## Abstract

We compute new solutions of the Skyrme model with massive pions. Concentrating on baryon numbers which are multiples of four, we find low-energy Skyrmion solutions that are composed of charge four subunits, as in the α-particle model of nuclei. We summarize our current understanding of these solutions, and discuss their relationship to configurations in the α-particle model.

## 1. Introduction

Skyrme's vision was that the three pion fields form a nonlinear semiclassical medium that dominates the interior of nuclei. He proposed what is now called the Skyrme model (Skyrme 1961, 1962), whose Lagrangian is a version of the nonlinear sigma model, in which the pion fields *π* are combined into an *SU*(2)-valued scalar field(1.1)

where the three components of ** τ** are the Pauli matrices and the sign of the square root is chosen to give a smooth field. There is an associated current with spatial components

*R*

_{i}=(∂

_{i}

*U*)

*U*

^{†}. Restricting to static fields, the energy in the Skyrme model is given by(1.2)and the vacuum field is

*U*=1. In the above expression, the energy and length units (which must be fixed by comparison to real data) have been scaled away, leaving only the pion mass parameter

*m*. This parameter is proportional to the (tree-level) pion mass, in scaled units. Motivated by the results from Battye

*et al*. (2005) and Battye & Sutcliffe (2006), we shall set

*m*=1 for most of our study.

The model has a conserved, integer-valued topological charge *B*, which is identified with baryon number. This is the degree of the mapping , which is well defined, provided *U*→1 at spatial infinity. In the above units, there is the Faddeev–Bogomolny energy bound, *E*≥|*B*|. Remarkably, for a purely pionic theory, Skyrme showed that there are topological soliton solutions—Skyrmions—that can be identified with baryons (for a review, see Manton & Sutcliffe 2004). The baryons are therefore self-sustaining structures in the pion field and explicit nucleonic sources are not needed.

Classically, the Skyrmions have no spin, but have rotational and isorotational collective coordinates which need to be quantized. In this way the basic, *B*=1 Skyrmion acquires spin and isospin, and its lowest energy states can be identified with spin 1/2 protons and neutrons. The next lowest states are identified with spin 3/2 delta resonances. The masses of the nucleons and deltas can be used to calibrate the Skyrme model, and it is then found that other physical properties of quantized Skyrmions are in reasonable, but not excellent, agreement with data (Adkins *et al*. 1983).

Skyrme and later workers hoped that multi-Skyrmion solutions, with baryon numbers greater than 1, could similarly be quantized and identified with nuclei. This programme has had some success (Braaten & Carson 1988; Braaten *et al*. 1990; Carson 1991; Walhout 1992; Battye & Sutcliffe 1997, 2001, 2002; Irwin 2000). Minimal-energy solutions of the Skyrme field equation, with baryon numbers *B*=2, 3, 4 and 6, have the right properties to model the deuteron, the isospin pair ^{3}H and ^{3}He, the α-particle ^{4}He and the nucleus ^{6}Li. In each case, the collective coordinate quantization is constrained by the symmetries of the classical solution and the need to impose Finkelstein–Rubinstein (FR) constraints (Finkelstein & Rubinstein 1968), which encode the requirement that the quantized *B*=1 Skyrmion is a spin 1/2 fermion. The resulting lowest energy states for *B*=2,3,4 have spin/parity, respectively, *J*^{P}=1^{+}, 1/2^{+}, 0^{+} and 1^{+} (with isospin *T*=0 in the even *B* cases and *T*=1/2 for *B*=3), agreeing with those of real nuclei. However, in the standard parametrization of the model, the binding energies are too large and the sizes too small.

All the classical solutions have interesting shapes (Kopeliovich & Stern 1987; Manton 1987; Verbaarschot 1987; Braaten *et al*. 1990; Battye & Sutcliffe 1997, 2001, 2002); they are not spherical like the basic *B*=1 Skyrmion. The *B*=2 Skyrmion is toroidal and the *B*=3 Skyrmion tetrahedral. The *B*=4 Skyrmion is cubic and can be obtained by bringing together two *B*=2 toroids along their common axis (with one flipped upside down, which breaks the axial symmetry). Finally, the *B*=6 solution has *D*_{4d} symmetry and can be formed from three toroids stacked one above the other. These structures, though very different from what one might expect from other models of nuclei, have some phenomenological support. Forest *et al*. (1996) have determined by Monte Carlo methods the wave functions of a number of light nuclei, regarded as bound states of individual nucleons, and have shown that the tensor forces imply that two-nucleon pairs predominantly form *T*=0, *J*=1 states (and exactly this for the deuteron) and that the spatial wave function is concentrated in a toroidal region with the *z*-axis as symmetry axis for *J*_{z}=0. The apparent dumb-bell shape that occurs for *J*_{z}=±1 can be interpreted as a toroid with its symmetry axis in the *x*–*y* plane, spinning about the *z*-axis. Furthermore, pairs or triples of these tori, which occur in larger nuclei like ^{4}He and ^{6}Li, tend to have their axes lined up, producing stacks of tori.

