The skyrmion-skyrmion interaction

Nuclear Physics A432 (1985) 567-609 @ North-Holland Publishing Company

THE SKYRMIONSKYRMION INTERACTION*

A. JACKSON and A.D. JACKSON

Department of Physics, State University of New York, Stony Brook, New York I1 794, USA

and

V. PASQLJIER

Service de Physique l”hTorique, Cenlre d’Etudes Nuclhires de Saclay, F-91 191 Gif-sur-Yvette, France

Received 31 January 1984 (Revised 16 August 1984)

Abstmet: Using the Skyrme effective lagrangian, baryons emerge as topological solitons. This effective lagrangian is used in adiabatic calculations aimed at providing an essentially parameter-free model of the interaction between such solitons. The resulting skyrmion-skyrmion interaction can be understood as terms simulating the exchange of T-, p-, and w-mesons between the solitons. It is readily transformed (by essentially projective techniques) into a low-energy nucleon-nucleon potential. Comparisons of this potential with the best available semiphenomenological nucleon- nucleon interactions are found to be successful at the 30% level. Similarities between the Skyrme model and the quark chiral-bag model are discussed.

1. Introduction

The description of baryons as chiral solitons was first proposed more than twenty years ago by Skyrme ‘). His original papers predate both quantum chromodynamics and even the notions of chiral invariance. Skyrme’s approach, which we shall follow, was to adopt a non-linear chiral lagrangian which does not contain baryons explicitly. When supplemented by a (relatively ad hoc) fourth-order term required to ensure stability, this lagrangian was found to support non-trivial solutions (at the classical level) with a topological quantum number. Skyrme offered suggestive arguments that this quantum number should be identified with the baryon number. Given the simplifying assumption of spherical symmetry of the chiral angle, the solution with B = 1 was to be identified with a suitable linear combination of the nucleon and A (1236 MeV) nucleon resonance. In short, Skyrme demonstrated that baryons could emerge spontaneously from a field theory containing only mesons. The few parameters of this model were readily adjusted to yield a sensible B = 1 mass. The B = 2 energy is almost exactly three times the B = 1 energy (independent of the parameters of the model), which led to the suggestion that the interaction between baryons at zero separation should be characterized by a short-range repulsion of roughly one baryon mass in magnitude. In this model the interaction between well-separated

567

568 A. Jackson et al. / Skyrmion-skyrmion interaction

solitons has the form of a simple one-pion-exchange interaction. Further, since the familiar second-order chiral lagrangian is dominant at large distances, this model rigorously obeys the Goldberger-Treiman relation which connects the pion decay constant, the soliton axial vector coupling constant, and the soliton-pion coupling constant.

Interest in Skyrme’s model has recently been rekindled within the context of QCD. t’Hooft and subsequent authors have shown that the low-energy, large-color limit of the mesonic sector of QCD should be an effective lagrangian involving mesons only ‘). More recently, Witten has argued that baryons should emerge as solitons in large-N, QCD [ref. ‘)I. Given the overwhelming evidence in support of chiral symmetry at low energies 4V5) and Skyrme’s demonstration that baryon-like solitons do emerge from his mesonic lagrangian, it would seem natural that something like the Skyrme model be obtained from QCD in these limits if QCD is correct. Note, however, that neither the Skyrme nor any other lagrangian has yet been derived from QCD. Witten’s careful work has led to a far more convincing demonstration that the topological quantum number in the Skyrme model is the baryon number 6,7). Further, it has been possible to construct operators for a number of baryon observables including the baryon number density. This density leads to an rms baryon number radius of 0.5 fm for the B = 1 soliton 778).

In order to study other properties of baryons, such as the NA mass splitting and the electromagnetic properties of baryons, it is necessary to obtain physical baryons from these solutions. Witten offered a projection procedure for this construction which bears striking similarities to techniques familiar from the description of deformed nuclei ‘). In the latter case, the dominance of quadrupole-quadrupole interactions justifies the use of a deformed Hat-tree-Fock wave function which sacrifices the total angular-momentum quantum number. States of good angular momentum are then obtained (after energy minimization) by simple projection pocedures which involve an average over orientations of the intrinsic deformed state weighted with rotation matrices. Such a procedure becomes exact as the deformation, as measured by the moment of intertia, becomes large. For the soliton the relevant correlations are presumed to be between spin and isospin lo). It remains to be seen whether these correlations are of sufficient strength to justify the technique adopted or whether Skyrme’s spherical ansatz for the chiral angle is merely a mathematical convenience. We hope to offer some guidance on this point. Nonetheless, this approach leads to a reasonable description of the NA mass splitting, nucleon magnetic moments, nucleon charge radii and the coupling constant ratio for nNN to TNA [ref. ‘)I.

The description of these quantities is usually regarded as one of the successes of the MIT bag model which is also intended as an approximate realization of QCD [ref. ‘I)]. In the bag model the mass of the nucleon is a consequence of the bag volume energy and the kinetic energy of the confined quarks. Coupling-constant ratios and magnetic moments are a simple matter of book-keeping. Mean square

A. Jackson et al / Skyrmion-sky&on interaction 569

radii of all sorts are related to the radius of the bag. The NA mass splitting is a

consequence of gluon effects. One expects that bag and soliton models are describing precisely the same physics in two rather different langauges. Their connection is made clearer in the chiral bag model 10,12*13). In this model, the interior bag of quarks is jointed, through suitable boundary conditions, to an external pion field described, again, by a chiral lagrangian. This hybrid model has a number of appealing features for low-energy physics. Not least is the fact that the pionic presence admits a more natural description of the interaction between baryons at large distances through the exchange of one or more pions. There is ample evidence in low-energy nucleon- nucleon scattering for the importance of such mechanisms ‘) which are not readily introduced in more pristine bag models.

Chiral-bag models have a variety of other virtues. The pressure of the pions on the bag membrane shrinks the bag radius from the large value of roughly 1.0 fm to half the size. As two bags are brought together their combined pion fields shrink the individual bag radii and reduce the distance at which bags fuse. This extends the range of validity of meson exchange pictures of baryon-baryon interactions well into the region which is normally presumed to be dominated by two-pion-exchange processes. Chiral-bag models also provide an explanation of the short-range repulsion between baryons r”*13). In this case the coupling of pions to the bag breaks the degeneracy of the twelve quark spin-color-flavor states corresponding to each spatial mode. In particular, the pion coupling induces a correlation between spin and isospin of the quarks and recommends the introduction of a new quantum number K which is the vector sum of quark spin and isospin. The three single-quark states with K = 0 lie substantially lower in energy than the nine states with K = 1. For zero separation between two nucleons, three quarks are forced to occupy the higher-energy states with K = 1 and a repulsion of roughly one. baryon mass in magnitude results. As we shall show, the spin-isospin correlations which characterize this quark hedgehog model are also those of the Skyrme model with the assumption of spherical symmetrry for the chiral angle.

Connections between hedgehog and Skyrme models became even more compelling when it was realized that the hedgehog solution in the pionic region also has a solitonic character lo). In particular, it has been shown that the pion field carries a portion of the baryon number while the effects of negative-energy quark states result in a corresponding depletion of the baryon number in the quark sector 10*14*15). Indeed, the baryon number is rigorously maintained in this two-phase model although the distribution of baryon number between quark and meson sectors depends on the bag radius. In the limit of large bag radius, the baryon number is associated purely with the quark sector. In the limit of zero bag radius, the baryon number is associated with the mesonic sector only as in the Skyrme model. Calcula- tions of the B = 1 energy in such a two-phase model reveal a similar insensitivity to the bag radius. This energy is found to vary by only 20% as the bag radius is changed from 0 to 1 fm [ref. I”)].

570 A. Jackson et al. / Sky~ion-sky~ion interuction

These results lead us to believe that either a chiral bag model or its zero-radius limit as a Skyrme model should be suitable for studying the low-energy properties of nucleons and the nucleon-nucleon interaction. Since differences between quark hedgehog and Skyrme models lie largely in the relization of color confinement (which seems of limited importance for low-energy properties), we expect that the low-energy predictions of these models should be rather similar although their language may be rather different. Although we recognize the two-phase nature of

QCD, the explicit incorporation of quark degrees of freedom may not be essential in the description of low-energy phenomena. Given the extreme simplicity of the Skyrme model, we shall adopt it in the present study of the bacon-baton interaction. We emphasize that the real QCD effective lagrangian has not yet been derived. Our adoption of the Skyrme lagrangian must be regarded as an interim, but not unreasonable, guess. The real QCD effective lagrangian could well include higher- order terms involving T- and a-fields. It could also include other elementary meson fields. We shall see, however, that such additional ingredients do not appear to be essential for obtaining a description of the low-energy nucleon-nucleon interaction at the low level of accuracy (roughly 30%) claimed for the present calculations.

In sect. 2 we shall review the Skyrme model and the manner in which we implement it. This is desirable since our approach to the two parameters in this model differs from that of some other authors. In sect. 3 we describe our adiabatic calculation of the sky~ion-skyrmion interaction. We shall report results as a function of the separation between skyrmions and also as a function of the Euler angles of the orthogonal transformation of the Skyrme approach. This angular dependence represents the last vestige of the spin-isospin structure of the baryon-baryon interaction and will be of importance in transforming the skyrmion-skyrmion interaction into the nucleon-nucleon interaction of interest. In order to facilitate this transformation and to further demonstrate the connection between quark hedgehog and Skyrme models, sect. 4 will be devoted to obtaining the Euler-angle dependence of the

interaction in a pure quark hedgehog model. We shall see that the quark hedgehog picture leads to the same asymptotic interaction as the Skyrme model and that the Euler-angle dependence suggested by the quark hedgehog is that found in Skyrme- model calculations. We shall use the quark hedgehog model to demonstrate that these mesons which can couple to hedgehogs are just the mesons which are known to have strong couplings to nucleons. These results suggest that the spin-isospin correlations frequently adopted in calculations with the Skyrme model have some empirical support and are not merely of calculational convenience. In sect. 5 we shall compare the results of our calculations with the semiphenomenological Paris nucleon-nucleon interaction. The success of this comparison will be offered as evidence that this simple and, for these purposes, parameterless model provides a reasonable description of the low-energy nucleon-nucleon interaction. A variety of conclusions will be drawn in sect. 6.

