1. Dmitri Diakonov Petersburg Nuclear Physics Institute
QNP-09, Beijing
Sep 24, 2009
Quark Nuclear Physics and Exotic Pentaquark as a Gamov-Teller Resonance
Dmitri Diakonov
Petersburg Nuclear Physics Institute

2. How does baryon spectrum look like at ?
(imagine number of colors is not 3 but 1003)
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
Examples: NJL, P-NJL models
The mean field is classical
Baryons are heavy objects, with mass
One-particle excitations in the mean field have energy
Collective excitations of a baryon as a whole have energy

3. 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
which one has to insert into Dirac Hamiltonian for quarks, with all 5 Fermi variants, in general:
For three light flavors u,d,s there are more variants for the mean field.
Important question: how to treat or what is smaller
Answer: so we can forget splitting inside SU(3) multiplets,
as well as mixing of multiplets, for the time being.

4. 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
SU(2) symmetry of simultaneous spin and isospin rotations
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.

5. A list of structures compatible with the SU(2) symmetry:
isoscalar
acting on u,d quarks.
One-particle wave functions
are characterized by
where K=T+J, J=L+S.
isovector
acting on s quarks.
One-particle wave functions
are characterized by
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.

6. Ground-state baryon and lowest resonances [Diakonov, JETP Lett
We assume confinement (e.g ) meaning that the u,d and s spectra are discrete.
Some of the components of the mean field (e.g ) are C,T-odd, meaning that the two
spectra are not symmetric with respect to
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 (8, 1/2+) and 1382(10, 3/2+)

7. 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 [B.-S. Zou (2008)]
N(1440, ½+) (uud) and (½+) ( )
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.]

8. 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…]
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.

9. Parity-minus rotational bands
1615(8,1/2-), 1710(8,1/2-),
1680(8,3/2-)
1758(10,1/2-),
1850(10,3/2-),
(1930,5/2-)?
1895(8,3/2-),
1867(8,5/2-),…?

10. Parity-plus rotational bands
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+)?

11. To summarize: … seem to be capable of explaining 2 excited levels
nicely all baryon multiplets < 2 GeV,
and predict a couple of new ones,
but not as many as the old quark
model.
2 excited levels
for u,d quarks
&
for s quarks …

12. Conclusions
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, Then the number of multiplets
and their (non) degeneracy is approximately right.
3. This scheme predicts the existence of as a “Gamov – Teller” excitation,
in particular,

13. History
The «exotic» baryon that cannot be made of 3 quarks but minimally of 5
(the “pentaquark”), has been predicted in 1997 by Diakonov, Petrov and Polyakov
[Zeit. Phys. A359 (1997) 305] as a light and narrow resonance:
This prediction initiated two independent searches, and at the end of 2002
the LEPS group lead by T. Nakano (Osaka) and the DIANA group lead by
Dolgolenko (ITEP) announced seeing the resonance with the predicted
mass and very narrow width. We suggested the name
In about 40 experiments have been carried out searching for this
and other pentaquarks; in most experiments there were no statistically significant
signals seen.
In 2005 г. CLAS collaboration (Jefferson Lab) did not confirm its own discovery of
2003, and obtained an upper limit for the pentaquartk production cross section (which, however, was 3 times higher than theoretical expectations…)
Since then it is widely believed that pentaquarks “do not exist”.

14. Experiments after 2005
А. Dolgolenko et al. (ITEP) have nearly doubled the statistics of the
events. The observed spectrum of :
The only “formation” (as opposed to “production”) experiment to date!
Bow and arrows can be more precise than a gun

15. 2. A. Aleev et al. [SVD-2, Protvino] studied @ 70 GeV.
A strong signal seen in two independent samples:

16. 3. LEPS collaboration (SPring-8, Osaka), T. Nakano et al. (2008):
Remarkably, LEPS does see the resonance in the same reaction and at the same energy where CLAS
does not see a signal. However, LEPS detector registers particles in the forward direction, while CLAS
registers everything except in the forward direction:

17. To see Theta+ from interference [Amarian, Diakonov, Polyakov (PRD, 2008)]
Resonance production cross section is quadratic in the (small) amplitude, whereas
the interference cross section is linear!
To amplify the signal further, one has to look for Theta+ produced with a small momentum transfer!

18. Theory
From the traditional view on hadrons as “made of” constituent quarks with mass ~350 MeV,
it is unclear
Why pentaquarks should exist in the first place
Why is so light (1530 МeV, and not 350 x ~1900 МeV)
Why is it so narrow (~ 1 MeV, whereas normally it should be ~100 МeV)
What is ignored in the standard quark models?
А) Quantum Field Theory saying that baryons are actually superpositions of Fock states
with 3,5,7,… quarks – it’s only a question of probabilities
B) Spontaneous Breaking of Chiral Symmetry saying that constituent quarks have to
interact strongly with pion and kaon fields
Skyrme model is very rough but at least it accommodates both А and B.

19. K+n elastic cross section in the Skyrme model
[I. Klebanov et al. + D. Diakonov and V. Petrov]
Scattering amplitude has a pole at
The Skyrme model predicts a light exotic baryon resonance, but it is too “strong”!
In a more realistic model one gets a very marrow width ~ 1 МэВ without any fitting parameters
[D. Diakonov and V. Petrov (2005), C. Lorce (2006), T. Ledwig, H.-C. Kim and K. Goeke (2008)].
The reason of the small width: pentaquark decays into the 5-quark component of the nucleon.
Therefore it is suppressed to the extent the 5-quark component of the nucleon is suppressed!