So far there is no direct phenomenological evidence for the hollow, cubically symmetric structure of the *B*=4 Skyrmion, but this may show up in a more refined analysis of the nucleon position correlations in ^{4}He wave functions, or possibly, by considering a superposition of 0^{+} and 4^{+} spin states (Manko & Manton 2006).

Great progress has been made in determining the classical solutions of the Skyrme model, at zero pion mass, for all baryon numbers from *B*=7 to 22 (Battye & Sutcliffe 1997, 2001, 2002), and various larger values of *B* (Battye *et al*. 2003). All these solutions are of the form of hollow polyhedra, rather like carbon fullerenes. The baryon density is concentrated in a shell of roughly constant thickness, surrounding a region whose volume increases like *B*^{3/2} and where the energy and baryon density is very small. Such a hollow structure clearly disagrees with the approximately constant baryon density observed in the interior of real nuclei, which implies that nuclear volume increases like *B*. (The Skyrmions with *B* between 2 and 6 also have zero baryon density at the centre, but this is perhaps consistent with the known dips of nuclear density seen in the cores of some small nuclei, which are not easy to understand (Anagnostatos *et al*. 1999).) Another difficulty for the Skyrme model at larger *B* is the disagreement between the spin/parity assignments of the lowest energy states and those of real nuclei. In particular, for *B*=7 the classical Skyrmion has a beautiful dodecahedral symmetry, but collective coordinate quantization leads to a lowest spin of *J*=7/2 for *T*=1/2 (Irwin 2000; Krusch 2003), disagreeing with the experimental values *J*=3/2 for the ground states of the isospin doublet ^{7}Li and ^{7}Be. This suggests that the *B*=7 dodecahedral solution is too symmetric to be the ground state and it would be preferable if a less symmetric solution existed, which could have a larger classical energy than the dodecahedron, but be quantized with a lower spin.

However, a crucial discovery has been made recently (Battye & Sutcliffe 2005, 2006). For larger baryon numbers, *B*≥8, the value of the pion mass has an important qualitative effect. The hollow polyhedral solutions discussed earlier were obtained in the limit where the pion mass vanishes. It is tempting to work in this limit because the Skyrme model has greater symmetry there, and careful numerical work has shown that the classical solutions for *B* less than about 8 are fairly insensitive to a reduction of the pion mass from its physical value to zero (Battye & Sutcliffe 2005; Houghton & Magee 2006). But it has now been established (Battye & Sutcliffe 2005, 2006) that the hollow polyhedral Skyrmions for *B*≥8 do not remain stable when the pion mass is set at a physically reasonable value (Adkins & Nappi 1984; Battye *et al*. 2005). The result, in retrospect, is not surprising. In the interior of the hollow polyhedra, the Skyrme field is very close to *U*=−1 (i.e. the antipode to the vacuum value), and here the pion mass term gives the field a maximal potential energy. Consequently, the polyhedra become unstable to squashing modes. The interior region where *U* is close to −1 tends to pinch off and separate into smaller subregions.

New Skyrmion solutions have been found (Battye & Sutcliffe 2006) for 10≤*B*≤16 which have a planar character. They can be interpreted as bound arrangements of lower charge clusters in a planar layer, with *B*=4 cubes and *B*=3 tetrahedra often occurring. A cluster structure is an encouraging development because it is known since the 1930s that many nuclei with isospin zero and with *B* a multiple of four may be described as arrangements of α-particles (Blatt & Weisskopf 1952). The planar Skyrmions found so far are local minima of the Skyrme energy, but it is probable that there are a number of different local minima, and it is not known which will be the global minima. It would be interesting if, over a larger range of baryon numbers that are multiples of four, there exist Skyrmions which are composed of *B*=4 subunits, as in the α-particle model of nuclei. In this paper, we report the results of a search for such solutions. We summarize our current understanding of these solutions and discuss their relationship to configurations in the α-particle model.