A. Jackson et al. / Skynnion-skyrmion interaction 571

2. The Skyrme model

Twenty years ago Skyrme ‘) proposed a model in which low-mass baryons emerge from a non-linear field theory as topological solitons; the topological quantum number being identified with the baryon number. This model remained a curiosum until it was recognized that it might be regarded as a suitable low-energy, large-color limit of QCD. This point of view has been advanced forcefully by Witten 3*6*7). Particular attention has been paid to demonstrating that the skyrmion is a fermion and that the identification of the topological charge with the baryon number is legitimate “v6). Th e model has been shown to lead to reasonable predictions for a variety of baryon properties including the nucleon mass and mean square radii, the A-nucleon mass splitting, the magnetic moment ratio ,+‘)(I,, and the coupling-

constant ratio g+JgrrNN [ref. ‘)I. The fact that the meson field carries a baryon

number suggests a natural way to merge the Skyrme model with the quark bag model lo). In view of these successes it seems reasonable to see what the Skyrme model has to say about the low-energy interaction between baryons.

Tbe basis of Skyrme’s model is a lagrangian appropriate for the SU(2) xSU(2) group:

L!? = -if: Tr [I&,] - $E’ Tr [L,, Ly]’ (2.1) with

L, = Va,u, (2.2)

U=[(i(x)+i~* 74x)]/&, UfU=l. (2.3)

Heref, is the pion decay constant (with the empirical value of 93 MeV), E a constant whose determination will be discussed, (T is the scalar meson field, and r is the isovector triplet pion field. The energy for a given choice of static fields can be written as

E=-

J d3x Z(x) (2.4)

and can be minimized with respect to functional variation of (T and 7~. This leads to a virtually intractable Euler equation. Skyrme avoided this obstacle by invoking the hedgehog assumption i)

V(r) = exp [ire %(r)]. (2.5)

The explicit forms of u and rr in this approximation can be recovered from eqs. (2.3) and (2.5). Although this approximation was introduced for convenience, its image appears in chiral bag models as a physically motivated approximation aimed at extracting maximum correlation from the coupling of quarks to pions at the bag surface “*13).

With the hedgehog assumption, eq. (2.4) assumes the more ‘manageable form

J m

E =2-&y. 32?rsZ *

dr e=[8’+2 sin2 @I+- dr e-= sin’ @[b2 +$ sin2 S] , (2.6) -m J r. -m


where we have adopted the convenient variable r = In (r/r,,) where r. is a redundant

but nonetheless useful scale factor. The related Euler equation is readily obtained

and solved (a pocket calculator is sufficient) to determine the chiral angle e(r)

[ref. “)I. The structure of the Euler equation demands that e(r) equal a, exp (-27)

for large T and [&r + CY~ exp (r)] for large negative T (i.e. small r). Here, B is an

integer which can be identified with the baryon number while a, and (YB are constants

determined solely by the Euler equation (and which depend only on the single ratio

A defined as 16E2/f:ri which can assume any positive value). Minimization of only

the quadratic term in eq. (2.6) with respect to the scale r. yields the trivial result

r. = 0 emphasizing that the quartic term is essential for the existence of a soliton.

The energy resulting from these operations is

EB=~‘~w~+B, (2.7)

where IB is independent of the various parameters in the Euler equation. We have

previously found 1B-r to be 8.2040 and lB=2 to be 24.446. If we choose to regard

EB=z as the energy of two B = 1 solitons at the same location, we can use these

results to make a statement about the skyrmion-skyrmion interaction at zero

separation without making any reference to f= and l 2:

V~(~)=EB=~-~E~,,-EB=, . (2.8)

The Skyrme model leads to a finite repulsion between two skyrmions at zero

separation of roughly the mass of a single baryon.

There are a variety of ways of determining the input parameters fx and Ed. One

could adopt the physical value off= and turn to the pion P-wave scattering length

for a determination of E’ [ref. ‘“)I. Unfortunately, the resulting limits of 7 x lop3 <

e2 < 5 x 1O-2 lead to an uncomfortably large variation in the mass of the B = 1

baryon of 1.6 GeV< n&1 < 4.3 GeV. A second approach, adopted by Adkins et

al. ‘), was to develop projection procedures allowing for the independent determina-

tion of the nucleon and A (1236) masses. The parameter values fr = 64.5 MeV and

E’ = 0.00424 were found to reproduce these masses and lead to a skyrmion mass of

mBEl = 0.866 GeV. This approach fails to reproduce the rNN coupling constant

giving a value of g,,, which is roughly 30% too small. This would result in a 50%

underestimate of the asymptotic interaction between nucleons. Instead, we adopt

the procedure of ref. “) which attempts to maintain the spirit of a low-energy limit

by fixingf, at its empirical value and requiring that the axial-vector coupling constant

g, for nucleons also be correct.

We can adjust g, by a suitable choice of the asymptotic form of the chiral angle

(through the scale factor ro) with the aid of the Goldberger-Treiman relation.

Specifically,

(2.9)

A. Jackson et al. / Skymion-skyrmion interaction 573

where fh is the ratio of axial vector coupling constants of the hedgehog and the nucleon. The projection arguments of Adkins et al. ‘) indicate that this ratio is precisely 3. The same result can be obtained from an explicit quark model in the NC + cc limit as shown in sect. 4. The solution to the Euler equation reveals that the ratio (a,,,/A) has the value 1.078 independent of A so that the value fh = 3

leads immediately to l ‘=0.00919 and mBE1 = 1.84 GeV. As we shall show in sect. 4, the quark model for the physically interesting value of NC = 3 yields fh = 2. At the very least, the compa~son of these values provides a measure of the nature of finite NC corrections. [Precisely this factor enters into the determination of g,& and g,,, in ref. 7), and this estimate of finite NC effects can help calibrate the substantial discrepancies found there.] We shall take this result more seriously and prefer to sacrifice the purity of the Skyrme model (and its implicit NC -+cx) limit) and adopt the value fh =$ We believe it is appropriate to implement such book-keeping corrections for finite NC and shall do so whenever they arise. Thus, we are led to the parameter values fY = 93 MeV and ez = 0.00552 which we shall adopt throughout this work. These parameters yield r&#=, = 1425 MeV. While this number is somewhat larger than the masses of the nucleon and A( 1236) which the skyrmion is assumed to describe, the agreement is nonetheless impressive in view of the extreme simplicity of the Skyrme model. Of course, this approach has been designed to describe gA for the nucleon. Through the Goldberg-Treiman relation, it will also reproduce the empirical value of g,.&N. Thus, this approach guarantees that the eventual nucleon- nucleon interaction will reduce to the correct one-pion-exchange potential at large distances.

The form of the chiral angle for B = 1 which will be important for our subsequent calculations is shown in fig. 1 along with the B = 1 baryon density. The baryon density is given as

&3(r) = 1 sin* eB d& _---

2~~ r2 dr ’ (2.10)

It is evident that eq. (2.9) is a perfect differential so that the volume integral of the baryon density is precisely the integer B which describes the value of the chiral angle in the vicinity radius of 0.48 fm.

of r =O. This baryon density yields an rms baryon number

3. The skyrmion-skyrmion interaction

In this section we shall describe an approximate adiabatic calculation of the interaction between baryons within the framework of the Skyrme model. Ideally, we would like to construct a general unitary transfo~ation describing a system of baryon number two and calculate its energy directly from eqs. (2. i)-(2.4). This B = 2 energy could be minimized with respect to the functional form of this unitary transformation subject to the constraint that the two baryons remain at fixed


m 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I.0 r Urn)

Fig. 1. The baryon density p”(r) in arbitrary units and the chiral angle e’(r) for baryon number one

with l * = 0.00552.

separation. With this minimum B = 2 energy in hand, one could adopt the prescrip-

tion of eq. (2.8) and define the potential energy for this separation as the B = 2

energy less twice the energy of an isolated B = 1 skyrmion. Unfortunately, such a

scheme is difficult to implement and leads to variational problems of unreasonable

numerical difficulty. Instead, we shall largely avoid the variational problem and

prescribe the B = 2 unitary transformation from the outset.

In general, the product of any two unitary tranformations each having B = 1 is

a unitary transformation with B = 2. Such a product form for U is expected to be

correct in the limit of infinite separation between the skyrmions ‘) provided that the

individual terms describe isolated solitons with baryon number one at locations r,

and r2. This approximation assumes the form

ULJ=,(x, Tlr r2) = V,(x) qx) . (3.1)

Although eq. (3.1) is rigorously valid only in the limit of infinite separation, we

shall adopt it for all separations between the two baryons. We shall make the

acditional assumption that the individual B = 1 unitary transformations appearing

in eq. (3.1) are of the hedgehog form given by eq. (2.5). This approximation is also

correct in the limit of infinite separation. Further, it generates a B = 2 transformation

of the hedgehog form in the limit of zero separation between the skyrmions. These

two limits suggest that it may be legitimate to make one additional approximation

and set the chiral angle implicit in eq. (3.1) to the value found from the solution

A. Jackson et al. / Skymion-skytmion interaction 575

of the Euler equation for a single B = 1 soliton which is shown in fig. 1. This apparently strenuous approximation, which is again legitimate in the limit of large separations, is also remarkably safe for zero separation where it merely involves the replacement of r?&+(x) by 2&,(x). Clearly, for x =0 both lead to a chiral angle of 2~r. The coefficient (r&2 deviates by only 2% from 2ogG1 indicating a near equivalence for small x. Both &2(X) and 26&I(X) pass through n at precisely the same point. Differences appear Only for large x where oB=2 iS seen to be 3oB, ,. As a result, the B = 2 energy for zero separation obtained with this approximation is only 2.5% higher than the exact B = 2 hedgehog energy. This corresponds to an error of roughly 7.5% in the potential energy for zero separation. This error seems acceptable in view of our largely qualitative expectations and the considerable simplification which results from the use of eq. (3.1). There is, however, no guarantee that this form will be equally suitable for regions in which the two solitons overlap substantially (i.e. roughly 0.6 fm to 1.2 fm). We shall consider this point in sect. 5 where we argue that this approximation has probably not introduced significant error in the calculated interaction.

Before proceeding, it is necessary to generalize eq. (2.5). Specifically, we write ‘)

U(X)=exp [iT&j[(YP~]ZZjO(X)], (3.2)

where eii[oPr] is a fixed orthogonal transformation describing a rotation through Euler angles [o&l, Adkins et al. ‘) write this generalization differently as

U(a,x)=A-‘(a)U(x)A(a) (3.3)

with

A(a)=a,+h*a, (3.4)

where a, and a are fixed and subject to the constraint that

iioai=l. (3.5)

In eq. (3.3), U(x) is the unrotated hedgehog of eq. (2.5). These two representations may be connected through the relation

eJc@y]=[t&(a~-a. a)+2a,aj+2aoaketjk]. (3.6)

The general form of eti[cu/3r] is given, for example, in eq. (4.8). It is clear that the unrotated hedgehog corresponds to the choice a0 = 1 or alternatively eJO] = S,

It has been noted previously that, since the energy of the soliton of eq. (2.4) depends only on the derivative of U, rotations of the kind introduced by eq. (3.2) will result in single hedgehog configurations of equal energy. This will not be the case when we consider more general unitary transformations of the form of eq.


