# Analytic studies of static and transport properties of (gauged) Skyrmions

## Abstract

We study static and transport properties of Skyrmions living within a finite spatial volume in a flat (3+1)-dimensional spacetime. In particular, we derive an explicit analytic expression for the compression modulus corresponding to these Skyrmions living within a finite box and we show that such expression can produce a reasonable value. The gauged version of these solitons can be also considered. It is possible to analyze the order of magnitude of the contributions to the electrons conductivity associated to the interactions with this Baryonic environment. The typical order of magnitude for these contributions to conductivity can be compared with the experimental values of the conductivity of layers of Baryons.

## 1 Introduction

The appearance of Skyrme theory [1, 2, 3] disclosed very neatly the fundamental role of topology in high energy physics (see for instance [4, 5, 6, 7, 8, 9]). First of all, the low energy QCD is very well described by the Skyrme theory [10, 11]. Secondly, the solitons of this Bosonic theory (*Skyrmions*) describe Baryons. Thirdly, the Baryon charge is the winding number of the configuration (see [10, 11, 12, 13, 14, 15, 16, 17, 18, 19] and references therein).

These arguments are more than enough to justify a profound analysis of the Skyrme model. Indeed, extensive studies of the latter can be found in literature (as the previous references clearly show). Not surprisingly,^{1} the Skyrme field equations are a very hard nut to crack and, until very recently no analytic solution was available. Nevertheless, many numerical studies have shown that the Skyrme model provides results in good agreement with experiments.

Despite the success of the model and the existence of several solutions among different contexts, the analysis of their phenomenological aspects seldom can be carried out in an analytic manner. For an analytic solution and a relevant study in compact manifolds see [20].

The gauged Skyrme model (which describes the coupling of a *U*(1) gauge field with the Skyrme theory) has also very important applications in the analysis of electromagnetic properties of Baryons, in the decay of nuclei in presence of defects (see [10, 11, 21, 22, 23, 24, 25] and references therein). Obviously, from the point of view of constructing analytic solutions, the *U*(1) gauged Skyrme model is even worse than the original Skyrme theory. Until very recently, no explicit topologically non-trivial solution was available. Thus, topological configurations of this theory have been deeply analyzed numerically (see [26, 27] and references therein).

Here we list three relevant problems in the applications of (gauged) Skyrme theory to high energy phenomenology which will be the focus of the present paper.

(1) *Finite density effects and the compression modulus*: Finite density effects (and, in general, the phase diagrams) in the Skyrme model have been historically a very difficult topic to analyze with analytic methods. The lack of explicit solutions with topological charge living within a finite flat box with the spherical Skyrme ansatz is the origin of the problem. Some numerical results with the use of the spherical Skyrme ansatz are presented in [28, 29, 30, 31, 32] and references therein. Due to the fact that both finite volume effects and isospin chemical potential break spherical symmetry it is extremely difficult to improve the pioneering results in [28, 29, 30, 31, 32] without changing the original Skyrme ansatz. The main problem in this group is certainly the *compression modulus* [37, 38, 39] (to be defined precisely in the next section) which, roughly speaking, has to do with the derivative of the total energy of the Skyrmions with respect to the volume. The experimental value is different from the value derived using the original spherical hedgehog ansatz. The usual way to compute the compression modulus is to assume the Derrick rescaling for the reaction of nuclear matter to the action of external pressure (see the detailed discussion in [40]). The resulting value is higher than the experimental value.^{2} A closely related technical difficulty is that, if one uses the original hedgehog ansatz for the Skyrmion, it is very unclear even *how to define* the compression modulus since the original Skyrme ansatz describes a spherical Skyrmion living within an infinite volume so that to compute the derivatives of the energy with respect to the volume becomes a subtle question. The best way out of this difficulty would be, of course, to have a consistent ansatz for a Skyrmion living within a finite volume. Relevant numerical results in the literature on that problem are presented in [33, 34, 35, 36] where non-spherical ansätze have been considered.

(2) *Existence of Skyrmion*–*antiSkyrmion bound states/resonances*: Multi-Skyrmionic bound states of Baryon charge higher than 1 are known to exist and they have been successfully constructed numerically (see, for instance, [13] and references therein). However, until very recently, the problem of the existence of Skyrmion–antiSkyrmion bound states and resonances did not possess the place it deserved in the literature on the Skyrme model and despite its importance. We can refer to an early work on the subject in [41]. Here we shall study analytic results over the properties of such configurations. Experimentally, Baryon–antiBaryon bound states and resonances do exist [42, 43, 44, 45, 46]: these should correspond to Skyrmion–antiSkyrmion bound states. Such bound states are very difficult to find since the corresponding classical solutions are not static. Indeed, at a semi-classical level, Skyrmion–antiSkyrmion bound states should look like time-periodic solutions in which a Skyrmion and an antiSkyrmion moves periodically around the center of mass of the system. These kinds of time-dependent configurations are difficult to analyze even numerically.

