Knots emerge in various fields of mathematics and physics today. We show that knots indeed appear as stable solitons in a realistic extension of the standard model of particle physics that provides the QCD axion and right-handed neutrinos. This result suggests that, during the early Universe, a “knot dominated era” may have existed, where knots were a dominant component of the Universe, and this scenario can be tested through gravitational wave observations. Furthermore, we propose that the end of this era involves the collapse of the knots via quantum tunneling, leading to the generation of matter-antimatter asymmetry in the Universe.

Japan

Minoru Eto

Germany

Yu Hamada

Muneto Nitta

Tying Knots in Particle Physics

We work out the Hopfion description of glueballs by inclusively comparing the energy spectra obtained by quantizing Hopfions with experimental data and lattice QCD. Identifying a Hopfion carrying a unit topological charge as <math altimg='si19.svg'><mrow><msub><mi>f</mi><mn>0</mn></msub><mrow><mo>(</mo><mn>1500</mn><mo>)</mo></mrow></mrow></math>, the Hopfions with the topological charge two are classified as glueballonia, i.e., two glueballs are bound together. We find a tightly and a loosely bound glueballonia complying with <math altimg='si20.svg'><mrow><msub><mi>f</mi><mn>0</mn></msub><mrow><mo>(</mo><mn>2470</mn><mo>)</mo></mrow></mrow></math> and a novel scalar particle carrying the mass around 2814 MeV, respectively, and calculate their binding energies. By the rigid body quantization of Hopfions, we predict a characteristic multiplet structure of tensor glueball states. Some of them are missing in the current experimental data and can be verified in future measurements.

Yuki Amari

Poland

Chihiro Sasaki

Kenta Shigaki

Satoshi Yano

Shigehiro Yasui

Glueballonia as Hopfions

Low-energy dynamics of QCD can be described by pion degrees of freedom in terms of the chiral perturbation theory. A chiral soliton lattice, an array of solitons, is the ground state due to the chiral anomaly in the presence of a magnetic field larger than a certain critical value at finite density. Here, we show in a model-independent and fully analytic manner (at the leading order of the chiral perturbation theory) that the chiral soliton lattice phase transits to a “domain-wall skyrmion phase” when the chemical potential is larger than the critical value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mo>=</mo><mn>16</mn><mi>π</mi><msubsup><mrow><mi>f</mi></mrow><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>/</mo><mn>3</mn><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>∼</mo><mn>1.03</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math> with the pion’s decay constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mi>π</mi></msub></math> and mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>π</mi></msub></math>, which can be regarded as the nuclear saturation density. There spontaneously appear stable two-dimensional skyrmions or lumps on a soliton surface, which can be viewed as three-dimensional skyrmions carrying even baryon numbers from the bulk. They behave as superconducting rings with persistent currents due to a charged pion condensation. This phase is in scope of future heavy-ion collider experiments.

Kentaro Nishimura

Domain-Wall Skyrmion Phase in Dense QCD at Strong Magnetic Fields Using Leading-Order Chiral Perturbation Theory

The ground state of QCD in sufficiently strong magnetic field at finite baryon density is an inhomogeneous state consisting of an array of solitons, called the chiral soliton lattice (CSL). It is, however, replaced in a region with higher density and/or magnetic field by the so-called domain-wall Skyrmion (DWSk) phase where Skyrmions are created on top of the CSL. This was previously proposed within the Bogomol’nyi-Prasad-Sommerfield (BPS) approximation neglecting a gauge field dynamics and taking into account its effect by a flux quantization condition. In this paper, by taking into account dynamics of the gauge field, we show that the phase boundary between the CSL and DWSk phases beyond the BPS approximation is identical to the one obtained in the BPS approximation. We also find that domain-wall Skyrmions are electrically charged with the charge one as a result of the chiral anomaly.

Domain-wall Skyrmion phase of QCD in magnetic field: gauge field dynamics

The color-flavor locked phase of high density QCD admits non-Abelian vortices that are superfluid vortices with confined color-magnetic fluxes. Vortex solutions were thus far constructed in an axially symmetric Ansatz with common vortex winding in the left- and right-handed diquark condensates. In this paper, we explore vortex solutions without any Ansatz. In addition to axially symmetric configurations known before, we find that the axial symmetry is broken in certain parameter regions; in one case a single vortex is split into two chiral non-Abelian vortices, i.e., vortices with winding only in the left- or right-handed diquark condensate, and they are connected by one or two domain walls forming a non-Abelian vortex molecule. In the other case, a chiral non-Abelian vortex molecule is confined inside a domain wall elongated to spatial infinity.

