Landau Gauge Research Articles

Consistency of Lorentz-violating Yang-Mills theories: Gauge invariance and nonperturbative effects

Previous investigations on the renormalizability properties of Lorentz-violating Yang-Mills (LVYM) theories in the Landau gauge have pointed out the necessity of the inclusion of a masslike term for the gauge fields. If one aims at generalizing the theory to a more complicated gauge, such a masslike term can bring severe issues regarding gauge dependence of correlation functions of observables. We propose a Lorentz-violating Yang-Mills theory supplemented by a gauge-invariant mass term which generalizes the model in the Landau gauge. We discuss the foundational aspects of the model and highlight how it simplifies in the Landau gauge. Following the recent literature in pure Yang-Mills theories, we remark how the proposed massive extension of LVYM theories can be used as an effective model to access infrared properties within perturbation theory. Finally, we present a Becchi-Rouet-Stora-Tyutin-invariant formulation of the so-called refined Gribov-Zwanziger action in the presence of Lorentz-violating terms in linear covariant gauges and the underlying tree-level gauge-field propagator which is enriched by the nonperturbative information carried by the elimination of Gribov copies. Published by the American Physical Society 2024

Impact of gauge fixing precision on the continuum limit of nonlocal quark-bilinear lattice operators

We analyze the gauge fixing precision dependence of some nonlocal quark-bilinear lattice operators interesting in computing parton physics for several measurements, using five lattice spacings ranging from 0.032 to 0.121 fm. Our results show that gauge-dependent nonlocal measurements are significantly more sensitive to the precision of gauge fixing than anticipated. The impact of imprecise gauge fixing is significant for fine lattices and long distances. For instance, even with the typically defined precision of Landau gauge fixing of 10−8, the deviation caused by imprecise gauge fixing can reach 12%, when calculating the trace of Wilson lines at 1.2 fm with a lattice spacing of approximately 0.03 fm. Similar behavior has been observed in ξ gauge and Coulomb gauge as well. For both quasi-parton-distribution functions (quasi-PDFs) and quasi-transverse-momentum-dependent PDFs operators renormalized using the regularization independent momentum subtraction (RI/MOM) scheme, convergence for different lattice spacings at long distance is only observed when the precision of Landau gauge fixing is sufficiently high. To describe these findings quantitatively, we propose an empirical formula to estimate the required precision. Published by the American Physical Society 2024

Infrared gluon propagator in the refined Gribov-Zwanziger scenario at one-loop order in the Landau gauge

The refined Gribov-Zwanziger (RGZ) action in the Landau gauge provides a local and renormalizable framework to account for the existence of infinitesimal Gribov copies in the path integral together with other relevant infrared effects such as the formation of condensates. The properties of the tree-level gluon propagator obtained in this setup have been thoroughly investigated over the past decade. It accommodates important properties seen in lattice simulations such as a finite value at vanishing momentum and positivity violation. Yet a comprehensive study about the stability of such properties against quantum corrections was lacking. In this work, we compute the gluon propagator in the RGZ scenario at one-loop order and implement an appropriate renormalization scheme in order to compare our findings with lattice data. Remarkably, the qualitative properties of the tree-level gluon propagator are preserved. In particular, the fits with lattice data show evidence for positivity violation and the existence of complex poles for SU(2) and SU(3) gauge groups. We comment on the results for the ghost propagator as well. Published by the American Physical Society 2024

Lattice determination of the Batalin-Vilkovisky function and the strong running interaction

The Batalin-Vilkovisky function is a central component in the modern formulation of the background field method and the physical applications derived from it. In the present work we report on novel lattice results for this particular quantity, obtained by capitalizing on its equality with the Kugo-Ojima function in the Landau gauge. The results of the lattice simulation are in very good agreement with the predictions derived from a continuum analysis based on the corresponding Schwinger-Dyson equations. In addition, we show that an important relation connecting this function with the ghost propagator is fulfilled rather accurately. With the aid of these results, we carry out the first completely lattice-based determination of the process-independent strong running interaction, employed in a variety of phenomenological studies.

On the conformal limit of a QED-inspired model

A conformal invariant QED-inspired model is solved for a general covariant linear gauge using the Dyson–Schwinger equations for the propagators assuming a pure vector like interaction. The leading corrections to the asymptotic solutions and the exponents, that characterize the corrections to each of the two fermion propagator functions, are computed as a function of the coupling and gauge fixing parameter ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}. For the scalar component of the fermion propagator our findings generalizes for linear covariant gauges previous results found in the literature and reproduce the outcome of the perturbative analysis of quenched QED in the Landau gauge. Our solution for the exponent associated with vector component of the fermion propagator is new and, in the weak coupling regime, agrees with the estimation based on the perturbative analysis of quenched QED. Of the two critical exponents describing the conformal limit of the vector interaction, one of them is, in QED, associated with the regime where chiral symmetry is broken dynamically, which demands one mass scale, namely the Miransky scaling. A second mass scale has to be introduced at larger coupling constants and is associated with a change on the nature of the fermion wave function. This provides one example, that it is possible to find two interwoven cycles in Quantum Field Theory, albeit in a truncated framework, as it is known in the quantum few-body problem in the limit of a zero-range interaction.

