aa r X i v : . [ h e p - ph ] D ec Published in Chinese Physics Letters 25 (2008) 3920-3923.
Simple Classification of Light Baryons
Yan Chen and Bo-Qiang Ma ∗ School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China (Dated: October 30, 2018)We introduce a classification number n which describes the baryon mass information in a fuzzymanner. According to n and J p of baryons, we put all known light baryons in a simple table in whichsome baryons with same ( n , J p ) are classified as members of known octets or decuplets. Meanwhile,we predict two new possible octets. PACS numbers: 14.20.-c, 11.30.Na, 14.20.Gk, 14.20.Jn
The idea of the quark structure of hadrons appearedfirst in the papers of Gell-Mann [1] and Zweig [2]. It wasshown that the SU(3) octet symmetry can be realized onthe basis of a fundamental triplet of some hypothesizedparticles, called quarks by Gell-Mann. At the beginning,quarks were take as a mathematical expression of theSU(3) properties of hadrons, but soon it was recognizedthat hadrons are bound systems of quarks: Baryon= qqq and Meson= q ¯ q . The light baryons are made up of u , d , s quarks, which implies an approximate SU (3) flavor sym-metry. Up to now, a number of approaches have beendeveloped for describing the baryon mass spectrum, suchas the SU(6) model [3, 4], the bag model [5], the Skyrmemodel [6, 7], the non-relativistic quark model [8, 9] andso on. These models incorporate partly the dynamicsof quantum chromodynamics (QCD) and achieve quanti-tative description. However, many predicted baryons ofthese models have been experimentally observed whilesome baryons have not been found. In addition, thepredicted baryon masses of different models are not thesame, i.e., the predictions are model-dependent. Al-though QCD has been widely accepted as a basic the-ory of strong interaction, it is still a big challenge tocompute the baryon mass spectrum from first principledue to the complicated non-perturbative and non-linearproperties. Therefore a comprehensive classification ofall known baryons would be very helpful for a funda-mental understanding of strong interaction and baryonstructure.In this Letter, we try to propose a simple method whichdirectly classifies the observed baryons to their possiblemultiplets according to a fuzzy concept of mass range,rather than the precise mass values. First, we introducea classification number n which describes baryon massrange information. Secondly, according to n and J p , weput all light baryons into a table. We find that manybaryons with same n and J p can make up of a multipletwhich is an octet or a decuplet listed in the booklet ofParticle Data Group (PDG) [10] based on the quark-model or in Ref. [11] based on SU(3) symmetry. We alsopredict two new possible octets. The classification ofknown multiplets and the new predictions may suggest ∗ Electronic address: [email protected] the feasibility of the classification method although it isphenomenological and rough.We now introduce the classification number n , whichis an integer determined by two parameters. One is thecenter mass denoted as M B n c for a kind of baryons withsame isospin and hypercharge, the other is the mass bandwidth denoted as M B n w corresponding to the center mass M B n c . We define M B n +1 c − M B n c = M B n w . If a baryonmass M B satisfies M B n c − M B n − w / ≤ M B < M B n c + M B