Nonperturbative structure of the ghost-gluon kernel

The ghost-gluon scattering kernel is a special correlation function that is intimately connected with two fundamental vertices of the gauge sector of QCD: the ghost-gluon vertex, which may be obtained from it through suitable contraction, and the three-gluon vertex, whose Slavnov-Taylor identity contains that kernel as one of its main ingredients. In this work we present a detailed nonperturbative study of the five form factors comprising it, using as starting point the `one-loop dressed' approximation of the dynamical equations governing their evolution. The analysis is carried out for arbitrary Euclidean momenta, and makes extensive use of the gluon propagator and the ghost dressing function, whose infrared behavior has been firmly established from a multitude of continuum studies and large-volume lattice simulations. In addition, special Ans\"atze are employed for the vertices entering in the relevant equations, and their impact on the results is scrutinized in detail. Quite interestingly, the veracity of the approximations employed may be quantitatively tested by appealing to an exact relation, which fixes the value of a special combination of the form factors under construction. The results obtained furnish the two form factors of the ghost-gluon vertex for arbitrary momenta, and, more importantly, pave the way towards the nonperturbative generalization of the Ball-Chiu construction for the longitudinal part of the three-gluon vertex.


INTRODUCTION
The nonperturbative behaviour of the fundamental Green's functions of QCD, such as propagators and vertices, has received considerable attention in recent years , and is believed to be essential for acquiring a deeper understanding of the strong interactions. In this particular quest, the combined efforts between various continuum approaches [1,6,7,9,27,46], and large-volume lattice simulations [51][52][53][54][55][56][57][58][59][60][61] have furnished a firm control on the infrared structure of the two-point sector of the theory (gluon, ghost, and quark propagators).
The case of the three-point functions (vertices) represents currently a major challenge, because, while their knowledge is considered to be crucial for both theory and phenomenology, their first-principle determination by means of conventional approaches is technically rather involved. In particular, such vertices possess, in general, rich tensorial structures, and their form factors contain three independent momenta. In order to determine the momentum dependence of vertex form factors, one may perform lattice simulations [62][63][64][65][66][67][68][69][70] or resort to continuum methods such as Schwinger-Dyson equations (SDEs) [13,30,32,33,49,[71][72][73][74][75] or the functional renormalization group [76][77][78]. Within these latter formalisms, the dynamical equations governing the momentum evolution of the vertices are derived and solved, under a variety of simplifying assumptions that reduce the inherent complexity of these calculations.
In a series of recent works [25,31,39,41,79,80], the aforementioned approaches have been complemented by an alternative procedure, which exploits the Slavnov-Taylor identities (STIs) satisfied by a given vertex, and constitutes a modern version of the so-called "gauge technique" [81][82][83][84]. The main upshot of this method is to determine the non-transverse part of the vertex 1 , in terms of the quantities that enter in the STIs, such as two-point functions and the so-called "ghost scattering kernels". These kernels correspond to the Fourier transforms of composite operators, where a ghost field and a quark or a gluon are defined at the same space-time point. In the case of the quark-gluon vertex considered in the recent literature, the quantity in question is the "ghost-quark" kernel; its form factors have been reconstructed from the corresponding SDE in [31,41], and certain special kinematic configurations have been computed in [25,79].
In the present work we turn our attention to the ghost-gluon kernel, to be denoted by H abc νµ (q, p, r) = −gf abc H νµ (q, p, r). The main objective is to compute from an appropriate SDE [see Fig. 1] the five form factors comprising this quantity, to be denoted by A i (q, p, r) (i = 1, ..., 5), for arbitrary Euclidean values of the momenta.
