Kei-Ichi Kondo

We give a review on hyperbolic magnetic monopoles and hyperbolic vortices obtained in the unified way through the conformal equivalence by the dimensional reduction from the symmetric instantons with various spatial symmetries in the four-dimensional Euclidean Yang-Mills theory. They are used to understand quark confinement in the sense of the area law of the Wilson loop average in a semi-classical picture from a unified treatment of Atiyah's hyperbolic magnetic monopole and Witten-Manton's hyperbolic vortex. In this way quark confinement is shown to be realized by the the non-perturbative vacuum disordered by these topological defects. For this purpose we start from the 4-dim. Euclidean Yang-Mills theory and require the conformal equivalence between the 4-dim. Euclidean space and the possible curved spacetimes with some compact dimensions. This requirement forces us to restrict the gauge configurations of 4-dim.Yang-Mills instantons to those with some spatial symmetries (called symmetric instantons) which are identified with magnetic monopoles and vortices living in the lower-dimensional curved hyperbolic spacetime with constant negative curvature through the dimensional reduction. At the same time, this scheme caused by the dimensional reduction give a holographic description of hyperbolic magnetic monopole dominance on AdS3 in the rigorous way without any further assumptions, which does not hold in the flat Euclidean case. This unified treatment of two topological defects is shown to give the semi-classical picture for quark confinement in the sense of Wilson. We give the understanding of the result from the viewpoint of the gauge-covariant Cho-Duan-Ge-Faddeev-Niemi decomposition for the gauge field.

In the lattice gauge-scalar model with a single scalar field in the fundamental representation of the gauge group SU(2), we have quite recently found that there exists a gauge-independent transition line separating the Confinement and Higgs phases without contradicting the well-known Osterwalder-Seiler-Fradkin-Shenker analyticity theorem between the two phases by performing numerical simulations without any gauge fixing. This was achieved by examining the correlation between the original fundamental scalar field and the so-called color-direction field constructed from the gauge field through the gauge-covariant decomposition due originally to Cho-Duan-Ge-Shabanov and Faddeev-Niemi. In this presentation, we give further numerical evidence for the gauge-independent separation between the Confinement and Higgs phases in the above model to establish their physical origin. For this purpose, we investigate the separation line precisely. We also investigate the contributions of magnetic monopoles to examine their role in confinement %and mass generation (mass gap) from the viewpoint of the dual superconductor picture.

The lattice gauge-scalar model with the scalar field in the adjoint representation of the gauge group has two completely separated confinement and Higgs phases according to the preceding studies based on numerical simulations which have been performed in the specific gauge fixing based on the conventional understanding of the Brout-Englert-Higgs mechanism. In this talk, we reexamine this phase structure in the gauge-independent way based on the numerical simulations for the model with SU(2) gauge group performed without any gauge fixing which is motivated to confirm the recently proposed gauge-independent Brout-Englert-Higgs mechanics for the mass of the gauge field without relying on any spontaneous symmetry breaking. For this purpose we investigate correlation functions between gauge-invariant operators obtained by combining the original adjoint scalar field and the new field called the color-direction field constructed from the gauge field based on the gauge-covariant gauge-field decomposition due to Cho-Duan-Ge-Shabanov and Faddeev-Niemi. We reproduce gauge-independently the transition line separating confinement and Higgs phase, and discover surprisingly a new transition line that divides the confinement phase into two parts. Finally, we discuss the physical meaning of the new transition and implications to confinement mechanism. This talk is based on the preprint [1].

According to the preceding studies, the lattice SU(2) gauge-scalar model with a single scalar field in the fundamental representation of the gauge group has a single confinement-Higgs phase where confinement and Higgs regions are subregions of an analytically continued single phase and there are no thermodynamic phase transitions, which is a well-known consequence of the Osterwalder-Seiler-Fradkin-Shenker theorem. In this paper, we show that we can define new types of gauge-invariant operators by combining the original fundamental scalar field and the so-called color-direction field which is obtained by change of field variables based on the gauge-covariant decomposition of the gauge field due to Cho-Duan-Ge-Shabanov and Faddeev-Niemi. By performing the numerical simulations on the lattice without any gauge fixing, we reproduce the conventional thermodynamic transition line in the weak gauge coupling, and moreover we find a new transition line detected by the new gauge-invariant operators which separates the confinement-Higgs phase into two parts, confinement phase and the Higgs phase, in the strong gauge coupling. All results are obtained in the gauge-independent way, since no gauge fixing has been imposed in the numerical simulations. Moreover, we discuss a physical interpretation for the new transition from the viewpoint of the realization of a global symmetry.

