Novel sum rules for the three-point sector of QCD

For special kinematic configurations involving a single momentum scale, certain standard relations, originating from the Slavnov-Taylor identities of the theory, may be interpreted as ordinary differential equations for the ``kinetic term'' of the gluon propagator. The exact solutions of these equations exhibit poles at the origin, which are incompatible with the physical answer, known to diverge only logarithmically; their elimination hinges on the validity of two integral conditions that we denominate ``asymmetric'' and ``symmetric'' sum rules, depending on the kinematics employed in their derivation. The corresponding integrands contain components of the three-gluon vertex and the ghost-gluon kernel, whose dynamics are constrained when the sum rules are imposed. For the numerical treatment we single out the asymmetric sum rule, given that its support stems predominantly from low and intermediate energy regimes of the defining integral, which are physically more interesting. Adopting a combined approach based on Schwinger-Dyson equations and lattice simulations, we demonstrate how the sum rule clearly favors the suppression of an effective form factor entering in the definition of its kernel. The results of the present work offer an additional vantage point into the rich and complex structure of the three-point sector of QCD.

As is well-known, the fundamental Slavnov-Taylor identities (STIs) [81,82] impose crucial constraints between the two-and three-point sectors of the theory [83][84][85][86][87]. In the present study we offer a novel point of view inspired by these profound relations, which, for the special kinematic conditions that we consider, give rise to two relatively simple sum rules.
The starting point of our considerations are certain special projections of Γ αµν (q, r, p), denoted here by L(q, r, p) [see Eq. (3.1)], which have been frequently employed in lattice studies [59][60][61][88][89][90][91][92]. These functions may be evaluated in special kinematic limits, with the final upshot of replacing their three momentum scales by a single one. In particular, in the so-called "asymmetric" and "symmetric" configurations, the resulting quantities, denoted by L asym (q 2 ) and L sym (s 2 ), respectively, have been computed on the lattice, both in quenched [60][61][62][63]92], and unquenched simulations [59,93]. The basic quantity evaluated in these cases is A a α (q) A b µ (r) A c ν (p) , where A a α are the SU(3) gauge fields in Fourier space, with the average · denoting functional integration over the gauge space.
From the continuous standpoint, L sym (s 2 ) and L asym (q 2 ) may be written as combinations of the form factors appearing in the tensorial decomposition of Γ αµν (q, r, p) [see Eq. (2.4)]; in particular, L asym (q 2 ) contains only longitudinal form factors, X i (q, r, p), while L sym (s 2 ) involves both longitudinal and transverse form factors, Y i (q, r, p) [94][95][96]. The nonperturbative extension [58] of the Ball-Chiu (BC) procedure [94], in turn, allows one to relate the X i with the following quantities: (i) the "kinetic term", J(q 2 ), of the gluon propagator, (ii) the ghost dressing function, F (q 2 ), and (iii) three of the five form factors, A i (q, p, r), comprising H µν [69,94,95]. Of course, this STI-based approach leaves the Y i completely undetermined, since they form the "automatically conserved" part of the three-gluon vertex. Then, the full implementation of this method gives rise to Eqs. (3.20) and (4.23) [58,59]. Past this point, one introduces theoretical information for the ingredients entering on the r.h.s. of these equations, thus obtaining definite predictions about L asym (q 2 ) and L sym (s 2 ), which are subsequently compared with the corresponding lattice results [59,63].
However, one may reverse this point of view entirely, and consider Eqs. (3.20) and (4.23) as relations that furnish J(q 2 ), once the lattice results for L asym (q 2 ) or L sym (s 2 ) have been used as inputs. If this alternative perspective is adopted, it becomes immediately clear that Eqs. (3.20) and (4.23) may be viewed as a first order linear differential equations for J(q 2 ), whose solution may be written in exact closed form.
It turns out that the general solution of the differential equation (3.20) displays a simple pole at the origin, while that of Eq. (4.23) exhibits a double one. However, it is well known that the physical J(q 2 ) does not possess any type of pole at q 2 = 0; instead, as it has been established in a series of works, the massless ghost loop entering into the SDE of the J(q 2 ) forces it to diverge logarithmicaly as q 2 → 0 [58,59,97].