Ideally, one would like to work with several values of the (scaled) pion mass, as the quantitative results are likely to be sensitive to its value. One should then recalibrate the Skyrme model to fit, if possible, the masses and sizes of a few nuclei like ^{12}C and ^{16}O, but this can only be done once one has the correct Skyrmion solutions. (One would lose the fit to the delta resonance mass, but this would be no great loss, as the delta resonance is broad and highly excited on the usual energy scale of nuclear states.) Unfortunately, it is computationally too expensive to perform numerical simulations for a range of pion masses, so most of our results are for *m*=1. However, the current evidence (Battye & Sutcliffe 2006; Houghton & Magee 2006) suggests that the qualitative properties of a particular Skyrmion are not very sensitive to variations in the pion mass over a range of values, though as mass thresholds are crossed some stable solutions may cease to exist and new local minima can appear.

We have used the same methods described in detail by Battye & Sutcliffe (1997, 2001, 2002) to numerically relax field configurations to static solutions of the Skyrme model with *m*=1. Most of the results presented in this paper used grids containing 100^{3} points with a lattice spacing Δ*x*=0.1, though larger grid sizes and smaller lattice spacings were also used. The energy computations by Battye & Sutcliffe (1997, 2001, 2002) are for the Skyrme model with *m*=0 and appear to be accurate to within around 0.1%. For *m*=1, there is a more rapid spatial variation of the fields, and consequently, we have found it more difficult to accurately compute energies. We estimate that the errors could be as large as 0.5%, though we expect relative energies to be more accurate than this, as we discuss later.

## 2. Skyrmions and α-particles

We shall now discuss Skyrmion solutions with baryon number *B* a multiple of four, with the aim of making contact with the α-particle model (Blatt & Weisskopf 1952; Brink *et al*. 1970). The α-particle model has considerable success describing the ground and excited states of nuclei with baryon number a multiple of four and isospin 0 (i.e. equal, even numbers of protons and neutrons). In the model, α-particles are treated as point particles subject to a phenomenological attractive potential which becomes repulsive at short range, and in their quantized states they form ‘molecules’.

### (a) *B*=8

We recall first that in the massless pion limit *m*=0, the *B*=8 Skyrmion is a hollow polyhedron with *D*_{6h} symmetry (a ring of 12 pentagons capped by hexagons above and below), with no obvious relation to a pair of cubic *B*=4 Skyrmions. However, motivated by the α-particle model, we expect that at *m*=1, the lowest energy solution is more likely to be a ‘molecule’ of two *B*=4 Skyrmions.

By considering the interaction of two *B*=4 Skyrmions in the massless pion limit, we will see why a connection with the α-particle model is difficult to achieve in this case, but the situation is improved by the introduction of massive pions. With massless pions, the leading order asymptotic fields of a *B*=1 Skyrmion resemble a triplet of orthogonal pion dipoles, and hence the interactions of well-separated Skyrmions can be understood in terms of scalar dipole–dipole forces (Manton & Sutcliffe 2004). The *B*=4 cube has no dipoles, only a doublet of quadrupoles (Manton 1994), so two *B*=4 cubes interact rather weakly by Skyrme model standards. Two *B*=4 cubes placed in the same orientation and next to each other weakly attract, as can be confirmed by calculating the quadrupole–quadrupole interaction. There is an associated Skyrme configuration, but it is only a saddle point, not a local energy minimum. Owing to a significant short-range octupole interaction in the single pion field component that has no quadrupole moment, it is better to twist one cube by 90° relative to the other around the axis joining them. Such an initial configuration can be constructed using Skyrme's product ansatz (see appendix A), and is presented in figure 1*a*. With the stronger attraction the cubes now merge, resulting in the stable solution displayed in figure 1*b*. This is a hollow polyhedral Skyrmion which has cubic symmetry *O*_{h}, and the associated polyhedron is the truncated octahedron, a figure with 14 faces (eight hexagons on the vertices of a cube and six squares on the faces) whose face centres correspond to the holes in the Skyrmion.

The truncated octahedron solution can be constructed using the rational map ansatz that was introduced by Houghton *et al*. (1998) and is reviewed in appendix A, which has been found to be a most effective analytical approximation for all the hollow polyhedral solutions at zero pion mass. The rational map for a *B*=8 truncated octahedron field is(2.1)

Together with a suitable radial profile, this gives a starting ansatz for the Skyrme field *U*. The true solution is obtained by relaxing this, while preserving *O*_{h} symmetry. For *m*=0 it is a low-energy local minimum, slightly higher in energy than the Skyrmion with *D*_{6d} symmetry.