According to the preceding studies, the lattice SU(2) gauge-scalar model with a single scalar field in the fundamental representation of the gauge group has a single confinement-Higgs phase where confinement and Higgs regions are subregions of an analytically continued single phase and there are no thermodynamic phase transitions, which is a well-known consequence of the Osterwalder-Seiler-Fradkin-Shenker theorem. In this paper, we show that we can define new types of gauge-invariant operators by combining the original fundamental scalar field and the so-called color-direction field which is obtained by change of field variables based on the gauge-covariant decomposition of the gauge field due to Cho-Duan-Ge-Shabanov and Faddeev-Niemi. By performing the numerical simulations on the lattice without any gauge fixing, we reproduce the conventional thermodynamic transition line in the weak gauge coupling, and moreover we find a new transition line detected by the new gauge-invariant operators which separates the confinement-Higgs phase into two parts, confinement phase and the Higgs phase, in the strong gauge coupling. All results are obtained in the gauge-independent way, since no gauge fixing has been imposed in the numerical simulations. Moreover, we discuss a physical interpretation for the new transition from the viewpoint of the realization of a global symmetry. Published by the American Physical Society 2024

The lattice gauge-scalar model with the scalar field in the fundamental representation of the gauge group has a single confinement-Higgs phase which is well-known as the Fradkin-Shenker-Osterwalder-Seiler analytic continuity theorem: Confinement and Higgs regions are subregions of an analytically continued single phase and there are no thermodynamics phase transitions between them. In this talk, however, we show that we can define new type of operators which enable to separate completely the confinement phase and the Higgs phase. In fact, they are constructed in the gauge-invariant procedure by combining the original scalar field and the so-called color-direction field which is obtained by change of field variables based on the gauge-covariant decomposition of the gauge field due to Cho-Duan-Ge-Shabanov and Faddeev-Niemi. We perform the numerical simulations for the model with SU(2) gauge group without any gauge fixing and find a new transition line which agrees with the conventional thermodynamic transition line in the weak gauge coupling and divides the confinement-Higgs phase into two separate phases, confinement and the Higgs, in the strong gauge coupling. All results are obtained in the gauge-independent way, since no gauge fixing has been imposed in the numerical simulations. Moreover, we give a physical interpretation for the new transition from the viewpoint of the spontaneous breaking of a global symmetry. This talk is based on the preprint [1].

All colored particles including dynamical quarks and gluons are confined if the color confinement criterion proposed by Kugo and Ojima is satisfied. The criterion was obtained under the gauge fixing of the Lorenz type. However, it was pointed out that the Kugo-Ojima criterion breaks down for the Maximal Abelian gauge, which is quite strange in view of the fact that quark confinement has been verified according to the dual superconductivity caused by magnetic monopole condensations. In order to make a bridge between color confinement due to Kugo and Ojima and the dual superconductor picture for quark confinement, we investigate a generalization of the color confinement criterion to obtain the unified picture for confinement. We show that the restoration of the residual local gauge symmetry which was shown in the Lorenz gauge by Hata to be equivalent to the Kugo-Ojima criterion indeed occurs in the Maximal Abelian gauge for the SU(N) Yang-Mills theory in two-, three- and four-dimensional Euclidean spacetime once the singular topological configurations of gauge fields are taken into account. This result indicates that the color confinement phase is a disordered phase caused by non-trivial topological configurations irrespective of the gauge choice. As a byproduct, we show that the compact U(1) gauge theory can have the disordered confinement phase, while the non-compact U(1) gauge theory has the deconfined Coulomb phase.

Propagators of confined particles, especially the Landau-gauge gluon propagator, may have complex singularities as suggested by recent numerical works as well as several theoretical models, e.g., motivated by the Gribov problem. In this paper, we study formal aspects of propagators with complex singularities in reconstructing Minkowski propagators starting from Euclidean propagators by the analytic continuation. We derive the following properties rigorously for propagators with arbitrary complex singularities satisfying some boundedness condition. The two-point Schwinger function with complex singularities violates the reflection positivity. In the presence of complex singularities, while the holomorphy in the usual tube is maintained, the reconstructed Wightman function on the Minkowski spacetime becomes a nontempered distribution and violates the positivity condition. On the other hand, the Lorentz symmetry and locality are kept intact under this reconstruction. Finally, we argue that complex singularities can be realized in a state space with an indefinite metric and correspond to confined states. We also discuss consequences of complex singularities in the Becchi-Rouet-Stora-Tyutin formalism. Our results could open up a new way of understanding a confinement mechanism, mainly in the Landau-gauge Yang-Mills theory.