Curci-Ferrari Model Research Articles

Ghost-gluon vertex in the presence of the Gribov horizon: General kinematics

Correlation functions are important probes for the behavior of quantum field theories. Already at tree-level, the refined Gribov-Zwanziger (RGZ) effective action for Yang-Mills theories provides a good approximation for the gluon propagator, as compared to that calculated by nonperturbative methods such as lattice field theory and Dyson-Schwinger equations. However, the study of higher correlation functions of the RGZ theory is still at its beginning. In this work we evaluate the ghost-gluon vertex function in Landau gauge at one-loop level, in d=4 space-time dimensions for the gauge groups SU(2) and SU(3). More precisely, we extend the analysis conducted in [1] for the soft-gluon limit to an arbitrary kinematic configuration. We introduce renormalization group effects by means of a toy model for the running coupling and investigate the impact of such a model in the ultraviolet tails of our results. We find that RGZ results match fairly closely those from lattice simulations, Schwinger-Dyson equations and the Curci-Ferrari model for three different kinematic configurations. This is compatible with RGZ being a feasible theory for the strong interaction in the infrared regime. Published by the American Physical Society 2024

Four-gluon vertex from the Curci-Ferrari model at one-loop order

We compute the four-gluon vertex from the Curci-Ferrari model at one-loop order for a collinear configuration. Our results display a good agreement with the first lattice data for this vertex, released very recently []. A noteworthy novelty of our work is that we can provide analytical expressions for the four-gluon vertex in collinear configurations, together with a renormalization scheme that allows us to perform reliable perturbative computations even in the infrared regime. We observe an infrared suppression in the form factor associated with the tree-level four-gluon tensor with a possible zero crossing in the deep infrared which demands new lattice investigations to be confirmed. Moreover, we report an infrared divergence in the completely symmetric tensor form factor due to the ghost-loop contributions. These results come as predictions, since previous two-point correlations fix all the available parameters of the model, up to an overall constant factor. Published by the American Physical Society 2024

Dynamically massive linear covariant gauges: Setup and first results

We discuss the possibility to obtain a massive Landau gauge, based on the local composite operator effective action framework combined with the Zimmermann reduction of couplings prescription. As a way to deal with the gauge ambiguity, we check that the ghost propagator remains positive, a necessary condition for gluon field configurations beyond the Gribov region to be negligible. We pay attention to the Becchi-Rouet-Stora-Tyutin invariance of the construction, allowing for a future generalization to a class of massive linear covariant gauges. As a litmus test, we compare our predictions to the lattice data for the two-point functions in Landau gauge introducing the “dynamically infrared-safe” renormalization scheme, including the renormalization group optimization of both the gap equation and the two-point functions. We also discuss the relation to and differences with the Curci-Ferrari model, the usefulness of which in providing an effective perturbative description of nonperturbative Yang-Mills theories became clear during recent years. Published by the American Physical Society 2024

Two-loop three-gluon vertex from the Curci-Ferrari model and its leading infrared behavior to all loop orders

We evaluate the three-gluon vertex with one vanishing external momentum within the Curci-Ferrari (CF) model at two-loop order and compare our results to Landau-gauge lattice simulations of the same vertex function for the SU(2) and SU(3) gauge groups in four dimensions. This extends previous works [J. A. Gracey et al., Phys. Rev. D 100, 034023 (2019); N. Barrios et al., Phys. Rev. D 102, 114016 (2020)] that considered similarly the two-loop ghost and gluon two-point functions as well as the two-loop ghost-antighost-gluon vertex (with vanishing gluon momentum). With the parameters of the model being adjusted by fitting the two-point functions to available lattice data, our evaluation of the three-gluon vertex arises as a pure prediction. We find that two-loop corrections systematically improve the agreement between the model and the lattice data as compared to earlier one-loop calculations, with a better agreement in the SU(3) case as already seen in previous studies. We also analyze the renormalization scheme dependence of our calculation. In all cases, this dependence diminishes when two-loop corrections are included, which is consistent with the perturbative CF paradigm. In addition, we study the low momentum regime of the three-gluon vertex in relation with the possibility of zero crossing. Within the CF model, we show that the leading infrared behavior of the exact vertex is given by the same linear logarithm that arises at one-loop order, multiplied by the all orders cubic ghost dressing function at zero momentum (we provide similar exact results for other vertex functions). We argue that this property remains true within the Faddeev-Popov framework under the assumption that the resummed gluon propagator features a decoupling behavior. This shows that the zero crossing is a property of the exact three-gluon vertex function. Within the CF model, we find however that the scale of the zero crossing is considerably reduced when going from one- to two-loop order. This seems consistent with some recent lattice simulations [G. T. R. Catumba et al., EPJ Web Conf. 258, 02008 (2022)]. Finally, our analysis also allows us to support recent claims about the dominance of the tree-level tensor component [F. Pinto-G\'omez et al., arXiv:2208.01020].