If we now consider this truncated octahedron Skyrmion as an initial configuration in the Skyrme model with pion mass *m*=1, then we find that it is unstable. A small squashing, along an axis not aligned with any of the cubic symmetry axes, followed by a numerical relaxation, produces the stable solution displayed in figure 1*c*, which has *D*_{4h} symmetry. In the process, the central eightfold degenerate point where *U*=−1 splits symmetrically into two clusters near the centres of the two cubes. (It appears that in the middle of each cube, *U*=−1 occurs with twofold degeneracy at a pair of slightly separated points.)

The new solution is clearly a bound configuration of two *B*=4 cubic solutions, and closely resembles the configuration shown in figure 1*a*. This matches the known physics that ^{8}Be is an almost bound state of two α-particles. The spectrum of ^{8}Be resonances is well enough known that this nucleus can be treated as a molecule of two α-particles in an attractive potential, not quite strong enough to produce a bound state. The *B*=8 Skyrmion should be thought of as the classical solution corresponding to the minimum of the potential. The classical energy required to break the solution into two well-separated *B*=4 clusters is small. If kinetic and Coulomb effects were included, the solution might naturally unbind. The FR constraints allow the lowest quantum state for this solution to be a 0^{+} state, with zero isospin, consistent with the quantum numbers of ^{8}Be.

Note that in Battye & Sutcliffe (2006) it was shown that taking as an initial condition the *m*=0 minimal-energy *D*_{6d} polyhedral Skyrmion and relaxing it (after a small symmetry breaking perturbation) in the Skyrme model with *m*=1 yields essentially the same polyhedral solution back again. The *D*_{6d} symmetry is restored and the only significant change is a small squashing in the direction of the main symmetry axis. Thus, we have found two very different local minima in the massive pion model and there could be others.

For *m*=1, we compute that, to three decimal places, the energies per baryon of both the *D*_{6d} and new *D*_{4h} solution are equal to *E*/*B*=1.294. For comparison, the energy per baryon of the *B*=4 cube is *E*/*B*=1.307 for *m*=1. These values were obtained on a grid containing 100^{3} lattice points with a lattice spacing Δ*x*=0.1. We have also computed the energies of these solutions on a grid with 200^{3} points with Δ*x*=0.05 and obtained energies of *E*/*B*=1.297 for both solutions. Thus, we estimate that numerical errors could be as large as 0.5% in our energy computations, though we expect relative energies to be more accurate than this. To four decimal places, on the grid with 200^{3} lattice points, we found *E*/*B*=1.2970 and 1.2967 for the *D*_{6d} and the *D*_{4h} solutions, respectively. This suggests that the *D*_{4h} solution might have the lower energy but as the energy difference is much less than the numerical accuracy to which we can reliably compute, we are unable to make any conclusive statement about which (if either) of these solutions is the global minimum.

### (b) *B*=12

In the α-particle model, the classical minima of the potential energy for three or four α-particles occur, respectively, for an equilateral triangle and a regular tetrahedron (Blatt & Weisskopf 1952; Brink *et al*. 1970; Wuosmaa *et al*. 1995). In this and the following subsection §2*c*, we shall investigate whether Skyrmion analogues of these, and other, results can be found.

Motivated by the α-particle model of ^{12}C, we have sought a triangular *B*=12 solution in the Skyrme model, composed of three *B*=4 cubes, oriented so that they attract. A configuration with approximate *D*_{3h} symmetry can be obtained by using the product ansatz to place cubes with their centres on the vertices of an equilateral triangle and meeting at a common edge. Each cube is related to its neighbour by a spatial rotation through 120° followed by an isorotation by 120°. The isorotation cyclically permutes the values of the pion fields on the faces of the cube, so that these values match on touching faces. This produces a field in which each pair of cubes is a slightly bent version of the new *B*=8 solution described earlier. However, it is fairly easy to see that around the centre of the triangle the field has a winding equivalent to a single Skyrmion, and it is unstable to a perturbation (included automatically by the product ansatz) that breaks the up–down reflection symmetry in the plane of the triangle. The instability results, upon relaxation, in the single Skyrmion at the centre of the triangle moving down and merging with the bottom face of the triangle; in fact it fills what was a hole in the baryon density isosurface. Apart from this difference, the resulting relaxed solution, which has only a *C*_{3} symmetry, is very similar to the initial field formed from the three cubes. This *C*_{3} symmetric Skyrmion is presented in figure 2, where views from both the top and the bottom are displayed, so that the up–down symmetry breaking is clearly demonstrated. We compute that *E*/*B*=1.288.