These unphysical poles may be eliminated from the solution for J(q 2 ) by means of an appropriate expansion around the origin, provided that certain integral conditions hold exactly.
These conditions, given in Eqs. (4.15) and (4.32), will be referred to as the "asymmetric" and "symmetric" sum rule, respectively. At this point, we postpone the determination of J(q 2 ) from the corresponding solutions, and focus instead on the content and potential applications of these sum rules.
In general, when different sets of ingredients are used as inputs, the sum rules will be satisfied at a varying degree of accuracy, thus providing a quantitative indication on the veracity of the approximations employed for obtaining these ingredients. In that sense, the sum rules may be used as a means of discriminating approximations or truncations schemes, offering hints for their systematic improvement. Such a possibility, in turn, may be especially useful in the field of SDEs, where the absence of a concrete expansion parameter complicates the task of assigning errors to the results obtained or the simplifications implemented.
Expanding on the previous point, it should be clear that, since the sum rules are deduced from the differential equations for J(q 2 ), the quantities to be probed must be determined from any approach other than the BC solutions themselves. For example, restricting our-selves to the asymmetric case, one may opt for a purely SDE-based analysis, computing both A i and X i from the SDEs of H µν [69] and Γ αµν [52,54,55,67,[98][99][100] respectively, and then plug the X i into Eq. (3.3) to obtain L asym SDE (q 2 ). Alternatively, one may use a combined approach, deriving the A i as before, but using lattice data for L asym (q 2 ) [63,64] ; this latter procedure will be followed in the analysis carried out in Sec. V.
Since the integrals appearing in Eqs. (4.15) and (4.32) are evaluated within the interval [0, µ 2 ], the sum rules explore the quantities entering in them over a wide range of momenta.
Note that the symmetric sum rule involves the contributions from the transverse form factors, Y i , comprising the term L sym T (s 2 ), which enters into the functionf 2 (s 2 ) [see Eqs. (4.22) and (4.26), respectively]. This fact reduces its effectiveness, at least within the confines of our approach, because the lattice does not furnish L sym L (s 2 ) and L sym T (s 2 ) separately, but only their sum. In addition, as can be seen in Eq. (4.32), the integrand of the sum rule contains an additional factor of t, with respect to its asymmetric counterpart; as a result, the support of the ingredients comprising the kernel is suppressed in the low energy regime, which is the most interesting from the nonperturbative point of view. Given the above limitations, for the purposes of this introductory presentation, the numerical analysis will be restricted to the case of the asymmetric sum rule only.
The article is organized as follows. In Sec. II we present a brief summary of the main ingredients, originating from the two-and three-point sectors of the theory, that are extensively used in this work. Sec. III is dedicated to the detailed derivation of Eq. (3.20), placing particular emphasis on the origin of the special function W(q 2 ). Sec. IV contains the main results of this study. In particular, after identifying the differential equations and specifying their corresponding solutions, we proceed to the detailed derivation of the two sum rules.
In Sec. V we demonstrate with a concrete example the possibilities that the asymmetric sum rule offers for constraining one of the ingredients that enter in the expressions defining the function W(q 2 ). In Sec. VI we summarize our results and discuss future applications of the ideas and techniques presented here. We conclude with two Appendices: in the first, we implement the transition from the Taylor scheme to the MOM-type scheme used in the lattice simulations of L sym (q 2 ); in the second, we present the steps necessary for the one-loop dressed determination of the function W(q 2 ). II.

BRIEF REVIEW OF THE MAIN THEORETICAL INGREDIENTS
In this section we introduce the necessary notation and summarize certain basic properties of the two-and three-point correlation functions entering in this work. We emphasize that we restrict ourselves to a "quenched" version of QCD, namely a SU(3) Yang-Mills theory with no dynamical quarks; note, in particular, the absence of quark propagators and quark-gluon vertices.
The diagrammatic representations of these three quantities are given in Fig. 1; all momenta are incoming, q + p + r = 0.
