Ordinary and exotic baryons, strange and charmed, in the relativistic mean field approach Dmitri Diakonov Petersburg Nuclear Physics Institute Kyoto, Feb 8, 2010 D.D., Nucl. Phys. A (2009), JETP Lett. (2009), arXiv:

Theoretical approaches to hadron physics Limitations: Perturbative QCD: only processes with large momentum transfer Chiral perturbation theory: only small momenta Lattice: chiral physics so far inaccessible, processes in Minkowski space inaccessible Holographic QCD

Chiral physics is critical to understand hadrons, as seen from the uncertainty principle: When one attempts to measure a quark position to an accuracy better than the pion Compton wave length of 1.4 fm, one produces a pion, i.e. a pair. Hence, viewing baryons as “made of three quarks” contradicts the uncertainty principle [Landau and Peierls (1931)] The statement “nucleons are made of three quarks” is kindergarten physics, and we are immediately punished if we attempt to take it seriously. For example, 1) “spin crisis”: only of the nucleon spin is carried by three valence quarks 2) “mass crisis”: only ¼ of the nucleon s term is carried by three valence quarks: Both paradoxes are explained by the presence of additional pairs in baryons.

[models with massive quarks and (confining) gluon interactions contradict chiral symmetry, as they violate invariance under chiral rotation

In general, quarks at low momenta are governed by a Dirac Hamiltonian, with all 5 Fermi terms: The scalar (S), pseudoscalar (P), vector (V), axial (A), and tensor (T) fields have, generally, non-local couplings to quarks, and are fluctuating quantum fields. However, in the limit they can be replaced by the non-fluctuating mean field. its minimum determines the mean field. equal-time Green function It’s helpful to consider the limit! At least, general properties are not lost!

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 Fierz-transformed into quarks interacting (possibly non-locally) with mesonic fields, whose quantum fluctuations are suppressed as. Examples: instanton-induced interactions, NJL model, …

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? Variant I (maximal symmetry): 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 !! In addition, the pion field is strong, and at large Nc must be classical, but there is no way to write the classical pion field in an SU(3) symmetric way! 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) and SO(3) separately 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 `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.

Ground-state baryon and lowest resonances 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: 1152(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 or G-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 [Diakonov, JETP Lett. 90, 451 (2009)] 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.] Sum rule: valid up to 1/Nc corrections

Theory of rotational bands above one-quark excitations [Victor Petrov and D.D.] 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…] Consider a flavor rotation of the mean field by an SU(3) matrix R, and a space rotation by an SU(2) matrix S. If R,S are constant matrices all energy levels remain the same. If R,S are slowly depending on time, one obtains the rotation Lagrangian are moments of inertia. is the vector sum of the grand spin of u,d quarks involved is the vector sum of the spin of s quarks involved. One needs to quantize the rotational Lagrangian.

Introduce angular momenta, flavor (F) and spin (J): The rotational Hamiltonian Casimir operator for an SU(3) multiplet isospin of baryons with given Y’

All one-quark excitations entail their own rotational levels. Some rotational bands are short, some are long. Some rotational levels are degenerate in the leading order, some are calculably split. Finally, the rotational energy of the SU(3) multiplet r containing baryons with a given hypercharge Y’ (whose isospin is T’ ) is The possible spins J of the multiplets are found from the vector addition law above. SU(3) multiplets are degenerate in spin J. The degenaracy is lifted in the next order Splitting inside SU(3) multiplets is Mixing of baryons with identical quantum numbers belonging to different multiplets is, leading to the mass shifts

Example: what is the rotational band about the s quark transition? hypercharge Y’ = 1/3 + 1/3 + 1/3 = 1. Octets: (T’ = ½) J = ½, ½, 3/2 – all degenerate up to 1/Nc^2 corrections Decuplets: (T’=3/2) J=1/2, 3/2, 3/2, 5/2 – all degenerate up to 1/Nc^2 corrections Parity = –. Splitting of multiplets due to rotation: The moment of inertia is the same ! (meaning the large-Nc logic works well !)

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-),…? [ no rotational band for this excitation ] [ no rotational band for this excitation ] (should be degenerate)

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+)? (should be degenerate)

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: Important: exciting P = -1 levels for u,d quarks is prohibited by Fermi statistics!

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!

In fact, there must be three almost degenerate multiplets, Neglecting mixing between particles with identical quantum numbers, one can write formulas for the splitting inside the anti-decapenta-plet: 6 masses, 3 parameters 3 “Gell-Mann – Okubo” relations: Six baryons at the corners of the anti-decapenta weight diagram are explicitly exotic (i.e. cannot be composed of three quarks), the rest are crypto-exotic, and mix up with A similar anti-decapenta-plet exists, containing the b quark, the lightest being the doublet with mass

Big question: What is the production rate of Expected production rate at LHC [Yu. Shabelsky + D.D.] : Decays: LHCb: typical Somewhat less number but still a considerable amount of events expected at LHC ! Should decay mainly as

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 confirms the existence of as a “Gamov – Teller” excitation, in particular, 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!

antiquark distribution at low virtuality (DD, Petrov, Polyakov, Pobylitsa and Weiss, 1996) [solid] vs. phenomenological GRV distribution. If we scan nucleons with a resolution q ~ 600 MeV, only ~65% of nucleons are made of 3 quarks, ~25% are made of 5 quarks, and ~10% of more than 5 quarks. Nevertheless, the fraction of momentum carried by antiquarks is very small: Appendix

Additional conclusions 1. “Baryons are made of three quarks” contradicts the uncertainty principle. In fact, already at low resolution ~65% of nucleons are made of 3 quarks, ~25% of 5 quarks, and ~10% of more than The 5-quark component of baryons is rather well understood, and should be measured directly 3. Pentaquarks (whose lowest Fock component has 5 quarks) are not too “exotic” – just Gamov-Teller excitations. In addition to the narrow there is a new prediction of charmed (and bottom) pentaquarks which decay only weakly.