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Exotic pentaquarks as Gamov-Teller resonances

If the number of colors Nc is taken large, baryons and their excitations can be considered in a mean-field approach. We argue that the mean field in baryons breaks spontaneously the spherical and SU(3) flavor symmetries, but retains the SU(2) symmetry of simultaneous rotations in space and isospace. The one-quark and quark-hole excitations in the mean field, together with the SU(3) rotational bands about them determine the spectrum of baryon resonances, which turns out to be in satisfactory accordance with reality when one puts Nc=3. A by-product of this scheme is a confirmation of the light pentaquark baryon Theta+ = uudds-bar as a typical Gamov-Teller resonance long known in nuclear physics. An extension of the same large-Nc logic to charmed (and bottom) baryons leads to a prediction of a anti-decapenta (15-bar)-plet of charmed pentaquarks, two of which, Beta++_c = cuuds-bar and Beta+_c = cudds-bar, may be light and stable with respect to strong decays, and should be looked for.


I. RELATIVISTIC MEAN FIELD
It has been argued 30 years ago by Witten [2] that if the number of colors N c is large, the N c quarks of a baryon can be viewed as moving in a mean field. It is helpful to understand how baryons look like in the large-N c limit, before 1/N c corrections are considered.
At the microscopic level quarks experience only color interactions, however large N c do not suppress gluon fluctuations: the mean field can be only 'colorless'. An example how originally color interactions are Fierz-transformed into interactions of quarks with mesonic fields are provided by the instanton liquid model [3].
We shall thus assume that quarks in the large-N c baryon obey the Dirac equation in a background mesonic field since there are no reasons to expect quarks to be non-relativistic, especially in excited baryons. In a most general case the background field couples to quarks through all five Fermi variants. If the background field is stationary in time, it leads to the eigenvalue equation for the u, d, s quarks in the background field: where S, P, V, A, T are the mean fields that are matrices in flavor. In fact, the one-particle Dirac Hamiltonian (1) is generally nonlocal, however that does not destroy symmetries in which we are primarily interested. We include dynamically-generated quarks masses into the scalar term S.
The key issue is the symmetry of the mean field. From the large-N c point of view, the current strange quark mass is very small, m s = O(1/N 2 c ) [4], therefore a good starting point is exact SU(3) flavor symmetry. A natural assumption, then, would be that the mean field is flavor-symmetric, and spherically symmetric. This assumption, however, leads to too many "missing resonances" in the spectrum. In addition, we know that baryons are strongly coupled to pseudoscalar mesons (g πN N ≈ 13). It means that there is a large pseudoscalar field inside baryons; at large N c it is a classical mean field. There is no way of writing down the pseudoscalar field that would be compatible with the SU(3) flav × SO(3) space symmetry.
This Ansatz breaks the SU(3) flav symmetry. Moreover, it breaks the symmetry under independent space SO(3) space and isospin SU(2) iso rotations, and only a simultaneous rotation in both spaces remains a symmetry, since a rotation in the isospin space labeled by a, can be compensated by the rotation of the space axes. Therefore, the Ansatz (2) breaks spontaneously the original SU(3) flav × SO(3) space symmetry down to the SU(2) iso+space symmetry.
It is analogous to the spontaneous breaking of spherical symmetry by the ellipsoid form of many nuclei.