The top view in figure 2 is very similar to the minimal-energy *B*=11 Skyrmion with *m*=0 (Battye & Sutcliffe 1997, 2001, 2002), and suggests that the initial arrangement of three cubes with an approximate *D*_{3h} symmetry can also be viewed as the *m*=1 version of this *B*=11 Skyrmion with a single Skyrmion placed inside at the origin. Such a field configuration can be constructed with exact *D*_{3h} symmetry using the double rational map ansatz (Manton & Piette 2001; see appendix A). This involves a *D*_{3h} symmetric outer map of degree 11, *R*^{out}, and a spherically symmetric degree 1 inner map, *R*^{in}, together with an overall radial profile function. Explicitly, the maps are(2.2)(2.3)where the real constants are *a*=−2.47, *b*=−0.84 and *c*=−0.13. Note that the orientation of the inner map has to be chosen so that the *D*_{3h} subgroup is realized in a way compatible with the degree 11 map. Numerically relaxing the field of the double rational map ansatz for *U* yields a solution which is three cubes with exact *D*_{3h} symmetry, but as we discussed earlier, this is only a saddle point and not a local energy minimum.

In Battye & Sutcliffe (2006), a *B*=12 solution with *C*_{3} symmetry was found, but this is different from the *C*_{3} symmetric solution described earlier. The solution in Battye & Sutcliffe (2006) is composed of three *B*=3 tetrahedra on the vertices of an equilateral triangle and three single Skyrmions on the vertices of the dual triangle. Its energy was found to be *E*/*B*=1.289. Once again, the energies of these two different local minima are too close to have any confidence which of the two solutions has the true lowest energy. It is a general observation of this work that rearrangements of clusters have only a tiny effect on the energy of a Skyrmion, so as *B* increases one expects an increasingly large number of local minima with extremely close energies.

Rearranged solutions are analogous to the rearrangements of the α-particles which model excited states of nuclei. An example is the Skyrme model analogue of three α-particles in a chain configuration for the 7.65 MeV excited state of ^{12}C with spin/parity 0^{+} (Morinaga 1956; Friedrich *et al*. 1971). We have found a suitable stable local minimum to describe three α-particles in a chain. It is a generalization of the *B*=8 Skyrmion and consists of three cubes placed next to each other in a line. The two end cubes have the same orientation, but the middle cube is rotated relative to the other two by 90° around the axis of the chain. This *B*=12 Skyrmion is displayed in figure 3 and has energy *E*/*B*=1.285. Thus, it appears to have the lowest energy of the *B*=12 solutions we have found, but this is not certain given our numerical accuracy. Note that the energy difference we are trying to understand, 7.65 MeV, is less than 0.1% of the total energy of a ^{12}C nucleus, and is therefore smaller than the present accuracy of our numerical energy computations.

To conclude, for *B*=12 we have found two new solutions in addition to the one already known from Battye & Sutcliffe (2006). These new solutions are similar to those found in the α-particle model and all three have equal energy within the level of accuracy we can achieve.

### (c) *B*=16

We have found a tetrahedrally symmetric *B*=16 solution which is an arrangement of four *B*=4 cubes. It was created using rational maps to provide a tetrahedrally symmetric initial condition. One again needs to use the double rational map ansatz as a starting point. This involves an outer map of degree 12, *R*^{out}, and an inner map of degree 4, *R*^{in}, with compatible symmetries, and an overall radial profile function. Essentially one is filling a hollow *B*=12 Skyrmion with a *B*=4 Skyrmion. There is a *T*_{d} symmetric map that approximates the *m*=0 Skyrmion with *B*=12 (Battye & Sutcliffe 1997, 2001, 2002), and this can be combined with the *O*_{h} symmetric map familiar from the *B*=4 Skyrmion, giving *T*_{d} symmetry overall. The maps are(2.4)(2.5)where and the real constants are *a*=−0.53 and *b*=0.78. Starting from the double rational map ansatz for *U*, one lets the field relax numerically, preserving the *T*_{d} symmetry. The result is the solution displayed in figure 4*a*. Initially, *U*=−1 on a whole spherical surface, but after relaxation, *U*=−1 occurs at 16 points, clustered into groups of four points close to the centre of each cube.

The energy of this solution is *E*/*B*=1.288 and the four cubes are clearly visible in a tetrahedral arrangement. The cubes are surprisingly distinct, in comparison with the earlier solutions in which cubes merge to a reasonable extent.

