Spin and flavor projection operators in the $SU(2N_f)$ spin-flavor group

The quadratic Casimir operator of the special unitary $SU(N)$ group is used to construct projection operators, which can decompose any of its reducible finite-dimensional representation spaces contained in the tensor product of two and three adjoint spaces into irreducible components. Although the method is general enough, it is specialized to the $SU(2N_f) \to SU(2)\otimes SU(N_f)$ spin-flavor symmetry group, which emerges in the baryon sector of QCD in the large-$N_c$ limit, where $N_f$ and $N_c$ are the numbers of light quark flavors and color charges, respectively. The approach leads to the construction of spin and flavor projection operators that can be implemented in the analysis of the $1/N_c$ operator expansion. The use of projection operators allows one to successfully project out the desired components of a given operator and subtract off those that are not needed. Some explicit examples in $SU(2)$ and $SU(3)$ are detailed.


I. INTRODUCTION
The concept of symmetry, and specially gauge symmetry, is crucial in elementary particle physics. Early analyses of atomic spectra successfully implemented the use of SU (2) representation theory to study the spin of particles. Further analyses in nuclear physics struggled to find out how protons and neutrons interact via a strong force to bind together into nuclei. Promptly it was discovered that the strong force had an SU (2) invariance; it was called isospin symmetry and its irreducible representations (irreps) were labeled by isospin 1/2, 1, . . . A well-known example is the two-dimensional isospin-1/2 representation made up by the proton and neutron.
In the early decade of the sixties of the past century, a large number of new strongly interacting particles had been discovered so it was imperative to classify them. Gell-Mann first suggested that they could be accommodated into irreps of SU (3), so he proposed an organizational scheme for hadrons. It was called the eightfold way [1]; this peculiar name, presumably, is closely related to the fact that Gell-Mann mainly used the eight dimensional adjoint representation of SU (3).
Eventually, it was evident that the SU (3) symmetry found by Gell-Mann was due to the existence of the three light quarks, u, d, s, which fit into the fundamental three-dimensional representation of SU (3). This symmetry has been since referred to as SU (3) flavor symmetry. Hadrons were thus organized into SU (3) representation multipletsoctets and decuplets-of roughly the same mass.
The special unitary group also plays a role in the local SU (3) ⊗ SU (2) ⊗ U (1) gauge symmetry, which defines the modern standard model (SM) of particles and their interactions. Roughly, the three factors of the gauge symmetry give rise to the three fundamental interactions. Quantum chromodynamics (QCD), the theory of the strong interactions, is the SU (3) component of the SM. It is a gauge theory of fermions-the quarks-and gauge bosons-the gluonsand stems from the fact that each quark comes in three completely identical states called colors; the symmetry is thus referred to as SU (3) color symmetry. Unlike flavor symmetry, which is an approximate symmetry due to the relatively small masses of the three light quarks and plays a marginal role in the SM, color symmetry is exact and does play a preponderant role. At low energies, the running coupling constant of the theory is large, and the colored quarks and gluons must clump together to form colorless hadrons.
Various attempts have been made so far to construct grand unified theories of the weak, strong, and electromagnetic interactions. These approaches mostly use Lie groups. Common examples are SU (5) in the simplest grand unification theory, SO (10), and E6. Further applications of SU (N ) can also be found in shell models of nuclear and atomic physics [2,3], the worldline approach to non-Abelian gauge fields [4][5][6], to name but a few.
It should be stressed that, despite the tremendous progress achieved in the understanding of the strong interactions with QCD, the analytical calculation of the structure and interactions of hadrons directly in terms of the underlying quark-gluon dynamics is not possible because the theory is strongly coupled at low energies. Soon after the advent of QCD, 't Hooft pointed out that gauge theories based on the SU (N c ) group simplify in the limit N c → ∞, where N c is the number of color charges [7]. Baryons in large-N c QCD were first studied by Witten [8]. Later, it was shown that in the large-N c limit the baryon sector has an exact contracted SU (2N f ) spin-flavor symmetry, where N f is the number of light quark flavors [9][10][11][12]. Physical quantities are then considered in this limit, where corrections emerge at relative orders 1/N c , 1/N 2 c , and so on; this sequence originates the 1/N c expansion of QCD. The 1/N c expansion turns out to be quite useful for studying the interactions and properties of large-N c color-singlet baryons at low energies. The construction of the 1/N c expansion of any QCD operator transforming according to a given spin⊗flavor representation is expressed in terms of n-body operators O n , which can be written as polynomials of homogeneous degree n in the spin-flavor generators. The operators O n make up a complete and independent operator basis [13]. It should be emphasized that for baryons at large finite N c , the 1/N c operator expansion only extends to N c -body operators in the baryon spin-flavor generators. Although straightforward in principle, the reduction of higher-order operator structures to the physical operator basis turns out to be quite tedious due to the considerable amount of group theory involved. The fact that the operator basis is complete and independent makes those reductions possible.