II. QUARKS IN THE 'HEDGEHOG' MEAN FIELD
We shall call the SU(2) iso+space symmetry of the mean field the "hedgehog symmetry".
What mesonic fields S, P, V, A, T in Eq. (1) are compatible with this symmetry? Since SU (3) symmetry is broken, all fields can be divided into three categories: I. Isovector fields acting on u, d quarks pseudoscalar : P a (x) = n a P 0 (r), vector : V a i (x) = ǫ aik n k P 1 (r), axial : A a i (x) = δ ai P 2 (r) + n a n i P 3 (r), tensor : T a ij (x) = ǫ aij P 4 (r) + ǫ bij n a n b P 5 (r).
III. Isoscalar fields acting on s quarks scalar : vector : V 0 (x) = R 1 (r), All the rest fields and components are zero as they do not satisfy the SU(2) symmetry and/or the needed discrete C, P, T symmetries. The 12 'profile' functions P 0,1,2,3,4,5 , Q 0,1,2 and R 0,1,2 should be eventually found self-consistently from the minimization of the mass of the ground-state baryon. However, even if we do not know those profiles, there are important consequences of this Ansatz for the baryon spectrum.
Given the Ansatz, the Hamiltonian (1)  All energy levels, both positive and negative, are probably discrete owing to confinement.
Indeed, a continuous spectrum would correspond to a situation when quarks are free at large distances from the center, which contradicts confinement. [One can model confinement by forcing the effective quark masses to grow at infinity, e.g.
According to the Dirac theory, all negative-energy levels, both for s and u, d quarks, have to be fully occupied, corresponding to the vacuum. It means that there must be exactly N c quarks antisymmetric in color occupying all (degenerate) levels with J 3 from −J to J, or K 3 from −K to K; they form closed shells that do not carry quantum numbers. Filling in the lowest level with E > 0 by N c quarks makes a baryon [4,5], see Fig. 1.  The mass of a baryon is the aggregate energy of all filled states, and being a functional of the mesonic field it is proportional to N c since all quark levels are degenerate in color.
Therefore quantum fluctuations of mesonic field in baryons are suppressed as 1/N c so that the mean field is indeed justified.
Quantum numbers of the lightest baryons are determined from the quantization of the rotations of the mean field, leading to specific SU(3) multiplets that reduce at N c = 3 to the octet with spin 1 2 and the decuplet with spin 3 2 , see e.g. [6]. Witten's quantization condition Y ′ = Nc 3 [7] follows trivially from the fact that there are N c u, d valence quarks each with the hypercharge 1 3 [8]. Therefore, the ground state shown in Fig. 1  At N c = 3 it is a pentaquark state uudds, precisely the exotic Θ + baryon predicted in Ref. [10] from other considerations. The quantization of its rotations produces the antidecuplet (10, 1 2 + ). In our original prediction the O(1) gap between Θ + and the nucleon was due to the rotational energy only, whereas here the main O(1) part of that gap is due to the one-particle levels, while the rotational energy is O(1/N c ). Methodologically, it is more satisfactory.
In nuclear physics, excitations generated by the axial current j ± µ 5 , when a neutron from the last occupied shell is sent to an unoccupied proton level or v.v. are known as Gamov-Teller transitions [12]. Thus our interpretation of the Θ + is that it is a Gamov-Teller-type resonance long known in nuclear physics.
An unambiguous feature of our picture is that the exotic pentaquark is a consequence of the three well-known resonances and must be light. Indeed, the Θ + mass can be estimated from the sum rule [4]: m Θ ≈ 1440 + 1535 − 1405 ≈ 1570 MeV, however there are O(m s ) corrections to this equation.
To account for higher baryon resonances one has to assume that there are higher oneparticle excitations, both in the u, d-and s-quark sectors, shown in Fig. 2. It is easy to obtain that order of levels under mild assumptions about the profile functions (3)-(5).

IV. BARYON RESONANCES FROM ROTATIONAL BANDS
The original SU(3) flav × SO(3) space symmetry is restored when flavor and space rotations are accounted for. Each transition in Fig. 2 generally entails "rotational bands" of SU (3) multiplets with definite spin and parity. The short recipe of getting them is: Find the hypercharge Y ′ from the number of u, d, s quarks involved; only those multiplets are allowed that contain this Y ′ . Take an allowed multiplet and read off the isospin(s) T ′ of particles at this value of Y ′ . The allowed spin of the multiplet obeys the angular momentum addition law: J = T ′ + J 1 + J 2 + K 1 + K 2 where J 1,2 and K 1,2 are the initial and final momenta of the s and u, d shells involved in the transition, respectively. The mass of the center of a multiplet does not depend on J but only on T ′ according to the relation [11] where C 2 (p, q) = 1 3 (p 2 + q 2 + pq) + p + q is the quadratic Casimir eigenvalue of the multiplet, and Ω c (2770) presumably form (6, 3/2 + ). There are3's and 6's with parity minus arising from exciting the 1/2 − s-quark level. The lightest are the degenerate singlets, presumably Λ c (2595, 1/2 − ) and Λ c (2625, 3/2 − ?).
Our new observation is that there is a Gamov-Teller-type transition when axial current annihilates a strange quark in the 1 2 + shell, and creates an u or d quark in the 0 + shell, like in the case of the Θ + . In heavy baryons it is even more simple as there is a hole in the u, d 0 + valence shell from the start. Filling in this hole means making charmed pentaquarks which we name "beta baryons", B + c = cudds and B ++ c = cuuds. Quantizing rotations tells us that these pentaquarks are members of the anti-decapenta-plet (15, 1/2 + ), Fig. 4. In fact, there must be two additional (nearly degenerate) multiplets, one with spin 1/2 + and the other with spin 3/2 + .
Charmed pentaquarks have been considered by Wu and Ma in another approach [13]; however, they get far larger masses and in addition pentaquarks withc quarks appear almost degenerate with those made of c quarks. In our picture the lightestc pentaquarks Θ c probably arise from putting the fourth (s) quark at the 1 2 − level; they form a quadruplet, have parity minus, and are much heavier.
Since we know the separation between the 1/2 + level for s quarks and the 0 + level for u, d detector, and should be looked for at LHC, Fermilab and B-factories. A cautionary remark, though, is that the production rate is expected to be quite low.
A detailed elaboration of the ideas presented here will be published elsewhere.
I am grateful to Victor Petrov, Maxim Polyakov and Alexei Vladimirov for their help.
I thank Ben Mottelson and Semen Eidelman for useful discussions and Harry Lipkin for a correspondence. This work has been supported in part by Russian Government grants