(3.1). The energy of the B = 2 system will depend on the Euler angles describing

the rotations of the individual solitions (which need not be the same) as well as on

their separation. Just as the interaction depends only on the relative distance between

the solitons, it depends only on their relative Euler angles. (We shall illustrate this

point below in connection with a discussion of the asymptotic form of the interac-

tion.) Thus, there is no loss of generality in setting one set of Euler angles equal to

zero and considering the interaction as a function of the separation r,* and the

remaining set of Euler angles [@y]. It is useful to note that a similar rotation arises

in the quark hedgehog model in which one exploits a new operator which is the

sum of spin and isospin operators. Results for a single baryon are invariant under

rotations of the isosopin quantization axis with respect to the spin quantization axis.

The orthogonal transformation relating these two axes is identical to the eV[@y]

introduced above up to a factor of two in the Euler angles. We shall return to this

point in sect. 4.

The skyrmion-skyrmion potential is, thus, a function of four variables: one relative

distance and three relative Euler angles. Although the latter dependence complicates

the numerical calculations, it is welcome. The Euler angle dependence of the

interaction represents the only sign of the spin and isospin-dependence of the

underlying baryon-baryon interaction which persists in the hedgehog problem. In

sect. 4 we shall see that it permits us to recover the dominant spin-isospin dependence

of the nucleon-nucleon interaction.

We have calculated the skyrmion-skyrmion interaction numerically subject to the

above approximations and constraints. The unitary transformation of eq. (3.1) was

constructed for one unrotated soliton and one soliton rotated through Euler angles

[(Y&] at a fixed separation r12 along the z-axis. Eqs. (2.1)-(2.4) were used to express

the B = 2 energy. The energies of the two isolated B = 1 solitons, which are known

with some precision from sect. 2, were subtracted as integrals of similar form so

that the substantial cancellations expected especially for large separations could be

realized in the integrands. Since the resulting integral over x describing the inter-

action involves the (numerically known) eB=, for two distinct arguments, it involves

two dimensions which must be treated numerically. In cases with p = 0, illustrated

by fig. 2a, the integral has an obvious cylindrical symmetry which admits trivial

analytic treatment. The remaining planar integration was performed on a suitable

rectangular mesh of points. The range and mesh were chosen to permit a 1%

evaluation of the integrals as indicated by their stability against changes in these

quantities and measured by the accurate reproduction of the energy of an isolated

B = 1 soliton. The case for non-zero p is slightly more complicated since the integrand

no longer has cylindrical symmetry. This is illustrated in fig. 2b for the case cx = p = $r

and y = 0. Although the +-integration can (in principle) still be done analytically,

it proved to be considerably simpler to treat this case as a three-dimensional

numerical integration. Again, integration ranges and meshes were chosen to provide

a 1% evaluation of the integrals.

A. Jackson et al. / Skywnion-skyrmion interaction

A 0 I I I I -r-

/t

/’

571

(b)

Fig. 2. The spheres represent contours of equal chiral angle while the arrows show the isospin direction determined by eij[c&] for (a) the case of two unrotated solitons separated by a distance r and (b) the

case when the isospin axes for soliton B have been rotated by [&] = [$&TO].

In figs. 3-5 we show the results for the following three sets of Euler angles:

V,(r)=V(r,[cu=p=~=o]),

Ve(r)= V(r,[a=$v,p=y=O]),

V,(r)=V(r,[cu=O,p=7r,y=o]). (3.7)

In sect. 4 we shall demonstrate that these three calculations are sufficient to determine the Euler-angle dependence of the interaction. These results permit several checks of our numerical procedure. When r Iz equals 0, VA possesses complete spherical

578 A. Jackson et al. / Senior-sk~rmion interaction

r (fm)

I 2

Fig. 3. The potential energy for two unrotated solitons V,.,(r) as a function of their separation as calculated from eqs. (2.4), (3.1) and (3.2). The break at r = 1.3 fm merely indicates a scale change by a factor of ten.

symmetry and can be calculated far more precisely from a one-dimensional radial integration. Agreement at the level claimed was obtained.

For large separations, the dominant contributions to the interaction come from regions where it is legitimate to use the asymptotic forms of both of the chiral angles. The following asymptotic form of the interaction results for two arbitrarily rotated

I t 1 I I I L I I I

I 2 r lfml

Fig. 4. The potential energy V,(r), for two solitions as a function of separation. In this case one soliton has been rotated through Euler angles [&TOO]. The crosses indicate the expected asymptotic behavior,

+202/? MeV. The break at r = 1.4 fm indicates a scale change by a factor of ten.

A. Jackson et al. / Skynnion-skynnion interaction 519

Fig. 5. The potential energy V,-(r) when one soliton has been rotated through Euler angles [OlrO]. The crosses indicate the expected asymptotic behavior, -202/r3 MeV.

hedgehogs I):

Without loss of generality we can perform the [~Y’p”y”] rotation by first rotating through [cu’/3’r’] and then rotating through a third set of Euler angles [cuPr] to the desired result. This can be written formally as

e,j[CY”P”~“]= ep,[a’p’y’]eAj[af3y]. (3.9)

Inserting this result in eq. (3.8) and exploiting the orthogonality of the transformations for equal Euler angles, we find

(3.10)

which is a specific realization of the general result that the interaction depends only on the relative Euler angles [aBy] just as it depends only on the relative separation of the skyrmions. Putting in the values of a, A, E’ and fT from sect. 2 we find

li~i V( r, [o$r]) = (101 MeV * fm’)eJc&] --& & -lr . 1 J 0

(3.11)

Using the explicit form of eJcr&] given in eq. (4.8), we are led to

heir V[r,[apy])=(lOl MeV*fm3)[cosP(2-cos(a+y))-cos(a+y)]/r3.

(3.*12)

The result indicates that V, of fig. 3 should vanish faster than re3 in the limit of large r. It actually vanishes exponentially for large r. The terms V, and V, should

580 A. Jackson et a/. / Skyrmion-skyrmion interaction

have the large-r limit of A202 MeV . fm/ r3, respectively. This asymptotic behaviour, which can be seen from figs. 4 and 5, is respected to better than 5% even at separations as large as 4 fm where the interactions are less than 0.5% of their magnitudes at zero separation. This would appear to provide a perfectly satisfactory check of our numerical methods.

There is little to be said about the skyrmion-skyrmion interaction at this juncture; a more detailed interpretation will be provided when it is converted to a nucleon- nucleon interaction in subsequent sections. We note that the asymptotic form of eq. (3.12) is precisely that expected from the exchange of a massless pion. We also note that the interaction between skyrmions at zero separation is repulsive for any choice of Euler angles and bounded by the values of 800 MeV and 1500 MeV shown in figs. 5 and 3. This universal repulsion of order 1 GeV is expected to persist in any baryon-baryon interaction which can be projected from these results.

4. Quark hedgehog modeIs

In this section we wish to address the interaction between solitons in a quark language. The purpose of this exercise is fourfold. First, this will help emphasize certain ~imila~ties between the two approaches. Second, it will suggest a parameterization of the Euler-angle dependence of the skyrmion-skyrmion interaction which correctly describes the results of sect. 3. Third, it is as simple to construct nucleon (or nucleon isobar) wave functions from hedgehog wave functions in a quark language, and it is thus straightforward to convert the previous skyrmion-skyrmion interaction into a nucleon-nucleon interaction. (Clearly, NA and AA interactions as well as a variety of transition potentials can be obtained in the same fashion although we shall not report such results.) This approach to the construction of the nucleon-nucleon interaction will be seen to be equivalent to a direct projection approach in the limit as NC tends to infinity. Finally, these constructions can be carried out for any number of colors and, thus, provide us with at least a minimal measure of the errors introduced by taking the NC + OC, limit.

In the chiral bag model ‘0913) it is conventional to abandon real baryons (having good spin and isospin) in favor of hedgehogs. One introduces a new quantum number

K=S+T (4.1)

and argues that, due to the coupling of quarks to the pion field at the bag surface, single-quark states with K = 0 are of lowest energy. To the extent that the quark spin-isospin correlations induced by the pion field are important, it may be desirable to sacrifice good spin and isospin in the interest of incorporating such correlations economically ‘“*‘3). This is, of course, the analogue of sacrificing good total angular momentum in favor of providing a simple description of the spatial correlations which lead to macroscopic deformations of nuclei. In both cases, it is ultimately possible to recover the desired good quantum numbers by projection after energy

A. Jackson et al. / Skyrmion-skyrmion interaction 581

minimization. This approach is expected to be valid if the corresponding correlations are large. The spin-isospin wave functions for single quarks with K = 0 are given as

Ih) = 4 []ut> - kU>l . (4.2)

The hedgehog wave function in the quark model is then obtained as a simple Slater determinant of NC such individual quark wave functions having the same spatial wave function, a spin-isospin wave function given by eq. (4.2) and the various NC colors. All antisymmetry resides explictly in the color wave function. This wave function, which possesses neither good spin nor good isospin, is a linear combination of physically interesting states. In the case NC = 3 it is a combination of nucleons and nucleon isobars in their various S, and I’, states. For larger values of N,-, states with values of S and T larger than f are also included. In the real world, where NC = 3, such states are of no physical interest.

It is an elementary matter to use such a wave function to calculate the axial-vector coupling constant gA for a quark hedgehog:

NC g$X(HI c a1’)71’)\H), (4.3)

i=l

where the sum in eq. (4.3) extends over all NC quarks in the hedgehog. (We shall, by and large, use “h” to denote single hedghog quarks and “H” to denote the set of NC such quarks intended to represent a baryon.) This matrix element is trivial and tells us that gA is precisely NC times the fundamental axial vector coupling constant for quarks. We shall make use of this result later.

Although the wave function built from eq. (4.2) is perfectly suitable for the calculation of the properties of single baryons, it is not the most general K = 0 wave function. In particular, in eq. (4.1) we have arbitrarily chosen the same axis for the quantization of spin and isospin. In greater generality, we could have chosen to quantize isospin along an axis obtained from the spin quantization axis by a rotation through a set of Euler angles. The corresponding generalization of eq. (4.2) is

[~[cY, /I, 71) =A {-sin /3 e-i’*-Y’lu~) +cos /3 ei(u+Y)luJ)

-cosp e -‘((I+Y)ldT) _ sin p ei(a-Y)(dJ)} . (4.4)

As we shall see, the use of such generalized hedgehogs will allow us to retain some vestige of the spin and isospin structure of the underlying baryon interactions which is largely lost in the hedgehog approximation.

