arXiv subjects

Daniel Zwanziger

Publications and source records attributed to Daniel Zwanziger.

At least 19 recordsLinked to original sources

arXiv2018

Schwinger-Dyson Equations in Coulomb Gauge Consistent with Numerical Simulation

In the present work we undertake a study of the Schwinger-Dyson equation (SDE) in the Euclidean formulation of local quantum gauge field theory, with Coulomb gauge condition $\partial_i A_i = 0$. We continue a previous study which kept only instantaneous terms in the SDE that are proportional to $δ(t)$ in order to calculate the instantaneous part of the time component of the gluon propagator $D_{A_0 A_0}(t, R)$. We compare the results of that study with a numerical simulation of lattice gauge theory and find that the infrared critical exponents and related quantities agree to within 1\% to 3\%. This raises the question, "Why is the agreement so good, despite the systematic neglect of non-instantaneous terms?" We discovered the happy circumstance that all the non-instantaneous terms are in fact zero. They are forbidden by the symmetry of the local action in Coulomb gauge under time-dependent gauge transformations $g(t)$. This remnant gauge symmetry is not fixed by the Coulomb gauge condition. The numerical result of the present calculation is the same as in the previous study; the novelty is that we now demonstrate that all the non-instantaneous terms in the SDE vanish. We derive some elementary properties of propagators which are a consequence of the remnant gauge symmetry. In particular the time component of the gluon propagator is found to be purely instantaneous $D_{A_0 A_0}(t, R) = δ(t) V(R)$, where $V(R)$ is the color-Coulomb potential. Our results support the simple physical scenario in which confinement is the result of a linearly rising color-Coulomb potential, $V(R) \sim σR$ at large $R$.

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arXiv2016

Instantaneous Dynamics of QCD

We start from the observation that, in the confining phase of QCD, the instantaneous color-Coulomb potential in Coulomb gauge is confining. This suggests that, in the confining phase, the dynamics, as expressed in the set of Schwinger-Dyson equations, may be dominated by the purely instantaneous terms. We develop a calculational scheme that expresses the instantaneous dynamics in the local formulation of QCD that includes a cut-off at the Gribov horizon.

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arXiv2016

Some Developments in Gribov's Approach to QCD

We review several developments in the formulation of QCD provided by the GZ action. These include the GZ-action at finite temperature, the relation of the horizon condition and the Kugo-Ojima confinement criterion, the relation of the horizon condition and the dual-Meisssner effect, the alternative derivation of the GZ action provided by the Maggiore-Schaden shift, and the spontaneous breaking of BRST symmetry. We conclude with a proposal for the definition of physical states in the presence of BRST breaking.

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arXiv2016

Local QCD Action at Finite Temperature

In this article, we carefully derive the horizon condition at finite temperature. Only the zero Matsubara mode is affected, and this result is consistent with the suitably modified Maggiore-Schaden shift, which takes into account temporal periodicity. We thus obtain a local, BRST-invariant action for QCD at finite temperature. The large-N limit and other calculational schemes for the magnetic mass and its relation to the Gribov mass are also discussed.

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arXiv2016

Origin of Confining Force

In this article we present exact calculations that substantiate a clear picture relating the confining force of QCD to the zero-modes of the Faddeev-Popov (FP) operator $\mathcal{M}(A) = - \partial \cdot D(A)$. This is done in two steps. First we calculate the spectral decomposition of the FP operator and show that the ghost propagator $\mathcal{G}(k; A) = \langle \vec{k}| \mathcal{M}^{-1}(A) | \vec{k} \rangle$ in an external gauge potential $A$ is enhanced at low $k$ in Fourier space for configurations $A$ on the Gribov horizon. This results from the new formula in the low-$k$ regime $\mathcal{G}^{ab}(k,A) = δ^{ab} λ_{|\vec{k}|}^{-1}(gA)$, where $λ_{|\vec{k}|}(gA)$ is the eigenvalue of the FP operator that emerges from $λ_{|\vec{k}|}(0) = \vec{k}^2$ at $A$ = 0. Next we derive a strict inequality signaling the divergence of the color-Coulomb potential at low momentum $k$ namely, $\widetilde{\mathcal{V}}(k) \geq k^2 G^2(k)$ for $k \to 0$, where $\widetilde{\mathcal{V}}(k)$ is the Fourier transform of the color-Coulomb potential $\mathcal{V}(r)$ and $G(k)$ is the ghost propagator in momentum space. The first result holds in the Landau and Coulomb gauges, whereas the second holds in the Coulomb gauge only. We propose a new numerical lattice gauge fixing that should be closer to the present analytic approach than other numerical gauges.

