近藤 慶一

We show that the non-Abelian magnetic monopole defined in a gauge-invariant way in SU(3) Yang-Mills theory gives a dominant contribution to confinement of the fundamental quark, in sharp contrast to the SU(2) case.

Recently we have proposed a new reformulation of Yang-Mills (YM) theory based on new variables on a lattice by extending the Cho-Faddeev-Niemi-Shabanov decomposition. Our reformulation allows options discriminated by the stability group H of the gauge group G. When H agrees with the maximal torus group H , it reduces to a manifestly gauge-independent reformulation of the conventional Abelian projection in the maximal Abelian gauge. Within this framework, a non-Abelian Stokes theorem enables us to express the Wilson loop operator in the fundamental representation by the “Abelian" variable extracted in association with the stability group in the minimal option, and to rewrite the Wilson loop operator using a non-Abelian magnetic monopole defined in a manifestly gauge-independent way. For G = SU(3), two options are possible: minimal one with H = U(2) and maximal one with H = U(1) ×U(1). In this talk we summarize the results of Monte Carlo simulations for SU(3) in the minimal option. Especially, we compare three Wilson loop averages defined by the “Abelian" variable, the monopole part and the original YM field. We confirm that the quark-antiquark confining potential is reproduced by the “Abelian" variable (“Abelian" dominance), and that the string tension is reproduced by the non-Abelian magnetic monopole (magnetic monopole dominance). Moreover, we mention the behaviors of correlation functions for new variables.

We show that the non-Abelian magnetic monopole defined in a gauge-invariant way in SU(3) Yang-Mills theory gives a dominant contribution to confinement of the fundamental quark, in sharp contrast to the SU(2) case. © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.

我々はSU(2)Yang-Mills(YM)場のCho-Faddeev-Niemi-Shabanov分解の格子上の定式化,及びSU(N)への拡張を提唱した.YM場のリンク変数の分解をゲージに依存することなく与え,双対超電導描像を整合する配位を直接抽出し解析することができる.ゲージ不変な磁気的モノポールがこの分解された変数によって定義され,クォーク間ポテンシャルのstring tensionを再現する(モノポールドミナンス).本研究では,閉じ込めに重要な役割を果たすとされる,meron,インスタントンやカロロンなど位相的配位とモノポールとの関係をこの新しい新しい定式化を適用し,モノポールループを直接抜き出し,これらの配位を特徴づけるゲージ不変な位相的指数,トポロジカルチャージ(密度)及び磁気的チャージ(密度)の関係を調べる.また,閉じ込め相及び非閉じ込め相の配位に対しても適用し解析を進める.

We prove that the Faddeev-Popov ghost dressing function in the Yang-Mills theory is nonzero and finite in the limit of vanishing momenta and hence the ghost propagator behaves like the free propagator in the deep infrared regime, within the Gribov-Zwanziger framework of the D-dimensional SU(N) Yang-Mills theory in the Landau gauge for D>2. This result implies that the Kugo-Ojima color confinement criterion is not satisfied in its original form. We point out that the result crucially depends on the explicit form of the nonlocal horizon term adopted. The original Gribov prediction in the Landau gauge should be reconsidered in connection with color confinement.

Recently, we have developed a reformulation of the lattice Yang-Mills theory based on the change of variables a la Cho-Faddeev-Niemi combined with a non-Abelian Stokes theorem. In this talk, we give a new procedure (called reduction) for obtaining the color field which plays the central role in this reformulation. In the 4D SU(2) lattice Yang-Mills theory, we confirm the gaugeindependent "abelian" dominance and gauge-independent magnetic-monopole dominance in the string tension extracted from the Wilson loop in the fundamental representation.

We have given a new description of the lattice Yang-Mills theory a la Cho-Faddeev-Niemi-Shabanov, which has enabled us to confirm in a gauge-independent manner "Abelian"-dominance and magneticmonopole dominance in the Wilson loop average, yielding a gauge-independent dual superconductor picture for quark confinement. In particular, we have given a new procedure (called reduction) for obtaining a gauge-independent magnetic monopole from a given Yang-Mills field. In this talk, we demonstrate how some of known topological configurations in the SU(2) Yang-Mills theory such as merons and instantons generate closed loops of magnetic-monopole current as the quark confiner, both of which are characterized by the gauge-invariant topological index, topological charge (density) and magnetic charge (density), respectively. We also try to detect which type of topological configurations exist in the lattice data involving magnetic-monopole loops generated by Monte Carlo simulation. Here we apply a new geometrical algorithm based on "computational homology" to discriminating each closed loop from clusters of magnetic-monopole current, since the magnetic-monopole current on a lattice is integer valued.