n w / , (1)the baryon belongs to a group of baryons with classifica-tion number n .To obtain reasonable M B n c and M B n w , we briefly re-view the relevant properties of the nonrelativistic poten-tial model [12, 13]. The ground-state baryon mass is M = X m i + 2 α s π (cid:10) δ ( r ij ) (cid:11) X i 280 MeV and m s ∼ 440 MeV, and alsotaking m B ∼ N/ ∆,Λ / Σ, Ξ, Ω, we assume M B c s of N/ ∆, Λ / Σ, Ξ, Ω be-ing 1000, 1150, 1270 and 1370 MeV respectively. Thedecrease of m B with the increase of the strange quarknumber of a baryon is inspired by the m i m j terms inthe denominators of Eq.(2). However, we should takethe above values as from assumptions rather than fromderivations. We need also to mention that ∆1 and Ω1are assumed to be imaginary particles with spin S = 1 / N 1, Λ1, Σ1, andΞ1.The idea of the mass band width M B n w comes fromthe zeroth order energies E = ( N + 3 / ω = (2 n + l +3 / ω [9, 13], where n is the number of radial nodes, l is the orbital angular momentum, and the non-strangeharmonic oscillator level spacing ω ≃ 520 MeV. Themass band width M B n w is somewhat similar to ω . There-fore, we suppose that M B n w and ω are of the same order.More explicitly, we suppose that M B w is 400 MeV and M B n ′ w ( n ′ > 1) is 300 MeV.Using the relation of baryon masses and the classifi-cation number n , i.e., Eq. (1), we put baryons N, Λ, Σ,Ξ, ∆ and Ω in Tables I, II and III respectively. The un-derlined baryons mean that their spins and parities arenot confirmed yet. We try to put them in these tablesaccording to their masses. It is noted that an underlinedbaryon might be putted in another place in these tablesas long as its mass satisfies Eq. (1).Then according to n and J p , we put all light baryonslisted in PDG [10] in Table IV. In this table, there are 38groups with same n and J p , noting that baryons Σ, Ξ inone group may belong to an octet or a decuplet. We findthat many observed baryons with same n and J p aredirectly classified to their possible multiplets, thirteenof which are the same with the multiplets in PDG [10]and seven of which are the same with the multiplets inRef. [11]. There are also some baryons which can not beclassified to a multiplet clearly.From Table IV, we predict two new possible octetsmarked with ⊙ . One is (N1900, Λ2000, Σ, Ξ) of J p =3 / + , the other is (N2000, Λ2110,Σ, Ξ) of J p = 5 / + .The Σ and Ξ mass ranges of these two octets are 2000-2300 MeV and 2100-2400 MeV respectively. We calculatethe baryon decay widths to check our prediction.For the decay process of a baryon B ∗ to a baryon Band a pseudoscalar meson M B ∗ → B + M, (3)the calculation of decay widths can be performed in theframework of Rarita-Schwinger formalism.The parity-conserving Lagrangian and decay widths ofthe process B ∗ / + → B / + + M are [14] L = g B ∗ BM m π ¯ΨΦ µ ∂ µ φ, (4)Γ = g B ∗ BM P cm [( m ∗ B + m B ) − m ]24 π ( m m π ) . (5)Those of the process B ∗ / + → B / + + M are L = i g B ∗ BM m π ¯Ψ γ Φ µ µ ∂ µ ∂ µ φ, (6)Γ = g B ∗ BM P cm [( m ∗ B − m B ) − m ]30 π ( m m π ) , (7)where P cm = { [ m ∗ B − ( m B + m ) ][ m ∗ B − ( m B − m ) ] } / m ∗ B , (8) with P cm being the c.m. momentum of final particles, g B ∗ BM being the universal coupling constant [15], m ∗ B and m B being the baryons masses, and m being the me-son mass. The results are listed in Table V, from whichwe notice that most predicted decay