The interest in H νµ and its form factors is mainly related with the two fundamental Yang-Mills vertices shown in Fig. 2 [85]. First, as was shown in the classic work of Ball and Chiu (BC) [86], the "longitudinal" part of the three-gluon vertex, Γ αµν (q, r, p), may be fully reconstructed from the set of STIs that it satisfies [see Eq. (2.4)]. The ingredients entering in the BC "solution" are the gluon propagator, the ghost dressing function, and three of the form factors of H νµ (q, p, r). Thus, in order to obtain reliable information on the infrared behaviour of Γ αµν (q, r, p) by means of this method, the nonperturbative structure of the ghost-gluon kernel must be firmly established. Second, by virtue of an exact relation [see Eq. (2. 3)], the ghost-gluon vertex, Γ µ (q, p, r), which constitutes an important ingredient for a variety of SDE studies, is completely determined from the contraction of H νµ (q, p, r) by q ν . Thus, knowledge of the A i (q, p, r) furnishes both form factors of Γ µ (q, p, r) [87].
The methodology used for the computation of the A i (q, p, r) may be described as follows. The diagrammatic definition of H νµ (q, p, r) shown in Fig. 1 involves the one-particle irreducible A µ A ρc c kernel (grey ellipse), whose skeleton expansion will be approximated by the "one-loop dressed" diagrams, depicted in Fig. 3; the basic quantities entering at this level are the gluon and ghost propagators, and the fully dressed vertices Γ αµν and Γ µ . The individual form factors of H νµ may then be isolated from the resulting equations by means of an appropriate set of projection operators. In the final numerical treatment we use the results of large-volume lattice simulations as input for the propagators, while for the vertices we resort to certain simplified Ansätze.
We next list the main highlights of our analysis: (i) we determine the form factors A i for general values of the Euclidean momenta, presenting the results in 3-D plots, where q 2 and p 2 will be varied, for fixed values of the angle θ between them; (ii) the nonperturbative results obtained for A i are compared with their one-loop counterparts in three special kinematic limits; (iii) with the help of a constraint imposed by the STI [see Eqs. (2.9) and (2.10)], we quantify the accuracy and veracity of our truncation scheme; (iv) as a direct application, the various A i are fed into the Euclidean version of Eq. (2.8), giving rise to both form factors of the ghost-gluon vertex, for arbitrary momenta.
The article is organized as follows. In section II we introduce the notation and set up the relevant theoretical framework. In section III, we discuss the truncation scheme employed and we define the set of projectors necessary for the derivation of the dynamical equations governing the form factors A i . In section IV we present the inputs and the additional approximations necessary for the numerical calculation of the A i . Then, in section V we present the numerical solution of the A i for general Euclidean momenta, and compare them with the one-loop results for some special kinematic limits. Next, in section VI we discuss how the constraint imposed by the STI may help us optimize the quality of the inputs used for the computation of the A i . In section VII we construct the two form factors of the ghost-gluon vertex, carry out a comparison with the results of various approaches in the literature, and study their impact on the SDE of the ghost propagator. In section VIII we present our discussion and conclusions. Finally, in two Appendices we present the one-loop results for the A i in some special kinematic limits, and certain lengthy expressions appearing in the derivation of the A i .

II. THEORETICAL BACKGROUND
In this section we introduce the basic concepts and ingredients necessary for the study of H νµ , and elucidate on its connection with the ghost-gluon and three-gluon vertices. In addition, we introduce a particular relation, which is a direct consequence of the STI that H νµ satisfies [86,88], and provides a nontrivial constraint on a combination of its form factors. We emphasize that throughout this article we work in the Landau gauge, where the gluon propagator ∆ ab µν (q) = δ ab ∆ µν (q) assumes the completely transverse form, The ghost-gluon scattering kernel H abc νµ (q, p, r) = −gf abc H νµ (q, p, r) is diagrammatically depicted in Fig. 1. The most general tensorial decomposition of H νµ (q, p, r) is given by [86,89] H νµ (q, p, r) = A 1 g µν + A 2 q µ q ν + A 3 r µ r ν + A 4 q µ r ν + A 5 r µ q ν , (2.2) where the momentum dependence, A i ≡ A i (q, p, r), has been suppressed for compactness.
