Three-gluon Vertex Research Articles

Nonlinear Schwinger mechanism in QCD, and Fredholm alternatives theorem

We present a novel implementation of the Schwinger mechanism in QCD, which fixes uniquely the scale of the effective gluon mass scale and streamlines considerably the procedure of multiplicative renormalization. The key advantage of this method stems from the nonlinear nature of the dynamical equation that generates massless poles in the longitudinal sector of the three-gluon vertex. An exceptional feature of this approach is an extensive cancellation involving the components of the integral expression that determines the gluon mass scale; it is triggered once the Schwinger–Dyson equation of the pole-free part of the three-gluon vertex has been appropriately exploited. It turns out that this cancellation is driven by the so-called Fredholm alternatives theorem, which operates among the set of integral equations describing this system. Quite remarkably, in the linearized approximation this theorem enforces the exact masslessness of the gluon. Instead, the nonlinearity induced by the full treatment of the relevant kernel evades this theorem, allowing for the emergence of a nonvanishing mass scale. The numerical results obtained from the resulting equations are compatible with the lattice findings, and may be further refined through the inclusion of the remaining fundamental vertices of the theory.

Magnetic corrections to the QCD coupling: Strong field approximation

We compute the one-loop vertex function of the QCD coupling in the presence of an ultraintense magnetic field. From the vertex function, we extract the effective coupling and show that it grows with increasing magnetic field. We consider the quark-gluon vertex and the three-gluon vertex, accounting for the propagators of charged particles within the loops using the lowest Landau level approximation in order to satisfy the condition where the magnetic field is the largest energy scale. Under this approximation, we find that the contribution from the three-gluon vertex vanishes. Therefore, this result arises from the competition between the color charge associated to gluons and to quarks as well, with the former being larger than the latter. The behavior of the QCD coupling as a function of the magnetic field strength is analogous to that exhibited by the light-quark condensate, indicating the magnetic catalysis occurs. This increasing behavior stems from the dominant contribution of color charge associated to gluons in the vertex function. Published by the American Physical Society 2024

Complete analysis of the Landau-gauge three-gluon vertex from lattice QCD

Several continuum and lattice investigations of the QCD three-gluon vertex have recently exposed its key properties, some intimately connected with the low-momentum behavior of the two-point gluon Green’s function and especially relevant for the emergence of a mass scale in this latter, via the Schwinger mechanism. In the present study, we report on a lattice determination of the Landau-gauge, transversely projected three-gluon vertex, particularly scrutinizing an outstanding one of these properties, termed , exploring its implications and capitalizing on it to gain further insight on the low-momentum running of both the three-gluon vertex and its associated strong coupling. Published by the American Physical Society 2024

Schwinger displacement of the quark–gluon vertex

The action of the Schwinger mechanism in pure Yang–Mills theories endows gluons with an effective mass, and, at the same time, induces a measurable displacement to the Ward identity satisfied by the three-gluon vertex. In the present work we turn to Quantum Chromodynamics with two light quark flavors, and explore the appearance of this characteristic displacement at the level of the quark–gluon vertex. When the Schwinger mechanism is activated, this vertex acquires massless poles, whose momentum-dependent residues are determined by a set of coupled integral equations. The main effect of these residues is to displace the Ward identity obeyed by the pole-free part of the vertex, causing modifications to its form factors, and especially the one associated with the tree-level tensor. The comparison between the available lattice data for this form factor and the Ward identity prediction reveals a marked deviation, which is completely compatible with the theoretical expectation for the attendant residue. This analysis corroborates further the self-consistency of this mass-generating scenario in the general context of real-world strong interactions.

Schwinger poles of the three-gluon vertex: symmetry and dynamics

The implementation of the Schwinger mechanism endows gluons with a nonperturbative mass through the formation of special massless poles in the fundamental QCD vertices; due to their longitudinal character, these poles do not cause divergences in on-shell amplitudes, but induce detectable effects in the Green’s functions of the theory. Particularly important in this theoretical setup is the three-gluon vertex, whose pole content extends beyond the minimal structure required for the generation of a gluon mass. In the present work we analyze these additional pole patterns by means of two distinct, but ultimately equivalent, methods: the Slavnov–Taylor identity satisfied by the three-gluon vertex, and the nonlinear Schwinger–Dyson equation that governs the dynamical evolution of this vertex. Our analysis reveals that the Slavnov–Taylor identity imposes strict model-independent constraints on the associated residues, preventing them from vanishing. Approximate versions of these constraints are subsequently recovered from the Schwinger–Dyson equation, once the elements responsible for the activation of the Schwinger mechanism have been duly incorporated. The excellent coincidence between the two approaches exposes a profound connection between symmetry and dynamics, and serves as a nontrivial self-consistency test of this particular mass generating scenario.