We consider a schematic picture of the interaction between quark hedgehogs aimed only at describing its Euler-angle dependence. We imagine two quark bags which can emit pions. These pions can be absorbed by the same bag and lead to the hedgehog correlations of eqs. (4.2) and (4.4). They can be absorbed by the second bag and lead to an exchange interaction. Pion interactions can, for this limited purpose, be approximated by the exchange of a variety of mesons with well-defined spin, parity and isospin.

582 A. Jackson et al / Skynnion-skyrmion interaction

It is convenient to characterize the various possible terms in the resulting interaction by the mesons whose exchange would iead to the same Euler-angle dependence. The lagrangian remains that of eq. (2.1), and there are no elementary p- or o-mesons in our description. There is ample precedent for such shorthand notation which should cause no confusion.

Such an interaction will have the structure of a scalar product of two vertices describing the production of such a meson through the interaction with a single quark in one hedgehog and its eventual absorption by a single quark in the other. There will be an additional factor describing the propagation of the meson:

Wb’ll c @‘ImmH[a]l c @j)lH[(Y])V(q) , (4.5)

where the sums extend over all quarks comprising each hedgehog, and q represents the momentum transferred between the hedgehogs. Of course, V(q) contains important dynamical information which we do not attempt to describe. A list of those form of 0 which we consider is given in table 1 along with the related physical mesons. Without loss of generality we restrict the Euler angles [a’fl’r’] to be [O].

The matrix elements required in eq. (4.5) vanish in precisely half of the cases. Specifically, non-zero results will be obtained only if the operator in question contains neither cr nor T or if the operator contains both u and T. Thus, V( T = 0),

m, o (vector coupled), and p (tensor coupled) mesons can be exchanged between hedgehogs. Similarly, neither a( T = i), 7,~ (tensor coupled) nor p (vector coupled) mesons can couple to hedgehogs. If the spin-isospin correlations emphasized by the hedgehog approximation are important in physical baryons, we would expect that this catalogue of allowed and forbidden meson exchanges would also apply to nucleons. Indeed, it does. Pion exchange and the state-independent central attraction (indicative of T = 0 a-meson exchange) are two of the hallmarks of the nucleon- nucleon interaction. The large isovector anomalous magnetic moments and the analysis of the pseudophysical NR+ OTT amplitudes provide ample evidence that the tensor coupling constant of p-mesons to nucleons is five to six times larger than the vector coupling constant. Similarly, the small isoscalar anomalous magnetic

TABLE 1

Various possible couplings between hedgehogs (or nucleons) and mesons (the spin and parity and related physical mesons are indicated)

J” @(T=O) Meson @I(?-= 1) Meson

0+ 1 C7 7

0- Ys+(fl.q) 1) Y5T+ (a. 4)T P

1- 7, + 4, 0 Y&&T+ CT P (vector)

1- (P +p’), + (u x 4) 0 (P+P’)&&~‘(~xq)~ P

(tensor)

A. Jackson et al / Skytmion-skynion interaction 583

moment of nucleons indicates the strong dominance of vector coupling of o-mesons to nucleons. There is neither phenomenological evidence nor theoretical arguments indicating important contributions of either 77 or CT ( T = 1) mesons in the nucleon- nucleon interaction ‘). Given the uniqueness of the S = T = f nucleon wave function, it is not surprising that a similar catalogue arises in the NC + cc limit of the quark model without the hedgehog approximation. It is more difficult to arrive at a similar conclusion in purely mesonic descriptions. Thus, we regard these qualitative observa- tions as strong albeit indirect evidence that the spin-isospin correlations embodied in the hedgehog approximation to the Skyrme model are physically significant.

Turning more quantitatively to the interaction between hedgehogs, we first consider the exchange of pions. We find immediately

vm(q) = N2,(h[~(yllai7~Ih[~(rl)(h[~a’l)ai~~IhE~cY’I) x Wj”m(q) * (4.6)

The single-quark matrix elements are readily evaluated using eq. (4.4) and found to be

where e,i[a] is the following general orthogonal transformation:

cos/?cosacos y-sinasiny cos j3 sin a cos y +cos ff sin y

e,i[~l= -cos/CIcosasiny-sinacosy cosacosy-ccsBsinasiny sin /? cos a sin j3 sin a

(4.7)

-sin /3 cos y sin /3 sin y .

cos p )

(4.8)

Thus, the exchange of a pion between hedgehogs leads to an interation of the form

C”‘(q) = NZ,e,i[LYle,i[(Y’lqi4iU,tq) - (4.9)

With the assumption (reasonable on the basis of the absence of a pion mass in the Skyrme model) that UT(r) is proportional to l/r at large distances, we find

(4.10)

This is precisely the form of the asymptotic interaction found by Skyrme years ago I). This form was previously derived working exclusively with Skyrme’s model. The point of the present operation is to demonstrate that the same Euler-angle dependence emerges from an explicit quark hedgehog model. Of course, given the low level at which we are implementing the quark hedgehog model in this section, it provides us with no information regarding the functions U(r). The Euler-angle dependence, which is provided, is all we seek.

To be more specific, we can use the explicit form of eq. (4.8) to write V,(r, [a])

in the case [a’fi’y] equal to [O]. This yields

V*( r, [a]) = 2dU

-cos (0 + y) co? g3 ; y 1 1 d=lJ - cos p $

I , (4.11)

584 A. Jackson et al. / Skyrnrion-skyrmion i~ierucfion

where we have chosen the z-axis to lie in the direction of the separation between the two hedgehogs. Similar results can be found for (vector coupled) w-exchange, (T = 0) a-exchange and (tensor coupled) p-exchange. Recognizing that the term involving y. is dominant in the exchange of a (vector coupled) vector meson, we find that w- and ( T = 0) c-exchange lead to precisely the same Euler-angle dependence or, more properly, independence:

(K+KJ)(~,[~l)= u,+u,* (4.12)

The evaluation of the p-exchange potential is equally straightforward and yields

V,(r* [CX]> = -2 cos* $3 cos (CX + y) 1 dU d2U ; -$+$

I -2 cos /3 $3. (4.13)

Regarding these three ingredients as a sufficient description of the interaction, we find

v~(r,[~]~=(u~+u~~-2cos(~+y)cos2~~ 1 -cos p (4.14)

whose total Euler-angle dependence is contained in the three terms shown explicitly. We could have considered other forms of the vertex such as ysyw (describing the exchange of an axial vector particle) or a&” (describing the exchange of a genuine tensor particle). Such terms do not yield new Euler-angle dependence. This is not surprising since our task is essentially to construct a scalar (ie. the energy) from, at most, one factor of e&a] and two factors of qi. There are only three possible scalars having Euler-angle dependence I, eii[a], and qieJa]e. These are precisely the forms reflected in eq. (4.14). There is no evidence suggesting important contributions to the nucleon-nucleon interaction from either the exchange of axial vector or tensor mesons.

For any given hedgehog separation, the coefficients in eq. (4.14) are fixed. We have studied our caIculation of !-&(I; [a]) as obtained from the Sky~e-model calculations of sect. 2 at several values of r and find that eq. (4.14) provides a good description of the Euler-angle dependence within numerical uncertainties. An example at r = 0.8 fm is provided in table 2. It should be noted that we have chosen this separation because it leads to the worst discrepancies between the calculated interaction and the form of eq. (4.14). This does not constitute a proof that eq. (4.14) is the most general form of the interaction which can arise in the Skyrme model. (Such proof is easily made for those parts of the interaction coming from the quadratic term in the Skyrme lagrangian. The quartic term is more difficult.) However, table 1 indicates that eq. (4.14) is sufficient in practice. It also provides a convenient mechanism for separating contributions to the interaction into terms behaving like the exchange of (o+w)-, p-, and m-mesons.

A. Jackson et al. / Skymion-skymion interaction 585

TABLE 2

Comparison of the actual Euler-angle dependence of the skyrmion

interaction with that of eq. (4.14) for a separation of 0.8 fm (the rms

deviation of 6 MeV is consistent with the numerical accuracy of the interaction calculations)

a P Calc. Fit

Y [Meal [Meal

0 0 440 441

0 0 430 426

0 0 398 390 0 0 353 354

0 0 331 339

277 0 388 391

4n 0 266 273 ;n 0 152 156

?r 0 107 107 $7 rl 243 231 P tr 107 107

V,(r)=V,(r,[CY=&r,p=~=0]),

V,(r) = V,(r, [p = lr, a = y = 01)) (4.15)

Given the computational time involved in exploring the Euler-angle and radial dependence of the interaction, it is expedient to consider only three cases,

V,(r)=V,(r,[a=p=r=o]),

and three particularly useful linear combinations,

(4.16)

These combinations are readily integrated numerically to yield U,(r) and U,(r) which may then be used for the construction of the separate p- and r-contributions to the coefficients in eq. (4.14) if desired. The functions VA, V,, and V, have been shown in figs. 3-5, and we give the functions VI, V2, and V, in fig. 6. The derivatives appearing in the expressions for V, and V, are of some interest in themselves. To the extent that V, and/or VP can be described by Yukawa potentials, we see that

= p2 evCLr/r, (4.17)


I 2

Fig. 6. The various ~mbinat~ons of V,, V,, and V, leading to perfect differentials according to eqs. (4.14) and (4.16). (a) Shows the central combination, V,(r). (b) Shows the spin-spin combination, b;(r).

(c) Shows the tensor combination, V,(r). The break again indicates a tenfold scale increase.

r~[t$(e-~r,r)]=r’e-r?,,[l+~+~]. (4.18)

The first of these forms, eq. (4.17), is precisely that associated with the usual spin-spin force between nucleons. The equation for V, shows the expected result that GT- and p-exchange contributions to this component of the interaction add. Eq. (4.18) yields the familiar radial form associated with the tensor force, and the above equations for V, indicates that T- and p-exchange contributions in this case tend to cancel. This result is also the expected one 4*‘9).