Notice that, at tree-level, H νµ (q, p, r) = g νµ , so that the form factors assume the values A As mentioned in the Introduction, our interest in the dynamics of H νµ stems mainly from its connection to two of the most fundamental Yang-Mills vertices [85], namely the ghost-gluon vertex, Γ abc µ (q, p, r) = −gf abc Γ µ (q, p, r), and the three-gluon vertex, Γ abc αµν (q, r, p) = gf abc Γ αµν (q, r, p), where g denotes the gauge coupling, and q + r + p = 0; both vertices are shown diagrammatically in Fig. 2. In particular, H νµ and the aforementioned vertices are related by the followings STIs, and where q 2 J(q) denotes the "kinetic term" of the gluon propagator which is defined as where m 2 (q) is the dynamical gluon mass [7,43,90,91]. In addition, F (q) stands for the ghost dressing function, which is obtained from the ghost propagator, D ab (q) = δ ab D(q), Evidently, the contraction of Γ αµν (q, r, p) with respect to q α or p ν leads to cyclic permutations of the STI in Eq. (2.4).
Employing the standard tensorial decomposition of Γ µ (q, p, r), where, at tree-level, B Thus, knowledge of the form factors of H νµ determines fully the corresponding form factors of the ghost-gluon vertex Γ µ (q, p, r).
Moreover, the STI of Eq. (2.4), together with its two cyclic permutations, permits the reconstruction of the "longitudinal" part of Γ αµν (q, r, p) [86]. The nonperturbative realization of the BC "solution" depends not only on the infrared behaviour of ∆ and F , which is rather well-known both from lattice simulations [54,55] and functional methods [7,91,92], but also on the details of A 1 , A 3 , and A 4 , which are largely unexplored.
Quite interestingly, the BC construction for the longitudinal part of Γ αµν hinges on the validity of a special relation between A 1 , A 3 , and A 4 , which in the original work of [86] was shown to hold at the one-loop level (in the Feynman gauge). Subsequently, this relation was derived from the fundamental STI that H νµ satisfies when contracted by the momentum of the incoming gluon [88], and is therefore exact both perturbatively to all orders as well as nonperturbatively. The relation in question may be expressed in terms of the ratio and states simply that, by virtue of the aforementioned STI, one must have 2 R(q 2 , p 2 , r 2 ) = 1 , (2.10) for any value of q, r, and p.
As we will see in sections IV and VI, the constraint of Eq. (2.10) is particularly useful for optimizing the form of the ingredients entering into the computation of the A i , and for quantifying the veracity of the truncations and approximations employed.

III. GHOST-GLUON KERNEL AT THE ONE-LOOP DRESSED LEVEL
In this section we derive the expressions for the form factors A i within the one-loop dressed approximation. In particular, the four point ghost-gluon scattering amplitude, entering in the diagrammatic definition of H µν (q, r, p) in Fig. 1, is approximated by its lowest order contributions, including the one-gluon and one-ghost exchange terms, which are subsequently "dressed" as shown in Fig. 3. Thus, the approximate version of the SDE that we employ reads where C A is the eigenvalue of the Casimir operator in the adjoint representation, and It is obvious from Eq. (3.2) that in the soft ghost limit, i.e. p → 0, the one-loop dressed corrections vanish, i.e. H νµ (q, p, r) = g νµ . This result is valid to all orders, independently of the truncation scheme adopted (see, e.g., Eqs. (6.24) and (6.25) of [95]), being a plain manifestation of Taylor's theorem [96].
The renormalization of Eq. (3.1) proceeds through the replacements [71] where Z A , Z c , Z 1 , Z 3 , and Z g are the corresponding renormalization constants. Within the momentum subtraction (MOM) scheme that we employ, propagators assume their tree-level values at the subtraction point µ, while an analogous condition is imposed on the vertices, usually implemented at a common value of all their momenta ("symmetric" point).
where the Z 1 originates from the renormalization of the H νµ (q, p, r) on the l.h.s. The subscript "R" will be subsequently suppressed to avoid notation clutter.
