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The Nucleon-Nucleon Potential in the 1/N_c Expansion

The nucleon-nucleon potential is analysed using the 1/N_c expansion of QCD. The NN potential is shown to have an expansion in 1/N_c^2, and the strengths of the leading order central, spin-orbit, tensor, and quadratic spin-orbit forces (including isospin dependence) are determined. Comparison with a successful phenomenological potential (Nijmegen) shows that the large-N_c analysis explains many of the qualitative features observed in the nucleon-nucleon interaction. The 1/N_c expansion implies an effective Wigner supermultiplet symmetry for light nuclei. Results for baryons containing strange quarks are presented in an appendix.


Introduction
The two-body nucleon-nucleon interaction is the basic ingredient that is used in calculating the properties of nuclei. There are a number of phenomenologically successful models for the interaction, typically constructed using meson exchange contributions. Unfortunately there is no direct connection between these models and the underlying theory of the strong interactions, QCD. Until recently, the only way that QCD and the physics of nucleon interactions could be rigorously related has been through symmetry arguments.
By making use of symmetry and effective field theory, one can calculate the low energy dynamics using a small number of parameters that are fitted to the experimental data.
Such theories can be quite predictive, but still remain somewhat remote from QCD -one would like a priori arguments for the sizes of the phenomenological parameters.
Recently, there has been significant progress in better understanding the implications of QCD for hadronic physics by exploiting the "hidden" expansion parameter of QCD -1/N c , where N c = 3 is the number of colors [1][2][3][4][5]. In the large N c limit, one finds that meson-baryon interactions respect an SU (4) spin-flavor symmetry -the same symmetry found in the nonrelativistic quark model quark model [1,6]. It has also been shown that the 1/N c expansion can provide information about the nucleon-nucleon potential [7]. In particular, ref. [7] analyzed the central potential for N N scattering and showed that the 1/N c expansion gave a qualitative understanding of its spin and isospin structure.
In this paper we extend the analysis of [7] to include the entire N N potential. Naively, one might think that the N c → ∞ limit is not relevant for analyzing nuclear physics.
Nuclear matter forms a classical crystal at N c = ∞, and so there must be a phase transition between N c = 3 and N c = ∞. While the 1/N c expansion is not reliable for studying bulk properties of nuclear matter, it does allow one to analyze the spin and isospin dependence of the nuclear force. One expects that the symmetry properties of the N N interaction will be independent of the phase of the many body groundstate.
(1 .2)  TABLE 1 Isospin The four terms V i 0 , V i σ constitute the central potential, while V i T , V i LS and V i Q are the tensor interaction, the spin-orbit interaction, and the quadratic spin-orbit interaction respectively; the ten functions V i a can in general be velocity dependent. The main result of this paper is that the strength of the ten functions V i a can be determined in the 1/N c expansion, and are as given in Table 1. As is apparent from this Table, the actual expansion parameter is not 1/N c but 1/N 2 c . Thus even though the actual value N c = 3 is not very large, an expansion in 1/N 2 c can be quite predictive. The organization of this paper is as follows: in §2 we briefly review general properties of baryons in the large N c limit. In §3 we derive the results given in Table 1. These results are compared in §4 with the "Nijmegen potential" of references [8,9] -a phenomenologically successful model of the N N interaction; we show that the the hierarchy of Table 1 is evident in N N phenomenology. §5 extends the discussion of ref. [7] concerning how the Wigner supermultiplet symmetry might arise in light nuclei as a consequence of the 1/N c expansion. This is followed by conclusions, and an appendix in which we extend our analysis to hyperon interactions (i.e, including the s quark).

