Occupation number formalism for arbitrary Nc baryons

A general method is presented for computing matrix elements of quark operators on baryonic states with low strangeness and arbitrary number of colors Nc. These results are useful in applications of the large Nc expansion to baryons and exotics. As an application we compute the matrix elements of strangeness changing operators contributing to kaon couplings to ground state baryons and pentaquarks, in broken SU(3).

A general method is presented for computing matrix elements of quark operators on baryonic states with low strangeness and arbitrary number of colors Nc. These results are useful in applications of the large Nc expansion to baryons and exotics. As an application we compute the matrix elements of strangeness changing operators contributing to kaon couplings to ground state baryons and pentaquarks, in broken SU (3). Introduction. In the large N c limit, with N c the number of colors, a new symmetry emerges in the baryon sector of QCD. This is the contracted symmetry SU (2F ) c , with F the number of light quark flavors [1,2]. This symmetry can be used to organize the 1/N c expansion of any quantity as an operator expansion. The nonrelativistic quark model is a convenient bookkeeping tool for implementing this expansion [3,4]. The baryons are constructed by placing the N c quarks into one-body states. For example, taking F = 3 a possible basis of one-body states consists of transforming in the fundamental representation of SU (6). The operators representing physical quantities such as masses, axial currents, etc. can be constructed from quark operators annihilating the basis states in Eq. (1). The building blocks are bilinears of the form q † Λ A q with Λ A the generators of SU(2F) One important technical problem in the implementation of this program is the computation of the matrix elements of these operators on quark model states. Various methods have been discussed in the literature for this purpose. The computation of these matrix elements for F ≥ 3 and arbitrary N c turns out to be rather involved. For F = 2 a method for computing with any N c was discussed in Refs. [5,6,7]. On the other hand, keeping N c = 3 (or other small values) other methods are available, such as the holomorphic representation of the harmonic oscillator discussed in [8]. The difficulty is connected with the necessity of manipulating complicated expressions involving SU(3) Clebsch-Gordan (CG) coefficients. Although not insurmontable (applica-tions of the large N c expansion in SU(3) have been presented in Refs. [9] and analytic expressions for some arbitrary N c CG coefficients have been computed recently in Refs. [10]), extracting the form of the result for arbitrary N c remains a challenge.
In practice, we are interested only in baryon states with at most a few strange or heavy quarks. Therefore it is natural to expect that the large N c dependence comes from the large number of light quarks in the u, d sector.
In this paper we give the details required for constructing explicitly any such states and computing matrix elements between them. The advantage of our method is that only usual SU(2) CG coefficients coefficients are ever required, and closed form expressions are found for all matrix elements for arbitrary N c .