widths are consistentwith the experimental data.However, in Table IV, there is a puzzle, i.e., severalbaryons of a same kind may have the same n and J p .There are 5 such cases, which are groups 3 and 4, groups6, 7 and 8, groups 12 and 13, groups 20 and 21, andgroups 25, 26 and 27. There are several possible solu-tions. Maybe when more baryons are observed, baryonswith the same n and J p may belong to different mul-tiplets. It was suggested in Ref. [16] that Σ(1480) andΞ(1620) might be members of a new light octet, whoseN member is predicted to have the mass around 1100MeV and the vanishingly small total width. Maybe somebaryons have the same n and J p can mix with each other,for example, the two Λ’s of the groups 20 and 21 maymix, and so do the Λ’s of the groups 25 and 26. Maybesome baryons belong to a multiplet with exotic baryons.For example, Σ(1770) makes it a potential candidate forthe Σ member of the antidecuplet [11]. Four groupsof baryons Σ(1560), Σ(1690), Σ(1580), ∆(2000) have nosolution and need more study.Although the introduced center mass M B n c and massband width M B n w are arbitrary in some sense, our simplemethod can directly classify the observed baryons to theirpossible multiplets known in the literatures. The classifi-cation of known multiplets and predictions of new possi-ble multiplets may suggest the feasibility of the introduc-tion of the classification number n which is based on thefuzzy concept of mass range instead of exact mass val-ues. The simple classification of all known light baryonsin Table IV might be inspiring for both experimental andtheoretical studies: experimentalists may search for pos-sible missing baryons by looking at vacancies in the tableand theorists may seek for better classification schemes ofbaryons and reveal more fundamental relations betweendifferent baryons. Of course, our simple method shouldbe only considered as a roughly phenomenological at-tempt to classify all light baryons. A more comprehen-sive and elegant classification should be searched for frommore fundamental and profound considerations.We are grateful to Qihua Zhou and Bin Wu for usefuldiscussions. This work is partially supported by NationalNatural Science Foundation of China (Nos. 10421503,10575003, 10528510), by the Key Grant Project of Chi-nese Ministry of Education (No. 305001), by the Re-search Fund for the Doctoral Program of Higher Edu-cation (China). [1] M.Gell-Mann, Phys. Lett. , 214 (1964).[2] G.Zweig, preprints CERN-TH-401 and CERN-TH-412(1964), published in Developments in the Quark The- ory of Hadrons, Volume 1. Edited by D.B. Lichtenbergand S.P. Rosen, Hadronic Press, Noantum, Mass., 1980.[3] F.G¨ursey and L.A.Radicati, Phys. Rev. Lett. , 173 TABLE I: The n number and masses of N, ∆ baryons n N n ∆ M Nc (MeV) 1000 1400 1700 2000 2300 2600 M ∆ c (MeV) 1000 1400 1700 2000 2300 2600 2900 J + N masses J + ∆ masses1/2 939 1440 1710 2100 1/2 1750 19103/2 1720 1900 3/2 1232 1600 19205/2 1680 2000 5/2 1905/20007/2 1990 7/2 1950 23909/2 2220 9/2 230011/2 11/2 242013/2 2700 13/215/2 15/2 2950 J − J − n number and masses of Λ, Σ baryons n Λ n Σ M Λ c (MeV) 1150 1550 1850 2150 2450 M Σ c (MeV) 1150 1550 1850 2150 2450 2750 3050 J + Λ masses J + Σ masses1/2 1116 1600 1810 1/2 1193 1660 1880/17703/2 1890 2000 3/2 1385/1560/1690 1840 20805/2 1820 2110 5/2 1915 20707/2 2020 7/2 20309/2 2350 9/2 245511/2 2585 11/215/2 15/2 3170 J − J − n number and masses of Ξ, Ω baryons n Ξ M Ξ c (MeV) 1270 1670 1970 2270 2570 J + Ξ masses1/2 1318 16903/2 1530 22505/2 20307/2 2120 J − M Ω c (MeV) 1370 1770 2070 2370 2670 J + Ω masses3/2 1672 24705/2 23807/2 2250 (1964).