In what follows we will set Z 1 = 1. This particular choice is exact in the case of the soft ghost limit, being strictly enforced by the validity of Taylor's theorem [96]. For any other MOM-related prescription, Z 1 deviates only slightly (a few percent) from unity, for the subtraction point µ = 4.3 GeV that we employ. For example, as we have explicitly confirmed from our results, in the case where the MOM prescription is imposed at the symmetric point The relation between H νµ and Γ µ , given by Eq. (2.3), prompts a final adjustment, which permits us to preserve the ghost-anti-ghost symmetry at the level of the approximate SDE that we consider 3 . Specifically, the form factor B 1 (q, p, r) of the ghost-gluon vertex is symmetric under the exchange of the ghost and anti-ghost momenta, p and q, respectively.
However, the truncated SDE of Fig. 3 does not respect this special symmetry, because the vertex where the ghost leg is entering is "dressed" while that of the anti-ghost is bare. A simple expedient for restoring this property is to "average" the SDEs dressed on either leg [32,33,97], which amounts to substituting into Eq. (3.2) In general, the individual A i may be projected out from H νµ (q, p, r) by means of a set of suitable projectors, T µν i (q, r). In particular, and Clearly, since in the present work H νµ (q, p, r) is approximated by Eq. (3.1), the corresponding form factors will be obtained through The implementation of the above projections may be carried out using an algebraic manipulation program, such as the Mathematica Package-X [98,99]; the rather lengthy expressions produced from these projections are presented in Appendix B.

IV. INPUTS AND APPROXIMATIONS
For the evaluation of Eq. (3.2) we need the following ingredients: (i) the gluon propagator ∆(q) and its "kinetic" term J(q), (ii) the ghost dressing function F (q), (iii) the three-gluon vertex, entering in (d 2 ) νµ , (iv) the ghost-gluon vertex, entering in both (d 1 ) νµ and (d 2 ) νµ , and (v) the value of the strong coupling α s ≡ g 2 /4π at the renormalization scale µ. The corresponding input quantities will be denoted by ∆ in (q), J in (q), F in (q), Γ in µαβ , and B in 1 (Q), respectively. It is important to comment already at this point on a characteristic feature shared by inputs (i)-(iv), which is implemented in order for the resulting A i to satisfy Eq. (2.10) as accurately as possible. In particular, in the deep ultraviolet all aforementioned quantities will be forced to tend to their tree-level values, i.e., their one-loop perturbative corrections (logarithms and/or constants) will be suppressed. This, in turn, will guarantee that, for large values of the momenta, the emerging A i will correctly capture their one-loop perturbative behavior [see also discussion in section VI]. In what follows we briefly review how the above input quantities are obtained.
(i) and (ii): As was done in a series of previous works [25,41,100], for ∆ in (q) and F in (q) we employ fits to the numerical solutions of the corresponding SDEs, which are in excellent agreement with the quenched SU(3) lattice data of [54], subject to the particular ultraviolet adjustments mentioned above. Below we consider the individual cases (i) and (ii) separately.
(i): The fit for ∆ in (q) (in Euclidean space) is given by [92] where the kinetic term has the form while the effective gluon mass m 2 (q) obeys a power-law running [91] 4 , respectively (blue continuous curves). The fits for ∆(q) and J(q) follow the same functional de- The lattice data is from Ref. [54].
with the adjustable parameters given by τ 1 = 12.68, τ 2 = 1.05 GeV 2 , m 2 0 = 0.15 GeV 2 , ρ 2 m = 1.18 GeV 2 and ρ l = 102.3. On the left panel of Fig. 4 we show the lattice data for ∆(q) (circles) [54], together with the corresponding fit (blue continuous curve) given by the combination of Eqs. On the right panel of Fig. 4 we present the J in (q) of Eq. (4.2); the reason for displaying it in isolation is that it constitutes the main ingredient in the approximation implemented for the three-gluon vertex in item (iii), see Eqs. (4.5) and (4.6). Notice that the J in (q) contains both massive and massless logarithms, which are crucial for triggering three characteristic features, namely its suppression with respect to its tree-level value (J (0) (q) = 1) for a wide range of physically relevant momenta, the reversal of its sign (zero-crossing), and its logarithmic divergence at the origin [34,68]. These features, in turn, will be inherited by the components of the three-gluon vertex constructed in (iii). Even though J in (q) contains these logarithms, for large q 2 it tends to 1, in compliance with the requirement discussed above, due to the inclusion of the function τ 1 /(q 2 + τ 2 ); note that this function becomes 1 in the "bona-fide" fit for J(q), which is also displayed in Fig. 4, for direct comparison.
