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

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). 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 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

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.

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.

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

Ground-state baryon and lowest resonances [Diakonov, JETP Lett. 90, 451 (2009)] 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+) 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.

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 neutrons 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…] 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.

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-),…?

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+)?

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. To summarize:

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,. 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,