---

SKYRMIONS AND THE NUCLEAR FORCE

arXiv:nucl-th/9210015v1 20 Oct 1992SKYRMIONS AND THE NUCLEAR FORCENiels R. Walet and R. D. AmadoDepartment of Physics, University of Pennsylvania, Philadelphia PA 19104-6396The derivation of the nucleon-nucleon force from the Skyrme model is reexam-ined. Starting from previous results for the potential energy of quasistatic solutions,we show that a calculation using the Born-Oppenheimer approximation properlytaking into account the mixing of nucleon resonances, leads to substantial centralattraction.

We obtain a potential that is in qualitative agreement with phenomeno-logical potentials. We also study the non-adiabatic corrections, such as the velocitydependent transition potentials, and discuss their importance.21.30.+y, 13.75.Cs, 12.40.-y, 11.10.LmTypeset Using REVTEX1

I. INTRODUCTIONA central problem of strong interaction theory is to derive the nucleon-nucleon interactionfrom more fundamental principles. It is well established that quantum chromodynamics(QCD) is the correct theory of strong interactions.

The phenomenology of the nucleon-nucleon interaction suggests a description in terms of nucleons and the exchange of mesons,while QCD is a non-abelian gauge theory of quarks of three colors interacting by exchange ofgluons. These two pictures, one of quarks and gluons and the other of nucleons exchangingmesons are difficult to reconcile.

The notions of asymptotic freedom and its counterpartconfinement suggest that quarks and gluons are the appropriate degrees of freedom for shortdistances and high energy, but are not for the large distances and relatively low energiesof the nucleon interaction, where the quark picture becomes strongly non-perturbative.The problem is to make a theory, starting from QCD, that obtains the nucleons and theirmeson exchange. If such a theory agrees with the vast body of nucleon-nucleon interactionphenomenology, it will have, in some sense, “derived” traditional nuclear physics from QCD.This paper reports the details of an effort based on the Skyrme model [1,2] to make thatconnection.

A brief summary of these results have been presented elsewhere [3].The Skyrme model describes a nonlinear, classical field theory of interacting pions. Thisseems far from QCD, but in fact the Skyrme Lagrangian has been shown to be an effectivetheory of QCD in the large distance (long wave length) non-perturbative regime in the limitof a large number of colors (NC).

There is no unique path from such a classical theory backto quantum chromodynamics with three colors, but attempts to construct alternate classicalpictures have shown that there is surprisingly little leeway [4], and work to extract nucleonphenomenology from the Skyrme Lagrangian has been relatively successful with results thatare fairly insensitive to variations in details of the theory. Hence the Skyrme Lagrangianprovides a robust, QCD-based starting point for studies of nucleon and nucleon-nucleonproperties in the long wavelength, non-perturbative regime.The Skyrme Lagrangian possesses a topologically conserved charge, B, that Skyrme [1]identified with baryon number.

The B = 1 solutions of the Skyrme Lagrangian yield anappealing picture of the nucleon and its excited states, but since the theory is a large NCapproximation, corrections for real nucleon observables are expected to be of order 1/NCor ∼30%. An interesting and important problem area is the calculation of these 1/NCcorrections.

These are first quantum corrections, and their calculation faces the standarduncertainties found in going from any classical theory, particularly a non-linear field theory,to a quantum theory.The relative success of the Skyrme picture for the B = 1 system invites its extension toB = 2 and in particular to the problem of the nucleon-nucleon interaction. The phenomenol-ogy of the NN interaction suggests that, with the possible exception of the repulsive core,the interaction is well described in terms of multiple pion exchange.Thus the Skyrmemodel which expresses the physics in terms of interacting pions, seems to be a good startingpoint.

Skyrme himself [1] noted that for very large separation, the NN interaction reducesto one pion exchange in his model. The key question then is whether the model can alsogive the central mid-range attraction that is responsible for nuclear binding.There arethree problems that must be overcome in addressing this question.

The first is the solutionof the classical B = 2 field equations to obtain and describe B = 2 configurations. The2

second is the problem of obtaining nucleon-nucleon interactions from Skyrmion-Skyrmioninteractions. The third is the introduction of 1/NC corrections.

As we shall see the lasttwo are inexorably mixed. The need for careful 1/NC corrections is much greater in theB = 2 problem than for B = 1, because the only way presently known to calculate theinteraction energy is to calculate the energy of the total B = 2 system and then subtractthe energy of two free B = 1 Skyrmions.

This is a difficult, delicate, and dangerous processsince the interaction energy is a very small fraction of the total. In this paper we present aprocedure for carrying out the three steps.

Our procedure yields a NN interaction closelyresembling standard phenomenology. The interaction has a one pion exchange tail at longdistance, a short range repulsion, a tensor force and, best of all, a strong, mid-range centralattraction–all in substantive agreement with phenomenology.The problem of solving or even of describing the B = 2 interacting Skyrmion systemis complex.

There are only two known stable (time-independent) B = 2 configurations.One is the trivial one of two infinitely separated B = 1 Skyrmions, and the other is acompact bound state of toroidal baryon density in which the identity of the two individualSkyrmions is lost. To obtain an interaction energy one must define a set of intermediate,unstable, static Skyrmion configurations at finite separation.

Each configuration defines apoint on an adiabatic, six dimensional, collective manifold, called the unstable manifoldby Manton [5]. Each point on the manifold is stabilized by imposing a constraint.

Thedimensionality of this manifold makes it virtually impossible to calculate the interactionenergy at every point.Fortunately much of the interaction energy can be described bystudying only a very small part of that manifold. But on that sub-manifold the descriptionmust be accurate.As we emphasized above, the interaction energy is calculated as thedifference between large numbers.

Hence seemingly small inaccuracies in configurations canlead to large discrepancies in interaction energy. It is this problem that has plagued earlyattempts to study the NN interaction in the Skyrme model with the product ansatz.

Thisansatz, also first given by Skyrme [1], describes the B = 2 system as the product of twoB = 1 systems. The configurations so obtained are not solutions of the constrained non-linear field equations, and hence the energy they yield is above the true energy.

For verylarge separation these differences are not serious, but for intermediate and short distancesthey are fatal. For example the product ansatz cannot describe the toroidal lowest B = 2state.

Early efforts [6,7] to study the NN system with the product ansatz developed muchof the vocabulary and sense of the problem, but in the end failed to give any mid-rangeattraction. Surprisingly enough one finds a repulsive core in this ansatz, but it has beenshown that this does not have the correct g-parity [8].

A far better approach to the B = 2system is direct numerical solution of the constrained, non-linear, field equations. RecentlyWalhout and Wambach (WW) [9] have produced just such solutions.

They form the startingpoint of our work.How does one go from the Skyrmion-Skyrmion interaction energy calculated on theinteresting sub-manifold of the unstable manifold to the NN interaction?A commonlyused prescription is to project the Skyrmions to nucleons at each separation. This methodhas been used extensively in the early product ansatz work.Using the same approachWW find weak central attraction.