[4] B.Sakita, Phys. Rev. , B1756 (1964).[5] A.Chodos, R.L.Jaffe, K.Johnson, C.B.Thorn, andV.Wesskopf, Phys. Rev. D , 3471 (1974); A.Chodos,R.L.Jaffe, K.Johnson, and C.B.Thorn, Phys. Rev. D , 2599 (1974); T.DeGrand, R.L.Jaffe, K.Johnson, andJ.Kiskis, Phys. Rev. D , 2060 (1975).[6] A.Hayashi and G.Holzwarth, Phys. Lett. B , 175(1984).[7] M.P.Mattis and M.Karliner, Phys. Rev. D , 2833(1985).[8] N. Isgur and G. Karl, Phys. Lett. B , 109 (1977); N.Isgur, G. Karl, and R. Koniuk, Phys. Rev. Lett. ,1269 (1978); N. Isgur and G. Karl, Phys. Lett. B ,353 (1978); Phys. Rev. D , 4187 (1978).[9] N. Isgur and G. Karl, Phys. Rev. D , 2653 (1979).[10] Particle Data Group, W.-M.Yao, et al. , J. Phys. G , 1(2006).[11] V. Guzey and M.V. Polyakov, arXiv:hep-ph/0512355.[12] A. De R´ujula, H. Georgi, and S.L. Glashow, Phys. Rev. TABLE IV: Mass table of light baryons classified by ( n , J P ) (in units of MeV)Group (n, J p ) N Λ Σ Ξ ∆ Ω Singlet Octet DecupletΛ NΛΣΞ ∆ΣΞΩ1 (1 , / + ) 939 1116 1193 1318 ⋆ , / + ) 1440 1600 1660 1690 ∗ , / + ) 1710 1810 1880 ⋆ , / + ) 1770 1750 ? ?5 (4 , / + ) 2100 1910 ? ?6 (2 , / + ) 1385 1530 1232 1672 ⋆ , / + ) 1560 ? ?8 (2 , / + ) 1690 ? ?9 (3 , / + ) 1720 1890 1840 1600 ∗ ⋆ 10 (4 , / + ) 1900 2000 2080 1920 2470 ⊙ ∗ 11 (3 , / + ) 1680 1820 1915 2030 ⋆ 12 (4 , / + ) 2000 2110 2070 2250 1905 2380 ⊙ ∗ 13 (4 , / + ) 2000 ?14 (4 , / + ) 1990 2020 2030 2120 1950 2250 ? ∗ 15 (5 , / + ) 2390 ?16 (5 , / + ) 2220 2350 2455 2300 ⋆ ?17 (5 , / + ) 2585 2420 ? ? ⋆ 18 (6 , / + ) 2700 ?19 (7 , / + ) 3170 2950 ? ?20 (2 , / − ) 1535 1670 1620 1620 ⋆ ⋆ 21 (2 , / − ) 1405 ⋆ 22 (3 , / − ) 1650 1800 1750 ⋆ 23 (4 , / − ) 2090 2000 1900 ? ?24 (5 , / − ) 2150 ?25 (2 , / − ) 1520 1690 1670 1820 ⋆ 26 (2 , / − ) 1520 1580 ⋆ ? ?27 (2 , / − ) 1480 1620 ? ?28 (3 , / − ) 1700 1940 1700 ∗ ⋆ 29 (4 , / − ) 2080 2250 2370 1940 ? ?30 (5 , / − ) 2325 ? ?31 (3 , / − ) 1675 1830 1775 1950 ∗ 32 (4 , / − ) 1930 ?33 (5 , / − ) 2200 2350 ? ?34 (4 , / − ) 2100 2100 ? ?35 (5 , / − ) 2190 2500 2200 ? ?36 (5 , / − ) 2250 2400 ⋆ ?37 (6 , / − ) 2600 2620 ? ?38 (7 , / − ) 3000 2750 ? ? The underlined baryons mean that their spins and parities are unknown and we put them in the table according to their classificationnumbers n artificially. ⋆ means that the baryons in the group belong to an octet or a decuplet listed in PDG, ∗ means the baryonsbelong to a multiplet listed in Ref. [11], and ⊙ means that the baryons belong to our prediction of a new octet, and ? means that thebaryons are still unknown to a multiplet. TABLE V: The masses and widths of baryons (in units of MeV)PDG width Decay mode Branching ratio Γ i (exp) Γ i (th) N (1900) 420-576 Nπ Nη K √ Γ N ¯ K Γ Σ π − − − . − . − . α = − . A =10.5 N (2000) 180-800 Nπ N ¯ K π α = − . A =4.0 α , A are the parameters of the universal coupling constants for the → + decays. D , 147 (1975).[13] For a review, see, i.e., S.Capstick and W.Roberts, Prog.Part. Nucl. Phys. , 241 (2000).[14] J.G. Rushbrooke, Phys. Rev. , 1345 (1966).[15] N.P. Samios, M. Goldberg, and B.T. Meadows, Rev. Mod. Phys. , 49 (1974).[16] Ya.I. Azimov, R.A. Arndt, I.I. Strakovsky and R.L.Workman, Phys. Rev. C68