SU(3) systematization of baryons. Vadim Guzey. Theory Center, Jefferson Lab

Transcription

1 SU(3) systematization of baryons Vadim Guzey Theory Center, Jefferson Lab In collaboration with M.V. Polyakov: V. Guzey, hep-ph/05176 V. Guzey and M.V. Polyakov, hep-ph/ Cake seminar, Theory Group, JLab, 04/09/008

2 Outline 1. Approximate flavor SU(3) symmetry of the strong interactions. Methods of flavor SU(3) symmetry for systematization of baryons and results 3. Present SU(3) analysis of baryons: Examples and main results 4. Conclusions

3 Approximate flavor SU(3) symmetry of strong interactions In order to understand the spectrum of strongly interacting particles (hadrons), Gell-Mann and Neeman (1961) proposed the hypothesis of the approximate flavor SU(3) symmetry of strong interactions (The Eightfold way): All hadrons are built from three fictitious particles (u, d and s constituent quarks, in modern terms), which form the fundamental 3 and 3 representations of the flavor SU(3) group: Mesons : 3 3 = Baryons : = All hadrons can be grouped into 1, 8 and 10 SU(3) representations (multiplets). The flavor SU(3) symmetry is an approximate symmetry of the strong interactions. The symmetry is broken by the mass of the strange quark Gell-Mann Okubo mass-splitting relations (see later).

4 In a given SU(3) multiplet, each state is characterized by isospin I, I 3 and hypercharge Y = B + S The most spectacular SU(3) prediction for the ground-state decuplet is the existence of Ω with J P = 3/ + (until now J P of Ω is not measured, but assigned using SU(3))

5 Methods of flavor SU(3) symmetry for systematization of baryons We want to group known baryons into SU(3) singlets, octets and decuplets. We use the following two criteria: Gell-Mann Okubo mass relations: octet : decuplet : 1 (m N + m Ξ ) = 1 4 (3m Λ + m Σ ) m Σ m = m Ξ m Σ = m Ω m Ξ Work with a few percent accuracy! Assumption that the coupling constants for two-body hadronic decays B 1 B + P are SU(3)-symmetric. Also, members of the same multiplet should have the same spin and parity.

6 Gell-Mann Okubo mass formulas Use tensor notations (Okubo, 196; deswart, 1963) Each baryon in a given irreducible representation µ is represented by the basis vector ψ µ ν=i,i 3,Y Each operator acting on the baryon has the same tensor form and transforms similarly under SU(3), T µ ν Matrix elements are calculated using the Wigner-Eckart theorem (ψ µ 3 ν 3,T µ ν ψ µ 1 ν 1 ) = γ ( µ1 µ ν 1 ν ) µ γ C ν γ 3 Here (...) is the known SU(3) Clebsch-Gordan SU(3) coefficients; C γ is unknown constant (reduced matrix element); γ labels different µ 3 in the tensor product µ 1 µ.

7 SU(3) is broken by the m s ss term in the strong Hamiltonian, which corresponds to operator is T Expansion in powers of the SU(3)-violating operator for the baryon mass: M µ B = Mµ 0 + γ ( µ 8 Y I 00 µ γ Y I ) C γ M µ 0 is the mass of the multiplet in the SU(3) limit For octets, 8 8 = 1 8 S 8 A M 8 B = M 0 + C S ( 8 8 Y I 00 8 S Y I ) + C A ( 8 8 Y I 00 8 A Y I ) 4 masses (ignore isospin breaking), 3 free constants one GMO relation: 1 (m N + m Ξ ) = 1 4 (3m Λ + m Σ )

8 For decuplets, 8 10 = M 10 B = M 0 + C ( 10 8 Y I Y I ) 4 masses, free constants two GMO relations (the equal spacing rule): m Σ m = m Ξ m Σ = m Ω m Ξ

9 SU(3) predictions for two-body hadronic decays Consider B 1 B + P decays (B and P are ground-state octets of baryons and pseudoscalar mesons). Assumption: SU(3) is exact for the decay coupling constants and is broken only by non-equal physical masses of the involved hadrons (which enter the predicted partial decay width via the phase space). For decays: two free constants ( 8 8 g B1 B P = A S Y I Y P I P 8 S Y 1 I 1 ) ( A A Y I Y P I P 8 A Y 1 I 1 ) For the decays: one free constant ( 10 8 g B1 B P = A 8 Y I Y P I P 8 Y 1 I 1 ) Many partial decay widths of a given multiplet can be related and fitted using normally -3 free parameters (mixing introduces additional free parameters) test of SU(3).