A. Jackson et aL / Skymion-skyrmion interaction 587

This simple description of the interaction offers one way to convert the skyrmion-

skyrmion interaction into a nucleon-nucleon interaction. It is merely necessary to replace the hedgehog wave functions of eq. (4.5) by nucleon wave functions (in a quark model). These wave functions are readily constructed for arbitrary NC. Starting with NC = 1, it is obvious, for example, that

IPf NC = 1) = lut> * (4.19)

For NC = 3, we add two quarks with T, = S, = 0. Symmetrizing the resulting wave function with respect to spin and isospin we find

IP?Nc = 3) = &?uTdS.) +lutdJu?> + k-GM>1

+P[bW> + bWIu&) + b.lO-Q> + h-Vu?) + bW + kWu?>l . (4.20)

The requirement that this three-quark system should have S = T = 4 can be imposed by requiring that the coefficients (Y and j3 be chosen in such a way that S, (or T+)

applied to this state gives zero. This leads immediately to (Y/P = -2. Normalization then yields the familiar result that CY =$J2 and p = -2J2. This procedure can be extended to arbitrary NC. It is then easy to show that, for example,

(NN,( T l”‘IN’N,) = N&J,, i=l

and that

(NN,I pl a;‘TyN’Nc) (NIV, = llcr&~N’N~ = 1)

=4(N, +2). (4.21)

These are the only spin-isospin matrix elements which are required for the construction of the nucleon-nucleon interaction.

One immediate observation is that eq. (4.21) provides the information required for the construction of the nucleon axial-vector coupling constant. From the related hedgehog matrix element of eq. (4.3), we see that

& &+2 -=- g: 3N, .

(4.22)

This result has two consequences. First, in the limit NC -*cc, eq. (4.22) yields gz/gg = $. This is precisely the ratio obtained by Adkins et al. ‘). Second, for the physically interesting case of NC = 3, this ratio becomes $ which suggests that finite NC corrections are of substantial importance. This leads us to include precisely those finite NC correction in the value of gA employed in the Goldberger-Treiman relation of eq. (2.9). We note that finite NC corrections are even more important when transforming r- and p-exchange pieces of the skyrmion interaction into a nucleon-nucleon interaction. In this case, a factor equal to the square of eq. (4.22)

588 A. Jackson et al. / Skymion-skyrmion interaction

is required. The situation is somewhat more subtle if the N, ratio is used for both the Goldberger-Treiman relation and the interaction calculation. If this is done, the asymptotic value of the r-exchange interaction between nucleons remains unaltered. The effects of this change are then limited to a change in strength of the quartic term in the Skyrme lagrangian. This would, in turn, modify the short-range piece of the interaction. Specifically, going from NC = 3 to NC + cc would replace the factor of g in eq. (2.9) by 3. The resulting increase in E* would result in a 30% increase in both the B = 1 energy and the magnitude of the repulsion between two skyrmions at zero separation. Changes of this magnitude are of interest, although comparable in magnitude to l/N,. While such “book-keeping” effects by no means exhaust the finite NC corrections to the Skyrme model, they are easy and, we believe, appropriate to include.

Before closing this digression, one more piece of evidence regarding finite NC effects is in order. It is equally simple to follow the steps leading to eq. (4.20) to construct the A(1236) wave function for arbitrary Nc.. (In this case, of course, we must start from the trivial NC = 3 case.) This allows us to calculate the matrix elements of eq. (4.21) in the case where one or both of the nucleons is replaced by A( 1236). These matrix elements are of importance in determining the coupling- constant ratios glrdd/grrNN and gnNd/gVNN. The former is independent of No. The latter is found to be

H

&,A * 9 (N,+5)(N,-1) =? (N,+2)(N,+2)’

(4.23) &NN

This ratio has the value $ in the physical case of NC = 3 and the value g in the limit NC + co. The latter ratio is precisely that obtained by Adkins et al. ‘) from projections from the Skyrme model. These authors regarded this increased coupling constant as a success of the Skyrme model (since it is in substantially better agreement with the empirical value as determined by the width of the A (1236).) The present analysis suggests that this “improvement” is actually an artifact of the NC + cc limit which would be eliminated by the inclusion of finite NC effects.

Returning now to the interactions, we wish to assume that, for any Nc, the radial and color wave functions of quarks in nucleons (and nucleon isobars) are identical to those in the hedgehog. The only differences reside in the spin-isospin wave functions. The nucleon-nucleon interaction may thus be obtained from the hedgehog interaction by replacing the spin-isospin matrix elements in eq. (4.6). In this sense our approach is similar in spirit to the projection of physical baryon states from the skyrmion as performed in ref. ‘). We shall return to this point. The spin-isospin matrix elements of 1 are, of course, the same for nucleons and hedgehogs. Thus, the term U, + U, is unaltered in going to the nucleon-nucleon interaction. Both r- and p-exchange involve matrix elements of ur and behave in much the same way. It is clear from eq. (4.21) that matrix elements of 1 u(~)T(~) are simply proportional to matrix elements of UT (where these operators refer to total nucleon spin and

A. Jackson et aL j Sky~io~-sky~~o~ interaction 589

isospin). The proportionality factor is simply given by eq. (4.21). Thus, the matrix elements of e.g. the one-pion-exchange interaction can be written as

where ai and ri are now nucleon operators, and it is understood that one is to take matrix elements between two-nucleon states. Some rearrangement is useful to cast eq. (4.24) in a more familiar form. We find

d Id --$,ZTl * 723- - - u,

[ 1 dr r dr (4.25)

As noted above, when U,(r) has a Yukawa form, eq. (4.25) looks precisely like the usual one-pion-exchange potential between nucleons. We note that in the present case of a massless pion, the spin-spin term will vanish (asymptotically) when U,(r) is proportional to l/r. This feature of the present calculation could be remedied with the introduction of a symmet~-breaking term in the original lagrangian to generate a non-zero pion mass. We shall return to this point, and a less elegant patch-up, in the following section. Similar manipulations lead to an w/o exchange interaction for nucleons identical to that for skyrmions. Construction of the p- exchange interaction completes the present picture of the nucleon-nucleon interaction:

(4.26)

Eq. (4.26) requires some comment. We note that the radial information required is just that obtained from the skyrmion interaction of sect. 3. There is, in particular, no need to integrate these forms to determine the potentials U,(r) and U,,(r)

separately unless we are curious to see these contributions independently. As we shall see, U,(r) and U,,(r) have the same sign, and eq. (4.26) retlects the general wisdom that p-exchange tends to cancel the P-exchange tensor force and enhance the spin-spin force ‘).

Although eq. (4.26) has been derived through the use of an explicit quark hedgehog model, we believe it to be of more general utility since we can associate the various radial quantities directly with the results of a Skyrme-model calculation. We emphasize that the role of the quark model is purely passive. It has shown how to transform the Euler-angle dependence of a hedgehog interaction into the more familiar spin- isospin algebra of the nucleon-nucleon interaction. As we shall show below, this task can equally well be performed by projection techniques (in the NC + a3 limit) without any reference to quarks. The quark-model arguments do have two virtues.

590 A. Jachon et al. / Skyrmion-skyrmion interaction

First, they cast the problem in a language more familiar to nuclear theorists. Second, they provide a straightforward vehicle for the introduction of certain finite Nc corrections which are not accessible from the pure Skyrme model.

We now relate these manipulations to the kinds of operations performed in ref. ‘) using the Skyrme model alone. In the case of one-body operators, these authors construct matrix elements for an arbitrarily rotated soliton and then obtain various baryon matrix elements by a projection with “baryon wave functions” constructed as suitable functions of the Euler angles.

Such connections are most easily made by writing the hedgehog wave function for a single quark, corresponding to eq. (4.2), as

lh[a])= 1 (fm,~m,lOO)(s =f, m,)lt =;, m,). (4.27) m, m,

The rotated hedgehog state is readily obtained by rotating the quantization.axis for isospin with aid of the usual rotation matrix L&(c@~) [ref. ‘“)I:

The full spin-isospin wave function for the hedgehog is simply the product of NC such wave functions:

(4.29)

It is a tedious business, but elementary, to recouple the NC spins to a total spin of S and a total isospin of T. Of course, S and T must be equal since we have started from a state which manifestly has K = 0. Specifically,

II-&l)= C c&T- M,TM,lOO)~,f,,,,(cypy)lS = TM,)(TM,) . (4.30) 71 MS MT

Note that only a single state exists for each S (and T) given the overall symmetry of the spin-isospin wave function so that no additional quantum numbers are required to uniquely label the No-particle states of a given S (or T). The rotation matrices satisfy a simple orthogonality relation:

This relation allows us to determine the states IS = TM,TM,) by a simple and completely rigorous projection:

c7( T-M,SM,100)ISM,TM,)=~ I

4al g%s,,,, (bl)bb1>. (4.32)

591

+W+113 64~~

dIa3 dEdl ~T1\3s~,(T~‘J)~~~~=(fal) x~~~~‘l~~{~~ 1 (4.33)

Again, this relation is exact, However, it invites a simplifying approximation in the limit as NC tends to infinity. The hedgehog matrix element (H[cr’][N[ar]) is the product of NC factors each of which is less than or equal to one. Measuring the Euler angles [a@y] with respect to [a’/?‘-$], these factors are

cos $p cos (fc# +$y) 1 (4.34)

Thus, the matrix element in eq. (4.33) is strongly peaked when [a&-j equals [a’j3’$J. In the NC + a3 limit, we are safe in replacing ~~~~~~~~~~~) by SJ&&[~]). This enables us to write

where N1( NC) is of order l/NC. The remaining integral is simply the orthogonality integral so that

(4.36)

Calculations of one-body matrix elements can be performed along similar lines, Specifically,

(SMsTMrl F BiISM’,T’M~) i-l

= (2?r+ 1)92T’+ 1)3’2 I&

64~~ C7&

x J dCa1 @#a’1 ~~~~~*~r~~~~- f;?;~~AC~72(Hfali~~=~~~r~~~ > (4.37)

where we have used the explicit form of the ~~ebsch-Jordan coe~~ie~ts and have made use of the fact that matrix elements of each @ are identical, As in the approximate calculation of c,, the hedgehog matrix element involves one non-trivial term and (NC - 1) equal factors which are less than or equal to one. Again, the matrix element is peaked for equal Euler angles, and we are encouraged to make the approximation

(ST1 C OilS’T’)= (2T-t 1)“2(2T+ 1)“’

87r2 r

592 A. Jackson et al. / Skynnion-skynnion interaction

where S = T and S’= T’. The coefficient Jz(N,) is, like X,(NC), of order l/N, in the large No limit. More to the point, the ratio [JVJ N,)/X,( No)] is one in this limit. (This can be shown by tiresome but simple calculations which do not seem essential to reproduce.) The point of this exercise is that matrix elements of operators between baryon states can be obtained directly from diagonal matrix elements of hedgehog states (in the sense of equal Euler angles). It is precisely such information which is accessible in Skyrme-model calculations. Indeed, eq. (4.38) is essentially the approach adopted in ref. ‘) and, as noted, is valid in the large No limit. Corrections to this expression can be seen to be one order lower in No. This result is expected and the resulting corrections are merely the book-keeping corrections previously noted.