China

Sven Bjarke Gudnason

Chiral non-Abelian vortex molecules in dense QCD

We numerically construct a stable three-string junction of a Y shape in a flag manifold nonlinear sigma model on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>/</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>.

Toshiaki Fujimori

Keisuke Ohashi

String junctions in flag manifold sigma models

The ground state of QCD with two flavors (up and down quarks) at finite baryon density in sufficiently strong magnetic field is in a form of either a chiral soliton lattice(CSL), an array of solitons stacked along the magnetic field, or a domain-wall Skyrmion phase in which Skyrmions are spontaneously created on top of the CSL. In the latter, one 2D (baby) Skyrmion in the chiral soliton corresponds to two 3D Skyrmions (baryons) in the bulk. In this paper, we study spin statistics of topological solitons by using the following two methods: the conventional Witten’s method by embedding the pion fields of two flavors into those of three flavors with the Wess-Zumino-Witten (WZW) term, and a more direct method by using the two-flavor WZW term written in terms of a spin structure. We find that a chiral soliton of finite quantized size called a pancake soliton and a hole on a chiral soliton are fermions or bosons depending on odd or even quantizations of their surface areas, respectively, and a domain-wall Skyrmion is a boson. We also propose surgeries of topological solitons: a domain-wall Skyrmion (boson) can be cut into a pancake soliton (fermion) and a hole (fermion), and a chiral soliton without Skyrmions can be cut into a pancake soliton (fermion) and a hole (fermion).

Ryo Yokokura

Spin statistics and surgeries of topological solitons in QCD matter in magnetic field

This paper explores how magnetic fields affect the Casimir effect within the context of a simple quasi-1D interacting fermionic system. A novel phenomenon emerges, resulting from the interaction between external magnetic fields and boundary conditions, which alters the ground state in complex ways and leads to first-order phase transitions among various ground states, specifically inhomogeneous solutions with differing node counts. We calculate the Casimir force, which exhibits discontinuities (jumps) at the transition points between the different ground states.

Antonino Flachi

Satoshi Takada

Ryosuke Yoshii

Fermion Casimir effect and magnetic Larkin-Ovchinnikov phases

We study topological lumps supported by the second homotopy group $\textit{π}$ ($\textit{S}$ ) ⋍ $\textit{ℤ}$ in a gauged $\textit{O}$(3) model without any potential term coupled with a (non)dynamical U(1) gauge field. It is known that gauged-lumps are stable with an easy-plane potential term but are unstable to expand if the model has no potential term. In this paper, we find that these gauged lumps without a potential term can be made stable by putting them in a uniform magnetic field, irrespective of whether the gauge field is dynamical or not. In the case of the non-dynamical gauge field, only either of lumps or anti-lumps stably exists depending on the sign of the background magnetic field, and the other is unstable to shrink to be singular. We also construct coaxial multiple lumps whose size and mass exhibit a behaviour of droplets. In the case of the dynamical gauge field, both the lumps and anti-lumps stably exist with different masses; the lighter (heavier) one corresponds to the (un)stable one in the case of the nondynamical gauge field. We find that a lump behaves as a superconducting ring and traps magnetic field in its inside, with the total magnetic field reduced from the background magnetic field.

Topological solitons stabilized by a background gauge field and soliton-anti-soliton asymmetry

A solitonic ground state called a chiral soliton lattice (CSL) is realized in a supersymmetric theory with background magnetic field and finite chemical potential. To this end, we construct, in the superfield formalism, a supersymmetric chiral sine-Gordon model as a neutral pion sector of a supersymmetric two-flavor chiral Lagrangian with a Wess-Zumino-Witten term. The CSL ground state appears in the presence of either a strong magnetic field and/or large chemical potential, or a background fermionic condensate in the form of a fermion bilinear consisting of the gaugino and a superpartner of a baryon gauge field.