Here is precisely where the aim of the present paper can be delineated: To present a general procedure to construct projection operators in SU (N ) out of the corresponding Casimir operators. The projection operators so obtained act on tensor operators that belong to tensor products of adjoint representation spaces, decomposing them into different operators with specific quadratic Casimir eigenvalues. The applicability to the 1/N c operator expansion is immediate. The cases of physical interest for N f = 2 and N f = 3 are worked out to show the usefulness of the resultant projectors. In passing, it can be pointed out that the method is not limited to the 1/N c expansion but it can also be used in shell models of atomic and nuclear physics; in this case, the projector method allows one to construct tensor operators which, with the aid of the Wigner-Eckart theorem, can be used to calculate transition amplitudes. The worldline approach to non-Abelian gauge fields is also another area where the projector method can be adapted to fit there. All in all, the method shows some potential applicability in areas where the SU (N ) group is involved.
The organization of the paper is as follows. In Sec. II some theoretical aspects of the SU (N ) group are briefly summarized, starting with some rather elementary concepts and definitions, which are provided to set notation and conventions. A key feature in the analysis is the definition of the adjoint space and the tensor space formed by the product of n adjoint spaces. The latter can always be decomposed into subspaces labeled by a specific eigenvalue of the quadratic Casimir operator of the algebra of SU (N ). The procedure to do so is discussed at the end of this section, and the defining general expression of the projection operator is provided. In Sec. III, the projection operators for the tensor product space of two adjoint spaces are constructed explicitly. The properties that by definition projection operators are demanded to fulfilled are rigorously verified. The particular case N = 2 is also discussed at the end of this section. In Sec. IV the projection operators previously defined are specialized to the SU (2N f ) spin-flavor symmetry group, which breaks to its spin and flavor groups SU (2) ⊗ SU (N f ). Consequently, the spin and flavor projection operators are constructed and readily applied to the 1/N c operator expansion. In Sec. V the method is outlined for the tensor product space of three adjoint spaces. In this case, the explicit construction of projection operators becomes a rather involved task, so only a few examples are detailed. Some closing remarks and conclusions are provided in Sec. VI. The paper is complemented by two appendices, where some supplemental material is provided.

II. PROJECTOR TECHNIQUE FOR SU (N ) ADJOINT TENSOR OPERATORS
To start with, a salient definition is that of a Lie group. It is defined as a group in which the elements are labeled by a set of continuous parameters with a multiplication law that depends smoothly on the parameters themselves [14]. A compact Lie group, on the other hand, is a Lie group in which the parametrization consists of a finite number of bounded parameter domains; otherwise the group is referred to as noncompact [15]. The SU (N ) group of all complex unitary matrices of order N with determinant 1 and the SO(N ) group of all real orthogonal matrices of order N with determinant 1 are two well-known examples of connected compact Lie groups.
The elements of a Lie group can be written as where β a , a = 1, . . . , N , are real numbers and X a are linearly independent Hermitian operators. Hereafter, and unless explicitly noticed otherwise, the sum over repeated indices will be implicit. The X a are referred to as the generators of the Lie group and they satisfy the commutation relations The f abc are referred to as the structure constants of the Lie group. The vector space β a X a , together with the commutation relations (2), define the Lie algebra associated with the Lie group. The generators satisfy the Jacobi identity, which in terms of the structure constants becomes The quadratic Casimir operator is defined as so that As for the SU (N ) group, let T a be operators that generate the Lie algebra of the group. There are N 2 − 1 of such operators, which serve as a basis for the set of traceless Hermitian N × N matrices. The generators satisfy the commutation relations where a, b, c run from 1 to the dimension of the Lie algebra of SU (N ), i.e., from 1 to N 2 − 1.
In the fundamental representation of SU (N ), the normalization convention usually adopted for the generators reads so in this convention the f abc are totally antisymmetric with respect to the interchange of any two indices. Let T a A define a set of operators such that, i.e., the structure constants themselves constitute a matrix representation of the operators. The representation generated by the structure constants is called the adjoint representation. An SU (N ) adjoint operator Q a can thus be defined, such that, The operators Q a can make up a basis for the carrier space where the generators of the Lie algebra of SU (N ) in the adjoint representation act [15]. If T a A are taken as the generators in the adjoint representation, relation (10) is equivalent to Hereafter, the carrier space generated by the operators Q a will be referred to as the adjoint space and will be denoted by adj = {Q a }.
