近藤 慶一

In this letter, we perform Monte-Carlo simulation of SU (2) Yang-Mills theory under the Cho-Faddeev-Niemi decomposition. Various quantities concerning magnetic condensate are studied. We also define CFN monople on the lattice and estimate a monopole density. The difference between usual DT monopole and CFN monople are explained.

We discuss the renormalization of a Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST invariant composite operator of mass dimension 2 in Yang-Mills theory with general BRST and anti-BRST invariant gauge-fixing terms of Lorentz type. The interest of this study stems from a recent claim that the nonvanishing vacuum condensate of the composite operator in question can be an origin of mass gap and quark confinement in any manifestly covariant gauge, as proposed by one of the authors. First, we obtain the renormalization group flow of the Yang-Mills theory. Next, we show the multiplicative renormalizability of the composite operator and that the BRST and anti-BRST invariance of the bare composite operator is preserved under the renormalization. Third, we perform the operator product expansion of the gluon and ghost propagators and obtain the Wilson coefficient corresponding to the vacuum condensate of mass dimension 2. Finally, we discuss the connection of this work with previous works and argue the physical implications of the obtained results.

We propose a vacuum condensate of mass dimension 2 consisting of gluons and ghosts in the framework of the manifestly covariant gauge fixing of the SU(N) Yang-Mills theory. This quantity is both BRST- and anti-BRST-invariant for any gauge. It includes the ghost condensation C-a(C) over bar (a) proposed first in the modified Maximal Abelian gauge and reduces to the gluon condensates (Att)2 of mass dimension 2 proposed recently in the Landau gauge of the Lorentz gauge fixing. The vacuum condensate of dimensions 2 can provide the effective mass for gluons and g-hosts. The possible existence of such condensations is demonstrated by calculating the operator product expansion of the gluon and ghost propagators in both gauges. Its implications to quark confinement are also discussed in consistent with the previous works. (C) 2001 Published by Elsevier Science B.V.

We show that all effective Abelian gauge theory can be obtained as a renormalizable theory from QCD in the maximal Abelian gauge. The derivation improves in a systematic manner the previous version that was obtained by one of the authors and was referred to as the Abelian-projected effective gauge theory. This result supports the view that we can construct an effective Abelian gauge theory from QCD without losing characteristic features of the original non-Abelian gauge theory. In fact, it, is shown that the effective coupling constant in the resulting renormalizable theory has a renormalization-scale dependence governed by the beta -function that is exactly the same as that of the original Yang-Mills theory, irrespective of the choice of gauge fixing parameters of the maximal Abelian gauge and the parameters used for identifying the dual variables. Moreover, we evaluate the anomalous dimensions of the fields and parameters in the resultant theory. By choosing the renormalized parameters appropriately, we call switch the theory into all electric or a magnetic theory.

By making use of the background field method, we derive a novel reformulation of the Yang-Mills theory which was proposed recently by the author to derive quark confinement in QCD. This reformulation identifies the Yang-Mills theory with a deformation of a topological quantum field theory. The relevant background is given by the topologically nontrivial field configuration, especially, the topological soliton which can be identified with the magnetic monopole current in four dimensions. We argue that the gauge fixing term becomes dynamical and that the gluon mass generation takes place by a spontaneous breakdown of the hidden supersymmetry caused by the dimensional reduction. We also propose a numerical simulation to confirm the validity of the scheme we have proposed. Finally we point out that the gauge fixing part may have a geometric meaning from the viewpoint of global topology where the magnetic monopole solution represents the critical point of a Morse function in the space of field configurations.