with the operator product expansion (see also [101]). In particular, m 2 (q) = m 2  with the corresponding lattice data; its functional form is given by with σ 1 = 0.70 GeV 2 and σ 2 = 0.39 GeV 2 . Again, in the limit of large q 2 , the above expression recovers the tree-level result, i.e. F in (q) = 1. On that same plot, the red dashed line corresponds to the fit of F (q) introduced in Eq. (6.1), which corresponds to the typical solution of the SDE for F (q) [71], and, as such, contains the appropriate perturbative logarithms. Evidently, the difference between the two fits becomes relevant in the deep ultraviolet, where the F (q) of Eq. (6.1) deviates gradually from unity, approaching eventually zero at a logarithmic rate. system. However, in order to reduce the complexity of our analysis, we will employ instead a set of approximations for these two vertices. We next analyze (iii) and (iv) separately.
(iii): Let us first consider the three-gluon vertex, entering in (d 2 ) νµ , and set t = −(ℓ + r).  [86] yields an "abelianized" version for Γ µαβ (r, t, ℓ), to be denoted by Γ in µαβ , which is used as a "seed" for obtaining the one-loop dressed approximation for H νµ . In particular, for Γ in µαβ we retain only its tree-level tensorial structure, namely Notice that at tree-level X in 1 = 1, and Eq. (4.5) reduces to (iv): Turning to the ghost-gluon vertex, as mentioned right after Eq. (3.2), two out of the three vertices have been naturally replaced by their B 1 components, and only the In what follows we will set (by hand) The approximation used for B 1 (q, p, r) is obtained as follows. We start by carrying out the first iteration of Eq. (3.2), using for B 1 its tree-level value. This furnishes the first approximation for the A i (q, p, r), which, by means of the first relation in Eq. (2.8), yields the next approximation for B 1 (q, p, r). At this point we isolate from B 1 (q, p, r) the "slice" that corresponds to the "totally symmetric" configuration shown on the right panel of Fig. 5 (red dashed line). Then, to get B in 1 (Q) we adjust the "tail" of the curve, such that it reaches the tree-level value 1 for large Q; the resulting functional form may be fitted by where the parameters τ 1 = 2.21 GeV −2 , τ 2 = 2.50 GeV −2 and λ = 1.68. Past this point, the iterative procedure described above is discontinued, and the B in 1 (Q) of Eq. (4.9) is fixed as the final input in Eq. (3.2). (4.10) (v): All quantities will be renormalized at µ = 4.3 GeV, where α s has been estimated to assume the value α s = 0.22.

V. RESULTS FOR THE FORM FACTORS OF THE GHOST-GLUON KERNEL
In this section we present the results for the five form factors A i . We will first present 3-D plots in general Euclidean kinematics, and then take a closer look at three special kinematic limits.
In what follows we will express all relevant form factors as functions of q 2 , p 2 and the angle θ, namely A i (q, p, r) → A i (q 2 , p 2 , θ). Note also that since the quantities entering in the integrals do not depend on the angle φ 3 , the last integral in (5.3) furnishes simply a factor of 2π. In the Figs. 6 and 7, we present a typical set of results for the form factors A i , for θ = 0 and θ = π. It is important to notice that all form factors exhibit the following common features: (i) they are all infrared finite; (ii) in the infrared they all display considerable departures from their tree-level values, approaching their expected one-loop results in the ultraviolet, and (iii) in general they display a mild dependence on the angle θ.

B. Special kinematics limits
In this subsection we extract three special kinematic configurations from the general solutions for A i reported above, and compare them with the corresponding perturbative results computed at one loop (see Appendix A).
