# Infrared facets of the three-gluon vertex

A. C. Aguilar, F. De Soto, M. N. Ferreira, J. Papavassiliou, J. Rodríguez-Quintero

IInfrared facets of the three-gluon vertex

A. C. Aguilar a , F. De Soto b , M. N. Ferreira a , J. Papavassiliou c , J. Rodríguez-Quintero d a University of Campinas - UNICAMP, Institute of Physics “Gleb Wataghin,” 13083-859 Campinas, São Paulo, Brazil b Dpto. Sistemas Físicos, Químicos y Naturales, Univ. Pablo de Olavide, 41013 Sevilla, Spain c Department of Theoretical Physics and IFIC, University of Valencia and CSIC, E-46100, Valencia, Spain d Dpto. Ciencias Integradas, Centro de Estudios Avanzados en Fis., Mat. y Comp., Fac. Ciencias Experimentales, Universidad de Huelva, Huelva 21071, Spain

Abstract

We present novel lattice results for the form factors of the quenched three-gluon vertex of QCD, in two special kinematic conﬁg-urations that depend on a single momentum scale. We consider three form factors, two associated with a classical tensor structureand one without tree-level counterpart, exhibiting markedly di ﬀ erent infrared behaviors. Speciﬁcally, while the former displaythe typical suppression driven by a negative logarithmic singularity at the origin, the latter saturates at a small negative constant.These exceptional features are analyzed within the Schwinger-Dyson framework, with the aid of special relations obtained from theSlavnov-Taylor identities of the theory. The emerging picture of the underlying dynamics is thoroughly corroborated by the latticeresults, both qualitatively as well as quantitatively. Keywords:

QCD, Three-gluon vertex, Lattice QCD, Schwinger-Dyson Equations

1. Introduction

The three-gluon vertex is a central component of QCD [1–3],being intimately linked to a variety of fundamental nonpertur-bative phenomena, and the scrutiny of its properties has re-ceived considerable attention in recent years [4–23]. A partic-ularly noteworthy feature of this vertex in the Landau gauge isthe infrared behavior of the form factors associated with theclassical (tree-level) tensorial structures. Speciﬁcally, as thespace-like momenta decrease from the ultraviolet to the infraredregime, the size of these form factors is gradually reduced, dis-playing the so-called “infrared suppression” [5–22]. This sup-pression culminates with the manifestation of a logarithmic di-vergence at the origin, which drives the form factors to negativeinﬁnity [9, 16, 19–22]As has been explained in detail in the recent literature, thisspecial behavior of the vertex originates from the interplay be-tween dynamical e ﬀ ects occurring in the two-point sector ofQCD [20–22]. In particular, while the gluon acquires dynam-ically an e ﬀ ective mass [24–28], responsible for the infraredsaturation of the Landau-gauge gluon propagator [29–47], theghost remains massless even nonperturbatively [36, 37, 48–50].As a result, loop diagrams containing ghost propagators furnishinfrared divergent logarithms, while gluonic loops, being “pro-tected” by the mass, are infrared ﬁnite.The formalism obtained from the fusion of the Pinch Tech-nique [24, 51–53] with the Background Field Method (PT-BFM) [54], known as “PT-BFM” scheme [35, 53, 55], is par-ticularly suitable for exposing this interplay, by combining theSchwinger-Dyson equations (SDEs) with the Slavnov-Tayloridentity (STI) satisﬁed by the three-gluon vertex [1–3, 56]. In the present work we employ this scheme to scrutinize fur-ther this dynamical picture, through the analysis of new resultsfrom quenched lattice simulations for three vertex form factors,deﬁned in two special kinematic conﬁgurations that involve asingle momentum scale.In particular, in the case of the two “classical” form factorssimulated, a considerable increase in the statistics permits usto obtain a cleaner signal of the infrared divergences that theydisplay, and accurately determine their strength. This new in-formation, in turn, enables us to probe more stringently, at thequantitative level, the underlying mechanisms associated withtheir emergence.In addition, we present for the ﬁrst time lattice results for aform factor that has no classical analogue. The dynamics ofthis purely quantum contribution may be described by meansof the corresponding SDE, and, in contradistinction to the clas-sical form factors, does not display any infrared divergences.The data obtained corroborate this prediction, being completelycompatible with a ﬁnite rather than a divergent contribution atlow momenta.

