近藤 慶一

Abstract 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 a 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 Kugo–Ojima color confinement and the dual superconductor picture for quark confinement, we investigate a generalization of the color confinement criterion to obtain a 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 by-product, we show that the compact U(1) gauge theory can have a disordered confinement phase, while the non-compact U(1) gauge theory has a deconfined Coulomb phase.

<title>Abstract</title> We show that the color $N$-dependent area law falloffs of the double-winding Wilson loop averages for the $SU(N)$ lattice gauge theory obtained in previous works are reproduced from the corresponding lattice Abelian gauge theory with the center gauge group $Z_N$. This result indicates the center group dominance in quark confinement.

<title>Abstract</title>In the previous paper, we have shown the existence of magnetic monopoles in the pure <italic>SU</italic>(2) Yang–Mills theory with a gauge-invariant mass term for the gluon field being introduced. In this paper, we extend our previous construction of magnetic monopoles to obtain dyons with both magnetic and electric charges. In fact, we solve under the static and spherically symmetric ansatz the field equations of the <italic>SU</italic>(2) “complementary” gauge-scalar model, which is the <italic>SU</italic>(2) Yang–Mills theory coupled to a single adjoint scalar field whose radial degree of freedom is eliminated. We show that the novel dyon solution can be identified with the gauge field configuration of a dyon with a minimum magnetic charge in the massive Yang–Mills theory. Moreover, we compare the dyon of the massive Yang–Mills theory obtained in this way with the Julia–Zee dyon in the Georgi–Glashow gauge-Higgs scalar model and the dyonic extension of the Wu–Yang magnetic monopole in the pure Yang–Mills theory. Finally, we identify the novel dyon solution found in this paper with a dyon configuration on <inline-formula><alternatives><tex-math>$$S^1 \times {\mathbb {R } }^3$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula> space with nontrivial holonomy and propose to use it to understand the confinement/deconfinement phase transition in the Yang–Mills theory at finite temperature, instead of using the dyons constituting the Kraan–van Baal–Lee–Lu caloron.

<title>Abstract</title>In order to understand the confining decoupling solution of the Yang–Mills theory in the Landau gauge, we consider the massive Yang–Mills model which is defined by just adding a gluon mass term to the Yang–Mills theory with the Lorentz-covariant gauge fixing term and the associated Faddeev–Popov ghost term. First of all, we show that massive Yang–Mills model is obtained as a gauge-fixed version of the gauge-invariantly extended theory which is identified with the gauge-scalar model with a single fixed-modulus scalar field in the fundamental representation of the gauge group. This equivalence is obtained through the gauge-independent description of the Brout–Englert–Higgs mechanism proposed recently by one of the authors. Then, we reconfirm that the Euclidean gluon and ghost propagators in the Landau gauge obtained by numerical simulations on the lattice are reproduced with good accuracy from the massive Yang–Mills model by taking into account one-loop quantum corrections. Moreover, we demonstrate in a numerical way that the Schwinger function calculated from the gluon propagator in the Euclidean region exhibits violation of the reflection positivity at the physical point of the parameters. In addition, we perform the analytic continuation of the gluon propagator from the Euclidean region to the complex momentum plane towards the Minkowski region. We give an analytical proof that the reflection positivity is violated for any choice of the parameters in the massive Yang–Mills model, due to the existence of a pair of complex conjugate poles and the negativity of the spectral function for the gluon propagator to one-loop order. The complex structure of the propagator enables us to explain why the gluon propagator in the Euclidean region is well described by the Gribov–Stingl form. We try to understand these results in light of the Fradkin–Shenker continuity between confinement-like and Higgs-like regions in a single confinement phase in the complementary gauge-scalar model.

<title>Abstract</title> We investigate the type of dual superconductivity responsible for quark confinement. For this purpose, we solve the field equations of the <italic>U</italic>(1) gauge-scalar model to obtain a single static vortex solution in the whole range without restricting to the long-distance region. Then we use the resulting magnetic field of the vortex to fit the gauge-invariant chromoelectric field connecting a pair of quark and antiquark which was measured by numerical simulations for <italic>SU</italic>(2) Yang–Mills theory on a lattice. This result improves the accuracy of the fitted value for the Ginzburg–Landau parameter to reconfirm the type I dual superconductivity for quark confinement which was claimed by preceding works based on the fitting using the Clem ansatz. Moreover, we calculate the Maxwell stress tensor to obtain the distribution of the force around the flux tube. This result suggests that the attractive force acts among chromoelectric flux tubes, in agreement with the type I dual superconductivity.