The nucleons only requirement is too strong, however.Phenomenologically all that is required is that the particles be nucleons asymptotically.When they approach and interact they can deform and be excited, for instance to the ∆3

state, as the dynamics dictates. This state mixing must be very important, since in thesolutions for the interacting Skyrmions found by WW, the individual Skyrmions deformstrongly as they interact, finally loosing their individuality completely in the toroidal state.If the time scales (or equivalently energy scales) for the radial motion and the excitation ofDeltas separate, the contribution of this distortion to the energy can be calculated in theBorn-Oppenheimer approximation.

Finite NC effects are critical in making the time scalesappropriately distinct. It is this combination of finite NC effects and Born-Oppenheimer thatis crucial to obtaining the phenomenological potential.

Each idea has been used before [10],but it is the combination of exact solution of the classical field equations, state mixing usingthe Born-Oppenheimer approximation and finite NC effects that together makes physicalsense and gives the phenomenologically correct potential as we shall see below. We shallfirst treat the effects of the distortions perturbatively, where it is clear that the distortionmakes an attractive contribution.

Since the contribution of the distortions becomes large atsmall distances, we turn to diagonalization of the Born-Oppenheimer Hamiltonian to checkthe perturbative results. Once we make the Born-Oppenheimer separation we can calculateexplicit corrections to the adiabatic picture.

These take the from of transition potentialsbetween the different adiabatic channels, as well as some velocity dependent potentialssimilar to Berry’s phase terms.In Section II we set out some key notions.We review the Skyrme Lagrangian, theadiabatic manifold for interacting B = 2 Skyrmion systems, review the algebraic method forprojecting baryon states from Skyrmions and for introducing finite NC effects, and show howthese algebraic methods permit a description of the Skyrmion-Skyrmion interaction on theunstable manifold. This section brings together the known tools we need for the problem.Section III presents our derivation of the NN interaction from the Skyrmion interaction forfinite NC.

It gives the adiabatic interaction derived first in perturbation theory and thenusing the Born-Oppenheimer approximation. The formalism for obtaining the transitionpotentials is also given here.

Our results are presented in Section IV. As we have discussedabove, the potentials are close to static NN potentials obtained from the data.

Section Vgives a brief summary and conclusions. Some of the algebraic details of the perturbationtreatment are presented in the appendix.II.

SOME KEY NOTIONSA. The Skyrme modelThe Skyrme model is a non-linear field theory that can be realized in terms of an SU(2)-valued matrix field U, with Lagrangian densityL = f 2π4 Tr[∂µU(x)∂µU†(x)] +132g2Tr[U†∂µU, U†∂νU]2.

(1)The model is covariant, as well as invariant under global SU(2)-rotations that are identifiedwith the isospin symmetry. As was discovered by Skyrme the model has a topologicallyconserved quantum number, which is identified as the baryon number B.

The U field isinterpreted as a combination of a scalar σ field and an isovector pion field, U = σ + i⃗τ · ⃗π.The σ field is not an independt physical field due to the unitarity constraint on U.4

The standard time-independent solution to the classical field equations for B = 1 is thedefensive hedgehog, where the pion field points radially outward,U1(⃗r) = exp(i⃗τ · ˆrf(r)). (2)The baryon number of this state is given by B = (f(0)−f(∞))/π = 1.

This solution breakstranslational invariance, as well as the O(4) spin-isospin symmetry. If we perform a globalSU(2) isorotation on the state,U1(⃗r|A) = A†U1(⃗r)A,(3)we obtain a state of the same energy.

One can easily show that an isorotation A has thesame effect on U as performing an ordinary rotation R[A−1] on ˆr, so that the grandspin K,the sum of spin and isospin, is a good quantum number, with eigenvalue 0,⃗K = ⃗S + ⃗I. (4)In the B = 2 system we shall frequently use the product ansatz.

This ansatz makesuse of the fact that the product of two B = 1 solutions has baryon number two. The mostgeneral ansatz we can therefore construct from two hedgehogs consists of the product of tworotated hedgehogs, where the centers have been translated,U2(r|⃗RAB) = A†U1(⃗r −⃗R/2)AB†U1(⃗r + ⃗R/2)B= U2(r|⃗RCD)= D†C1/2†U1(⃗r −⃗R/2)CU1(⃗r + ⃗R/2)C1/2†D.

(5)In the last of (5) we have introduced the matrix D, that describes the rigid isorotation ofthe whole system, as well as a relative isorotation C. When R is very large changing C orD does not change the energy of the solution. For smaller R, D still generates a zero-mode(corresponding to broken isospin symmetry), but the energy will depend on C. Again, theenergy is also invariant under spatial rotation, due to the conservation of angular momentum⃗J = ⃗L + ⃗S.B.

Determination of manifoldAfter having constructed all the constrained B = 2 solutions, we construct a manifoldthat contains all these solutions.This method, which is well know in nuclear structurephysics as the adiabatic large amplitude collective motion [11], seems first to have beenintroduced in the theory of interacting Skyrmions by Manton [5], under the name unstablemanifold. His discussion is based on the deep geometric results by Atiyah and Hitchin forthe interaction of BPS monopoles [12], which make an identification of the structure of themanifold very clear but do not give many tools for constructing it.One of the ways to solve for the adiabatic collective manifold would be to solve a localfluctuation equation.

This is a very complex problem for a field theory, but even whenwe reduce the complication by selecting only a finite number of degrees of freedom as inthe Atiyah-Manton ansatz [13,14] this remains a formidable (but interesting) task. The5

minimal set of degrees of freedom can be identified without performing any calculation,however. There are six trivial ones, consisting of the three center of mass coordinates, aswell as global rotations of the whole system (in both space and isospace), which we shallignore in the present discussion.

For large separations we find that an additional six degreesof freedom are necessary to describe the product ansatz (5): the relative separation ⃗R (threecoordinates) and three more coordinates describing the relative orientation (as discussedin the previous subsection usually represented by the U(2) matrix C = c4I + i⃗c · ⃗τ, wherec24 + ⃗c2 = 1). The minimal degrees of freedom are just the continuously deformed set ofsix parameters as we take R to smaller values.

The meaning of the word “adiabatic” inadiabatic manifold is that the motion in the manifold consists of the slowest modes. Thisonly makes sense if at any point all the unstable directions lie within the collective space,so that no lower energy mode lies outside the manifold.

An analysis of the unstable B = 2hedgehog by Manton [5] shows that it is plausible that this state has six unstable modes,again corresponding to relative separation (three modes) and a relative rotation (anotherthree). Since this hedgehog is supposed to be part of the collective surface, we again find aminimum of six collective modes.Once one determines the collective surface there exist standard techniques [11] to de-termine the collective Hamiltonian, with the restriction that it is quadratic in momenta.Schematically it takes the formH = K(R, ˆR, C, pR, Li, pCj) + 12IiMijIj + 12JiNijJj + V (R, ˆR, C)(6)Here pR is the momentum associated with R, L the orbital angular momentum and pC isshorthand for the momentum associated with C. The symbols Ii and Ji denote the intrinsiccomponents of the (conserved) isospin and angular momentum, respectively.