(3) *Conductivities*: The analysis of electrons transport through gauged Skyrmions is a very interesting open issue. At semi-classical level, one should solve the Dirac equation for the electron in the background of the gauged Skyrmion and, from the solution of the Dirac equation, one could compute the conductivity. It would be especially interesting to be able to describe complex structures assembled from neutrons and protons interacting with electromagnetic fields (such as slabs of Baryons interacting with the corresponding Maxwell field). In nuclear physics and astrophysics these structures are called *nuclear pasta* and they are very relevant in a huge variety of phenomena (see, for instance, [47, 48, 49, 50] and references therein). On the other hand, there are very few “first principles” computations of the transport properties of these complex structures (see [51] and references therein). At a first glance, one could think that this kind of complex structure is beyond the reach of the gauged Skyrme model.

In order to achieve a deeper understanding of the above open issues, it is mandatory to be able to construct analytic examples of gauged multi-Skyrmionic configurations.

In [52, 53, 54, 55, 56, 57, 58, 59, 60] a strategy has been developed to generalize the usual spherical hedgehog ansatz to situations without spherical symmetry both in Skyrme and Yang-Mills theories (see [61, 62, 63] and references therein). Such a framework also allows to analyze configurations living within a finite region of space.

As far as the three open issues described above are concerned, this tool (which will be called here “generalized hedgehog ansatz”) gave rise to the first derivation not only of the critical isospin chemical potential beyond which the Skyrmion living in the box ceases to exist, but also of the first explicit Skyrmion–antiSkyrmion bound states. Thus, this approach appears to be suitable to deal with the problems mentioned previously.

Interestingly enough, the generalized hedgehog ansatz can be adapted to the *U*(1) gauged Skyrme model [64, 65]: it allowed the construction of two types of gauged solitons. Firstly, gauged Skyrmions living within a finite volume. Secondly, smooth solutions of the *U*(1) gauged Skyrme model whose periodic time-dependence is protected by a topological conservation law (as they cannot be deformed to static solutions).

Here we demonstrate that by using this strategy it is possible to derive an explicit expression of the compression modulus. The transport properties of these gauged Skyrmions can also be analyzed. In this work we also present a simple estimate of the order of magnitude of the correction to the electron conductivities due to the interactions of the electrons with the baryonic environment. As far as transport properties are concerned, we will work at the level of approximation in which the electrons perceive the gauged Skyrmions as a classical background. Large **N** arguments strongly suggest that this is a very good approximation^{3} (see for a detailed review chapter 4 and, in particular, section 4.2 of the classic reference [66]).

This paper is organized as follows: in Sect. 2 the action for the gauged Skyrme model and our notations will be introduced. In Sect. 3, the method to deal with Skyrmions at finite density will be described: as an application, a closed formula for the compression modulus of Skyrmions living within a cube will be derived. In Sect. 4, the gauged Skyrmions at finite density will be considered. In Sect. 5, the transport properties associated to electrons propagating in the Baryonic environment corresponding to the finite-density Skyrmions are analyzed. In Sect. 6, we draw some concluding ideas.

## 2 The *U*(1) gauged Skyrme model

*U*(1) gauged Skyrme model in four dimensions with global

*SU*(2) isospin internal symmetry and we will follow closely the conventions of [64, 65]. The action of the system is

*K*and \(\lambda \) are fixed experimentally, \(\kappa \) the coupling for the

*U*(1) field and \(\sigma _{j}\) are the Pauli matrices. In our conventions \(c=\hbar =\mu _{0}=1\), the space-time signature is \((-,+,+,+)\) and Greek indices run over space-time. The stress-energy tensor is

*gauged Skyrmions*and

*gauged time-crystals*will be terms describing to the two different kinds of gauged topological solitons appearing as solutions of the coupled system expressed by Eqs. (5) and (6).

The aim of the present work is to show that the Skyrme model and its gauged version are able to give good predictions for important quantities such as the compression modulus and the conductivity.

### 2.1 Topological charge

*U*(1) gauge potential has been constructed in [21] (see also the pedagogical analysis in [26]):

*W*is the Baryon charge. In fact it has been recently shown [64, 65] that it is very interesting to also consider cases in which \(\Sigma \) is time-like or light-like. Indeed, (whether \(\Sigma \) is light-like, time-like or space-like) configurations with \(W\ne 0\) cannot decay into the trivial vacuum \(U=\mathbb {\mathbf {I}}\). Hence, if one is able to construct configurations such that \(W\ne 0\) along a time-like \(\Sigma \), then the corresponding gauged soliton possesses a topologically protected time-dependence as it cannot be continuously deformed into static solutions (since all the static solutions have \(W=0\) along a time-like \(\Sigma \)). The natural name for these solitons is “(gauged) time-crystals” [64, 65].