We derive a new version of the non-Abelian Stokes theorem for the Wilson loop in the SU(N) case by making use of the coherent state representation on the coset space SU(IV)IU(1)(N-1) = FN-1, the flag space. We consider the SU(N) Yang-Mills theory in the maximal Abelian gauge in which SU(N) is broken down to U(1)(N-1). First, we show that the Abelian dominance in the string tension follows from this theorem and the Abelian-projected effective gauge theory that was derived by one of the authors. Next (but independently), combining the non-Abelian Stokes theorem with a novel reformulation of the Yang-Mills theory recently proposed by one of the authors, we proceed to derive the area law of the Wilson loop in four-dimensional SU(N) Yang-Mills theory in the maximal Abelian gauge. Owing to dimensional reduction, the planar Wilson loop at least for the fundamental representation in four-dimensional SU(N) Yang-Mills theory can be estimated by the diagonal (Abelian) Wilson loop defined in the two-dimensional CPN-1 model. This derivation shows that the fundamental quarks are confined by a single species of magnetic monopole. The origin of the area la cy is related to the geometric phase of the Wilczek-Zee holonomy for U(N - 1). The calculations are performed using the instanton calculus (in the dilute instanton-gas approximation) and using the large N expansion (in the leading order).

We show that off-diagonal gluons and off-diagonal ghosts acquire their masses dynamically in QCD if the maximal Abelian gauge is adopted, This result strongly supports the Abelian dominance in low-energy region of QCD. The mass generation is shown to occur due to ghost-anti-ghost condensation caused by attractive quartic ghost interactions within the Abelian projected effective gauge theory (derived by one of the authors). In fact, the quartic ghost interaction is indispensable for the renormalizability due to nonlinearity of the maximal Abelian gauge. The ghost-anti-ghost condensation is associated with the spontaneous breaking of global SL(2, R) symmetry recently found by Schaden at least for SU(2) case. Moreover we write down a new extended BRS algebra in the maximal Abelian gauge which should be compared with that of Nakanishi-Ojima for the Lorentz gauge. Finally, we argue that the mass generation may be related to the spontaneous breaking of a supersymmetry OSp(4/2) hidden in the maximal Abelian gauge. (C) 2000 Published by Elsevier Science B.V.

We derive a new version of SU(3) non-Abelian Stokes theorem by making use of the coherent state representation on the coset space SU(3)/(U(1) x U(1)) = F-2, the flag space. Then we outline a derivation of the area law of the Wilson loop in SU(3) Yang-Mills theory in the maximal Abelian gauge (the detailed exposition wilt be given in a forthcoming article). This derivation is performed by combining the non-Abelian Stokes theorem with the reformulation of the Yang-Mills theory as a perturbative deformation of a topological field theory recently proposed by one of the authors. Within this framework, we show that the fundamental quark is confined even if G = SU(3) is broken by partial gauge fixing into H = U(2) just as G is broken to H = U(1) x U(1). An origin of the area law is related to the geometric phase of the Wilczek-Zee holonomy for U(2). Abelian dominance is an immediate by-product of these results and magnetic monopole plays the dominant role in this derivation.

We give another derivation of quark confinement in QCD from the viewpoint of the low-energy effective Abelian gauge theory of QCD obtained via Abelian projection. It is based on the recently discovered Berezinskii-Kosterlitz-Thouless phase transition in the four-dimensional Abelian gauge theory. Moreover, we show that there exists a critical gauge coupling constant in QCD, above which confinement set in and below which there is no confinement. Tn a SU(N) gauge theory with N-f flavored fermions, we argue that this leads to a critical value of fermion flavors N-f(c) for the confinement as well as the chiral symmetry breaking, which separates the deconfining and chiral symmetric phase from the confining and chiral symmetry breaking phase. A finite-temperature deconfinement transition is also discussed briefly. (C) 1999 Published by Elsevier Science B.V. All rights reserved.

We prove Abelian magnetic monopole dominance in the string tension of QCD. Abelian and monopole dominance in low-energy physics of QCD has been confirmed for various quantities by recent Monte Carlo simulations of lattice gauge theory. In order to prove this dominance, we use the reformulation of continuum Yang-Mills theory in the maximal Abelian gauge as a deformation of a topological field theory of magnetic monopoles. which was proposed in the previous article by the author. This reformulation provides an efficient way for incorporating the magnetic monopole configuration as a topological nontrivial configuration in the functional integral. We derive a version of the non-Abelian Stokes theorem and use it to estimate the expectation value of the Wilson loop. This clearly exhibits the role played by the magnetic monopole as an origin of the Berry phase in the calculation of the Wilson loop in a manifestly gauge-invariant manner. We show that the string tension derived from the diagonal (Abelian) Wilson loop in topological field theory (studied in the previous article) converges to that of the full non-Abelian Wilson loop in the limit of a large Wilson loop. Therefore, within the above reformulation of QCD, this result (together with the previous result) completes the proof of quark confinement in QCD based on the criterion of the area law of the full non-Abelian Wilson loop. [S0556-2821(98)08420-3].