In order to clarify the mechanism of quark confinement in the Yang-Mills theory with mass gap, we propose to investigate the massive Yang-Mills model, namely, Yang-Mills theory with “a gauge-invariant gluon mass term”, which is to be deduced from a specific gauge-scalar model with a single radially-fixed scalar field under a suitable constraint called the reduction condition. The gluon mass term simulates the dynamically generated mass to be extracted in the low-energy effective theory of the Yang-Mills theory and plays the role of a new probe to study the phase structure and confinement mechanism. In this talk, we first explain why such a gauge-scalar model is constructed without breaking the gauge symmetry through the gauge-independent description of the Brout-Englert-Higgs mechanism which does not rely on the spontaneous breaking of gauge symmetry. Then we discuss how the numerical simulations for the proposed massive Yang-Mills theory can be performed by taking into account the reduction condition in the complementary gauge-scalar model on a lattice. By using the reweighting method, we have investigated the effect of the gluon mass term to the Wilson loop (the static potential) and the dynamically generated mass. Moreover, we point out that the adjoint case would gives an alternative understanding for the physical meaning of the gauge-covariant decomposition for the Yang-Mills field known as the Cho-Duan-Ge-Faddeev-Niemi decomposition, while the fundamental case would give a novel decomposition which has been overlooked so far.

We investigate the type of dual superconductivity responsible for quark confinement. For this purpose, we solve the field equations of the U(1) Abelian-Higgs model to obtain the static vortex solution in the whole range without restricting to the long-distance region. Then we use the resulting magnetic field of the vortex to fit the gauge-invariant chromoelectric field connecting a pair of quark and antiquark which was measured by numerical simulations for SU(2) and SU(3) Yang-Mills theories on a lattice. This result improves the accuracy of the fitted value for the Ginzburg-Landau parameter to reconfirm the type I dual superconductivity for quark confinement, which was claimed by preceding works based on an approximate method based on the Clem ansatz. Moreover, we calculate the Maxwell stress tensor for the fitted model to obtain the distribution of the force around the flux tube. This suggests that the attractive force acts on the surface perpendicular to the chromoelectric flux tube, in agreement with the type I dual superconductivity.

The Abelian dominance for the string tension was shown for the fundamental sources in MA gauge in the lattice simulations. For higher representations, however, it is also known that the naive “Abelian” Wilson loop, which is defined by using the diagonal part of the gauge field, does not reproduce the correct behavior. To solve this problem, for an arbitrary representation of an arbitrary compact gauge group, we propose to redefine the “Abelian” Wilson loop. By using this redefined operator, we demonstrate the “Abelian” dominance for sources in the adjoint representation and the sextet representation of SU(3) gauge group in lattice simulations.

The dual superconductivity is a promising mechanism of quark confinement. In the preceding works, we have given a non-Abelian dual superconductivity picture for quark confinement, and demonstrated the numerical evidences on the lattice. In this talk, we focus on the the confinement and deconfinement phase transition at finite temperature in view of the dual superconductivity. By using our new formulation of lattice Yang-Mills theory and numerical simulations on the lattice, we extract the dominant mode for confinement by decomposing the Yang-Mills field, and we investigate the Polyakov loop average, static quark potential, chromoelectric flux, and induced monopole current for both Yang-Mills field and decomposed restricted field in both confinement and deconfinement phase at finite temperature. We further discuss the role of the chromomagnetic monopole in the confinement/deconfinement phase transition.

Dual superconductor picture is one of the most promising scenarios for quark confinement. We have proposed a new formulation of Yang-Mills theory on the lattice so that the so-called restricted field obtained from the gauge-covariant decomposition plays the dominant role in quark confinement. This framework improves the Abelian projection in the gauge-independent manner. For quarks in the fundamental representation, we have demonstrated some numerical evidences for the dual superconductivity. However, it is known that the expected behavior of the Wilson loop in higher representations cannot be reproduced if the restricted part of the Wilson loop is extracted by adopting the Abelian projection or the field decomposition naively in the same way as in the fundamental representation. In this talk, therefore, we focus on confinement of quarks in higher representations. By virtue of the non-Abelian Stokes theorem for the Wilson loop operator, we propose suitable operators constructed from the restricted field only in the fundamental representation to reproduce the correct behavior of the original Wilson loop in higher representations. Moreover, we perform lattice simulations to measure the static potential for quarks in higher representations using the proposed operators. We find that the proposed operators well reproduce the expected behavior of the original Wilson loop average, which overcomes the problem that occurs in naively applying Abelian-projection to the Wilson loop operator for higher representations.