It is customary to decompose the vertex Γ αµν (q, r, p) in two distinct pieces [94,95,117], where the "longitudinal" part, Γ αµν L (q, r, p), saturates the standard STIs satisfied by the vertex [see Eq. (2.9)], while the totally "transverse" part, Γ αµν T (q, r, p), is annihilated when contracted by q α , r µ , or p ν . The tensorial decomposition of these two terms reads [94][95][96]117] Γ αµν where the explicit expressions of the basis elements ℓ αµν i and t αµν i are given in Eqs. (3.4) and (3.6) of [58], respectively. At tree level, Regarding Γ µ (q, p, r) and H νµ (q, p, r), note first that they are related by the STI q ν H νµ (q, p, r) = Γ µ (q, p, r). Their respective tensorial decompositions are given by [69,94,95] Γ µ (q, p, r) = q µ B 1 (q, p, r) + r µ B 2 (q, p, r) , where the argument (q, p, r) of the A i has been suppressed for compactness. At tree-level, In addition, it is convenient to introduce the short-hand notation Finally, of central importance for the ensuing analysis is the STI and its cyclic permutations [58,94], relating the two-and three-point sectors of the theory.
The validity of this set of STIs, in conjunction with the nonperturbative generalization of the standard BC construction [94], allows one to express the longitudinal form factors, X i , in terms of F (q 2 ), J(q 2 ), A 1 (q, r, p), A 3 (q, r, p), and A d (q, r, p) [58], but leaves the transverse ones, Y i , completely undetermined.

Of central interest in what follows is the quantity
employed in numerous lattice simulations of the three-gluon vertex [59,63]. W αµν (q, r, p) represents specific tensors, which project out particular components of the Γ α ′ µ ′ ν ′ (q, r, p), evaluated in special kinematic limits [59,63].
In the present work we will focus on the asymmetric limit, corresponding to the kinematic configuration p → 0 , r = −q . As has been shown in [58,59], L asym (q 2 ) is given by since it contains no transverse form factors, Y i , L asym (q 2 ) is fully determined from the BC solution for X 1 and X 3 . The A i entering in this solution appear in two different kinematic configurations, A i (q, −q, 0) and A i (q, 0, −q), corresponding to the soft gluon and soft ghost limits, respectively; we will employ the short-hand notation Note that, by virtue of Taylor's theorem [81], in the Landau gauge, The case of X 1 (q, −q, 0) can be read off directly from [58]; specifically, after conversion to Euclidean space, The form factor X 3 (q, r, p) is given by (Minkowski space) Evidently, due to the vanishing of the denominator, the p = 0 limit requires an appropriate expansion. Using that q 2 − r 2 = −2 q · p + O(p 2 ), and expanding J(r 2 ) in the numerator, In order to evaluate G 3 (q 2 ) further, note that A 1 (q, p, r) is obtained from H νµ (q, p, r) through the projection A 1 (q, p, r) = T µν 1 (q, r)H νµ (q, p, r), where the projector T µν 1 (q, r) satisfies with t = q, r, p [see Eq. (3.8) of [69]]. Moreover, in the Landau gauge, all quantum corrections to H νµ (q, p, r) are proportional to p, such that [118] H νµ (q, p, r) = g µν + p ρ K νµρ (q, p, r) . (3.11) Hence, the G 2 (q, r, p) of Eq. (3.9) may be cast in the form (3.12) Evidently, the limit p = 0 may be taken directly in the expression in square brackets, yielding Next, K νµρ (q, 0, −q) may be written as where the ellipses denote terms proportional to g µρ q ν , g ρν q µ , and q µ q ν q ρ , which do not contribute to G 2 (q, r, p), by virtue of Eq. (3.10). Hence, we have which, upon substitution into Eq. (3.13), leads to G 3 (q 2 ) = −W(q 2 )/q 2 . Using the last expression into Eq. (3.8), and passing to Euclidean space (q 2 → −q 2 ), we finally get The above equation may be simplified considerably by resorting to the exact relation which is a direct consequence of a fundamental STI satisfied by H νµ [69,119]. In particular, setting in it p = 0 and r = −q, we obtain The substitution of the above result into Eq. (3.17) eliminates all dependence on A 1 (q 2 ), , and A 4 (q 2 ), yielding the compact result It is clear from Eq. (3.20) that the logarithmic divergence displayed by J(q 2 ) in the deep infrared is transferred to L asym (q 2 ). In particular, from Eq. (B9) follows that W(0) = 0 [for more details, see the end of Appendix B]; moreover, Appendix A], while the term lim is subleading, contributing a finite constant. Thus, the leading contribution of Eq. (3.20) is given by relating the rates of divergence of L asym (q 2 ) and J(q 2 ) at the origin.

