Euclidean Yang-Mills Theories Research Articles

A State Space for 3D Euclidean Yang–Mills Theories

It is believed that Euclidean Yang–Mills theories behave like the massless Gaussian free field (GFF) at short distances. This makes it impossible to define the main observables for these theories—the Wilson loop observables—in dimensions greater than two, because line integrals of the GFF do not exist in such dimensions. Taking forward a proposal of Charalambous and Gross, this article shows that it is possible to define Euclidean Yang–Mills theories on the 3D unit torus as ‘random distributional gauge orbits’, provided that they indeed behave like the GFF in a certain sense. One of the main technical tools is the existence of the Yang–Mills heat flow on the 3D torus starting from GFF-like initial data, which is established in a companion paper. A key consequence of this construction is that under the GFF assumption, one can define a notion of ‘regularized Wilson loop observables’ for Euclidean Yang–Mills theories on the 3D unit torus.

Refined Gribov-Zwanziger theory coupled to scalar fields in the Landau gauge

The Refined Gribov-Zwanziger (RGZ) action in the Landau gauge accounts for the existence of infinitesimal Gribov copies as well as the dynamical formation of condensates in the infrared of Euclidean Yang-Mills theories. We couple scalar fields to the RGZ action and compute the one-loop scalar propagator in the adjoint representation of the gauge group. We compare our findings with existing lattice data. The fate of BRST symmetry in this model is discussed, and we provide a comparison to a previous proposal for a non-minimal coupling between matter and the RGZ action. We find good agreement with the lattice data of the scalar propagator for the values of the mass parameters that fit the RGZ gluon propagator to the lattice. This suggests that the non-perturbative information carried by the gluon propagator in the RGZ framework provides a suitable mechanism to reproduce the behavior of correlation functions of colored matter fields in the infrared.

Toward background field independence within the Gribov horizon

We introduce a background gauge akin to the Landau-DeWitt gauge but deformed by the presence of a gauge parameter for the quantization of Euclidean Yang-Mills theories. In the limit where the background field vanishes, standard linear covariant gauges are recovered. This gauge allows for an explicit investigation of the effects of infinitesimal Gribov copies and their impact to background and gauge parameter dependence of physical correlators. Similarly to linear covariant gauges, the introduction of gauge-invariant dressed fields is essential to restore the Becchi-Rouet-Stora-Tyutin (BRST) symmetry. Hence, we construct a BRST symmetric action in linear covariant background gauges which eliminates regular infinitesimal Gribov copies in analogy to the recently introduced BRST invariant (refined) Gribov-Zwanziger action. The issue of background dependence and its relation to gauge parameter dependence is discussed in the light of nonperturbative effects driven by the elimination of Gribov copies.

The universal character of Zwanziger's horizon function in Euclidean Yang–Mills theories

In light of the recently established BRST invariant formulation of the Gribov–Zwanziger theory, we show that Zwanziger's horizon function displays a universal character. More precisely, the correlation functions of local BRST invariant operators evaluated with the Yang–Mills action supplemented with a BRST invariant version of the Zwanziger's horizon function and quantized in an arbitrary class of covariant, color invariant and renormalizable gauges which reduce to the Landau gauge when all gauge parameters are set to zero, have a unique, gauge parameters independent result, corresponding to that of the Landau gauge when the restriction to the Gribov region Ω in the latter gauge is imposed. As such, thanks to the BRST invariance, the cut-off at the Gribov region Ω acquires a gauge independent meaning in the class of the physical correlators.

Accessing the topological susceptibility via the Gribov horizon

The topological susceptibility, $\chi^4$, following the work of Witten and Veneziano, plays a key role in identifying the relative magnitude of the $\eta^{\prime}$ mass, the so-called $U(1)_{A}$ problem. A nonzero $\chi^4$ is caused by the Veneziano ghost, the occurrence of an unphysical massless pole in the correlation function of the topological current. In a recent paper (Phys.Rev.Lett.114 (2015) 24, 242001), an explicit relationship between this Veneziano ghost and color confinement was proposed, by connecting the dynamics of the Veneziano ghost, and thus the topological susceptibility, with Gribov copies. However, the analysis is incompatible with BRST symmetry (Phys.Rev.D 93 (2016) no.8, 085010). In this paper, we investigate the topological susceptibility, $\chi^4$, in SU(3) and SU(2) Euclidean Yang-Mills theory using an appropriate Pad\'e approximation tool and a non-perturbative gluon propagator, within a BRST invariant framework and by taking into account Gribov copies in a general linear covariant gauge.

Renormalizability of the refined Gribov-Zwanziger action in linear covariant gauges

The refined Gribov-Zwanziger framework takes into account the existence of equivalent gauge field configurations in the gauge-fixing quantization procedure of Euclidean Yang-Mills theories. Recently, this setup was extended to the family of linear covariant gauges giving rise to a local and Becchi-Rouet-Stora-Tyutin (BRST)-invariant action. In this paper, we give an algebraic proof of the renormalizability of the resulting action to all orders in perturbation theory.

