A multishell solution in the Skyrme model

The Skyrme model [1] is a non-linear sigma model of pseudoscalar meson fields with quartic selfinteraction. This model allows for static compact solutions with finite energy which may be considered as baryons [2]. The baryon charge $B$ is naturally identified with the topological charge number in the Skyrme model. The solution with $B=1$ provides a satisfactory description of a nucleon, see, e.g., Ref. [3] for a review.

Solutions with $B>1$ are usually considered as systems of interacting nucleons. In Ref. [4], spherically symmetric skyrmions with $B=2$ and $B=3$ were studied numerically. It was shown that these solutions are unstable, and thus should decay into spatially separated solutions with $B=1$.

Recent developments in the Skyrme model were devoted to constructing multi-$B$ solutions with reduced symmetry which could model atomic nuclei. In Ref. [5, 6, 7], an axially symmetric solution in the Skyrme model was found for $B=2$. In Refs. [8, 9, 10], solitonic configurations with $B\leq 9$ were constricted. Notably, crystal-like solutions with $B\to\infty$ were found in Refs. [11, 12].

The solutions of the Skyrme model proposed in Refs. [9, 10] within the rational map ansatz have a shell-like structure. The number of units of baryon charge per shell is equal to the degree of the rational map. In Refs. [13, 14] it was conjectured that such multishell solutions with a large number of units of baryon charge per shell may describe skyrmion stars, analogs of neutron stars. The authors of these works managed to study numerically only large-$B$ solutions with two shells and approached multishell configurations semianalytically using ramp-profile ansatz that may give overestimated value of the mass.

In this note, we focus on multishell solutions in the Skyrme model which have a large number of shells and arbitrary number of units of baryon charge per shell. In Sec. II, we show that equations for stationary skyrmion configurations within the rational map ansatz are linearized when the number of shells goes to infinity. As a result, we find a simple analytic solution describing multishell skyrmion configurations which may be used for studying skyrmion stars or compact composite objects playing role of dark matter particles in some dark matter models [15, 16, 17, 18]. We show that the mass of this solution scales as $B^{2}$ while its root mean square radius grows as $B^{1/2}$. Thus, the obtained solution is unstable against decays into single-shell or single-skyrmion configurations, and it may be realized only under external conditions preventing them from the decay. In Sec. IV we discuss the results and speculate that adding the pion mass term to the Skyrme model may help stabilize multishell solutions.

In this paper, natural units with $\hbar=c=1$ are used.

In this section, we provide a background information about the Skyrme model, which is necessary for completeness of our presentation in this paper.

At low energy, baryons and mesons represent the effective degrees of freedom emerging from QCD. The mesons are dominated by pseudoscalar pion fields $\vec{\pi}$ which are described by a non-linear sigma model based on the $SU(2)_{L}\times SU(2)_{R}$ group. These fields constitute the following matrix $U(x)=f_{\pi}^{-1}(\mathds{1}\sigma+i\vec{\tau}\cdot\vec{\pi})\in SU(2)$, where $f_{\pi}=93$ MeV is the pion decay constant and $\sigma$ is the scalar meson with the constraint $\sigma^{2}+\vec{\pi}^{2}=f_{\pi}^{2}$. The field $U(x)$ transforms in the $(\frac{1}{2},\frac{1}{2})$ representation of $SU(2)_{L}\times SU(2)_{R}$: $U(x)\to AUB^{-1}$, $A\in SU(2)_{L}$ and $B\in SU(2)_{R}$.

At any fixed time, $U(x)$ maps the space $\mathds{R}^{3}$ onto the group manifold $S^{3}\simeq SU(2)$. Since the pion fields vanish at spatial infinity, the space $\mathds{R}^{3}$ may be compactified to the sphere $S^{3}$, and the above map turns into $U(x):S^{3}\to S^{3}$. This map is charaterized by a topological charge density $B^{0}$ arising as a 0-component of a conserved 4-current

$$B^{\mu}=\frac{\varepsilon^{\mu\nu\rho\sigma}}{24\pi^{2}}\text{tr}[(U^{\dagger}\partial_{\nu}U)(U^{\dagger}\partial_{\rho}U)(U^{\dagger}\partial_{\sigma}U)]\,,$$

(1)

with $\varepsilon_{0123}=-\varepsilon^{0123}=1$.

The Skyrme model is a non-linear sigma-model of the pion fields $\vec{\pi}$ with the Lagrangian

$${\cal L}=\frac{1}{4}f_{\pi}^{2}\text{tr}(\partial_{\mu}U\partial^{\mu}U^{\dagger})+\frac{1}{32e^{2}}\text{tr}[(\partial_{\mu}U)U^{\dagger},(\partial_{\nu}U)U^{\dagger}]^{2}\,,$$

(2)

where $e$ is a dimensionless coupling constant. A skymion is a static solution in this model of the form

$$U(x)=\exp[iF(r)\vec{\tau}\cdot\vec{n}]\,,$$

(3)

where $\vec{n}$ is a unit vector and $F(r)$ is a spherically symmetric function with boundary conditions

$$F(r)|_{r=0}=b\pi\,,\qquad F(r)|_{r\to\infty}\to 0\,.$$

(4)

Here $b$ is an integer.

In the original Skyrme model [1], the unit vector $\vec{n}$ is chosen in a spherically symmetric way, $\vec{n}=\vec{r}/r$. This case is suitable for studying solution with $b=1$, which is usually interpreted as a models of nucleon. Solutions with $b>1$ were shown to be unstable [4].