IV. DERIVATION OF THE SUM RULES
It is clear that the STI-derived relation given in Eq. (3.20) may be regarded as a first order linear differential equation for J(q 2 ), whose solution allows one to express J(q 2 ) in terms of all other functions. It turns out that this particular point of view, when appropriately explored, leads to two novel constraints, whose detailed derivation is the focal point of this section.

A. Asymmetric sum rule
Let us consider Eq. (3.20), set x = q 2 , and define Then, Eq. (3.20) may be cast in the "canonical" form of a linear differential equation with where λ(x) is the "integrating factor", given by and µ 2 is the point where the initial condition is chosen.
At this point, it is natural to opt for an initial condition dictated by the physics, identifying µ 2 with the subtraction point where J(x) has been renormalized. Specifically, in the momentum subtraction scheme (MOM) usually employed, we have that J(µ 2 ) = 1, so that Eq. (4.4) becomes The particularity of this solution originates from the presence of the x in the denominator of P (x), which, in general, introduces in the answer a pole divergence at x = 0.
To see this property in its most rudimentary manifestation, set into Eq. (4.1) W(x) = 0 (tree-level value), so that f 1 (x) = 1, while f 2 (x) is kept arbitrary. Then, the integrating factor becomes simply and the solution reads Now, let us suppose that we know from independent considerations that J(x) does not diverge as a pole at x = 0, but rather as a logarithm [58,59,97]. Then, the question that arises naturally is how to reconcile this information with the form of Eq. (4.8).
Perhaps the most direct approach for answering this question is to consider the Taylor expansion around x = 0 of the expression in square brackets on the r.h.s. of Eq. (4.8).
Specifically, one has with Clearly, in order for the solution not to possess a pole at the origin, we must have a 0 = 0.
This condition amounts to the integral constraint which must be obeyed by the function f 2 (t) within the interval of integration [0, µ 2 ].
If the above condition is satisfied, then the solution of the differential equation at x = 0 yields J(0) = f 2 (0), which is none other than the leading term of Eq. Having fixed the ideas, let us next consider the complete case, where the function f 1 (x) retains its full structure. Noting that W(0) = 0, we have that f 1 (0) = 1; it is then convenient to define with u(0) = 0. Then, Eq. (4.5) yields , which drops out when forming the ratio that defines u(x). Consequently, both u(x) and σ(x) are "renormalization group invariant"(µ-independent) quantities.
With the definitions introduced in Eq. (4.13), the solution in Eq. (4.6) becomes 14) and the Taylor expansion around x = 0 can be carried through as in Eq. (4.9). At this point, the requirement of eliminating from the solution the pole at x = 0 imposes the constraint which is a central result of this work, to be referred to as the "asymmetric sum rule".
Next, assuming the constraint of Eq. (4.15) to be satisfied, the asymptotic behavior of J(q 2 ) is obtained from the first nonvanishing term of the Taylor expansion, yielding exactly the leading relation reported in Eq. (3.21).

B. Alternative derivation
Let us return to Eq. (3.20), but, instead of solving it for J(x), use Eq. (2.2) to set ]/x, and convert it into a differential equation for m 2 (x), treating ∆(x) as an input known from lattice simulations. Then, straightforward algebra yields It is clear at this point that the integrating factor, λ m (x), is given by , which allows us to write Then, choosing the boundary condition at the origin, namely m 2 (0) = ∆ −1 (0), and using that σ(0) = 1, the solution for m 2 (x) is given by The direct comparison between the last line of Eq.   We emphasize that Eqs. (4.20) and (4.14) are completely equivalent, as can be seen immediately by using x 0 in Eq. (4.14) and subsequently employing Eq. (4.15). Furthermore, note that once the solution for J(x) has been cast in the form of Eq. (4.20), its pole may be explicitly removed by means of the simple change of variables t = xy.

C. Symmetric sum rule
Another special version of the L(q, r, p) defined in Eq. (3.1), to be denoted by L sym (s 2 ), corresponds to the totally symmetric limit [59,63], with the appropriate projector W αµν (q, r, p) given by Eq. (2.18) of [59].
