# The quark-gluon vertex and the QCD infrared dynamics

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## Abstract

The Dyson–Schwinger quark equation is solved for the quark-gluon vertex using the most recent lattice data available in the Landau gauge for the quark, gluon and ghost propagators, the full set of longitudinal tensor structures in the Ball-Chiu vertex, taking into account a recently derived normalisation for a quark-ghost kernel form factors and the gluon contribution for the tree level quark-gluon vertex identified on a recent study of the lattice soft gluon limit. A solution for the inverse problem is computed after the Tikhonov linear regularisation of the integral equation, that implies solving a modified Dyson–Schwinger equation. We get longitudinal form factors that are strongly enhanced at the infrared region, deviate significantly from the tree level results for quark and gluon momentum below 2 GeV and at higher momentum approach their perturbative values. The computed quark-gluon vertex favours kinematical configurations where the quark momentum *p* and the gluon momentum *q* are small and parallel. Further, the quark-gluon vertex is dominated by the form factors associated to the tree level vertex \(\gamma _\mu \) and to the operator \(2 \, p_\mu + q_\mu \). The higher rank tensor structures provide small contributions to the vertex.

## 1 Introduction

The interaction of quarks and gluons is described by Quantum Chromodynamics [1, 2, 3, 4], a renormalisable gauge theory associated to the color gauge group SU(3). Of its correlation functions, the quark-gluon vertex has a fundamental role in hadron phenomenology, in the understanding of chiral symmetry breaking mechanism and the realisation of confinement. Despite its relevance for strong interactions, our knowledge of the quark-gluon vertex from first principles calculations is relatively poor. At the perturbative level, only recently a full calculation of the twelve form factors associated to this vertex was published [5] but only for some kinematical configurations, namely the symmetric configuration (equal incoming, outgoing quark and gluon squared momenta), the on-shell configuration (quarks on-shell with vanishing gluon momentum) and what the authors called its asymptotic limit. In particular, the vertex asymptotic limit was used to investigate *ansätze* that can be found in the literature [6, 7, 8, 9, 10, 11] with the aim to test their description of the ultraviolet regime.

At the non-perturbative level, the quark-gluon vertex has been studied within continuum approaches to QCD by several authors [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Typically, the computation is performed after writing the vertex in terms of other QCD vertices and propagators and taking into account its perturbative tail. Most of the computations include only a fraction of the twelve form factors required to fully describe the quark-gluon vertex. In [17, 22, 23] the authors look for a first principle determination of the vertex by solving the theory at the non-perturbative level, gathering information on the vertex from QCD symmetries and relying on one-loop dressed perturbation theory. The vertex has also been investigated perturbatively within massive QCD, i.e. using the Curci-Ferrari model [18], and all its (perturbative) tensor structures form factors accessed for some kinematical configurations.

Lattice simulations, both for quenched [24, 25, 26] and full QCD [27], were also used to investigate the quark-gluon vertex. Again, only a limited set of kinematical configurations were accessed and, in particular, its the soft gluon limit, defined by a vanishing gluon momenta, was mostly explored. For full QCD so far only a single form factor, that associated with the tree level tensor structure, was measured on lattice simulation in the soft gluon limit.

One can also find in the literature attempts to combine continuum non-perturbative QCD equations with lattice simulations to study the quark-gluon vertex. Indeed, in [28] a generalised Ball-Chiu vertex was used in the quark gap equation, together with lattice results for the quark, gluon and ghost propagators to investigate the quark-gluon vertex. In [29], the full QCD lattice data for \(\lambda _1\) was studied relying on continuum information about the vertex.

The use of continuum equations with results coming from lattice simulations requires high quality lattice data to feed the continuum equations that should be solved for the vertex. In this approach, the computation of a solution of the continuum equations requires assuming some type of functional dependence for various propagator functions. In recent years, there has been an effort to improve the quality of the lattice data, in the sense of being closer to the continuum and producing simulations with large statistical ensembles, both for propagators and for vertex functions. This approach that combines lattice information and continuum equations relies strongly on the effort to access high precision lattice simulations.

For the practitioner oftentimes it is sufficient to have a good model of the vertex that should incorporate the perturbative tail to describe the ultraviolet regime, some “guessing” for the infrared region and, hopefully, comply with perturbative renormalisation [5, 6, 7, 8, 31]. A popular and quite successful model was set in [32], named the Maris-Tandy model, that assumes a bare quark-gluon vertex and introduces an effective gluon propagator that is strongly enhanced at infrared scales and recovers the one-loop behaviour at higher momentum. This model simplifies considerably the momentum dependence of the combined effective gluon propagator and quark-gluon vertex and assumes that the dominant momentum dependence is associated only with the gluon momentum. Such type of vertex that appears in the Dyson–Schwinger and the Bethe–Salpeter equations can be seen as a reinterpretation of the full vertex tensor structure, after rewriting its main components in a way that formally can be associated with the effective gluon propagator. Although the Maris-Tandy model is quite successful for phenomenology, it is not able to describe the full set of hadronic properties and fails to explain the mass splittings of the \(\rho \) and \(a_1\) parity partners, underestimates weak decay constants of heavy-light mesons and cannot reproduce simultaneously the mass spectrum and decay constants of radially excited vector mesons to point out some known limitations. For a more complete description see, for example, [30, 33, 34, 35] and references therein. Several authors have tried to improve the Maris-Tandy model either by studying its dependence on the various parameters, see e.g. [34], or by changing its functional dependence at low momenta, see e.g. [36], to achieve either a better description of Nature or good agreement with the results from lattice simulations.