*SU*(2)-valued scalar \( U(x^{\mu }) \)

*U*(1) field) reads \(\rho _{B}=12\sin ^{2}C\sin F\ dC\wedge dF\wedge dG\). If we want a non-vanishing topological charge in this setting we have to demand \(dC\wedge dF\wedge dG\ne 0\).

## 3 Skyrmions at finite volume

In the present section, the Skyrmions living within a finite flat box constructed in [64] will be slightly generalized. These explicit Skyrmionic configurations allow the explicit computations of the total energy of the system and, in particular, of its dependence on the Baryon charge and on the volume. Hence, among other things, one can arrive at a well-defined closed formula for the compression modulus.

*SU*(2) group is the starting point of the analysis

*SU*(2) [71] it follows that

*H*will be discussed below; in any case, its range is in the segment \(H \in [0,\frac{\pi }{2}]\), while for

*r*we assume \(0\le r\le 2 \pi \). With the parametrization introduced by (12) and (13) the

*SU*(2) field assumes the form

*U*throughout all the range of the variables \(\gamma \) and \(\phi \), which makes it a continuous function of the latter.

### 3.1 Skyrmions in a rectangular cuboid

We can extend the results presented in [64] by considering a cuboid with three different sizes along the three axis instead of a cube. Thus, we will use three – different in principle – fundamental lengths characterizing each direction, \(l_{1}\), \(l_{2}\) and \(l_{3}\), inside the metric.

^{4}\(H=H(r)\). We note that in this section we do not take into account the effects of an electromagnetic field, hence we have \(A_{\mu }=0\) in the relations of the previous sections.

*r*and \(\tilde{H}(r)\) which are the boundaries of the two integrals. Of course we consider \(\tilde{I}_0 > 0\). As a starting point for the integration we take \(r=0\), \(\tilde{H}(0)=0=H(0)\), although we could also set \(r=0\), \(\tilde{H}=\pi \) (\(H(0)=\frac{\pi }{2}\)). The difference between the two boundary choices is just in the sign of the topological charge. These boundary values, for

*H*and those that we have seen in (14) for \(\gamma \) and \(\phi \) lead to a topological charge \(W= p q\) in (8) (for \(A_\mu =0\)).

In the special case when \(l_1=l_2=l_3=l\) we obtain the particular case which was studied in [64]. Here, we give emphasis to this general case and, especially, we want to study the most energetically convenient configurations and the way in which they are affected by the anisotropy in the three spatial directions. In Fig. 1 we see a schematic representation of the finite box we are considering for this Skyrmionic configuration with a baryon number \(B =p q\).

*p*and

*q*Skyrmionic layers a particular form of this matter that is encountered in nature. The dimensions of the configuration are governed by the three numbers \(l_{1}\), \(l_{2}\) and \(l_{3}\). Of course we do not expect the binding energies of such a configuration to be at the same level with those produced by the usual spherically symmetric ansatz. This is something that we examine thoroughly in the next section.

#### 3.1.1 The energy function

*r*we obtain

*x*is related to \(I_0\) through

*H*with the help of (18) and obtain – in principle – the energy as a function of the \(l_i\)’ s,

*p*and

*q*. However, due to the fact that relation (26) cannot be straightforwardly inverted so as to substitute \(I_0\) as a function of \(l_1\) (through (26) and (27)) we choose to express the energy function in terms of

*x*instead of \(l_1\). In what follows, we assume the values \(K=2\) and \(\lambda =1\) for the coupling constants [14], so that lengths are measured in fm and the energy in MeV. In this manner we get

*x*, as we discussed, is linked – with the help of the boundary conditions of the problem – through (26) to \(l_1\). If we fix all variables apart from

*x*and plot the energy as a function of the latter we get what we see in Fig. 2. In this graph, we observe that the minimum of the energy is “moving” to smaller values of

*x*as the box is being enlarged in the two directions of \(l_2\) and \(l_3\). However, we have to keep in mind that the other of the lengths, namely \(l_1\), depends also on the values of \(l_2\) and \(l_3\) through (26). For the particular set of values used in the figure we can see that as \(l_2\) and \(l_3\) rise, \(l_1\) is also relocated to larger values. In the next section we study more thoroughly the function \(E(x,l_2,l_3,p,q)\) and its derivatives near the values that correspond to the most energetically convenient configurations.

#### 3.1.2 The energy as a function of the three \(l_i\)’s

*p*and

*q*to specific values. In the Table 1 we can observe the location of the minimum of the energy for specific values of

*p*and

*q*.