The kineticenergy is closely related to the geometric structure of the collective surface, and can bevery complicated. One example that has been studied extensively is the case of the BPSmonopole, where one has only kinetic and no potential energy.It has been proven [12]that the underlying geometry is reasonably complicated and allows for non-trivial scatteringof monopoles.

In the current work we concentrate on the potential energy, however. Weassume that the kinetic terms (including the contributions of I and J) take the formK =12M ⃗p2R + 12I (C12 + C22),(7)where Ci2 denotes the O(4) Casimir operator 12(S2i + I2i ).

The only correction to this thathas been studied is the R and C dependence of M [20], but we ignore that in this paper. Itwill be interesting to return to this problem in the context of the NN interaction.To calculate the manifold beyond the product ansatz we use the fact that for specialchoices of C, due to symmetry, there exist three stable situations [14] – the stable attractivechannel, the marginally stable hedgehog-hedgehog and repulsive channels – where a pathof the collective surface is traced out for fixed C. In each of these channels (labeled byi for convenience) one calculates the classical potential energy Vi.The identification ofthe symbols R in each of the three Hamiltonians as one and the same radial coordinateis a bit precarious.There is no apparent reason why we cannot separately redefine theradial coordinate in each channel (say, by a point transformation).

Here the topology comes6

into play however. Since each channel corresponds to a definite choice of C, the three pointtransformations can be embedded in a single transformation that depends on both R and C,so that we are just making a change of coordinate transformation on the manifold, by whichwe do not gain much.

In order to sensibly identify the R’s in the same channel we shoulduse geometric notions. First note that each of the three channels i seems to correspond to ageodesic line on the collective surface, due to the extra symmetry of the pion field.

In orderto be able to make a meaningful identification of points on the three lines as having the sameR the three points should lie on a single geodesic surface on the manifold, that crosses thethree geodesic lines i orthogonally. This is of course extremely difficult to implement, andwe do not know of a good way of doing this.

In most discussions of the Skyrmion-Skyrmioninteraction, including the present one, these problems are ignored, and a special definitionof R is made that is then used to identify the different channels.C. The algebraic modelHaving determined the functions Vi we need to understand their relation to the nucleon-nucleon potential.

The usual approach is to return to the product ansatz, where one makesvery strong assumptions about the form of the two-Skyrmion solution. Again, this form isexact in the large-R limit.

The advantage of this form is that it guides us into a partic-ular expansion of V in terms of functions ⃗c · ˆR, c4 (the quaternion c specifies the relativeorientation). This expansion is closely related to an algebraic model of the Skyrmion [15],which in the large-N limit behaves in exactly this manner.

There is no a priori reason whythis form should be restricted to the product ansatz, however. As we have argued abovethe coordinates of the collective surface are in one-to-one correspondence with those of theproduct ansatz, so that the current problem can again be modeled by a similar form [17].Let us now discuss the algebraic model [18,15] that we use.

Originally it was introducedto give an algebraic alternative to the rotational zero-mode quantization of the Skyrmion.Since the Skyrmion breaks SO(4) ∼SUI(2) × SUS(2) symmetry, but is invariant under aSO(3) subgroup, we find that we can describe each of the equivalent configurations by apoint of on the sphere S3 ∼SO(4)/SO(3) [19].We can now introduce a bosonic model with dynamical symmetry SU(4), which in thelimit of large boson number is also described by a point on the sphere S3. Without discussingthe hamiltonian in any detail, we can draw some interesting conclusions.

For any N (thenumber of bosons) we find that, as in the Skyrme model, all states have I = S.Thespectrum of the algebraic model is cut offat I = N/2, however. Since we believe that inthe real world, where NC = 3, we have a cut-offat isospin 3/2 for the delta, we impose thecondition that the number of bosons N is equal to the number of colors NC.

If we makethis identification one can immediately show that for N = NC = 3 the algebraic model isidentical to the non-relativistic quark model without radial excitations. It has been shownthat this model can be used to extract finite NC corrections to observables.

Of course wequantize only a few coordinates and we can only calculate a very limited set of finite NCcorrections, corresponding to the zero-mode quantization. One may hope, however, thatthrough a judicious choice of collective coordinates we have found the most important, andmaybe even the largest, contributions.In this paper we are not interested in the single Skyrmion case, but rather in the the case7

of two Skyrmions. This has been discussed in quite some detail for the case of the productansatz in Ref.

[15]. The algebraic model used consists of two sets of u(4) algebras, one foreach Skyrmion, as well as a radial coordinate ⃗R.

The interaction of the two Skyrmions canbe expanded in terms of three operators, the identity I, and the operators W and Z:W = T αpiT βpi/N2C,Z = T αpiT βpj[3 ˆRi ˆRj −δij]/N2C. (8)Here α and β label two different sets of bosons, used to realize the u(4) algebras, and T isa one-body operator with spin and isospin 1.

The semi-classical (large-NC) limit of theseoperators can be given in terms of ˆR and C = c4I + ⃗c · ⃗τ as [15]:Wcl = 3c24 −⃗c2,Zcl = 6(⃗c · ˆR)2 −2⃗c2. (9)If we expand the Skyrmion-Skyrmion interaction we findv(R) = v1(R) + v2(R)Wcl + v3(R)Zcl + higher order.

(10)Fortunately studies of the B = 2 system with the product ansatz [6,7,15], the Atiyah-Mantonansatz [13,14] and with direct numerical integration of the equations [9] have shown thatmuch information about the B = 2 energy as a function of separation R is contained in theparticular choice of three paths discussed earlier, corresponding to three definite choices ofthe relative orientation, or equivalently to a special symmetry of the pion field. This allowsus to calculate an approximate form for the interaction energy of two Skyrmions whichconsist of the first three term of (10) (the form used in Ref.

[9] is fully equivalent), whichcorrespond to an expansion up to first order only in terms of the two scalar operators Wand Z. One of the important consequences of the explicit realization of these operators isthat now algebraic methods [15,18] can be used to obtain their explicit finite NC realization.In this way we can include some (arguably the most important) finite NC corrections inthe calculation.

This includes the well-known amplification of the nucleon-nucleon matrixelement of T by the factor (NC + 2)/NC.III. THE ADIABATIC INTERACTIONThe problem we address here is how to obtain a nucleon-nucleon interaction which in-cludes some finite NC effects, from (10).

To date most workers have simply sandwiched(10) between states containing two nucleons for each value of R and called that the interac-tion. This is definitely not the nucleon-nucleon interaction determined phenomenologicallyfrom scattering phaseshifts, where we can only require that the state contains two nucle-ons at large separation.

For shorter distances, when the hadrons are interacting they canbe (virtually) what ever the dynamics requires, for example ∆’s. We know [14] that thebaryon density is strongly deformed for moderate R – a deformation that finally results inthe toroidal shape with its 70 MeV of binding.