The goal of the present work is to explore further the quark-gluon vertex in the non-perturbative regime from first principles calculations combining continuum methods with results coming from lattice simulations. Our approach follows the spirit of the calculation performed in [28] that solves the quark Dyson–Schwinger equation for the vertex. In [28] the quark-gluon vertex was described as a generalised Ball-Chiu vertex and single unknown form factor, function only of the gluon momentum, was considered. The current work goes beyond this approximation and includes the full set of longitudinal form factors that appear in the Ball-Chiu vertex. In this work we disregard any contribution due to the transverse form factors and consider the Landau gauge quark-gluon vertex, to profit from the recent high quality lattice data for the quark, the gluon and the ghost propagators. Moreover, our computation also incorporate the recent analysis of the full QCD lattice simulation for the quark-gluon vertex in the soft gluon limit that identifies an important contribution, for the infrared vertex, associated with the gluon propagator [29]. As in other studies, we rely on a Slavnov–Taylor identity to write the vertex longitudinal form factors as a function of the quark wave function, the running quark mass, the quark-ghost kernel form factors and the ghost propagator. The normalisation of the quark-ghost kernel form factors \(X_0\) (see below for definitions) derived in [17] for the soft gluon limit is also taken into account when solving the quark gap equation for the vertex. The normalisation of \(X_0\) played an important role in the analysis of the full QCD lattice data analysis for \(\lambda _1\), the form factor associated with the tree level tensor structure \(\gamma _\mu \), performed in [29] that identified an important contribution for \(\lambda _1\) coming from the gluon propagator.

Our solution for the quark-gluon vertex returns a \(X_0\) that deviates only slightly from the normalisation condition referred above. However, the longitudinal form factors describing the quark-gluon vertex are strongly enhanced in the infrared region. The enhancement of the four longitudinal form factors occurs for quark and gluon momentum below 2 GeV and can be traced back to ghost contribution introduced by the Slavnov–Taylor identity and the gluon dependence of the ansatz. At high momentum the form factors seem to approach their perturbative values. The matching with the perturbative tail is not perfect and this result can be understood partially due to the regularisation for the mathematical problem, i.e. the Tikhonov regularisation, and partially to the parametrisation of the vertex. Indeed, by calling the gluon propagator to describe the various form factors, the inversion of the Dyson–Schwinger equations is quite sensitive to the low momentum scales, where the gluon propagator is much larger, and less sensitive to the ultraviolet regime. In order to overcome this problem, we considered a relatively large cutoff in the inversion and in this way add information on the perturbative tail.

The computed quark-gluon vertex is a function of the angle between the quark four momentum *p* and the gluon four momentum *q* that, clearly, favours kinematical configurations where *p* and *q* of the order of 1 GeV or below. The enhancement occurs preferably at momenta of \(\sim \, \varLambda _{QCD}\). Furthermore, the vertex is enhanced when all momenta entering the vertex (see Fig. 1) tends to be parallel in pairs, solving in this way the compromise that the momenta are restricted to a region around \(\varLambda _{QCD}\). Within our solution for the quark-gluon vertex, the dominant form factors are associated with the tree level vertex \(\gamma _\mu \) and the operator \(2 \, p_\mu + q_\mu \). The higher rank tensor structures give sub-leading contributions to the vertex.

In the current work, the vertex is written using the Ball-Chiu construction. It is known that the Ball-Chiu vertex has kinematical singularities for the Landau gauge [37] that are associated to the transverse form factors (see definitions below). These singularities can be avoided by considering a different tensor basis for the full vertex as described, for example, in [37]. However, the singularities are not associated to the longitudinal form factors and our calculation only takes into account this class of form factors.

*ansatz*and perform a scaling analysis of the integral equations. In Sect. 5 we give the details of the lattice data used in the current work for the various propagators and on the functions that parametrise the lattice data. The kernels for the Euclidean space DSE are discussed in Sect. 6, together with the solutions for the vertex of the gap equation. The quark-gluon vertex form factors are reported in Sect. 7 for several kinematical configurations. Finally, on Sect. 8 we summarise and conclude.

## 2 The quark gap equation and the quark-gluon vertex

*g*is the strong coupling constant and \(t^a\) are the color matrices in the fundamental representation.

### 2.1 QCD symmetries and the quark-gluon vertex

*H*and \(\overline{H}\) are associated to the quark-ghost kernel. As discussed in [38], these functions can be parametrised in terms of four form factors as

*A*,

*B*, \(X_i\) and \(\overline{X}_i\). This is a particularly important point when modelling the vertex.

## 3 Decomposing the Dyson–Schwinger equation into its scalar and vector components

The Dyson–Schwinger equation for the quark propagator is written in (4), with the quark self-energy being given by (5). This equation can be projected into its scalar and vector components by taking appropriate traces.

## 4 The Dyson–Schwinger equations in Euclidean space

*q*limit it follows that

*q*

*q*analysis should take into account the logarithmic corrections coming from the gluon, the ghost and the quark propagators that for \(N_f = 2\) result in \(\gamma = \gamma _{glue} + \gamma _{ghost} + \gamma _{quark} = -137/116\). Then, assuming a large

*q*behaviour as in (51) times the log correction, the integration function at high momenta becomes

*q*for the vector component of the Dyson–Schwinger equations.

The dynamics of QCD generates infrared mass scales for the quark and gluon propagators, see Sects. 5.1 and 5.2, that eliminate possible infinities associated to the low momentum limit in the integral of the quark gap equation and the analysis of the infrared limit does not add any new constraints.

*a*and

*b*are constants, that in terms of \(Y_1\) and \(Y_3\) translates into

*q*limit gives \(X_1 \sim Y_1 (q^2)/q^2\) and regularises the ultraviolet behaviour in agreement with the discussion summarised in (51). Similarly, Eq. (55) also suggests

*q*momenta a \(X_3 \sim \tilde{X}_3 (q^2) / q^2\) and, in this way, the ultraviolet problems referred in (51) are solved. Furthermore, for large quark momentum the ansatz (56) and (57) give

*p*.

## 5 Preparing to solve the Euclidean Dyson–Schwinger equations

The computation of a solution of the Dyson–Schwinger equations requires parameterising either the quark-gluon vertex, if one aims to look at the quark propagator, or the quark propagator functions to extract information on the quark-gluon vertex. In both these cases, a complete description of the gluon and ghost propagators is assumed explicitly.

