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Spectrum of the excited Nucleon and Delta baryons in a relativistic chiral quark model

Spectrum of the excited Nucleon and Delta baryons in a relativistic chiral quark model E.M. Tursunov , INP, Tashkent with S. Krewald , FZ, Juelich J. Phys. G:Nucl. Part. Phys., 31 (2005) 617-629. J. Phys. G:Nucl. Part. Phys., 36 (2009) 095006.

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Spectrum of the excited Nucleon and Delta baryons in a relativistic chiral quark model

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  1. Spectrum of the excited Nucleon and Delta baryons in a relativistic chiral quark model • E.M.Tursunov, INP, Tashkent • with S. Krewald, FZ, Juelich • J. Phys. G:Nucl. Part. Phys., 31 (2005) 617-629. • J. Phys. G:Nucl. Part. Phys., 36 (2009) 095006. • J. Phys. G: Nuc. Part . Phys., 37(2010) 105013 • arXiv (hep-ph): 1103.3661 (2011) • arXiv (hep-ph): 1204.0412 (2012)

  2. Outline • Motivation • Chiral quark potential model (ChQPM) • Selection rules for quantum numbers: connection • with the strong decay of excited baryons • with orbital structure (1S)2(nlj) • Center of mass correction for the zero-order • energy values of the N and Delta states • Numerical estimation of the ground and excited • Nucleon and Delta mass spectrum within ChQPM • Conclusions

  3. Motivation CiralQuark Models have been extensively used to study the structure of the ground state N(939) S. Theberge, A.W. Thomas and G.A. Miller, Phys. Rev. D22, 2838 (1980); A.W. Thomas, S. Theberge and G.A. Miller, Phys. Rev. D24, 216 (1981). K. Saito, Prog. Theor. Phys. V71, 775 (1984). • E. Oset, R. Tegen, W. Weise Nucl. Phys. A426, 456 (1984) • Th. Gutsche & D. Robson . Phys.Lett. B229, 333 (1989) A.W. Thomas, Prog.Part.Nucl.Phys. 61, 219 (2008); F. Myhrer and A.W. Thomas, Phys.Lett. B663, 302 (2008).

  4. Excited baryon spectroscopy: problems within Constituent Quark Models • relativistic effects v ≈ c; • the “missing resonances” problem • a number of fitting parameters (5-10) • what is the most important exchange mechanism between quarks: • one gluon exchange ? (Isgur & Karl, Phys. Let. B72, 109 (1977); Phys. Rev. D21, 779(1980) • π, K, η exchange?(Glozman & Riska. Phys. Rep. 268 (1996) 263) OR • g (mq=330-350 MeV) • π, K, η N* (∆*) N* (∆*)

  5. Spectrum of N* in the CQM (2000 г.)(PPNP, 45, 241)

  6. Spectrum of ∆* in the CQM (2000 )(PPNP, 45, 241)

  7. Chiral quark potential model • Effective chiralLagrangian • (based on the linearizedσ-model) • E. Oset, R. Tegen, W. Weise Nucl. Phys. A426, 456 (1984) • Th. Gutsche & D. Robson • Phys.Lett. B229, 333 (1989) N* (∆*)

  8. The confinement and Coulomb potentials The Dirac equation (variational method on a harmonic oscillator basis)

  9. Field operators for the quark

  10. Field operators for the pion

  11. Propagators (Green functions)

  12. Estimation of the energy spectrum At zeroth order: Higher orders (Gell-Mann & Low ):

  13. Contribution of the self-energy diagramms (π)

  14. 2-nd order Feynman diagrams of the self energy term due-to pion field

  15. Final expression for the contribution of the 2-nd order self-energy diagrams due-to pion fields

  16. Contribution of the 2-nd order self-energy diagrams due-to gluon fields

  17. 2-nd order self-energy Feynman diagrams due-to gluon fields

  18. Final expression for the contribution of the 2-nd order self-energy diagrams due-to gluon to the energy spectrum of baryons