10 A 8 = 15/10 A S + 3/6 A A, α = 3/6A S /A A Decay mode g B1B P g B1B P g B1B P N Nπ 3 A8 Nη [(4 α 1)/ 3] A 8 ΣK 3 ( α 1) A8 ΛK [(α + 1)/ 3] A 8 π / 5 A 8 Σ K 1/ 5 A 8 Λ NK /3 ( α + 1) A8 1/ A 1 Σπ (α 1) A 8 6/4 A1 Λη / 3 (α 1) A 8 ( /4) A 1 ΞK /3 (4 α 1) A8 1/ A 1 Σ π 15/5 A 8 Ξ K 10/5 A 8 Σ Σπ α A 8 Λπ / 3 (α 1) A 8 NK ( α 1) A8 Ση / 3 (α 1) A 8 ΞK A 8 K 30/15 A 8 Σ π 30/15 A 8 Σ η 5/5 A 8

11 SU(3) systematization of hadrons is an attempt to form as many SU(3) multiplets as possible by subjecting candidate states to the two tests: the GMO mass relations SU(3) predictions for two-hadron partial decay widths vs. measured partial decay widths (χ -analysis) To the best of our knowledge, the most recent systematization is done by Samios, Goldberg and Meadows (1974). Main conclusion: The detailed study of mass relationships, decay rates, and interference phenomena shows remarkable agreement with that expected from the most simple unbroken SU(3) symmetry scheme. Samios et al. established 10 baryon multiplets and 3 meson nonets. Adding the results of Kokkedee (1969) and several earlier works, the following picture of SU(3) multiplets emerges.

12 octets=(n, Λ, Σ, Ξ); decuplets=(, Σ, Ξ, Ω) 1 (8, 1 + ) (939, 1115, 1189, 1314) (10, 3 + ) (13, 1385, 1530, 167) (56, L = 0) 3 (8, 1 + ) (1440,...,...,...) 4 (8, 1 + ) (1710,...,...,...) 5 (10, 3 + ) (1600,...,...,...) 6 (1, 1 ) Λ(1405) 7 (1, 3 ) Λ(150) 8 (8, 3 ) (150, 1690, 1670, 180) (70, L = 1) 9 (8, 1 ) (1535, 1670, 1750, 1835) 10 (10, 1 ) (160,...,...,...) 11 (8, 3 ) (1700,...,...,...) 1 (8, 5 ) (1675, 1830, 1775, 1950) 13 (10, 3 ) (1700,...,...,...) 14 (8, 1 ) (1650,...,...,...) 15 (8, 5 + ) (1680, 180, 1915, 030) 16 (10, 3 + ) (190,...,...,...) (56, L = ) 17 (8, 3 + ) (170,...,...,...) 18 (10, 5 + ) (1905,...,...,...) 19 (10, 1 + ) (1910,...,...,...) 0 (10, 7 + ) (1950, 030, 10, 50)

13 Present SU(3) analysis of baryons The main purpose of our work was to update/complete the 1974 picture of SU(3) baryon multiplets using the present knowledge of baryon masses (we considered masses MeV) and decays using the same two SU(3) criteria. Main source of information is The Review of Particle Physics 004. In this respect, note that Since 1974, 9 new baryons in the considered mass range were discovered. There were 13 baryons, which were known in 1974 but were not used by Samios et al. because of their weak status or because they corresponded to incomplete multiplets. We also wanted to understand the pattern of decays of the antidecuplet containing the Θ +.

14 Example of our SU(3) analysis of well-established octet (8, 5/ + )=(1680, 180, 1915, 030) All members of the octet are well-established: N, Λ and Σ have four stars; Ξ has three stars and J P 5/?. The Gell-Mann Okubo mass relation: 1 (m N + m Ξ ) = 1855 MeV vs. 1 4 (m Λ + m Σ ) = 1847 MeV The χ -fit to the measured partial decay widths, which we predict using SU(3)-symmetric coupling constants: A 8 = 5.0 ± 1.3, α = 0.39 ± 0.0, [α SU(6) = 0.4], χ /d.o.f. = 7.85/5 A 8 = 19. ± 3.4, χ /d.o.f. = 3.44/ Since the two checks have been successful, we conclude that SU(3) works for the considered octet.