In the current work, we aim to solve the gap equation for the quark-gluon vertex and, therefore, the knowledge of the various propagators over all range of momenta appearing in the integral equation is required. This is achieved fitting the Landau gauge lattice propagators with model functions that are compatible with the results of 1-loop renormalisation group improved perturbation theory. In this way, it is ensured that the perturbative tails are taken into account properly in the parameterisation of the propagators. The parameterisations considered here are compared to those of [28] in “Appendix B”. As can be seen on Fig. 44, the differences between the two sets of curves are more quantitative than qualitative.

### 5.1 Landau gauge lattice gluon and ghost propagators

The lattice data for the Landau gauge gluon dressing function \(p^2 D (p^2)\), renormalised in the MOM-scheme at the mass scale \(\mu = 3\) GeV and the fit associated to Eq. (63) can be seen on the top part of Fig. 4.

### 5.2 Lattice quark propagator

For the quark propagator we consider the result of a \(N_f = 2\) full QCD simulation in the Landau gauge [27, 42] for \(\beta = 5.29\), \(\kappa = 0.13632\) and for a \(32^3 \times 64\) lattice. For this particular lattice setup, the corresponding bare quark mass is 8 MeV and the pion mass reads \(M_\pi = 295\) MeV.

*p*. In order to be compatible with perturbation theory, we identify the region of momenta where \(Z(p^2)\) is constant and, for momenta above this plateaux, we replace the lattice estimates of \(Z(p^2)\) by constant values, i.e. the higher value of the quark wave function belonging to the plateaux. The original lattice data and the ultraviolet corrected lattice data can be seen on top of Fig. 5. The UV corrected lattice data is then fitted to the rational function

The removal of the lattice artefacts for the running quark mass is more delicate when compared to the evaluation of the quark wave function lattice artefacts [24, 42, 43]. The lattice data published in [27, 42] and reported on Fig. 5 (bottom) was obtained using the so called hybrid corrections to reduce the lattice effects [24] . The hybrid method results in a smoother mass function when compared to the one obtained by applying the multiplicative corrections. The differences on the corrected running mass between the two methods occur for momenta above 1 GeV, with the multiplicative corrected running mass being larger than the corresponding hybrid estimation; see Appendix on [42]. The running mass provided by the two methods, corrected for the lattice artefacts, seems to converge to the same values at large momentum.

*p*is given in GeV and \(m_q(p^2)\) and \(M(p^2)\) are given in MeV.

## 6 Solving the Dyson–Schwinger equations

*q*momentum integration does not change the outcome reported on Figs. 6, 7 and 8 and the main difference being that the associated numerical values are considerably smaller.

The major contributions of the \({\mathscr {N}}^{(0)}_{B}\) and \({\mathscr {N}}^{(0)}_{A}\) kernels occur in a well defined momentum region where \(p \lesssim 1.5\) GeV and \(q \lesssim 1.5\) GeV. Further, for *p*, \(q \gtrsim 2\) GeV the kernels become marginal. The results for \({\mathscr {N}}^{(0)}_{B}\) and \({\mathscr {N}}^{(0)}_{A}\) reproduce the corresponding behaviour observed in [28]. It follows that the integration over momentum in Eqs. (40) and (41) associated to the kernels \({\mathscr {N}}^{(0)}_{B}\) and \({\mathscr {N}}^{(0)}_{A}\) kernels, that are coupled to \(X_0(q^2)\), is finite.

*q*. The requirement of a finite integration over

*q*demands that \(X_1\) and \(X_3\) should approach zero fast enough to compensate the increase with

*q*of these kernel functions; see the discussion of the kernels ultraviolet limit in Sect. 4. The ansatz (59)–(61) adds a multiplicative gluon propagator term that is just enough to regularize the high momentum associated to \({\mathscr {N}}^{(1)}_{B} (p, q)\), \({\mathscr {N}}^{(3)}_{B} (p, q)\) and \({\mathscr {N}}^{(3)}_{A} (p, q)\). Indeed if one takes into account the multiplicative gluon propagator contribution to the kernels, those who are divergent become well behaved. This can be seen on Figs. 9 and 10 where the kernels, now including the multiplicative gluon propagator term, are reported. The new versions of \({\mathscr {N}}^{(1)}_{B} (p, q)\), \({\mathscr {N}}^{(3)}_{B} (p, q)\) and \({\mathscr {N}}^{(3)}_{A} (p, q)\) mimic the pattern observed for \({\mathscr {N}}^{(0)}_{B}\), \({\mathscr {N}}^{(0)}_{A}\) and \({\mathscr {N}}^{(1)}_{A} (p, q)\) and, once more, their main contribution to the integral equations happens for \(p \lesssim 2\) GeV and \(q \lesssim 2\) GeV. The inclusion of the gluon propagator in the kernels makes the integration over

*q*finite.

### 6.1 One-loop dressed perturbation theory for \(X_0(q^2)\)

The four longitudinal quark-gluon form factors were parametrised in terms of the three quark-ghost kernel form factors \(X_0\), \(Y_1\) and \(Y_3\). However, the quark gap equation provides only two independent equations and, therefore, it is not possible to compute all the form factors at once for the full range of momenta.

A first look at the quark-ghost kernel form factors is possible if one computes \(X_0\) within one-loop dressed perturbation theory with a simplified version of a quark-ghost kernel where one sets \(Y_1 = Y_3 = 0\) and, then, solve the gap equation to estimate \(Y_1\) and \(Y_3\). The way the solutions of the Dyson–Schwinger equations for \(Y_1\) and \(Y_3\) are built also illustrates the numerical procedure used to solve the integral equations.

*k*. The introduction of a Gauss–Legendre quadrature reduces the integral Eq. (70) to a linear system of equations that was solved using the QR decomposition of the matrix appearing in the linear system.