We propose a reformulation of Yang-Mills theory as a. perturbative deformation of a novel topological (quantum) field theory. We prove that this reformulation of four-dimensional QCD leads to quark confinement in the sense of an area law of the Wilson loop. First, Yang-Mills theory with a non-Abelian gauge group G is reformulated as a deformation of a novel topological field theory. Next, a special class of topological field theories is defined by both Becchi-Rouet-Stora-Tyupin (BRST) and anti-BRST exact actions corresponding to the maximal Abelian gauge leaving the maximal torus group H of G invariant. Then we find topological field theory (D>2) has a hidden supersymmetry for a choice of maximal Abelian gauge. As a result, the D-dimensional topological field theory is equivalent to the (D-2)-dimensional coset G/H nonlinear sigma model in the sense of the Parisi-Sourlas dimensional reduction. After maximal Abelian gauge fixing, the topological property of the magnetic monopole and antimonopole of four-dimensional Yang-Mills theory is translated into that of an instanton and anti-instanton in a two-dimensional equivalent model. It is shown that the linear static potential in four dimensions follows fi om the instanton-anti-instanton gas in the equivalent two-dimensional nonlinear sigma model obtained from the four-dimensional topological field theory by dimensional reduction, while the remaining Coulomb potential comes from the perturbative part in four-dimensional Yang-Mills theory. The dimensional reduction opens a path for applying various exact methods developed in two-dimensional quantum field theory to study the nonperturbative problem in low-energy physics of four-dimensional quantum field theories. [S0556-2821(98)04920-0].

We show that four-dimensional U(1) gauge theory in the continuum formulation has a confining phase (exhibiting the area law of the Wilson loop) in the strong coupling region above a critical coupling g(c). This result is obtained by taking into account topological nontrivial sectors in U(1) gauge theory. The derivation is based on the reformulation of gauge theory as a deformation of topological quantum field theory and a subsequent dimensional reduction of the D-dimensional topological quantum field theory to the (D-2)-dimensional nonlinear cr model; The topological quantum field theory part of four-dimensional U(1) gauge theory is exactly equivalent to the two-dimensional O(2) nonlinear sigma model. The confining (r. Coulomb) phase of U(1) gauge theory corresponds to the high- (r, low-) temperature phase of the O(2) nonlinear sigma model and the critical point g(c) is determined by the Berezinskii-Kosterlitz-Thouless phase transition temperature. The quark (charge) confinement in the strong coupling phase is caused by vortex condensation. Thus the continuum gauge theory has direct correspondence to the compact formulation of lattice gauge theory. [S0556-2821(98)05718-X].

Starting from SU(2) Yang-Mills theory in 3+1 dimensions, we prove that the Abelian-projected effective gauge theories are written in terms of the maximal Abelian gauge held and the dual Abelian gauge field interacting with magnetic monopole current. This is performed by integrating out all the remaining non-Abelian gauge field belonging to SU(2)/U(1). We show that the resulting Abelian gauge theory recovers exactly the same one-loop beta function as the original Yang-Mills theory. Moreover, the dual Abelian gauge field becomes massive if the monopole condensation occurs. This result supports the dual superconductor scenario for quark confinement in QCD. We give a criterion of dual superconductivity and point out that the magnetic monopole condensation may be estimated from the classical instanton configuration. Therefore there can exist an effective Abelian gauge theory which shows both asymptotic freedom and quark confinement based on the dual Meissner mechanism. The inclusion of an arbitrary number of fermion flavors is straightforward in this approach. Some implications to the lower dimensional case will also be discussed.

