QNP-09, Beijing
Sep 24, 2009
Quark Nuclear Physics
and

Exotic Pentaquark  as a
Gamov-Teller Resonance
Dmitri Diakonov
Petersburg Nuclear Physics Institute
How does baryon spectrum look like at
(imagine number of colors is not 3 but 1003)
Nc   ?
Witten (1979): Nc quarks in a baryon can be considered in a mean field
(like electrons in a large-Z atom or nucleons in a large-A nucleus).
Color field fluctuates strongly and cannot serve as a mean field,
but color interactions can be Fiertz-transformed into quarks
interacting (possibly non-locally) with mesonic fields, whose
quantum fluctuations are suppressed as O(1/ Nc ) .
Examples: NJL, P-NJL models
The mean field is classical
Baryons are heavy objects, with mass O( Nc )
.
One-particle excitations in the mean field have energy O(1)
Collective excitations of a baryon as a whole have energy O(1/ Nc )
What is the symmetry of the mean field ?
Expect maximal – spherical – symmetry !
Had there been only 1 flavor, the maximal-symmetry mean field compatible with P, T
symmetries would be
 ( x)  P1 (r ),
V0 ( x)  P2 (r ),
T0i ( x)  ni P3 (r )
which one has to insert into Dirac Hamiltonian for quarks, with all 5 Fermi variants, in general:
H   0 (ii i    i 5  V   A  5  iT [   ])
For three light flavors u,d,s there are more variants for the mean field.
Important question: how to treat
Answer:
ms /   O(1/ N c2 )
ms
or what is smaller
ms

or
1
?
Nc
so we can forget splitting inside SU(3) multiplets,
as well as mixing of multiplets, for the time being.
Two variants of the mean field :
Variant I : the mean field is SU(3)-flavor- and SO(3)-rotation-symmetric,
as in the old constituent quark model (Feynman, Isgur, Karl,…) In principle, nothing wrong
about it, except that it contradicts the experiment, predicting too many excited states !!
Variant II : the mean field for the ground state breaks spontaneously SU(3) x SO(3)
symmetry down to SU(2) symmetry of simultaneous space and isospin rotations,
like in the hedgehog Ansatz
breaks SU(3) but supports
a
a
4,5,6,7,8
SU(2) symmetry of simultaneous
4
spin and isospin rotations
  n P (r ), a  1, 2,3;

0
There is no general rule but we know that most of the heavy nuclei (large A) are not
spherically-symmetric. Having a dynamical theory one has to show which symmetry
leads to lower ground-state energy.
Since SU(3) symmetry is broken, the mean fields for u,d quarks, and for s quark are
completely different – like in large-A nuclei the mean field for Z protons is different from
the mean field for A-Z neutrons.
Full symmetry is restored when one SU(3)xSO(3) rotates the ground and one-particle excited
states  there will be “rotational bands” of SU(3) multiplets with various spin and parity.
A list of structures compatible with the SU(2) symmetry:
  P1 (4)
V0  P2 (r )
isoscalar
acting on u,d quarks.
One-particle wave functions
P
are characterized by K
where K=T+J, J=L+S.
T0i  ni P3 (r )
 a  n a P4 (r )
Vi a  òaik nk P5 (r )
isovector
Aia   ai P6 (r )  na ni P7 (r )
Tija  òaij P8 (r )  òbij na nb P9 (r )
  Q1 (r )
V0  Q2 (r )
T0i  ni Q3 (r )
acting on s quarks.
One-particle wave functions
P
are characterized by J
where J=L+S.
12 functions P(r), Q(r) must be found self-consistently if a dynamical theory is known.
However, even if they are unknown, there are interesting implications of the symmetry.
Ground-state baryon and lowest resonances
[Diakonov, JETP Lett. 90, 451 (2009)]
We assume confinement (e.g.  ~ r) meaning that the u,d and s spectra are discrete.
Some of the components of the mean field (e.g. V0 ) are C,T-odd, meaning that the two
spectra are not symmetric with respect to E  E
One has to fill in all negative-energy levels
for u,d and separately for s quarks, and the
lowest positive-energy level for u,d.
This is how the ground-state baryon N(940,1/2+) looks like.
SU(3) and SO(3) rotational excitations of this filling scheme form the lowest baryon
multiplets - 1155(8, 1/2+) and 1382(10, 3/2+)
The lowest resonances beyond the rotational band
are
(1405, ½-), N(1440, ½+) and N(1535, ½-). They are one-particle excitations:
(1405, ½-) and N(1535, ½-) are two different
ways to excite an s quark level. N(1535, ½-) is
in fact a pentaquark uudss [B.-S. Zou (2008)]
N(1440, ½+) (uud) and
(½+) ( uudds )
are two different excitations of the same level of
u,d quarks.
is an analog of the Gamov-Teller
excitation in nuclei! [when a proton is excited
to the neutron’s level or vice versa.]
Theory of rotational bands above one-quark excitations
SU(3)xSO(3) symmetry is broken spontaneously by the ground-state mean field,
down to SU(2). The full symmetry is restored when one rotates the ground-state baryon
and its one-particle excitations in flavor and ordinary spaces. [cf. Bohr and Mottelson…]
I1 3
I2
a
a 2
Lrot   (   ) 
2 a 1
2
H rot
7
A 2
8
a
a
a