15 (8, 5/ + )=(1680, 180, 1915, 030) Mass and width (MeV) Observables Experiment (MeV) SU(3) pred. (MeV) N(1684) Γ Nπ 97.3 ± Γ = 139 ± 8 Γ π 19.5 ± Γ Nη 0.00 ± Λ(183) Γ NK 44.7 ± Γ = 77 ± 5 ΓNK Γ Σπ 1.6 ± ΓNK Γ Λη 6.9 ± Γ Σ(1385)π 3.7 ± Γ Σπ 19.8 Σ(190) Γ NK 3.9 ± Γ = 130 ± 10 ΓNK Γ Σπ 4.7 ± ΓNK Γ Λπ 11.7 ± Γ Σπ 37.4 Γ Λπ 6.3 Ξ(05) Γ ΣK 46.9 Γ = 1 ± 6 Γ Ξπ 4.1

16 Example of our SU(3) analysis of an incomplete octet (8, 1/ + )=(1710, 1810, 1880,...) N, Λ have three stars; Σ has two stars and known J P ; the Ξ member is missing. We estimate the mass of the missing Ξ from Gell-Mann Okubo mass relation: m Ξ = 1950 MeV The χ -fit to the measured partial decay widths, which we predict using SU(3)-symmetric coupling constants: A 8 = 14.9 ± 1.1, α = 0.3 ± 0.03, [α SU(6) = 0.4], χ /d.o.f. = 1.73/ A 8 = 44.9 ± 14.4, χ /d.o.f. =.0/1 χ -values are low SU(3) works for the considered octet.

17 (8, 1/ + )=(1710, 1810, 1880, 1950) Mass and width (MeV) Observables Experiment (MeV) SU(3) pred. (MeV) N(1717) Γ Nπ 43. ± Γ = 480 ± 30 Γ π 187. ± Γ Nη 8.8 ± Λ(1841) Γ NK 39.4 ± Γ = 164 ± 0 ΓNK Γ Σπ 39.4 ± Γ Σ(1385)π.1 ± Γ Σπ 31.6 Γ Λη 3.9 Σ(186) Γ NK 5.1 ± Γ = 85 ± 15 Γ Σπ 13.6 Γ Λπ 13.0 Γ Σ(1385)π 7.3 Ξ(1950) Γ Ξπ 5.9 Γ ΣK 3.4 Γ Ξ(1530)π 9.1

18 Example of our SU(3) analysis of an octet missing Λ (8, 3/ )=(1700,..., 1940,...) A very remarkable octet the only octet with a missing Λ. All other 11 Λ s required for our scheme are very well-established. For example, negative parity Λ s: Mass J P RPP status Total width (MeV) **** **** **** **** *** **** 95 Review of Particle Physics does not contain any other Negative parity Λ s until Λ(000) no candidate Λ

19 N(1700) and Σ(1940) have three stars. Since two states are missing, we cannot use Gell-Mann-Okubo formulas to estimate the mass of the missing Λ. Instead, assume by analogy with other unmixed octets that m Λ m N 150 MeV m Λ = 1850 ± 50 MeV The mass of the missing Ξ is found from GMO: m Ξ = 045 MeV The χ -fit to the measured decays: A 8 = 8.3 ± 3.5, α = 0.70 ± 0.54, [α SU(6) = 1/], χ /d.o.f. = 0.4/1 A 8 = 67. ± 31.0, χ /d.o.f. = 0.8/1

20 (8, 3/ )=(1700, 1850, 1940, 045) Mass and width (MeV) Observables Experiment (MeV) SU(3) pred. (MeV) N(1700) Γ Nπ 10.0 ± Γ = 100 ± 50 Γ Nη 0 ± 1.0 Γ π, D-wave 14.4 ± Λ(1850) Γ NK 0. Γ Σπ 17.8 Γ Λη 1. Γ Σ(1385)π 1.8 Σ(1940) Γ NK Γ Σπ 4.0 ± Γ = 300 ± 80 ΓNK Γ Λπ 18.0 ± 10.. Γ K, D-wave 47. ± Γ Σπ 9.1 Γ Λπ 13.1 Γ NK 37.4 Γ Σ(1385)π 5.4 Ξ(045) Γ Ξπ 39.5 Γ ΛK 1.3 Γ ΣK 4.8 Γ Ξ(1530)π 6.7