Based on the path integral formalism, we rederive and extend the transverse Ward-Takahashi identities (which were first derived by Yasushi Takahashi) for the vector and the axial vector currents and simultaneously discuss the possible quantum anomaly for them. Subsequently, we propose a new scheme for writing down and solving the Schwinger-Dyson equation in which the transverse Ward-Takahashi identity together with the usual (longitudinal) Ward-Takahashi identity are applied to specify the fermion-boson vertex function. Within this framework, we give an example of exactly soluble truncated Schwinger-Dyson equation for the fermion propagator in an Abelian gauge theory in arbitrary dimension when the bare fermion mass is zero. It is especially shown that in two dimensions, it becomes the exact and closed Schwinger-Dyson equation which can be exactly solved.

In the normal phase (where no dynamical fermion mass generation occurs) of the D-dimensional quantum electrodynamics with N-f flavors of fermions, we derive an integral equation which should be satisfied by (the inverse of) the wave function renormalization of the fermion in the Landau gauge. For this we use the inverse Landau-Khalatnikov transformation connecting the nonlocal gauge with the Landau gauge. This leads to a similar equation for the running flavor number in the framework of the 1/N-f resumed Schwinger-Dyson equation. Solving the equation analytically and numerically, we study the infrared behavior and the critical exponent of the 3-dimensional QED (QED(3)). This confirms that the flavor number in QED(3) runs according to the beta function which is consistent with the asymptotic freedom as that in 4-dimensional QCD. (C) 1997 Elsevier Science B.V.

We propose the gauged Thirring model as a natural gauge-invariant generalization of the Thirring model with a four-fermion interaction of current-current type. In the strong gauge-coupling limit, the gauged Thirring model reduces to the recently proposed reformulation of the Thirring model as a gauge theory. We pay special attention to;the effect coming from the kinetic term for the gauge boson field which was originally the auxiliary field without the kinetic term. In 3 + 1 dimensions, we find nontrivial phase structure for the gauged Thirring model, based on the Schwinger-Dyson equation for the fermion propagator and a gauge-invariant effective potential for the chiral order parameter. Within this approximation, we study the renormalization group flows (lines of constant physics) and find a signal for nontrivial continuum limit with non-vanishing renormalized coupling constant and large anomalous dimension for the gauged Thirring model in 3 + 1 dimensions, at least for a small number of flavors N-f. Finally we discuss the (perturbatively) renormalizable extension of the gauged Thirring model.

We discuss how to define and obtain the running coupling of a gauge theory in the approach of the Schwinger-Dyson (SD) equation, in order to perform a nonperturbative study of the theory, For this purpose, we introduce the nonlocally generalized gauge fixing into the SD equation, which is used to define the running coupling constant (this method is applicable only to a gauge theory). Some advantages and the validity of this approach are exemplified in QED(3). This confirms the slowing down of the rate of decrease of the running coupling and the existence of the nontrivial infrared fixed point (in the normal phase) of QED(3), claimed recently by Aitchison and Mavromatos, without so many of their approximations. We also argue that the conventional approach is recovered by applying the (inverse) Landau-Khalatnikov transformation to the nonlocal gauge result.

We investigate the critical behavior of the gauged NJL model (QED plus four-fermion interaction) and the gauged Yukawa model by the use of the inversion method. By calculating the gauge-invariant chiral condensate in the inversion method to the lowest order, we derive the critical line which separates the spontaneous chiral-symmetry breaking phase from the chiral symmetric one. The critical exponent for the chiral order parameter associated with the second order chiral phase transition is shown to take the mean field value together with possible logarithmic correction to the mean-field prediction. All the above results are gauge-parameter independent and are compared with the previous results obtained from the Schwinger-Dyson equation for the fermion propagator.

Non-perturbative renormalization is given for both symmetric and spontaneously broken phases of the chiral symmetry in the gauged Nambu-Jona-Lasinio model. We explicitly obtain beta-function having non-trivial uv fixed line for the renormalized coupling as well as the bare one. In both phases the anomalous dimension is very large (> 1) near the fixed line without discontinuity.