A direct solution of \(B = {\mathscr {N}} X\) results in a meaningless result, with the components of *X* oscillating over very large values due to the presence of very small eigenvalues of the matrix \({\mathscr {N}}\), that translates the ill defined problem in hands. The linear system can be solved using the Tikhonov regularisation [50] that replaces the original linear system by a minimisation of the functional \(|| B - {\mathscr {N}} X ||^2 + \epsilon ||X||^2\), where \(\epsilon \) is a small parameter to be determined in the inversion. This functional favours solutions that solve approximately the linear system but whose norm is small. For real symmetric matrices, Tikhonov regularisation replaces the original system by its normal form \( {\mathscr {N}}^T B = ( {\mathscr {N}}^T {\mathscr {N}} + \epsilon )X\). Although in our case \({\mathscr {N}}\) is not a symmetric matrix, we will solve the system as given in its normal form.

The determination of the optimal \(\epsilon \) is done by solving \( {\mathscr {N}}^T B = ( {\mathscr {N}}^T {\mathscr {N}} + \epsilon )X\) for various values of \(\epsilon \) and look at how \(|| B - {\mathscr {N}} X ||^2\) and \(||X||^2\) behave as a function of the regularisation parameter \(\epsilon \). The outcome of the inversions for different \(\epsilon \) can be seen on Fig. 13. For smaller values of \(\epsilon \), i.e. when one is closer to the original ill defined problem, the corresponding solution of the linear system results on \(Y_1\) and \(Y_3\) with larger norms. The larger values of the regularisation parameter \(\epsilon \) are associated to solutions of the modified linear system with smaller \(Y_1\) and \(Y_3\) norms. The optimal value of \(\epsilon \) is given by the solution whose residuum, i.e. the difference between the lhs and the rhs of the original equations, is among the smallest values just before the norms of \(Y_1\) and \(Y_3\) start to grow but without changing the residuum. On the above figure we point out three solutions in the region where \(\epsilon \) takes approximately its optimum value.

have the same \(m^{b.m.}\) and \(Z_2\) as the previous ones and \(||X_0 - 1||^2 = 0.45283\). In both cases, the norms of \(X_0\), \(Y_1\) and \(Y_3\) are the norms of the corresponding part of the vector that appear in the linear system .

The quark-ghost kernel form factors \(Y_1\) and \(Y_3\) computed for the various \(\epsilon \) associated to the solutions I (TL) – III (TL) and I (Enh) – III (Enh) can be seen on Fig 16. For \(Y_1\) the outcome of resolving the integral equations using either the tree level or the enhanced ghost-gluon vertex result on essentially the same function. Further, the various solutions provide essentially the same \(Y_1(p^2)\), with the exception of III (TL) and III (Enh) that return a suppressed form factor relative to the other solutions. For \(Y_3\) the situation is similar, with the form factor associated to the solutions I (TL) and I (Enh) being enhanced at momentum above 2 GeV. Looking at Fig. 15, one can observe that solutions II (TL) and II (Enh) are those with smaller relative errors over the full range of momentum considered. So, from now on we will take these solutions as our best solutions associated to the perturbative \(X_0\) form factor. Note that the scalar equation is solved with a relative error \(\lesssim \) 8% and the vector equation is solved with a relative error \(\lesssim \) 6%.

The quark-ghost kernel form factors \(X_0\), \(X_1\) and \(X_3\) were also computed in [23], see their Fig. 4, combining the quark gap equation with a one-loop dressed perturbation theory for the quark-ghost kernel. The comparison between the results of the two calculations is not straightforward. Indeed, if in our calculation \(X_0\) is assumed to be a function only of the gluon momentum, in [23] the authors take into account its full momentum dependence and evaluate how it changes with the quark momentum, the gluon momentum and the angle between these two momenta. There calculation results on a \(X_0\) that is always close to its tree level value \(X_0 = 1\) and whose values are in the range [1, 1.12]. A qualitative comparison with the here reported form factor, shows that, from the point of view of the dynamical range of values, our \(X_0\) computed using the enhanced ghost-gluon vertex is closer to that reported in [23]. In both cases \(X_0\) is always close to its tree level value and differs from unit by, at most, 10%.

Our estimations for \(X_1\) and \(X_2\) for the solutions referred previously and for the particular kinematics \(p = 0\), sometimes called the soft quark limit, can be seen on Fig. 17. The first comment about these form factors is that they seem to be independent of the ghost-gluon vertex and, indeed, the form factors associated to the solution using a tree level ghost-gluon vertex, named (TL) in the figure, are essentially indistinguishable from those computed using the enhanced ghost-gluon vertex, named (Enh) in the figure. The quark-ghost kernels form factors \(X_1\) and \(X_3\) differ from there perturbative values at low momenta, i.e. for \(q \lesssim 1\) GeV for \(X_1\) and for \(q \lesssim 2\) GeV for \(X_3\). According to [23], these two form factors increase (in absolute value) for momenta below \(\sim 1\) GeV, a result that is in good qualitative agreement with our estimations. From their Fig. 4, it is not clear if \(X_3 \ne 0\) extends over a wider range of momenta, when compared to \(X_1\). Our calculation returns a \(X_3\) that differ from zero on larger range of momenta, when compared with \(X_1\). Furthermore, the \(X_1\) and \(X_3\) computed in [23] are monotonic increase functions (absolute values) when the zero momentum limit is approached. The form factors reported on Fig. 17 show a pattern of maxima, with \(X_1\) having a single maximum for \(q \sim 350\) MeV and \(X_3\) showing several maxima (in absolute value) at momenta \(q \sim 250\) MeV, \(\sim 600\) MeV and \(\sim 1\) GeV. Note that the zero crossing for \(X_3\) occur for momenta that are of the same order of magnitude of \(\varLambda _{QCD}\), the gluon mass (or twice \(\varLambda _{QCD}\)) and the usual considered as a non-perturbative mass scale (1 GeV). It is not obvious why the zero crossing of \(X_3\) occur for such mass scales and why the crossing is not seem for \(X_1\). If the form factors reported on Fig. 4 of [23] never cross the zero value, that is not the case of the form factors represented on Fig 17. \(X_1\) shows various zeros that, curiously, seem to disappear for the solution with the smaller norm. On the other hand, \(X_3\) cross zero for \(q \approx 363\) MeV and 835 MeV for solutions II and III. This is a major difference between the two sets of solutions under discussion. Another important difference being the dynamical range of values. Our \(X_1\) is within the range of values [0, 3.5] \(\hbox {GeV}^{-1}\), while the same form factor computed in [23] is within [0, 0.2] \(\hbox {GeV}^{-1}\) which represents a factor of \(\sim 20\) smaller than our estimation. However, the two calculations report a \(|X_3|\) within the range [0, 0.5] \(\hbox {GeV}^{-2}\). Our result overestimates \(X_1\) relative to [23] but returns a \(|X_3|\) within the same dynamical range of values. Another major difference being that the maxima of the form factors does not occur at vanishing momenta as in [23] but at finite and small momenta, \(\sim \varLambda _{QCD}\) for \(X_1\) and \(\sim 2 \, \varLambda _{QCD}\) for the absolute maxima of \(|X_3|\).