(i) The soft gluon limit, which means that r = 0; then, the momenta q and p have the same magnitude, |p| = |q|, and are anti-parallel i.e., θ = π. Our results are expressed in terms of the momentum q. In this kinematic limit only A 1 and A 2 survive [see Eq.(2.2)], and are shown in Fig. 8.
A 1 (q, −q, 0) (left panel) displays only a mild deviation from its tree-level value in the entire range of momenta. The maximum deviation is of the order of 5%, and happens around q ≈ 1 GeV. Notice that A 1 (0, 0, 0) = 1; this particular value is recovered again for higher values of q, as expected from the one-loop calculation of Eq. (A3). Clearly, for large values of q we see a qualitative agreement between both curves (blue and purple); evidently, the small ultraviolet deviation between them is expected, and can be attributed to the higher order corrections that A 1 (q, −q, 0) contains in it.
On the right panel of Fig. 8 we show the dimensionless combination q 2 A 2 (q, −q, 0), which in the ultraviolet tends towards the constant value predicted by the one-loop result given by Eq. (A3). Once again, the maximum deviation from its tree-level value is located around q ≈ 1 GeV.
(ii) The soft anti-ghost limit, in which q = 0 and the momenta |p| = |r|; evidently, |q||p| cos θ = 0, and any dependence on the angle θ is washed out .
In this limit only the form factors A 1 and A 3 survive, and their functional dependence is expressed in terms of the momentum r. In the Fig. 9, we can see that both form factors,   (iii) The totally symmetric limit, defined in Eq. (4.8).
In Fig. 10   one-loop result, the case where the Γ (0) µαβ of Eq. (4.7) is used as input in (d 2 ) µν , and the case where the Γ in µαβ of Eq. (4.5) is used instead. We clearly see that the A i obtained with either vertex display a sizable deviation from their tree-level value in the region of Q ≈ 1 − 2 GeV, while for large values of Q they recover the ultraviolet behaviour expected from one-loop perturbation theory, given by Eqs. (A6). Interestingly enough, except for q 2 A 5 (Q), the use of Γ in µαβ yields A i that are more suppressed.

VI. THE CONSTRAINT FROM THE STI
The next item of our analysis is dedicated to the STI-derived constraint of Eq. (2.10).
The way this particular constraint becomes relevant for our considerations is two-fold. First, a considerable degree of hindsight gained from this equation has already been used in section IV, in order to optimize the ultraviolet features of the input functions. Second, as we will see below, the amount by which the calculated value for R deviates from unity favors the use of dressed rather than bare vertices in the graphs (d 1 ) νµ and (d 2 ) νµ .
With respect to the first point, note that the relation of Eq. (2.10), being a direct consequence of the BRST symmetry, is satisfied exactly at any fixed order calculation in perturbation theory. However, in general, our truncation procedure does not reduce itself to a fixed order perturbative result, for any limit of the kinematic parameters. This happens because certain of the (higher order) terms, generated after the integration of all ingredients, ought to cancel/combine with contributions stemming from two-and higher-loop dressed diagrams of H νµ , which, evidently, have been omitted from the outset. The resulting mismatches, in turn, affect unequally the different kinematic configurations entering in R, thus distorting the subtle balance that enforces Eq. (2.10).
A concrete manifestation of the underlying imbalances occurs when one uses input propagators and vertices containing perturbative information (e.g., are of the general form 1 + cα s log q 2 /µ 2 ). Since one may not intervene in the actual numerical evaluation and discard "by hand" terms of O(α 2 s ) and higher, the final answer contains a certain amount of unbalanced contributions. The clearest manifestation of this effect occurs when evaluating R for asymptotically large momenta: contrary to what one might expect, the "tails" of R deviate markedly from unity; in fact, the deviation increases as the momenta grow.