Based on the Cornwall-Jackiw-Tomboulis effective potential, we extensively study nonperturbative renormalization of the gauged Nambu-Jona-Lasinio model in the ladder approximation with standing gauge coupling. Although the pure Nambu-Jona-Lasinio model is not renormalizable, presence of the gauge interaction makes it possible that the theory is renormalized as an interactirig continuum theory at the critical line in the ladder approximation. Extra higher dimensional operators (''counter terms'') are not needed for the theory to be renormalized. By virtue of the effective potential approach, the renormalization (''symmetric renormalization'') is performed in a phase-independent manner both for the symmetric and the spontaneously broken phase of the chiral symmetry. We explicitly obtain beta function having a nontrivial ultraviolet fixed line for the renormalized coupling as well as the bare one. In both phases the anomalous dimension is very large (greater-than-or-equal-to 1) without discontinuity across the fixed line. Operator product expansion is explicitly constructed, which is consistent with the large anomalous dimension owing to the appearance of the nontrivial extra power behavior in the Wilson coefficient for the unit operator. The symmetric renormalization breaks down at the critical gauge coupling, which is cured by the generalized renormalization scheme (''MBAR-dependent renormalization''). Also emphasized is the formal resemblance to the four-fermion theory in less than four dimensions which is renomalizable in 1/N expansion.

Under a novel ansatz for the vertex function, the Schwinger-Dyson equation for the fermion propagator in the cutoff QED is solved in the arbitrary gauge, taking account of the vacuum polarization in the photon propagator. For any ultraviolet cutoff A, there exists a bifurcation point e(c)(LAMBDA) of the bare coupling constant above which die trivial fermion-mass function for massless QED bifurcates to another, nontrivial massive solution. With a proper choice of the transverse vertex function and the longitudinal vertex that respects die Ward-Takahashi identity, the critical point e(c)(infinity) and the critical scaling behavior in the vicinity of the critical point are shown to be gauge-independent. In the arbitrary gauge, it is shown that the quenched, planar QED obeys Miransky's scaling of the essential-singularity type and that the unquenched QED exhibits the mean-field critical behavior with classical critical exponents.

For a lattice regularized chiral-invariant SU(2)L x SU(2)R fermion-scalar model with a Yukawa coupling y and a Wilson-Yukawa coupling w, we investigate the phase structure and show in particular the existence of the multicritical line, in the strong Yukawa and/or Wilson-Yukawa coupling region, at which four phases meet. The result is in good agreement with the Monte Carlo simulation. This analytical result is derived from the effective scalar model obtained by integrating out the fermion field where the action is explicitly obtained from the hopping parameter expansion up to next-to-leading order. For estimates on the correlation function of the scalar field we apply the mean-field method.

Dynamical chiral-symmetry-breaking in massless QED with N fermion species is studied through the numerical solution of the coupled Schwinger-Dyson (SD) equation. We have taken into account the fermion loop effect (at the 1-loop level) in the SD equation for the photon propagator through the vacuum polarization function PI(k2), with and without the standard approximation: PI((P-q)2) almost-equal-to PI(max (p2, q2)). We have found that the scaling law is unchanged by this approximation and that, irrespective of the fermion flavor N, the dynamical fermion mass and chiral order parameter obey the same mean-field type scaling, while the quenched planar QED without the vacuum polarization (N = 0 limit) obeys the Miransky scaling with the essential singularity.

In the strong coupling region of QED, we have obtained a numerical solution to the simultaneous Schwinger-Dyson equation for the fermion and the photon propagators. It is shown that there is a critical point separating the spontaneous-chiral-symmetry-breaking (strong coupling) phase and the weak coupling phase. This critical point is consistently interpreted as the second order phase transition point at which the continuum limit of cutoff QED may be taken. The vacuum polarization function obtained in this frame-work work exhibits essentially the same asymptotic uv behavior as that predicted from the one-loop calculation. The scaling behavior is not inconsistent with the mean-field result as predicted from the one-loop case, which favors the triviality of QED.