Finally, it is useful to illustrate the use of eq. (4.38) for the specific case when Oi is equal to u(zi)r?) which is appropriate for the calculation of gA and the coupling constant ratio of eq. (4.23). In this case the hedgehog matrix element has a particularly simple form:

w[+l”dyw4) = m+z444) * (4.39)

This matrix element can be expressed conveniently in terms of the rotation matrices:

(~bll4c4) = %o(bl) . (4.40)

In this case, eq. (4.38) reduces to

NC (SM,TM,I C a!‘b~)~S’M;T’M’T) =

(2T+1)1’2(2T+1)“2

i=l 87r2

The remaining integral is a familiar one and leads to the result (4.41)

(SM,TM,I “c’ a!“~~)~S’M~T’M’7) = (2T+1)1’2 (SM,lO~S’M~)(TM,lO~T’M’,). i=l (2T’+ 1)1’2

(4.42)

It can be verified that this expression reproduces the previous results for gA and

grrl.LJglrrW in the large-color limit. Similar results can be obtained for other operators which describe the interaction

between two hedgehogs. In this case the interaction is expressed microscopically as a sum of scalar products of two single-quark operators. Arguments analogous to those leading to eq. (4.38) can be obtained to permit the determination of the interaction between two baryons, given the interaction between hedgehogs, from a similar projection procedure. In this case, we require matrix elements of the form of eq. (4.5) and two projections similar to eq. (4.38) are required. The necessary matrix elements come directly from the Skyrme calculation without ever mentioning quarks. This approach leads again to eq. (4.26) for the description of the nucleon-

A. Jackson et al / Skytmion-skynnion interaction 593

nucleon interaction in the NC -*cc limit provided that the skyrmion-skyrmion interaction is described by eq. (4.14). As indicated by table 1, this is the case.

We wish to emphasize that the primary purpose of this section has been didactic. We have stressed similarities between the Skyrme approach to the nucleon-nucleon interaction and more conventional approaches involving quarks and the exchange of mesons. Given the skyrmion interaction as a function of Euler angles and separation between the particles, these approaches lead to the same statement regarding the nature of the nucleon-nucleon interaction provided that one is content with taking the large color limit. The quark hedgehog model is not without more active virtues. It has suggested a useful parameterization of the Euler-angle dependence of the skyrmion interaction. It suggests certain finite NC corrections which can be important. In the case of g,,Jg,,, it would appear that such corrections explain the apparent difference between Skryme- and quark-model results. In the case of the nucleon-nucleon interaction, as we have constructed it, such corrections have no influence on the interaction at those large distances dominated by one-pion exchange. At shorter distances, they suggest roughly 30% corrections to the interaction. Although we do not claim that these corrections are exhaustive, they do provide some measure of the reliability of Skyrme-model calculations.

5. The nucleon-nucleon interaction

The aim of this section is to implement the techniques of sect. 4 to obtain a nucleon-nucleon interaction from the skyrmion results of sect. 3 in a form which is suitable for comparison with a realistic (semiphenomenological) nucleon-nucleon potential. For purposes of comparison, we shall adopt the Paris potential ‘I). This potential has a dominant theoretical piece which contains one pion exchange and a calculation of two-pion-exchange effects obtained with the aid of dispersion theoretic techniques and suitable (analytically continued) IAN scattering data. At short distances (i.e. internucleon separations of less than 0.8 fm), this theoretical potential is jointed smoothly to a phenomenological potential which permits a full quantitative fit to low-energy nucleon-nucleon scattering data. We believe that the Paris potential represents the most satisfactory compromise between theoretical and phenomenological potentials currently available.

To begin with we show the central, (u, - +)(T~ - T*), and tensor components of the nucleon-nucleon interaction. These have been obtained from eq. (4.26) with the radial functions given by eq. (4.16) being given by the Skyrme-model calculations of sect. 3. Here, we have used NC = 3. Given our implementation of the Goldberger- Treiman relation, eq. (2.9), the use of NC + 00 would be inconsistent and would result in spurious behaviour of the nucleon-nucleon interaction for large r. These results are shown in figs. 7-9 in which we have adopted the simplified notation

(5.1)


Fig. 7. The central nucleon-nucleon potential V,(r) as deduced from the skyrmion-skyrmion potentials

using eqs. (4.26) and (5.1). The dashed line indicates the asymptotic form of eq. (5.2).

The effects of one-pion exchange are confined to V, and VT,.. Since our initial

lagrangian assumes a pion mass of zero, the asymptotic contribution to V,, from

the pion is zero. This result has been noted in connection with eq. (4.17) above.

The tensor force does have a non-zero asymptotic pion contribution and, for large

r, is proportional to l/r3. The various large-r limits are found to be approximately

V,(r) + u, e-+c’/ r , v, = 2500 MeV - fm , p, = 3.05 fm-’

V,,(r)+v,,e-@‘-T’/r, a,,=-194MeV*fm, pu, = 2.93 fm-’

V&r)+ UTI/r3+21T2[3/r3+3CLT/r2+CL~/r]e-’”T’

V -rI=30.95MeV.fm3, t+=-15.154MeV.fm3, pT = 2.725 fm-’ (5.2)

The value of uT1 is consistent with the value which would be expected on the basis

of the asymptotic form of the interaction given by eq. (3.12). The discrepancies

between the two numbers is less than 1% which is better than the accuracy which

we would claim for our numbers. By construction, this number is also consistent

with the usual pion-exchange potential with the empirical coupling constant.

It is of some interest to see the contributions to V,, and VT, from P-exchange

and p-exchange separately. These contributions can be obtained from eq. (4.16) by

integrating V,(r) and V3( r) twice numerically and then determining U,(r) and

U,(r). From the form of the differential operators, it is clear that U,, +2U, is

determined up to a constant term and a term proportional to l/r. The constant term

can be determined by the requirement that U, +2 U,, should vanish for large r. The l/r term is determined by the observation that the pion is the only zero mass

particle in the calculation, and such a l/r term would thus appear only in U, Thus,

the pion contribution to V, and VT, are shown in figs. 8 and 9. Fig. 9 reveals that

there is a substantial cancellation between 7~ and (tensor coupled) p-meson

A. Jackson et al / S~~ion-s~~jon interaction

-I

t 1 1

V,:(rl(MeV)

(pT=0.7F-‘l

595

I .u z.u r ffm)

Fig. 8. (a) The nucleon-nucleon spin-spin potential VToTAL (r) showing also the contribution from the (massless) pion atone, V;:(r). (b) The contribution to V,(r) from a massive pion according to eq. (5.3).

The break indicates a tenfold scale change.

596 A. Jackson et al. / Skyrmion-skynnion interaction

Fig. 9. The nucleon-nucleon tensor potential VT,(r) showing the total potential and the contribution

from the pion alone, V;,(T).

exchange contributions to V&r). This result is familiar from the folk-lore of one-boson-exchange potenials 4*‘9,2’).

The expected coherence of T- and p-exchange contributions to V_(r) at large distances is not realized in fig. 8. This is due to our use of a zero-mass pion as its asymptotic contribution to V,(r) is precisely zero. The use of zero mass pions will also complicate comparison with phenomenological potentials, and it seems useful to modify the results of figs. 8 and 9 to include finite-pion-mass effects. The most satisfactory approach to this problem would be to introduce a suitable symmetry- breaking term in the lagrangian, eqs. (2.1)-(2.3). Skyrme suggests an additional term of the form

which seems satisfactory. Such a term would require solution of a new Euler equation to determine the B = 1 chiral angle for the massive case. The strength of the phenomenological quartic term would require adjustment to guarantee that the Goldberger-Treiman relation is again satisfied. Using the new chiral angle, the interaction calculations of sect. 3 could be repeated. Such calculations have been performed and will be reported elsewhere. A far simpler expedient, which seems adequate for our purposes, is to identify the asymptotic power-law term coming from massless pion exchange and to replace it by a suitable Yukawa form. This replacement would appear to be legitimate for large r where we can perform a rigorous analysis of the interaction analogous to that leading to eq. (3.10). At small distances, the results of the more formally correct calculations indciate that the form

A. Jackson et al / Skyrmion-skyrmion interaction 597

of the interaction is no more singular than in the massless case “). Thus, our replacement terms should include suitable short-range modifications aimed at avoid- ing all l/r singularities at short distances. Thus, we obtain approximate results for massive pions by the following simple modifications:

Ur; = U, + (5.055 MeV * fm)[eePwr - e-4pJl f, (5.3)

(5.4)

In both equations p,, is set equal to the empirical value of 0.7 fm-‘. The use of eqs. (5.3) and (5.4) does not materially alter the cancellation of T- and p-exchange contributions to V,. It does reinstate the expected coherence of T- and p-exchange contributions to V,. At shorter distances, the introduction of finite-pion-mass effects results in changes in the interaction on the order of 20%. These components of the calculated nucleon-nucleon interaction are shown in figs. 11 and 12.

Before making a detailed comparison of Skyrme model and phenomenological potentials, it is useful to make some comparison of the asymptotic forms of the various potentials. For the pion exchange potential we find that the Paris potential yields the asymptotic coefficients 5.129 MeV * fm and 3 1.44 MeV - fm which are to be compared with the values 5.055 MeV - fm and 30.95 MeV * fm in eqs. (5.3) and (5.4). We have repeatedly stressed that our calculation was designed to reproduce precisely these asymptotic propertes of the nucleon-nucleon interaction, and this comparison is a comforting numerical check. Assuming that V,(r) is completely

dominated by (vector-coupled) o-meson exchange, we can use the asymptotic form of eq. (5.2) to determine the o-meson mass and coupling constant. The range of this term is consistent with an w-meson of mass 610 MeV which is only 20% less than the empirical value. Using the results of ref. 19) we find a coupling constant, g’w of 12.5. This value is somewhat smaller than the results obtained from an analysis of nucleon isoscalar electromagnetic form factors 23) but is roughly the same as is conventionally adopted in more purely phenomenological one-boson-exchange models of the nucleon-nucleon interaction. We can determine the asymptotic properties of the effective p-meson from either V,, or VT, given by eq. (5.2) by removing the pion exchange contributions as outlined above. The p-meson mass is found to be roughly that of the w-meson, 590 MeV. The (tensor) coupling constant g’, is found to be approximately 15. This value is roughly twice as large as that obtained from nucleon isovector ellectromagnetic form factors but only 15% larger than the value obtained from the analysis of (analytically continued) VN scattering data. The quality of the various asymptotic forms of the Skyrme-model interaction is remarkably high. Given the nature of the analysis of ref. r8), it was perhaps to be expected that something p-meson-like would emerge from the present calculations.