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arXiv2015

Poincare' Symmetry of the GZ-Model

Due to internal symmetries of its ghost sector, the Poincare' generators of the GZ-model are not unique. The model apparently has two linearly independent symmetric and conserved energy momentum tensors. We show that these energy-momentum tensors are physically equivalent and differ by unobservable conserved currents only. There is a single physical energy-momentum operator that is invariant under all symmetries of the ghost sector, including BRST. This resolves concerns about Poincare' invariance raised by the explicit $x$-dependence of the BRST operator. The energy, momentum and angular momentum of physical states are well-defined quantities that vanish for the ground state of this theory. We obtain and discuss the physical Ward identities resulting from Poincare' invariance.

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arXiv2015

BRST Cohomology and Physical Space of the GZ Model

We address the issue of BRST symmetry breaking in the GZ model, a local, renormalizable, non-perturbative approach to QCD. Explicit calculation of several examples reveals that BRST symmetry breaking apparently afflicts the unphysical sector of the theory, but may be unbroken where needed, in cases of physical interest. Specifically, the BRST-exact part of the conserved energy-momentum tensor and the BRST-exact term in the Kugo-Ojima confinement condition both have vanishing expectation value. We analyze the origin of the breaking of BRST symmetry in the GZ model, and obtain a useful sufficient condition that determines which operators preserve BRST. Observables of the GZ theory are required to be invariant under a certain group of symmetries that includes not only BRST but also others. The definition of observables is thereby sharpened, and excludes all operators known to us that break BRST invariance. We take as a hypothesis that BRST symmetry is unbroken by this class of observables. If the hypothesis holds, BRST breaking is relegated to the unphysical sector of the GZ theory, and its physical states are obtained by the usual cohomological BRST construction. The fact that the horizon condition and the Kugo-Ojima confinement criterion coincide assures that color is confined in the GZ theory.

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arXiv2015

Phase structure and the gluon propagator of SU(2) gauge-Higgs model in two dimensions

We study numerically the phase structure and the gluon propagator of the SU(2) gauge-Higgs model in two dimensions. First, we calculate gauge-invariant quantities, in particular the static potential from Wilson Loop, the W propagator, and the plaquette expectation value. Our results suggest that a confinement-like region and a Higgs-like region appear even in two dimensions. In the confinement-like region, the static potential rises linearly, with string breaking at large distances, while in the Higgs-like region, it is of Yukawa type, consistent with a Higgs-type mechanism. The correlation length obtained from the W propagator has a finite maximum between these regions. The plaquette expectation value shows a smooth cross-over consistent with the Fradkin-Shenker-Osterwalder-Seiler theorem. From these results, we suggest that there is no phase transition in two dimensions. We also calculate a gauge-dependent order parameter in Landau gauge. Unlike gauge invariant quantities, the gauge non-invariant order parameter has a line of discontinuity separating these two regions. Finally we calculate the gluon propagtor. We infer from its infrared behavior that the gluon propagator would vanish at zero momentum in the infinite-volume limit, consistent with an analytical study.

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arXiv2014

Two-dimensional phase structure of SU(2) gauge-Higgs model

We study the phase structure of SU(2) gauge-Higgs model in two dimensions using lattice simulations. We show the result for the plaquette expectation value, static potential, and W propagator. Our results suggest that a confinement-like region and a Higgs-like region appear even in two dimensions. The behavior of the plaquette expectation value is consistent with a smooth cross-over in accordance with the Fradkin-Shenker-Osterwalder-Seiler theorem. In the confinement-like region, the static potential seems to rise linearly with string breaking at large distances, while in the Higgs-like region there seems to be a massive behavior which means that the BEH mechanism occurs. The correlation length obtained from the W propagator has a finite maximum between these phases, which supports no second-order phase transition. Based on these results, we suggest that there is no phase transition in two dimensions.