Using weak and strong Yukawa coupling expansions combined with the mean field theory, we determine the phase structure of Yukawa models with Z(2), O(2), O(4) scalars. We have taken into account all the terms in the relevant expansions up to eighth order. In the leading order, all the Yukawa models have the four phases and do not possess the ferrimagnetic (FI) phase. This feature persists for Z(2) and O(2) Yukawa models to all orders. In the O(4) Yukawa model, on the other hand, the Fl phase appears in the next to leading order and all higher orders, and two symmetric phases are completely separated by the FI phase. The result holds irrespective of the choice of fermions; naive or staggered.

In the leading order of the 1/N expansion for the vacuum polarization, we obtain the chiral-symmetry-breaking solutions of the Schwinger-Dyson equation in three-dimensional QED with N four-component Dirac fermions. Special attention is paid on the critical number of the fermion flavor N(c) and the scaling law for the dynamical fermion mass. In this framework we introduce the infrared cutoff as well as the ultraviolet one. We show the relation between the vertex ansatz and the scaling law which is consistent with our numerical results that the scaling law is affected by the infrared cutoff. Especially in the case of the Pennington and Webb's vertex ansatz, the mean-field type scaling law and cutoff-dependent finite N(c) are obtained in the large infrared cutoff, while the exponential type scaling and infinite N(c) are given in the sufficiently small infrared cutoff. The connection with the Monte Carlo simulation is discussed.

We investigate the phase structure of (QCD-like) gauged Nambu-Jona-Lasinio model (QCD-like gauge theories plus four-fermion interactions) based on the ladder Schwinger-Dyson equation with one-loop running gauge coupling. Through analytical and numerical studies, we establish two-fixed points structure, one with a large anomalous dimension gamma-m congruent-to 2 and the other with a small one gamma-m congruent-to 0. We further obtain the power critical exponents through the equation of state, which, as they stand, imply that the former fixed point is a Gaussian fixed point. We emphasize that logarithmic corrections due to the gauge interaction is crucial to obtaining an interacting continuum theory at this fixed point.

The critical behavior of strongly coupled QED with a chiral-invariant four-fermion interaction (gauged Nambu-Jona-Lasinio model) is investigated through the unquenched Schwinger-Dyson equation including the fermion loop effect at the one-loop level. It is shown that the critical exponents satisfy the (hyper)scaling relations as in the quenched case. However, the respective critical exponent takes the classical mean-field value, and consequently unquenched QED belongs to the same universality class as the zero-charge model. On the other hand, it is pointed out that quenched QED violates not only universality but also weak universality, due to continuously varying critical exponents. Furthermore, the renormalization group flow of constant renormalized charge is given. All the results are consistent with triviality of QED and the gauged Nambu-Jona-Lasinio model in the unquenched case.

The chiral-symmetry-breaking phase transition of massless QED with chiral-invariant four-fermion interaction is studied through the Schwinger-Dyson equation, taking into account the vacuum polarization effect of the photon propagator. We determine the structure of the phase diagram, the critical line and the scaling relation for the dynamical fermion mass and the chiral order parameter in the neighborhood of the critical line. Our results are consistent with the mean-field behavior and there is no compelling evidence for an interacting continuum limit.

In the presence of vacuum polarization, non-trivial solutions of the Schwinger-Dyson (SD) equation for the fermion propagator are obtained in massless QED with N fermion flavors. We derive the critical line and scaling relation associated with the chiral phase transition which separates the weak coupling phase from the non-perturbative phase where chiral symmetry is spontaneously broken. It is shown that the dynamically generated fermion mass and the chiral order parameter obeys the same scaling of the mean-field (MF) type near the critical point with the same critical exponent nu-m = nu-psi-psi = 1/2, irrespective of the fermion flavor N except for N = O. The anomalous dimension gamma-m of the composite operator psiBAR-psi is also calculated to result in gamma-m = O irrespective of N on the critical line of the phase diagram (beta, N), in contrast with the quenched planar case, gamma-m = 1.

We have found a novel spontaneous-chiral-symmetry-breaking solution to the ladder Schwinger-Dyson equation for QED plus a chiral-invariant four-fermion interaction. The critical line is explicitly obtained in the plane of two coupling constants of gauge and four-fermion interactions. The existence of a dilaton pole has also been examined on the full critical line. © 1989 The American Physical Society.