Minimum of the energy for values of *p* and *q*

\(E_{min}\) (MeV) | | | \(l_1\) (fm) | \(l_2\) (fm) | \(l_3\) (fm) |
---|---|---|---|---|---|

167 | 1 | 1 | 0.251 | 0.413 | 0.413 |

334 | 1 | 2 | 0.251 | 0.413 | 0.826 |

669 | 2 | 2 | 0.251 | 0.826 | 0.826 |

835638 | 100 | 50 | 0.251 | 41.306 | 20.653 |

835638 | 50 | 100 | 0.251 | 20.653 | 41.306 |

First, we have to note that the interchange of *p* and *q* makes no significant difference, so weather you take \(p=100\) and \(q=50\) or \(p=50\) and \(q=100\), the only thing that happens is that the values of the corresponding lengths \(l_{2}\) and \(l_{3}\) are also interchanged. However, the arithmetic value that the energy assumes remains the same. Another thing that we have to notice is that, if we calculate the percentage difference of the minimum of the energy from the topological bound \(E_{0}=12\pi ^{2}|B|=12\pi ^{2}pq\); in all cases we get \(\Delta (\%)=\frac{E-E_{0}}{E_{0}}(\%)=41.11\%\). Thus, we see that the minimum of the energy \(E(l_{1},l_{2},l_{3})\) has a fixed deviation from the Bogomol’nyi bound irrespectively of the *p*, *q* configuration. We also observe that this most energetically convenient situation arises when the box has convenient lengths. In particular we see that the relation \(\frac{l_{2}}{l_{3}}=\frac{p}{q}\) is satisfied in all cases, while \(l_{1}\) remains fixed in a single “optimal” value. By comparing with the usual spherically symmetry Skyrmionic configuration in an infinite volume, this higher deviation from the Bogomol’nyi bound may be anticipated due to the “compression” of the system into a finite volume.

*E*in terms of

*x*which also involves \(l_{1}\), \(l_{2}\) and \(l_{3}\) we need to write

*x*after fixing \(l_{2}\) and \(l_{3}\) to their minimum value for various

*p*,

*q*configurations. We can see that \(\frac{\partial E}{\partial l_{2}}\) and \(\frac{\partial E}{\partial l_{3}}\) are indistinguishable when \(p=q\). On the other hand if \(q>p\) the \(\frac{\partial E}{\partial l_{3}}\) line runs closer to the vertical axis than \(\frac{\partial E}{\partial l_{2}}\) and vice versa when \(p>q\). Finally, before proceeding to study the energy as a function of

*p*and

*q*, we give in Fig. 5 its graph in terms of \( l_{2}\) and \(l_{3}\) when \(l_{1}\) assumes the value that corresponds to the minimum of the energy.

### 3.2 The energy of the symmetric configuration

*x*, \(l_{2}\) and \(l_{3}\), it is not straightforward from that expression to derive what happens in the case where one considers a symmetric box \(l_{1}=l_{2}=l_{3}=l\). In this section we treat this situation from the very beginning by setting all fundamental lengths as equal in Eq. (18). We have to note that throughout this section we also make use of the system of units \(K=2\), \(\lambda =1\). The expression relative to (25), from the resulting integral of motion, leads to

*x*is defined as in the previous section by relation (27), with \(l_{1}=l\). By following the exact same steps as before we are led to the following expression for the energy

*p*, \(q-1\), with an exception in the \(p=q=2\) case.

### 3.3 The compression modulus for the rectangular box

*V*: then the compression modulus is related to the second derivative of the total energy of the system with respect to

*V*. As it has been already mentioned, this requires to generalize the hedgehog ansatz to situations without spherical symmetry. On the other hand, if one insists in defining the compression modulus for the spherical hedgehog, it becomes a rather subtle issue (see the nice analysis in [40]) how to define the derivative of the energy with respect to the volume. Here we are using the generalized hedgehog ansatz [64, 65] which is well suited to deal with situations without spherical symmetry. In this way we can analyze Skyrmions living within a region of flat space-time of finite spatial volume avoiding all the subtleties mentioned above. In particular, in the present case the “derivative with respect to the volume” means, literally, the derivative (of the total energy of the system) with respect to the spatial volume of the region in which the Skyrmions are living.

Deviation from the topological bound for several values of *p* and *q*

| |
| \(\Delta (\%)\) |
---|---|---|---|

1 | 1 | 0.322 | 53 |

2 | 1 | 0.369 | 105 |

3 | 1 | 0.385 | 177 |

2 | 2 | 0.463 | 104 |

3 | 2 | 0.505 | 138 |

3 | 3 | 0.571 | 148 |

*B*is the baryon charge and

*V*the finite volume in which we confine the system; in our case this volume is \(V=16\pi ^{3}l_{1}l_{2}l_{3}\). The difference in the sign of (34) in comparison to other expressions in the literature [74] is owed to the metric signature that we follow here and which affects the derivation of

*E*from \(T_{00}\). In order to express the energy that we obtain from (30) as a function of the volume, we introduce the following reparametrization of the \(l_{i}\)’s into three new variables

*V*and substituting to the second we obtain the energy as a pure function of

*x*which is associated through (26) with the volume

*V*. We can thus calculate the first and second derivatives of the energy with respect to the volume by just taking \(\frac{dE}{dV} =\left( \frac{dV}{dx}\right) ^{-1}\frac{dE}{dx}\) and \(\frac{d^{2}E}{dV^{2}} =\left( \frac{dV}{dx}\right) ^{-1}\frac{d}{dx}\left[ \left( \frac{dV}{dx} \right) ^{-1}\frac{dE}{dx}\right] \).