L sym (s 2 ) receives contributions from both longitudinal and transverse form factors, .
The main difference between Eq. (4.28) and Eq. (4.14) is that now the would-be pole in the solution is not simple but double. Therefore, one needs to consider one more term in the corresponding Taylor expansion, namely withā Clearly, in order for the solution not to have a pole (double or simple) at the origin, we must haveā 0 = 0 andā 1 = 0. The first condition amounts to the integral constraint which constitutes the second major result of the present study, to be referred to as the "symmetric sum rule".

NUMERICAL ANALYSIS
In this section, we focus on the sum rule of Eq. (4.15) and analyze in detail how it can be used to restrict the form of V (ℓ 2 ), which is one of the main ingredients entering in the one-loop dressed approximation of the function W(q 2 ), given by Eqs. (B9) and (B10).
A. Setting up the stage (i) It is convenient to cast the sum rule (4.15) into the equivalent form The quantity R(t) captures the net effect that different forms of σ(t) induce on the kernel K(t).
(iv) For B 1 (q 2 ), entering in Eq. (B10), we use the curve shown in Fig. 2, obtained from the numerical solution of the SDE governing the ghost-gluon vertex, evaluated in the "softghost" kinematic limit; see [69] for details. B 1 (q 2 ) can be accurately fitted by the functional form where the adjustable parameters acquire the following values: γ 1 = 1.128, γ 2 = 1.84, κ 2 1 = 0.101 GeV 2 , κ 2 2 = 1.59 GeV 2 , ω 1 = 0.379, and ω 2 = 1.071. (v) For L asym (q 2 ) we employ a rather good fit to the lattice data of [63,64] whose The curve is shown in Fig. 3, and its functional form is given by with Note that the above fit incorporates, by construction, the renormalization condition L asym (µ 2 ) = 1, corresponding to the "asymmetric" MOM scheme employed. In addition, the zero crossing of L asym (q 2 ) is located at 167 MeV.
Observe that the above functional form captures the expected infrared asymptotic be- (vi) To every set of ingredients entering into the sum rule we will assign the corresponding   We next present a concrete example of how the sum rule of Eq. (5.1), accompanied by a set of physically motivated assumptions, may restrict severely some of the ingredients  (5.9) in order to obtain the V i (ℓ 2 ) shown in Fig. 4. The tree-level case, V 3 = 1, is recovered by simply setting λ s = 0. In the last row we quote the percentage error ε, given by Eq. (5.7), when the sum rule (5.1) is computed using the σ i (q 2 ) obtained with the variations of the form factor V i (ℓ 2 ) shown in Fig. 4.
comprising its kernel. Specifically, we discuss in detail the impact that the variations of the form factor V (ℓ 2 ) have on W(q 2 ), and, eventually, through the form of σ(q 2 ), on the sum rule itself. This particular choice is prompted by the numerical exploration of the expressions for W d 1 (q 2 ) and W d 2 (q 2 ), given in Eq. (B10), whose upshot is that W d 2 (q 2 ) furnishes the dominant contribution, and that its value is considerably more sensitive to variations of V (ℓ 2 ) rather than of B 1 (q 2 ) [F (q 2 ) and ∆(q 2 ) are held fixed]. It is therefore reasonable to establish whether the sum rule is able to place nontrivial bounds on the form and main features exhibited by V (ℓ 2 ).
Let us emphasize at this point that V (ℓ 2 ) emerged at the final step of a series of approximations, described in Appendix B, whose purpose was to simplify the treatment of the equations defining W(q 2 ); it should be therefore interpreted as an "effective" form factor, capturing the collective action of the various X i and Y i comprising the three-gluon vertex, with their multitude of sizes and kinematic dependence. In this sense, there is no a priori guarantee that V (ℓ 2 ) will inherit from them their characteristic suppression in the intermediate region of momenta. Nonetheless, as we will see in what follows, the sum rule clearly favors a "suppressed" V (ℓ 2 ), imposing, at the same time, strict limits on the amount of its suppression.
