No planar degeneracy for the Landau gauge quark-gluon vertex

A nonperturbative calculation of quantum chromodynamics found that the Landau-gauge quark-gluon vertex cannot be approximated reliably by collapsing its angular dependence into a planar form. Although the vertex form factors vary only weakly with angle when expressed in carefully chosen variables, that variation substantially changes the self-consistent quark propagator and quantities derived from it. The result challenges an attractive computational shortcut: angular effects that appear negligible in the vertex itself can become important after integration through the coupled equations.

The study used Dyson-Schwinger equations to solve simultaneously for the quark propagator and the quark-gluon vertex in the chiral limit, where current quark masses are set to zero. It was a theoretical calculation in the quenched approximation, meaning that the Yang-Mills sector was supplied as input rather than being coupled back to the quarks. The authors employed a three-particle-irreducible, three-loop truncation of the vertex equation, containing a non-Abelian diagram with the three-gluon vertex and, in an approximate implementation, an Abelian diagram with three dressed quark-gluon vertices. The findings are model calculations rather than experimental measurements, and their principal limitations arise from the truncation, the supplied Yang-Mills functions and the simplified treatment of the three-gluon vertex.

Landau gauge reduces the vertex to eight transverse tensor structures. Four are chirally symmetric and four violate chiral symmetry, with a separate form factor describing each structure as a function of three Lorentz-invariant variables: the gluon momentum squared, the squared average momentum of the two quark lines and the cosine of the angle between the gluon and average quark momenta. This parametrization respects the exchange of the quark and antiquark legs and the charge-conjugation properties of the dressed vertex. It also makes the angular dependence easier to assess than alternative choices of kinematic variables, which can display a larger spread even when the underlying physical behavior is the same.

Numerically, the calculation required two coupled equations for the quark dressing functions and eight for the vertex form factors. The authors used fixed-point iteration, cubic-spline interpolation and Gaussian quadrature for radial and angular loop integrals. Repeated solutions with finer external momentum grids and internal integration points provided estimates of numerical uncertainty. Over the relevant momentum domain, individual vertex form factors carried estimated errors of about 5% or slightly more. The quark dressing function A(p²) had an estimated error of about 0.5%, the scalar dressing function B(p²) about 2%, and the quark mass function M(p²) about 2.5%. These figures assessed numerical precision only: systematic effects from the equation truncation, input data and possible artefacts could not be quantified from this one study.

The calculated form factors did show weak angular dependence across many kinematic regions. In the sub-GeV domain, much of that behavior could be represented by a constant plus a small coefficient multiplying the square of the angular variable. For most momentum combinations, the absolute value of that coefficient was below 0.1, while a few approached the upper bound of about 0.5. The tree-level form factor displayed its strongest angular variation at intermediate and multi-GeV momenta, where even powers of the angle were needed to describe it. The non-tree-level structures were most sensitive in the sub-GeV region but became very small at high momentum, where their absolute importance was correspondingly limited.

Weak did not mean dispensable. The authors compared the full three-variable solution with calculations that fixed the angular variable at one of 13 selected values. These planar-like approximations either suppressed important contributions to the quark propagator, produced an ultraviolet decline that was too weak, or produced both failures. Even restricting an already converged full vertex to one angle after solving the complete system caused sizeable changes in the infrared quark mass function, the pion decay constant estimated through the Pagels-Stokar approximation and the quark condensate. The angular dependence therefore remained weak in form, but its numerical impact was substantial. Much of the relevant kernel contribution originated near the endpoints of the angular interval, where the angular spread of the integrands carried disproportionate weight.

The calculation also identified a self-consistent role for chirality-violating vertex structures in dynamical chiral symmetry breaking. Chirally symmetric components supplied most of the contribution to A(p²), whereas chirality-violating components drove B(p²) and thus the dynamically generated scalar quark mass. In the sub-GeV region, the tree-level vertex contributed less than one quarter of B(p²) point by point, leaving the non-tree-level tensor structures with the dominant role. In this framework, the dynamically generated tensor coupling of gluons to quarks and the scalar part of the quark propagator reinforce one another. Their coupling changes the momentum dependence of the mass function and therefore affects the propagator’s analytic structure in a way that a truncated vertex description can miss.

A comparison of two classes of Yang-Mills input produced another notable result. The calculations used two decoupling solutions, in which the gluon propagator and ghost renormalisation function approach finite infrared values, and a scaling solution with power-law infrared behavior. With the quark-gluon vertex retained at full kinematic dependence, the resulting quark propagators agreed within the estimated numerical errors, although individual vertex form factors could differ noticeably. The agreement is consistent with the possibility that the competing deep-infrared Yang-Mills solutions reflect a nonperturbative gauge-fixing issue rather than distinct quark sectors. It does not settle that interpretation, and a fully iterated, computationally expensive Abelian diagram was not completed; the approximate inclusion of that diagram had only a limited effect on the propagator.

The authors also fitted the Euclidean mass and dressing functions to examine the quark propagator’s analytic structure. The two lowest poles remained stable under several fit parameterizations and were located on the real timelike axis. For the primary decoupling input, they occurred at p² = −0.136 GeV², corresponding to a magnitude of 0.369 GeV, and p² = −0.823 GeV², corresponding to 0.907 GeV. The scaling input produced closely similar values of −0.143 GeV² and −0.875 GeV². The lowest pole had a positive residue, while the second pole, near 0.9 GeV, had a negative residue and was described as a ghost excitation. A third real pole beyond 1 GeV² was more sensitive to the chosen parameterization. The fit also generated a complex-conjugate pair of poles, but the authors cautioned that this feature might or might not be an artefact of the approximations and truncations.

Overall, the calculation shows that a visually small angular dependence can have a large effect on self-consistent results. Its importance lies not only in the variation of the vertex form factors but also in how those variations enter the integrals that build the quark propagator, particularly through the chirality-violating structures that govern the scalar mass function. The authors report a numerical relation, Γ⁽⁵⁾ = −1.40 Γ⁽⁶⁾, over almost the entire studied kinematic range, alongside a less precise proportionality between Γ⁽⁷⁾ and Γ⁽⁸⁾. They suggest that an off-shell version of the Gordon identity might help explain such relationships, but that remains unresolved. Within the study’s stated truncation and numerical precision, the central conclusion is that the Landau-gauge quark-gluon vertex has no usable planar degeneracy: preserving its angular dependence is essential for a reasonably precise, self-consistent description of quark dynamics in the model.

Subject of Research: Nonperturbative quark-gluon vertex dynamics and planar degeneracy in quenched QCD

Article Title: No planar degeneracy for the Landau gauge quark-gluon vertex

Article References: Wieland, G., & Alkofer, R. (2026). No planar degeneracy for the Landau gauge quark-gluon vertex. The European Physical Journal C, 86(10), Article 1202. https://doi.org/10.1140/epjc/s10052-026-16422-4

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16422-4

Keywords: quantum chromodynamics, quark-gluon vertex, Landau gauge, Dyson-Schwinger equations, dynamical chiral symmetry breaking, quark propagator, planar degeneracy, nonperturbative physics, quenched QCD, planar, degeneracy, Landau