Fe b 20 20 Baryon quadrupole moment in the 1 / N c expansion of QCD

The quadrupole moments of ground state baryons are discussed in the framework of the 1/Nc expansion of QCD, where Nc is the number of color charges. Theoretical expressions are first provided assuming an exact SU(3) flavor symmetry, and then the effects of symmetry breaking are accounted for to linear order. The rather scarce experimental information available does not allow a detailed comparison between theory and experiment, so the free parameters in the approach are not determined. Instead, some useful relations among quadrupole moments are provided. The overall predictions of the 1/Nc expansion are quite enlightening.


I. INTRODUCTION
Understanding the structure of baryons is still a daunting task in Quantum Chromodynamics (QCD). The most interesting static properties of baryons, e.g., masses, magnetic moments, matter and charge radii, etc., fall in the nonperturbative regime of QCD so analytic calculations of these properties are not possible because the theory is strongly coupled at low energies, with no small expansion parameter.
On the other hand, the information about the higher-order electromagnetic moments (electric quadrupole and magnetic octupole moments) is less profuse. Analyses about the quadrupole moments of baryons have also been performed within the quark model (and its variants) [31][32][33], light-cone QCD sum rules [34,35], the Skyrme model [36], a QCD parameterization method [37,38], the 1/N c expansion [39][40][41], and chiral perturbation theory [26,42]. In contrast, the experimental data about quadrupole moments is rather scarce. The only experimental values reported are the helicity amplitudes for the process ∆ + → pγ [30], which can be used to extract the value of the ratio between the electric quadrupole (E2) and the magnetic moment (M 1), E2/M 1.
The present paper is focused on the computation of quadrupole moments of the ground state baryons in the context of the 1/N c expansion, which has proven to be quite effective in the calculation of static properties of baryons. For this purpose, in Sec. II a survey of the 1/N c expansion is presented to introduce the notation and conventions. In Sec. III the analysis starts with the construction of the most general spin-2, flavor octet operator which describes the baryon quadrupole moment. Next, the effects of flavor SU (3) symmetry breaking (SB) to linear order are accounted for in Sec. IV; the detailed construction of baryon operators which make up the series is described for each flavor representation present in the tensor product of the quadrupole moment and the perturbation to identify redundant operators. Once this task is completed, the full series is provided in Sec. V. The lack of experimental information does not allow to determine the free parameters of the theory so no attempt to predict any of the quadrupole moments numerically is made. Instead, various relations among them are provided. Some of them are valid in the limit of exact flavor symmetry and others are valid even in the presence of SB. Some closing remarks are provided in Sec. VI. The paper is complemented by Appendix A, where some useful baryon operator reductions are listed in order to discard redundant operators.
There are two papers that also address the evaluation of quadrupole moments of baryons within the 1/N c expansion [40,41]. In these papers, the minimal assumption of the single photon exchange ansatz is used, which implies that the photon probing these observables couples to only one quark line inside the baryon. This reduces the number of operators involved. This assumption is not used in the present analysis; instead, the full operator basis is used here. Although this might be counterproductive due to the larger number of free parameters introduced, in fact, it leads to interesting relations among quadrupole moments which can not be determined otherwise. Another noticeable difference between the approach implemented in Ref. [41] and the one used here concerns the way SB enters into play. In this reference, SB is accounted for by modifying the spin-spin terms with ratios between the constituent quark masses. In the present analysis, as it was pointed out above, SB enters perturbatively. More details on the subject are provided in the following sections.

II. A SURVEY OF THE 1/Nc EXPANSION OF QCD
The present analysis builds on the seminal work on large-N c baryons presented in Ref. [43], so only a few prominent facts will be highlighted here.
In the large-N c limit, the baryon sector exhibits a contracted SU (2N f ) spin-flavor symmetry, where N f is the number of light quark flavors [44,45]. This baryon representation decomposes under SU (2) × SU (N f ) into a tower of baryon states with spins J = 1/2, 3/2, . . . , N c /2. The present analysis is done for the special case N f = 3. Therefore, the ground state baryons transform as the completely symmetric product of three 6's of SU (6), which is the 56 dimensional representation. Three spin 1/2's added together can yield spin 1/2 or spin 3/2, so the 56 representation contains spin-1/2 and spin-3/2 baryons.