Another tensor space of interest is the one formed by the product of the adjoint space with itself n times. It is denoted by n i=1 adj⊗. This space can usually be decomposed into subspaces labeled by a specific eigenvalue of the quadratic Casimir operator C of the Lie algebra of SU (N ). The decomposition can be achieved by adapting the projector technique for decomposing reducible representations introduced in Ref. [16]. Following the lines of that reference, the sought projection operators P (m) are thus constructed as where k labels the number of different possible eigenvalues for the quadratic Casimir operator and c m are its eigenvalues given by [17], where n is the total number of boxes of the Young tableu for a specific representation, r i is the number of boxes in the i-th row and c i is the number of boxes in the i-th column. From the defining expression (12), it can be inferred that if satisfies the commutation relation (10), then where the tensorQ a1...an is an eigenstate for the quadratic Casimir C with eigenvalue c m , In the following sections, the decompositions of the tensor spaces adj ⊗ adj and adj ⊗ adj ⊗ adj will be carried out by using the projector technique described above.

III. PROJECTION OPERATORS IN THE TENSOR SPACE adj ⊗ adj
The Young tableau for the adjoint representation is given by The tensor space adj ⊗ adj decomposes as In the notation of Ref. [13], the above irreps are designated by so this convenient notation will also be used here.
The quadratic Casimir eigenvalues for each representation in the decomposition of adj ⊗ adj are obtained from Eq. (13), and are listed in the second column (from left to right) of Table I.

Rep
Eigenvalue Representation Since five different eigenvalues are available, the projectors in Eq. (12) are computed as where and A word of caution is in order here. The defining expression of P (m) , Eq. (12), and its subsequent version for the tensor space adj ⊗ adj, Eq. (17), impose the condition c m = c ni in order to avoid singularities. Particularly, note thatās andsa are complex-conjugated representations so they share the same eigenvalue of the Casimir operator, c 2 = 2N , according to Table I. For this reason, it is not only convenient but also necessary to construct a projection operator that comprises both representations, as it is described below.
On the other hand, a complete determination of C demands the evaluation of the generators T a 2A that act in the tensor space adj ⊗ adj. In terms of T A they are given by Therefore, Since T e A T e A is the quadratic Casimir operator for the adjoint representation, then by Schur's lemma, Thus, The action of C on a tensor operator Q b1 1 Q b2 2 yields, according to Eq. (11), Therefore, in components, C reads where the second equality follows from the identities listed in Appendix A. For the ease of notation, the symbols F a1a2b1b2 and D a1a2b1b2 have also been introduced; they read Additionally, let us also define the operator G acting on Q b1 1 Q b2 2 with components, Then, the quadratic Casimir C in Eq. (22) can be rewritten as, so that powers of C are straightforwardly obtained as By making use again of the identities listed in Appendix A, the powers of the operator G required in the analysis are explicitly given by and All the necessary powers of C involved in Eq. (12) are now explicitly determined so the projection operator P (m) , corresponding to eigenvalue c m of C, can be evaluated. For instance, for c 0 = 0, where and Thus, The procedure can be repeated for the remaining four eigenvalues, which yields the projection operators The above projection operators satisfy the properties which are demanded by definition. Also, notice that so they constitute a complete set of operators. Now, given two adjoints Q b1 1 and Q b2 2 , the action of projectors P (m) on the adjoint tensor operator Q b1 1 Q b2 2 yields The operators on the left-hand sides in Eqs. (41)-(45) are labeled by an index that indicates the space representation they belong to. Therefore, when projection operator P (m) acts on the tensor product of two adjoints, it projects out precisely the component of the representation it belongs to. Two simple examples for N = 2 and N = 3 suffice to illustrate the usefulness of the projection operators so far constructed. These examples are worked out in the following sections.
A. Projection operators for N = 2 SU (2) is the simplest non-Abelian Lie group. It appears in two scenarios in physics. One is as the spin double cover of the rotation SO(3) group, the other is as an internal symmetry relating types of particles. Explicit realizations of them are spin and isotopic spin symmetries. The generators J i and I a correspond to spin and isospin, respectively, and the corresponding conventional structure constants are ǫ ijk (i, j, k = 1, 2, 3) and ǫ abc (a, b, c = 1, 2, 3), which are totally antisymmetric.