Signatures of the Yang-Mills deconfinement transition from the gluon two-point correlator

We evaluate the longitudinal or (chromo-)electric Yang-Mills gluon propagator in the recently proposed center-symmetric Landau gauge at finite temperature [D. M. van Egmond et al., SciPost Phys. 12, 087 (2022).]. To model the effect of the Gribov copies in the infrared, we use the Curci-Ferrari model which, in turn, allows us to rely on perturbative calculations. At one-loop order in the SU(2) case, the so-obtained longitudinal gluon propagator provides a clear signature for ${Z}_{2}$ center-symmetry breaking with a singular behavior, characteristic of a continuous phase transition. This is in sharp contrast with what is found within the standard Landau gauge. We also identify various signatures for ${Z}_{3}$ center-symmetry breaking in the SU(3) case in the form of genuine order parameters. Among those, we find that the gluon propagator, although degenerate along the diagonal color directions in the confining phase, becomes nondegenerate in the deconfined phase. Our results open new ways of identifying the transition from correlation functions both within continuum approaches and on the lattice.

Polyakov loop, gluon mass, gluon condensate, and its asymmetry near deconfinement

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 ${T}_{c}\ensuremath{\approx}0.230\text{ }\text{ }\mathrm{GeV}$ in the SU(2) case and a first-order one at ${T}_{c}\ensuremath{\approx}0.164\text{ }\text{ }\mathrm{GeV}$ in the SU(3) case, values which are again rather close to those obtained within the Curci-Ferrari model at one-loop order.

One-loop unquenched three-gluon and ghost-gluon vertices in the Curci-Ferrari model

In this article we study the unquenched three-gluon and ghost-gluon vertex in the entire momentum range from the ultraviolet to infrared regime using the Curci-Ferrari model at one loop in Landau gauge, as an extension of the results presented in by Pelaez et al. [Phys. Rev. D 88, 125003 (2013)]. Results are compared with recent lattice data for SU(3) in the unquenched case. This calculation is a pure prediction of the model because it does not require fixing any parameter once two-point functions are fitted. An analysis of the influence of dynamical quarks at the position of the zero crossing of the three-gluon vertex is presented. Due to the recent improvements in infrared lattice data for the quenched three-gluon correlation function [Phys. Lett. B 818, 136352 (2021)], we also redo the comparison between our one-loop results in this limit and the lattice results, obtaining a very good match.

A novel background field approach to the confinement-deconfinement transition

We propose a novel approach to the confinement-deconfinement transition in Yang-Mills theories in the context of gauge-fixed calculations. The method is based on a background-field generalisation of the Landau gauge (to which it reduces at vanishing temperature) with a given, center-symmetric background. This is to be contrasted with most implementations of background field methods in gauge theories, where one uses a variable, self-consistent background. Our proposal is a bona fide gauge fixing that can easily be implemented on the lattice and in continuum approaches. The resulting gauge-fixed action explicitly exhibits the center symmetry of the nonzero temperature theory that controls the confinement-deconfinement transition. We show that, in that gauge, the electric susceptibility diverges at a second order transition [e.g., in the SU(2) theory], so that the gluon propagator is a clear probe of the transition. We implement our proposal in the perturbative Curci-Ferrari model, known for its successful description of various infrared aspects of Yang-Mills theories, including the confinement-deconfinement transition. Our one-loop calculation confirms our general expectation for the susceptibility while providing transition temperatures in excellent agreement with the SU(2) and SU(3) lattice values. Finally, the Polyakov loops above the transition show a more moderate rise, in contrast to previous implementations of the Curci-Ferrari model using a self-consistent background, and our SU(3) result agrees quite well with the lattice data in the range [0,2T_c][0,2Tc].