### 6.2 Solving the Dyson–Schwinger equations

*X*contains the form factor \(X_0\) defined in all the set of Gauss–Legendre points used in the integration over the loop momenta. The remaining components of the large

*X*vector are the form factors \(Y_1\) and \(Y_3\) defined at the lower first half set of Gauss–Legendre points used in the integration over the momentum. This means that the solution of the linear system (78) returns \(X_0(q^2)\) for \(q \in [ 0 , \, \varLambda ]\) and \(Y_1(q^2)\) and \(Y_3(q^2)\) for \(q \in [ 0 , \, \varLambda / 2]\). For \(Y_1\) and \(Y_3\) and for \(p > \varLambda /2\) the form factors will be assumed to vanish. In order to fulfil the boundary conditions for \(X_0 (q^2)\) we write \(X_0 (q^2) = 1 + \tilde{X}_0(q^2)\) and solve the linear system for \(\tilde{X}_0(q^2)\), rebuilding \(X_0 (q^2)\) at the end. The resulting linear system is then regularised using the Tikhonov regularisation and the corresponding \({\mathscr {N}}^T B = ( {\mathscr {N}}^T {\mathscr {N}} + \epsilon )X\) linear system is solved for various \(\epsilon \). The choice of the optimal regularisation parameter \(\epsilon \) follows the criteria discussed in Sect. 6.1. We have checked that by interchanging the roles of \(X_0\), \(Y_1\) and \(Y_3\) when building the large linear system the solutions are unchanged; more on that below. The differences only occur for those functions calculated only for \(q \in [ 0 , \, \varLambda / 2]\), compared to the version of the linear system were they are computed in the range \(q \in [ 0 , \, \varLambda ]\). In the first case, i.e. for the solutions computed only for \(q \in [ 0 , \, \varLambda / 2]\), there appears a discontinuity at \(q = \varLambda /2\) (recall that the form factors are set to zero for momenta above \(\varLambda /2\)) but for smaller

*q*the form factors of all versions of the linear system are indistinguishable.

On Fig. 18 the residuum squared of the scalar (top) and vector (bottom) components of the gap equation are shown against the norm of various form factors. Smaller values of \(\epsilon \) are associated to solutions with larger norms and appear at the right side of the plots, while larger values of \(\epsilon \) are associated to solutions with smaller norms that show up on the left side of the plots. As shown on the figure, the residuum for both equations has a stronger dependence on \(\epsilon \) and can take quite small values. For the solutions featured in the plot, the smallest residuum squared reaches values of the order of \(10^{-5}\) for the scalar equation and \(10^{-4}\) for its vector component. Similar values for the minimum residuum were also observed for the solutions computed using the perturbative estimation of \(X_0\).

The relative errors for the solutions I–IV of the regularised linear system are show on Fig. 19. In general the solution for the vector component of the equation is satisfactory, with the scalar component of the equation being more demanding and not all of the solutions I–IV resolve the scalar part of the gap equation with a relative error below 10%. Only solutions III and IV resolve the DSE equations with a relative error below 8%. In particular for these solutions the value of \(||X_0 - 1||\) is of the order of \(10^{-2}\) suggesting that the non-perturbative solution prefers having a \(X_0 \simeq 1\) and, in this sense and for this form factor, are close to the result from perturbation theory discussed in Sect. 6.1. The observed growth of the relative error for \(p \gtrsim 10\) GeV is probably related also to the missing components of \(Y_1\) and \(Y_3\) which are set to zero for this range of momenta.

*p*just above 1 GeV, and approaching its tree level value at high momentum from below. The differences between the non-perturbative \(X_0\) and its tree level value for \(p \gtrsim 10\) GeV are rather small.

*X*. For the so-called \(X_0 \, X_1 \, X_3\) the extended vector included \(X_0\) over the full set of Gauss–Legendre points with \(Y_1\) and \(Y_3\) being obtained only in the range \(p \in [ 0 \, , \, \varLambda /2]\). For the so-called \(X_1 \, X_0 \, X_3\) the extended vector included \(Y_1\) over the full set of Gauss–Legendre points with \(X_0\) and \(Y_3\) being obtained only in the range \(p \in [ 0 \, , \, \varLambda /2]\). For the so-called \(X_3 \, X_0 \, X_1\) the extended vector included \(Y_3\) over the full set of Gauss–Legendre points with \(X_0\) and \(Y_1\) being obtained only in the range \(p \in [ 0 \, , \, \varLambda /2]\). We call the readers attention to the stability of the solution of the various linear systems. Furthermore, the comparison of Fig. 16 from Sect. 6.1 and Fig. 23 show quite similar \(Y_1\) and \(Y_3\) suggesting, once again, that \(X_0\) almost does not deviates from its tree level value.