To construct the projection operators for N = 2, an important issue to be kept in mind is the fact that SU (2) does not admit representations for the eigenvalues c 2 = 2N and c 4 = 2(N − 1) of the quadratic Casimir operator in the space adj ⊗ adj listed in Table I, so the procedure to construct P (m) must be adapted accordingly because, in particular, P (1) of Eq. (35) as it stands is ill-defined for N = 2. Therefore, the procedure must be repeated accounting for the eigenvalues c 0 , c 1 , and c 3 only.
While the projector P (0) is easily obtained as P (1) is constructed as From C and C 2 given in Eqs. (27) and (28) for N = 2, it follows that so that Similarly, Now, given two adjoints Q b1 1 and Q b2 2 defined in spin space, for instance, the projectors P (0) , P (1) , and P (3) , given by Eqs. (46), (49), and (50), acting on the adjoint tensor operator Q b1 1 Q b2 2 , project out the J = 0, J = 1, and J = 2 spin components of that tensor product, respectively. Similar conclusions can be reached for isospin space, of course.
In the introductory section it was pointed out that the baryon sector of QCD has a contracted SU (2N f ) symmetry, where N f is the number of light quark flavors [9][10][11][12]. Under the decomposition SU (2N f ) → SU (2) ⊗ SU (N f ), the spin-flavor representation yields a tower of baryon flavor representations with spins J = 1/2, 3/2, . . . , N c /2 [11,13]. The spin-flavor generators of SU (2N f ) can be written as 1-body quark operators acting on the N c -quark baryon states, namely, Here q † α and q α constitute a set of quark creation and annihilation operators, where α = 1, . . . , N f denote the N f quark flavors with spin up and α = N f + 1, . . . , 2N f the N f quark flavors with spin down. Likewise, J k are the spin generators, T c are the flavor generators, and G kc are the spin-flavor generators. The SU (2N f ) spin-flavor generators satisfy the commutation relations listed in Table II [13].
The approach to obtain projection operators discussed in the previous sections can now be implemented to the SU (2N f ) spin-flavor symmetry to construct spin and flavor projection operators, which will act on well-defined n-body operators. For the ease of notation, throughout this section lowercase letters (i, j, . . .) will denote indices transforming according to the vector representation of spin and (a, b, . . .) will denote indices transforming according to the adjoint representation of the SU (N f ) flavor group.
Spin projection operators are easily adapted from Eqs. (46), (49), and (50) as As for flavor projection operators, the tensor product of two adjoints can be separated into an antisymmetric and a symmetric product, (adj ⊗ adj) A and (adj ⊗ adj) S , respectively. In the notation of Ref. [13], these products are written as and (adj ⊗ adj) S = 1 ⊕ adj ⊕ss ⊕āa.
Thus, the explicit forms of flavor projection operators read, It should be remarked that the first and second summands of Eq. (57) define the antisymmetric and symmetric components of P (adj) flavor a1a2b1b2 , respectively. Let us also notice that P (ss) Implicit forms of the projectors (58) and (61) can be inferred respectively from Eqs. (A13) and (A17) of Ref. [13].
Both approaches yield the same results.
A. Applications of spin and flavor projection operators in the 1/Nc operator expansion The way spin and flavor projection operators work can be better seen through a few examples. For definiteness, the analysis can be confined to the physically interesting case of N f = 3 light quark flavors; thus, the lowest-lying baryon states fall into a representation of the SU (6) spin-flavor group, which decomposes as SU (2) ⊗ SU (3).
For the SU (3) flavor group, the adj,ās +sa, andss representations are the 8, 10 + 10, and 27, respectively, while the representationāa does not exist. In consequence, it can be shown that for SU (3).
First, let us analyze the 2-body operator J j1 J j2 , which is a spin-2 object. It can be written as Projecting out the J = 0, J = 1, and J = 2 components of this product of operators is straightforwardly done with the help of projection operators (52), (53), and (54). The spin projections for the operator J j1 J j2 read and whereas the nonzero spin projections of the anticommutator and commutator in Eq. (63) read and where J 2 ≡ J i J i . The consistency between these relations can be checked by a simple inspection.