The Large N c QCD Analysis
The large N c limit is defined by taking the number of colors N c of QCD to be large while simultaneously rescaling the QCD coupling as g → g/ √ N c , keeping Λ QCD fixed [10].
The 1/N c expansion has proven to be a powerful tool for analyzing baryon properties, since baryon structure simplifies considerably in this limit. Antiquarks in the baryon are suppressed, and as baryons consist of N c quarks interacting with 1/N c strength, the Hartree approximation becomes exact in the large N c limit [11]. Although one cannot solve the Hartree equations due to the nonlinearity of glue interactions, one can nevertheless determine a number of useful properties of the spin and flavor properties of the baryons and their interactions. To analyze the flavor and spin structure of the Hartree Hamiltonian in the case of two light flavors, it is convenient to use as an operator basis the one-quark operators of the quark modelŜ where q = (u, d) and q † are the creation and annihilation operators for the u and d quark flavors, and σ i , τ a are the standard SU (2) Pauli matrices acting on spin and isospin respectively. The q and q † operators do not carry color, and are bosonic. The Hartree Hamiltonian can then be constructed as monomials of these operators. An important result from large N c QCD is that the Hartree Hamiltonian takes the form [3][4][5]: where the operators {Ŝ,Î,Ĝ} are given in eq. would have worked just as well [12].
The lowest lying eigenstates of H with baryon number B = 1 have isospin I and spin S satisfying I = S = 1 2 , 3 2 , 5 2 . . .. The first two states can be identified with the N and ∆; the higher spin states do not exist for N c = 3. To leading order, this baryon tower is degenerate with mass M ∼ N c . One can show that it transforms as an irreducible representation (the totally symmetric N c -index tensor) of an approximate SU (4) spin-flavor symmetry; this is the symmetry under which the quark operators u ↑, u ↓, d ↑, d ↓ transform as the four-dimensional fundamental representation [1,13]. For N c = 3 the {N, ∆} spin states transform as the 20 dimensional representation of SU (4), familiar from the quark model.
Additional information can be obtained by considering matrix elements of operators between baryon states B and B ′ restricted to have S = I ∼ 1. For example, matrix elements of the basis operators (2.1) satisfy Matrix elements of many body operators can be analyzed as well, using various relations among powers of the basic operatorsŜ i ,Î a , andĜ ia [1,3]. This allows one to greatly reduce the number of linearly independent terms in (2.2) at a given order in 1/N c . Using these techniques, one is able to show that the matrix element of a general n-quark operator O (n) I,S with B = 0, isospin I and spin S, is of size [3] (see also [7] for a derivation) The fact that the operators with the largest matrix elements have I = S was first observed in the Skyrme model [14], and is known as the "I t = J t rule".

The Nucleon-Nucleon Interaction
There are two independent three-momenta for baryon-baryon scattering in the center of mass frame, which can conveniently taken to be These momenta are to be considered independent of N c in the 1/N c expansion. To leading order in 1/N c , the entire baryon tower is degenerate and |p p in | = |p p out | for elastic scattering, up to O(1/N 2 c ) corrections, and so q q · k k = 0 to the same order. The general baryon-baryon interaction potential is then a matrix where H is the Hartree Hamiltonian (2.2) and α, . . . , δ denote internal quantum numbers of the baryons, such as spin, flavor and particle type (e.g, N or ∆). Throughout this paper we will define the N N potential as the above matrix element restricted to the space of nucleons. We do not consider second order effects due to virtual ∆'s, etc.
There are two ways that 1/N c factors can suppress terms in the potential.

∆L = 0: The Central Potential
This is the case analyzed in ref. [7]. The central force can be written as a sum of products of 1-quark operators as in eq. (2.2), where the operators act on either the N 1 or N 2 nucleon states, and the coefficients v stn are general scalar functions of |q q| and |k k|. It follows from eq. (2.3) that the leading contribution will have noŜ i /N c orÎ a /N c operators, since each of these implies a 1/N c suppression; instead it will consist solely of powers of G ia /N c . By rotational symmetry, since the v coefficients are scalars, theĜ ia operators must be contracted to form spin invariants. Similarly, isospin symmetry implies that thê G ia must be contracted to form isospin invariants. From these constraints, it is possible to show that at leading order in 1/N c , the most general form for the central potential is [7] In general, the coefficients v n are functions of both |q q| 2 , |k k| 2 , and obey the rule eq. (3.3). One can further restrict the powers ofĜ 1 ·Ĝ 2 in eq. (3.4) to be completely symmetric in theĜ 1 indices, and in theĜ 2 indices, before the two sets of indices are contracted.
It is straightforward to verify that (3.4) is the most general form of the leading order ∆L = 0 potential. We have argued that it can only involve powers of theĜ ia operators, on the basis of eq. (2.3); what must be shown is that the indices are contracted as above in eq. (3.4). By the operator reduction rule [3] any terms in which two indices ofĜ iaĜjb (where bothĜ's act on the same baryon) are contracted with each other by δ ij , δ ab , ǫ ijk or ǫ abc can be eliminated in favor of terms with fewer powers ofĜ Thus the only allowed invariants are obtained by contracting the indices ofĜ ia 1 with those ofĜ ia 2 , as in (3.4). More complicated contractions, such aŝ can be written as (3.6) The term with two ǫ symbols can be reduced toĜ 1 ·Ĝ 2 using the relations in [3], so that all contractions ofĜ 1 withĜ 2 can be written as powers ofĜ 1 ·Ĝ 2 . One can also restrict the indices on powers ofĜ 1 andĜ 2 to be completely symmetrized, since terms antisymmetric in the indices can be eliminated using the operator identities. The series in while As was noted in [7] and will be discussed in §5, the above relation implies that the central potential obeys an effective Wigner supermultiplet symmetry. 1 An easy way to see this is to normal order the operators. A normal ordered product involving with more than N c quark operators on a baryon vanishes, which gives the desired identity.

