We investigate the gluon, ghost and quark propagators in the Landau gauge with dynamic quarks. We perform a one-loop calculation in a model where the standard Faddeev-Popov Lagrangian is complemented by a mass term for the gluons which is seen as a minimal way of taking into account the effect of the Gribov copies. The analytic results are compared with lattice data obtained in four dimension and for two, three and four quark flavors. The gluon and ghost propagators are reproduced with a few percent accuracy in the whole range of accessible momenta. The scalar part of the quark propagator is found to be in good agreement with the lattice data. However, the quark renormalization is poorly described. We attribute this discrepancy to the fact that the one-loop corrections to this quantity are unusually small so that the two loop contribution can not be discarded. The results are expressed in terms of the coupling, the gluon mass and the light quark mass at 1 GeV.
I analyze the IR fixed point structure of Landau gauge QCD. Precisely the fixed point with a strong kinematic singularity of the quark-gluon vertex that proved crucial for the recently proposed confinement mechanism in the quenched approximation is absent in dynamical QCD. Therefore, the IR singularities do not induce asymptotic quark confinement but the long-range interaction is screened by unquenching loops at scales of the order of the quark mass. This provides the prerequisite for a microscopic description of deconfinement and string breaking. The fixed points determine the qualitative for…
We determine the two loop correction to Gribov's mass gap equation for quantum chromodynamics in the Landau gauge in the MSbar scheme by computing the two loop correction to the horizon condition derived from Zwanziger's local renormalizable Lagrangian which incorporates the Gribov parameter. We verify that with the explicit result, the two loop ghost propagator is enchanced in the infrared.
In lattice QCD the computation of one-particle irreducible (1PI) Green's functions with a large number (> 2) of legs is a challenging task. Besides tuning the lattice spacing and volume to reduce finite size effects, the problems associated with the estimation of higher order moments via Monte Carlo methods and the extraction of 1PI from complete Green's functions are limitations of the method. Herein, we address these problems revisiting the calculation of the three gluon 1PI Green's function.
We construct the local Gribov-Zwanziger action for SU(2) Euclidean Yang-Mills theories in the maximally Abelian (MA) gauge with U(1)$_3$ Landau gauge fixing based on the Zwanziger's work in the Landau gauge. By the restriction of the functional integral region to the Gribov region in the MA gauge, we give the nonlocal action. We localize the action with new fields and obtain the action with the shift of the new scalar fields, which has the terms, corresponding to the localized action of the horizon function in the MA gauge. The diagonal gluon propagator in the MA gauge at tree level behaves li…
We investigate the infrared behaviour of gluon and ghost propagators in Landau gauge QCD by means of an exact renormalisation group equation. We explain how, in general, the infrared momentum structure of Green functions can be extracted within this approach. An optimisation procedure is devised to remove residual regulator dependences. In Landau gauge QCD this framework is used to determine the infrared leading terms of the propagators. The results support the Kugo-Ojima confinement scenario. Possible extensions are discussed.
Correlation functions of Landau gauge Yang-Mills theory are calculated from their equations of motion. Solutions for the three- and four-gluon vertices and the role of two-loop diagrams in the gluon propagator equation are discussed.
We present first evidence for the Landau level structure of Dirac eigenmodes in full QCD for nonzero background magnetic fields, based on first principles lattice simulations using staggered quarks. Our approach involves the identification of the lowest Landau level modes in two dimensions, where topological arguments ensure a clear separation of these modes from energetically higher states, and an expansion of the full four-dimensional modes in the basis of these two-dimensional states. We evaluate various fermionic observables including the quark condensate and the spin polarization in this …
The construction outlined by Henyey is employed to provide examples of normalizable zero modes of the Faddeev-Popov operator in the Landau and maximal Abelian gauges in SU(2) Euclidean Yang-Mills theories in d=3 dimensions. The corresponding gauge configurations have all finite norm ||A||^2 < \infty. In particular, in the case of the Landau gauge, the explicit construction of an infinite class of normalizable zero modes with finite norm ||A||^2 is provided.
