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# Ghost and gluon dressing functions (left) and gluon propagator (right) from lattice [61] and DSE calculations [2] in the Landau gauge.

Source publication

We calculate gluon and ghost propagators in Yang-Mills theory in linear covariant gauges. To that end, we utilize Nielsen identities with Landau gauge propagators and vertices as the starting point. We present and discuss numerical results for the gluon and ghost propagators for values of the gauge parameter 0 < ξ ≤ 5. Extrapolating the propagators...

## Contexts in source publication

**Context 1**

... of the solutions is the one that agrees best with lattice results and has only a shallow, hardly visible maximum in the gluon propagator. As a second choice, we take a solution with a more pronounced maximum; see Fig. 2. It remains to specify the models used for the vertices Γ A Ã Ac and Γ c Ã cc . When using bare vertices, we found that in the infrared (IR) individual loop diagrams can qualitatively modify the IR solution for the gluon propagator. This is either resolved by cancellations between individual diagrams or by the IR behavior of the ...

**Context 2**

... all plots in this section, we used up to here the solution that is closest to lattice results. It is characterized by a very flat maximum of the gluon propagator and a ghost dressing function that is relatively small in the deep infrared. The different Landau gauge solution used next as a starting point for the NIs is shown in Fig. 2 in comparison to the previously used solution. The second solution has a pronounced maximum in the gluon propagator and shows a clear increase of the ghost dressing function at low momenta. The results for a selection of values of ξ are shown in Fig. 9. The typical features of the Landau gauge solution type are inherited by the ξ > 0 ...

## Citations

... (80) and (81) numerically. To do so, we use the resumed anomalous dimension of the gauge field in Eq. (82). In the left-hand side panel of Fig. 7, the renormalization group flow of the gauge coupling is presented. ...

... To study confinement phenomena, we need improvements of the approximation. This can be systematically done within the functional renormalization group with the vertex expansion scheme [71][72][73][74][75][76][77][78][79][80][80][81][82][83][84][85][86][87][88][89][90][91][92][93][94][95][96]. We can take into account vertex corrections of the gauge fields. ...

The SU(N) S U ( N ) Yang-Mills theory in \mathbb R^4\times S^1 ℝ 4 × S 1 spacetime is studied as a simple toy model of Gauge-Higgs unification. The theory is perturbatively nonrenormalizable but could be formulated as an asymptotically safe theory, namely a nonperturbatively renormalizable theory. We study the fixed point structure of the Yang-Mills theory in \mathbb R^4\times S^1 ℝ 4 × S 1 by using the functional renormalization group in the background field approximation. We derive the functional flow equations for the gauge coupling and the background gauge-field potential. There exists a nontrivial fixed point for both couplings at finite compactification radii. At the fixed point, gauge coupling and vacuum energy are both relevant. The renormalization group flow of the gauge coupling describes the smooth transition between the ultraviolet asymptotically safe regime and the strong interacting infrared limit.

... Although for QED substantial progress has been achieved, see, e.g., [34][35][36] and references therein, in the studies of the role of the fermion-photon vertex for gauge independence, the corresponding question in QCD, namely, on the impact of the quark-gluon vertex on the gauge (in-)dependence of hadron observables, has proven to be an extremely hard question. Even the much more humble question of how the different tensors of the quark-gluon vertex may depend on the gauge parameter within the class of linear covariant gauges and how this will effect the underlying mechanism for DχSB in this class of gauges seems beyond reach given the status of Dyson-Schwinger studies of the Yang-Mills sector in the linear covariant gauge, see, e.g., [37][38][39]. ...

... Therefore, although the question of whether DχSB in QCD is delicate and intricate only in the Landau gauge and might be a robust phenomenon in other gauges is highly interesting, it will likely remain unanswered in the next years. Nevertheless, in view of the insights which may be gained in studying the role of the quark-gluon vertex and its impact on DχSB in different gauges, an extension of the approach based on Nielsen identities (as performed in [39]) to the quark sector is certainly desirable. One might also apply the technique of interpolating gauges [40][41][42][43] to relate the existing Landau and Coulomb gauge results on DχSB. ...

Dynamical chiral symmetry breaking (DχSB) in quantum chromo dynamics (QCD) for light quarks is an indispensable concept for understanding hadron physics, i.e., the spectrum and the structure of hadrons. In functional approaches to QCD, the respective role of the quark propagator has been evident since the seminal work of Nambu and Jona-Lasinio has been recast in terms of QCD. It not only highlights one of the most important aspects of DχSB, the dynamical generation of constituent quark masses, but also makes plausible that DχSB is a robustly occurring phenomenon in QCD. The latter impression, however, changes when higher n-point functions are taken into account. In particular, the quark–gluon vertex, i.e., the most elementary n-point function describing the full, non-perturbative quark–gluon interaction, plays a dichotomous role: It is subject to DχSB as signalled by its scalar and tensor components but it is also a driver of DχSB due to the infrared enhancement of most of its components. Herein, the relevant self-consistent mechanism is elucidated. It is pointed out that recently obtained results imply that, at least in the covariant gauge, DχSB in QCD is located close to the critical point and is thus a delicate effect. In addition, requiring a precise determination of QCD’s three-point functions, DχSB is established, in particular in view of earlier studies, by an intricate interplay of the self-consistently determined magnitude and momentum dependence of various tensorial components of the gluon–gluon and the quark–gluon interactions.