This tetrahedral solution is only a saddle point. A perturbation which breaks the tetrahedral symmetry, followed by a numerical relaxation, reveals that it is energetically more favourable for the two cubes on a pair of opposite edges of the tetrahedron to open out, leading to the *D*_{2d} symmetric solution presented in figure 4*b*. This Skyrmion resembles a bent square with four cubes on the vertices and has a slightly lower energy, *E*/*B*=1.284. Possibly, Coulomb effects could return the tetrahedral solution to stability, and there is always the possibility that the energy ordering of these two competing configurations could be reversed for an increased value of the pion mass parameter *m*; however, an initial investigation reveals that this last possibility does not happen for *m*=2.

A stable tetrahedral solution would be preferable. It has been shown that the closed shell structure of ^{16}O, just slightly perturbed, is compatible with clustering into a tetrahedral arrangement of four α-particles. Moreover, the ground and the excited states at 6.1 and 10.4 MeV, with spin/parity 0^{+}, 3^{−} and 4^{+}, and some higher states, look convincingly like a rotational band for a tetrahedral intrinsic structure (Dennison 1954; Robson 1979).

It is interesting to note that a configuration similar to our bent square in figure 4*b* has been found as a low-energy intrinsic state using an α-cluster model, where it is termed a bent rhomb (Bauhoff *et al*. 1984). Furthermore, when Coulomb effects are not included, it has been found that the regular tetrahedron is not the ground state, but rather an elongated tetrahedron is preferred (Abulaffio & Irvine 1972). Thus, there is some support, using more traditional nuclear models, for the qualitative features of the solutions we have found.

There is a further solution of low energy, in which all the four *B*=4 cubes have the same orientation, and are connected together to form a flat square (see figure 4*c*). This solution has been obtained by using an initial condition derived from the product ansatz of four cubes, and also from a very different initial condition constructed using a rational map with *D*_{4h} symmetry. The solution has energy *E*/*B*=1.293, so one might expect that it is only a saddle point, having an unstable mode that bends the square to the solution of figure 4*b*. However, perturbations of this solution have failed to excite such a mode, so the current evidence suggests that it may be a local minimum.

Yet another tetrahedral configuration of four cubes exists. It is obtained from an initial condition using the single rational map , where *c* is an arbitrary positive real parameter and *R* is the cubic map (2.5) of degree 4. If *c*=1 then this map has cubic symmetry *O*_{h}, but for all other values of *c* the symmetry reduces to tetrahedral, *T*_{d}. Relaxing such a tetrahedral initial condition preserves the symmetry and yields the solution shown in figure 4*d*. It is again a tetrahedral arrangement of four cubes, but with different spatial orientations of the cubes than in the earlier solution shown in figure 4*a*. This solution resembles half of the *B*=32 crystal chunk (discussed in §2*g*) where alternate cubes have been removed. Its energy is *E*/*B*=1.295 and therefore higher than the earlier tetrahedral solution. A perturbation that breaks the tetrahedral symmetry again results in the bent square solution of figure 4*b*.

In Battye & Sutcliffe (2006), a planar *B*=16 solution was found that is not composed of cubes and has an energy *E*/*B*=1.288. This energy is very slightly higher than that of the bent square *E*/*B*=1.284, so our current belief is that for *m*=1 the bent square may be the global minimal-energy Skyrmion with *B*=16.

In summary, we have found four new *B*=16 solutions of the Skyrme model with *m*=1. Two of these have tetrahedral symmetry and are saddle points, and the other two appear to be local minima, with the bent square being a good candidate for the global minimum energy Skyrmion.

### (d) *B*=20

In the α-particle model, the classical minimum of the potential energy for five α-particles is a triangular bipyramid. An initial condition of five *B*=4 cubes, with the same spatial and isospin orientations, placed on the vertices of a triangular bipyramid is displayed in figure 5*a*. The relaxation of this starting configuration completely changes the shape and produces the solution presented in figure 5*b*. This solution is still composed of five cubes, but they are substantially deformed and create a more planar arrangement. It has a *C*_{2} symmetry and each cube is twisted slightly compared to its neighbours, in a manner similar to that of the *B*=16 bent square. The energy is *E*/*B*=1.283.

Motivated by the results for *B*=24, which are presented in §2*e*, we consider an initial condition (see figure 6*a*) consisting of four cubes in a square with a fifth cube placed on top of the square and in the centre. All five cubes have the same spatial orientation and the four cubes in the square have the same isospin orientation, but the fifth cube has been given an isospin rotation by 180°. This isospin rotation aids the attraction between the cubes and is motivated by trying to match the pion fields of the fifth cube with the pion fields at the centre of the square of cubes. Relaxation of this initial configuration produces the solution shown in figure 6*b*. It consists of the *B*=16 bent square Skyrmion of figure 4*b*, together with an extra cube joined to one of the corners of the bent square. This solution has energy *E*/*B*=1.285, again demonstrating that the rearrangement of cubes has only a tiny effect on the energy.