*E*(

*V*) with respect to the volume defines the pressure of the system, i.e. \(P=\frac{dE}{dV}\). In Fig. 7 we see the graphs of the pressure the compression modulus and the energy with respect to the volume for specific regions of the variable

*V*. Due to the complicated nature of the relation between

*x*and

*V*it is not easy to put in this parametric plot the behavior of

*P*and

*E*near the region where \( V\rightarrow 0\). However, one can calculate through the relations that as one shrinks the volume to zero, the pressure suddenly falls and changes sign becoming negative. The same happens to the compression modulus \({\mathcal {K}}\) as well, for even smaller values of

*V*, while the energy remains positive for all

*V*. Unfortunately the expressions are too cumbersome to present them analytically in this work, but the graphs in Fig. 7 demonstrate the general behavior. In the case of a finite cube with \( l_{1}=l_{2}=l_{3}\) the situation is a lot simpler as we can see in the following section.

#### 3.3.1 Compression modulus in the symmetric case

*V*with the variable

*x*on which the energy depends (33). In this manner we can get an analytical expression for the compression modulus of the cube in terms of the variable

*x*, which is

*p*and

*q*the behavior of the before mentioned quantities is described by the same graphs as given in Fig. 7.

*n*and the corresponding baryon numbers

*B*, whose compression modulus – as calculated with the help of (36) – is \({\mathcal {K}}\sim 230\) MeV. In all cases presented in the table we have considered \(p=q\), thus \(B=p^{2}\).

Examples of configurations corresponding to a compression modulus \( {\mathcal {K}}\sim 230\,\)MeV

| 144 | 196 | 225 | 324 |
---|---|---|---|---|

| 0.044 | 0.048 | 0.051 | 0.057 |

## 4 Gauged solitons

Here we will shortly describe (a slight generalization of) the gauged solitons constructed in [65].

### 4.1 Gauged Skyrmions

*M*and

*N*being

*H*(

*r*) can be solved explicitly. More importantly, the above algebraic conditions in Eq. (42) are consistent with the Maxwell equations written above. Indeed, if one plugs the two algebraic conditions in Eq. (42) into the three Maxwell equations one obtains a single Maxwell equation for \(b_{3}(r)\):

*H*(

*r*) we have a decoupled (from \(b_{3}\)) equation that reads

*r*

*H*(

*r*) can be found explicitly), however its integration is not a trivial task. In any case, integration of (43) that results in an expression for \(b_{3}\) makes trivial the determination of the other two components of \(A_{\mu }\) since both \(b_{1}\) and \(b_{2}\) are given algebraically in terms of \(b_{3}\) through conditions (42). Nevertheless, even without the explicit expressions, it is still possible to analyze the generic features of the transport properties electrons passing through the above gauged Skyrmions.

### 4.2 Gauged time-crystals

*M*and

*N*are zero.

*H*(

*r*). In this case the relative equation reads

### 4.3 Topological current for the gauged Skyrmion

### 4.4 Baryonic current for the time-crystal

## 5 On the conductivity of gauged solitons

*U*(1) field (the \(\kappa A_{\mu }\) in Eq. (69), see Sect. B.1 of Appendix B) and the other arising from the term produced from the baryon current (the \( G_F J_{\mu }^{B}\) in Eq. (69)) one needs to evaluate the relative strength of the

*U*(1) coupling with respect to the interactions with the Skyrmionic current. There are two competing factors in the interactions with the Skyrmionic current. The first factor is the electro-weak coupling constant (which is obviously weaker than the

*U*(1) coupling). The second factor is related with the Skyrmions profile

*H*and can be evaluated explicitly thanks to the present analytic solutions. Assuming that both \(\sin (2H)\) and \(H^{\prime }\) are of order 1 (since both quantities are adimensional and the solitonic solutions we are considering are smooth and regular) one can see that the effective adimensional coupling \(\widehat{g}\) measuring the strength of the contributions to the conductivity due to the interactions of the electrons with the Skyrmionic current is:

For completeness, in Sects. B.2 and B.3 of Appendix B we have included the Dirac equations for the electrons propagating in the gauged solitons background described above. Although these Dirac equations cannot be solved analytically (due to the fact that Eqs. (43) and (55) are not integrable in general), they can be useful starting points for numerical analysis of transport properties of the present gauged solitons.