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arXiv2014

Infrared Saturation and Phases of Gauge Theories with BRST Symmetry

We investigate the infrared limit of the quantum equation of motion of the gauge boson propagator in various gauges and models with a BRST symmetry. We find that the saturation of this equation at low momenta distinguishes between the Coulomb, Higgs and confining phase of the gauge theory. The Coulomb phase is characterized by a massless gauge boson. Physical states contribute to the saturation of the transverse equation of motion of the gauge boson at low momenta in the Higgs phase, while the saturation is entirely due to unphysical degrees of freedom in the confining phase. This corollary to the Kugo-Ojima confinement criterion in linear covariant gauges also is sufficient for confinement in general covariant gauges with BRST- and anti-BRST symmetry, maximal Abelian gauges with an equivariant BRST symmetry, non-covariant Coulomb gauge and in the Gribov-Zwanziger theory.

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arXiv2013

Analytic and numerical study of the free energy in gauge theory

We derive some exact bounds on the free energy W(J) in an SU(N) gauge theory, where J_mu^b is a source for the gluon field A_mu^b in the minimal Landau gauge, and W(J) is the generating functional of connected correlators, exp W(J) = . We also provide asymptotic expressions for the free energy W(J) at large J and for the quantum effective action Gamma(A) at large A. We specialize to a source J(x)=h cos(kx) of definite momentum k and source strength h, and study the gluon propagator D(k,h) in the presence of this source. Among other relations, we prove int_0^inf dh D(k,h)<=2^1/2 k, which implies lim_(k->0) D(k,h) = 0, for all positive h>0. Thus the system does not respond to a static color probe, no matter how strong. Recent lattice data in minimal Landau gauge in d =3 and 4 dimensions at h=0 indicate that the gluon propagator in the minimum Landau gauge is finite, lim_(k->0) D(k,0)>0. Thus these lattice data imply a jump in the value of D(k,h) at h=0 and k=0, and the value of D(k,h) at this point depends on the order of limits. We also present numerical evaluations of the free energy W(k,h) and the gluon propagator D(k,h) for the case of SU(2) Yang-Mills theory in various dimensions which support all of these findings.

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arXiv2013

Bounds on free energy in QCD

We derive some exact bounds on the free energy $W(J)$ in QCD, where $J_μ^b$ is a source for the gluon field $A_μ^b$ in the minimal Landau gauge, and $W(J)$ is the generating functional of connected gluon correlators. Among other results, we show that for a static source $J(x) = h$ the free energy vanishes, $W(h) = 0$, together with its first derivative, ${\partial W(h) \over \partial h} = 0,$ for all $h$, no matter how strong. Thus the system does not respond to a static color probe. We also present numerical evaluations of the free energy $W(J)$ and find that the bounds are well satisfied and in fact undersaturated.

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arXiv2012

Some exact properties of the gluon propagator

Recent numerical studies of the gluon propagator in the minimal Landau and Coulomb gauges in space-time dimension 2, 3, and 4 pose a challenge to the Gribov confinement scenario. We prove, without approximation, that for these gauges, the continuum gluon propagator $D(k)$ in SU(N) gauge theory satisfies the bound ${d-1 \over d} {1 \over (2 π)^d} \int d^dk {D(k) \over k^2} \leq N$. This holds for Landau gauge, in which case $d$ is the dimension of space-time, and for Coulomb gauge, in which case $d$ is the dimension of ordinary space and $D(k)$ is the instantaneous spatial gluon propagator. This bound implies that $\lim_{k \to 0}k^{d-2} D(k) = 0$, where $D(k)$ is the gluon propagator at momentum $k$, and consequently $D(0) = 0$ in Landau gauge in space-time $d = 2$, and in Coulomb gauge in space dimension $d = 2$, but D(0) may be finite in higher dimension. These results are compatible with numerical studies of the Landau-and Coulomb-gauge propagator. In 4-dimensional space-time a regularization is required, and we also prove an analogous bound on the lattice gluon propagator, ${1 \over d (2 π)^d} \int_{- π}^π d^dk {\sum_μ\cos^2(k_μ/2) D_{μμ}(k) \over 4 \sum_λ\sin^2(k_λ/2)} \leq N$. Here we have taken the infinite-volume limit of lattice gauge theory at fixed lattice spacing, and the lattice momentum componant $k_μ$ is a continuous angle $- π\leq k_μ\leq π$. Unexpectedly, this implies a bound on the {\it high-momentum} behavior of the continuum propagator in minimum Landau and Coulomb gauge in 4 space-time dimensions which, moreover, is compatible with the perturbative renormalization group when the theory is asymptotically free.