V,(r )(MeV)

Fig. 10. The solid line denotes the central potential V,(r) deduced from the Skyrme model and modified

to simulate massive pion exchange as indicated by eqs. (5.3) and (5.4). The crosses indicate the

corresponding component of the Paris potential. The break in the curves indicates a tenfold scale change.

We have no similar argument to predict the appearance of the w-meson. We do not claim that the Skyrme lagrangian actually generates p- and W-mesons, but there is no doubt that the resulting interaction contains components which behave like the exchange of p- and w-mesons with masses and coupling constants which are approximately correct.

These results for massive pions should be suitable for comparison with the Paris potential and such comparisons are made in figs. 10-12. The agreement between

2c

IO

V,,(r)(MeV)

Fig. 11. The nucleon-nucleon spin-spin potential V,(r) as in fig. 5.10.

A. Jackson et at. / Skyrmion-skyrmion interaction 599

Fig. 12.

I I ‘ 1 L f * t I1 I ) I I I .o 2.0

The nucleon-nucleon tensor potential V&r) as In fig. 10.

the values of V,(r) and l&(r) obtained from the present calculations and from the Paris potential is quite good for internucleon separations greater than 1 fm. We emphasize that this agreement is not a trivial consequence of the one pion exchange contributions to these components of the interaction (which are incorporated in both descriptions). At a distance of 1.3 fm, for example, one pion exchange represents one-third of the Vm component of the Paris potential. Thus, the non-one-pion- exchange component of the interaction differ by less than 30% from the corresponding pieces of the present interaction. Similar results obtain for VT, At the same separation the pion accounts for 145% of the Paris VT, The non-one-pion-exchange pieces of both the Paris tensor potential and the present calculation are negative and their discrepancy is less than 25%. At larger separations, these components of the interaction are more totally dominated by one pion exchange and their agreement is both better and, to some extent, misleading. For distances less than 1 fm, the Paris potential is dominated by the constraints of the fit to low-energy nucleon- nucleon scattering data. Since this piece of the interaction must repair possible inadequacies in the longer-range theoretical part of the interaction, the resulting interaction can possess an inordinate sensitivity to the details of the phase shifts. Compa~sons with the results of our Sk~me-model calculations are not meaningful in this region. Further, in practice the tensor potential must be used with two wave functions having at least two units of orbital angular momentum (between them). Matrix elements will thus have at least a factor of r2 at short distances further reducing the significance of comparisons in this region.

The most striking indication of a genuine inadequacy in the Skyrme calculatians is provided by fig. 10 which shows the purely central potential. The central short- range repulsion, required by the “saturation” of nuclear binding energies and qualitatively identified with w-meson exchange, is present in both caIculations.

600 A. Jackson et al. / Skyrmion-skytmion interaction

However, the central attraction of intermediate range, clearly present in the Paris potential at distances greater than 1.1 fm, is missing in our calculations. This attraction, usually identified with a phenomenological isoscalar scalar meson and more properly described as the exchange of a pair of interacting S-wave pions, is a crucial ingredient in the nucleon-nucleon interaction ‘). It is the primary glue which binds nuclei. Its absence in the present calculation is the most conspicuous fly in an ointment of otherwise surprisingly satisfactory quality. There are a number of possible explanations for its absence. Some represent fundamental inadequacies of the Skyrme model or of our implementation of it. These will be discussed in the following section. A simpler explanation might be that our sharply constrained variational calculation of the interaction is not adequate to reveal this important feature of the interaction. This point requires immediate attention.

We have investigated the effect on the potential energy of relaxing the constraint on the chiral angle, e(r) = OB=‘(r), used in eq. (3.2) by allowing for a localized radial scale change. This corresponds to the use of

6(r) =.P’(p(r)), (5.5)

where we have parameterized p(r) as either

h

h +cosh ((r-r,)/c> 1 (which expands the soliton) or

p(r) = r h

cosh((r-r,-,)/c) 1

(5.6)

(5.7)

(which shrinks it). We have varied the parameters h, r, and c to minimize the B = 2 energy separately for each set of Euler angles and each separation. The lowering of the B = 2 energies is not large. For example, at r = 0.8 fm V, is reduced by 12% ; for r = 1.4 fm it is reduced by 4%. Nowhere do we gain sufficient energy to make the central potential attractive for any soliton separation. Given our overall numerical expectations, we believe that the changes resulting from such scale transformations are sufficiently small to justify the use of the more sharply constrained eq. (3.2).

The phenomenological Paris potential has a variety of terms not present in the Skyrme model which leads only to the three terms of eq. (5.1). These terms, inaccessible in the present calculations, can assume a variety of forms of which we list some of the dominant ones:

(CT, * &Vi, (71 - dV7, &VT, L* SVLS, and L - S( T, * Q) V,, .

Roughly speaking, V, and V, are smaller than V, at short distances and smaller than V, at larger distances. For intemucleon separations on the order of 1 fm, however, they cannot be discarded. The tensor force arising from the exchange of an isoscalar particle VT is dramatically smaller than V,, for intermediate and large

A. Jackson et al. / Skynnion-sky&on interaction 601

distances which is generally understood in terms of the dominance of one pion exchange. The two spin-orbit forces are large and of considerable importance in the description of both low-energy nucleon-nucleon scattering and the properties of nuclei. Like the quadratic spin-orbit force not shown, the spin-orbit force is evidently inaccessible in adiabatic calculations such as ours. There are two possible origins of these terms. They can come from the exchange of other mesons which do not couple to hedgehogs (e.g. the n-meson or an isovector scalar meson) or from other couplings of the mesons effectively embodied in the present calculations (e.g. tensor coupled o-mesons or vector-coupled p-mesons). We can make no statement about such effects. Of greater interest, the mesons and couplings presently included can lead to all of the forms indicated when one allows for the effects of small (i.e. non-leading) terms in a more careful non-adiabatic treatment of one boson exchange 19). Again, these terms do not make contributions to the interaction between hedgehogs. However, if we are willing to take the picture of sect. 4 more seriously, we can estimate them by allowing the hedgehog interaction to determine the basic radial forms associated with +, p-, and o-exchange and relying on a one-boson- exchange picture to identify-the various spin-isospin dependences of the individual exchanges.

Let us illustrate this for the case of vector-coupled o-meson exchange. The various pieces of this interaction, in a one-boson-exchange model, have the form 19)

V,W = g: ePL-‘/ r , (5.8)

VZ= $$ g:e-+-'/r=$D,(V:), ( 1 (5.9)

VT = -(+$$I: e-Y-‘/r[&+&+ 1] =& &(v,w) , (5.10)

V& = s gt eecLur ’ [-

r 1 +cLw -1 r2 r =&QW:), (5.11)

where we have used the differential forms of eqs. (4.17) and (4.18) and defined a new differential operator

That V,O, V,O, and VF, represent small component effects is indicated by the presence of the nucleon mass, m, in the related denominators. Predictions for these force components can be obtained by identifying V,” with the central force term V, shown in figs. 7 and 10 and applying the various differential operators numerically. The results of such a hybrid calculation are shown in figs. 13-15. The qualitative behaviour of all three small components is correct at all distances and the comparison with V, and VU coming from the Paris potential is startlingly good. The qualitative nature of these results should come as no surprise. It has long been known that the


-6O- I

-70 - I I

-8O- x V,(r)(MeV) r hd

I I I I I I I I 2

Fig. 13. The solid line represents the (non-adiabatic) isoscalar spin-spin potential inferred from the

Skyrme model results using eqs. (5.8)-(5.11). The crosses represent the corresponding component of the Paris potential. The break indicates a tenfold scale increase.

nucleon-nucleon interaction displays the regularities of a boson-exchange picture which includes one- and two-pion exchange and w-meson exchange. These are essentially the ingredients in the present Skyrme-model calculations. For these components of the interaction, the use of adiabatic approximations in our Skyrme calculations would appear to more serious than the hedgehog approximation.

The situation is somewhat less satisfactory in the case of V,, and V,. One-pion exchange makes no contribution to these forces. Similar estimates of small components from (tensor coupled) p-meson exchange lead to a properly attractive VLsl

which is roughly half the magnitude of the corresponding term in the Paris potential.

IO x I I , Isoscalar Exchange _ 1

: VT(r) (MeV) _

\ I \ ?I 2 r h-d

0 3’3’:

-10 -

Fig. 14. As fig. 13 for the (non-adiabatic) isoscalar tensor potential, V,(r).

A. Jackson et al. / Skynnion-skymion interaction 603

f,,,,,,,,,,,

: Isoscolor Exchange

V,,(r) (MeVI -

r Mm) - I I 2

Fig. 15. As fig. 13 for the (non-adiabatic) isoscalar spin-orbit potential, V,(r).

The disagreement in VT is more substantial. At short distances both are attractive (although the Paris potential has considerably larger magnitude). For distances greater than 1 fm, our potential is small and positive while the corresponding component of the Paris potential remains negative and roughly twice as large. Since one-boson-exchange models do lead to a repulsive V. from the exchange of a (tensor coupled) p-meson, we regard this disagreement as an indication of the importance of mesons and meson couplings not present in the hedgehog approximation to the Skyrme model.

In sect. 4 the quark model with exchanged mesons was used for purely passive interpretative purposes or for the introduction of minimal book keeping corrections aimed at gauging the importance of l/N, corrections. The present calculation of “small” components in the nucleon-nucleon interaction places a far greater reliance on this analogy than we are prepared to defend. Such calculations are nontheless useful indicators of the limitations of the hedgehog approximation and adiabatic treatment which are important ingredients in our calculations.

6. Conclusions

Flying in the face of certain popular trends in nuclear physics, we have adopted the point of view that the correct low-energy, large-color limit of quantum chromodynamics is not a bag model but rather an effective lagrangian describing the interaction between effective mesons. Since the most important empirical aspect of this lagrangian would seem to be chiral symmetry, we have adopted the Skyrme model as a likely candidate for the low-energy limit of QCD. This choice is supported by

604 A. Jachon et al. / Skynnion-skyrmion interaction

the ability of this model to generate solutions which can be identified with baryons. We have performed an adiabatic and highly constrained variatioal calculation aimed at providing an estimate of the interaction between skyrmions. As Skyrme showed many years ago, this simple model does provide a correct description of the asymptotic interaction between skyrmions and, following projection, between nucleons in terms of the exchange of a single pion. (The strength of this asymptotic interaction is guaranteed to be correct only if the pion decay constant and the nucleon axial vector coupling constant are fixed at their empirical values.) While the asymptotic dominance of one pion exchange is a trivial consequence of the Skyrme model, it is far from easy to obtain in extreme versions of the bag model ‘l).