We show that the application of a novel gauge invariant truncation scheme to the Schwinger-Dyson equations of QCD leads, in the Landau gauge, to an infrared finite gluon propagator and a divergent ghost propagator, in qualitative agreement with recent lattice data.
The Landau background gauge, also known as the Landau-DeWitt gauge, has found renewed interest during the past decade given its usefulness in accessing the confinement-deconfinement transition via the vacuum expectation value of the Polyakov loop, describable via an appropriate background. In this Letter, we revisit this gauge from the viewpoint of it displaying gauge (Gribov) copies. We generalize the Gribov-Zwanziger effective action in a BRST and background invariant way; this action leads to a restriction on the allowed gauge fluctuations, thereby eliminating the infinitesimal background g…
Examples of normalizable zero modes of the Faddeev-Popov operator in SU(2) Euclidean Yang-Mills theories in the Landau gauge are constructed in d=2,3,4 dimensions.
Fixing a gauge in the non-perturbative domain of Yang-Mills theory is a non-trivial problem due to the presence of Gribov copies. In particular, there are different gauges in the non-perturbative regime which all correspond to the same definition of a gauge in the perturbative domain. Gauge-dependent correlation functions may differ in these gauges. Two such gauges are the minimal and absolute Landau gauge, both corresponding to the perturbative Landau gauge. These, and their numerical implementation, are described and presented in detail. Other choices will also be discussed. This investigati…
Complete gauge-fixing beyond perturbation theory in non-Abelian gauge theories is a non-trivial problem. This is particularly evident in covariant gauges, where the Gribov-Singer ambiguity gives an explicit formulation of the problem. In practice, this is a problem if gauge-dependent quantities between different methods, especially lattice and continuum methods, should be compared: Only when treating the Gribov-Singer ambiguity in the same way is the comparison meaningful. To provide a better basis for such a comparison the structure of the first Gribov region in Landau gauge, a subset of all …
We summarise an analysis of the infrared regime of Landau gauge QCD by means of a flow equation approach. The infrared behaviour of gluon and ghost propagators is evaluated. The results provide further evidence for the Kugo-Ojima confinement scenario. We also discuss their relation to results obtained with other functional methods as well as the lattice.
We report on the first self-consistent solution of the Dyson-Schwinger equation (DSE) for the three-gluon vertex. Based on earlier results for the propagators which match data from lattice Monte-Carlo simulations, we obtain results for the three-gluon vertex that are in very good agreement with available lattice data likewise. Feeding these results back into the propagator DSEs leads to some changes especially in the gluon propagator. These changes allow us to assess previously used models for the three-gluon vertex and to systematically estimate the influence of neglected two-loop diagrams wi…
Starting from the lattice Landau gauge gluon and ghost propagator data we use a sequence of Padé approximants, identify the poles and zeros for each approximant and map them into the analytic structure of the propagators. For the Landau gauge gluon propagator the Padé analysis identifies a pair of complex conjugate poles and a branch cut along the negative real axis of the Euclidean $p^2$ momenta. For the Landau gauge ghost propagator the Padé analysis shows a single pole at $p^2 = 0$ and a branch cut also along the negative real axis of the Euclidean $p^2$ momenta. The method gives precise es…
The coupled system of the quark-gluon vertex and quark propagator Dyson-Schwinger equations (DSEs) is investigated within Landau gauge QCD. The aim is to get a deeper insight into the mechanisms of quark confinement and dynamical chiral symmetry breaking and into a possible relation between these two phenomena. To this end an earlier study is extended by improving systematically on the truncations imposed on the quark-gluon vertex DSE. A clear infrared enhancement for all tensor structures of the quark-gluon vertex is obtained.
