New topological structures of Skyrme theory: baryon number and monopole number

The European Physical Journal C, Feb 2017

Based on the observation that the skyrmion in Skyrme theory can be viewed as a dressed monopole, we show that the skyrmions have two independent topology, the baryon topology \(\pi _3(S^3)\) and the monopole topology \(\pi _2(S^2)\). With this we propose to classify the skyrmions by two topological numbers (m, n), the monopole number m and the shell (radial) number n. In this scheme the popular (non spherically symmetric) skyrmions are classified as the (m, 1) skyrmions but the spherically symmetric skyrmions are classified as the (1, n) skyrmions, and the baryon number B is given by \(B=mn\). Moreover, we show that the vacuum of the Skyrme theory has the structure of the vacuum of the Sine-Gordon theory and QCD combined together, which can also be classified by two topological numbers (p, q). This puts the Skyrme theory in a totally new perspective.

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New topological structures of Skyrme theory: baryon number and monopole number

Eur. Phys. J. C New topological structures of Skyrme theory: baryon number and monopole number Y. M. Cho 0 1 2 Kyoungtae Kimm 4 J. H. Yoon 3 Pengming Zhang 2 0 School of Physics and Astronomy, Seoul National University , Seoul 151-742 , Korea 1 Konkuk University , Administration Building 310-4, Seoul 143-701 , Korea 2 Institute of Modern Physics, Chinese Academy of Science , Lanzhou 730000 , China 3 Department of Physics, Konkuk University , Seoul 143-701 , Korea 4 Faculty of Liberal Education, Seoul National University , Seoul 151-747 , Korea Based on the observation that the skyrmion in Skyrme theory can be viewed as a dressed monopole, we show that the skyrmions have two independent topology, the baryon topology π3(S3) and the monopole topology π2(S2). With this we propose to classify the skyrmions by two topological numbers (m, n), the monopole number m and the shell (radial) number n. In this scheme the popular (non spherically symmetric) skyrmions are classified as the (m, 1) skyrmions but the spherically symmetric skyrmions are classified as the (1, n) skyrmions, and the baryon number B is given by B = mn. Moreover, we show that the vacuum of the Skyrme theory has the structure of the vacuum of the SineGordon theory and QCD combined together, which can also be classified by two topological numbers ( p, q). This puts the Skyrme theory in a totally new perspective. - The Skyrme theory has played an important role in physics. It has been proposed as an effective field theory of pion physics in strong interaction where the baryons appear as the skyrmions, topological solitons made of pions [1–4]. This view has been very successful, and the rich topological structure of the theory has advanced our understanding of the extended objects greatly [5–7]. The construction of skyrmions as nuclei has a long history. A novel way to obtain non-spherically symmetric multiskyrmions was developed based on the rational map, and the solutions have been associated with and compared to real nuclei [8,9]. And a systematic approach to construct the skyrmions with large baryon number numerically which have the shell strucutre has been developed [10–13]. This, with the improved computational power has made people construct skyrmions with the baryon number up to 108 [14]. With the new development the Skyrme theory have had a remarkable progress recently. It has been able to provide a quantitative understanding of the spectrum of rotational excitations of carbon-12, including the excitation the Hoyle state which is essential for the generation of heavy nuclear elements in early universe [15–17]. And the spin-orbit interaction which is essential for the magic number of nuclei is investigated within the framework of Skyrme theory [18]. Moreover, a method to reduce the binding energy of skyrmions to a realistic level to improve the Skyrme model has been developed [19,20]. So by now in principle one could construct all nuclei as multi-baryon skyrmions and discuss the phenomenology of nuclear physics, although the experimental confirmation of the theory is still in dispute. But the Skyrme theory has multiple faces. In addition to the well known skyrmions it has the (helical) baby skyrmion and the Faddeev–Niemi knot. Most importantly, it has the monopole which plays the fundamental role [21–23]. In this view all finite energy topological objects in the theory could be viewed either as dressed monopoles or as confined magnetic flux of the monopole-antimonopole pair. The skyrmion can be viewed as a dressed monopole, the baby skyrmion as a magnetic vortex created by the monopole-antimonopole pair infinitely separated apart, and the Faddeev–Niemi knot as a twisted magnetic vortex ring made of the helical baby skyrmion. This confirms that the theory can be interpreted as a theory of monopole in which the magnetic flux of the monopoles is confines and/or screened. The fact that the skyrmion is closely related to the monopole has been appreciated for a long time. It has been well known that the skyrmions could actually be viewed as the monopoles regularized to have finite energy [21–23]. In fact it has been well appreciated that the rational map which plays the crucial role in the construction of the multiskyrmions is exactly the π2(S2) mapping which provides the monopole quantum number [10]. Nevertheless the skyrmions have always been classified by the baryon number given by π3(S3), not by the monopole number π2(S2). This was puzzling. The purpose of this paper is twofold. We first show that the skyrmions have two topology, the baryon topology and the monopole topology, so that they are classified by two topological numbers, the baryon number B and the monopole number M . Moreover, we show that the baryon number can be replaced by the radial (shell) number which describes the π1(S1) topology of radial excitation of multi-skyrmions. This is based on the observation that the SU(2) space S3 has the Hopf fibering S3 S2 × S1 and that (...truncated)


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Y. M. Cho, Kyoungtae Kimm, J. H. Yoon, Pengming Zhang. New topological structures of Skyrme theory: baryon number and monopole number, The European Physical Journal C, 2017, pp. 88, Volume 77, Issue 2, DOI: 10.1140/epjc/s10052-017-4655-6