That configuration corresponds to a greatdeal of mixing between all states of the individual Skyrmions. This admixture will lead toadditional attraction at intermediate distances.

The purpose of this paper is to calculate8

the effect of this mixing on the N-N interaction for NC = 3 starting with (10). The needfor state mixing has been realized before [21] in the context of the product ansatz, but thatansatz is a poor starting point.

Furthermore, as we state below, state mixing only makessense for finite NC. At the same time there are algebraic finite NC corrections to matrixelements that are difficult to include in the formalism of [21].

For these reasons, these earlyattempts to introduce state mixing did not find sufficient central attraction. We shouldremark here that a very different formalism, based on a study of fluctuations around theproduct ansatz [22] has been used to show that some attraction exists in two-pion range.Unfortunately the expansion underlying this approach fails in the region that is most crucialto central attraction; 1 fm < R < 2 fm.For large NC, one can distinguish two energy scales or reciprocally two time scales.

Theslower time scale is associated with the motion in the collective manifold, i.e., R and theorientation. The other time scale corresponds to the almost instantaneous response of thepion field to changes in R and the relative isospin orientation.

For large NC we cannotseparate the time scales for the two sets of adiabatic modes, as can be seen in the highlycorrelated doughnut. For NC equal to three the situation changes.

The R motion is typicallymuch slower than the rotational motion which leads to the separation of the nucleon and ∆states. We thus have three energy scales, that of the pion field, of the N −∆separation andof the R motion.

We can now calculate a Born-Oppenheimer potential for the R-motion,which constitutes the slowest degree of freedom.A. Perturbative approachAll of the effects of quick response of the pion field at each R are already in Eq.

(10). Itis the effect of the rotational states we want to include.

As a guide, we begin by studyingit perturbatively. Let us call the full potential between two nucleons, including the effectsof the rotational excited states, V .

In terms of the Skyrmion interaction v of (10) and tosecond order, V is given byV (R) = ⟨NN|v(R)|NN⟩+Xs′ ⟨NN|v(R)|s⟩⟨s|v(R)|NN⟩ENN(R) −Es(R). (11)(Here ENN is the two-nucleon energy and ES is the energy of the relevant excited state.

)The first term on the right is the direct nucleon-nucleon projection of v and is the termthat has appeared in the literature. The second term is the correction due to rotational orexcited states.

It is clear from the energy denominator that the second term is attractive.For NC = 3 the states |s⟩that can enter are |N∆⟩, |∆N⟩and |∆∆⟩. Except for a centrifugalterm we shall add later, the excitation energy is assumed to be independent of R, and thuscan be expressed in terms of the N-∆energy difference, 300 MeV.

It is here that we areneglecting the effect of the position dependence of the inertial parameters.The lowest order potential has been calculated by several groups using several levels ofapproximation to the Skyrme model [7,9,14]. The form found (or assumed), however, canalways be cast in the form of the algebraic model given by [15].

This allows us to study boththe large-NC limit, as well as to include finite NC effects explicitly in a systematic way. (Thereader should note that the results in [9] include corrections for the tensor and spin-spininteractions.

)9

The leading term in this expansion is given by the form (cf. (10))v(⃗R) = v1(r) + v2(R)W + v3(R)Z( ˆR).

(12)The algebraic operators W and Z have very simple expectation value for nucleons, (see [15])⟨NN| W |NN⟩= 19P 2N ⟨NN| σ1· σ2τ 1· τ 2 |NN⟩,⟨NN| Z |NN⟩= 19P 2N ⟨NN| (3σ1· ˆR σ2· ˆR −σ1· σ2)τ 1· τ 2 |NN⟩,(13)i.e., apart from a finite NC correction factor P 2N (PN = 1 + 2/NC) they represent the spin-spin and tensor interactions. If we use this form we find that the lowest order interaction inthe nucleon-nucleon channel is given byV (0) = v1 + v2P 2N9σ1· σ2τ 1· τ 2 + v3P 2N9(3σ1· ˆRσ2· ˆR −σ1· σ2)τ 1· τ 2.

(14)If we wish to evaluate the perturbation correction (11) we must evaluate the two setsof matrix elements ⟨NN| ˜v(⃗R) |N∆⟩⟨N∆| ˜v(⃗R) |NN⟩and ⟨NN| ˜v(⃗R) |∆∆⟩⟨∆∆| ˜v(⃗R) |NN⟩separately.Details are discussed in appendix A, but the final result for the first ordercorrection to the NN interaction isV (1)PT = −Q2Nδ {[ 13Q2NP τ0 + ( 1627P 2N + 527Q2N)P τ1 ](v22 + 2v23) +(σ1· σ2)[−118Q2NP τ0 + ( 1681P 2N −5162Q2N)P τ1 ](v22 −v23)(3σ1· ˆRσ2· ˆR −σ1· σ2)[−118Q2NP τ0 + ( 1681P 2N −5162Q2N)P τ1 ](v23 −v2v3)}. (15)Here QN is an additional finite NC correction factorQN =q(1 −1/NC)(1 + 5/NC),(16)δ is the N-∆energy difference, and P τT is a projection operator onto isospin T.B.

Diagonalization of the interactionThe perturbative expansion discussed in the previous subsection just gives a rough es-timate of the effect of state mixing in the Born-Oppenheimer approximation. To solve forthe Born-Oppenheimer Hamiltonian the total collective Hamiltonian is first divided into twoparts, an intrinsic part and a part that only describes the radial motion,H = −¯h22M R2∂RR−2∂R + ˜H(R).

(17)The hamiltonian ˜H(R) is nothing but the sum of the potential v(R) we used in the pertur-bative calculation, plus the kinetic term used to calculate the energy difference in the energydenominators,K = 12Λ(I21 + I22) +¯h22MR2L(L + 1). (18)10

We now introduce the instantaneous eigenstates of the intrinsic Hamiltonian. For fixedR we diagonalize ˜H,˜H(R) |i⟩R = Ei(R) |i⟩R .

(19)The standard Born-Oppenheimer approach is to start at large R, where the states are pureNN, and follow the continuous energy curve as R becomes smaller to define the adiabaticpotential energy. This is basically the procedure we follow, apart from a few additionalcomplications encountered if we have more than one NN state in any given channel.

Thisis solved by using a model Hamiltonian to determine the unmixed adiabatic eigenstates, aswill be discussed below.Having defined the adiabatic eigenstates, the mathematically correct way to derive theadiabatic Hamiltonian is to decompose a general time-dependent solution of the Schr¨odingerequation in adiabatic eigenstates,|Ψ(t)⟩=Xiai(R, t) |i⟩R . (20)If we substitute this in the time-dependent Schr¨odinger equation, and multiply from the leftwith R⟨i|, we findi˙aiδij = δij −¯h22M1R2∂RR2∂R + Ei(R)!ai+−¯h2MXjR ⟨i| ∂R |j⟩R ∂Raj(R, t)+XjR ⟨i| −¯h22M1R2∂RR2∂R |j⟩R aj(R, t).