Comparisons with semi-theoretical potentials (which should be reliable for large internucleon separations) indicate substantial agreement in the dominant components of the nucleon-nucleon interaction. Viewed in the light of bag-model descriptions of the nucleon-nucleon interaction, this would appear to be a singular success. From the viewpoint of more traditional nuclear physics, these results would appear merely as meeting minimal conditions for an acceptable model of the interaction: the basic nature of the results indicating only the known importance of chiral symmetry (as manifested in one- and two-pion exchange potentials) and the important role of the exchange of vector-coupled w-mesons. From this point of view the present calculations suffer the serious defect of not providing any indication of the central, intermediate-range attraction arising from the exchange of an S-wave pair of pions. This absence is disturbing and merits some discussion.

In microscopic models of the two-pion-exchange potential, the most important ingredient in obtaining reasonable statements regarding the range and strength of an effective “sigma” meson is the quality of the description of the S-wave TQT phase shifts 24). The longest-range piece of this interaction is dominated by low-energy ~7r phase shifts and, thus, the S-wave scattering length. One of the initial successes of chiral (i.e. sigma) models was their ability to describe this S-wave scattering length. The quadratic term in the Skyrme lagrangian is the usual non-linear sigma model, and it has been shown that the Skyrme model leads to a satisfactory description of both the S- and P-wave scattering lengths. The latter indicates the presence of the p-meson in some approximation. It is, thus, surprising that our present calculations show no indication of m-meson effects. There are several possible reasons for this. The first is that our calculation of the interaction may have been too sharply constrained. The form we have chosen for the B = 2 unitary transformation is correct in the limit of large soliton separations and should be more than adequate in the limit of small separations. Its use in regions where the two solitons have substantial overlap (i.e. 0.6 fm to 1.2 fm) is more questionable. Studies with a somewhat more general form, described in sect. 5 do not reveal the missing attraction. Given the modest energy changes realized in these more general calculations, we do not anticipate that better variational calculations will have materially greater success.

A. Jackson et al. / Skymion-skyrmion interaction 605

In more phenomenological models of the nucleon-nucleon interaction the u- meson has a coupling constant which is substantially smaller than that of the o-meson. In such models it is visible only because its mass is somewhat less than that of the o-meson making its identification possible at large distances. This is trivially realized in phenomenological models where the a-meson mass is an adjust- able parameter. In the case of the Skyrme model the mass of a-meson would emerge from the calculations much as we have seen similar masses appear for p- and w-mesons. If the Skyrme model contained a a-meson of mass greater than or equal to that of the w-meson, we would not see it in the interaction. This possibility could be checked by a study of the skyrmion-antiskyrmion interaction. In this case the exchange of both u- and w-mesons would lead to attraction independent of the Euler angles (due to the fact that the o-meson has negative G-parity). The comparison of baryon-baryon and baryon-antibaryon interactions would thus permit the separation of u- and w-meson contributions.

The most likely explanation is that the Skyrme lagrangian, like the non-linear sigma-model lagrangian it contains, involves a o-meson of infinite mass. In this event even the correct description of the TT scattering length is not sufficient to guarantee the finite-range contribution to the central interaction which we seek. This notion is supported by the fact that both analytic and numerical arguments show that, in the present scheme, the quadratic term in the Skyrme lagrangian (which should contain a-meson effects) makes a contribution to the central interaction which is rigorously zero. Repair of this problem would require the evaluation of vacuum fluctuation corrections to the present classical calculations. While such corrections are known to vanish in the No + 00 limit, their effects can be significant for No = 3. Corrections to the “u-meson propagator”, such as shown in fig. 16, would give the a-meson a finite effectve mass and would lead to the missing attraction. Such repair is not easily implemented since the Skyrme model is, at best, an effective lagrangian. It is not clear which vacuum fluctuation corrections, if any, may be legitimately considered. Even the technical problem of introducing such corrections into the non-linear sigma model is not a trivial one. A satisfactory interim solution might be to supplement the present results with a more traditional estimate of S-wave two-pion-exchange effects ‘“). A more draconian solution would be to supplement the Skyrme lagrangian with an elementary u-meson of finite mass. While

H H

Fig. 16. A sigma-meson exchange process with a vacuum correction to the sigma-meson propagator.


this possibility cannot be excluded by the present calculations, it is rather far from

their spirit.

Again adopting a traditional view of the nucleon-nucleon interaction, the interest-

ing feature of the Skyrme-model calculations is the nature of its predictions regarding

the interaction at short distances. This is a region in which such traditional pictures,

based on the exchange of one or two mesons, must admit to a questionable theoretical

foundation. Although the phenomenological constraints of low-energy nucleon-

nucleon phase shifts have frequently been used to determine this portion of the

interaction, such determinations are suspect. This is due to the fact that the correction

of small inadequacies in the theoretical description of the interaction at larger

distances can lead to major modifications of the interaction at short-distances. The

suggestion of the Skyrme model that the short-range repulsion in the nucleon-

nucleon interaction is on the order of one baryon mass can be of real value in

constraining phenomenological interactions.

The theoretical short-coming of one- and two-boson exchange models of the

nucleon-nucleon interaction lies in the fact that, at short distances, it is no longer

correct to assume that the range of a given process decreases simply with the number

of pions exchanged. There is no reason to believe that the perturbative inclusion

of multi-pion exchange processes will converge. One must rather seek a non-

perturbative approach which can provide a satisfactory approximation to the com-

plete class of multi-pion-exchange diagrams. This is one way of looking at the

Skyrme-model calculations which can be regarded as an approximation to the sum

of tree diagrams which should be valid in the low-energy (and large-color) limit.

Chiral bag models make rather similar predictions regarding the nature of the

interaction at short distances. At zero separation, three quarks can occupy the K = 0

state of lowest energy. The remaining three quarks are forced into the lowest K = 1

state which is some 300 MeV higher in energy. Again, a repulsion of roughly one

baryon mass results. Of course, chiral bag models also allow for a mesonic region

outside the bag in which the interactions are again described by the second-order

lagrangian of the Skyrme model. Such models have been shown to have similar

solitonic behaviour in the meson sector. The primary difference is that, in chiral-bag

models, stability of the soliton is provided by the bag of quarks while, in the Skyrme

model, stability is provided by the phenomenological fourth-order term in the

lagrangian 16). We feel confident that a calculation of the nucleon-nucleon interac-

tion using the chiral bag model would lead to results similar to those obtained here

in regions in which the two bags do not overlap. At zero separation, as noted, both

models are expected to lead to similar statements regarding the magnitude of the

repulsion. What remains to distinguish such calculations is, then, the precise manner

in which the two models interpolate between roughly the same values for the

interaction at - 1 fm and zero separation. We find it unlikely that differences arising

in such parallel calculations will lead to significant differences in the low-energy

nucleon-nucleon interaction. Thus, although we do not question the two-phase

A. .Iackson et al. / Skynnion-skynnion interaction 607

description of nucleons which is the most appealing feature of the chiral bag model, it seems likely that the description of the nucleon-nucleon interaction at low energies may not be sensitive to the explicit inclusion of quark degrees of freedom. A similar insensitivity of low-energy single-baryon observables to the bag radius has been demonstrated recently in the context of the chiral-bag model 16).

Although we do not believe quark bag models to be essential in understanding the nucleon-nucleon interaction, it has nonetheless proved useful to refer to them in the present calculation. This helped us express the Skyrme results in more familiar language. More practically, we used quark models to suggest the Euler-angle dependence of the hedgehog interaction and found this description to be consistent with our numerical results. Quark-model considerations were also useful in seeing how to transform the hedgehog interaction into an interaction between nucleons and how, in the NC + 00 limit, this leads to results identical to those obtained by projection from a pure Skyrme model. More importantly, we have seen how hedgehog spin-isospin correlations lead to a catalogue of the mesons which couple to baryons and, in the case of vector mesons, the nature of their coupling. The correspondence between this list and the mesons which actually coupling strongly to nucleons provides additional evidence for the physical legitimacy of the hedgehog assumption in the Skyrme model. These goals could have been reached without explicit references to quarks. Explicit quark models are important, however, if we wish to assess the accuracy of the large color limit which is an unavoidable part of the Skyrme model.

We have seen that the physically interesting case of NC = 3 can lead to substantial changes in coupling constants. The quark model, for an arbitrary number of colors, can provide a useful measure of the possible magnitude of such corrections. (In the case of the ~NA/TNN coupling constant ratio, for example, such an analysis suggests that the apparent improvement obtained with the Skyrme model is precisely an artifact of the large-color limit.) Finite NC corrections can also affect the nucleon-nucleon interaction. As noted, if care is used in the implementation of the Goldberger-Treiman relation, the asymptotic form of the interaction will be independent of the number of colors. The NC dependence can, however, result in roughly 30% modifications in the interaction at short distances as well as similar uncertainties in the masses of single baryons. We have made the obvious book-keeping corrections for finite color suggested by the quark model in the results reported in sect. 5. Under the self-serving assumption that the 30% corrrections noted above represent the bulk of the finite NC corrections, it is unlikely that the additional corrections would make a substantial correction to the calculation of low-energy two-nucleon observables which are not sensitive to relatively small changes in the magnitude of the repulsion at short distance.

It is useful to keep in mind that the results obtained here depended on only two parameters which were, in fact, exhausted a priori by the pion decay constant and the ~TN coupling constant. The remaining results of sect. 5 should be viewed as

608 A. Jackson et al. / Skyrmion-skynnion interaction

predictions of the Skyrme model. The semi-quantitative success of our calculations is all the more intriguing in view of the extraordinary economy of the model from which they were obtained. Finally, we emphasize that the Skyrme model has been adopted as little more than an enlightened guess for the true effective lagrangian for QCD. Nonetheless, at the l/N, level of reliability expected for the present calculations, the nucleon-nucleon interaction does not indicate irreparable defects in the Skyrme lagrangian.

We would like to thank G.E. Brown, M. Rho, V. Vento and, particularly, L. Castillejo for any number of helpful discussions and suggestions. This work was supported in part by the US Department of Energy under contract no. DE-AC02- 76ER13001.

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The skyrmion-skyrmion interaction