### 6.3 Solving the DSE for \(X_0 = 1\)

The form factors \(Y_1(p^2)\) and \(Y_3(p^2)\) associated to the solutions I–IV are reported on Fig. 26 and reproduce the same patterns as the solutions computed in the previous sections.

### 6.4 Full form factors and comparison of solutions

*H*and \(\overline{H}\), see Eq. (15), should be multiplied by the gluon propagator at the proper kinematical configuration. Herein, we aim to compare the various solutions found in previous section and, in this way, provide an estimation of the systematics associated to our ansatz, and also to provide the form of the full functions appearing on

*H*and \(\overline{H}\). Looking at the relative errors on the Dyson–Schwinger equations and at the convergence of the form factors at higher momenta, the comparison will be done using Sol. II computed using the perturbative \(X_0\) and the tree level ghost-gluon vertex, Sol. III computed when the gap equation is solved for the full set of form factors and Sol. IV when the gap equation is solved for \(X_0 = 1\).

The form factor \(Y_1(p^2)\) can be seen on Fig. 28 for all the solutions. Note that all solutions reproduce essentially the same function of the gluon momentum, with \(Y_1(p^2)\) being small for \(p\gtrsim 1.5\) GeV and showing a sharp peak at \(p \simeq 400\) MeV. \(Y_1(p^2)\) is positive defined except for a small range of momenta \(p \in [ 0.75 \, , \, 1.4]\) GeV where it takes small negative values.

The form factor \(Y_3(p^2)\) can be seen on Fig. 29 for all the solutions. Surprisingly, \(Y_3(p^2)\) seems to have a relative large tail that appears in all the solutions. Up to momenta \(p \simeq 3\) GeV the solutions reproduce essentially the same function. However for \(p \simeq 3\) GeV the solution associated to \(X_0 = 1\) is enhanced relative to all the others, with the solutions associated to the one-loop perturbative \(X_0\) being slightly enhanced relative to the non-perturbative solution obtained from inverting the gap equation. \(Y_3(p^2)\) shows a maxima at \(p \simeq 200\) MeV, an absolute maxima at \(p \simeq 1.4\) GeV and an absolute minima at \(p \simeq 650\) MeV. This form factor is positive defined at infrared momenta \(p \lesssim 350\) MeV and the high momenta \(p \gtrsim 900\) MeV taking negative values in \(p \in [ 0.35 \, , \, 0.9]\) GeV.

In summary, Figs. 26, 27, 28 and 29 resume the computations of the quark-ghost kernel form factors performed so far.

### 6.5 Tunning \(\alpha _s\)

The results for the relative errors on the scalar and vector components of the Dyson–Schwinger equation seen on Figs. 15, 19, 25 show a relative error that for \(p \gtrsim 10\) GeV grow with *p* and take its maximum value \(\sim 10\)% at the cutoff. This can be viewed in many ways and one of them being that our choice for the strong coupling constant is not the best one. In our approach we mix quenched lattice results with dynamical simulations and, in order to be able to solve the gap equation for the quark-ghost kernel, the renormalization constant \(Z_1\), see Eq. (5), is set to identity.^{1} Although the original integral equation is linear on the form factors \(X_0\), \(Y_1\) and \(Y_3\), the regularized system that is solved introduces an extra parameter that needs to be fixed in the way described above and, therefore, changing the strong coupling constant changes the balance between the regularizating parameter \(\epsilon \) and the various form factors, allowing for adjustments on the solutions. Therefore, the relative errors on the integral equations can be adjusted by changing the strong coupling constant.

## 7 The Quark-Gluon vertex form factors

*p*,

*q*and of the angle between the quark and gluon momenta. The angular dependence appears associated to the scalar products (

*pq*) and also on the argument of the gluon propagator \(D\left( (p^2 + k^2)/2 \right) \).

*ansatz*for the vertex takes into account the dependence between the angle of the incoming quark momentum and the incoming gluon momentum.

The overall picture of the various form factors when the angle between the incoming quark momentum *p* and the incoming gluon momentum *q* is \(\theta = 0\) can be seen on Fig. 36. On Fig. 37 the \(\lambda _1\) to \(\lambda _4\) are given for a \(\theta = 2 \pi /3\). The form factors \(\lambda _1\) to \(\lambda _4\) are finite for all *p* and *q* and approach asymptotically their perturbative values. Further, for our definition of the operators \(L^{(1)}_\mu \) – \(L^{(4)}_\mu \), see Eq. (11) for their definition in Minkowski space, the corresponding form factors are essentially positive defined. The exception being \(\lambda _4\) that takes both positive and negative values and whose maximum absolute value is negative and appears for small *p* and *q*. The relative magnitude of the \(\lambda _i\) suggest that the quark-gluon vertex is essentially saturated by \(\lambda _1\) and \(\lambda _3\), with \(\lambda _2\) and \(\lambda _4\) playing a minor role, i.e. the tensor structures of the longitudinal part of the vertex seem to be sub-leading; see also the discussion for the soft quark limit, defined by a vanishing quark momentum, and the symmetric limit below.

Our result differs significantly from the perturbative estimation of the form factors [5], where all the strength appears associated to \(\lambda _1\). For example, for the kinematical configuration defined by \(p^2 = (p-q)^2\) at vanishing *p* they have \(\lambda _1 \approx 1.1\) and \(\lambda _2 \approx 0.12\) \(\hbox {GeV}^{-2}\) and \(\lambda _3 \approx 0.18\) \(\hbox {GeV}^{-1}\) for a current mass \(m_q = 115\) MeV, a renormalisation scale \(\mu = 2\) GeV and for \(\alpha _s = 0.118\). Of course, one should look to the relative values of the various \(\lambda \)’s and not to their absolute values. For the comparison of the contributions from the various form factors one can use the non-perturbative momentum scale of 1 GeV to build dimensionless quantities. Then, as seen on Figs. 36 and 37 the scales for \(\lambda _1\) and \(\lambda _3\) are similar, while the maximum of \(\lambda _2\) is about 10% relative to the maxima of \(\lambda _1\) and \(\lambda _3\) and the maximum for \(\lambda _4\) is about half of that for \(\lambda _2\).