The use of input functions that tend to their tree-level values ameliorates the situation substantially, because, in this way, the A i computed display at least their correct one-loop behavior. This improvement, in turn, must be combined with a judicious choice for the F (p) and F (r) appearing explicitly in R [see Eq. (2.9)]; in particular, the function used must display asymptotically the logarithmic behaviour dictated by one-loop perturbation theory. Specifically, we use the standard fit [31]  Then, after these adjustments, the "tails" of R display only a minuscule deviation from unity, which decreases slowly as the momenta increase.
We next turn to the second point, and consider what the STI constraint suggests regarding the vertices used in the calculation.
Clearly, for any kinematic configuration where |p| = |r|, the numerator and the denominator of Eq. (2.9) become equal, and Eq. (2.10) is trivially satisfied. In particular, this is precisely what happens in the "soft anti-ghost" and "totally symmetric" limits, presented in the previous subsection.
Notice that in both cases we evaluate R(q 2 , p 2 , r 2 ) using two different approximations:  In the kinematic configuration presented on the right panel, we notice that the deviations are milder. Specifically, the maximum deviation appears in the momentum range 0.8 -1.1 GeV, and is less than 3% when bare vertices are used, dropping to less than 1% for dressed vertices. In the ultraviolet the deviation from unity is of the order of 0.1%. In Fig. 12 and Fig. 13 we show, respectively, the form factors B 1 and B 2 as functions of  q 2 , p 2 and θ. In order to appreciate their angular dependence, we present two representative cases: θ = 0 and θ = π. As we can see, the angular dependence of B 1 is relatively weak, whereas B 2 is clearly more sensitive to changes in θ. Note also that both form factors tend to their perturbative behaviour whenever one of the ghost (p) or anti-ghost (q) momenta becomes large. In addition, for p 2 = q 2 = 0 they revert to their tree-level values, due to the fact that the one-loop dressed contributions to H νµ vanish at the origin. Moreover, we may visually verify that B 1 (q 2 , p 2 , θ) is symmetric under the exchange q 2 ↔ p 2 , for any θ, as required by the ghost-anti-ghost symmetry. It is clear that B 1 and B 2 will depend through the A i on our choice for Γ µαβ . In order to study this effect, we employ the results presented in the section V B, where the A i were computed using as input for Γ µαβ either the Γ µαβ is used. Next, in Fig. 15, we compare our results for B 1 in the soft gluon configuration with those obtained in earlier works [48][49][50]; this configuration is the most widely explored in the literature, and the only one simulated on lattice for SU(3) [63,64]. The green dasheddotted curve represents the results for B 1 (q, −q, 0), obtained from the approach developed in [48], based on the infrared completion of expressions derived using operator product expansion techniques. In the case of [49], B 1 was determined in general kinematics, using a system of coupled SDEs, while in [50] the B 1 was determined exclusively in the soft gluon (green dashed-dotted), [49] (red dashed), and [50] (magenta dotted). The lattice data (circles) are from [63,64].
configuration. It is interesting to notice that all analytical studies display the characteristic peak and converge to unity at the origin. Moreover, all of them are in qualitative agreement with the lattice data (note, however, that the error bars are quite sizable).
Finally, in Fig. 16, we illustrate the impact that the full structure of B 1 (q 2 , p 2 , θ) has on the SDE of the ghost dressing function. To that end, we explore two scenarios: (i) we couple the entire momenta dependence of B 1 to the SDE for F (q), carrying out the additional angular integration [see Eq. (2.14) of [71]], and (ii ) we fix its momentum dependence to the soft ghost configuration [47,71]. We observe that with mild adjustments to the value of α s , both scenarios reproduce the standard lattice results of [54] rather accurately; in particular, while for the case (i) α s = 0.25, for (ii) we obtain α s = 0.24.
The reason for this small difference in the values of α s can be easily understood. As mentioned in section V B, in the region of momenta of about two to three times the QCD mass scale, the soft ghost configuration maximizes the deviation from the tree-level value.
Therefore, when we approximate the entire momentum dependence of Γ µ just by this configuration (instead of integrating over all of them), we slightly overestimate the contribution of the ghost-gluon vertex to the ghost SDE.

