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PATTERNS IN THE NONSTRANGE BARYON SPECTRUM
P. González, J. Vijande, A. Valcarce, H. Garcilazo
INDEX
i) The baryon spectrum: SU(3) and SU(6) x O(3).
ii) The Quantum Number Assignment Problem.
iii) Screened Potential Model for Nonstrange Baryons.
iv) SU(4) x O(3) : Spectral predictions up to 3 GeV.
v) Conclusions.
What is the physical content of the baryon spectrum?
The richness of the baryon spectrum tells us about the existence,
properties and dynamics of the intrabaryon constituents.
How can we extract this physical content?
The knowledge of spectral patterns is of great help.
The Eightfold Way: SU(3)
The pattern of multiplets makes clear the existence of quarks with
“triplet” quantum numbers and the regularities in the spectrum.
From the spectral regularities one can make predictions and obtain
information on the dynamics (SU(3) breaking terms).
SU(3 ) : Quarks (3 x 3 x 3 = 10 + 8 + 8 + 1) Baryons
JP 
1
2

Strange quark mass splitting?
3
J 
2
P
I prediction by GellMann

Quarks with Spin : SU(6) i SU(3) x SU(2)
6  6  6  56  70  70  20
56  4 10  2 8
Quarks with Spin in a Potential : SU(6) x O(3)
( N , LP )  (56, 0 )
3 1 1 3
S z   , , ,
2 2 2 2
1 1
S z   ,
2 2

SU(6) Breaking : Strange quark mass + Hyperfine (OGE) splitting
Vij HS
c2    

i . j  i . j
mi m j
The Baryon Quantum Number Assignment Problem
( N , , , , , ) : (56, 0 )  ( 2 8  4 10, 0 )
( N , , , , , )* : ??
JP

1
2

1
2

3
2

3
2

5
2

5
2
L
0,1,2
1,2
0,1,2,3
1,2,3
1,2,3,4
1,2,3,4
The Baryon Quantum Number Assignment, determined by QCD,
requires in practice the use of dynamical models (NRQM,…).
Regarding the identification of resonances the experimental
situation for nonstrange baryons is (though not very precise)
more complete.
From a simple NRQM calculation we shall show that SU(4) x O(3)
is a convenient classification scheme for non-strange baryons in
order to identify regularities and make predictions.
NRQM for Baryons
Lattice QCD : Q-Q static potential
(G. Bali, Phys. Rep. 343 (2001) 1)
Quenched approximation (valence quarks)
1

Vst (r )  ( r  )
2
r
The Bhaduri Model
The Missing State Problem
E > 1.9 GeV: many more predicted states than observed resonances.
The observed resonances seem to correspond to predicted
states with a significant coupling to pion-nucleon channels
(S. Capstick, W. Roberts PRD47, 1994 (1993)).
Lattice QCD : Q-Q static potential
Unquenched (valence + sea quarks)
(DeTar et al. PRD 59 (1999) 031501).
String breaking
The saturation of the potential is a consequence of the
opening of decay channels.
The decay effect can be effectively taken into account
through a saturation distance in the potential providing
a solution to the quantum number assignment.
Screened Potential Model
V (r ij )  VBhaduri(r ij )
if rij  rsat
V (r ij )  VBhaduri(r sat ) if rij  rsat
(N, ) Ground States : SU(4) x O(3)
4  4  4  20 S  20 M  20 M  4
P  () L
20 S  4 4  2 2
20M  4 2  2 4  2 2
For J>5/2 :
i)
( E ) J  2  ( E ) J  400  500 MeV
For J>5/2 :
ii )
N (J
,
)  ( J
,
) for
4n  3
J
2
with n  1,2,3...
Dynamical Nucleon Parity Series
For J>5/2 :


iii ) N ( J )  N ( J ) for
Negative parity
4n  1
J
with n  1,2,3...
2

Positive parity
K  1 L  1
(Bigger Repulsion)
S  1
(Bigger Attraction )
(N, ) First Nonradial Excited States
Our dynamical model (absence of spin-orbit and tensor forces)
suggests the following rule satisfied by data at the level of the 3%
5
J
2
The first nonradial excitation of N,  (J) and the ground
state of N,  (J+1) respectively are almost degenerate.
For radial as well as for higher excitations the results are much
more dependent on the details of the potential.
Spectral Pattern Rules
For J>5/2 the pattern suggests the following dynamical regularities
i)
ii )
( E ) J  2  ( E ) J  400  500 MeV
N (J
,
)  ( J

,
) for

iii ) N ( J )  N ( J ) for
iv) ( N , ) J  ( N , ) J 1
*
4n  3
J
2
with n  1,2,3...
4n  1
J
with n  1,2,3...
2
Conclusions
i)
The use of a NRQM containing a minimal screened dynamics
provides an unambiguous assignment of quantum numbers to
nonstrange baryon resonances, i. e. a spectral pattern.
ii) The ground and first non-radial excited states of N’s and ’s are
classified according to SU(4) x O(3) multiplets with hyperfine
splittings inside them.
iii) The spectral pattern makes clear energy step regularities, N-
degeneracies and N parity doublets.
iv) Ground and first non-radial excited states for N’s and ’s, in the
experimentally quite uncertain energy region between 2 and 3
GeV, are predicted.
THE END
(N, ) Ground States : SU(4) x O(3)
4  4  4  20 S  20 M  20 M  4
P  () L
20 S  4 4  2 2
20M  4 2  2 4  2 2
For J>5/2 the pattern suggests the following dynamical regularities
i)
ii )
( E ) J  2  ( E ) J  400  500 MeV
N (J
,

)  ( J

,
) for
iii ) N ( J )  N ( J ) for
4n  3
J
2
with n  1,2,3...
4n  1
J
with n  1,2,3...
2