Next, we introduce a concrete functional form for V (ℓ 2 ), whose variations will generate distinct versions of this quantity. Specifically, we employ the Ansatz with η 2 (ℓ 2 ) given by Eq. (5.5). The parameters that will be varied throughout this anal-  Table I. ysis are ξ and τ 0 ; all others are kept fixed at the values λ s = 0.22, τ 1 = 11.6 GeV 2 , and τ 2 = 0.0856 GeV 2 . In Table I we Table I].
Since ε changes sign when switching from V (ℓ 2 ) = V 1 (ℓ 2 ) to V (ℓ 2 ) = V 4 (ℓ 2 ), it is reasonable to expect that there will be an "intermediate" curve, to be denoted by V ⋆ (ℓ 2 ), for which the sum rule will be fulfilled exactly (ε = 0). To determine it, we employ Eq. (5.9) in order to produce a sequence of variations for V (ℓ 2 ), determining each time the corresponding ε; note that the modifications to V (ℓ 2 ) are implemented in the transition region between nonperturbative and perturbative regimes, i.e., approximately from 300 MeV to and infrared regimes; specifically, in the ultraviolet they recover the one-loop results for X 1 (ℓ 2 ) = X 4 (ℓ 2 ) = X 7 (ℓ 2 ) [94,95], while in the infrared they diverge at a common logarithmic rate [58].
In Table I we show the values for ε when the sum rule is evaluated using the σ(q 2 ) obtained with the various V i (ℓ 2 ) shown in Fig. 4. Evidently, the sum rule favors clearly a V (ℓ 2 ) = V ⋆ (ℓ 2 ) that is suppressed with respect to the tree-level value captured by V 3 , but certainly less so than the initial V 4 (ℓ 2 ).
It is rather instructive to analyze in some detail how the above result emerges. To that end, denote by [W 1 (q 2 ), σ 1 (q 2 )], [W ⋆ (q 2 ), σ ⋆ (q 2 )], and [W 4 (q 2 ), σ 4 (q 2 )], the corresponding quantities obtained when V (ℓ 2 ) = V 1 (ℓ 2 ), V (ℓ 2 ) = V ⋆ (ℓ 2 ), and V (ℓ 2 ) = V 4 (ℓ 2 ), respectively. As can be seen in Fig. 5, the suppression induced to V (ℓ 2 ) through the transition V 1 (ℓ 2 ) → V 4 (ℓ 2 ) leads to an enhancement of W(q 2 ) in the region of 800 MeV -2 GeV, where clearly W 4 (q 2 ) > W 1 (q 2 ). This difference is transmitted to the σ 1 (q 2 ) and σ 4 (q 2 ), which, as depicted in Fig. 5, satisfy the relation σ 4 (q 2 ) > σ 1 (q 2 ) in the entire range of momenta. Notice that, even though W 1 (q 2 ) and W 4 (q 2 ) merge into each other past q = 4 GeV, σ 1 (q 2 ) and σ 4 (q 2 ) remain clearly separated in the same region of momenta; in fact, as can be seen in the inset, they reach their maximum difference precisely at the end point of the momentum interval. This, in turn, indicates that the ultraviolet behavior of σ(q 2 ) is particularly sensitive to the low energy structure of the corresponding W(q 2 ). Quite interestingly, as can be seen in Fig. 5, when the corresponding ratios R 1 (q 2 ) and R 4 (q 2 ) are formed, the initial hierarchy σ 4 (q 2 ) > σ 1 (q 2 ) is inverted: now R 4 (q 2 ) < R 1 (q 2 ). As a result, the suppressed V 4 (ℓ 2 ) gives rise to the kernel K 4 (t) that is itself suppressed, with respect to K 1 (t), in the momentum interval between the zero crossing (at approximately 0.03 GeV 2 ) and µ 2 , which practically accounts for the entire value of the sum rule integral. Indeed, the support between the origin and the zero crossing is completely negligible; for example, in the case of K ⋆ (t), it contributes to the full answer a mere −0.017. Consequently, the logarithmic divergence encoded into L asym (q 2 ) is practically undetected by the sum rule.
Up until this point, the entire analysis has been carried out using a fixed L asym (q 2 ), namely the one given by Eq. (5.3), with the values of the parameters quoted below Eq. (5.5). It is clearly important to examine the stability of the result obtained for V ⋆ (ℓ 2 ) when variations in the form of L asym (q 2 ) are introduced. To that end, we repeat the previous procedure, using for L asym (q 2 ) the two limiting curves that demarcate the green shaded area in Fig. 3; the corresponding V ⋆ (ℓ 2 ) turn out to be particularly close to the original one, forming the narrow blue band shown in Fig. 6.