... is characterized by a nonperturbative transverse dressing function, Δ ξ ðμ 2 Þ ¼ 1=μ 2 , and was studied with different lattice and functional approaches, e.g., Refs. [29,31,32]. The most general Poincaré-covariant form of the solutions to Eq. (1) is written in terms of covariant scalar and vector amplitudes: ...

We study the gauge dependence of the quark propagator in quantum chromodynamics by solving the gap equation with a nonperturbative quark-gluon vertex which is constrained by longitudinal and transverse Slavnov-Taylor identities, the discrete charge conjugation and parity symmetries and which is free of kinematic singularities in the limit of equal incoming and outgoing quark momenta. We employ gluon propagators in renormalizable Rξ gauges obtained in lattice QCD studies. We report the dependence of the nonperturbative quark propagator on the gauge parameter, in particular we observe an increase, proportional to the gauge-fixing parameter, of the mass function in the infrared domain, whereas the wave renormalization decreases within the range 0≤ξ≤1 considered here. The chiral quark condensate reveals a mild gauge dependence in the region of ξ investigated. We comment on how to build and improve upon this exploratory study in future in conjunction with generalized gauge covariance relations for QCD.

... Typically, Green's QCD function is defined within the quantization scheme obtained by implementing the linear covariant (R ξ ) gauges [182]. The corresponding SDEs are derived and solved within this same quantization scheme, particularly in the Landau gauge (ξ = 0), where lattice simulations are almost exclusively performed; for studies away from the Landau gauge, see e.g., [55,58,66,74,75,110,114,120,[183][184][185][186][187][188][189][190][191]. A great deal may be learned, however, by considering Green's functions and corresponding SDEs formulated within the "PT-BFM" scheme [109,192], namely the framework that arises from the fusion of the pinch technique (PT) [14,96,100,[193][194][195] with the background field method (BFM) [196][197][198][199][200][201][202][203][204][205][206]. ...

The dynamics of the QCD gauge sector give rise to non-perturbative phenomena that are crucial for the internal consistency of the theory; most notably, they account for the generation of a gluon mass through the action of the Schwinger mechanism, the taming of the Landau pole, the ensuing stabilization of the gauge coupling, and the infrared suppression of the three-gluon vertex. In the present work, we review some key advances in the ongoing investigation of this sector within the framework of the continuum Schwinger function methods, supplemented by results obtained from lattice simulations.

... Typically, the Green's function of QCD are defined within the quantization scheme obtained by implementing the linear covariant (R ξ ) gauges [183]. The corresponding SDEs are derived and solved within this same quantization scheme, and in particular in the Landau gauge (ξ = 0), where lattice simulations are almost exclusively performed; for studies away from the Landau gauge, see e.g., [55,61,67,75,76,111,115,121,[184][185][186][187][188][189][190][191][192]. A great deal may be learned, however, by considering the Green's functions and corresponding SDEs formulated within the "PT-BFM"scheme [110,193], namely the framework that arises from the fusion of the pinch technique (PT) [14,97,101,[194][195][196] with the background field method (BFM) [197][198][199][200][201][202][203][204][205][206][207]. ...

The dynamics of the gauge sector of QCD give rise to nonperturbative phenomena that are crucial for the internal consistency of the theory; most notably, they account for the generation of a gluon mass through the action of the Schwinger mechanism, the taming of the Landau pole and the ensuing stabilization of the gauge coupling, and the infrared suppression of the three-gluon vertex. In the present work, we review some key advances in the ongoing investigation of this sector within the framework of the continuum Schwinger function methods, supplemented by results obtained from lattice simulations.

... The same holds for the unstable Nielsen-Olesen mode in constant magnetic fields affecting the effective action. Moreover, our formalism straightforwardly accommodates nonperturbative information about propagators in the Landau gauge in the form of the so-called decoupling solution featuring a massive gluon and a massless ghost [21,[73][74][75][76][77][78][79][80][81][82][83][84][85][86][87]. Using this nonperturbative input together with a self-dual background, a simple one-loop computation already provides evidence for the formation of a gluon condensate for sufficiently large coupling in agreement with results from nonperturbative functional RG studies [48,88]. ...

... Though this removes the instability encountered in the preceding subsection, the treatment of the zero mode still requires some care. As a second ingredient, we choose the ghost mass term to vanishm gh ¼ 0 while we keep a finite gluon mass m > 0. In fact, this mimics the so-called decoupling solution known from the nonperturbative study of gluon and ghost propagators [21,[73][74][75][76][77][78][80][81][82][83][84][85][86][87]. A parametrization of these nonperturbative propagators in terms of massive gluons but massless ghosts works surprisingly successfully in phenomenological applications [8][9][10][11][12][13][14]. ...