(21)This expression exhibits both the adiabatic Hamiltonian (the first term on the right-handside), and the corrections. If we assume there are no diagonal terms of ∂R between adiabaticeigenstates (since there are no closed circuits in R space, it seems a reasonable assumption toignore Berry’s phase), the first correction describes a velocity dependent transition potential.The next term gives a non-adiabatic correction to the diagonal adiabatic potential, as wellas a velocity independent transition potential.1.

The potential in angular momentum coupled formTo evaluate (11) it is convenient to introduce states |I1I2LSJT⟩labeled by the conserved(total) isospin T and angular momentum J, as well as the NN channel orbital angularmomentum L and spin S. Since we wish to evaluate the matrix elements of ˜H(R) in thebasis |I′1I′2LSJT⟩we need to know the matrix elements of the potential v in this basis. Afterperforming some standard angular momentum algebra, we find that⟨I1I2LSJT| v |I′1I′2LSJT⟩= v1δSS′δLL′δI1I′1δI2I′211

+v29 (−1)S+TδSS′δLL′(I1 I2 SI′2 I′1 1)(I1 I2 TI′2 I′1 1)⟨I1||| T (11) |I′1⟩⟨I2||| T (11) |I′2⟩+v39√30(−1)L+L′+S+J+T+I2+I′1 ˆLˆL′ ˆS ˆS′× L 2 L′0 0 0! (SL JL′ S′ 2)(I1 I2 TI′2 I′1 1)I1 I2 SI′1 I′2 S′112⟨I1||| T (11) |I′1⟩⟨I2||| T (11) |I′2⟩.

(22)The relevant reduced matrix elements of T are⟨N||| T (11) |N⟩= −10,(23)⟨∆||| T (11) |∆⟩= −20,(24)⟨∆||| T (11) |N⟩= −8√2. (25)The matrix elements of the kinetic part are taken to be very simple,⟨I1I2LSJT| K |I′1I′2L′S′JT⟩= δI1I′1δI2I′2δLL′δSS′ 300[I1 + I2 −1/2] + L(L + 1)2MI1I2R2!.

(26)To account for the effects of centripetal repulsion at small R, we have added the centrifugalenergy, ¯h2L(L + 1)/2MI1I2R2, to channel energies. We take MI1I2 to be the reduced mass inthe relevant channel,MI1I2 =MI1MI2MI1 + MI2,(27)where M1/2 = 932 MeV and M3/2 = 1232 MeV.For sake of comparison we shall also need the matrix elements of a model nucleon-nucleoninteraction in the same channel (we have suppressed the spin and isospin projection quantumnumbers),⟨NNLSJT| V Tc + V Ts σ1 · σ2 + V Tt σ1i σ2j(3 ˆRi ˆRj −δij) |NNL′S′JT⟩= δSS′δLL′ V Tc + V Ts (−1)1+S6(1/2 1/2 S1/2 1/2 1)!+6√30V Tt (−1)L+L′+S+J ˆL ˆL′ ˆS ˆS′ L 2 L′0 0 0!

(LS JS′ L′ 2)1/2 1/2 S1/2 1/2 S′112. (28)Here we have chosen to use a parametrization identical to the one used in the discussion ofperturbation theory.In order to determine an effective potential from the analysis given above we perform acalculation that is standard in the Born-Oppenheimer approximation.

We assume that ateach R we can diagonalize the potential. We then follow a state from large R, where it ispurely made up out of nucleons, to small R where we have mixing.

This gives the usualadiabatic potential curves of molecular physics. A complication will be that we may havemore than one nucleon state for given quantum numbers.

We shall see that we can still fitthe interaction in that case.12

The model space splits into two parts that are either symmetric or antisymmetric underinterchange of the particles [23]. We select the antisymmetric part where the NN statessatisfy the standard selection rule L + S + T is odd.

Of course, in contrast to the lowestorder perturbative result, we will now find a channel dependent potential.The fact that we have a different reduced mass in each channel will also lead to a positiondependent effective mass. Suppose that c2I1I2(R) gives the percentage of the I1I2 state inthe diagonalization of the Born-Oppenheimer Hamiltonian.

In that case the average radialkinetic energy isXI1I2−¯h22MI1I2c2I1I2(R) 1R2∂RR2∂R,(29)and we can define a position dependent mass byM(R) = 1/XI1I2c2I1I2(R)MI1I2.(30)2. Derivation of the potentialLet us first consider the case T = 0.

A little analysis shows that in this case the twochannels Jπ = 1+ and Jπ = 1−give three two-nucleon states, enough to determine the threeindependent functions in (28). (There are no NN states with T = 0 and J = 0.

)For Jπ=1+ we have the two possible states |NNL = 0S = 1J = 1T = 0⟩and|NNL = 2S = 1J = 1T = 0⟩. The matrix elements of (28) in this space take the form V 0c + V 0s2√2Vt2√2VtV 0c + V 0s −2V 0t!,(31)with eigenvaluesE1 = V 0c + V 0s + 2V 0t ,E2 = V 0c + V 0s −4V 0t .

(32)These numbers should be fitted to the two lowest eigenvalues of the diagonalization ofthe algebraic potential in the complete nucleon-delta space. If we also look at the state|NNL = 1S = 0J = 1T = 0⟩, with matrix element of (28) equal toE3 = V 0c −3V 0s ,(33)(again, this is equated to the lowest eigenvalue of the complete diagonalization) we findV 0c = E1/2 + E2/4 + E3/4,(34)V 0s = E1/6 + E2/12 −E3/4,(35)V 0t = (E3 −E2)/6.

(36)A similar calculation for T = 1 involves the three channels Jπ = 0+, 0−, 1−(againL + S + T =odd). The relevant matrix elements of (28) are13

E′1 ≡⟨NNL = 0S = 0J = 0T = 1| V |NNL = 0S = 0J = 0T = 1⟩= V 1c −3V 1sE′2 ≡⟨NNL = 1S = 1J = 1T = 1| V |NNL = 1S = 1J = 1T = 1⟩= V 1c + V 1s + 2V 1tE′3 ≡⟨NNL = 1S = 1J = 0T = 1| V |NNL = 1S = 1J = 0T = 1⟩= V 1c + V 1s −4V 1t . (37)ThusV 1c = E′1/4 + E′2/2 + E′3/4,V 1s = −E′1/4 + E′2/6 + E′3/12,V 1t = (E′2 −E′3)/6.

(38)We subtract the centrifugal force from our potentials, since in general this is not taken tobe included in the potential itself. We had to include it in our kinetic term, since we makethe Born-Oppenheimer separation between the radial motion and everything else, includingthe orbital motion.IV.

RESULTSAs stated before, for each total isospin, T = 0, 1 we write V in the |NNLSJT⟩space as⟨NNLSJT|V Tc + V Ts σ1 · σ2 + V Tt σ1i σ2j (3 ˆRi ˆRj −δij)|NNL′SJT⟩. (39)It is the V Tc , V Tsand V Tt that we study.