## 6 Conclusions and perspectives

In the present paper we have studied (gauged) Skyrmionic configurations in a finite box. We provided the reduced field equations under the adopted ansatz and distinguished the conditions over the potential functions \(A_\mu \) for which the aforementioned equations can be characterized as integrable. Additionally, we have presented analytic expressions for the energy and studied its general behaviour in relation to the baryon number and the possible sizes of the box under consideration. We also managed to demonstrate and analyze the cases where the more energetically convenient configurations emerge in relations to these variables.

What is more, we have derived an explicit analytic expression for the compression modulus corresponding to Skyrmions living within a finite volume in flat space-times. This is the first case in which one can derive an analytic formula (Eqs. (36) and (37) in the previous section) for such an important quantity in a highly interacting theory such as the low energy limit of QCD. This expression produces a reasonable value with a correct order of magnitude. The gauged version of these solitons living within a finite volume can be also considered. Using these gauged solitons, it is possible to analyze the contributions to the electrons conductivity associated to the interactions with this Baryonic environment (which represents a slab of baryons which can be very large in two of the three spatial directions). To the best of authors knowledge, the present is the first concrete setting in which it is possible to perform analytic computations of these relevant quantities in the original version of the Skyrme model (and its gauged version).

## Footnotes

- 1.
At least taking into account that it is reasonable to expect that the theory describing the low energy limit of QCD should be a quite complicated one.

- 2.
The following analysis suggests that this “uniform rescaling” assumption could be too strong. Indeed, the results at the end of Sect. 3 shows that Skyrme theory, when analyzed at finite density, provides with values of the compression modulus which are close to the experimental one.

- 3.
In the leading ’t Hooft approximation, in meson-Baryon scattering, the heavy Baryon (the Skyrmion in our case) is unaffected and, basically, only the meson can react. This is even more so in the electron-Baryon semiclassical interactions due to the huge mass difference between the Skyrmion and the electron. In this approximation, electrons perceive the Skyrmions as an effective medium.

- 4.
On the other hand, when the coupling with Maxwell field is neglected, the profile can depend on time as well. In this case, one gets an effective sine-Gordon theory for the profile

*H*(*t*,*r*) [64]. - 5.
On the other hand, the gauge potential \(A_{\mu }\) and the Baryonic current \( J_{\mu }^{B}\) are the ones corresponding to the gauged Skyrmion and gauged time-crystal described in the previous section.

- 6.
The gauge potential \(A_{\mu }\) and the Baryon current \(J_{\mu }^{B}\) in the interaction Hamiltonian are the ones corresponding to the gauged Skyrmion and to the gauged time-crystal defined in the previous section.

## Notes

### Acknowledgements

The authors would like to thank A. Zerwekh for useful discussions. This work has been funded by the Fondecyt grants 1160137 and 3160121. The Centro de Estudios Científicos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt.

