Physics of the hadronic matter with finite density and temperature has been a hot topic for a few decades [

Chinese Physics C, Volume. 44, Issue 1, (2020)

Skyrmion stability at finite isospin chemical potential and temperature *

The skyrmion stability at finite isospin chemical potential

Keywords

1. Introduction

Physics of the hadronic matter with finite density and temperature has been a hot topic for a few decades [

There have been considerable developments concerning the phase diagram. The lattice QCD has provided much information about the finite temperature transitions, such as the value of the critical temperature and the form of the equation of state (EOS) [

As a consequence, various alternative effective models have been used, such as the Nambu-Jona-Lasinio (NJL) model [

In this work, we resort to the effective model which is expressed in terms of mesons and baryons that arise as topological objects — the Skyrme model [

This paper is organized as follows. In Sec. 2, we study the skyrmion properties as function of isospin chemical potential at zero temperature. We find a critical isospin chemical potential

2. Finite isospin chemical potential and the skyrmion stability

We start from the Skyrme model with the pion mass term [

where
*e* is a numerical parameter,
*U* is the

The isospin chemical potential can be introduced through the substitution [

where

The Skyrme model including the isospin chemical potential then becomes

where,

To calculate the effect of the isospin chemical potential on the skyrmion properties, we take the probe approximation of the profile function, that is the following hedgehog ansatz for the skyrmion solution which is still valid

where

For convenience of calculations, we define the following dimensionless variables

We then obtain the dimensionless soliton mass as

where *M* is the soliton mass. The minimization of

Let us now analyze the behavior of the profile function for large distances. By considering

From this equation one can easily obtain the critical value for

We plot in

Figure 1.(color online) Hedgehog profiles

We also study the soliton mass as a function of the isospin chemical potential

Figure 2.(color online) The soliton mass as a function of the isospin chemical potential

We study next the relation between the other relevant physical quantities and the isospin chemical potential. It is well know that the Skyrme model allows the existence of different conserved currents and respective charges, which can be used to define several effective radii. Thus, we can study these radii for finite isospin chemical potentials, which give additional information about the behavior close to the phase transition.

Here, we consider the isoscalar rms radius

where the baryon number density

When the isospin chemical potential is turned on, we find from the Euler-Lagrange equation for the soliton profile

Therefore, substituting the solution of Eq. (8) into Eq. (10) and Eq. (11), we obtain the dependence of the skyrmion radius on the isospin chemical potential.

We plot in

Figure 3.(color online)

Figure 4.(color online)

Another physical quantity which indicates a phase transition occurs is the distribution of the baryon number density

Figure 5.(color online) The baryon number density

We would like to clarify here that, as is known, the skyrmion is a topological soliton, which is in fact a topological defect. Above the critical isospin chemical potential, the stable skyrmion solution cannot exist. This is to say that the topological defect is not present anymore and that a deconfinement phase transition occured. Similarly to the Kosterlitz-Thouless (KT) phase transition [

3. Finite temperature effects and the skyrmion stability

The procedure to construct a skyrmion-like configuration from an

where

In terms of the dimensionless quantities given in Eq. (6), Eq. (15) can be rewritten as

We plot in

Figure 6.(color online) The skyrmion thermal profile

We study the effect of temperature on the skyrmion mass by substituting the profile
*T _{c}* where

Figure 7.(color online) The soliton mass

In

Figure 8.(color online) The dependence of the critical temperature

We would like to point out that the deconfinement phase transition in the skyrmion model cannot be described by the Landau symmetry breaking theory. Thus, the order of deconfinement phase transition cannot be investigated by the traditional Landau phase theory, which is different from the Polyakov loop. It is well known that although the center symmetry is explicitly broken by quarks in the fundamental representation, the Polyakov loop is still used to study the deconfinement phase transition in the Polyakov loop extended NJL model, which is similar to the chiral phase transition with explicitly broken term for finite current quark mass. Furthermore, the deconfinement phase transition is described in this study by the topological defect, so that the order of the phase transition cannot be defined.

4. Summary and conclusions

In this work, we have studied the behavior of skyrmion properties on the isospin chemical potential

*We would like to thank Zhu-Fang Cui and Yong-LiangMa for their constructive comments and reading through the manuscript. Z.-F.C. also helpedto conceive the idea.*

[23] **. Phys. Rev. D, 67, 014505(2003)**.

[28] **. World Scientific(2016)**.

[37] **. Physics Bulletin, 34, 29(1983)**.

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Wen-Li Yuan, Zhen-Ni Xu, Jin-Li Zhang, Hong-Shi Zong. Skyrmion stability at finite isospin chemical potential and temperature *[J]. Chinese Physics C, 2020, 44(1):

Paper Information

Category: Nuclear physics

Received: Jul. 20, 2019

Accepted: --

Published Online: Sep. 29, 2020

The Author Email: Yuan Wen-Li (wlyuan7@gmail.com), Xu Zhen-Ni (jennie.phys@gmail.com), Zhang Jin-Li (jlzhang@njnu.edu.cn), Zong Hong-Shi (zonghs@nju.edu.cn)