Two-loop corrections to the QCD propagators within the Curci-Ferrari model

We evaluate all two-point correlation functions of the Curci-Ferrari (CF) model in four dimensions and in the presence of mass-degenerate fundamental quark flavors, as a natural extension of an earlier investigation in the quenched approximation. In principle, the proper account of chiral symmetry breaking ($\ensuremath{\chi}\mathrm{SB}$) and the corresponding dynamical generation of a quark-mass function within the CF model requires one to go beyond perturbation theory [M. Pel\'aez, U. Reinosa, J. Serreau, M. Tissier, and N. Wschebor, Phys. Rev. D 103, 094035 (2021)]. However, it is interesting to assess whether a perturbative description applies to correlation functions that are not directly sensitive to $\ensuremath{\chi}\mathrm{SB}$, such as the gluon, ghost and quark dressing functions. We compare our two-loop results for these form factors to QCD lattice data in the two-flavor case for two different values of the pion mass, one that is relatively far from the chiral limit, and one that is closer to the physical value. Our results confirm that the QCD gluon and ghost dressing functions are well described by a perturbative approach within the CF model, as already observed at one-loop order [M. Pel\'aez, M. Tissier, and N. Wschebor, Phys. Rev. D 90, 065031 (2014)]. Our new main result is that the quark dressing function is also well captured by the perturbative approach, but only starting at two-loop order, as also anticipated [M. Pel\'aez, M. Tissier, and N. Wschebor, Phys. Rev. D 90, 065031 (2014)]. The quark-mass function predicted by the CF model at two-loop order is in good agreement with the data if the quarks are not too light but it shows some clear tension with respect to the two-loop CF dressing functions in the close to physical case, as expected. Interestingly, however, we find that there is much less tension between the nonperturbative quark-mass function, as it can be obtained from lattice simulations or from M. Pel\'aez, U. Reinosa, J. Serreau, M. Tissier, and N. Wschebor, Phys. Rev. D 103, 094035 (2021), and the two-loop CF dressing functions, which confirms the perturbative nature of the latter.

Spontaneous chiral symmetry breaking in the massive Landau gauge: Realistic running coupling

We investigate the spontaneous breaking of chiral symmetry in QCD by means of a recently proposed approximation scheme in the Landau-gauge Curci-Ferrari model, which combines an expansion in the Yang-Mills coupling and in the inverse number of colors, without expanding in the quark-gluon coupling. The expansion allows for a consistent treatment of ultraviolet tails via renormalization group techniques. At leading order, it leads to the resummation of rainbow diagrams for the quark propagator, with, however, a trivial running of both the gluon mass and the quark-gluon coupling. In a previous work, by using a simple model for a more realistic running of these parameters, we could reproduce the known phenomenology of chiral symmetry breaking, including a satisfactory description of the lattice data for the quark mass function. Here, we get rid of this model-dependence by taking our approximation scheme to next-to-leading order. This allows us to consistently include the realistic running of the parameters and to access the unquenched gluon and ghost propagators to first nontrivial order, which we can compare to available lattice data for an even more stringent test of our approach. In particular, our results for the various two-point functions compare well with lattice data while the parameters of the model are strongly constrained.

Effects of a quark chemical potential on the analytic structure of the gluon propagator

We perform complex analyses of the gluon propagator at nonzero quark chemical potential in the long-wavelength limit, using an effective model with a gluon mass term of the Landau-gauge Yang-Mills theory, which is a Landau-gauge limit of the Curci-Ferrari model with quantum corrections being included within the one-loop level. We mainly investigate complex poles of the gluon propagator, which could be relevant to confinement. Around typical values of the model parameters, we show that the gluon propagator has one or two pairs of complex conjugate poles depending on the value of the chemical potential. In addition to a pair similar to that in the case of zero chemical potential, a new pair appears near the real axis when the chemical potential is roughly between the effective quark mass and the effective gluon mass of the model. We discuss possible interpretations of these poles. Additionally, we prove the uniqueness of analytic continuation of the Matsubara propagator to a class of functions that vanish at infinity and are holomorphic except for a finite number of complex poles and singularities on the real axis.