We examine how the average of double-winding Wilson loops depends on the number of color N in the SU(N) Yang-Mills theory. In the case where the two loops C₁ and C₂ are identical, we derive the exact operator relation which relates the doublewinding Wilson loop operator in the fundamental representation to that in the higher dimensional representations depending on N. By taking the average of the relation, we find that the difference-of-areas law for the area law falloff recently claimed for N = 2 is excluded for N ⩾ 3, provided that the string tension obeys the Casimir scaling for the higher representations. In the case where the two loops are distinct, we argue that the area law follows a novel law (N − 3)A₁/(N − 1) + A₂ with A₁ and A₂(A₁ &lt; A₂) being the minimal areas spanned respectively by the loops C1 and C2, which is neither sum-ofareas (A₁ + A₂) nor difference-of-areas (A₂ − A₁) law when (N ⩾ 3). Indeed, this behavior can be confirmed in the two-dimensional SU(N) Yang-Mills theory exactly.

We study the double-winding Wilson loops in the SU(N) Yang-Mills theory on the lattice. We discuss how the area law falloff of the double-winding Wilson loop average is modified by changing the enclosing contours C1 and C2 for various values of the number of color N. By using the strong coupling expansion, we evaluate the double-winding Wilson loop average in the lattice SU(N) Yang-Mills theory. Moreover, we compute the double-winding Wilson loop average by lattice Monte Carlo simulations for SU(2) and SU(3). We further discuss the results from the viewpoint of the Non-Abelian Stokes theorem in the higher representations.

First, we give a gauge-independent definition of chromomagnetic monopoles in SU(N) Yang-Mills theory which is derived through a non-Abelian Stokes theorem for the Wilson loop operator. Then we discuss how such magnetic monopoles can give a nontrivial contribution to the Wilson loop operator for understanding the area law of the Wilson loop average. Next, we discuss how the magnetic monopole condensation picture are compatible with the vortex condensation picture as another promising scenario for quark confinement. We analyze the profile function of the magnetic flux tube as the non-Abelian vortex solution of U(N) gauge-Higgs model, which is to be compared with numerical simulations of the SU(N) Yang-Mills theory on a lattice. This analysis gives an estimate of the string tension based on the vortex condensation picture, and possible interactions between two non-Abelian vortices.

In the preceeding works, we have given a non-Abelian dual superconductivity picture for quark confinement, and demonstrated the numerical evidences on the lattice. In this talk, we discuss the confinement and deconfinement phase transition at finite temperature in view of the dual superconductivity. We investigate chromomagnetic monopole currents induced by chromoelectric flux in both confinement and deconfinement phase by the numerical simulations on a lattice at finite temperature, and discuss the role of the chromomagnetic monopole in the confinement/deconfinement phase transition.

The dual superconductivity is a promising mechanism for quark confinement. We have presented a new formulation of the Yang-Mills theory on the lattice that enables us to change the original non-Abelian gauge field into the new field variables such that one of them called the restricted field gives the dominant contribution to quark confinement in the gauge independent way. We have pointed out that the SU(3) Yang-Mills theory has another reformulation using new field variables (minimal option), in addition to the way adopted by Cho, Faddeev and Niemi (maximal option). In the past lattice conferences, we have shown the numerical evidences that support the non-Abelian dual superconductivity using the minimal option for the SU(3) Yang-Mills theory. This result should be compared with Abelian dual superconductivity obtained in the maximal option which is a gauge invariant extension of the conventional Abelian projection method in the maximal Abelian gauge. In this talk, we focus on discriminating between two reformulations, i.e., maximal and minimal options of the SU(3) Yang-Mills theory from the viewpoint of dual superconductivity for quark confinement. We investigate the confinement/deconfinement phase transitions at finite temperature in both options, which are compared with the original Yang-Mills theory. For this purpose, we measure the distribution of Polyakov-loops and the Polyakov-loop average, the correlation function of the Polyakov loops and the distribution of the chromoelectric flux connecting a quark and antiquark in both confinement and deconfinement phases.