Less trivial examples are found when spin and flavor are simultaneously involved so the corresponding projectors can act in conjunction. For example, the operator X (j1b1)(j2b2) = {G j1b1 , G j2b2 } + {G j2b1 , G j1b2 } is a spin-2 object and transforms as a flavor 27. Projecting out the spin J = 0, J = 1, and J = 2 components of this operator yields and (68) Now, the flavor 1, 8, 10 + 10, and 27 components of X (j1b1)(j2b2) , for each spin, can be straightforwardly projected out. The J = 0 projections read the J = 1 projections vanish, and the J = 2 projections become In particular, both sides of Eq. (76) are spin-2 objects and transform purely as flavor 27 tensors, i.e., their spin 0 and 1 components and their flavor singlet and octet components have been properly subtracted off by using the appropriate spin and flavor projectors. Operators of this kind appear in the analysis of baryon quadrupole moments [19].

V. PROJECTION OPERATORS EXTENDED TO THE TENSOR SPACE adj ⊗ adj ⊗ adj: A FEW EXAMPLES
Projection operators defined in the tensor space adj ⊗ adj ⊗ adj can be obtained by extending the approach used in the construction of the corresponding ones in the tensor space adj ⊗ adj. The starting point is the decomposition of the tensor product adj ⊗ adj into the irreps indicated in Eq. (16), so the tensor product of the adjoint representation and each of these irreps can be evaluated.
The simplest construction is the tensor product of the adjoint and the singlet representation 1, i.e., 1 ⊗ adj = adj. Therefore, the projector acting on the tensor operator Q b1 which transforms as an adjoint operator. Increasing complexity can be found in the tensor productss ⊗ adj, which can be represented by Let T a 3A denote the generators for the tensor product spacess ⊗ adj. These generators are given by where T a 2A are defined in Eq. (19). Accordingly, the quadratic Casimir operator reads whose explicit form in components becomes  which follows from the use of the identity along with Eqs. (21) and (24). The eigenvalues of the quadratic Casimir operator for each representation are displayed in Table III. Following relation (12) and gathering together partial results, the corresponding projection operators arẽ where the coefficientsα i readα The powers of C required in Eq. (83) are obtained as and with a = 3N + 2 and and Here E 0 represents the identity for the tensor product space under consideration and the tensor [T ] a1a2a3b1b2b3 is defined as The final expression for the projectorsP (m) can be cast into the compact form where the coefficients h m and e (m) n are listed in Appendix B. A long and tedious but otherwise standard calculation is required to prove that A. An example of projection operators in SU (2) In this case the projector P (3) corresponds to the representation with spin-2 given in Eq. (50). Therefore, the projectorP (5) which, according to Table III, corresponds to an adjoint representation (spin-1 with three indices) and is given byP Using the expressions for E i given in (86a)-(86f) for N = 2,P (5) in components can be rewritten as B. An example of projection operators in SU (3) Formally, given three SU (3) adjoints Q a1 1 , Q a2 2 , and Q a3 3 , the tensor product between them, Q a1 1 Q a2 2 Q a3 3 , possesses all flavor 1, 8, 10 + 10, 27, 35 + 35, and 64 components. Operators transforming in the flavor 64 representation, for instance, are relevant in the analysis of baryon mass splittings of the spin-1/2 octet and spin-3/2 decuplet baryons in the 1/N c expansion combined with perturbative flavor breaking at order O(ǫ 2 ), where ǫ ∼ m s is a (dimensionless) measure of SU (3) breaking [18].
In order to subtract off all but the flavor 64 component, the projection operatorP (m) , for m = 0, is constructed following the lines of Eq. (88); this procedure leads toP (64) flavor . The eigenvalue of the Casimir operator is c 0 = 3(N f + 2) and the corresponding Young tableau can easily be obtained from the corresponding one depicted in Table  III for N f = 3.
Let Q (64) be the operator that transforms as a genuine flavor 64. It is thus given by The projection operatorP flavor itself has a rather involved form, containing several hundreds of terms. Because of the length and unilluminating nature of the resultant expression, it is more convenient to list a few components of Q (64) . For instance, In this section some relations between the structure constants of the Lie algebra of SU (N ) used repeatedly in the present analysis are provided. The list by no means is exhaustive, but it ranges from the Jacobi identity up to the product of 8 f 's. The relations read, F a1e1a2e2 F b1e1b2e2 = δ a1a2 δ b1b2 + 1 2 δ a1b1 δ a2b2 + 1 2 δ a1b2 δ a2b1 + N 4 (D a1a2b1b2 + F a1a2b1b2 ), F a1e1a2e2 F e1e3e2e4 F b1e3b2e4 = N δ a1a2 δ b1b2 + N 2 8 (D a1a2b1b2 + F a1a2b1b2 ) + 1 2 (F a1b1a2b2 + F a1b2a2b1 ), (A8)