectrum, extrapolation to real world is not possible in thisway. Rigid (soft as well) rotator model and the bound state model coincide in the first twoorders of 1 /N c , but differ in the next orders. Configuration mixing is important, as it followsfrom the RRM results.There is correspondence of the chiral soliton model (RRM) and the quark modelpredictions for pentaquarks spectra in the negative strangeness sector of the { } and { } -plets: the effective mass of strange quark is about 135 − M eV , slightly smaller for { } . Our estimates are in reasonable agreement with calculations made from differentpoints, see e.g. [25] where the effective strange quark mass is ontained from analysis ofthe m s -corrections to Cabibbo-suppressed tau lepton decays within the perturbative QCDdynamics.For positive strangeness components the link between the CSA and QM requiresstrong dependence of the effective ¯ s mass on particular SU (3) multiplet. Configurationmixing slightly pushes spectra towards the simplistic model with equal masses of strangequark and antiquark, but explanation of this remarkable consequence of the configurationmixing is absent still.In spite of these problems, the chiral soliton models, based on few principles andingredients incorporated in the effective lagrangian, allow to describe qualitatively, in somecases even quantitatively, various chracteristics of baryons and nuclei — from ordinary ( S =0) nuclei to known hypernuclei. This suggests that predictions of pentaquark states shouldbe considered seriously. Existence of PQ by itself is without any doubt, although very narrowPQ may not exist. Wide, even very wide PQ should exist.In view of existing theoretical uncertainties, further experimental investigations ofbaryon spectrum, in particular, searches for exotic baryons - strange, charmed or beautifal,wide or narrow - are of great importance. References
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