  19. Wave functions of the SU(2) baryons Contribution of the exchange diagrams (pion)

  20. Feynman pionexchange diagrams

  21. Pionexchange operators π () () • ℓβ • ℓβ± • ℓα • ℓα±

  22. One-gluon exchange operators

  23. Feynman gluon-exchange diagrams

  24. Selection rules for quantum numbers: connection with the strong decay of an excited baryons N* (J,T) and ∆*(J,T) -the orbital configuration of the SU(2) baryon 1S π • ℓ • ℓ± ( ) • 0 • 1 () π (nlj) N* (∆*) (J,T) Chiral constraints: π Ng.s.(1/2+)

  25. Consequences of chiral constraints For the fixed orbital configuration (band) the number of N* and ∆* states decreases by 1 • (lj)=P1/2 : l=1; Lπ=l’=0 • S0=0 ; J=1/2 (N*) • S0=1 ; J=1/2 (N*, ∆*) • 2 (N*) + 1 (∆*) • (lj)=P3/2 : l=1; Lπ = l’=2 • S0=0: J=3/2 (N*) • S0=1: J=3/2, 5/2 (N*, ∆*) • 3 (N*) + 2 (∆*) • (lj) ≠ P1/2 : • 3 (N*) + 2 (∆*)

  26. Center of mass correction for the zero-order energy values of the g.s. N and Delta (Moshinsky transformation) K. Shimizu, et al. Phys. Rev. C60, 035203(1999) [R=0 method] D. Lu, et al. Phys. Rev. C57, 2628 (1998) [P=0] R. Tegen, et al., Z. Phys. A307 (1982), 339 [LHO] L. Wilets “Non topological solitons”, World Scientific, Singapoure).1989 R=0: P=0: LHO: Normalization:

  27. Center of mass correction for the zero-order energy values of the excited N* and Delta* states Fixed orbital configuration: (degenerate at zero order) With spin coupling:

  28. Scalar-vector oscillator potential (exact separation in Jacobi coordinates) If S0=0

  29. Simple solution of the two-body bound state Dirac equation

  30. Test: Positronium1S0 (singlet) (bound state of e+e-) • V(r)= α/r +2 βr me • E(1S0 ) SchrÖdinger: 6.803 eV Dirac: 6.806 eV • E(21S0 - 11S0 ) SchrÖdinger: 5.10 eV Dirac: 4.99 eV

  31. Linear scalar and vector Coulomb potentials (in Jacobi coordinates) Expansion over multipols: ( for ρ/3 < r/2)

  32. First approximation (freediquark+ quark)

  33. Numerical estimation of the ground and excited Nucleon and Delta mass spectrum within ChQPM(condition of the calculations) М.T.Kawanai & S. Sasaki, PPNP, 67(2012)130 МэВ Th. Gutsche, Ph.D. thesis. 1987 M. Luescher, Nucl. Phys. B130 (1981) 317 αS=0.65

  34. Self energy of the valence quark due-to pion fields as a function of the intermediate quark(antiquark) total momentum (convergence)

  35. Self energy of the valence quark states due-to color-magnetic gluon fields (convergence)

  36. Ground state nucleon N(939)energy values in MeV CM correction : K. Shimizu, et al. Phys. Rev. C60, 035203(1999) [100] D. Lu, et al. Phys. Rev. C57, 2628 (1998) [101] R. Tegen, et al., Z. Phys. A307 (1982), 339 [102] L. Wilets “Non topological solitons”, World Scientific, Singapoure).1989

  37. Test of the CM correction for the g.s. N and Delta First approximation (free scalar diquark+ quark) EQ=632 (di-q)+419(q)=1051MeV Modification (fit to g.s. N): EQ=394+546=940 MeV

  38. Spectrum of N* (our estimation) Not presented in PDG2012 Exp. Data from: E. Klempt & J.M. Richard, Rev.Mod. Phys. 82 (2010) 1095

  39. Spectrum of ∆*in our model Exp. Data from: E. Klempt & J.M. Richard, Rev.Mod. Phys. 82 (2010) 1095

  40. Conclusions • For fixed orbital band • of the SU(2) baryon states • a)Chiral constraints (selection rules) • b) Connection with the strong decay 2. A way to decrease the number of baryon resonances. Possible way to the solution of the “missing resonances” problem (!?) • 3. a) Simple solution of the 2-body bound-state Dirac equation • b) New method for the CM correction for E Q (N*; Δ*) • 4. Without fitting parameters the spectrum of N* and ∆* are described at the CQM level !

  41. THANKS !!!

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