Corrections to the large-N c limit are expressed in terms of 1/N c -suppressed operators with well-defined spin-flavor transformation properties [45]; this approach leads to the 1/N c expansion of QCD.
Concretely, the 1/N c expansion of a QCD operator at leading order reads [43] where the sum is over all possible operators O n , n = 0, . . . , N c , which are polynomials in the spin-flavor generators of total order n. O n are thus referred to as n-body operators. The spin-flavor generators can be written as 1-body quark operators acting on the N c -quark baryon states, namely, where J k are the spin generators, T c are the flavor generators , and G kc are the spin-flavor generators. The operator coefficients c n also have power series expansions in 1/N c beginning at order unity. On the other hand, q † α and q α are operators that create and annihilate states in the fundamental representation of SU (6) and σ k and λ c are the Pauli spin and Gell-Mann flavor matrices, respectively. Because the baryon matrix elements of the spin-flavor generators (2) can be taken as the values in the nonrelativistic quark model, this convention is usually referred to as the quark representation [43].

III. THE QUADRUPOLE MOMENT OPERATOR IN THE LIMIT OF EXACT SU (3) FLAVOR SYMMETRY
The electromagnetic current operator can be expanded in a power series of the photon momentum k γ (the multipole expansion); the series can be expressed as [39] where µ ia is the magnetic moment operator [12] and Q (ij)a is the quadrupole moment operator. Q (ij)a is a spin-2 object and a flavor octet, so it transforms as (2,8) under SU (2) × SU (3). Q (ij)a is a symmetric, traceless tensor in the spin indices i and j. The electromagnetic current is T-odd, so that Q (ij)a is T-even. The physical interpretation of the operators in the multipole expansion (3) can readily be seen through their matrix elements between SU (6) symmetric baryon states. For example, the magnetic moment for a spin-1/2 baryon B p , for a spin-3/2 baryon B ′ q , or for transitions B p → B q and B ′ p → B q , can be generically denoted by where the spin and flavor indices, i and a, have been set to 3 and Q ≡ 3 + (1/ √ 3)8, respectively. Hereafter, any operator of the form X Q should be understood as X 3 + (1/ √ 3)X 8 , where X 3 and X 8 denote the isovector and isoscalar components of the operator X a , respectively.
Similarly, the zero component of the rank-2 tensor (in spin space) Q (ij)a , or equivalently, the ℓ = 2, m ℓ = 0 component for i = j = 3 and a = Q, is usually referred to as the spectroscopic quadrupole moment [37] for baryon B ′ q , which reads, Angular momentum selection rules forbid a spin-1/2 baryon from having a spectroscopic quadrupole moment. Similar definitions can also be given for transitions B ′ p → B q . For instance, for the radiative decay ∆ + → pγ, the only multipoles that contribute are the magnetic moment (M 1) and electric quadrupole (E2), which are defined as [39] and E2 = 1 12 On the other hand, to extract information about the shape of a spatially extended particle, its intrinsic quadrupole moment Q 0 , given by is generally used. Q 0 is defined with respect to the body-fixed frame. For a charge density concentrated along the z-direction, Q 0 is positive and the particle is prolate. For a charge density concentrated in the equatorial plane perpendicular to z, Q 0 is negative and the particle is oblate [37]. The intrinsic quadrupole moment must be distinguished from the spectroscopic one measured in the laboratory frame. Although a spin-1/2 baryon does not have a spectroscopic quadrupole moment, it may have an intrinsic one. Indeed, within various models, the proton and ∆ + are found to possess a prolate and an oblate deformation, respectively [37].
The present analysis is focused on the calculation of the spectroscopic quadrupole moment of baryons; it will be loosely referred to as the quadrupole moment hereafter. For this task, the quadrupole moment operator Q (ij)a is constructed in the framework of the 1/N c expansion, which requires to seek all the operator structures that transform as (2,8) under SU (2) × SU (3), written as polynomials in the spin-flavor generators J i , T a , and G ia [43]. Symmetry requirements are used in order to eliminate those structures which are not either T-even or symmetric or traceless in the spin indices. In this regard, operator structures should contain an even number of either J, G or a combination of them, or an odd number of J, G or a combination of them, along with a factor of if abc or iǫ ijk to yield hermitian operators. Of course, spin and flavor indices must be properly saturated to have (2, 8) resultant operators. The 1-body operator iǫ ijk G ka , for instance, is T-even and traceless but antisymmetric in i and j, so it is discarded.