## References

- 1.T. Skyrme, Proc. R. Soc. Lond. A
**260**, 127 (1961)ADSMathSciNetCrossRefGoogle Scholar - 2.T. Skyrme, Proc. R. Soc. Lond. A
**262**, 237 (1961)ADSMathSciNetCrossRefGoogle Scholar - 3.T. Skyrme, Nucl. Phys.
**31**, 556 (1962)MathSciNetCrossRefGoogle Scholar - 4.T.S. Walhout, Nucl. Phys. A
**531**, 596 (1991)ADSCrossRefGoogle Scholar - 5.C. Adam, M. Haberichter, A. Wereszczynski, Phys. Rev. C
**92**, 055807 (2015)ADSCrossRefGoogle Scholar - 6.J.-I. Fukuda, S. Žumer, Nat. Commun.
**2**, 246 (2011)CrossRefGoogle Scholar - 7.H. Stefan et al., Nat. Phys.
**7**, 713 (2011)CrossRefGoogle Scholar - 8.D. Fostar, S. Krusch, Nucl. Phys. B
**897**, 697 (2015)ADSCrossRefGoogle Scholar - 9.M. Gillard, Nucl. Phys. B
**895**, 272 (2015)ADSCrossRefGoogle Scholar - 10.E. Witten, Nucl. Phys. B
**223**, 422 (1983)ADSCrossRefGoogle Scholar - 11.E. Witten, Nucl. Phys. B
**223**, 433 (1983)ADSCrossRefGoogle Scholar - 12.D. Finkelstein, J. Rubinstein, J. Math. Phys.
**9**, 1762–1779 (1968)ADSCrossRefGoogle Scholar - 13.N. Manton, P. Sutcliffe,
*Topological Solitons*(Cambridge University Press, Cambridge, 2007)zbMATHGoogle Scholar - 14.V.G. Makhanov, Y.P. Rybakov, V.I. Sanyuk,
*The Skyrme Model*(Springer, Berlin, 1993)CrossRefGoogle Scholar - 15.D. Giulini, Mod. Phys. Lett. A
**8**, 1917–1924 (1993)ADSCrossRefGoogle Scholar - 16.A.P. Balachandran, A. Barducci, F. Lizzi, V.G.J. Rodgers, A. Stern, Phys. Rev. Lett.
**52**, 887 (1984)ADSCrossRefGoogle Scholar - 17.G.S. Adkins, C.R. Nappi, E. Witten, Nucl. Phys. B
**228**, 552–566 (1983)ADSCrossRefGoogle Scholar - 18.E. Guadagnini, Nucl. Phys. B
**236**, 35–47 (1984)ADSCrossRefGoogle Scholar - 19.G. Dvali, A. Gußmann, Nucl. Phys. B
**913**, 1001 (2016)ADSCrossRefGoogle Scholar - 20.A.D. Jackson, N.S. Manton, A. Wirzba, Nucl. Phys. A
**495**, 499 (1989)ADSCrossRefGoogle Scholar - 21.C.G. Callan Jr., E. Witten, Nucl. Phys. B
**239**, 161–176 (1984)ADSCrossRefGoogle Scholar - 22.J.M. Gipson, H.C. Tze, Nucl. Phys. B
**183**, 524 (1981)ADSCrossRefGoogle Scholar - 23.J. Goldstone, F. Wilczek, Phys. Rev. Lett.
**47**, 986 (1981)ADSMathSciNetCrossRefGoogle Scholar - 24.E. D’Hoker, E. Farhi, Nucl. Phys. B
**241**, 109 (1984)ADSCrossRefGoogle Scholar - 25.V.A. Rubakov, Nucl. Phys. B
**256**, 509 (1985)ADSCrossRefGoogle Scholar - 26.B.M.A.G. Piette, D.H. Tchrakian, Phys. Rev. D
**62**, 025020 (2000)ADSMathSciNetCrossRefGoogle Scholar - 27.E. Radu, D.H. Tchrakian, Phys. Lett. B
**632**, 109–113 (2006)ADSCrossRefGoogle Scholar - 28.I. Klebanov, Nucl. Phys. B
**262**, 133 (1985)ADSCrossRefGoogle Scholar - 29.A. Actor, Phys. Lett. B
**157**, 53 (1985)ADSMathSciNetCrossRefGoogle Scholar - 30.H.A. Weldon, Phys. Rev. D
**26**, 1394 (1982)ADSCrossRefGoogle Scholar - 31.M. Loewe, S. Mendizabal, J.C. Rojas, Phys. Lett. B
**632**, 512 (2006)ADSCrossRefGoogle Scholar - 32.J.A. Ponciano, N.N. Scoccola, Phys. Lett. B
**659**, 551 (2008)ADSCrossRefGoogle Scholar - 33.M. Kugler, S. Shtrikman, Phys. Rev. D
**40**, 3421 (1989)ADSCrossRefGoogle Scholar - 34.M. Kugler, S. Shtrikman, Phys. Lett. B
**208**, 491 (1988)ADSCrossRefGoogle Scholar - 35.M. Harada, Y.L. Ma, H.K. Lee, M. Rho, in
*Fractionized Skyrmions in Dense Compact-Star Matter. The Multifaceted Skyrmion*, 2nd edn., ed. by M. Rho, I. Zahed (World Scientific, Singapore, 2016)Google Scholar - 36.Y.-L. Ma, M. Rho,
*Effective Field Theories for Nuclei and Compact-Star Matter*(World Scientific, Singapore, 2018)CrossRefGoogle Scholar - 37.L.C. Biedenharn, Y. Dothan, M. Tarlini, Phys. Rev. D