In Landau gauge QCD the infrared behavior of the propagator of transverse gluons can be analytically determined to be a power law from Dyson-Schwinger equations. This propagator clearly shows positivity violation, indicating the absence of the transverse gluons from the physical spectrum, i.e. gluon confinement. A simple analytic structure for the gluon propagator is proposed capturing all important features. We provide arguments that the Landau gauge quark propagator possesses a singularity on the real timelike axis. For this propagator we find a positive definite Schwinger function.
Yang-Mills theory can be formulated for any semi-simple Lie algebra, and thus any semi-simple Lie group. In principle, the dynamics could be different for each one. However, functional studies predict that the propagators in Landau gauge depend only quantitatively on the gauge algebra. In particular, genuine non-perturbative effects should be present even in the large N-limit for su(N) gauge algebras. Lattice gauge theory is used to investigate this in detail. The propagators are determined for the gauge groups SU(2), SU(3), SU(4), SU(5), SU(6) and G2, in two and three dimensions. In accordanc…
We study analytic structures of the gluon, quark, and ghost propagators in the Landau-gauge QCD and general properties from the existence of unusual singularities. First, we investigate analytic structures of the QCD propagators using the massive Yang-Mills model, in which the one-loop gluon and ghost propagators are in good agreement with the numerical lattice results in the Landau gauge. We find that both gluon and quark propagators in this model have complex poles that invalidate the usual spectral representation. Second, we discuss general properties of propagators in the presence of such …
We apply the Local Composite Operator method to construct the three loop effective potential for the dimension two operator $\frac{1}{2} { A_μ^a }^2$ in the Landau gauge in Quantum Chromodynamics. For $SU(3)$ we show that the three loop value of the effective mass of the gluon is similar to the two loop estimates when the number of massless quarks is strictly less than five for $SU(3)$.
We apply the Bogoliubov compensation principle to QCD. The non-trivial solution of compensation equations for a spontaneous generation of the anomalous three-gluon interaction leads to the determination of parameters of the theory, including behavior of the gauge coupling $α_s(Q^2)$ without the Landau singularity, the gluon condensate $V_2\,\simeq\,0.01\,GeV^4$, mass of the lightest glueball $M_G\,\simeq\,1500\,MeV$ in satisfactory agreement with the phenomenological knowledge. The results strongly support the applicability of N.N. Bogoliubov compensation approach to gauge theories of the Stan…
In this note, we consider the Landau gauge in the continuum formulation. Our purposes are twofold. Firstly, we try to work out the consequences of the recently derived Cucchieri-Mendes bounds on the inverse Faddeev-Popov operator at the level of the path integral quantization. Secondly, we give an explicit renormalizable prescription to implement the so-called Landau B-gauges as introduced by Maas.
We analyse the complete algebraic structure of the background field method for Yang--Mills theory in the Landau gauge and show several structural simplifications within this approach. In particular we present a new way to study the IR behavior of Green functions in the Landau gauge and show that there exists a unique Green function whose IR behaviour controls the IR properties of the gluon and the ghost propagators.
Expanding the Landau gauge gluon and ghost two-point functions in a power series we investigate their infrared behavior. The corresponding powers are constrained through the ghost Dyson-Schwinger equation by exploiting multiplicative renormalizability. Without recourse to any specific truncation we demonstrate that the infrared powers of the gluon and ghost propagators are uniquely related to each other. Constraints for these powers are derived, and the resulting infrared enhancement of the ghost propagator signals that the Kugo-Ojima confinement criterion is fulfilled in Landau gauge Yang-Mil…
We construct the effective potential for the dimension two composite operator 1/2 A^{a 2}_μin QCD with massless quarks in the Landau gauge for an arbitrary colour group at two loops. For SU(3) we show that an estimate for the effective gluon mass decreases as N_f increases.