21 Predicted decay properties of Λ(1850) The new Λ(1850) has a vanishingly small coupling to the NK state because g ΛNK = 3 (α + 1)A 8 and our χ -fit gave α = 0.70 ± 0.54 [α SU(6) = 1/] The weak coupling to the NK state explains why Λ(1850) has not been seen in the the PWA of KN scattering! SU(3) predicts that the Λ(1850) has significant branching ratios into the Σπ and Σ π final states Λ(1850) can be searched for in production reactions using the Σπ and Σ π invariant mass spectrum.

22 Comparison to the constituent quark model predictions for Λ(1850) Isgur, Karl, 1978 Use the (70,L = 1) supermultiplet to construct negative-parity states by construction, predict all 7 negative-parity Λ s as required by our SU(3) counting. For Λ(1850) predict: m Λ = 1880 MeV Br(NK) 0 Br(Σπ) and Br(Σ π) dominant (Isgur, Koniuk, 1980)

23 Löring, Metsch, Petry, 001 Also predict all 7 negative-parity Λ s Mass predictions: m Λ = 1775 MeV (preferred) or m Λ = 1900 MeV Predict that Br(N K) 0. Glozman, Plassas, Varga, Wagenbrunn, 1998 Correct number of negative-parity Λ s Mass predictions: m Λ = 1780 MeV Decays are not considered

24 Summary of our SU(3) analysis Completed the picture of twenty+one SU(3) baryon multiplets. Used up all 4- and 3-star baryons and almost all - and 1-star resonances with the mass MeV. Five presently known baryons cannot be fitted in our picture Σ(1480)* and Ξ(160)*: members of superlight octet (Azimov et al., 003) Σ(1770)*: a candidate for the antidecuplet Σ-member (1750)*, J P = 1/ + Σ(1580)**: not seen in most of the partial wave analyses (PWA) of NK data In order to complete multiplets, we predicted 18 strange baryons (hyperons). The most spectacular prediction: a new Λ baryon with J P = 3/ and m 1850 MeV. This Λ stands out very dramatically: all other eleven required Λ s are well-established and have 3- and 4-star status. The Review of Particle Physics does not have any negative parity Λ s until Λ(000) with J P =??.

25 Final list of SU(3) multiplets 1 (8, 1 + ) (939, 1115, 1189, 1314) (10, 3 + ) (13, 1385, 1530, 167) (56, L = 0) 3 (8, 1 + ) (1440, 1600, 1660, 1690) 4 (8, 1 + ) (1710, 1810, 1880, 1950) 5 (10, 3 + ) (1600, 1690, 1900, 050) 6 (1, 1 ) Λ(1405) 7 (1, 3 ) Λ(150) 8 (8, 3 ) (150, 1690, 1670, 180) (70, L = 1) 9 (8, 1 ) (1535, 1670, 1560, ) 10 (10, 1 ) (160, 1750, 1900, 050) 11 (8, 3 ) (1700, 1850, 1940, 045) 1 (8, 5 ) (1675, 1830, 1775, 1950) 13 (10, 3 ) (1700, 1850, 000, 150) 14 (8, 1 ) (1650, 1800, 160, ) 15 (8, 5 + ) (1680, 180, 1915, 030) 16 (10, 3 + ) (190, 080, 40, 470) (56, L = ) 17 (8, 3 + ) (170, 1890, 1840, 035) 18 (10, 5 + ) (1905, 070, 50, 380) 19 (10, 1 + ) (1910, 060, 10, 360 ) 0 (10, 7 + ) (1950, 030, 10, 50) 1 (10, 1+ ) (1540, 1670, 1760, 186)

