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**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 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 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). 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).

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**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= MeV) π, K, η N* (∆*) N* (∆*)

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

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

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**Chiral quark potential model**

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

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**The confinement and Coulomb potentials**

The Dirac equation (variational method on a harmonic oscillator basis)

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**Field operators for the quark**

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**Field operators for the pion**

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**Propagators (Green functions)**

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**Estimation of the energy spectrum**

At zeroth order: Higher orders (Gell-Mann & Low ):

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**Contribution of the self-energy diagramms ( π )**

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

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

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

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

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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

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**Contribution of the exchange diagrams (pion)**

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

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**Feynman pion exchange diagrams**

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**Pion exchange operators**

π ( ) ( ) ℓβ ℓβ± ℓα ℓα±

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**One-gluon exchange operators**

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**Feynman gluon-exchange diagrams**

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**-the orbital configuration of the SU(2) baryon**

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± ( 1 ) ( ) π (nlj) N* (∆*) (J,T) Chiral constraints: π Ng.s.(1/2+)

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**For the fixed orbital configuration (band)**

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/ (N*) S0=1 ; J=1/ (N*, ∆*) 2 (N*) + 1 (∆*) (lj)=P3/2 : l=1; Lπ = l’=2 S0=0: J=3/ (N*) S0=1: J=3/2, 5/2 (N*, ∆*) 3 (N*) + 2 (∆*) (lj) ≠ P1/2 : 3 (N*) + 2 (∆*)

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**Center of mass correction for the zero-order energy values of the g. s**

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, (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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**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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**Scalar-vector oscillator potential **

(exact separation in Jacobi coordinates) If S0=0

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

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**Test: Positronium 1S0 (singlet) (bound state of e+e-)**

V(r)= α/r +2 βr me E(1S0 ) SchrÖdinger: eV Dirac: eV E(21S0 - 11S0 ) SchrÖdinger: 5.10 eV Dirac: 4.99 eV

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**Linear scalar and vector Coulomb potentials (in Jacobi coordinates)**

Expansion over multipols: ( for ρ/3 < r/2)

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**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) М.T. Kawanai & S. Sasaki, PPNP, 67(2012)130 МэВ Th. Gutsche, Ph.D. thesis M. Luescher, Nucl. Phys. B130 (1981) 317 αS=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)

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

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**Ground state nucleon N(939) energy values in MeV**

CM correction : K. Shimizu, et al. Phys. Rev. C60, (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

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**Test of the CM correction for the g.s. N and Delta**

First approximation (free scalar diquark+ quark) EQ=632 ( di-q)+419(q)=1051 MeV Modification (fit to g.s. N): EQ= =940 MeV

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**Spectrum of N* (our estimation)**

Not presented in PDG2012 Exp. Data from: E. Klempt & J.M. Richard, Rev.Mod. Phys. 82 (2010) 1095

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**Spectrum of ∆* in our model**

Exp. Data from: E. Klempt & J.M. Richard, Rev.Mod. Phys. 82 (2010) 1095

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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 !

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THANKS !!!

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