The comparison of our results with those reported in [17, 22, 23] is difficult to perform but in these works \(\lambda _1\) clearly dominates. On [17] \(\lambda _2\) reaches at most 16% of the maximum value of \(\lambda _1\), while \(\lambda _3\) seems to have the possibility of taking large values. On [22, 23], \(\lambda _2\) and \(\lambda _3\) take, at most, a numerical value that is about 23% of the maxima of \(\lambda _1\), with \(\lambda _4\) being essentially negligible. Our solution shows a vertex dominated by \(\lambda _1\) and \(\lambda _3\) with these form factors reaching numerical values of the same order of magnitude – see also Fig. 42.

As seen on Figs. 36 and 37 the quark-gluon form factors are significantly enhanced for low values of *p* and *q*. The momentum region where one observes the enhancement of the \(\lambda _1\) to \(\lambda _4\) happens for \(p \lesssim 1 \) GeV and \(q \lesssim 1\) GeV, with its maximum values showing up for \(p \approx q \approx \varLambda _{QCD}\) – see, also, the discussion below on the angular dependence.

The infrared enhancement of \(\lambda _1\) to \(\lambda _4\) with the gluon momentum is a direct consequence of using the Slavnov–Taylor identity (13) to rewrite the form factor. Indeed, as can be seen on Eqs. (80)–(83), all the form factors have, as a global factor, the ghost dressing function \(F(q^2)\). The ghost dressing function is enhanced, roughly by a factor of three, in the infrared, see Fig. 4, implying the increase of the \(\lambda _i\) as \(q = 0\) is approached.

*ansatz*that relies on the analysis of the soft gluon limit of the Landau gauge lattice data for \(\lambda _1\) performed in [29]. Indeed, this work identified a dependence of \(\lambda _1\) on the gluon propagator that was incorporated in the

*ansatz*, making the quark-ghost kernel form factors \(X_1\) and \(X_3\) proportional to \(D ( (p^2 + (p-q)^2 / 2)\). This term is crucial to have well behaved kernels in the integral equations, i.e. to ensure that the Dyson–Schwinger equations are finite, and it introduces an additional dependence on the angle between the quark and the gluon momenta. The gluon propagator is a decreasing function of its argument and, therefore, for a given

*q*and angle between the quark and gluon momenta, the terms proportional to \(X_1\) and \(X_3\) increase as

*p*decreases. This explains, in part, the observed enhanced of the quark-gluon form factors together with the increase of the ghost dressing function.

The dependence of the quark-gluon form factors in the angle between *p* and *q* can be seen on Figs. 38, 39, 40 and 41. These figures also provide a clear picture of the maxima of the various form factors as functions of the gluon momenta. For \(\lambda _1\) and \(\lambda _3\) the maxima are for \(q \approx 300\) MeV, while for \(\lambda _2\) the maximum is at \(q \approx 600\) MeV. \(\lambda _4\) seems to be a more complicated function of *p*, *q* and \(\theta \). Indeed, this later form factor shows various maxima of the same order of magnitude for different *p*, *q* and \(\theta \) values.

All the form factors appear to be monotonous decreasing functions of the angle between the incoming quark and incoming gluon momenta \(\theta \). If the pattern of the *q* dependence of \(\lambda _1\), \(\lambda _2\) and \(\lambda _3\) seems to be independent of \(\theta \), \(\lambda _4\) seems to reverse is behaviour relative to the \(q-axis\) for \(\theta \gtrsim \pi /3\). Clearly, the maximum values for all the form factors occurs for \(\theta = 0\), i.e. the quark-gluon vertex favours the kinematical configurations with small values of *p* and *q* and also of the angle between the quark and gluon momentum.^{2} It follows that the quark-gluon vertex favours small quark and gluon momenta and parallel four-vectors *p* and *q*.

^{3}Clearly, our ansatz underestimates \(\lambda _1\) in the infrared region. As discussed in [29], in the soft gluon limit

*p*is the quark incoming momenta, and in our notation

*q*in the Dyson–Schwinger equations, these form factor are multiplied by \(q^3\) that, possibly, prevent the inversion to resolve correctly \(Y_1(q^2)\) and \(Y_3(q^2)\) in the deep infrared region. If in the calculation of the soft gluon limit one assumes that \(Y_1(0)\) deviates from zero by a small quantity, the agreement with the lattice data is considerably improved both in the infrared and in the ultraviolet. This is represented by the two full curves in colour of Fig. 43 where \(Y_1(0)\) is set to a small value. The colour curves suggest a \(Y_1(0) \sim 0.05\)–0.07 GeV. Further, the agreement in the ultraviolet region can also be improved if \(Y_3(0) \) assumes small and positive values; recall that \(Y_3(q^2)\) approaches zero from the above when \(q^2 \rightarrow 0\) as can be seen on Fig. 35.

## 8 Summary and conclusions

In this work we investigated the non-perturbative regime of the Landau gauge quark-gluon vertex (QGV), taking into account only its longitudinal components, and relying on lattice results for the quark, gluon and ghost propagators, together with continuum exact relations, namely a Slavnov–Taylor identity and the quark propagator Dyson–Schwinger equation. Furthermore, we incorporate the exact normalisation condition for the quark-ghost kernel form factor \(X_0\) derived in [17]. In addition, we take into account an empirical relation that links the gluon propagator and the soft gluon limit of the form factor \(\lambda _1\) checked against full QCD lattice simulations [29]. The full set of the quark-ghost kernel tensor structures are taken into account to build an *ansatz* for the longitudinal quark-gluon vertex that is a function of both the incoming quark *p* and gluon *q* momenta, and the angle between *p* and *q*.