We have extracted the necessary values of v1, v2, v3from the work of Walhout and Wambach [9].This is an uncertain process for large Rsince the potentials are quite small in this region and are calculated in [9] on a lattice byfirst calculating the total interaction energy of the B = 2 system and then subtracting therest energy of two free Skyrmions. We wish to compare the Skyrme based potential witha realistic nucleon-nucleon interaction.

One cannot relate our result to modern effectivepotentials, such as the Bonn [24] and Paris [25] potentials, since their central parts containexplicit momentum dependent terms. For that reason we compare our potentials to the Reidsoft core (RSC) potential [26].In Figs.

1 and 2 we show the central potentials V Tc calculated from Eq. (11), first fromthe first term on the right of (11) only (this is the result of Ref.

[9]) and then with theBorn-Oppenheimer method.Note we only show the result for intermediate R, 1 fm

The nucleons onlyresult from [9] is independent of T and shows a weak attraction. The perturbation result(15) gives considerably more attraction, which is in turn somewhat reduced by the full Born-Oppenheimer (BO) diagonalization.

This reduction comes mostly from the centrifugal termsin the kinetic energy which were not included in the perturbative calculations, and partlyfrom the higher order effects included in the diagonalization. The centrifugal terms reducethe mixing at small R, and hence lead to less central attraction.

All the results, nucleonsonly, perturbation theory and BO diagonalization agree at large R. This reflects the factthat state mixing involves multiple pion exchange and hence has intermediate range. Also14

shown in Figs. 1 and 2 are the T = 0 and T = 1 components of the Reid soft core potential.They have less central attraction than we find.

As we shall see that reflects a rather differentshuffling of attraction between the central and the spin dependent interactions between theRSC and the Skyrme approach. But even at this level the fact that the combination of carefulsolution of the B = 2 sector, state mixing and finite NC effects gives too much attraction isa welcome change from the many previous attempts to derive the NN interaction from theSkyrme approach which always found too little central attraction.In Figs.

3 and 4 we show the T = 0 and T = 1 spin-dependent potentials. In thesecases the nucleons-only potential, the perturbative results and the full diagonalization areall quite similar, but now the RSC is stronger than the Skyrme result, rather than weaker asit was in the central potential.

This is the shuffling we referred to above. Figs.

5 and 6 showthe tensor force for both values of isospin. Except for T = 1 at small R all the Skyrmioncalculations and even the RSC agree.

This reflects the dominance of one pion exchange forthe tensor potential.We see that though the effect of the Born-Oppenheimer corrections are modest for largeR they become large for intermediate R in the central channel, but remain small in the otherchannels. The reason for this is that the corrections are small for all channels compared tothe typical size of the potential, but that the lowest order central potential is much smallerthan this typical size.We saw in Figs.

1–4 that the RSC gives less central spin-independent attraction than wefind but a stronger spin-dependent potential. It is therefore instructive to compare the morephysical S-channel L = 0 potentials.

In Fig. 7 and 8 we show the 3S1 (T = 0) and the 1S0(T = 1) potentials.

Now we see that the full Born-Oppenheimer potential looks quite similarto the RSC while the one obtained without mixing does not resemble the RSC. It is in thissense that we say that the present results give a credible account of the nucleon-nucleoninteraction at intermediate range.We now turn to a more detailed analysis of the Born-Oppenheimer approximation, theadiabatic approximation and non-adiabatic corrections.

We analyze a representative case,the J = 1−, T = 1 channel, since it mixes NN, N∆and ∆∆states. The relevant statesin the diagonalization are |NN, L = 1, S = 1⟩, |N∆, L = 1, S = 1⟩, |N∆, L = 1, S = 2⟩,|∆∆, L = 1, S = 1⟩and |∆∆, L = 1, S = 3⟩.

In Fig. 9 we plot the energy of the adiabaticeigenstates, which remain distinct for all separations, except for a narrow avoided crossingnear 1.2 fm.

For this channel all the energies are repulsive. In Fig.

10 we show the percent-age of each component in the lowest adiabatic eigenstate. It is this state that was used inthe determination of the NN potential.

One can see that the NN component is dominant,and we have only small admixtures of the other components. For small R it is the thirdadiabatic eigenstate that admixes most strongly.In Fig.

11 we show the effective mass of the lowest adiabatic eigenstate as defined byEq. (30).

Since the only effect on the masses we include is the centrifugal term, the massincreases with decreasing R. Preliminary calculations of the mass parameters for the classicalB = 2 system give mass parameters that decrease with decreasing R, at least in the attractivechannel [20]. Thus the results of Fig.

11 should be taken only to remind us that the Skyrmeapproach leads to a picture of nucleons that distort and change their mass as they interact.Corrections to the adiabatic approximation lead to transition potentials among the adi-abats. The relevant matrix elements can be calculated exactly using standard algebraic15

techniques:R ⟨i| ∂R |j⟩R =R ⟨i| (∂R ˜H(R)) |j⟩R /(Ei(R) −Ej(R)). (40)We further assume that the (imaginary) diagonal matrix element is zero, which correspondsto a consistent real choice of |j⟩R.

The second derivatives can now be evaluated using asum over intermediate states. As argued in the previous section we have two terms, a localtransition potential and a velocity dependent potential.

We continue to study the J=1−,T = 1 channel and consider the transition from the NN adiabat to each of the five others.The local potentials for these transitions are shown in Fig. 12 as a function of R. They areon average much smaller than the difference between the relevant adiabatic eigenvalues, asthey should be if the adiabatic approximation works.

Near R = 1.2 fm there is a spectacularexcursion of the velocity dependent potential in Fig. 12.

This can be traced back to thenarrowly avoided level crossing in Fig. 9.

This picture would suggest that at high NN energythere should be the onset of strong ∆production. The theory is still too crude to makea detailed prediction, but this aspect of NN dynamics suggested by the Skyrme approachdeserves further investigation.

The velocity dependent potential are a little more difficult tointerpret. We have chosen to plot, in Fig.

13, the potential such that we have to multiply itby −i times the scaled radial velocity −iβ = v/c =¯hMc∂R. As can be seen this potential isvery small for non-relativistic velocities.V.

SUMMARY AND CONCLUSIONSHere we have shown that the Skyrme model can give strong mid-range nucleon-nucleonattraction that is in qualitative agreement with phenomenological potentials. This is in sharpcontrast to early results based on the product ansatz.

Two components play a key role inobtaining the attraction. First one must pay attention to the non-linear nature of the SkyrmeLagrangian, as shown in Refs.

[9,14], but second one must also include configuration mixingat intermediate distances as we have stressed here. This mixing is not easily formulatedin the large NC limit, but is easily included for NC = 3, and hence brings with it someother finite NC effects.The combined effect of the careful treatment of the non-linearequations and the configuration mixing is to give substantial central mid-range attractionfor the NN system that is in qualitative agreement with the data.