VIII. CONCLUSIONS
We have presented a detailed nonperturbative study of the form factors, A i , comprising the ghost-gluon kernel, H νµ , using the "one-loop dressed" approximation of its dynamical equation, for general Euclidean momenta. The results obtained have been presented in 3-D plots, and certain "slices", corresponding to special kinematic limits, have been singled out and inspected in detail. The A i obtained have been subsequently used for the determination of the two form factors, B 1 and B 2 , of the ghost-gluon vertex.
The ingredients entering in the calculations are the gluon and ghost propagators, and the vertices Γ αµν and Γ µ . Given that the H νµ itself is intimately connected to both these vertices, a strictly self-consistent treatment would require to couple the dynamical equation governing H νµ to the equations relating it to both Γ αµν and Γ µ , and proceed to the solution of the entire coupled system. Instead, we have treated the problem at hand by employing simplified versions of these vertices, whose use in recent studies [91,103] yielded satisfactory results. Moreover, as has been explained in detail, there exists a subtle interplay between the truncation of the equations employed, the ultraviolet behavior of the ingredients used for their evaluation, and the accuracy with which the resulting A i satisfy the STI constraint of Eq. (2.10). Note in particular that while our input expressions for the two-point functions are in excellent agreement with the lattice data of [54] for infrared and intermediate momenta, their ultraviolet tails have been adjusted to their tree-level values.
We have paid particular attention to the impact that the structure of Γ αµν may have on the results. All our findings indicate that the use of a dressed Γ αµν , corresponding to the so-called "minimal BC solution", Γ in µαβ , induces an appreciable suppression with respect to the results obtained by merely resorting to Γ (0) αµν . This happens because the form factor X in 1 is itself suppressed in the infrared, due to the form of the functions J(q) that enter its definition [see Eq. (4.6)]. This special feature of the three-gluon vertex, in turn, appears to be favored by the STI-derived constraint, in the sense that the results obtained with Γ in µαβ are considerably closer to unity (see Fig. 11).
The information obtained on the structure of the ghost-gluon kernel opens the way towards the systematic nonperturbative construction of the 10 form factors comprising the "longitudinal" part of the three-gluon vertex, using the BC construction [86] as a starting point. The detailed knowledge of these form factors, in turn, may have considerable impact on the study of the dynamical formation of gluon dominated bound states, such as glueballs and hybrids (see, e.g., [104], and references therein). We hope to be able to present results on this topic in the near future.

Appendix A: One-loop results for special kinematic configurations
In this Appendix we present the one-loop results for the various A i in the three special kinematic configurations considered in subsection V B [89,105].

Soft gluon limit:
To derive this configuration, we set r → 0 directly into Eqs. (2.2) and (A1). It is straightforward to see that in this limit the tensorial structure of H (1) νµ (q, 0) given by (2.2) reduces to and the form factors become Then, the one-loop result for B 1 (q, −q, 0) may be directly obtained using Eq. (2.8).
2. Soft anti-ghost limit: This limit is obtained by setting q = 0. The one-loop expression for H (1) νµ becomes where A 3. Symmetric configuration: This kinematic limit is defined in (4.8); in this case all form factors survive, and are given by where I is a constant [105] defined as with ψ 1 (z) being the "trigamma function', expressed in terms of the standard Γ(z) function as and it has the following special values We write the A i as the sum of their tree-level value and the contributions from (d 1 ) νµ 1 = 1 and A (0) i = 0 for i = 2, 3, 4, 5. We then introduce new kinematic variables s = q − ℓ , t = −ℓ − p , u = −p − q , and v = −ℓ + p + q, the inner products a 1 = ℓ·p , a 2 = ℓ·q , and a 3 = p·q , together with the combinations , Moreover, as a short-hand expedient, we will denote the arguments of several functions as a super/subscript, i.e., f (x, y, z) = f xyz or f (x, y, z) = f xyz .
Finally, it is understood that, for the numerical evaluation of the above expressions, all relevant quantities are to be replaced by their "input" expressions, namely ∆ in (q), X in 1 (r, t, ℓ), F in (q), and B in 1 (Q), introduced in section IV.