Four-gluon Vertex Research Articles

Nonperturbative four-gluon vertex in soft kinematics

We present a nonperturbative study of the form factor associated with the projection of the full four-gluon vertex on its classical tensor, for a set of kinematics with one vanishing and three arbitrary external momenta. The treatment is based on the Schwinger-Dyson equation governing this vertex, and a large-volume lattice simulation, involving ten thousand gauge field configurations. The key hypothesis employed in both approaches is the “planar degeneracy”, which classifies diverse configurations by means of a single variable, thus enabling their meaningful “averaging”. The results of both approaches show notable agreement, revealing a considerable suppression of the averaged form factor in the infrared. The deviations from the exact planar degeneracy are discussed in detail, and a supplementary variable is used to achieve a more accurate description. The effective charge defined through this special form factor is computed within both approaches, and the results obtained are in excellent agreement.

Four-gluon vertex in collinear kinematics

To date, the four-gluon vertex is the least explored component of the QCD Lagrangian, mainly due to the vast proliferation of Lorentz and color structures required for its description. In this work we present a nonperturbative study of this vertex, based on the one-loop dressed Schwinger–Dyson equation obtained from the 4PI effective action. A vast simplification is brought about by resorting to “collinear” kinematics, where all momenta are parallel to each other, and by appealing to the charge conjugation symmetry in order to eliminate certain color structures. Out of the fifteen form factors that comprise the transversely-projected version of this vertex, two are singled out and studied in detail; the one associated with the classical tensorial structure is moderately suppressed in the infrared regime, while the other diverges logarithmically at the origin. Quite interestingly, both form factors display the property known as “planar degeneracy” at a rather high level of accuracy. With these results we construct an effective charge that quantifies the strength of the four-gluon interaction, and compare it with other vertex-derived charges from the gauge sector of QCD.

Perturbative unitarity and the four-point vertices in the constructive standard model

We find a complete set of four-point vertices in the constructive Standard Model (CSM). This set is smaller than in Feynman diagrams as the CSM does not need or allow any additional four-point vertices (or “contact” terms) beyond what is present in Feynman diagrams and, furthermore, it does need or allow a four-point vertex for Z,Z,W¯,W, W,W,W¯,W¯, γ,Z,W,W¯ or γ,γ,W,W¯, in addition to the already known absence of the four-gluon vertex. We show that with this set of four-point vertices, perturbative unitarity is satisfied in the CSM. Additionally, we show that many constructive diagrams are not Feynman diagrams rewritten in spinor form. In fact, we show that there is a significant rearrangement of contributions from the diagrams in constructive calculations relative to Feynman diagrams for some processes. In addition to the already known or expected rearrangement in diagrams involving external photons, we also find that diagrams involving four-vector bosons are also significantly different from their Feynman counterparts. 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

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

Simplifying QCD event generation with chirality flow, reference vectors and spin directions

The chirality-flow formalism, combined with good choices of gauge reference vectors, simplifies tree-level calculations to the extent that it is often possible to write down amplitudes corresponding to Feynman diagrams immediately. It has also proven to give a very sizable speedup in a proof of concept implementation of massless tree-level QED in MadGraph5_aMC@NLO. In the present paper we extend this analysis to QCD, including massive quarks. We define helicity-dependent versions of the gluon vertices, derive constraints on the spinor structure of propagating gluons, and explore the Schouten identity to simplify the four-gluon vertex further. For massive quarks, the chirality-flow formalism sheds light on how to exploit the freedom to measure spin along any direction to shorten the calculations. Overall, this results in a clear speedup for treating the Lorentz structure at high multiplicities.

Color-kinematics relation from the Feynman diagram perspective

Feynman diagrams for gluon tree amplitudes are studied in the Feynman gauge and in any number of spacetime dimensions. The color-kinematics combinations $\mathrm{\ensuremath{\Delta}}={n}_{s}\ensuremath{-}{n}_{t}\ensuremath{-}{n}_{u}$ of numerators are explicitly calculated for $N=4$, 5, 6 gluons to see whether the color-kinematics relation $\mathrm{\ensuremath{\Delta}}=0$ is satisfied. This is a tedious task because of the presence of four-gluon vertices, and the large number of Feynman diagrams, numerators, and $\mathrm{\ensuremath{\Delta}}$ combinations involved, especially when $N=6$. For on-shell amplitudes, it is found that $\mathrm{\ensuremath{\Delta}}=0$ for $N=4$, but $\mathrm{\ensuremath{\Delta}}\ensuremath{\ne}0$ for $N=5$ and $N=6$ owing to the presence of the four-gluon vertex. However, a local generalized gauge transformation can bring about $\mathrm{\ensuremath{\Delta}}=0$ for $N=5$, but not for $N=6$. This raises the question whether gluon amplitudes satisfying the color-kinematics relation contain nonlocal interactions.

