Quark Nuclear Physics or A theory of baryon resonances at large N c Dmitri Diakonov, Victor Petrov and Alexey Vladimirov Petersburg Nuclear Physics Institute, Bochum University April 17, 2012, Palaiseau D.D., Nucl. Phys. A (2009), JETP Lett. (2009), Chin. Phys. C (2010), Prog Theor Phys (2010), D.D., Victor Petrov and Alexey Vladimirov, Theor & Math Physics (2012) and in preparation

How does baryon spectrum look like at ? (imagine number of colours 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 Splittings inside octets, decuplets are Colour field fluctuates strongly and cannot serve as a mean field, but colour interactions can be Fierz-transformed into quarks interacting (possibly non-locally) with mesonic fields, whose quantum fluctuations are suppressed as. Examples: instanton-induced interactions, NJL model, holographic QCD… E. Witten E. Jenkins and A. Manohar T. Cohen and R. Lebed …

Important Q.: if what is smaller, the answer: splitting inside SU(3) multiplets is, numerically ~140 MeV splitting between the centers of multiplets is, numerically ~ 230 MeV. Hence, meaning that one can first put, obtain the degenerate SU(3) multiplets, and only at the final stage account for nonzero, leading to splitting inside multiplets, and mixing of SU(3) multiplets.

What is the symmetry of the mean field? The pion field is strong, and at large Nc must be classical, but there is no way to write the classical pion field in SU(3)- and spherically-symmetric way! The best one can do is to assume the ``hedgehog Ansatz’’ Such 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! 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 `hedgehog’ SU(2) symmetry: isoscalar isovector Mean fields acting on u,d quarks. One-particle wave functions are characterized by where K = T + J, J = L + S. Mean fields acting on s quarks. One-particle wave functions are characterized by where J = L + S. 12 functions P, Q, R must be found self-consistently from a dynamical theory. However, even if they are unknown, there are interesting implications of the symmetry.

Dirac energy levels (example): u,d quark levels background field

Ground-state baryons SU(3) and SO(3) rotational excitations of this filling scheme form the lowest baryon multiplets: 1152(8, 1/2+) and 1382(10, 3/2+) We assume confinement (e.g. ) meaning that the u,d and s spectra are discrete. 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 looks like:

Lowest one-quark excitations The two lowest baryons resonances that do not belong to the ground-state (8, ½), (10, 3/2), are SU(3) singlets They can be explained if there are two s-quark levels with In the large N c limit, these excitations induce two `towers’ or `bands’ of rotational states, but at N c = 3 they reduce to two single rotational states: (1, 1/2-) and (1, 3/2-). “Gamov – Teller” transitions

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. [ similar to Bohr and Mottelson theory of collective excitations in elliptic nuclei, but more interesting.. ] [ like atomic level splitting in the magnetic field, the Zeeman effect] When one u,d quark is excited to a level with grand spin K, it is in fact (2K+1)-degenerate. Degeneracy is lifted by rotations One-quark excitations induce “rotational bands”, a bunch of octets and decuplets with definite spin, whose centers of mass are given by excitation generates (8, 1/2), (10, 3/2) excitation generates (8, 1/2), (8, 3/2), (10, 1/2), (10, 3/2), (10, 5/2) excitation generates (8, 3/2), (8, 5/2), (10, 1/2), (10, 3/2), (10, 5/2), (10, 7/2) NB: A few multiplets are in fact spurious: they are artifacts of the mean-field approximation but contradict Fermi statistics when center-of-mass motion is taken into account.

All known baryon multiplets fit into this scheme: All resonances below ~2 GeV are accounted for, but there is one extra state which is a prediction.

Splittings inside SU(3) multiplets, as due to nonzero m s We parameterize masses of individual members of all octets and all decuplets as such that the Gell-Mann – Okubo relations are satisfied automatically. All splittings are expressed through only four basic quantities, so there are many model-independent relations between splittings in various multiplets belonging to one rotational band, f.e. For the band stemming from 2+ excitation: For the band stemming from 1- excitation: 1% accuracy ! 2% accuracy !

Charmed baryons from the large-Nc perspective If one of the Nc u,d quarks is replaced by c or b quark, the mean field is still the same, and all the levels are the same! Therefore, charmed baryons can be predicted from ordinary ones! standard charmed baryons mean masses: The difference 2570 – 2408 = 162 MeV =. On the other hand, can be found from the octet-decuplet splittingIt is a check that the mean field and the position of levels do not change much from light to charmed baryons! degenerate up to

anti-decapenta-plet exotic 5-quark charmed baryons Exotic 5-quark charmed baryons are light (~2420 MeV) and can decay only weakly: clear signature, especially in a vertex detector. Life time There is also a Gamov-Teller-type transition: “Beta-sub-c” NB: is another pentaquark, hypothetized by Lipkin and Karliner; in our approach it must be ~350 MeV heavier!

Conclusions 2. Hierarchy of scales: baryon mass ~ Nc one-quark excitations ~ 1 splitting between multiplets ~ 1/Nc mixing, and splitting inside multiplets ~ m s < 1/Nc 3. The key issue is the symmetry of the mean field : the number of states, degeneracies follow from it. The original maximal possible SU(3) x SO(3) symmetry is spontaneously broken to. Then the number of multiplets and their splittings come out correct. 3. All baryon multiplets below ~2 GeV find their explanation. One extra resonance is predicted: 4. An extension of the same idea, based on large Nc, to charmed (bottom) baryons leads to a prediction of anti-decapenta-plets of pentaquarks. The lightest and are exotic and stable under strong decays, and should be looked for! Problem for future: Many relations between couplings, widths, branching ratios,… 1. “Baryons made of three quarks” contradicts the uncertainty principle and is a poor approximation. In fact, already at low resolution ~65% of nucleons are made of 3 quarks, ~25% of 5 quarks, and ~10% of more than 5 quarks.

Appendix: New observation of the Theta+ pentaquark from the analysis of the same old g11 data by CLAS collaboration, that was used to show that there is no Theta.

old analysisnew analysis, with kinematical “vertex cut”

old analysis -- structureless spectrum new analysis the upper limit for Theta+ production was estimated from this distribution