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

Deconfinement, center symmetry and the ghost propagator in Landau gauge pure SU(3) Yang-Mills theory

The temperature dependence of the Landau gauge ghost propagator is investigated in pure SU(3) Yang-Mills theory with lattice QCD simulations. Its behavior around the confined-deconfined phase transition temperature, Tc ∼ 270 MeV, is investigated. The simulations show that in the deconfined phase, the ghost propagator is enhanced for small momenta, ≲ 1 GeV. Furthermore, the analysis of the spontaneous breaking of center symmetry on the ghost propagator is studied. Similarly as observed for the gluon propagator, the simulations result in a decoupling of the sectors where the phase of the Polyakov loop is either 0 or ±2π/3 sectors, with the latter remaining indistinguishable. The results point to the possible use of the ghost propagator as an “order parameter” for the confined-deconfined phase transition.

Dependence of the Landau gauge ghost-gluon-vertex on the number of flavors

The gauge-boson, ghost and fermion propagators as well as the gauge-boson-ghost vertex function are studied for SU(N), Sp(2N), and SO(N) gauge groups. We solve a set of coupled Dyson-Schwinger equations in Landau gauge for a variable, fractional number Nf of massless fermions in the fundamental representation. For large Nf we find a phase transition from a chirally broken into a chirally symmetric phase that is consistent with the behavior expected inside the conformal window. Even in the presence of fermions the gauge-boson-ghost-vertex dressing remains small. In the conformal window this vertex shows the expected power law behavior. It does not assume its tree-level value in the far infrared, but the respective dressing function is a constant greater than 1. Published by the American Physical Society 2024

Spinning pairs: Supporting P03 quark-pair creation from Landau-gauge Green’s functions

Abundant phenomenology suggests that strong decays from relatively low-excitation hadrons into other hadrons proceed by the creation of a light quark-antiquark pair with zero total angular momentum, the so called P03 mechanism originating from a scalar bilinear. Yet the quantum chromodynamics (QCD) interaction is perturbatively mediated by gluons of spin one, and QCD presents a chirally symmetric Lagrangian. Such scalar decay term must be spontaneously generated upon breaking chiral symmetry. We attempt to reproduce this with the help of the quark-gluon vertex in Landau gauge, whose nonperturbative structure has been reasonably elucidated in the last years, and insertions of a uniform, constant chromoelectric field. This is akin to Schwinger pair production in quantum electrodynamics (QED), and we provide a comparison with its two field-insertions diagram. We find that, the symmetry being cylindrical, the adequate quantum numbers to discuss the production are rather Σ30, Σ31, and Π30 as in diatomic molecules, and we indeed find a sizeable contribution of the third decay mechanism, which may give a rationale for the P03 phenomenology, as long as the momentum of the produced pair is at or below the scale of the bare or dynamically generated fermion mass. On the other hand, ultrarelativistic fermions are rather ejected with Σ31 quantum numbers. In QED, our results suggest that Σ30 dominates, whereas the constraint of producing a color singlet in QCD leads to Π30 dominance at sub-GeV momenta. Published by the American Physical Society 2024

Four-gluon vertex from lattice QCD

A lattice QCD calculation for the four-gluon one-particle irreducible Green function in the Landau gauge is discussed. Results for some of the associated form factors are reported for kinematical configurations with a single momentum scale. Our results show that the computation of this Green function requires large statistical ensembles with 10 K or larger number of gauge configurations. The simulations considered herein have a clear Monte Carlo signal for momenta up to ∼1 GeV. The form factors show an hierarchy, with the form factor associated with the tree level Feynman rule being dominant and essentially constant for the range of momenta accessed. The remaining form factors seem to increase as the momentum decreases, suggesting that a possible log divergence may occur. The computed form factors are, at least, in qualitative agreement with the results obtained with continuum approaches to this vertex, when available. Published by the American Physical Society 2024

A Charged Particle with Anisotropic Mass in a Perpendicular Magnetic Field–Landau Gauge

The loss of any symmetry in a system leads to quantum problems that are typically very difficult to solve. Such a situation arises for particles with anisotropic mass, like electrons in various semiconductor host materials, where it is known that they may have an anisotropic effective mass. In this work, we consider the quantum problem of a spinless charged particle with anisotropic mass in two dimensions and study the resulting energy and eigenstate spectrum in a uniform constant perpendicular magnetic field when a Landau gauge is adopted. The exact analytic solution to the problem is obtained for arbitrary values of the anisotropic mass using a mathematical technique that relies on the scaling of the original coordinates. The characteristic features of the energy spectrum and corresponding eigenstate wave functions are analyzed. The results of this study are expected to be of interest to quantum Hall effect theory.