Correlation functions of Landau gauge Yang-Mills theory

Correlation functions of Yang-Mills theory in the Landau gauge are calculated from their equations of motion. The employed setup is completely parameter free and leads, within errors, to good quantitative agreement with corresponding lattice results for the ghost and gluon propagators as well as the ghost-gluon and three-gluon vertices. Also the four-gluon vertex is calculated. The present setup allows for the first time for a unique subtraction of quadratic divergences in the gluon propagator Dyson-Schwinger equation. Thus, there are no ambiguities which can arise due to the use of models or auxiliary workarounds. In addition, several self-tests of the results are described that allow assessing the truncation error in a self-consistent way. This enables a new perspective on how to identify limitations of the present setup and develop future improvements.

Off-shell Ward identities for N-gluon amplitudes

Off-shell Ward identities in non-abelian gauge theory continue to be a subject of active research, since they are, in general, inhomogeneous and their form depends on the chosen gauge-fixing procedure. For the three-gluon and four-gluon vertices, it is known that a relatively simple form of the Ward identity can be achieved using the pinch technique or, equivalently, the background-field method with quantum Feynman gauge. The latter is also the gauge-fixing underlying the string-inspired formalism, and here we use this formalism to prove the corresponding form of the Ward identity for the one-loop N-gluon amplitudes.

Numerical solutions to Giovannini’s QCD parton branching processes

In 1979, Giovannini [1] formulated multiparticle production as a manifestation of four QCD processes: A: g → gg (gluon bremsstrahlung), A-: q → qg (quark bremsstrahlung), B: g → qq (quark pair creation) and C: g → ggg (four-gluon vertex). At present, only approximate solutions exist, such as the Generalized Multiplicity Distribution (GMD) which excludes pro- cesses B and C [2, 3]. These approximate solutions have been very successful in describing charged particle multiplicities at lower energies, at least until the appearance of a shoulder-like structure at √s ≈ 200 GeV and 900 GeV first reported by the UA5 collaboration [4], also known as “KNO-scaling violation”. Here, we extend the work of Sakai [5] and attempt a numerical solution that incorporates all four QCD processes, and see if the addition of processes B and C can explain the shoulder-like structure.

Gluonic Vertices and the Gluon Propagator in Landau Gauge Yang--Mills Theory

Correlation functions of Landau gauge Yang-Mills theory are calculated from their equations of motion. Solutions for the three- and four-gluon vertices and the role of two-loop diagrams in the gluon propagator equation are discussed.

Gluon mass scale through nonlinearities and vertex interplay

We present a novel analysis of the gluon gap equation, where its full nonlinear structure is duly taken into account. In particular, while in previous treatments the linearization of this homogeneous integral equation introduced an indeterminacy in the scale of the corresponding mass, the current approach determines it uniquely, once the value of the gauge coupling at a given renormalization point is used as input. A crucial ingredient for this construction is the ``kinetic term'' of the gluon propagator, whose form is not obtained from the complicated equation governing its evolution, but is rather approximated by suitable initial Ans\"atze, which are subsequently improved by means of a systematic iterative procedure. The multiplicative renormalization of the central equation is carried out following an approximate method, which is extensively employed in the studies of the standard quark gap equation. This approach amounts to the effective substitution of the vertex renormalization constants by kinematically simplified form factors of the three- and four-gluon vertices. The resulting numerical interplay, exemplified by the infrared suppression of the three-gluon vertex and the mild enhancement of the four-gluon vertex, is instrumental for obtaining positive-definite and monotonically decreasing running gluon masses. The resulting gluon propagators, put together from the gluon masses and kinetic terms obtained with this method, match rather accurately the data obtained from large-volume lattice simulations.

Off-shell gluon amplitudes in QCD

In this contribution we discuss off-shell gluon amplitudes from worldline formalism perspective. We use the Bern-Kosower master formula and two IBP methods to extract the form factor of the off-shell three-and four-gluon vertices. For the four-gluon case we also discus its representation in = 4 SYM.