In Gribov-Zwanziger scenario of color confinement, Zwanziger proposed two types of horizon function in lattice Landau gauge, the first one and the second one of which foundation was discussed in detail. The second type horizon function is focussed on in the present study. Its derivation and the horizon condition are briefly reviewed along the line of Zwanziger, and it is also reviewed that this horizon condition and Kugo-Ojima color confinement criterion coincide in the continuum limit. In case of free boundary condition in contrast to periodic boundary condition, it was shown that the horizon…
The infrared behaviour of gluon and ghost propagators, ghost-gluon vertex and three-gluon vertex is investigated for both the covariant Landau and the non-covariant Coulomb gauge. Assuming infrared ghost dominance, we find a unique infrared exponent in the d=4 Landau gauge, while in the d=3+1 Coulomb gauge we find two different infrared exponents. We also show that a finite dressing of the ghost-gluon vertex has no influence on the infrared exponents. Finally, we determine the infrared behaviour of the three-gluon vertex analytically and calculate it numerically at the symmetric point in the C…
Some features of the solutions to the truncated Dyson-Schwinger equations(DSEs) for the propagators of QCD in Landau gauge are summarized. In particular, the Kugo-Ojima confinement criterion is realized, and positivity of transverse gluons is manifestly violated in these solutions. In Landau gauge, the gluon-ghost vertex function offers a convenient possibility to define a nonperturbative running coupling. The infrared fixed point obtained from this coupling which determines the 2-point interactions of color-octet quark currents implies the existence of unphysical massless states which are nec…
In view of the expectation that the existence of complex poles is a signal of confinement, we investigate the analytic structure of the gluon, quark, and ghost propagators in the Landau gauge QCD and QCD-like theories by employing an effective model with a gluon mass term of the Yang-Mills theory, which we call the massive Yang-Mills model. In this model, we particularly investigate the number of complex poles in the parameter space of the model consisting of gauge coupling constant, gluon mass, and quark mass for the gauge group $SU(3)$ and various numbers of quark flavors $N_F$ within the as…
It is shown that the commonly accepted relationship between the Landau singularity in the running coupling constant of QED or QCD and the renormalon singularities in the Borel sums of perturbation theory expansions is only a particular feature of the restriction of the perturbative $β$--function to the one loop level.
A solvable systematic truncation scheme for the Dyson-Schwinger equations of Euclidean QCD in Landau gauge is presented. It implements the Slavnov-Taylor identities for the three-gluon and ghost-gluon vertices, whereas irreducible four-gluon couplings as well as the gluon-ghost and ghost-ghost scattering kernels are neglected. The infrared behavior of gluon and ghost propagators is obtained analytically: The gluon propagator vanishes for small spacelike momenta whereas the ghost propagator diverges stronger than a massless particle pole. The numerical solutions are compared with recent lattice…
A detailed discussion of the renormalization properties of a class of gauges which interpolates among the Landau, Coulomb and Maximal Abelian gauges is provided in the framework of the algebraic renormalization in Euclidean Yang-Mills theories in four dimensions.
The non-perturbative BRST quartet mechanism in infrared Landau gauge QCD is presented. It is demonstrated that positivity violation for transverse gluons allows to identify the gluon's non-perturbative BRST quartet. To describe the respective BRST-daughter state a truncated Bethe-Salpeter equation for the gluon-ghost bound state is investigated. An analogous construction for quarks yields a truncated Bethe-Salpeter equation for the quark-ghost bound state. The gluon-ghost bound state equation in two space-time dimensions has been numerically solved.
The infrared behaviour of QCD Green's functions in Landau gauge has been focus of intense study. Different non-perturbative approaches lead to a prediction in line with the conditions for confinement in local quantum field theory as spelled out in the Kugo-Ojima criterion. Detailed comparisons with lattice studies have revealed small but significant differences, however. But aren't we comparing apples with oranges when contrasting lattice Landau gauge simulations with these continuum results? The answer is yes, and we need to change that. We therefore propose a reformulation of Landau gauge on…
We prove that three-dimensional Yang-Mills theories in the Landau gauge supplemented with a infrared regulating, parity preserving mass term are ultraviolet finite to all orders. We also extend this result to the Curci-Ferrari gauge.