In Fig. 6 one may appreciate how the sum rule maps the original uncertainty in L asym (q 2 ) (green band) into a corresponding uncertainty in V ⋆ (ℓ 2 ) (blue band). In particular, note that, through the action of the function σ(q 2 ), the maximum separation is shifted from the deep infrared to the physically more interesting range of 300 MeV − 1.3 GeV, where the three V ⋆ (ℓ 2 ) reach their maximum mutual discrepancy of about 4%.

VI. DISCUSSION AND CONCLUSIONS
We have presented a couple of new sum rules, which originate from the STIs that connect the two-and three-point sectors of quenched QCD, in the Landau gauge. The key observations that are crucial for their derivation may be summarized as follows. In the context of two special kinematic configurations that involve a single momentum scale, the nonperturbative BC solutions may be interpreted as exactly solvable ordinary differential equations for the function J(q 2 ), known as the "kinetic" term of the gluon propagator. The general solutions of these differential equations predict the presence of poles (simple or double) at the origin; however, as dictated by its own SDE, J(q 2 ) must diverge at the origin only logarithmically. Nonetheless, as an appropriate expansion reveals, these two descriptions may be eventually reconciled, provided that certain integral conditions (sum rules) are exactly satisfied. These sum rules are referred to as "asymmetric" or "symmetric", depending on the kinematic configuration that has served as the starting point for their derivation.
The only element from the two-point sector that enters in these sum rules is the ghost dressing function; its behavior is particularly well-established, thanks to large-volume lattice simulation as well as a variety of continuous approaches. All remaining ingredients are related to the three-point sector, whose quantitative exploration, despite the considerable advances mentioned in the Introduction, remains a major technical challenge for the QCD practitioners. The sum rules may serve as a complementary tool in this ongoing quest, furnishing useful constraints for the various components entering in them.
For the purposes of the present work, we have opted for a mixed approach, where certain of the quantities, such as L asym (q 2 ), were obtained from the lattice, while the components related to the ghost-gluon kernel from the corresponding SDEs. Alternatively, one may resort to an entirely SDE-based analysis, along the lines of [52,54,55,67,[98][99][100], or to an exclusive functional renormalization group treatment, in the spirit of [14,38,57], such that all relevant quantities are computed within a self-contained framework, and their quality is subsequently assessed by means of the corresponding sum rule. In that vein, one may also envisage a purely lattice-driven approach, especially in the context of the asymmetric sum rule; such an effort would entail the simulation of the ghost-gluon kernel, and the extraction of the function W(q 2 ).
In the numerical study of Sec. V we have used the asymmetric sum rule in order to constrain the effective form factor V (ℓ 2 ), entering in the SDE that determines W(q 2 ). In that sense, the sequence V (ℓ 2 ) → W(q 2 ) → ε was considered, and the "perfect" V ⋆ (ℓ 2 ) (ε = 0) has been determined, which displays a distinct suppression (with respect to unity) in the intermediate range of momenta. It is important to emphasize that no initial bias regarding the suppression of V (ℓ 2 ) was built in; in fact, cases with notable enhancement have been considered (V 1 , V 2 ) , which, however, were clearly disfavored by the sum rule, as was the case with excessive suppression (V 4 ). Note finally, that, as mentioned at the end of Sec. V, the sum rules are completely insensitive to the presence of the zero crossing, and the subsequent logarithmic divergence, displayed by all the L asym (q 2 ) considered.
Within the confines of the present approach, the lack of information on L sym  (ii) Since J(q 2 ), F (q 2 ), H νµ , and Γ αµν are related by the STI of Eq. (2.9), the number of renormalization conditions that may be chosen freely is reduced down to three. Usually, in addition to J(µ 2 ) = 1, one imposes the condition F (µ 2 ) = 1, because lattice simulations of two point functions opt for this natural condition, and the various functional treatments comply, in order to facilitate the comparison of the results. Moreover, the lattice data for the special projection of Γ αµν that we consider have been renormalized such that L asym (µ 2 ) = 1; we will refer to this particular scheme as "asymmetric" MOM. Thus, at this point, within this particular scheme, the value of the H νµ at the renormalization point µ is completely fixed by the aforementioned STI.