In the limit of exact SU (3) flavor symmetry, the 1/N c expansion of Q (ij)a actually starts with the 2-body operator At 3-body operator level, several possibilities emerge, for instance, The use of operator identities [43] restricts the number of linearly independent operators. In Appendix A some useful operator reductions are listed. Accordingly, a convenient 3-body operator to include in the series is and all additional 3-body operators are redundant and can be ignored.
On the other hand, 4-body operators can be constructed as products of four G's, two J's and two T 's, two J's and two G's, three G's and one J, and three J's and one G, with the spin and flavor indices properly saturated. Among these structures, a convenient 4-body operator is whereas the others, according to the operator reductions listed in Appendix A, can be expressed in terms of either (12) and the lower-order operators (9) and (11), or in terms of (9) itself and its anticommutator with J 2 . Hence, there is a second 4-body operator obtained asÕ Now, 5-body operators can be constructed out of the tensor products of the 4-body operators listed above and T a , with the proper contraction of spin and flavor indices. Following the operator reductions of Appendix A, it can be concluded that there is a single 5-body operator given by Therefore, without loss of generality, starting from O (ij)a m (m = 2, 3, 4), higher-order operators can be obtained as anticommutators of these operators with There are additional higher-order operators computed asÕ (ij)a n }, for n even, n ≥ 4. Thus, in the limit of exact SU (3) flavor symmetry, the 1/N c expansion of Q (ij)a is written as where k m ,k n are unknown parameters which also have a 1/N c expansion beginning at order unity; these parameters are multiplied by a characteristic hadronic quadrupole size (in fm 2 ). The vanishing trace condition can be easily verified as For N c = 3 the series can be truncated after the first two terms, i.e., which is valid up to corrections of relative order 1/N 3 c . The quadrupole moments of baryons in the limit of exact SU (3) flavor symmetry, Q

SU(3) B
, can be obtained from the matrix elements of the baryon operators that make up Q (ij)a , Eq. (17). These matrix elements are listed in Tables I, II, and III, for octet baryons, decuplet baryons, and decuplet-octet transitions. Although these matrix elements are provided for the special case N c = 3, its overall dependence on N c can be better seen from the matrix elements of the 1-body operators T a and G ia , a = 3 and a = 8, which occur quite often in the analysis. They can be rewritten in terms of the quark number and spin operators [43] where The leading N c counting of the matrix elements of operators (18) is deduced as follows: . By using these counting rules and recalling that the operator coefficients k i are order unity, the isovector leading term in the series (17) which is the sum of two null quantities (isovector and isoscalar quadrupole moments) and not the cancellation of two equal in magnitude but opposite in sign quantities. This is a completely expected (and consistent) result. For decuplet baryons the quadrupole moments read where q B ′ q is the electric charge of decuplet baryon B ′ q given by A few comments are in order here. Equation (20) has been purposely written in a way to exhibit that the effects of higher-body operators can readily be accounted for without altering the basic structure of the equation itself. Thus one would be prompted to express Eq. (20) in terms of a single parameter, let us say k, so that which would be valid to all orders in the 1/N c expansion. This approach, however, is not entirely correct. The reasons can be better seen when computing the transition quadrupole moments, Q and Due to the fact that the operator coefficients k i participate differently in Q q Bp , there is not a unique way of recombining these operator coefficients to reduce the number of free parameters. Therefore, retaining up to 3-body operators in the 1/N c expansion of quadrupole moments implies the existence of two free independent parameters, k 2 and k 3 .
In the SU (3) limit, two relations become evident, namely.