(

)

Y



(
K

J

u ,d
s )
A 4
2 2
C2 ( r )  Y 
 1
1 
3

 T (T   1) 


2I2
2
I
2
I
 1
2 
All one-quark excitations entail their own rotational levels.
Some rotational bands are short, some are long.
Some rotational levels are degenerate, some are calculably split.
J  T  K u ,d  J s
Parity-minus rotational bands
0

u ,d


1

2s

1
1

2s
2s
0


u ,d
3

2s


1
3

2s
2s
 1 
1, 
 2 
(1405,1/ 2 )
 1   1   3 
 8,  ,  8,  ,  8, 
 2   2   2 


 1 

3  
5 
 10,  , 2   10,  ,  10, 
2 
2  
2 


 3 
 1, 
 2 
1615(8,1/2-), 1710(8,1/2-),
1680(8,3/2-)
1758(10,1/2-),
1850(10,3/2-),
 (1930,5/2-)?
(1520,3 / 2 )
 1 
 3   5 
 8,  , 2   8,  ,  8, 
 2 
 2   2 
1895(8,3/2-),
1867(8,5/2-),…?
Parity-plus rotational bands
0u,d  0u,d
0u,d  2u,d

 1  
3 
 8,  ,  10, 
2 
 2  
 1   3   5 
 8,  ,  8,  ,  8, 
 2   2   2 





1  
3  
5  
7 
 10,  ,  10,  ,  10,  ,  10, 
2  
2  
2  
2 


1
 0u,d
2s


1 
 10, 
2 

1630(8,1/2+),
1732(10,3/2+)
1845(8,1/2+),
1865(8,3/2+),
1867(8,5/2+)
2060(10,1/2+),
2087(10,3/2+),
2071(10,5/2+),
 (1950,7/2+)?
1750(anti-10,1/2+)?
To summarize:
2 excited levels
for u,d quarks
&
2 excited levels
for s quarks …
… seem to be capable of explaining
nicely all baryon multiplets < 2 GeV,
and predict a couple of new ones,
but not as many as the old quark
model.
Conclusions
1. Hierarchy of scales:
baryon mass ~ Nc
one-quark excitations ~ 1
splitting between multiplets ~ 1/Nc
mixing, and splitting inside multiplets ~ m_s Nc < 1/Nc
2. The key issue is the symmetry of the mean field : the number of states, degeneracies
follow from it. I have argued that the mean field in baryons is not maximal but
next-to-maximal symmetric, SU (3)  SO(3)  SU (2) . Then the number of multiplets
and their (non) degeneracy is approximately right.


1 
3. This scheme predicts the existence of  10,  as a “Gamov – Teller” excitation,
2 

in particular,