(iii) The practical upshot of these considerations is that for the numerical evaluation of the sum rules one may not use the A i obtained in [69] together with the lattice results for L asym (q 2 ). This is so because in [69] the renormalization was carried out in the Taylor scheme [121], which requires that H νµ (q, p, r) collapses to its tree level value, g νµ in the softghost kinematics (p = 0); however, this value does not coincide with the one obtained from the STI when the asymmetric MOM scheme is employed. To remedy this inconsistency, the A i of [69] must undergo an appropriate rescaling, which will render them compatible with all other inputs.
(iv) As is well-known, the choice of renormalization scheme affects the finite part of the various cutoff-dependent renormalization constants, Z i , and the transition of the Green's functions from one scheme to the next may be conveniently described as the action of additional finite renormalizations constants. In that sense, the H νµ calculated in the Landau gauge is special, because its quantum corrections are known to be finite, and do not require an infinite renormalization. Therefore, the corresponding renormalization constant, Z 1 , is finite, and its numerical value may be directly used to describe the transition between the various schemes. Note in particular, that in the Taylor scheme, which will serve as our point of departure, Z 1 acquires the special value Z 1 = 1 [81]. Consequently, the transitions to other schemes will manifest themselves as deviations of Z 1 from unity. Clearly, the conversion of the A i into the asymmetric MOM scheme requires the use of such a constant, to be denoted by Z asym 1 .
(iv) We next determine approximately the value of Z asym 1 . Reserving the notation A 1 (q 2 ) and W(q 2 ) for the quantities defined in the asymmetric scheme, and denoting by A T 1 (q 2 ) and W T (q 2 ) their counterparts in the Taylor scheme, we have where we used the special result of the Taylor scheme, A T 1 = 1. Then, we set q 2 = µ 2 in Eq. (3.20), and evaluate it in the asymmetric and Taylor schemes; denoting the corresponding results by L asym (µ 2 ) andL asym (µ 2 ), respectively, we obtain Since, by definition, L asym (µ 2 ) = 1, taking the ratio of the two sides of Eq. (A2) yields Next, we assume that the renormalization point µ = 4.3 GeV used in the lattice simulations is sufficiently large for perturbation theory to be a reasonable approximation for Z asym 1 . Then, the one-loop results for the X i (q, −q, 0) of [58,95], which were computed in the Taylor scheme, can be used to approximateL asym (µ 2 ) bȳ where C A is the Casimir eigenvalue in the adjoint representation [N for SU(N)]. Substituting the above expression in Eq. (A3), yields (α s = 0.27) (v) An exactly analogous discussion holds for the symmetric configuration. The condition L sym (µ 2 ) = 1, employed for the lattice data in [59,63], defines a scheme that is distinct to the asymmetric MOM, mentioned above; by analogy, it is referred to as the "symmetric" MOM scheme. The conversion of the A i from the Taylor to this latter scheme proceeds by means of a finite renormalization constant, to be denoted by Z sym 1 , whose numerical value is not required in the present work.
Appendix B: Computing the function W(q 2 ) In this Appendix we provide details related with the determination of W(q 2 ). The one-loop dressed approximation of H νµ is obtained from the two diagrams of Fig. 7 [Eq. (3.1) of [69]], which are proportional to the ghost momentum p, as can be easily established by contracting the structures (ℓ + p) α and (ℓ − q) α by the Landau gauge propagators ∆ αβ (ℓ) and ∆ αβ (ℓ + r), respectively; their remainder defines the quantity K νµρ (q, p, r), appearing in the Eq. (3.11), where we set directly p = 0. Then, from Eq. (3.14) it is straightforward to deduce that W(q 2 ) can be extracted from K νµρ (q, 0, −q) through the projection W(q 2 ) = − 1 3 q ρ P µν (q) K νµρ (q, 0, −q) .
The implementation of Eq. (2.6) at the level of the three ghost-gluon vertices entering in the two diagrams of Fig. 7 reveals that only the form factors B 1 survive the projection in Eq. (B1). Therefore, the function B 2 does not contribute to W(q 2 ).