We propose the reformulations of the SU(N) Yang-Mills theory toward quark confinement and mass gap. In fact, we have given a new framework for reformulating the SU(N) Yang -Mills theory using new field variables. This includes the preceding works given by Cho, Faddeev and Niemi, as a special case called the maximal option in our reformulations. The advantage of our reformulations is that the original non-Abelian gauge field variables can be changed nib the new field variables such that one of them called the restricted field gives the dominant contribution to quark confinement in the gauge independent way. Our reformulations can be combined with the SU(N) extension of the Diakonov-Petrov version of the non -Abelian Stokes theorem for the Wilson loop operator to give a gauge-invariant definition for the magnetic monopole in the SU(N) Yang -Mills theory without the scalar field. In the so-called minimal option, especially, the restricted field is non Abelian and involves the non -Abelian magnetic monopole with the stability group U(N - 1). This suggests the non -Abelian dual superconductivity picture for quark confinement. This should be compared with the maximal option: the restricted field is Abelian and involves only the Abelian magnetic monopoles with the stability group U (1)(N-1), just like the Abelian projection. We give some applications of this reformulation, e.g., the stability for the homogeneous chromomagnetic condensation of the Savvidy type, the large N treatment for deriving the dimensional transmutation and understanding the mass gap, and also the numerical simulations on a lattice which are given by Dr. Shibata in a subsequent talk.

In order to investigate quark confinement, we give a new reformulation of the 80(N) Yang -Mills theory on a lattice and present the results of the numerical simulations of the SU(3) Yang -Mills theory on a lattice. The numerical simulations include the derivation of the linear potential for static interquark potential, i.e., non-vanishing string tension, in which the "Abelian" dominance and magnetic monopole dominance are established, confirmation of the dual Meissner effect by measuring the chromoelectric flux tube between quark-antiquarlt pair, the induced magnetic -monopole current, and the type of dual superconductivity, etc.

We have pointed out that the SU(3) Yang-Mills theory has a new way of reformulation using new field variables (minimal option), in addition to the conventional option adopted by Cho, Faddeev and Niemi (maximal option). The reformulation enables us to change the original non-Abelian gauge field into the new field variables such that one of them called the restricted field gives the dominant contribution to quark confinement in the gauge-independent way. In the minimal option, especially, the restricted field is non-Abelian U(2) and involves the non-Abelian magnetic monopole. In the preceding lattice conferences, we have accumulated the numerical evidences for the non-Abelian magnetic-monopole dominance in addition to the restricted non-Abelian field dominance for quark confinement supporting the non-Abelian dual superconductivity using the minimal option for the SU(3) Yang-Mills theory. This should be compared with the maximal option which is a gauge invarient version of the Abelian projection in the maximal Abelian gauge: The restricted field is Abelian U(1)×U(1) and involves only the Abelian magnetic monopole, just like the Abelian projection. In this talk, we focus on discriminating between two reformulations, i.e., maximal and minimal options of SU(3) Yang-Mills theory for quark confinement from the viewpoint of dual superconductivity. For this purpose, we measure the distribution of the chromoelectric flux connecting a quark and an antiquark and the induced magnetic-monopole current around the flux tube.

We give an analytical derivation of the confinement/deconfinement phase transition at finite temperature in the SU(N) Yang-Mills theory in the D-dimensional space time for D &gt 2. For this purpose, we use a novel reformulation of the Yang-Mills theory which allows the gauge-invariant gluonic mass term, and calculate analytically the effective potential of the Polyakov loop average concretely for the SU(2) and SU(3) Yang-Mills theories by including the gauge-invariant dynamical gluonic mass M. For D = 4, we give an estimate on the transition temperature Td as the ratio Td=M to the mass M which has been measured on the lattice at zero temperature and is calculable also at finite temperature. We show that the order of the phase transition at Td is the second order for SU(2) and weakly first order for SU(3) Yang-Mills theory. We elucidate what is the mechanism for quark confinement and deconfinement at finite temperature and why the phase transition occurs at a certain temperature. These initial results are obtained easily based on the analytical calculations of the "one-loop type" in the first approximation. We discuss also how these results are improved to eliminate the artifacts obtained for some thermodynamic observables.