Our search for a solution of triangular bipyramidal form with *D*_{3} symmetry was not successful.

### (e) *B*=24

The results we have found so far with *B*=16 and 20 suggest that the classical minimum of the potential energy for point α-particles is not a good guide to predict the cluster arrangements of *B*=4 Skyrmions in energy minimizing solutions. Nonetheless, the point particle approximation does suggest reasonable starting configurations, which can be created using the product ansatz, ensuring that any unwanted symmetry is only approximate and can therefore be destroyed by the relaxation process. For *B*=24, we begin with six *B*=4 cubes on the vertices of an octahedron, but all with the same spatial orientation, so there is no octahedral symmetry even approximately (see figure 7*a*). The two cubes above and below the four in a square have been given an isospin rotation by 180°, as this aids the attraction between the cubes. As in §2*d*, the relaxed solution, shown in figure 7*b*, is still formed from *B*=4 subunits, but it is very different from the initial condition. It can clearly be seen that this *B*=24 solution is the *B*=16 bent square Skyrmion of figure 4*b*, joined with the *B*=8 Skyrmion of figure 1*c*. The two cubes of the *B*=8 Skyrmion lie across the diagonal of the *B*=16 bent square, so a *C*_{2} rotational symmetry is preserved. The fact that these *B*=16 and 8 solutions appear as sub-structures in the *B*=24 solution is further evidence that these Skyrmions are the minimal-energy arrangements of *B*=4 cubes. The energy of this *B*=24 Skyrmion is *E*/*B*=1.282.

### (f) *B*=28

For both *B*=20 and 24, we have found low-energy solutions which contain the *B*=16 bent square as a sub-structure. It therefore seems probable that a similar solution might exist for *B*=28. In an attempt to find such a solution we used the starting configuration shown in figure 8*a*, with four cubes in a square and three extra cubes placed above the square, but directly over holes in the square that do not correspond to faces of the cubes below. All the cubes have the same space and isospace orientations. We do not expect this to be a very attractive arrangement, and indeed this is one of the motivations for choosing this initial condition, since we expect the cubes to rearrange into a low-energy solution.

Figure 8*b* displays the solution which results from the relaxation. Note that each of the three cubes on top of the square has indeed aligned with a cube in the four below, but that these four cubes have not formed the bent square of figure 4*b* but rather remain in the flat square of figure 4*c*. The presence of the three connected cubes above appears to suppress the tendency of the square of cubes to bend. The cubic *B*=32 Skyrmion (discussed in detail in §2*g*) has eight cubes on the vertices of a larger cube and this *B*=28 solution clearly resembles the *B*=32 Skyrmion with one of the eight cubes removed. The energy of this solution is *E*/*B*=1.279, which is slightly lower than might have been expected, given the previous energies for *B*=16, 20, 24. This solution therefore appears to be a good candidate for the minimal-energy *B*=28 Skyrmion with *m*=1.

### (g) *B*=32

In Battye & Sutcliffe (2006), it was shown that, even for relatively small values of the pion mass parameter *m*, the energy of a *B*=32 cubic Skyrmion is lower than that of the minimal-energy, hollow polyhedral fullerene-type Skyrmion. This makes the *B*=32 cubic Skyrmion a candidate for the minimal-energy Skyrmion at this baryon number. It may be thought of as eight *B*=4 cubic Skyrmions placed on the vertices of a cube, each with the same spatial and isospin orientations. Alternatively, it may be created by cutting out a cubic *B*=32 chunk from the infinite, triply periodic Skyrme crystal (Baskerville 1996). However, both these constructions appear a little artificial in that the final solution does not differ greatly from the initial condition.

A more convincing method to obtain the *B*=32 crystal chunk is to begin with an initial condition given by the double rational map ansatz. We place a *B*=4 cube inside a *B*=28 Skyrmion with cubic symmetry using the maps(2.6)(2.7)where *a*=0.33 and *b*=1.64, and *p*_{±}(*z*) are as before. The initial condition is displayed in figure 9*a*. The numerical relaxation yields the solution presented in figure 9*b*, which is indeed the *B*=32 crystal chunk, with energy *E*/*B*=1.274. We have also performed a similar simulation in which the initial double rational map Skyrme field is perturbed to break the cubic symmetry, and the result is again the cubic *B*=32 solution, suggesting that it is a stable local minimum. Furthermore, the cubic *B*=32 solution also arises from a simulation in which the initial conditions were created using the product ansatz to place eight cubes in an arrangement where seven of the cubes are as shown in figure 8*a*, and the eighth cube is placed in the obvious missing location to create a cross of four cubes.