We propose a new version of SU(N) Yang-Mills theory reformulated in terms of new field variables which are obtained by a nonlinear change of variables from the original Yang-Mills gauge field. The reformulated Yang-Mills theory enables us to study the low-energy dynamics by explicitly extracting the topological degrees of freedom such as magnetic monopoles and vortices to clarify the mechanism for quark confinement. The dual superconductivity in Yang-Mills theory is understood in a gauge-invariant manner, as demonstrated recently by a non-Abelian Stokes theorem for the Wilson loop operator, although the basic idea of this reformulation is based on the Cho-Faddeev-Niemi decomposition of the gauge potential.

We propose a new description of the SU (N) Yang-Mills theory on a lattice, which enables one to explain quark confinement based on the dual superconductivity picture in a gauge independent way. This is because we can define gauge-invariant magnetic monopoles which are inherent in the Wilson loop operator. For SU (3) there are two options: the minimal option with a single type of non-Abelian magnetic monopole characterized by the maximal stability subgroup H = U (2) = SU (2) ×U (1), and the maximal one with two types of Abelian magnetic monopoles characterized by the maximal torus subgroup H = U (1) ×U (1). The maximal option corresponds to a gauge independent reformulation of the Abelian projection represented by the conventional MAG. In the minimal option, we have successfully performed the numerical simulation of the SU (3) Yang-Mills theory on a lattice. We give preliminary numerical results showing the dominance of the non-Abelian magnetic monopole in the string tension obtained from the Wilson loop in the fundamental representation, and the infrared dominance of a decomposed field variable for correlation functions after demonstrating the preservation of color symmetry which was explicitly broken by the conventional MAG.

Dual superconductivity is believed to be a promising mechanism for quark confinement and has been investigated on a lattice effectively by a particular gauge called the maximal Abelian (MA) gauge. We propose a new formulation of SU(3) Yang-Mills theory on a lattice based on a non-linear change of variables where the new field variables are expected to reduce to those of the Cho-Faddeev-Niemi-Shabanov decomposition in the continuum limit. By introducing a new variable, say color field, carrying the color direction with it, this formulation enables us to restore and maintain color symmetry that was lost in the conventional MA gauge due to the naive separation of the gauge potential into diagonal and off-diagonal components. An advantage of this formulation is that we can define gauge-invariant magnetic monopoles without relying on specific gauges to investigate quark confinement from the viewpoint of dual superconductivity. In this talk, we will present the relevant lattice formulation to realize the above advantages and preliminary results of numerical simulations to demonstrate the validity of this formulation. This SU(3) formulation is an extension of the SU(2) version already proposed by us in the previous conference.

We propose an approach which enables one to obtain simultaneously the glueball mass and the gluon mass in the gauge-invariant way to shed new light on the mass gap problem in Yang-Mills theory. First, we point out that the Faddeev (Skyrme-Faddeev-Niemi) model can be induced through the gauge-invariant vacuum condensate of mass dimension 2 from SU( 2) Yang-Mills theory. Second, we obtain the glueball mass spectrum by performing the collective coordinate quantization of the topological knot soliton in the Faddeev model. Third, we demonstrate that a relationship between the glueball mass and the gluon mass is obtained, since the gauge-invariant gluon mass is also induced from the relevant vacuum condensate. Finally, we determine physical values of two parameters in the Faddeev model and give an estimate of the relevant vacuum condensation in Yang-Mills theory. Our results indicate that the Faddeev model can play the role of a low-energy effective theory of the quantum SU( 2) Yang-Mills theory.