Planar degeneracy of the three-gluon vertex

We present a detailed exploration of certain outstanding features of the transversely-projected three-gluon vertex, using the corresponding Schwinger–Dyson equation in conjunction with key results obtained from quenched lattice simulations. The main goal of this study is the scrutiny of the approximate property denominated “planar degeneracy”, unveiled when the Bose symmetry of the vertex is properly exploited. The planar degeneracy leads to a particularly simple parametrization of the vertex, reducing its kinematic dependence to essentially a single variable. Our analysis, carried out in the absence of dynamical quarks, reveals that the planar degeneracy is particularly accurate for the description of the form factor associated with the classical tensor, for a wide array of arbitrary kinematic configurations. Instead, the remaining three form factors display considerable violations of this property. In addition, and in close connection with the previous point, we demonstrate the numerical dominance of the classical form factor over all others, except in the vicinity of the soft-gluon kinematics. The final upshot of these considerations is the emergence of a very compact description for the three-gluon vertex in general kinematics, which may simplify significantly nonperturbative applications involving this vertex.

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

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

Evidence of the Schwinger Mechanism from Lattice QCD

In quantum chromodynamics (QCD), gluons acquire a mass scale through the action of the Schwinger mechanism. This mass emerges as a result of the dynamical formation of massless bound-states of gluons which manifest as longitudinally coupled poles in the vertices. In this contribution, we show how the presence of these poles can be determined from lattice QCD results for the propagators and vertices. The crucial observation that allows this determination is that the Schwinger mechanism poles induce modifications, called “displacements”, to the Ward identities (WIs) relating two- and three-point functions. Importantly, the displacement functions correspond precisely to the Bethe–Salpeter amplitudes of the massless bound-states. We apply this idea to the case of the three-gluon vertex in pure Yang–Mills SU(3). Using lattice results in the corresponding WI, we find an unequivocal displacement and show that it is consistent with the prediction based on the Bethe–Salpeter equation.

Schwinger mechanism for gluons from lattice QCD

Continuum and lattice analyses have revealed the existence of a mass-scale in the gluon two-point Schwinger function. It has long been conjectured that this expresses the action of a Schwinger mechanism for gauge boson mass generation in quantum chromodynamics (QCD). For such to be true, it is necessary and sufficient that a dynamically-generated, massless, colour-carrying, scalar gluon+gluon correlation emerges as a feature of the dressed three-gluon vertex. Working with results on elementary Schwinger functions obtained via the numerical simulation of lattice-regularised QCD, we establish with an extremely high level of confidence that just such a feature appears; hence, confirm the conjectured origin of the gluon mass scale.

Gauge Sector Dynamics in QCD

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.

How to Determine the Branch Points of Correlation Functions in Euclidean Space II: Three-Point Functions

The analytic structure of elementary correlation functions of a quantum field is relevant for the calculation of masses of bound states and their time-like properties in general. In quantum chromodynamics, the calculation of correlation functions for purely space-like momenta has reached a high level of sophistication, but the calculation at time-like momenta requires refined methods. One of them is the contour deformation method. Here we describe how to employ it for three-point functions. The basic mechanisms are discussed for a scalar theory, but they are the same for more complicated theories and are thus relevant, e.g., for the three-gluon or quark-gluon vertices of quantum chromodynamics. Their inclusion in existing truncation schemes is a crucial step for investigating the analytic structure of elementary correlation functions of quantum chromodynamics and the calculation of its spectrum from them.

Lattice three-gluon vertex in extended kinematics: Planar degeneracy

We present novel results for the three-gluon vertex, obtained from an extensive quenched lattice simulation in the Landau gauge. The simulation evaluates the transversely projected vertex, spanned on a special tensorial basis, whose form factors are naturally parametrized in terms of individually Bose-symmetric variables. Quite interestingly, when evaluated in these kinematics, the corresponding form factors depend almost exclusively on a single kinematic variable, formed by the sum of the squares of the three incoming four-momenta, q, r, and p. Thus, all configurations lying on a given plane in the coordinate system (q2,r2,p2) share, to a high degree of accuracy, the same form factors, a property that we denominate planar degeneracy. We have confirmed the validity of this property through an exhaustive study of the set of configurations satisfying the condition q2=r2, within the range [0,5GeV]. This drastic simplification allows for a remarkably compact description of the main bulk of the data, which is particularly suitable for future numerical applications. A semi-perturbative analysis reproduces the lattice findings rather accurately, once the inclusion of a gluon mass has cured all spurious divergences.

Patterns of gauge symmetry in the background field method

The correlation functions of Yang–Mills theories formulated in the background field method satisfy linear Slavnov–Taylor identities, which are naive generalizations of simple tree level relations, with no deformations originating from the ghost-sector of the theory. In recent years, a stronger version of these identities has been found to hold at the level of the background gluon self-energy, whose transversality is enforced separately for each special block of diagrams contributing to the gluon Schwinger–Dyson equation. In the present work we demonstrate by means of explicit calculations that the same distinct realization of the Slavnov–Taylor identity persists in the case of the background three-gluon vertex. The analysis is carried out at the level of the exact Schwinger–Dyson equation for this vertex, with no truncations or simplifying assumptions. The demonstration entails the contraction of individual vertex diagrams by the relevant momentum, which activates Slavnov–Taylor identities of vertices and multi-particle kernels nested inside these graphs; the final result emerges by virtue of a multitude of extensive cancellations, without the need of performing explicit integrations. In addition, we point out that background Ward identities amount to replacing derivatives of propagators by zero-momentum background-gluon insertions, in exact analogy to standard properties of Abelian gauge theories. Finally, certain potential applications of these results are briefly discussed.