∆L = 1: The Spin-Orbit Potential
The ∆L = 1 baryon interaction amplitude contains the spin-orbit coupling term; it is obtained from the general Hartree Hamiltonian eq.
This can be derived by arguments similar to those in the previous subsection. Time reversal and parity invariance requires the coefficients v i m,n in (3.9) to be proportional to (q q × k k) times an arbitrary function of q q 2 and k k 2 . In position space, a contribution of the form U (q q 2 ) (q q × k k) · (S S 1 + S S 2 ) is of the form (∇ ∇U (r) × k k) · S S, which is the usual spin-orbit force.
There is a hidden suppression factor of 1/N c (which follows from eq. (3.3)) in the spin-orbit force which is not manifest in eq. (3.9), since the ∆L = 1 interaction necessarily involves both q q and k k.
The Wigner-Eckart theorem implies that there are only two distinct operators when the expression (3.9) for the ∆L = 1 amplitude is restricted to the nucleon sector. These are the two spin-orbit terms appearing in eq. (1.1). Thus we find The spin-orbit force is O(1/N 2 c ) in strength relative to the central force, and it is of comparable strength in the two isospin channels.

∆L = 2: The Tensor and Quadratic Spin-Orbit Potentials
The ∆L = 2 amplitude is obtained by requiring that the coefficients v in eq. (2.2) transform under rotations as ∆L = 2. The leading order amplitude is a polynomial in thê G's that transforms as S = 2, I = 0. One can obtain an amplitude that does not violate the I t = J t rule on each baryon line by combining I = S = 1 amplitudes on each baryon into total I = 0 and total ∆L = 2. The general form of the leading order amplitude is where the coefficient v ij n is a symmetric traceless tensor that depends on q q and k k. Time reversal invariance requires the coefficients to have the form where v n is a scalar function of q q 2 , k k 2 and (q q · k k) 2 .
If one restricts the interaction eq. (3.11) to the nucleon sector, one gets Terms with n > 1 in eq. (3.11) can be dropped, because two spin-1/2 nucleons can only give non-zero matrix elements for operators with spin ≤ 1. Comparing with eq. (1.1), we see that The other term in the tensor potential, V 0 T , has |I − S| = 1 at each nucleon line, and so by eq. (2.4) (3.14) A similar and straightforward analysis for V Q gives the results listed in Table 1.

The N N potential and the ∆
One flaw in our discussion that should be eventually improved upon is the treatment of the ∆. The Hartree Hamiltonian (2.2) implicitly acts on the entire I = S baryon tower, including both nucleons and ∆'s, all of which are degenerate in the N c → ∞ limit. In our discussion of the N N potential, we have simply projected H to the nucleon sector. A more sophisticated treatment would be to integrate the ∆'s out of the theory (keeping track of the 1/N c mass splitting) and to construct an effective theory for nucleons alone. This is a subtle analysis (see, for example, [15]) and beyond the scope of this paper.