The Coulomb gauge in QCD is the only explicitly unitary gauge. But it suffers from energy-divergences which means that it is not rigorously well-defined. One way to define it unambiguously is as the limit of a gauge interpolating between the Landau gauge and the Coulomb gauge. This interpolating gauge is characterised by a parameter theta and the Coulomb gauge is obtained in the limit theta tends to zero. We study the renormalization of this theta-gauge for all values of theta, and note some special features of it.
Using a spin-charge separation of the gluon field in the Landau gauge we show that the SU(2) Yang-Mills theory in the low-temperature phase can be considered as a nematic liquid crystal. The ground state of the nematic crystal is characterized by the A^2 condensate of the gluon field. The liquid crystal possesses various topological defects (instantons, monopoles and vortices) which are suggested to play a role in non-perturbative features of the theory.
We propose a generalisation of the Faddeev-Popov trick for Yang-Mills fields in the Landau gauge. The gauge-fixing is achieved as a genuine change of variables. In particular the Jacobian that appears is the modulus of the standard Faddeev-Popov determinant. We give a path integral representation of this in terms of auxiliary bosonic and Grassman fields extended beyond the usual set for standard Landau gauge BRST. The gauge-fixing Lagrangian density appearing in this context is local and enjoys a new extended BRST and anti-BRST symmetry though the gauge-fixing Lagrangian density in this case i…
The local composite operator A^2 is analysed in pure Yang-Mills theory in the Landau gauge within the algebraic renormalization. It is proven that the anomalous dimension of A^2 is not an independent parameter, being expressed as a linear combination of the gauge beta function and of the anomalous dimension of the gauge fields.
The Faddeev-Popov determinant of Landau gauge in d dimensions and Coulomb gauge in d+1 dimensions is calculated in the heat-kernel expansion up to next-to-leading order. The UV-divergent parts in d=3,4 are isolated and the counterterms required for a non-perturbative treatment of the Faddeev-Popov determinant are determined.
The existence of a SL(2,R) symmetry is discussed in SU(N) Yang-Mills in the maximal Abelian Gauge. This symmetry, also present in the Landau and Curci-Ferrari gauge, ensures the absence of tachyons in the maximal Abelian gauge. In all these gauges, SL(2,R) turns out to be dynamically broken by ghost condensates.
We uniquely determine the infrared asymptotics of Green functions in Landau gauge Yang-Mills theory. They have to satisfy both, Dyson-Schwinger equations and functional renormalisation group equations. Then, consistency fixes the relation between the infrared power laws of these Green functions. We discuss consequences for the interpretation of recent results from lattice QCD.
A gauge-invariant reformulation of QCD in three spacetime dimensions is presented within a Hamiltonian formalism, extending previous work to include fermion fields in the adjoint and fundamental representations. A priori there are several ways to define the gauge-invariant versions of the fermions; a consistent prescription for choosing the fermionic variables is presented. The fermionic contribution to the volume element of the gauge orbit space and the gluonic mass-gap is computed exactly and this contribution is shown to be closely related to the mechanism for induction of Chern-Simons term…
We construct a $W_{\infty}$ gauge field theory of electrons in the lowest Landau level. For this purpose we introduce an external gauge potential $\cal A $ such that its $W_{\infty}$ gauge transformations cancel against the gauge transformation of the electron field. We then show that the electromagnetic interactions of electrons in the lowest Landau level are obtained through a non-linear realization of $\cal A$ in terms of the $U(1)$ gauge potential $A^{\m}$. As applications we derive the effective Lagrangians for circular droplets and for the $\n =1$ quantum Hall system.