Note that slicing the *B*=32 crystal chunk in half produces the square *B*=16 solution of figure 4*c*, whereas if one removes every alternate cube then the result is the tetrahedral *B*=16 solution of figure 4*d*. As we remarked in §2*f*, removing a single cube produces the *B*=28 solution of figure 8*b*.

Recall that a Skyrme field created using the original rational map ansatz with a degree *B* map has a polyhedral structure with 2*B*−2 holes in the baryon density (Houghton *et al*. 1998). As can easily be seen from figure 9*b*, the *B*=32 crystal chunk contains 54 exterior holes and this corresponds to a degree 28 map. This was one of the main motivations for considering the above double rational map construction using maps of degrees 28 and 4. The next crystal chunk with cubic symmetry contains 27 cubes and therefore has *B*=108. To create this from a triple rational map ansatz, by wrapping a third shell around the *B*=4 and 28 shells, would therefore require a map with degree *B*=76. A degree 76 map has 150 holes and it is easy to see that this is precisely the correct number of exterior holes for the *B*=108 crystal chunk. It therefore seems probable that a triple rational map ansatz exists that would provide suitable initial conditions, with exact cubic symmetry, which relaxes to the *B*=108 crystal chunk. However, there is a six-parameter family of *O*_{h} symmetric degree 76 maps and it is not clear how to obtain an appropriate member of this family with a suitable distribution of the 150 holes. In the simpler case of the required degree 28 cubic map it turns out that the energy minimizing map is suitable, but we have verified that for the degree 76 map the energy minimizing map does not have the required distribution of holes.

### (h) *B*=7

*B*=7 Skyrme fields with cluster subunits also look interesting. A *D*_{3d} symmetric deformation of small energy of the *B*=7 dodecahedral Skyrmion solution (Battye & Sutcliffe 1997, 2001, 2002) looks like a pair of *B*=4 cubes, sharing one *B*=1 Skyrmion, and it should be possible to quantize this with a lower spin than *J*=7/2. A further, less symmetric, deformation should split this into two clusters, a *B*=4 cube and a *B*=3 tetrahedron. It is also probable that if a *B*=1 Skyrmion collides (classically) with this field configuration, the resulting structure will break into two *B*=4 cubic Skyrmions, thereby modelling the reaction p+^{7}Li→^{4}He+^{4}He.

We have attempted to construct a stable *B*=7 Skyrmion solution which is less symmetric than the dodecahedral solution, for example, by using initial conditions containing a *B*=3 tetrahedron and a *B*=4 cube, but so far all our attempts have failed, in that the dodecahedral solution is always recovered.

## 3. Conclusion

This work has shown that the Skyrme model with positive pion mass has qualitatively new solutions for baryon numbers *B*≥8. These are not hollow polyhedra, but more dense structures with clear clustering into *B*=4 cubes, the Skyrme model analogue of α-particles. This gives confidence that the model is a true competitor to other successful models of nuclei. New solutions for *B*=8, 12, 16, 20, 24 and 28, and also the crystal chunk for *B*=32 discovered earlier, which is a cubic arrangement of eight *B*=4 cubes, are the Skyrme model analogues of α-particle ‘molecules’. Their quantization should give sensible rotational bands. Methods that may be useful in the quantization of these solutions have been developed recently (Krusch 2006). We have found that the cubic *B*=4 Skyrmions can be rearranged with quite small energy changes. These rearranged solutions are analogous to the rearrangements of the α-particles which model excited states of nuclei.

Greater numerical precision would be helpful to determine the Skyrmion energies and the energy gaps between nearby solutions. We hope then to recalibrate the Skyrme model's three parameters by fitting to, say, the ^{12}C and ^{16}O masses and sizes, together perhaps with the excitation energy of the lowest 3^{−} state in ^{16}O. We expect the parameters to be considerably different from those which emerge by fitting the proton mass, proton size and proton–delta mass difference, but believe they will be more suitable for modelling nuclei.

## Acknowledgments

Many thanks to Steffen Krusch for useful discussions. This work was supported by the PPARC special programme grant ‘Classical Lattice Field Theory’. The parallel computations were performed on COSMOS at the National Cosmology Supercomputing Centre in Cambridge.

## Footnotes

- Received June 8, 2006.
- Accepted August 1, 2006.

- © 2006 The Royal Society