The quark-ghost kernel requires four scalar form factors \(X_0\), \(X_1\), \(X_2\), \(X_3\) [38]. For the construction of the quark-ghost kernel a perfect symmetry between incoming and outgoing quark momentum is assumed, which simplified the description of the QGV in terms of \(X_0\), \(X_1 = X_2\) and \(X_3\). Charge conjugation demands that for the soft gluon limit, defined by \(q = 0\), \(\lambda _4 = 0\) and our construction implements such constraint. Noteworthy to mention that our *ansatz* goes beyond the Ball-Chiu type of vertex [6] and includes it as a particular case, when \(X_1 = X_3 = 0\) and \(X_0 = 1\).

The Dyson–Schwinger equations are solved for the quark-gluon vertex that are written in terms of the unknown functions \(X_0\), \(X_1\) and \(X_3\). From the point of view of the quark-ghost kernel form factors, these are linear integral equations. The corresponding mathematical problem is ill defined and needs to be regularised in order to obtain a meaningful solution. The original integral equations for the scalar and vector components of the quark gap equation are transformed into a set of linear system using Gauss–Legendre quadratures to perform the integrations and after doing the angular integration. In our approach we rely on the Tikhonov linear regularisation that is equivalent to minimize \(|| B - {\mathscr {N}} X ||^2 + \epsilon ||X||^2\). The solutions are found numerically after writing the regularised linear system in its normal form. The small parameter \(\epsilon \) is set by looking at the balance between the associated error on the Dyson–Schwinger equations, i.e. the difference between the l.h.s and the r.h.s. \(|| B - {\mathscr {N}} X ||^2\), and the norm of the corresponding quark-ghost form factors, i.e. \(||X||^2\), for each solution of the regularised linear system.

The resulting quark-gluon vertex form factors \(\lambda _1\)–\(\lambda _4\) show a strong enhancement in the infrared region and deviate significantly from their tree level results for quark and gluon momenta below \(\sim 2\) GeV. At high momentum the form factors approach their perturbative values. In what concerns the gluon momentum, the observed infrared enhancement for the QGV form factors can be traced back to the multiplicative contribution of the ghost dressing function introduced through the Slavnov–Taylor identity. Recall that the gluon dressing function peaks at \(q = 0\) and, therefore, favours that the incoming and outgoing quark momentum to be parallel. On the other hand, the infrared enhancement associated to the quark momentum is linked to the gluon dependence that was observed on the analysis of the soft gluon limit of the QGV and, clearly, favours small quark momentum \(p \sim 0\) and also *p* parallel to *q*; see Eqs. (60), (61) and (80)–(83) and, in particular, the argument appearing on the gluon propagator term. The maxima of the computed form factors are essentially at the maxima of \(X_0\), \(X_1\) and \(X_3\) and they appear for momenta \(p, \, q \sim \varLambda _{QCD}\), which again seems to set the appropriate non-perturbative momentum scale. Recall that the momentum scale comes from the use of lattice data for the propagators. Further, we find that the quark-gluon vertex is dominated by the form factors associated to the tree level vertex \(\gamma _\mu \) and to \(2 \, p_\mu + q_\mu \), with the higher rank tensor structures giving small contributions. Overall, our findings are in qualitative agreement with previous works both with phenomenological approaches, as in the case of the effective vertex introduced in [32], and those based on first principles *ab initio* continuum methods, see e.g. [23] and references therein.

The high momentum behaviour of the quark-gluon vertex form factors reproduces their perturbative values. However, the matching between the computed form factors and their perturbative tail is not yet implemented. In addition, we verified that for the soft gluon limit, \(\lambda _1\) is not able to reproduce quantitatively the lattice data from full QCD simulations, apart the qualitative momentum behaviour. This can be traced back to the poor resolution of the kernel in the deep infrared region, due to the \(q^3\) factor coming from the momentum integration. As we have verified, a small tuning of \(X_1\) and \(X_3\) at \(q = 0\) is enough to reproduce the soft gluon limit lattice data within the present framework. This two challenging problems, together with inclusion of the transverse part of the vertex, call for an improvement of the approach devised herein and are to be tackled in a future work. Despite of that, we expect that the present results can help understanding the non-perturbative dynamics of quarks and gluons in the infrared region and that can motivate further applications to the study of hadron phenomenology based on quantum field theoretical approaches as those using Bethe–Salpeter and/or Faddeev equations – see e.g. [51] and references therein.

## Footnotes

- 1.
It can also be viewed as been included in the definition of the various form factors.

- 2.
For example, for \(p = 0.5\) GeV \(\lambda _1\) and \(\lambda _3\) there is an enhancement of a factor of \(\sim 2.7\) and \(\sim 2.0\), respectively, between the maxima values for \(\theta = 0\) relative to \(\theta = \pi \). For \(\lambda _2\) and \(\lambda _4\) this enhancement is \(\sim 3.5\) and \(\sim 3.2\). The corresponding factors for an incoming quark momentum \(p = 1\) GeV are \(\sim 1.5\) for \(\lambda _1\), \(\sim 4.4\) for \(\lambda _2\), \(\sim 2.3\) for \(\lambda _3\) and a suppression by a factor of \(\sim 0.6\) for \(\lambda _4\). Also, for \(\lambda _1\) and \(\lambda _3\) the maxima at \(\theta = \pi /2\) is about half of the maxima at \(\theta = 0\).

- 3.
The normalisation essentially removes the

*F*(0) that can appear in the expression for \(\lambda _1\); see Eq. (80).

## Notes

### Acknowledgements

This work was partly supported by the Fundação de Amparo à Pesquisa do Estado de São Paulo [FAPESP Grant No. 17/05660-0], Conselho Nacional de Desenvolvimento Científico e Tecnológico [CNPq Grant 308486/2015-3] and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) of Brazil. This work is a part of the project INCT-FNA Proc. No. 464898/2014-5. O. Oliveira acknowledge support from CAPES process 88887.156738/2017-00 and FAPESP Grant Number 2017/01142-4. JPCBM acknowledges CNPq, contracts 401322/2014-9 and 308025/2015-6. This work was part of the project FAPESP Temático, No. 2017/05660-0.

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