More generally, the unit vector in Eq. (3) may be chosen in the form [9]

$$\vec{n}=\frac{1}{1+|R|^{2}}(2\text{\,Re\,}(R),2\text{\,Im\,}(R),1-|R|^{2})\,,$$

(5)

where $R$ is a rational function on the complex plane with coordinates $z$ related to the spherical angles $\theta$ and $\phi$ of $\mathds{R}^{3}$ through the stereographic projection $z=\tan(\theta/2)e^{i\phi}$. It was shown in Ref. [9] that the total baryon charge $B$ within this ansatz is given by

$$B=kb\,,$$

(6)

where $b$ is an integer specifying the boundary condition (4) and $k$ is the degree of the rational map $R$. This ansatz allows for stable multi-$B$ solutions which satisfactory describe some atomic nuclei [19, 20].

Substituting the ansatz (5) into the Lagrangian (2) one finds the expression for the skyrmion mass

	$\displaystyle M=$	$\displaystyle 2\pi f_{\pi}^{2}\int_{0}^{\infty}\left[r^{2}(F^{\prime})^{2}+2k\sin^{2}F\right]dr$		(7)
		$\displaystyle+\frac{2\pi}{e^{2}}\int_{0}^{\infty}\left[\frac{{\cal I}(k)\sin^{4}F}{r^{2}}+2k(F^{\prime})^{2}\sin^{2}F\right]dr\,,$

with $F^{\prime}=dF/dr$ and

$${\cal I}(k)=\frac{1}{4\pi}\int\left(\frac{1+|z|^{2}}{1+|R|^{2}}\left|\frac{dR}{dz}\right|\right)^{4}\frac{2idzd\bar{z}}{(1+|z|^{2})^{2}}\,.$$

(8)

Numerical values of this function for various $k$ were studied in Ref. [9]. In particular, ${\cal I}=1$ for $k=1$, and

$${\cal I}(k)|_{k\to\infty}\to{\cal I}_{\infty}k^{2}\,,\qquad{\cal I}_{\infty}\approx 1.28\,.$$

(9)

Varying the expression (7) we obtain the equation for the function $F$:

		$\displaystyle\left[(r^{2}+2kr_{\text{sk}}^{2}\sin^{2}F)F^{\prime}\right]^{\prime}=\sin 2F\bigg{[}k$		(10)
		$\displaystyle+kr_{\text{sk}}^{2}(F^{\prime})^{2}+{\cal I}(k)\frac{r_{\text{sk}}^{2}}{r^{2}}\sin^{2}F\bigg{]}\,,$

where $r_{\text{sk}}=\frac{1}{ef_{\pi}}$.

The baryon charge density (1) acquires a simple expression within the ansatz (3):

$$\rho_{B}\equiv B^{0}=-\frac{k}{2\pi^{2}}\frac{\sin^{2}F}{r^{2}}F^{\prime}\,.$$

(11)

By construction, integrating this baryon charge density one finds the baryon charge number,

$$4\pi k\int_{0}^{\infty}\rho_{B}r^{2}\,dr=kb=B\in\mathds{Z}\,.$$

(12)

In this paper we focus on the case $b\gg 1$ with arbitrary $k$.

Equation (10) is non-linear, and its exact analytic solutions are not known. Numerical solutions of this equations were studied in Refs. [2, 4] for $B=1,2,3$. In this section we construct an explicit analytic solution of this equation for $B\gg 1$. More precisely, we consider a multishell solution with a large number of shells and arbitrary number of units of baryon charge per shell.

The boundary condition (4) shows that near the origin the solution is large, $F\gg 1$ if $b\gg 1$, and it drops rapidly with the distance. Thus, in this region the functions $\sin 2F$ and $\sin^{2}F$ are rapidly oscillating with the following average values

$$\langle\sin 2F\rangle=0\,,\qquad\langle\sin^{2}F\rangle=\frac{1}{2}\,.$$

(13)

With these values of the trigonometric functions, Eq. (10) is linearized,

$$\left[(r^{2}+kr_{\text{sk}}^{2})F^{\prime}\right]^{\prime}=0\,.$$

(14)

A solution of this equation subject to the boundary condition (4) reads

$$F(r)=2b\,\text{arccot}\frac{r}{\sqrt{k}r_{\text{sk}}}\,.$$

(15)

Note that a similar solution was found in a self-dual modified Skyrme model [21].

In this note, we constructed an explicit analytic multishell solution (15) in the Skyrme model. This solution appears due to the observation that equation (10) for the Skyrme field $F(r)$ is linearized for $F\gg 1$. Thus, this analytic solution provides an accurate description of multishell configurations in the Skyrme model in the interval $0<r<R$, in which $F(r)\gg 1$. The cut-off parameter $R$ is given in Eq. (22). For $r>R$, the solution (15) should be modified, because it falls off with distance slower than asymptotic solutions of Eq. (10). One possible long-distance modification of the solution (15) is proposed in Eq. (23).

The analytic solution (15) allows us to calculate exactly the mass of multishell configurations in the Skyrme model, see Eq. (16). In a particular case of one baryon charge per shell, $k=1$, the mass of such solution scales as $M\propto B^{2}$ with the baryon charge $B$. This scaling was first found in Ref. [22] whereas in this paper we determine exact coefficient in this relation. In the case of arbitrary number of units of baryon charge per shell we show that the mass per unit baryon charge grows at least as $\sqrt{k}$, and, thus, any multishell solution is unstable against decays into single-shell or single-skyrmion configurations.