26 Predicted baryons Particle J P (multiplet) Mass (MeV) Γ body (MeV) Γ tot (MeV) Large branchings Λ 3/ (11) Σπ, Σ π Σ 1/ + (1) Σπ, Σ π Σ 3/ (13) Λπ, NK, Σ π Σ 1/ + (19) Λπ, NK, Σπ Ξ 1/ (9) Ξπ, ΛK Ξ 1/ (14) ΣK, ΛK, Ξη Ξ 1/ (10) Ξπ, ΣK, ΛK, Ξ π Ξ 3/ + (5) Ξπ, ΛK, Ξ π Ξ 1/ + (4) ΣK, Ξ π Ξ 3/ (13) Σ K, Ξ π, ΛK, Ξπ Ξ 3/ + (17) ΣK Ξ 3/ (11) Ξπ, ΛK Ξ 1/ + (19) ΛK, Ξπ, ΣK Ξ 3/ + (16) ΛK, Ξπ, ΣK Ω 1/ (10) ΞK Ω 3/ + (5) ΞK, Ξ K Ω 3/ (13) ΞK, Ξ K Ω 1/ + (19) ΞK

27 Conclusions We successfully cataloged almost all known baryons with m MeV in twenty+one SU(3) multiplets using the methods of the approximate flavor SU(3) symmetry of strong interactions: GMO mass formulas and SU(3)-symmetric decay coupling constants. In order to have complete multiplets, we predicted the existence of 18 strange baryons The most remarkable among them is the Λ hyperon with J P = 3/, the mass around 1850 MeV, the total width approximately 130 MeV, significant branching into the Σπ and Σ(1385)π states and a very small coupling to the NK state. Model-independent confirmation of constituent quark model predictions. Decay patterns of the antidecuplet are understood by introducing its mixing with a non-exotic octet (details in the paper).

28 Back-up slides

29 How many low-lying SU(3) multiplets are there? Since different SU(3) multiplets have different J P, which is beyond flavor SU(3), we need to consider a larger group SU(6) O(3). The SU(6) O(3) symmetry does not need to hold (and it doesn t)- we only need it as a guiding principle to estimate the number of states! This estimate is done in several steps. Combine flavor SU(3) with spin SU(): SU(3) SU() SU(6). For three quarks in the fundamental 6 representation, = , Here 0 representation is totally antisymmetric, 56 is totally symmetric and 70 has mixed symmetry.

30 It is a phenomenological observation that most likely only the 56 and 70 representations of SU(6) are realized in Nature. Their SU(3) decomposition 56 = 70 = ( 8, 1 ) ( 1, 1 ) ( 10, ( 8, 1 ) 3, ) ( 8, ) 3 ( 10, ) 1, The second number in the parenthesis denotes the total spin S of the multiplet. Multiplets of different parities are obtained by coupling the total orbital moment of the three quarks L to their spin S. Assuming that the radial part of the baryon wave function is symmetric, 56 admits only even L (positive parity) and the 70 accepts all L (any parity in general, negative in our scheme).

31 The resulting decomposition in terms of SU(3) multiplets with different J P (56, L = 0) = (70, L = 1) = (56, L = ) = ( 8, ( 1, ( 10, ( 8, ( 10, + 1 ) ( 10, 1 ) ( 1, ) ( 10, + ) ( 8, + 7 ) ), ) ( 8, 3 ), + ) ( 10, 1 ) ( 8, + 1 ) ( 10, 3 ) ( 8, + 3 ) ( 10, 5 ) + 5 ) The minimal number of SU(3) multiplets is 17. The three additional multiplets are radial excitations.

32 SU(3) predictions for two-body partial decay widths Γ(B 1 B + P) = g B1 B P ( k M ) l ( k M 1 ) M where k is the momentum of B in the c.m.; l is the relative orbital moment of the B + P system; M 1 is the mass of B 1 and M = 1000 MeV. The kinematic factor is ambiguous: it can be multiplied by any function of the involved masses. This would correspond to different mechanisms of SU(3) symmetry breaking in the decays. For instance, Γ (B 1 B + P) = g B1 B P ( k M ) l ( k M 1 ) ( ) M M M 1 gives much better description of the decays of the ground-state decuplet. It is interesting that this issue has recently been rediscovered and hotly debated in connection to the total width of the Θ +, Jaffe vs. Diakonov, Petrov and Polyakov.

33 In essence: In the original Z. Phys. A (1997) paper by Diakonov, Petrov and Polyakov (DPP), the factor (M/M 1 ) 1 was present in the partial decay widths of both usual (13) and exotic Θ +. That was a misprint the factor is needed only for the from purely phenomenological grounds (Samios, 1974). Jaffe noticed that, taking the published expression at their face values, the factor (M/M 1 ) 1 increases Γ Θ + by at least a factor of 1.5.