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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) J. Phys. G:Nucl. Part. Phys., 36 (2009) J. Phys. G: Nuc. Part. Phys., 37(2010) arXiv (hep-ph): (2011) arXiv (hep-ph): (2012)

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Outline 2 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

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Motivation 3 Ciral Quark 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). A.W. Thomas, Prog.Part.Nucl.Phys. 61, 219 (2008); F. Myhrer and A.W. Thomas, Phys.Lett. B663, 302 (2008). 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)

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Excited baryon spectroscopy: problems within Constituent Quark Models 4 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) N* (*) π, K, η g N* (*) OR (m q = MeV)

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

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

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

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

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Field operators for the quark 9

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Field operators for the pion 10

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Propagators (Green functions) 11

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

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Contribution of the self-energy diagramms ( π ) 13

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

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

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

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

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

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Contribution of the exchange diagrams (pion) 20 Wave functions of the SU(2) baryons

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Feynman pion exchange diagrams 21

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Pion exchange operators 22 ( ) α α ± β β ± π

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One-gluon exchange operators 23

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Feynman gluon-exchange diagrams 24

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Selection rules for quantum numbers: connection with the strong decay of an excited baryons N* (J,T) and *(J,T) 25 -the orbital configuration of the SU(2) baryon (J,T) N g.s. (1/2 + ) π π ( ) 0101 ± π () N* (*) 1S (nlj) Chiral constraints:

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26 (lj)=P 1/2 : l=1; L π =l=0 S 0 =0 ; J=1/2 (N*) S 0 =1 ; J=1/2 (N*, *) 2 (N*) + 1 (*) (lj) P 1/2 : 3 (N*) + 2 (*) For the fixed orbital configuration (band) the number of N* and * states decreases by 1 (lj)=P 3/2 : l=1; L π = l=2 S 0 =0: J=3/2 (N*) S 0 =1: J=3/2, 5/2 (N*, *) 3 (N*) + 2 (*) Consequences of chiral constraints

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Center of mass correction for the zero-order energy values of the g.s. N and Delta (Moshinsky transformation) 27 K. Shimizu, et al. Phys. Rev. C60, (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:

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28 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:

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29 If S 0 =0 Scalar-vector oscillator potential (exact separation in Jacobi coordinates)

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30 Simple solution of the two-body bound state Dirac equation

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31 Test: Positronium 1 S 0 (singlet) (bound state of e + e - ) V(r)= α/r +2 β r m e E( 1 S 0 ) Schr Ö dinger: eV Dirac: eV E(2 1 S S 0 ) Schr Ö dinger: 5.10 eV Dirac: 4.99 eV

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

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33 First approximation (free diquark+ quark)

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Numerical estimation of the ground and excited Nucleon and Delta mass spectrum within ChQPM (condition of the calculations) 34 МэВ М.T. Kawanai & S. Sasaki, PPNP, 67(2012)130 M. Luescher, Nucl. Phys. B130 (1981) 317 Th. Gutsche, Ph.D. thesis α S =0.65

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Self energy of the valence quark due-to pion fields as a function of the intermediate quark (antiquark) total momentum (convergence) 35

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

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37 Ground state nucleon N(939) energy values in MeV [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 CM correction : K. Shimizu, et al. Phys. Rev. C60, (1999)

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

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

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

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Conclusions 43 For fixed orbital band of the SU(2) baryon states 2. A way to decrease the number of baryon resonances. Possible way to the solution of the missing resonances problem (!?) 1.a)Chiral constraints (selection rules) b) Connection with the strong decay 4. Without fitting parameters the spectrum of N* and * are described at the CQM level ! 3. a) Simple solution of the 2-body bound-state Dirac equation b) New method for the CM correction for E Q (N*; Δ*)

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44 THANKS !!!

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