Comparison of large N c QCD with a phenomenologically successful model
Our large-N c results for the general nucleon-nucleon potential of eq. (1.1) are displayed in Table 1 The results can be directly compared with nuclear potential models in momentum space. A particularly simple phenomenological model to compare with is the meson exchange model "Nijmegen potential" of references [8,9]. In this model, the N N potential is approximated in momentum space by a sum of Yukawa and Gaussian interactions times powers of momenta divided by masses, contracted with the spin and isospin Pauli matrices. The Yukawa potentials correspond to one-particle exchange of both real mesons(π, η, η ′ , ρ, ω, φ, a 0 , f 0 ) and an "effective meson" (ǫ), while the Gaussian potentials are labelled by P , f 2 , f ′ 2 and a 2 . The motivation for this form of the potential is unimportant here; it provides a phenomenologically successful parametrization for the N N potential that can be compared with Table 1. The N c dependence should appear in the relative strengths of the potentials, and the 1/M factors that appear when the potential is decomposed as in eq. (1.1). The strength of the contributions to the Nijmegen potential are simple to evaluate, since they are presented explicitly in momentum space, and we can treat all momenta and meson masses as ∼ 1 in the 1/N c expansion. The N c dependence must then reside in the strengths of the couplings used in the Nijmegen potential, as well as the explicit factors of the nucleon mass that appear in the formulas of ref. [8]. One finds for the strength of the various terms in the potential where I = 0, 1 correspond to the 1 · 1 and τ 1 · τ 2 isospin structures respectively, M is the nucleon mass, and Λ is a strong interaction scale characterizing the derivative expansion (denoted M in [8]). The parameters g IS correspond to the coupling constants of the model with t-channel (isospin, spin)= (I, S) in the nonrelativistic limit; in particular, the scalar coupling g S and vector couplings g V and f V of ref. [8] are given by g I0 , g I0 and g I1 respectively, where I is the meson isospin. (The pseudoscalar contributions are parametrized differently in [8] and are mentioned below). As far as the N c scaling goes, M ∼ N c , while the Λ and the masses of the exchange mesons are all ∼ 1. In eq. (4.1) we have omitted dimensionful quantities that do not scale with N c , such as the meson propagators 1/(q 2 + m 2 ). By comparing the expressions in eq. (4.1) with our results in Table 1, one sees that they are consistent provided that the couplings g IS scale with N c as This N c scaling can be compared with the numerical values given in ref. [9]. In fig. 3 we have plotted the couplings determined numerically in ref. [9], rescaled by their value for f ρ . Since f ρ is a g 11 coupling, eq. (4.2) implies that the leading large-N c prediction for the ratio isĝ As can be seen from fig. 3, there is good qualitative agreement between the large-N c prediction (4.3), and the g IS values used in the Nijmegen potential. Omitted from fig. 3 are the pseudoscalar couplings, which are not readily compared with heavy meson couplings, due to their special status as pseudo-Goldstone bosons. However, the pseudoscalar meson couplings are related to the axial current couplings, which have been analyzed in detail, and shown to agree with 1/N c predictions [16]. There are two couplings in the Nijmegen potential, the φ and a 2 coupling, that deviate significantly from the 1/N c pattern. The φ meson is a puress state, and only couples to the nucleons through quark loops. Its coupling is OZI suppressed, and should be of order 1/N c relative to the ω couplings. The Nijmegen fit has g φ /g ω ≈ 0.1, which is a factor of three smaller than the naive 1/N c prediction. The a 2 coupling is even somewhat smaller.
It must be stressed that the numerical parameters plotted in fig. 3 were obtained by treating the couplings as phenomenological parameters in the N N potential, chosen to provide the best fit to N N scattering data. There is no reason to assume that the N N force is actually due to single meson exchange; in fact, the ǫ and P contributions to the 3) contains more than the claim that meson-baryon couplings obey the I t = J t rule. We take fig. 3 to provide encouraging evidence that our large-N c analysis of the N N interaction describes the qualitative features seen in nature.

The Central Potential and Wigner Supermultiplet Symmetry
It was suggested in ref. [7] that the approximate Wigner supermultiplet symmetry observed in light nuclei could be explained by the 1/N c expansion of QCD. Under the Wigner symmetry SU (4) W , the four nucleon states p ↑, p ↓, n ↑ and n ↓ transform as the four dimensional fundamental representation. Note that SU (4) W is distinct from the quark model SU (4), and that the former cannot be realized as a symmetry at the quark level.
Nevertheless, ref. [7] argued that the 1/N c expansion explains how SU (4) W symmetry could emerge as an accidental symmetry in light nuclei. As that work only examined the central part of the N N potential, it is worth reexamining the argument.
Under SU (4) W symmetry, a two-nucleon state transforms like 4 × 4 = 6 A + 10 S , where the subscripts A and S denote the antisymmetric and symmetric combinations.
If the two nucleons are in an even partial wave, they must be in a totally antisymmetric spin ⊗ isospin state, so they are in a (0, 1) or (1, 0) state, i.e. in 6 A of SU (4) W . If the two nucleons are in an odd partial wave, they must be in a totally symmetric spin ⊗ isospin state, so they are in a (0, 0) or (1, 1) state, i.e. in 10 S of SU (4) W .
We have shown in §3 that the leading contributions to the N N potential (at strength T . The first two terms correspond to the operators 1, Nevertheless, there is reason to expect to see Wigner symmetry in light nuclei. The nucleons inside a nucleus have low momentum, so the dominant interaction is s-wave scattering, with higher partial waves being kinematically suppressed. Furthermore, the tensor mean field is small in nuclei. Therefore all of the leading order violations of SU (4) W may be expected to be small. So why is SU (4) W not evident in heavy nuclei? At subleading order (a relative 1/N 2 c ), the potentials V 0 σ , V 1 0 , V 0 T and V 0,1 LS all break the Wigner symmetry. The mean fields of all but the spin-orbit force are small in nuclei. However, the importance of the spin-orbit force grows like A 1/3 , proportional to the number of particles in the maximum angular momentum shell. Therefore, for large A, the spin-orbit force is expected to overcome the 1/N 2 c suppression and destroy the approximate Wigner supermultiplet symmetry. It may be interesting to pursue this further, to determine at what values of A one might expect SU (4) W symmetry to fail.