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arXiv2012

Minimizing gauge-functional for 2-d gravity and string theory

We show the existence of a minimizing procedure for selecting a unique representative on the orbit of any given Riemann surface that contributes to the string partition function. As it must, the procedure reduces the string path integral to a final integration over a particular fundamental domain, selected by the choice of the minimizing functional. This construction somehow demystifies the Gribov question.

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arXiv2012

The Gribov problem and QCD dynamics

In 1967, Faddeev and Popov were able to quantize the Yang-Mills theory by introducing new particles called ghost through the introduction of a gauge. Ever since, this quantization has become a standard textbook item. Some years later, Gribov discovered that the gauge fixing was not complete, gauge copies called Gribov copies were still present and could affect the infrared region of quantities like the gauge dependent gluon and ghost propagator. This feature was often in literature related to confinement. Some years later, the semi-classical approach of Gribov was generalized to all orders and the so-called GZ action was born. Ever since, many related articles were published. This review tends to give a pedagogic review of the ideas of Gribov and the subsequent construction of the GZ action, including many other toipics related to the Gribov region. It is shown how the GZ action can be viewed as a non-perturbative tool which has relations with other approaches towards confinement. Many different features related to the GZ action shall be discussed in detail, such as BRST breaking, the KO criterion, the propagators, etc. We shall also compare with the lattice data and other non-perturbative approaches, including stochastic quantization.

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arXiv2012

Exact bounds on the free energy in QCD

We consider the free energy $W[J] = W_k(H)$ of QCD coupled to an external source $J_μ^b(x) = H_μ^b \cos(k \cdot x)$, where $H_μ^b$ is, by analogy with spin models, an external "magnetic" field with a color index that is modulated by a plane wave. We report an optimal bound on $W_k(H)$ and an exact asymptotic expression for $W_k(H)$ at large $H$. They imply confinement of color in the sense that the free energy per unit volume $W_k(H)/V$ and the average magnetization $m(k, H) ={1 \over V} {\p W_k(H) \over \p H}$ vanish in the limit of constant external field $k \to 0$. Recent lattice data indicate a gluon propagator $D(k)$ which is non-zero, $D(0) \neq 0$, at $k = 0$. This would imply a non-analyticity in $W_k(H)$ at $k = 0$. We also give some general properties of the free energy $W(J)$ for arbitrary $J(x)$. Finally we present a model that is consistent with the new results and exhibits (non)-analytic behavior. Direct numerical tests of the bounds are proposed.

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arXiv2011

An improved model of color confinement

We consider the free energy $W[J] = W_k(H)$ of QCD coupled to an external source $J_μ^b(x) = H_μ^b \cos(k \cdot x)$, where $H_μ^b$ is, by analogy with spin models, an external "magnetic" field with a color index that is modulated by a plane wave. We report an optimal bound on $W_k(H)$ and an exact asymptotic expression for $W_k(H)$ at large $H$. They imply confinement of color in the sense that the free energy per unit volume $W_k(H)/V$ and the average magnetization $m(k, H) ={1 \over V} {\p W_k(H) \over \p H}$ vanish in the limit of constant external field $k \to 0$. Recent lattice data indicate a gluon propagator $D(k)$ which is non-zero, $D(0) \neq 0$, at $k = 0$. This would imply a non-analyticity in $W_k(H)$ at $k = 0$. We present a model that is consistent with the new results and exhibits (non)-analytic behavior. Direct numerical tests of the bounds are proposed.

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arXiv2010

A model of color confinement

A simple model is presented that describes the free energy $W(J)$ of QCD coupled to an external current that is a single plane wave, $J(x) = H \cos(k \cdot x)$. The model satisfies a bound obtained previously on $W(J)$ that comes from the Gribov horizon. If one uses this model to fit recent lattice data --- which give for the gluon propagator $D(k)$ a non-zero value, $D(0) \neq 0$, at $k = 0$ --- the data favor a non-analyticity in $W(J)$.

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