We consider a model which effectively restricts the functional integral of Yang--Mills theories to the fundamental modular region. Using algebraic arguments, we prove that this theory has the same divergences as ordinary Yang Mills theory in the Landau gauge and that it is unitary. The restriction of the functional integral is interpreted as a kind of spontaneous breakdown of the $BRS$ symmetry.
Gribov and Zwanziger proposed a modification of Yang-Mills theory in order to cure the Gribov copy problem. We employ field-dependent BRST transformations to generalize the Gribov-Zwanziger model from the Landau gauge to general R_xi gauges. The Gribov horizon functional is presented in explicit form, in both the non-local and local variants. Finally, we show how to reach any given gauge from the Landau one.
The infrared behavior of the gluon and ghost propagators is analyzed in Yang-Mills theories in the presence of dynamical mass generation in the Landau gauge. By restricting the domain of integration in the path-integral to the Gribov region $Ω$, the gauge propagator is found to be suppressed in the infrared, while the ghost propagator is enhanced.
The Landau-Khalatnikov-Fradkin transformations (LKFTs) represent an important tool for probing the gauge dependence of the correlation functions within the class of linear covariant gauges. Recently these transformations have been derived from first principles in the context of non-Abelian gauge theory (QCD) introducing a gauge invariant transverse gauge field expressible as an infinite power series in a Stueckelberg field. In this work we explicitly calculate the transformation for the gluon propagator, reproducing its dependence on the gauge parameter at the one loop level and elucidating th…
Non-perturbative formalism of the generalized effective action is used for deriving the Schwinger-Dyson equations. In order to clear the domain of integration in the functional integral from gauge copies a restriction to the Gribov horizon due to Zwanziger is implemented. In this approach an asymptotic behaviour of gluon propagator and propagator of the Faddeev-Popov ghosts at small momenta is studied. Such a behaviour is obtained as a result of solving coupled Schwinger-Dyson equations in zeroth and first-order approximation. The qualitative agreement of these results with the ones obtained b…
In Dyson-Schwinger studies of the Yang-Mills propagators the four-gluon interaction has been usually neglected due to the related technical difficulties with the associated two-loop terms and especially their renormalization. A possible scenario to renormalize these fully-dressed two-loop terms is presented. Preliminary results for the Landau gauge gluon propagator are shown. Implications for the gluon propagators in maximally Abelian gauge are discussed.
Ghost condensates of dimension two are analysed in pure SU(N) Yang-Mills theories by combining the local composite operators technique with the algebraic BRST renormalization.
We construct a vector gauge invariant transverse field configuration $V^H$, consisting of the well-known superfield $V$ and of a Stueckelberg-like chiral superfield. The renormalizability of the Super Yang Mills action in the Landau gauge is analyzed in the presence of a gauge invariant mass term $m^2 \int dV \mathcal{M}(V^H)$, with $\mathcal{M}(V^H)$ a power series in $V^H$. Unlike the original Stueckelberg action, the resulting action turns out to be renormalizable to all orders.
In a previous work, we presented a new method to account for the Gribov ambiguities in non-Abelian gauge theories. The method consists on the introduction of an extra constraint which directly eliminates the infinitesimal Gribov copies without the usual geometric approach. Such strategy allows to treat gauges with non-hermitian Faddeev-Popov operator. In this work, we apply this method to a gauge which interpolates among the Landau and maximal Abelian gauges. The result is a local and power counting renormalizable action, free of infinitesimal Gribov copies. Moreover, the interpolating tree-le…
Massive renormalizable Yang-Mills theories in three dimensions are analysed within the algebraic renormalization in the Landau gauge. In analogy with the four dimensional case, the renormalization of the mass operator A^2 turns out to be expressed in terms of the fields and coupling constant renormalization factors. We verify the relation we obtain for the operator anomalous dimension by explicit calculations in the large N_f. The generalization to other gauges such as the nonlinear Curci-Ferrari gauge is briefly outlined.