Summary and Conclusions
The 1/N c expansion has been shown elsewhere to be a useful tool in analyzing the properties of baryons 2 ; the analysis presented here and in ref. [7] shows that it also provides a useful tool for understanding qualitative features of the nuclear force. In particular, we have computed the relative strengths of the various components of the N N interaction in the 1/N c expansion ( Table 1) and argued that the predicted patterns are reproduced in phenomenological models of the N N force (fig. 3). We also extended the argument of ref.
[7] that the approximate Wigner supermultiplet symmetry observed in light nuclei (see [7] for examples and references) is in fact understandable in terms of the 1/N c expansion. We are aware of no other explanations for this peculiar SU (4) W symmetry.
Aside from obtaining directly from QCD a qualitative explanation for the spin, isospin and tensor structure of the N N potential, it is hoped that the 1/N c expansion could serve as a guide toward better understanding the interactions of baryons with strangeness, where the experimental data is much poorer. To this end we have included the three flavor analysis in Appendix A. Our hope is that this could prove useful for understanding hypernuclei, as well as matter in extreme conditions where strangeness may play a significant role, such as in heavy ion collisions, or dense matter with kaon condensation [19] or hyperons.

Acknowledgments
We would like to thank G. Bertsch J.

Appendix A. Three Flavors
The quark operator basis for three flavors is denoted bŷ where Q = (u, d, s) and Q † are the creation and annihilation operators for the three light quark flavors, i = 1, 2, 3 and A = 1, . . . , 8. T a are the standard SU (3) matrices in the fundamental representation, normalized so that trT A T B = δ AB /2. These 1-quark operators act on a baryon state which is the completely symmetric tensor product (in spin ⊗ flavor) of N c quarks.
It is convenient to break the operator basis (A.1) for the SU (6) generators by separating Q = (u, d, s) into q = (u, d) and s. Under this decompositionŜ i ,T A andĜ ia break up into linear combinations of andŶ iα † andK α † which are the hermitian conjugates of Y iα and K α . For baryons with N c quarks, and strangeness of order one,Ĝ ia is of order N c ,Ŷ iα andK α are of order √ N c , andŜ i ud ,Î a ,Ŝ i s andN s are of order one [3]. Note thatŶ iα andK α are strangeness changing operators.
The (properly normalized) SU (6) generators √ 2Ĝ iA ,T A / √ 2 andŜ i / √ 3 are collectively denoted byΛ M . The operator basis for two and three flavors are summarized in Table 2. An expansion using the operator basis (A.2) gives us the predictions of the 1/N c expansion for three flavors, without assuming SU (3) symmetry. One can also impose SU (3) symmetry, which places additional restrictions on the final result. The results for two flavors are obtained by using only the operators S i , I a , and G ia . The results of the paper can be generalized to the case of three light flavors. The analysis is more complicated because one also has to include operatorsN s ,Ŝ i s ,Ŷ iα andt α that involve the s quark. We will simply give the final results here.
The 1/N c ∆L = 0 interaction is The N s term violates SU (3) symmetry, so its coefficient is proportional to SU (3) breaking in the baryon sector, which is parameterized by ǫ, a dimensionless number of order 0.3. It is clear from Eq. (A.3) that the N s term violates SU (6) symmetry but respects SU (4) symmetry, so that SU (6) violation is of order ǫ/N c , but SU (4) violation is of order 1/N 2 c . The ∆L = 1 interaction for three flavors is which is the three-flavor generalization of (3.9). Time reversal and parity invariance requires the coefficients in (3.9) to be of the form where the coefficient f ij r is a symmetric traceless tensor that depends on p p in and p p out . This is the three-flavor generalization of (3.11).