Center-symmetric Landau gauge: Further signatures of confinement

In a recent article [D. M. van Egmond and U. Reinosa, ], we have identified new signatures for the Yang-Mills deconfinement transition, based on the finite-temperature or gluon propagator as computed in the center-symmetric Landau gauge. Here, we generalize these considerations into a systematic study of the center symmetry identities obeyed by the correlation functions in this gauge. Any violation of these constraints signals the breaking of center symmetry and can thus serve as a probe for the deconfinement transition. 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

Three-loop effective potential for softly broken supersymmetry

The effective potential has been previously calculated through three-loop order, in Landau gauge, for a general renormalizable theory using dimensional regularization. However, dimensional regularization is not appropriate for softly broken supersymmetric gauge theories, because it explicitly violates supersymmetry. In this paper, I obtain the three-loop effective potential using a supersymmetric regulator based on dimensional reduction. Checks follow from the vanishing of the effective potential in examples with supersymmetric vacua, and from renormalization-scale invariance in examples for which supersymmetry is broken, either spontaneously or explicitly by soft terms. As by-products, I obtain the three-loop Landau gauge anomalous dimension for the scalar component of a chiral supermultiplet, and the beta function for the field-independent vacuum energy. Published by the American Physical Society 2024

Interaction Potential between a Uniformly Charged Square Nanoplate and Coplanar Nanowire

We study a structure consisting of two electrostatically interacting objects, a uniformly charged square nanoplate and a uniformly charged nanowire. A straightforward motivation behind this work is to introduce a model that allows a classical description of a finite two-dimensional quantum Hall system of few electrons when the Landau gauge is imposed. In this scenario, the uniformly charged square nanoplate would stand for the neutralizing background of the system while a uniformly charged nanowire would represent the resulting quantum striped state of the electrons. A second important feature of this model is that it also applies to hybrid charged nanoplate-nanowire systems in which the dominant interaction has electrostatic origin. An exact analytical expression for the electrostatic interaction potential between the uniformly charged square nanoplate and coplanar nanowire is obtained by using a special mathematical method adept for this geometry. It is found that the resulting interaction potential is finite, monotonic and slowly-varying for all locations of the nanowire inside the nanoplate.

Dynamic and static properties of quantum Hall and harmonic oscillator systems on the non-commutative plane

We study two quantum mechanical systems on the noncommutative plane using a representation independent approach. First, in the context of the Landau problem, we obtain an explicit expression for the gauge transformation that connects the Landau and the symmetric gauge in noncommutative space. This lead us to conclude that the usual form of the symmetric gauge A⃗=−β2Ŷ,β2X̂, in which the constant β is interpreted as the magnetic field, is not true in noncommutative space. We also be able to establish a precise definition of β as a function of the magnetic field, for which the equivalence between the symmetric and Landau gauges is held in noncommutative plane. Using the symmetric gauge, we obtain results for the spectrum of the quantum Hall system, its transverse conductivity in the presence of an electric field, and other static observables. These results amend the literature on quantum Hall effect in the noncommutative plane in which the incorrect form of the symmetric gauge, in noncommutative space, is assumed. We also study the non-equilibrium dynamics of simple observables for this system. On the other hand, we study the dynamics of the harmonic oscillator in non-commutative space and show that, in general, it exhibits quasi-periodic behavior, in striking contrast with its commutative version. The study of dynamics reveals itself as a most powerful tool to characterize and understand the effects of non-commutativity.

The Three-Gluon Vertex from Quenched Lattice QCD in Landau Gauge

The Three-Gluon Vertex from Quenched Lattice QCD in Landau Gauge

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.

Parton decomposition of nucleon spin and momentum: Gluons from dressed quarks

The lowest two Mellin moments of hadronic Generalized Parton Distributions are explored within a model that allows investigation of the inter-related quark and gluon contributions. For light quarks their dynamical connection is strong due to quark dressing. Our principal focus is the angular momentum J of the nucleon. This work employs and extends dynamical insights obtained from our recent model results for quark and gluon momentum fractions 〈x〉q/g in both pion and nucleon. The employed model is based on the Rainbow-Ladder truncation of the Dyson-Schwinger equations of QCD. The special case of a 1-loop treatment of a single hadronic quark is used to motivate several insights and obtain initial estimates such as the Wilson line correction to the established Landau gauge model (−7% for both Jq and 〈x〉q), and the “binding gluon” contribution to 〈x〉g (≤10%). We obtain the proton J within 1% after inclusion of the pion cloud mechanism to produce the sea. The gluon second Mellin moments of the proton reflect similar dynamics; (〈x〉g, Jg) are (26%, 24%) at model scale, and (40%, 38%) at 2 GeV. We also find that Jtot is shared almost equally between the total orbital and total intrinsic spin contributions.

Renormalization of three-quark operators at two loops in the RI′/SMOM scheme

We consider the renormalization of the three-quark operators without derivatives at next-to-next-to-leading order in QCD perturbation theory at the symmetric subtraction point. This allows us to obtain conversion factors between the MS‾ scheme and the regularization invariant symmetric MOM (RI/SMOM, RI′/SMOM) schemes. The results are presented both analytically in Rξ gauge in terms of a set of master integrals and numerically in Landau gauge. They can be used to reduce the errors in determinations of baryonic distribution amplitudes in lattice QCD simulations.