The last relations have also been noticed within the chiral constituent quark model analysis of Ref. [33]. As a side remark, notice that Q Ξ * − Ξ − = 0, which is a well-known result derived in the quark model [33]. These transitions are forbidden by U -spin conservation if flavor symmetry is exact. Now, it is straightforward to test the combinations sensitive to I = 3 and I = 2 operators [41]. For the former case, and for the latter case, There is only a 0-body operator transforming as (2, 1) under SU (2) × SU (3), and a single 1-body operator none of which contributes to Q (ij)a by virtue of the symmetry or trace conditions discussed above. Thus, the only non-trivial operator found is the 2-body operator because even the 3-body operator is also redundant, according to Eq. (A3) of Appendix A. Therefore, the SB contribution to Q (ij)a from the (2, 1) representation reads 1 whereas higher-order operators are consecutively obtained as O The (2, 8) operators that generate SB corrections to Q (ij)a can be obtained in a close analogy with expressions (9) and (11) and read respectively for 2and 3-body operators, and Higher-order operators are obtained as O (ij)a n,8 } for n ≥ 2. Additional (2, 8) operators obtained by replacing the d ab8 symbol with the if ab8 symbol turn out to be T-odd so they are forbidden by time reversal invariance. 1 Hereafter, for the ease of notation, O (ij)a n,rep will stand for an n-body operator belonging to flavor representation rep.
C. (2,27) The analysis of (2, 27) operators is more involved than the ones previously discussed. In the present case, not only the symmetry of operators under the exchange of spin indices must be manifest, but also the symmetry of operators under the exchange of flavor indices, which is required for flavor-27 operators. Let S (ij){ab} + be one of such operators. The singlet and octet components are subtracted off S (ij){ab} + to obtain a genuine flavor-27 operator S (ij){ab} according to [43] The contribution of S (ij){ab} to Q (ij)a is actually obtained by setting b = 8 so the contribution is effectively S (ij){a8} . The 1/N c expansion for a (2,27) operator that contributes to Q (ij)a begins with a single 2-body operator which, after subtracting singlet and octet components in accordance with prescription (34), gets the form After a straightforward algebraic manipulation, the SB correction to Q (ij)a from this 2-body operator becomes As for 3-body operators, there is a single one given by which turns into by following prescription (34). Therefore, the SB correction to Q (ij)a from 3-body operators is then given by Next 4-body operators can be worked out. They can conveniently constructed as tensor products of a spin-0 and a spin-1, 2-body operators. There are two of such operators, namely, and The procedure to subtract singlet and octet components yields, for the former, and for the latter, Finally, the SB contributions from 4-body operators to Q (ij)a are given by There is an additional 4-body operator constructed as where, by construction With the above considerations, the series for the (2, 10 + 10) SB operators actually begins with a single 3-body operator where the octet component to be subtracted off reads This last term is particularly interesting. First, notice that it is T-odd, so in principle it is forbidden in the 1/N c expansion of Q (ij)a . And secondly, it vanishes under contraction of spin indices. It is necessary, though, to fulfill condition (49). Thus, the SB contribution to Q (ij)a from (2, 10 + 10) operators is where the octet component has been safely ignored. Higher-order operators are obtained as O n,10+10 }, for n ≥ 3.

V. A FULL EXPRESSION FOR QUADRUPOLE MOMENT
The final expression for the quadrupole moment operator to linear order in SB is where Q (ij)a is the operator whose matrix elements yield the SU (3) symmetric values; it is given in Eq. (15). In turn, δQ (ij)a includes all the operators due to first-order SB and its 1/N c expansion reads At the physical value N c = 3, the series is truncated as where the operators O       ). All in all, there are eight undetermined parameters. Unfortunately, the experimental information [30] is rather scarce, so at this time, it is not possible to perform a least-squares fit to compare theory and experiment and extract information on these parameters. Any other attempts of reducing the number of free parameters are fruitless. The only pieces of information known up to now are those corresponding to the transition ∆ + → p [30]. The difficulties in measuring quadrupole moments depend on many factors. For instance, except for Ω − , all of the decuplet baryons decay strongly, so couplings of the form decuplet-decuplet-γ are available only through virtual processes, which are difficult to measure [40].
To overcome the lack of experimental information on quadrupole moments, some relations among them are provided instead. Apart from the isospin relations (58) that must be satisfied by quadrupole moments in the presence of SB, other relations are also provided, (59), which can be quite useful in the future, when additional experiments are envisaged. In the meantime, the predictions of the 1/N c expansion are in accordance with expectations.