**31**, 649 (1985)ADSCrossRefGoogle Scholar - 38.U.-G. Meissner, I. Zahed, Adv. Nucl. Phys.
**17**, 143 (1986)Google Scholar - 39.T. Meissner, F. Grummer, K. Goeke, M. Harvey, Phys. Rev. D
**39**, 1903 (1989)ADSCrossRefGoogle Scholar - 40.C. Adam, C. Naya, J. Sanchez-Guillen, J.M. Speight, A. Wereszczynski, Phys. Rev. D
**90**, 045003 (2014)ADSCrossRefGoogle Scholar - 41.H. Walliser, A. Hayashi, G. Holzwarth, Nucl. Phys. A
**456**, 717 (1986)ADSCrossRefGoogle Scholar - 42.B.E.S. Collaboration, J.Z. Bai et al., Phys. Rev. Lett.
**91**, 022001 (2003)CrossRefGoogle Scholar - 43.BES Collaboration, M. Ablikim et al., Phys. Rev. D
**71**, 072006 (2005). arXiv:hep-ex/0503030 - 44.BES Collaboration, M. Ablikim et al., Phys. Rev. Lett.
**93**, 112002 (2004)Google Scholar - 45.S. Olsen, in
*Invited talk at the International Symposium on Hadron Spectroscopy, Chiral Symmetry and Relativistic Description of Bound States, Tokyo*(2003). arXiv:hep-ex/0305048 - 46.Z.-G. Wang, Eur. Phys. J. A
**47**, 71 (2011)ADSCrossRefGoogle Scholar - 47.D.G. Ravenhall, C.J. Pethick, J.R. Wilson, Phys. Rev. Lett.
**50**, 2066 (1983)ADSCrossRefGoogle Scholar - 48.D. Page, S. Reddy, Annu. Rev. Nucl. Part. Sci.
**56**, 327–374 (2006)ADSCrossRefGoogle Scholar - 49.W.G. Newton, Nat. Phys.
**9**, 396 (2013)CrossRefGoogle Scholar - 50.J.A. Pons, D. Viganò, N. Rea, Nat. Phys.
**9**, 431 (2013)CrossRefGoogle Scholar - 51.D.G. Yakovlev, MNRAS
**453**, 581 (2015)ADSCrossRefGoogle Scholar - 52.F. Canfora, H. Maeda, Phys. Rev. D
**87**, 084049 (2013)ADSCrossRefGoogle Scholar - 53.F. Canfora, Phys. Rev. D
**88**, 065028 (2013)ADSCrossRefGoogle Scholar - 54.F. Canfora, F. Correa, J. Zanelli, Phys. Rev. D
**90**, 085002 (2014)ADSCrossRefGoogle Scholar - 55.F. Canfora, A. Giacomini, S. Pavluchenko, Phys. Rev. D
**90**, 043516 (2014)ADSCrossRefGoogle Scholar - 56.S. Chen, Y. Li, Y. Yang, Phys. Rev. D
**89**, 025007 (2014)ADSCrossRefGoogle Scholar - 57.E. Ayon-Beato, F. Canfora, J. Zanelli, Phys. Lett. B
**752**, 201–205 (2016)ADSMathSciNetCrossRefGoogle Scholar - 58.F. Canfora, M. Di Mauro, M.A. Kurkov, A. Naddeo, Eur. Phys. J. C
**75**, 443 (2015)ADSCrossRefGoogle Scholar - 59.F. Canfora, G. Tallarita, Nucl. Phys. B
**921**, 394 (2017)ADSCrossRefGoogle Scholar - 60.F. Canfora, G. Tallarita, Phys. Rev. D
**94**, 025037 (2016)ADSMathSciNetCrossRefGoogle Scholar - 61.F. Canfora, G. Tallarita, Phys. Rev. D
**91**, 085033 (2015)ADSMathSciNetCrossRefGoogle Scholar - 62.F. Canfora, G. Tallarita, JHEP
**1409**, 136 (2014)ADSCrossRefGoogle Scholar - 63.F. Canfora, S.H. Oh, P. Salgado-Rebolledo, Phys. Rev. D
**96**, 084038 (2017)ADSMathSciNetCrossRefGoogle Scholar - 64.P.D. Alvarez, F. Canfora, N. Dimakis, A. Paliathanasis, Phys. Lett. B
**773**, 401–407 (2017)ADSCrossRefGoogle Scholar - 65.L. Aviles, F. Canfora, N. Dimakis, D. Hidalgo, Phys. Rev. D
**96**, 125005 (2017)ADSMathSciNetCrossRefGoogle Scholar - 66.H. Weigel,
*Chiral Soliton Models for Baryons. Lecture Notes in Physics*(Springer, Berlin, 2008)Google Scholar - 67.F. Wilczek, Phys. Rev. Lett.
**109**, 160401 (2012)ADSCrossRefGoogle Scholar - 68.A. Shapere, F. Wilczek, Phys. Rev. Lett.
**109**, 160402 (2012)ADSCrossRefGoogle Scholar - 69.F. Wilczek, Phys. Rev. Lett.
**111**, 250402 (2013)ADSCrossRefGoogle Scholar - 70.K. Sacha, J. Zakrzewski, Time crystals: a review. Rep. Prog. Phys.
**81**, 016401 (2018). arXiv:1704.03735 - 71.Y.M. Shnir,
*Magnetic Monopoles*(Springer, Berlin, 2005), p. 500CrossRefGoogle Scholar - 72.G.E. Brown, E. Osnes, Phys. Lett. B
**159**, 223–227 (1985)ADSCrossRefGoogle Scholar - 73.G. Co’, J. Speth, Phys. Rev. Lett.
**57**, 547–550 (1986)ADSCrossRefGoogle Scholar - 74.J.P. Blaizot, Phys. Rep.
**64**, 171–248 (1980)ADSMathSciNetCrossRefGoogle Scholar - 75.M.E. Caplan, C.J. Horowitz, Rev. Mod. Phys.
**89**, 041002 (2017)ADSCrossRefGoogle Scholar - 76.M. Dutra, O. Lourenço, J.S. Sà Martins, A. Delfino, J.R. Stone, P.D. Stevenson, Phys. Rev. C
**85**, 035201 (2012)ADSCrossRefGoogle Scholar - 77.M. Dressel, G. Grüner,
*Electrodynamics of Solids*(Cambridge University Press, Cambridge, 2002)CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}