Probing the tensor structure of lattice three-gluon vertex in Landau gauge

In this paper we test an approximate method that is often used in lattice studies of the Landau gauge three-gluon vertex. The approximation consists in describing the lattice correlator with tensor bases from the continuum theory. With the help of vertex reconstruction, we show that this "continuum" approach may lead, for general kinematics, to significant errors in vertex tensor representations. Such errors are highly unwelcome, as they can lead to wrong quantitative estimates for vertex form factors and related quantities of interest, like the three-gluon running coupling. As a possible solution, we demonstrate numerically and analytically that there exist special kinematic configurations for which the vertex tensor structures can be described exactly on the lattice. For these kinematics, the dimensionless tensor elements are equal to the continuum ones, regardless of the details of the lattice implementation. We ran our simulations for an $SU(2)$ gauge theory in two and three spacetime dimensions, with Wilson and $\mathcal{O}(a^2)$ tree-level improved gauge actions. Our results and conclusions can be straightforwardly generalised to higher dimensions and, with some precautions, to other lattice correlators, like the ghost-gluon, quark-gluon and four-gluon vertices.

Correlation functions of three-dimensional Yang-Mills theory from the FRG

We compute correlation functions of three-dimensional Landau-gauge Yang-Mills theory with the Functional Renormalisation Group. Starting from the classical action as only input, we calculate the non-perturbative ghost and gluon propagators as well as the momentum-dependent ghost-gluon, three-gluon, and four-gluon vertices in a comprehensive truncation scheme. Compared to the physical case of four spacetime dimensions, we need more sophisticated truncations due to significant contributions from non-classical tensor structures. In particular, we apply a special technique to compute the tadpole diagrams of the propagator equations, which captures also all perturbative two-loop effects, and compare our correlators with lattice and Dyson-Schwinger results.

Nonperturbative finite-temperature Yang-Mills theory

We present nonperturbative correlation functions in Landau-gauge Yang-Mills theory at finite temperature. The results are obtained from the functional renormalisation group within a self-consistent approximation scheme. In particular, we compute the magnetic and electric components of the gluon propagator, and the three- and four-gluon vertices. We also show the ghost propagator and the ghost-gluon vertex at finite temperature. Our results for the propagators are confronted with lattice simulations and our Debye mass is compared to hard thermal loop perturbation theory.

Landau gauge Yang-Mills correlation functions

We investigate Landau gauge $SU(3)$ Yang-Mills theory in a systematic vertex expansion scheme for the effective action with the functional renormalisation group. Particular focus is put on the dynamical creation of the gluon mass gap at non-perturbative momenta and the consistent treatment of quadratic divergences. The non-perturbative ghost and transverse gluon propagators as well as the momentum-dependent ghost-gluon, three-gluon and four-gluon vertices are calculated self-consistently with the classical action as only input. The apparent convergence of the expansion scheme is discussed and within the errors, our numerical results are in quantitative agreement with available lattice results.

QCD gluon vertices from the string-inspired formalism

The Bern–Kosower formalism, developed around 1990 as a novel way of obtaining QCD amplitudes as the limit of infinite string tension of the corresponding string amplitudes, was originally designed as an on-shell formalism. Building on early work by Strassler, the authors have recently shown that this “string-inspired formalism” is extremely efficient also as a tool for the study of off-shell amplitudes in QCD, and in particular for achieving compact form factor decompositions of the [Formula: see text]-gluon vertices. Among other things, this formalism allows one to achieve a manifestly gauge invariant decomposition of these vertices by way of integration-by-parts, rather than the usual tedious analysis of the non-abelian off-shell Ward identities, and to combine the spin zero, half and one cases. Here, we will provide a summary of the method, as well as its application to the three- and four-gluon vertices.

Correlation functions of three-dimensional Yang-Mills theory from Dyson-Schwinger equations

The two- and three-point functions and the four-gluon vertex of three-dimensional Yang-Mills theory are calculated from their Dyson-Schwinger equations and the 3PI effective action. Within a self-contained truncation various effects of truncating Dyson-Schwinger equations are studied. Estimates for the errors induced by truncations are derived from comparisons between results from different equations, comparisons with lattice results, and varying higher Green functions. The results indicate that the two-loop diagrams are important in the gluon propagator, where they are explicitly calculated, but not for the vertices. Furthermore, the influence of the four-gluon vertex on lower Green functions is found to be small.

A Non-perturbative Study of the Correlation Functions of Three-dimensional Yang--Mills Theory

Yang-Mills theory is studied in three dimensions using the equations of motion of the $1$PI and $3$PI effective actions. The employed self-contained truncation includes the propagators, the three-point functions and the four-gluon vertex dynamically. In the gluon propagator also two-loop diagrams are taken into account. The higher gluonic correlation functions show sizable deviations from the tree-level only at low momenta. Also the couplings derived from the vertices agree well down to a few GeV. In addition, different methods to subtract spurious divergences are explored.