We have proposed the non-Abelian dual superconductivity in SU(3) Yang-Mills theory for the mechanism of quark confinement, and we presented the numerical evidences in preceding lattice conferences by using the proposed gauge link decomposition to extract magnetic monopole in the gauge invariant way. In this talk, we focus on the dual Meissner effects in view of the magnetic monopole in SU(3) Yang-Mills theory. We measure the chromoelectric and chromomagnetic flux due to a pair of quark and antiquark source at finite temperature. Then, we measure the correlation function of Polyakov loops and Polyakov loop average at various temperatures, and investigate chromomagnetic monopole current induced by chromo-magnetic flux in both confinement and deconfinement phase. We will discuss the role of the chromoelectric monopole in confinement/deconfinement phase transition.

The dual superconductivity is a promising mechanism for quark confinement. We proposed the non- Abelian dual superconductivity picture for SU(3) Yang-Mills theory, and demonstrated the restricted field dominance (called conventionally "Abelian" dominance), and non-Abelian magnetic monopole dominance in the string tension. In the last conference, we have demonstrated by measuring the chromoelectric flux that the non-Abelian dual Meissner effect exists and determined that the dual superconductivity for SU(3) case is of type I, which is in sharp contrast to the SU(2) case: The border of type I and type II. In this talk, we focus on the confinement/deconfinemen phase transition and the non-Abelian dual superconductivity at finite temperature: We measure the chromoelectric flux between a pair of static quark and antiquark at finite temperature, and investigate its relevance to the phase transition and the non-Abelian dual Meissner effect.

We have presented non-Abelian dual superconductivity picture in the SU(3) Yang-Mills(YM) theory, and shown evidences such as the restricted U(2)-field dominance and the non-Abelian magnetic monopole dominance in the string tension. To establish the dual superconductivity picture, the dual Meissner effect in Yang-Mills theory must be examined, and we also presented the evidence of non-Abelian dual Meissner effect by measuring chromo-electric flux tube in the last lattice conferences. In this talk, by applying a new formulation of the YM theory on a lattice, the we further investigate the non- Abelian dual Meissner effect for SU(3) YM theory through correlation function. We examine non-abelian magnetic monopole currents as well as color flux created by the quark-antiquark source.

We have proposed the non-Abelian dual superconductivity picture for quark confinement in the SU(3) Yang-Mills (YM) theory, and have given numerical evidences for the restricted-field dominance and the non-Abelian magnetic monopole dominance in the string tension by applying a new formulation of the YM theory on a lattice. To establish the non-Abelian dual superconductivity picture for quark confinement, we have observed the non-Abelian dual Meissner effect in the SU(3) Yang-Mills theory by measuring the chromoelectric flux created by the quark-antiquark source, and the non-Abelian magnetic monopole currents induced around the flux. We conclude that the dual superconductivity of the SU(3) Yang-Mills theory is strictly the type I and that this type of dual superconductivity is reproduced by the restricted field and the non-Abelian magnetic monopole part, in sharp contrast to the SU(2) case: the border of type I and type II. © Copyright owned by the author(s).

We present recent results on quark confinement: in SU(3) Yang-Mills theory, confinement of fundamental quarks is obtained due to the dual Meissner effect originated from non-Abelian magnetic monopoles defined in a gauge-invariant way, which is distinct from the well-known Abelian projection scenario. This is achieved by using a non-Abelian Stokes theorem for the Wilson loop operator and a new reformulation of the Yang-Mills theory.

The dual Meissner effect is the promising mechanism for quark confinement. We have proposed a new formulation of SU(N) Yang-Mills (YM) theory on a lattice, which can extract the dominant mode for quark confinement in the gauge independent manner. In the last lattice conference, we have demonstrated by measuring the string tension from the Wilson loop average in the SU(3) YM theory that the restricted non-Abelian variable and the extracted non-Abelian magnetic monopoles play the dominant role in confinement of fundamental quarks (dominance in the string tension), in marked contrast to the Abelian projection. In this talk, we focus on the dual Meissner effect in SU(3) YM theory, which is examined by measuring the distribution of chromo-electric field strength created by a static quark-antiquark pair. We apply the new lattice formulation, and examine whether or not the non-Abelian dual superconductivity claimed by us is indeed a mechanism of quark confinement. We present a preliminary result of the direct evidence for the non-abelian dual Meissner effect, that is to say, restricted U(2)-field part of the flux tube plays the dominant role in the quark-antiquark potential.