Ghost-antighost-gluon vertex from the Curci-Ferrari model: Two-loop corrections

The Curci-Ferrari model has been shown to provide a good grasp on pure Yang-Mills correlation functions in the Landau gauge, already at one-loop order. In a recent work, the robustness of these results has been tested by evaluating the two-loop corrections to the gluon and ghost propagators. We pursue this systematic investigation by computing the ghost-antighost-gluon vertex to the same accuracy in a particular kinematic configuration that makes the calculations simpler. Because both the parameters of the model and the normalizations of the fields have already been fixed in a previous work, the present calculation represents both a pure prediction and a stringent test of the approach. We find that the two-loop results systematically improve the comparison to Monte-Carlo simulations as compared to earlier one-loop results. The improvement is particularly significative in the SU($3$) case where the predicted ghost-antighost-gluon vertex is in very good agreement with the data. The same comparison in the SU($2$) case is not as good, however. This may be due to the presence of a larger coupling constant in the infrared in that case although we note that a similar mismatch has been quoted in non-perturbative continuum approaches. Despite these features of the SU($2$) case, it is possible to find sets of parameters fitting both the propagators and the ghost-antighost-gluon vertex to a reasonable accuracy.

Gluon propagator in two-color dense QCD: Massive Yang-Mills approach at one loop

We study the Landau gauge gluon propagators in dense two-color QCD at quark chemical potential, $\mu_q$, in the range from 0.5 to 1.0 GeV not reachable by the perturbative method at weak coupling. In order to take into account the non-perturbative effects, at tree level we use a massive Yang-Mills model for the Yang-Mills theory (or the Curci-Ferrari model) which has successfully described the lattice results of the gluon and ghost propagators in the Landau gauge. We couple quarks to this theory and compute the one-loop polarization effects in medium. The presence of the gluon mass significantly tempers the medium effects and uncertainties associated with the strong coupling constant $\alpha_s$. The diquark condensate in two-color QCD is color-singlet, for which neither electric nor magnetic screening masses should appear at the scale less than the diquark gap. The presence of the gap helps to explain the lattice results which are not very sensitive to the quark density. Meanwhile we also found the limitation of the one-loop estimate as well as the lack of some physics in perturbative medium corrections.

Some remarks on the spectral functions of the Abelian Higgs model

We consider the unitary Abelian Higgs model and investigate its spectral functions at one-loop order. This analysis allows to disentangle what is physical and what is not at the level of the elementary particle propagators, in conjunction with the Nielsen identities. We highlight the role of the tadpole graphs and the gauge choices to get sensible results. We also introduce an Abelian Curci-Ferrari action coupled to a scalar field to model a massive photon which, like the non-Abelian Curci-Ferarri model, is left invariant by a modified non-nilpotent BRST symmetry. We clearly illustrate its non-unitary nature directly from the spectral function viewpoint. This provides a functional analogue of the Ojima observation in the canonical formalism: there are ghost states with nonzero norm in the BRST-invariant states of the Curci-Ferrari model.

Two-loop calculation of Yang-Mills propagators in the Curci-Ferrari model

The Landau-gauge gluon and ghost correlation functions obtained in lattice simulations can be reproduced qualitatively and, to a certain extent, quantitatively if a gluon mass is added to the standard Faddeev-Popov action. This has been tested extensively at one loop, for the two and three point correlation functions of the gluons, ghosts and quarks. In this article, we push the comparison to two loops for the gluon and ghost propagators. The agreement between lattice results and the perturbative calculation considerably improves. This validates the Curci-Ferrari action as a good phenomenological model for describing the correlation functions of Yang-Mills theory in the Landau gauge. It also indicates that the perturbation theory converges fairly well, in the infrared regime.