To go from this workto a theory that can be confronted with experiment in detail is a difficult challenge. Thereare R dependent corrections to inertial parameters to include, there are dynamical quantumcorrections, there are non-adiabatic effects that are particularly important at small R, andthere are other mesons to include in the Skyrme Lagrangian.

All these are under study. Thesuccess so far encourages us to proceed and to suggest that the results obtained so far willbe robust under these refinements.The success of the Born-Oppenheimer approach to the nucleon-nucleon interaction in theSkyrme model raises interesting questions about what “nucleons” are when they are interact-ing.

We find that the state that is two nucleons at large separation becomes, adiabatically, astate with delta components as the nucleons approach. For the Skyrmion-Skyrmion systemitself this distortion and deformation under interaction is even more dramatic.

What thenshould we say about the interacting nucleons? That there is a big delta component in their16

wave function? That they are distorted by the interaction?

In some sense this is true, butin the spirit of the Born-Oppenheimer approximation, it is not. The adiabatic potentialderived from the Born-Oppenheimer prescription involves state mixing, in fact much of itsattraction comes from that mixing, but that adiabatic potential is to be used as an effectivepotential between nucleons.

Think of the van der Waals potential between hydrogen atoms.It comes from virtual excitation of the atoms, principally to the first L = 1 state, but thepotential is to be used between atoms in their ground state. One usually does not talk aboutthe percent of atomic L = 1 state in the hydrogen molecule.

Following that example, weshould not talk about the percent of delta in the state of two interacting nucleons. On theother hand, one could take apart the Born-Oppenheimer calculation and quote such a per-cent.

It is really a matter of what basis one uses. Many discussion of delta’s in the nucleusare unclear about this ambiguity.

Our calculation helps to emphasize the perspective de-pendence of these discussions. We plan to return to this topic both for the nucleon-nucleonscattering system and for the deuteron in a subsequent paper, as well as to the question ofthe interaction effects on the nucleon mass parameter.

We have also seen that correctionsto the adiabatic approximation make it possible to compute transition potentials. Thesecan be used to calculate Delta production in NN collisions.

In this paper we have onlyscratched the surface of this interesting problem.In summary we have shown that including the effects of channel coupling (∆mixing)in the projection of the nucleon-nucleon interaction from recent studies of the potentialin baryon number two Skyrmion systems, substantially increases the strength of the mid-range central attraction bringing it into qualitative agreement with experiment. To obtainthis result we introduce some quantum corrections (equivalently corrections for the finitenumber of colors).

Many more sophisticated quantum corrections remain to be made. Butour results show that a non-perturbative approach to the problem of obtaining the “static”nucleon-nucleon interaction from QCD based on the Skyrme approach has great promise.We are therefore encouraged to follow up the first success reported here.ACKNOWLEDGMENTSWe wish to acknowledge Atsushi Hosaka for his contributions in the initial stages of thiswork.

We also would like to thank Jochem Wambach for stimulating discussions.This work was partially supported by the U.S. National Science Foundation.APPENDIX A: DETAILS OF THE PERTURBATIVE EXPANSIONWe exhibit details of the calculation for the first matrix element, for the part of ˜Hproportional to W. We again make use of an equation from [15], Eq. (30).⟨NN| W |N∆⟩⟨N∆| W |NN⟩= ⟨N| T αpi |N⟩⟨N| T αqj |N⟩⟨N| T βpi |∆⟩⟨∆| T βqj |N⟩= ⟨N| T αpi |N⟩⟨N| T αqj |N⟩(⟨N| T βpiT βqj |N⟩−⟨N| T βpi |N⟩⟨N| T βqj |N⟩)= ⟨NN| N4C9 τ 1p σ1i τ 1q σjP 2N{ 12Q2N( 23δij −i3ǫijlσ2l )( 23δpq −i3ǫpqrτ 2r )} |NN⟩17

= ⟨NN| N4CP 2NQ2N18(2 + 23σ1· σ2)(2 + 23τ 1· τ 2) |NN⟩. (A1)Here we have used the finite-NC correction terms PN, defined before, and QN≡q(1 −1/NC)(1 + 5/NC).

In this manner we find that⟨NN| ˜H(⃗R) |N∆⟩⟨N∆| ˜H(⃗R) |NN⟩= ⟨NN| P 2NQ2N(2 + 23τ 1 · τ 2){ 19(v22 + 2v23) + 127(v22 −v23)σ1· σ2 +127(v23 −v2v3)(3σ1· ˆRσ2· ˆR −σ1· σ2)} |NN⟩,(A2)and for an intermediate state with two ∆’s,⟨NN| ˜H(⃗R) |∆∆⟩⟨∆∆| ˜H(⃗R) |NN⟩= ⟨NN| Q4N( 43 −29τ 1 · τ 2){ 13(v22 + 2v23) −118(v22 −v23)σ1· σ2 −118(v23 −v2v3)(3σ1· ˆRσ2· ˆR −σ1· σ2)} |NN⟩. (A3)When we use the fact that the energy needed for excitation of a single ∆, which we denoteby δ, is half that for excitation of two ∆’s, we findV (2)exact = −Q2Nδ {[( 49P 2N + 29Q2N) + ( 427P 2N −127Q2N)τ 1· τ 2](v22 + 2v23) +(σ1· σ2)[( 427P 2N −127Q2N) + ( 481P 2N +1162Q2N)τ 1· τ 2](v22 −v23)(3σ1· ˆRσ2· ˆR −σ1· σ2)[( 427P 2N −127Q2N) + ( 481P 2 +1162Q2N)τ 1· τ 2](v23 −v2v3)}.

(A4)Of course this can be rewritten by introducing spin and isospin projection operators,P τ0 = 1 −τ1· τ24,P τ1 = 3 + τ1· τ24,(A5)τ 1· τ 2 = −3P τ0 + P τ1 .We thus get the form (subscripts now denote isospin projections)V (2)exact = −Q2Nδ {[ 13Q2NP τ0 + ( 1627P 2N + 527Q2N)P τ1 ](v22 + 2v23) +(σ1· σ2)[−118Q2NP τ0 + ( 1681P 2N −5162Q2N)P τ1 ](v22 −v23)(3σ1· ˆRσ2· ˆR −σ1· σ2)[−118Q2NP τ0 + ( 1681P 2N −5162Q2N)P τ1 ](v23 −v2v3)}. (A6)18

REFERENCES[1] T. H. R. Skyrme, Nucl. Phys.

31 (1962) 556[2] I. Zahed and G. E. Brown, Phys. Rep. 142 (1986) 1; K. F. Liu (ed.

), Chiral solitons,(World Scientific, Singapore, 1987). [3] N. R. Walet, R. D. Amado, and A. Hosaka, Phys.

Rev. Lett.

68, 3849 (1992). [4] A. D. Jackson, C. Weiss and A. Wirzba, Nucl.

Phys. A529, 741 (1991).

[5] N. S. Manton, Phys. Rev.