We give a new way of looking at the Cho-Faddeev-Niemi (CFN) decomposition of the Yang-Mills theory to answer how the enlarged local gauge symmetry respected by the CFN variables is restricted to obtain another Yang-Mills theory with the same local and global gauge symmetries as the original Yang-Mills theory. This may shed new light on the fundamental issue of the discrepancy between two theories for independent degrees of freedom and the role of the Maximal Abelian gauge in Yang-Mills theory. As a byproduct, this consideration gives new insight into the meaning of the gauge invariance and the observables, e.g., a gauge-invariant mass term and vacuum condensates of mass dimension two. We point out the implications for the Skyrme-Faddeev model.

Dual superconductivity is believed to be a promising mechanism for quark confinement. Indeed, that this picture is true has been confirmed in the maximal Abelian (MA) gauge. However, it is not yet confirmed in any other gauge, and the MA gauge explicitly breaks color symmetry. To remedy this defect, we propose to use our compact formulation of a non-linear change of variables (NLCV), called once by the Cho-Faddeev-Niemi (CFN) decomposition, on a lattice. This formulation has succeeded to extract the magnetic monopole with integer-valued magnetic charge in the gauge-invariant way. We present measurements of various correlation functions for the operators constructed from the NLCV in SU(2) Yang-Mills theory. Some of our results reproduce previous results obtained in MA gauge, e.g., DeGrand-Toussaint monopole, infrared Abelian dominance and off-diagonal gluon mass generation. These studies preserve color symmetry, which is sharp contrast to the conventional MA gauge. We argue the gauge fixing independence of these results and the implications for quark confinement.

We give a short review of recently obtained results on a new lattice formulation of the non-linear change of variables which was once called the Cho-Faddeev-Niemi decomposition in SU(2) Yang-Mills theory. Based on this formulation, we proposed a new gauge-invariant definition of the magnetic monopole current which guarantees the magnetic charge quantization. We also demonstrated the magnetic monopole dominance in the string tension in SU(2) Yang-Mills theory on a lattice. Our formulation enables one to reproduce in the gauge-invariant way remarkable results obtained so far only in the Maximally Abelian gauge.

We present the first implementation of the Cho-Faddeev-Niemi decomposition of the SU(2) Yang-Mills field on a lattice. Our construction retains the color symmetry (global SU(2) gauge invariance) even after a new type of maximally Abelian gauge, as explicitly demonstrated by numerical simulations. Moreover, we propose a gauge-invariant definition of the magnetic monopole current using this formulation and compare the new definition with the conventional one by DeGrand and Toussaint to exhibit its validity. (c) 2005 Elsevier B.V. All rights reserved.

We give a formal proof that the spacetime average of the vacuum condensate of mass dimension two, i.e., the vacuum expectation value of the squared potential A(mu)(2), is gauge invariant in the weak sense that it is independent of the gauge-fixing condition adopted in quantizing the Yang-Mills theory. This is shown at least for the small deformation from the generalized Lorentz and the modified maximal Abelian gauge in the naive continuum formulation neglecting Gribov copies. This suggests that the numerical value of the condensate could be the same no matter what gauge-fixing conditions for choosing the representative from the gauge orbit are adopted to measure it. Finally, we discuss how this argument should be modified when the Gribov copies exist. (c) 2005 Elsevier B.V. All rights reserved.

We show that the magnetic monopole promoted to the dyon due to the vacuum angle θ resolves the U(1) problem in the sense that the dyon obtained in this way gives a dominant contribution to the topological susceptibility. For this purpose, we derive an Abelian-projected effective gauge theory written in terms of Abelian degrees of freedom, which is obtained by integrating out all the off-diagonal degrees of freedom involved in the SU(2) Yang-Mills theory with the vacuum angle θ. We evaluate the topological susceptibility by estimating the classical part of the effective dyon action obtained by performing the duality transformation. The obtained result is consistent with the Veneziano-Witten formula. © 2005 The American Physical Society.

We give a short review of the recently obtained result that the magnetic monopole promoted to the dyon due to the vacuum angle θ resolves the U(1) problem in the sense that the dyon obtained in this way gives a dominant contribution to the topological susceptibility [1].