The Makeenko–Migdal Equation for Yang–Mills Theory on Compact Surfaces

We prove the Makeenko-Migdal equation for two-dimensional Euclidean Yang-Mills theory on an arbitrary compact surface, possibly with boundary. In particular, we show that two of the proofs given by the first, third, and fourth authors for the plane case extend essentially without change to compact surfaces.

Nonperturbative aspects of Euclidean Yang-Mills theories in linear covariant gauges: Nielsen identities and a BRST-invariant two-point correlation function

In order to construct a gauge invariant two-point function in a Yang-Mills theory, we propose the use of the all-order gauge invariant transverse configurations A^h. Such configurations can be obtained through the minimization of the functional A^2_{min} along the gauge orbit within the BRST invariant formulation of the Gribov-Zwanziger framework recently put forward in [1,2] for the class of the linear covariant gauges. This correlator turns out to provide a characterization of non-perturbative aspects of the theory in a BRST invariant and gauge parameter independent way. In particular, it turns out that the poles of <A^h A^h> are the same as those of the transverse part of the gluon propagator, which are also formally shown to be independent of the gauge parameter entering the gauge condition through the Nielsen identities. The latter follow from the new exact BRST invariant formulation introduced before. Moreover, the correlator <A^h A^h> enables us to attach a BRST invariant meaning to the possible positivity violation of the corresponding temporal Schwinger correlator, giving thus for the first time a consistent, gauge parameter independent, setup to adopt the positivity violation of <A^h A^h> as a signature for gluon confinement. Finally, in the context of gauge theories supplemented with a fundamental Higgs field, we use <A^h A^h> to probe the pole structure of the massive gauge boson in a gauge invariant fashion.

All-order bounds for correlation functions of gauge-invariant operators in Yang-Mills theory

We give a complete, self-contained, and mathematically rigorous proof that Euclidean Yang-Mills theories are perturbatively renormalisable, in the sense that all correlation functions of arbitrary composite local operators fulfil suitable Ward identities. Our proof treats rigorously both all ultraviolet and infrared problems of the theory and provides, in the end, detailed analytical bounds on the correlation functions of an arbitrary number of composite local operators. These bounds are formulated in terms of certain weighted spanning trees extending between the insertion points of these operators. Our proofs are obtained within the framework of the Wilson-Wegner-Polchinski-Wetterich renormalisation group flow equations, combined with estimation techniques based on tree structures. Compared with previous mathematical treatments of massless theories without local gauge invariance [R. Guida and Ch. Kopper, arXiv:1103.5692; J. Holland, S. Hollands, and Ch. Kopper, Commun. Math. Phys. 342 (2016) 385] our constructions require several technical advances; in particular, we need to fully control the BRST invariance of our correlation functions.

Non-perturbative BRST quantization of Euclidean Yang–Mills theories in Curci–Ferrari gauges

In this paper we address the issue of the non-perturbative quantization of Euclidean Yang-Mills theories in the Curci-Ferrari gauge. In particular, we construct a Refined Gribov-Zwanziger action for this gauge which takes into account the presence of gauge copies as well as the dynamical formation of dimension two condensates. This action enjoys a non-perturbative BRST symmetry recently proposed in \cite{Capri:2015ixa}. Finally, we give attention to the gluon propagator in different space-time dimensions.

Large-Nground state of the Lieb-Liniger model and Yang-Mills theory on a two-sphere

We derive the large particle number limit of the Bethe equations for the ground state of the attractive one-dimensional Bose gas (Lieb-Liniger model) on a ring and solve it for arbitrary coupling. We show that the ground state of this system can be mapped to the large-N saddle point of Euclidean Yang-Mills theory on a two-sphere with a U(N) gauge group, and the phase transition that interpolates between the homogeneous and solitonic regime is dual to the Douglas-Kazakov confimenent-deconfinement phase transition.

Gribov horizon and non-perturbative BRST symmetry in the maximal Abelian gauge

The non-perturbative nilpotent exact BRST symmetry of the Gribov–Zwanziger action in the Landau gauge constructed in [1] is generalized to the case of Euclidean Yang–Mills theories quantized in the maximal Abelian gauge. The resulting diagonal gluon propagator is evaluating in dimensions D=4,3,2. In D=4,3 a decoupling type behavior is found in the infrared region, while in D=2 a scaling type behavior emerges.

Dimension two condensates in the Gribov-Zwanziger theory in Coulomb gauge

We investigate the dimension two condensate $< {\bar \phi}^{ab}_i \phi^{ab}_i -{\bar \omega}^{ab}_i \omega^{ab}_i >$ within the Gribov-Zwanziger approach to Euclidean Yang-Mills theories in the Coulomb gauge, in both 3 and 4 dimensions. An explicit calculation shows that, at the first order, the condensate $< {\bar \phi}^{ab}_i \phi^{ab}_i -{\bar \omega}^{ab}_i \omega^{ab}_i >$ is plagued by a non-integrable IR divergence in 3D, while in 4D it exhibits a logarithmic UV divergence, being proportional to the Gribov parameter $\gamma^2$. These results indicate that in 3D the transverse spatial Coulomb gluon two-point correlation function exhibits a scaling behaviour, in agreement with Gribov's expression. In 4D, however, they suggest that, next to the scaling behaviour, a decoupling solution might emerge too.

