Pentaquark in the SU(3) Skyrme Model and Chiral Quark Soliton Model

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1 Pentaquark in the SU(3) Skyrme Model and Chiral Quark Soliton Model Michał Praszałowicz Phys. Lett. B575 (2003) 234 [hep-ph/ ] Talk at the Cracow Workshop on Skyrmions and Anomalies, Mogilany, Poland, Feb 20-24, 1987, World Scientific 1987, p.112. M. Smoluchowski Institute of Physics, Jagellonian University, Kraków, Poland and Nuclear Theory Group, BNL

2 Particle Data Group 1986 JLab, Nov.7, 2003 Michał Praszałowicz 2

3 Chiral Quark Model constituent quark mass ~ 350 MeV pions fermions integrate out quarks "Skyrme" Model only pion fields, kinetic term + interaction terms JLab, Nov.7, 2003 Michał Praszałowicz 3

4 SU(3) soliton: static solution hedgehog Ansatz: P(0)=nπ and P( )=0 n - winding number variational approach: P(r)=P(r/r 0 ), r 0 -soliton size Energy has minimum as a function of r 0 In the Skyrme model winding number is interpreted as a baryon number. JLab, Nov.7, 2003 Michał Praszałowicz 4

5 Quantizing SU(3) Skyrmion time-dependent rotation angular velocities: JLab, Nov.7, 2003 Michał Praszałowicz 5

6 Quantizing SU(3) Skyrmion every quantity is a power series in Ω no kinetic term for Ω 8 commutes with λ 8 JLab, Nov.7, 2003 Michał Praszałowicz 6

7 Quantizing SU(3) Skyrmion add small moment of inertia ε generalized "momenta": Hamiltonian: JLab, Nov.7, 2003 Michał Praszałowicz 7 constraint for ε 0

8 Wave functions analogy with a symmetric top m - momentum projection on z k - momentum projection z on z' z' QM textbook Landau, Lipschitz: QM sect. 103 x' x y' y ψ ~ N D (J)* (ϕ',θ,ϕ) m,k - angular momentum projections JLab, Nov.7, 2003 Michał Praszałowicz 8 mk In SU(3) J R = (p,q), m,k (Y,I,I 3 ) however, because of the constraint not all k's are allowed but only those which have k = (Y=1, I,I 3 )

9 Wave functions and allowed states I 3 Y JLab, Nov.7, 2003 Michał Praszałowicz 9

10 Mass formula octet-decuplet splitting exotic-nonexotic splittings first order perturbation in the strange quark mass: x JLab, Nov.7, 2003 Michał Praszałowicz 10

11 Gell-Man Okubo mass formulae three independent constants: F, C and D JLab, Nov.7, 2003 Michał Praszałowicz 11

12 Chiral model relations symmetry breaking Hamiltonian is too primitive: richer structure is needed JLab, Nov.7, 2003 Michał Praszałowicz 12

13 Yabu-Ando: higher orders in m s second order: H. Yabu, K. Ando, Nucl.Phys.B301 (1988) free parameters: M sol, I 1, I 2 and α, but now I 2 contributes to nonexotic splittings JLab, Nov.7, 2003 Michał Praszałowicz GMO YA fix α and then minimize χ 2 with respect to the remaining parameters

14 Ω mass [MeV] threshold Θ + Ξ * Σ * Ξ Σ 1100 Λ N α [MeV] JLab, Nov.7, 2003 Michał Praszałowicz 14

15 1700 (a) 1900 (b) mass [MeV] threshold Θ + Ω Ξ * Σ * Ξ Σ Λ mass [MeV] Ξ 3/2 Σ_ 10 N * Θ α [MeV] N α [MeV] the simplest Skyrme model predicts light Θ + prediction for Ξ matches NA49 data within model accuracy JLab, Nov.7, 2003 Michał Praszałowicz 15

16 Numerical values for all masses JLab, Nov.7, 2003 Michał Praszałowicz 16

17 χqm breaking hamiltonian calculate next-to-leading contributions to H' E. Guadagnini Nucl.Phys.B236 (1984) 35 equivalent to Guadagnini mass formula: O(N c )+O(1) O(1) O(1) all O(m s ) in the Skyrme model β, γ = 0 Diakonov, Petrov, Polyakov, Z.Phys A359 (97) 305 richer H': * no handle on I 2 * only 2 linear combinations of parameters α', β and γ enter nonexotic splittings splittings in

18 Width Diakonov, Petrov, Polyakov, Z.Phys A359 (97) 305 Weigel, hep-ph/ N c O(N c ) O(1) O(1) one can extract constants G 0,1,2 from known decuplet decays G 2 ~ 0, then _ G 10 = G 0 +G 1 /2 ~ 19 and G 10 = G 0 -G 1 ~ note that Γ R ~ G R ; that explains smallness of the Θ width _ exact result is: G 10 = G 0 -G 1 -G 2 /2 _ hence: G 10 = 0! in QM limit G 1 /G 0 = 4/5 and G 2 /G 0 = 2/5 JLab, Nov.7, 2003 Michał Praszałowicz 18

19 Wave functions and allowed states I 3 Y JLab, Nov.7, 2003 Michał Praszałowicz 19

20 Wave functions and allowed states G. Karl, J. Patera, S. Perantonis, Phys. Lett 172B (1986) 49, J. Bijnens, H. Sonoda, M. Wise, Can. J. Phys. 64 (1986) 1, Z. Duliński, M. Praszałowicz, Acta Phys.Pol. B18 (1988) JLab, Nov.7, 2003 Michał Praszałowicz 20

21 Mass formula O(N c ) O(N c,m s ) unknown corrections O(1) O(1) O(N c ) O(1/N c ) O(N c,m s ) JLab, Nov.7, 2003 Michał Praszałowicz 21

22 O(1/N c 2 ) O(1) O(N c 3 ) O(1/N c ) O(1/N c 2 ) O(N c 3 ) JLab, Nov.7, 2003 Michał Praszałowicz 22

23 chiral limit: nonzero meson masses: JLab, Nov.7, 2003 Michał Praszałowicz 23

24 Congres International sur le Rayonnement Cosmique, Bagneres de Bigorre, Juillet 1953 "The particles described in this conference are not entirely fictitious and every analogy with the particles really existing in nature is not entirely coincindental." JLab, Nov.7, 2003 Michał Praszałowicz 24