We present the first implementation of the Cho-Faddeev-Niemi decomposition of the SU(2) Yang-Mills field on a lattice. Our construction retains the color symmetry (global SU(2) gauge invariance) even after a new type of Maximally Abelian gauge, as explicitly demonstrated by numerical simulations.

We demonstrate that a clear physical content and relevance can be attributed to the on-shell BRST-invariant mixed gluon-ghost condensate of mass dimension two which was recently proposed by the author. We argue that a gauge invariant observable is associated with the mixed condensate. (C) 2003 Published by Elsevier B.V.

We propose a novel Ansatz to find the infrared and ultraviolet asymptotic solutions to the coupled Schwinger-Dyson equation for the gluon and Faddeev-Popov ghost propagators in Yang-Mills theory. We show that there does not exist a logarithmic correction to the infrared asymptotic solution with power behavior which has recently been found for the gluon and Faddeev-Popov ghost propagators in the Landau gauge. On the other hand, we show that the l/p2 power corrections to the ultraviolet asymptotic solutions are allowed as consistent solutions. This result supports the existence of the vacuum condensate 〈 Aμ2 〉 with mass dimension 2, as recently suggested by the operator product expansion and lattice simulations.

We examine the possibility that there may exist a logarithmic correction to the infrared asymptotic solution with power behavior which has recently been found for the gluon and Faddeev-Popov ghost propagators in the Landau gauge. We propose a new ansatz to find a pair of solutions for the gluon and ghost form factors by solving the coupled Schwinger-Dyson equation under a simple truncation. This ansatz enables us to derive the infrared and ultraviolet asymptotic solutions simultaneously and to understand why the power solution and the logarithmic solution is possible only in the infrared and ultraviolet limit, respectively. Even in the presence of the logarithmic correction, the gluon propagator vanishes and the ghost propagator is enhanced in the infrared limit, and the gluon-ghost-antighost coupling constant has an infrared fixed point (but with a different beta function). This situation is consistent with Gribov-Zwanziger confinement scenario and color confinement criterion of Kugo and Ojima. (C) 2002 Elsevier Science B.V. All rights reserved.

We outline a derivation of area law of the Wilson loop in SU(3) Yang-Mills theory in the maximal Abelian gauge. This is based on a new version of non-Abelian Stakes theorem and the novel reformulation of the Yang-Mills theory. Abelian dominance and monopole dominance of the string tension in SU(3) QCD are immediate cosequences of this derivation.

We outline a derivation of area law of the Wilson loop in SU(3) Yang-Mills theory in the maximal Abelian gauge. This is based on a new version of non-Abelian Stokes theorem and the novel reformulation of the Yang-Mills theory, Abelian dominance and monopole dominance of the string tension in SU(3) QCD are immediate cosequences of this derivation. We propose a Monte Carlo simulation to confirm these results on a lattice.

The fermion flavor Nf dependence of non-perturbative solutions in the strong coupling phase of the gauge theory is reexamined based on the interrelation between the inversion method and the Schwinger-Dyson equation approach. Especially we point out that the apparent discrepancy on the value of the critical coupling in QED will be resolved by taking into account the higher order corrections which inevitably lead to the flavor-dependence. In the quenched QED, we conclude that the gauge-independent critical point alpha(c) = 2pi/3 obtained by the inversion method to the lowest order will be reduced to the result alpha(c) = pi/3 of the Schwinger-Dyson equation in the infinite order limit, but its convergence is quite slow. This is shown by adding the chiral-invariant four-fermion interaction.

Based on the ladder Schwinger-Dyson equation, we investigate phase structure of the gauged Yukawa model possessing a global SU(2)_L×SU(2)_R symmetry and an unbroken (vector-like) gauge symmetry. We show that even when we tune the squared mass of the scalar boson in the Lagrangian to be positive, there still exists the dynamical chiral symmetry breaking due to fermion pair condensate (VEV of the composite scalar) triggered by the strong Yukawa coupling larger than a certain critical value. We find a "nontrivial ultraviolet fixed line" and "renormalized trajectories" in the three-dimensional coupling space of the Yukawa coupling, the gauge coupling and the "hopping parameter" of the elementary scalar field. Presence of the gauge coupling is crucial to existence of the fixed line. Implications of the result for the lattice calculation and the top quark condensate model are also discussed.