2.
States. Consider the spin-flavor group SU (6), which contains as a subgroup (see, e.g. Ref. [11]) The factors on the right-hand side correspond to spinisospin SU(4), the strange quarks' spin SU(2) K , and the U(1) associated with strangeness −n s . We would like to construct the decomposition of a baryon state containing N c quarks and transforming in the completely symmetric representation S Nc of spinflavor SU (6) into irreducible representations of the subgroup on the rhs of Eq. (3). This is given by We denoted the representations of SU(2) K by their multiplicity 2K + 1, with K the spin of the strange quarks. The terms written have K = |n s |/2, which corresponds to the maximally possible value of the strange quark spin. This is required by Fermi statistics as applied to the system of the strange quarks, assuming that they are all in a completely symmetric orbital wave function. This is satisfied by ground state baryons, but not by orbitally excited states. We will comment on this case below.
The wave function of a hadron containing N c quarks, of which n s are strange quarks, factors according to Eq. (4) into a product of wave functions for its components.
The nonstrange system has a symmetric spin-flavor wave function. For describing its state, it is convenient to use a Fock state formalism, familiar from the theory of manybody systems [12].
Since the orbital wave functions of all quarks are the same, we can label the state of a system of N c identical quarks by giving the occupation numbers of the onebody spin-flavor states in Eq. (1). We introduce the "6nsymbol" defined as with N = 6 i=1 n i . These states are normalized as Nonstrange hadrons have n 5 = n 6 = 0 and can be described by "4n-symbols" {n 1 , n 2 , n 3 , n 4 }. For simplicity we will use this notation when appropriate. The nonstrange states have spin and isopin satisfying I = J. Their spin-flavor symmetric wave functions can be given in closed form as [6,7] where N u,d are the number of up and down quarks, respectively, and N ud = N c − n s .
A few representative nonstrange J 3 = + 1 2 states are Strange quarks are also straightforwardly added We could have equally well written these states in terms of the "6n-symbol" introduced above, but we gave them here in a form which is not symmetrized under the exchange of the s quark with the u, d quarks. This is appropriate in broken SU (3) and for hadrons containing one heavy quark, with the replacement s → Q.
Exotic states containing both quarks and antiquarks can also be constructed. We consider here only positive parity pentaquark-type states, containing N c + 1 quarks and one antiquark. We take the N c + 1 quarks to contain only u,d quarks in a spin-flavor symmetric state, as in Ref. [13], while the antiquark can be a strange or heavy quark. Representative states with I = 0, 1 can be chosen as Next we consider the action of quark operators on these states. Any such operator can be constructed from onebody annihilation q i and creation q † i operators, where i = 1 − 6 denotes one of the basis states in Eq. (1). Their action is given explicitly as with all occupation numbers n j =i unchanged. They satisfy the usual commutation relations for bosonic operators [q i , q † j ] = δ ij . As an example of their application, consider the isospin lowering and raising operators. Their action can be obtained by first writing them in terms of quark operators followed by the application of the rules Eqs. (15). One finds  They are useful for obtaining the entire isospin multiplets from the representative states listed above.
between sectors of different n s . They can be chosen as The most general n-body operator can be constructed from the building blocks shown in Eq. (20). The matrix elements of the strangeness conserving operators I a , G ia , N s , J i s can be obtained using well-known SU(2) methods [5,6]. The matrix element of G ia on states with N ud = N c − n s up and down quarks are given by with X (N ud ) (I ′ , I) = X (N ud ) (I, I ′ ), given explicitly as For the matrix element of G ia on general states |JIn s containing N ud u,d quarks and n s strange quarks we obtain This matrix element has a 1/N c expansion of the form with the first two terms X 0,1 in agreement with the model-independent prediction following from the contracted symmetry [2]. In the following we compute also the matrix elements of the two strangeness lowering operators t α , Y iα (with raised indices), and the two strangeness raising operators t α ,Ỹ iα .
We define the reduced matrix elements as and similarly for Y iα , in terms of Y (I ′ J ′ K ′ , IJK). The reduced matrix elements of t α andt α are defined as and similarly for t(I ′ K ′ , IJK).
The action of the strangeness changing operators on quark states can be obtained straightforwardly using the rules Eq. (15). Typical relations are Repeated application of the s creation operators leads to states containing multiple strange quarks. Typical matrix elements that appear for example in the computation of Ξ → Σ matrix elements are (in the notation of Eq. (7) for the nonstrange states) We show in Tables I, II the results for the strangeness raising transitionsỸ (I ′ J ′ K ′ , IJK) andt(I ′ K ′ , IJK) for all ground state baryons.
In Table III we give also the results for the kaon decays of pentaquarks with symmetric spin-flavor wave function. The computation of matrix elements of operators containing antiquarks requires some care. The antiquark spin doublets has componentss β = (−s ↓ ,s ↑ ) β . This can be used to express the operators Y i α , t α in terms of quark and antiquark one-body operators as For example, the action of a typical strangeness changing operator on pentaquark states containing as quark is given bȳ The results for the reduced matrix elements Y,Ỹ and t,t take a simpler form in the large N c limit, and can be given in analytical form. Expanding these operators as and similarly for t and the strangeness raising operators, the reduced matrix elements at leading order are given by (with [I] ≡ 2I + 1) The result Eq. (36) was found in Ref. [2] using the method of the induced representations for the contracted symmetry; the result Eq. (37) is new. From Tables I, II and III we find c(1, 4. Discussion and extensions. The applications discussed so far were limited to symmetric spin-flavor states. The methods of this paper can be extended also to mixedsymmetric states, which are relevant for the application of the 1/N c expansion to orbitally excited baryons in the 70 − [9,15,16], and to negative parity exotic states [17,18]. The corresponding decomposition in broken SU(3) of a state transforming in the mixed-symmetric MS Nc representation of SU(6) is more complicated than that for the symmetric state Eq. (4). Keeping only terms with n s = 0, 1, this has the form MS Nc = (M S Nc , 1, 0) (38) ⊕ (M S Nc−1 , 2, 1) ⊕ (S Nc−1 , 2, 1) ⊕ · · · where the two terms with one strange quark have the wave function of the nonstrange system in a M S and S representation of SU(4), respectively. The states in (M S Nc−1 , 2, 1) and (S Nc−1 , 2, 1) that fall into MS Nc can be constructed by requiring that they are eigenstates of the quadratic SU(6) Casimir C 2 = A Λ A Λ A with the eigenvalue of the MS Nc representation.
The M S states of N c − 1 nonstrange quarks can be represented in the usual way [7,9,15] as tensor products of a spin-flavor symmetric "core" of N c − 2 quarks with one "excited" quark. This can be accomodated in our formalism in a similar way as we treated the strange quark. New annihilation and creation operators acting on this "excited" one-body state have to be introduced, from which transition operators can be constructed in the usual way.
In conclusion, we presented in this paper a new general method which simplifies computations of matrix elements for ordinary and exotic baryon states, containing both light and strange or heavy quarks, for arbitrary number of colors N c . The main advantage of the method is that only SU(2) Clebsch-Gordan coefficients need to be used at any stage, and closed form results can be obtained for all matrix elements. This work was supported by the DOE under cooperative research agreement DOE-FC02-94ER40818 (D.P.). The work of C.S. was supported in part by Fundación Antorchas.