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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) 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)

Outline

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• 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

Motivation

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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)

Excited baryon spectroscopy: problems within Constituent Quark Models

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• 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(mq=330-350 MeV)

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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)

Chiral quark potential model

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Effective chiral Lagrangian (based on the linearized σ-model)

N* (∆*)

E. Oset, R. Tegen, W. Weise Nucl. Phys. A426, 456 (1984)Th. Gutsche & D. RobsonPhys.Lett. B229, 333 (1989)

The confinement and Coulomb potentials

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

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

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At zeroth order:

Higher orders (Gell-Mann & Low ):

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)

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Wave functions of the SU(2) baryons

Feynman pion exchange diagrams

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Pion exchange operators

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( ) ( )ℓα

ℓα±

ℓβ

ℓβ±

π

One-gluon exchange operators

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Feynman gluon-exchange diagrams

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Selection rules for quantum numbers: connection with the strong decay of an excited baryons

N* (J,T) and ∆*(J,T)

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-the orbital configuration of the SU(2) baryon

(J,T)

Ng.s.(1/2+) π

π ( )0

1 ℓ

ℓ± π

( )N*

(∆*)

1S

(nlj)

Chiral constraints:

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(lj)=P1/2 : l=1; Lπ=l’=0 S0=0 ; J=1/2 (N*)

S0=1 ; J=1/2 (N*, ∆*) 2 (N*) + 1 (∆*)

(lj) ≠ P1/2 : 3 (N*) + 2 (∆*)

For the fixed orbital configuration (band)

the number of N* and ∆* states decreases by 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 (∆*)

Consequences of chiral constraints

Center of mass correction for the zero-order energy values of the g.s. N and Delta (Moshinsky transformation)

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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:

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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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If S0=0

Scalar-vector oscillator potential (exact separation in Jacobi coordinates)

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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: 6.803 eV

Dirac: 6.806 eV

E(21S0 - 11S0 ) SchrÖdinger: 5.10 eV

Dirac: 4.99 eV

32( for ρ/3 < r/2)

Linear scalar and vector Coulomb potentials (in Jacobi coordinates)

Expansion over multipols:

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First approximation (free diquark+ quark)

Numerical estimation of the ground and excited Nucleon and Delta mass spectrum within ChQPM

(condition of the calculations)

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МэВ

М.T. Kawanai & S. Sasaki, PPNP, 67(2012)130

M. Luescher, Nucl. Phys. B130 (1981) 317

Th. Gutsche, Ph.D. thesis. 1987

αS=0.65

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

[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, 035203(1999)

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

First approximation (free scalar diquark+ quark)

Modification (fit to g.s. N):

EQ=632 ( di-q)+419(q)=1051 MeV

EQ=394+546=940 MeV

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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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Spectrum of ∆* in our model

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

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Conclusions

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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 equationb) New method for the CM correction for E Q (N*; Δ*)

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THANKS !!!