Universal aspects of the phase diagram of QCD with heavy quarks

The flavor dependence of the QCD phase diagram presents universal properties in the heavy quark limit. For the wide class of models where the quarks are treated at the one-loop level, we show, for arbitrary chemical potential, that the flavor dependence of the critical quark masses-for which the confinement-deconfinement transition is second order-is insensitive to the details of the (confining) gluon dynamics and that the critical temperature is constant along the corresponding critical line. We illustrate this with explicit results in various such one-loop models studied in the literature: effective matrix models for the Polyakov loop, the Curci-Ferrari model, and a recently proposed Gribov-Zwanziger-type model. We further observe that the predictions which follow from this one-loop universality property are well satisfied by different calculations beyond one-loop order, including lattice simulations. For degenerate quarks, we propose a simple universal law for the flavor dependence of the critical mass, satisfied by all approaches.

Ghost-gluon vertex in the presence of the Gribov horizon

We consider Yang-Mills theories quantized in the Landau gauge in the presence of the Gribov horizon via the refined Gribov-Zwanziger (RGZ) framework. As the restriction of the gauge path integral to the Gribov region is taken into account, the resulting gauge field propagators display a nontrivial infrared behavior, being very close to the ones observed in lattice gauge field theory simulations. In this work, we explore a higher correlation function in the refined Gribov-Zwanziger theory: the ghost-gluon interaction vertex, at one-loop level. We show explicit compatibility with kinematical constraints, as required by the Ward identities of the theory, and obtain analytical expressions in the limit of vanishing gluon momentum. We find that the RGZ results are nontrivial in the infrared regime, being compatible with lattice Yang-Mills simulations in both SU(2) and SU(3), as well as with solutions from Schwinger-Dyson equations in different truncation schemes, Functional Renormalization Group analysis, and the renormalization group-improved Curci-Ferrari model.

Influence of Gribov ambiguities in a class of nonlinear covariant gauges

We consider Yang-Mills theories in a recently proposed family of nonlinear covariant gauges that consistently deals with the issue of Gribov ambiguities. Such gauges provide a generalization of the Curci-Ferrari-Delbourgo-Jarvis gauges which can be formulated as an extremization procedure and might be implemented in numerical calculations. This would allow for nonperturbative studies of Yang-Mills correlators in a broad class of covariant gauges continuously connected to the well-studied Landau gauge. We compute the ghost and gluon propagators in the continuum formulation at one-loop order in perturbation theory and we study their momentum dependence down to the deep infrared regime, with and without renormalization group improvement. In particular, we show that the theory admits infrared-safe renormalization-group trajectories with no Landau pole. Both the gluon and the ghost behave as massive fields at low energy, and the gluon propagator is transverse even away from the Landau gauge limit. We compare our results to those obtained in the usual Curci-Ferrari model, which allows us to pinpoint the specific effects arising from our treatment of Gribov copies.

Physical unitarity for a massive Yang-Mills theory without the Higgs field: A perturbative treatment

In a series of papers, we examine the physical unitarity in a massive Yang-Mills theory without the Higgs field in which the color gauge symmetry is not spontaneously broken and kept intact. For this purpose, we use a new framework proposed in the previous paper Kondo [arXiv:1208.3521] based on a nonperturbative construction of a non-Abelian field describing a massive spin-one vector boson field, which enables us to perform the perturbative and nonperturbative studies on the physical unitarity. In this paper, we present a new perturbative treatment for the physical unitarity after giving the general properties of the massive Yang-Mills theory. Then we reproduce the violation of physical unitarity in a transparent way. This paper is a preliminary work to the subsequent papers in which we present a nonperturbative framework to propose a possible scenario of restoring the physical unitarity in the Curci-Ferrari model.

Nonperturbative construction of massive Yang-Mills fields without the Higgs field

In order to understand the so-called decoupling solution for gluon and ghost propagators in QCD, we give a nonperturbative construction of a massive vector field describing a non-Abelian massive spin-one particle, which has the correct physical degrees of freedom and is invariant under a modified Becchi-Rouet-Stora-Tyutin transformation, in a massive Yang-Mills model without the Higgs field, i.e., the Curci-Ferrari model. The resulting non-Abelian massive vector boson field is written using a nonlinear but local transformation from the original fields in the Curci-Ferrari model. As an application, we write down a local mass term for the Yang-Mills field and a dimension-two condensate, which are exactly invariant under the modified Becchi-Rouet-Stora-Tyutin transformation, Lorentz transformation and color rotation.