Lett. 60, 1916 (1988).

[6] A. Jackson, A. D. Jackson and V. Pasquier, Nucl. Phys.

A432, 567 (1985). [7] R. Vinh Mau, M. Lacombe, B. Loiseau, W. N. Nottingham and P. Lisboa, Phys.

Lett.150B, 259 (1985). M. Lacombe, B. Loiseau, R. Vinh Mau, and W. N. Cottingham, ibid.161B, 31 (1985); ibid.

169B, 121 (1986); Phys. Rev.

Lett. 57, 170 (1986).

[8] This has been argued by Jackson et al [6] using the product ansatz. It is hard to assessthe significance of this statement since the product ansatz does not apply at these verysmall distances.

[9] T. S. Walhout and J. Wambach, Phys. Rev.

Lett. 67, 314 (1991).

[10] Some workers used state mixing in the context of the product ansatz [21], and many,including WW, used some finite NC effects. [11] A. Klein, N. R. Walet, and G. Do Dang, Ann.

Phys. (NY) 208, 90 (1991) and referencestherein.

[12] M. F. Atiyah and N. Hitchin, The geometric quantization of magnetic monopoles, (Prince-ton University Press, Princeton, 1988). [13] M. F. Atiyah and N. S. Manton, Phys.

Lett. 222B, 438 (1989); N. S. Manton, in: Geometryof Low-Dimensional Manifolds : Proceedings of the Durham Symposium, S. K. Donaldsonand C. B. Thomas, eds.

(Cambridge University Press, Cambridge, 1990). [14] A. Hosaka, M. Oka and R. D. Amado, Nucl.

Phys. A530, 507 (1991).

[15] M. Oka, R. Bijker, A. Bulgac, and R. D. Amado, Phys. Rev.

C 36, 1727 (1987). [16] V. B. Kopeliovic and B. E. Stern, JETP Lett.

45, 203 (1987); J. J. M. Verbaarschot, T.S. Walhout, J. Wambach, H. W. Wyld, Nucl.

Phys. A468, 520 (1987); N. S. Manton,Phys.

Lett. 192B, 177 (1988); J. J. M. Verbaarschot, Phys.

Lett. 195B, 235 (1987).

[17] It is not immediately clear whether one can describe the symmetries in the doughnut inthis way. We have studied this problem in some detail and have been able to prove thatthis is possible, but we have not yet been able to derive a sensible physical mechanism forthe partial symmetry restoration of the doughnut.

[18] R. D. Amado, R. Bijker and M. Oka, Phys. Rev.

Lett. 58, 654 (1986).

[19] Note the close analogy to an axially deformed nucleus, which can be represented by apoint on the sphere S2. In both cases this represents a state that is not the most generaldeformed body for the group under consideration, which leads to selection rules, Kz = 0for the nuclear case, and ⃗K = ⃗I + ⃗S = 0 for the case of the Skyrmion.

[20] J. Wambach, private communication; A. Hosaka, unpublished result (contributed paperto “Baryons ’92”). [21] A. Depace, H. M¨uther and A. Faessler, Phys.

Lett. 188B, 307 (1987); G. K¨albermannand J. M. Eisenberg, J. Phys.

G15, 157 (1989). [22] H. Yamagishi and I. Zahed, Phys.

Rev. D 45, 965 (1991); V. Thorsson and I. Zahed,Phys.

Rev. D 45 965 (1992); T. Otofuji, Y. Kondo, S. Saito and R. Seki, Phys.

Rev. D,in press.19

[23] This corresponds to assuming either symmetry or antisymmetry between the individualSkyrmions. In the semiclassical quantization of the doughnut a similar problem is en-countered, in terms of the Finkelstein-Rubinstein constraints (E. Braatten and L. Carson,Phys.

Rev. D 38, 3525 (1988)).

[24] C. Holinde, R. Machleidt and C. Elster, Phys. Rep. 149, 1 (1987).

[25] M. Lacombe, B. Loiseau, J. M. Richard, R. Vinh Mau, J. Cote, D. Pires, and R. DeTour-reil, Phys. Rev.

C 21, 861 6q(1980). [26] R. V. Reid jr., Ann.

Phys. (NY) 50, 411 (1968).20

FIGURESFIG. 1.

The central potential V Tcas a function of R in the region 1 to 2 fm for the T = 0channel. The solid line gives the nucleons only result of [9].

The longer dashed line is the result ofthe state mixing in perturbation theory and the shorter dashed line of the full Born-Oppenheimerdiagonalization. The dotted line is the Reid soft core potential in this channel.FIG.

2. Same as Figure 1 but for T = 1.FIG.

3. The spin dependent potential V Ts as a function of R in the region 1 to 2 fm for the T = 0channel.

The solid line gives the nucleons only result of [9]. The longer dashed line is the result ofthe state mixing in perturbation theory and the shorter dashed line of the full Born-Oppenheimerdiagonalization.

The dotted line is the Reid soft core potential in this channel.FIG. 4.

The same as Figure 3 but for the T = 1 spin-dependent potential.FIG. 5.

The tensor potential V Ttas a function of R in the region 1 to 2 fm for the T = 0channel. The solid line gives the nucleons only result of [9].

The longer dashed line is the result ofthe state mixing in perturbation theory and the shorter dashed line of the full Born-Oppenheimerdiagonalization. The dotted line is the Reid soft core potential in this channel.FIG.

6. The same as Figure 5 but for the T = 1 tensor potential.FIG.

7. The potential in the 3S1 (T = 0) channel as a function of R. The solid line gives thenucleons only result of [9].

The dashed line is the result of state mixing in a full Born-Oppenheimerdiagonalization, and the dotted line is the Reid soft core potential in this channel.FIG. 8.

Same as Figure 7 but for the 1S0 (T = 1) channel.FIG. 9.

The Born-Oppenheimer energy eigenvalues coming from diagonalization of the adia-batic energy in the J = 1−, T = 1 channel as a function of the adiabatic variable R. The lowestenergy state goes over to the NN state for large R. The next three states go over into the N∆states, and the upper two into the ∆∆states.FIG. 10.

The percentage admixture of configurations in the lowest (N −N ) adiabatic stateof the J = 1−T = 1 channel as a function of R. The solid line is the NN component, and theseecond largest component, given by the short-dashed line is the state |N∆, L = 1, S = 2⟩.FIG. 11.

The effective mass of the “nucleon” in the lowest J = 1−T = 1 channel as a functionof R. This is taken to be twice the reduced mass.21

FIG. 12.

The velocity independent transition potential from the lowest (NN) adiabatic stateof the J = 1−T = 1 channel as a function of R for transitions to each of the five other channels.The numbers in the legend refer to the order of the states in Fig. 9, where one is the lowest energyadiabatic state.FIG.

13. Same as Figure 12 but for the velocity dependent transition potential.

Each term isunderstood to be multiplied by −iβ where β = v/c.22

---

출처: arXiv:9210.015 • 원문 보기