Mixing and decays of the antidecuplet in the context of approximate - PDF document

Mixing and decays of the antidecuplet in the context of approximate SU(3) symmetry Vadim Guzey (Bochum) and M.V. Polyakov (Liege) 1. Motivation 2. Antidecuplet mixed with three octets: General expressions for 10 partial decay widths 3. Emerging picture of N 10 and Σ 10 decays 4. Discussion (influence of the mixing with 27-plet, ideal mixing) and conclusions Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium V. Guzey

Motivation • Approximate flavor SU(3) symmetry of strong interactions allows to group all hadrons into certain multiplets. Only singlets, octets ( 8 ) and decuplets ( 10 ) were believed to be realized in Nature. The discoveries of the Θ + and Ξ −− , if confirmed, mean the existence of a new physical multiplet: the antidecuplet ( 10 ). • Approximate SU(3) symmetry works surprisingly well: Mass splittings and partial decay widths of all baryon multiplets (singlets, octets, decuplets) are described and predicted with good accuracy: Gell-Mann and Ne’eman 1964; Kokkedee 1969; Samios, Goldberg, Meadows 1974 . • Approximate SU(3) suits best for establishing in a model- independent way the overall structure of a given SU(3) multiplet: Mass splittings, necessity of mixing with other multiplets due to SU(3) breaking, correlations between partial decay widths. Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium V. Guzey

• This is exactly what one needs for the antidecuplet: a reliable overall picture of 10 and its mixing with other multiplets and a way to systemize the present experimental info on the 10 decays. • Since SU(3) is broken, states from different multiplets with the same spin and parity can mix. Because of the small width of Θ + , even small mixing dramatically affects predictions for the 10 decays. At the same time, small mixing with 10 affects very little non-exotic multiplets. This means that one can use the results of SU(3) analysis of the non-exotic multiplets (three octets in our case) in the SU(3) analysis of 10 decays. • After the SU(3) picture of 10 is established using the scarce experimental info on 10 decays, one can make model- independent predictions for unmeasured decays and assess available models of the 10 mixing. Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium V. Guzey

Antidecuplet mixing with three octets We consider the scenario that 10 mixes with three J P = 1 / 2 + octets: the ground-state octet, the octet containing N (1440) Λ(1600) , Σ(1660) and Ξ(1690) , the octet containing N (1710) , Λ(1800) , Σ(1880) and Ξ(1950) . The mixing takes place through the N 10 and Σ 10 and the corresponding N and Σ octet states:  | N phys      � 1 0 0 sin θ 1 | N 1 � 1 | N phys � 0 1 0 sin θ 2 | N 2 �       2  =       | N phys 0 0 1 sin θ 3 | N 3 � �       3      | N phys − sin θ 1 − sin θ 2 − sin θ 3 1 | N 10 � � 10 • We assume that θ i mixing angles are small, θ i = O ( ǫ ) , where ǫ is a small parameter of SU(3) breaking. We systematically neglect O ( ǫ 2 ) terms. • The | N 1 � , | N 2 � and | N 3 � states can mix among themselves, i.e. they can belong to several different octets. Using the χ 2 fit to the measured decays, we find that | N 2 � and | N 3 � states are slightly mixed (it is legitimate to neglect 10 at this stage. Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium V. Guzey

After this is taken into account, it is sufficient to consider only the mixing of each individual | N phys � with | N phys � . i 10 • The mixing angles θ i and θ Σ i are related, � � � � N phys − N phys = sin θ Σ Σ phys − Σ phys sin θ i , i i i 10 10 which becomes θ i = θ Σ i ignoring O ( ǫ 2 ) terms. • Gell-Mann–Okubo mass formulas, which describe the mass splitting inside SU(3) multiplets, are not sensitive to small mixing � = N i + sin 2 θ i N 10 = N i + O ( ǫ 2 ) , N phys ≡ � N phys | ˆ M | N phys i i i It is not legitimate to estimate the mixing angles from the Gell-Mann–Okubo mass formula. Instead, one has to consider decays which contain both O (1) and O ( ǫ ) terms. • We assume that SU(3) symmetry is violated by non-equal masses inside a given multiplet and mixing and that SU(3) is exact in decay vertices → finite number of universal SU(3) coupling constants. Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium V. Guzey

General expressions for 10 couplings: Γ Θ + and G 10 In our analysis, Γ Θ + and Σ π N are external parameters, which are varied in the following intervals: 1 ≤ Γ Θ + ≤ 5 MeV; 45 ≤ Σ π N ≤ 75 MeV. Σ π N determines the θ 1 mixing angle with the ground state octet; Γ Θ + determines the G 10 and H 10 ( H 10 = 2 G 10 − 18 ) coupling constants using Praszalowicz, hep-ph/0402038; R.A. Arndt et al. , PRC 69 (2004) 035208 √ � � 1 5 g Θ+ N K = √ G 10 + sin θ 1 H 10 4 5 10 G 10 8 6 4 2 0 -2 -4 Γ Θ + =1 MeV -6 Γ Θ + =3 MeV Γ Θ + =5 MeV -8 -10 45 50 55 60 65 70 75 Σ π N , MeV Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium V. Guzey