Study of the all orders multiplicative renormalizability of a local confining quark action in the Landau gauge

The inverse of the Faddeev–Popov operator plays a pivotal role within the Gribov–Zwanziger approach to the quantization of Euclidean Yang–Mills theories in Landau gauge. Following a recent proposal (Capri et al., 2014), we show that the inverse of the Faddeev–Popov operator can be consistently coupled to quark fields. Such a coupling gives rise to a local action while reproducing the behaviour of the quark propagator observed in lattice numerical simulations in the non-perturbative infrared region. By using the algebraic renormalization framework, we prove that the aforementioned local action is multiplicatively renormalizable to all orders.

Two-dimensional Yang-Mills theory on recursive infinite genus surfaces

The partition function of Euclidean Yang-Mills theory on two dimensional surfaces is given by the Migdal formula. It involves the area and topological characteristics of the surface. We consider this theory on a class of infinite genus surfaces that are constructed recursively. We make use of this recursive structure to compute the partition functions (with or without additional Wilson loops) on such surfaces. Our method also works for the quantum deformed Yang-Mills theory.

Gribov-Zwanziger action in SU(2) maximally Abelian gauge withU(1)3Landau gauge

We construct the local Gribov-Zwanziger action for SU(2) Euclidean Yang-Mills theories in the maximally Abelian (MA) gauge with $\mathrm{U}(1{)}_{3}$ Landau gauge fixing based on Zwanziger's work in the Landau gauge. By restricting 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 like the propagator from the Gribov-Zwanziger action in the Landau gauge, and shows the violation of the Kallen-Lehmann representation. From our result, the validity of the Gribov-Zwanziger scenario in the MA gauge can be studied by comparing with the lattice result in two dimensions, which is qualitatively consistent with the result of the original Gribov-Zwanziger action in the Landau gauge.

Supersymmetric quiver gauge theories on the lattice

Abstract In this paper we detail the lattice constructions of several classes of supersymmetric quiver gauge theories in two and three Euclidean spacetime dimensions possessing exact supersymmetry at finite lattice spacing. Such constructions are obtained through the methods of topological twisting and geometric discretization of Euclidean Yang-Mills theories with eight and sixteen supercharges in two and three dimensions. We detail the lattice constructions of two-dimensional quiver gauge theories possessing four and eight supercharges and three-dimensional quiver gauge theories possessing eight supercharges.

Sphaleron solutions of the Skyrme model from Yang–Mills holonomy

We discuss how an approximation to the axially symmetric sphalerons in the Skyrme model can be constructed from the holonomy of a non-BPS Yang–Mills calorons. These configurations, both in the Skyrme model and in the Euclidean Yang–Mills theory, are characterized by two integers n and m, where ±n are the winding numbers of the constituents and the second integer m defines type of the solution, it has zero topological charge for even m and for odd values of m the corresponding chain has topological charge n. It is found numerically that the holonomy of the chains of interpolating calorons–anticalorons provides a reasonably good approximation to the corresponding Skyrmion–antiSkyrmion chains when the topological charge of the Skyrmion constituents is two times more than the Chern–Pontryagin index of the caloron.

Chromo-field flux sheets as confining gauge field configurations in the SU(N) Euclidean Yang-Mills theory in the Landau gauge

For the four-dimensional SU(N) Euclidean Yang-Mills theory in the Landau gauge, we present two sets of gauge field configurations that satisfy the Euclidean equations of motion. These configurations generate four-dimensional chromo-field flux sheets whose spatial cross sections are three-dimensional chromo-field flux tubes. In lattice simulations, they may be detected as center vortices. The first set of gauge field configurations generates chromo-electric flux tubes that should contribute to a chromo-electric flux tube between two static color charges. The string tension for two static color charges in representation r then naturally satisfies the Casimir scaling. Applying a gauge transformation to this set of gauge field configurations, we can transform them into those in the maximal Abelian gauge. These transformed configurations generate chromo-electric flux tubes that should contribute to those observed between two static quarks in lattice simulations performed in the maximal Abelian gauge. The second set of gauge field configurations generates chromo-magnetic flux tubes. When rotated in a plane that includes the temporal-axis and is perpendicular to the flux tube axis, the rotated gauge field configuration generates a chromo-electric flux tube and should contribute to the chromo-electric flux tubes observed in lattice simulations in the Landau gauge. We also argue that when regulated on a lattice, any of the flux sheet gauge field configuration with a finite flux sheet thickness is located on the Gribov horizon in the infinite lattice volume limit. We thus suggest that these sets of gauge field configurations contribute significantly to the low energy properties of QCD, particularly the quark confinement.

A study of the zero modes of the Faddeev–Popov operator in Euclidean Yang–Mills theories in the Landau gauge in d=2,3,4 dimensions

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.