We combine a recent construction of a Becchi-Rouet-Stora-Tyutin (BRST)-invariant, nonlinear massive gauge fixing with the background field formalism. The resulting generating functional preserves background-field invariance as well as BRST invariance of the quantum field manifestly. The construction features BRST-invariant mass parameters for the quantum gauge and ghost fields. The formalism employs a background Nakanishi-Lautrup field which is part of the nonlinear gauge-fixing sector and thus should not affect observables. We verify this expectation by computing the one-loop effective action and the beta function of the gauge coupling as an example. The corresponding Schwinger functional generating connected correlation functions acquires additional one-particle reducible terms that vanish on shell. We also study off-shell one-loop contributions in order to explore the consequences of a nonlinear gauge fixing scheme involving a background Nakanishi-Lautrup field. As an application, we show that our formalism straightforwardly accommodates nonperturbative information about propagators in the Landau gauge in the form of the so-called decoupling solution. Using this nonperturbative input, we find evidence for the formation of a gluon condensate for sufficiently large coupling, whose scale is set by the BRST-invariant gluon mass parameter.

... Actually, among other choices, it is the Landau gauge that was extremely popular and preferred in SDE studies previously. However, as is very well known (see for instance [71]) changes within varied gauge-fixing parameters are shown to be dramatic in the sector of unphysical gluons and ghost propagators. Hence, to understand confinement in QCD properly, one should be able to build gauge (fixing) invariant quantity in framework of SDEs. ...

We determine the gluonic spectral function SU(3) Yang-Mills theory as well as we found fermionic spectral functions in the strong quenched QED where we found a novel solution. Our developed technique provides solutions with the usual branch cut for propagators while not showing any pole structure at the first Riemann sheet (identical with entire complex plane) of the complex square of momentum. Implications and further utilization are briefly addressed for QCD and Standard model calculations.

... The identities are a direct consequence of the Becchi-Rouet-Stora-Tyutin (BRST) symmetry which is displayed by the Faddeev-Popov Lagrangian of QCD and Yang-Mills theories. There is a growing interest in the role of the Nielsen identities for determining the properties of the propagators in a generic covariant gauge [75] and for their explicit numerical evaluation [76]. ...

One-loop explicit expressions are derived for the gluon Nielsen identity in the formalism of the screened massive expansion for Yang-Mills theory. The gauge-parameter-independence of the poles and residues is discussed in a strict perturbative context and, more generally, in extended resummation schemes. No exact formal proof was reached by the approximate resummation schemes, but some evidence is gathered in favor of an exact invariance of the phase, consistently with previous numerical studies.

... Because the decoupling behavior as observed on the lattice extends beyond the Landau gauge to linear covariant gauges [59,60], it will be important to make sure that the dynamical mass generation mechanism applies independently of the gauge-fixing parameter. Moreover, for the dynamically generated mass to carry a physical significance, it should be associated to a BRST-invariant gluon condensate. ...

We consider a BRST-invariant generalization of the “massive background Landau gauge,” resembling the original Curci-Ferrari model that saw a revived interest due to its phenomenological success in modeling infrared Yang-Mills dynamics, including that of the phase transition. Unlike the Curci-Ferrari model, however, the mass parameter is no longer a phenomenological input, but it enters as a result of dimensional transmutation via a BRST-invariant dimension-2 gluon condensate. The associated renormalization constant is dealt with using Zimmermann’s reduction of constants program, which fixes the value of the mass parameter to values close to those obtained within the Curci-Ferrari approach. Using a self-consistent background field, we can include the Polyakov loop and probe the deconfinement transition, including its interplay with the condensate and its electric–magnetic asymmetry. We report a continuous phase transition at Tc≈0.230 GeV in the SU(2) case and a first-order one at Tc≈0.164 GeV in the SU(3) case, values which are again rather close to those obtained within the Curci-Ferrari model at one-loop order.

... Because the decoupling behavior as observed on the lattice extends beyond the Landau gauge to linear covariant gauges [56,57], it will be important to make sure that the dynamical mass generation mechanism applies independently of the gauge-fixing parameter. Moreover, for the dynamically generated mass to carry a physical significance, it should be associated to a BRST invariant gluon condensate. ...

We consider a BRST invariant generalization of the "massive background Landau gauge", resembling the original Curci-Ferrari model that saw a revived interest due to its phenomenological success in modeling infrared Yang-Mills dynamics, including that of the phase transition. Unlike the Curci-Ferrari model, however, the mass parameter is no longer a phenomenological input but it enters as a result of dimensional transmutation via a BRST invariant dimension two gluon condensate. The associated renormalization constant is dealt with using Zimmermann's reduction of constants program which fixes the value of the mass parameter to values close to those obtained within the Curci-Ferrari approach. Using a self-consistent background field, we can include the Polyakov loop and probe the deconfinement transition, including its interplay with the condensate and its electric-magnetic asymmetry. We report a continuous phase transition at Tc ~ 0.230 GeV in the SU(2) case and a first order one at Tc ~ 0.164 GeV in the SU(3) case, values which are again rather close to those obtained within the Curci-Ferrari model at one-loop order.