General expressions for N 10 couplings   √ � � 1 5 7 �  , gN 10 N π =  G 10 + sin θ 1 H 10 − G 8 sin θi gNiN π √ √ − 4 2 5 5 i =2 , 3   √ � � 1 5 1 �  , gN 10 N η =  − G 10 + sin θ 1 + sin θi gNiN η H 10 − G 8 √ √ 4 2 5 5 i =2 , 3   1 4 �  , gN 10Λ K =  G 10 + sin θ 1 G 8 + sin θi gNi Λ K √ √ 2 5 5 i =2 , 3   2 � gN 10∆ π =  sin θ 1 G 8 + sin θi gNi ∆ π √  5 i =2 , 3 • The g N i B P coupling constants are determined by the χ 2 fit to the measured decays of the octets; the θ 2 , 3 are left as free parameters. • Important correlation: Mixing with the octets can decrease g N 10 N π and simultaneously increase g N 10 N η . • The N 10 ∆ π decay is possible only due to mixing. • The partial decay widths are found from p | 3 | � M 2 Γ( B 1 → B 2 + P ) = 3 | g B 1 B 2 P | 2 2 π ( M 1 + M 2 ) 2 M 1 Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium V. Guzey

Part. decay width Γ (N * → N π ), MeV Γ Θ + =1 MeV, Σ π N =45 MeV Σ π N =75 MeV 35 90 30 80 70 25 60 20 50 15 40 30 10 20 5 10 0.2 0.2 0.1 0.2 0.1 0.2 0.1 0.1 0 0 0 0 -0.1 -0.1 sin θ 3 -0.1 sin θ sin θ 3 -0.1 sin θ -0.2 -0.2 2 2 -0.2 -0.2 Γ Θ + =5 MeV, Σ π N =45 MeV Σ π N =75 MeV 20 60 17.5 50 15 40 12.5 10 30 7.5 20 5 10 2.5 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0 0 0 0 -0.1 -0.1 sin θ 3 -0.1 sin θ sin θ 3 -0.1 sin θ -0.2 -0.2 2 2 -0.2 -0.2 Part. decay width Γ (N * → N η ), MeV Γ Θ + =1 MeV, Σ π N =45 MeV Σ π N =75 MeV 2.4 5 4.8 2.2 4.6 2 4.4 4.2 1.8 4 1.6 3.8 3.6 1.4 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0 0 0 0 sin θ 3 sin θ sin θ 3 sin θ -0.1 -0.1 -0.1 -0.1 2 2 -0.2 -0.2 -0.2 -0.2 Γ Θ + =5 MeV, Σ π N =45 MeV Σ π N =75 MeV 5.4 8 5.2 7.75 5 7.5 4.8 7.25 4.6 7 4.4 6.75 4.2 6.5 4 6.25 3.8 6 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0 0 0 0 sin θ 3 sin θ sin θ 3 sin θ -0.1 -0.1 -0.1 -0.1 2 2 -0.2 -0.2 -0.2 -0.2 Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium V. Guzey

Part. decay width Γ (N * → Λ K), MeV Γ Θ + =1 MeV, Σ π N =45 MeV Σ π N =75 MeV 1.6 3 1.4 1.2 2.5 1 2 0.8 0.6 1.5 0.4 1 0.2 0.2 0.2 0.1 0.2 0.1 0.2 0.1 0.1 0 0 0 0 -0.1 -0.1 sin θ 3 -0.1 sin θ sin θ 3 -0.1 sin θ -0.2 -0.2 2 2 -0.2 -0.2 Γ Θ + =5 MeV, Σ π N =45 MeV Σ π N =75 MeV 4.5 2.5 2.25 4 2 3.5 1.75 3 1.5 1.25 2.5 1 2 0.75 1.5 0.5 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0 0 0 0 -0.1 -0.1 sin θ 3 -0.1 sin θ sin θ 3 -0.1 sin θ -0.2 -0.2 2 2 -0.2 -0.2 Part. decay width Γ (N * → ∆π ), MeV Σ π N =45 MeV Σ π N =75 MeV 140 225 120 200 175 100 150 80 125 60 100 75 40 50 20 25 0.2 0.2 0.15 0.15 0.1 0.1 0.2 0.2 0.05 0.05 0.1 0.1 0 0 -0.05 0 -0.05 0 s s i i n -0.1 n -0.1 -0.1 -0.1 sin θ 2 sin θ 2 θ θ -0.15 -0.15 3 3 -0.2 -0.2 -0.2 -0.2 Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium V. Guzey

What is presently known about N 10 ? • The PWA analysis of R.A. Arndt et al. , PRC 69 (2004) 035208 gives two candidate states with masses 1680 MeV and 1730 MeV. Both states should have Γ N 10 → N π ≤ 0 . 5 MeV. • GRAAL observes a narrow nucleon resonance near 1670 MeV in the reaction γ n → n η V. Kuznetsov for the GRAAL Collab., hep-ex/0409032 . Interpretation: Γ N 10 → N η should not be too small. • STAR observes a narrow peak at 1734 MeV and only a weak indication of a narrow peak at 1693 MeV in the Λ K S invariant mass S. Kabana for the STAR Collab., hep-ex/0406032 . Interpretation: Γ N 10 → Λ K is possibly suppressed. We find that this picture of N 10 decays can be realized by suitable choice of θ i . In particular, we impose the Γ N 10 → N π ≤ 1 MeV cut and find unsuppressed Γ N 10 → N η and somewhat suppressed Γ N 10 → Λ K . Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium V. Guzey