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].

Theory and phenomenology of the three-gluon vertex

The three-gluon vertex is a fundamental ingredient of the intricate QCD dynamics, being inextricably connected to key nonperturbative phenomena, such as the emergence of a mass scale in the gauge sector of the theory. In this presentation, we review the main theoretical properties of the three-gluon vertex in the Landau gauge, obtained from the fruitful synergy between functional methods and lattice simulations. We pay particular attention to the manifestation and origin of the infrared suppression of its main form factors and the associated zero crossing. In addition, we discuss certain characteristic phenomenological applications that require this special vertex as input.

Ghost dynamics from Schwinger-Dyson equations

We discuss the coupled dynamics of the ghost dressing function and the ghost-gluon vertex through the Schwinger-Dyson equations that they satisfy. In order to close the system of equations, we combine recent lattice data for the gluon propagator and an approximate STI-derived Ansatz for the general kinematics three-gluon vertex. The numerical solution of the resulting coupled system exhibits excellent agreement to lattice data, for both the ghost dressing function and the ghost-gluon vertex, and allows the determination of the coupling constant. Next, in the soft gluon limit the full three-gluon vertex appearing in the ghost-gluon equation reduces to a special projection that is tightly constrained by lattice simulations. Specializing the ghost-gluon Schwinger-Dyson equation to this limit provides a nontrivial consistency check on the approximations employed for the three-gluon interaction and shows that the latter has an important quantitative effect on the ghost-gluon vertex. Finally, our results stress the importance of eliminating artifacts when confronting lattice data with continuum predictions.

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.

More on the three-gluon vertex in SU(2) Yang-Mills theory in three and four dimensions

The three-gluon vertex has been found to be a vital ingredient in non-perturbative functional approaches. We present an updated lattice calculation of it in various kinematical configurations for all tensor structures and multiple lattice parameters in three dimensions, and in a subset of those in four dimensions, for SU(2) Yang-Mills theory in minimal Landau gauge. In three dimensions an unambiguous zero crossing for the tree-level form-factor is established, and consistency for all investigated form factors with a power-like divergence towards the infrared is observed. Using very coarse lattices this is even seen towards momenta as low as about 15 MeV. The results in four dimensions are consistent with such a behavior, but do not yet reach deep enough into the infrared to establish it.

Exploring smoking-gun signals of the Schwinger mechanism in QCD

In Quantum Chromodynamics, the Schwinger mechanism endows the gluons with an effective mass through the dynamical formation of massless bound-state poles that are longitudinally coupled. The presence of these poles affects profoundly the infrared properties of the interaction vertices, inducing crucial modifications to their fundamental Ward identities. Within this general framework, we present a detailed derivation of the non-Abelian Ward identity obeyed by the pole-free part of the three-gluon vertex in the soft-gluon limit, and determine the smoking-gun displacement that the onset of the Schwinger mechanism produces to the standard result. Quite importantly, the quantity that describes this distinctive feature coincides formally with the bound-state wave function that controls the massless pole formation. Consequently, this signal may be computed in two independent ways: by solving an approximate version of the pertinent Bethe-Salpeter integral equation, or by appropriately combining the elements that enter in the aforementioned Ward identity. For the implementation of both methods we employ two- and three-point correlation functions obtained from recent lattice simulations, and a partial derivative of the ghost-gluon kernel, which is computed from the corresponding Schwinger-Dyson equation. Our analysis reveals an excellent coincidence between the results obtained through either method, providing a highly nontrivial self-consistency check for the entire approach. When compared to the null hypothesis, where the Schwinger mechanism is assumed to be inactive, the statistical significance of the resulting signal is estimated to be 3 standard deviations.

Three-gluon vertex in Landau-gauge from quenched-lattice QCD in general kinematics

We report on a novel and extensive lattice QCD analysis for the three-gluon vertex from quenched lattice-QCD simulations. Using standard Wilson action, we have computed the three-gluon vertex beyond the usual kinematic restriction to the symmetric (q2 = r2 = p2) and soft-gluon (p = 0) cases where it depends on a single momentum scale. The so-dubbed bisectoral case (r2 = q2 ≠ p2), where the transversely projected vertex can be cast in terms of three independent tensors, have been the object of a recent exhaustive scrutiny [1], also shown in this communication. Herein, beyond this special case, results for kinematic configurations with three different squared momenta are also presented. All data considered, the lattice estimate of the three-gluon vertex exhibits a clear dominance of the tree-level tensor form factor.