2. General considerations and theoretical setup

Our point of departure is the three-point correlation functioncomposed by SU(3) gauge ﬁelds, (cid:101) A a α ( q ), in Fourier space, G abc αµν ( q , r , p ) = (cid:104) (cid:101) A a α ( q ) (cid:101) A b µ ( r ) (cid:101) A c ν ( p ) (cid:105) = f abc G αµν ( q , r , p ) , (1)with q + r + p = G abc αµν ( q , r , p ) may be cast in the form G αµν ( q , r , p ) = g Γ αµν ( q , r , p ) ∆ ( q ) ∆ ( r ) ∆ ( p ) , (2) Preprint submitted to Physics Letters B February 10, 2021 a r X i v : . [ h e p - ph ] F e b here we have introduced the transversally projected ver-tex [21, 22], Γ αµν ( q , r , p ) = I Γ α (cid:48) µ (cid:48) ν (cid:48) ( q , r , p ) P α (cid:48) α ( q ) P µ (cid:48) µ ( r ) P ν (cid:48) ν ( p ) , (3)with I Γ denoting the usual one-particle irreducible (1PI) three-gluon vertex. In addition, g is the gauge coupling, and ∆ ( q )the scalar component of the gluon propagator, ∆ ab µν ( p ) = (cid:104) (cid:101) A a µ ( p ) (cid:101) A b µ ( − p ) (cid:105) = ∆ ( p ) δ ab P µν ( p ) , (4)with P µν ( p ) = g µν − p µ p ν / p , the standard transverse projector.Evidently, q α G αµν = r µ G αµν = p ν G αµν = L ( q , p , r ; λ ) = G αµν ( q , r , p ) λ αµν ( q , r , p ) λ αµν ( q , r , p ) λ αµν ( q , r , p ) , (5)where λ αµν is a transverse tensor whose form should be appro-priately chosen, depending on the kinematic conﬁguration em-ployed and the form factor that one wants to extract. In whatfollows we will focus our attention on the (i) totally symmetric and (ii) asymmetric conﬁgurations of the three-gluon vertex.In case (i) , the momenta conﬁguration is deﬁned by q = p = r : = s , such that q · r = q · r = r · p = − s / θ = (cid:98) qr = (cid:99) qp = (cid:98) rp = π/

3. The tensor structure of Γ is then re-duced down to [57, 58] Γ αµν ( q , r , p ) = Γ sym1 ( s ) λ αµν ( q , r , p ) +Γ sym2 ( s ) λ αµν ( q , r , p ) , (6)with the two tensors λ αµν ( q , r , p ) = Γ αµν ( q , r , p ) , (7a) λ αµν ( q , r , p ) = ( q − r ) ν ( r − p ) α ( p − q ) µ s ; (7b) Γ αµν ( q , r , p ) is the tree-level version of the vertex in Eq. (3).We next project out of L two particular combinations, de-noted by T sym i , each containing one of the Γ sym i , namely T sym i ( s ) : = g Γ sym i ( s ) ∆ ( s ) = L (cid:16) λ i (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) q = r = p : = s , (8)where λ αµν i ( q , r , p ) = (cid:88) j = β i j λ αµν j ( q , r , p ) , (9)with β = β = / β = /

11, and β =

1, such that λ i αµν ( q , r , p ) λ αµν j ( q , r , p ) = δ i j λ i αµν ( q , r , p ) λ αµν i ( q , r , p ) . (10)In case (ii) , the asymmetric conﬁguration corresponds to thekinematic limit p → r = − q and θ = (cid:98) qr = π . In these kine-matics we have [59] Γ αµν ( q , − q , = Γ asym3 ( q ) λ αµν ( q , − q , , (11)in terms of the single tensor, λ αµν ( q , − q , = q ν P αµ ( q ) , (12) which emerges after the implementation of the asymmetriclimit on the tensorial basis of the three-gluon vertex.A careful analysis reveals that the projection of Γ asym3 from L proceeds through contraction by λ αµν ( q , − q ,

0) itself, namely T asym3 ( q ) = g Γ asym3 ( q ) ∆ ( q ) ∆ (0) = L ( λ ) (cid:12)(cid:12)(cid:12) r = q ; p → . (13)Note that the limit p → path-independent , i.e. , does not depend on the angle formed between p and q .We next implement multiplicative renormalization by intro-ducing the standard renormalization constants, Z i , relating bareand renormalized quantities as ∆ R ( q ) = Z − A ∆ ( q ) , G R ( q , r , p ) = Z − / A G ( q , r , p ) , g R = Z / A Z − g , Γ R ( q , r , p ) = Z Γ ( q , r , p ) . (14)Within the momentum subtraction (MOM) scheme [60] thatwe use, the renormalized correlation functions must acquiretheir tree-level expressions at the subtraction point µ , e.g. , ∆ − ( µ ) = µ .Turning to the kinematic conﬁgurations (i) and (ii) , we im-pose, correspondingly, the MOM conditions Γ sym1 R ( µ ) = , Γ asym3 R ( µ ) = , (15)which deﬁne the symmetric and asymmetric MOM schemes,respectively [57–59, 61].Focusing on case (i) , we want to express Γ symR i ( s ) exclu-sively in terms of the bare lattice quantities ∆ and T sym i . Thismay be readily accomplished, since multiplicative renormaliza-tion entails that, for any correlation function G ( q ), the ratio G ( q ) / G ( q ) = G R ( q ) / G R ( q ) is a renormalization-group in-variant combination.In particular, forming the ratio T sym i ( s ) / T sym1 ( µ ) usingEq. (8), and employing the condition of Eq. (15), we ﬁnd Γ sym i R ( s ) = T sym i ( s ) T sym1 ( µ ) (cid:32) ∆ ( µ ) ∆ ( s ) (cid:33) with i = , . (16)Applying exactly analogous reasoning to the case (ii) , we obtain Γ asym3 R ( q ) = T asym3 ( q ) T asym3 ( µ ) (cid:32) ∆ ( µ ) ∆ ( q ) (cid:33) . (17)From this point on, we drop the subscript “R” from the renor-malized Γ i .

3. Connecting the two- and three-point sectors of QCD

In this section we present the salient features of PT-BFM ap-proach, pertinent to the the gluon propagator and three gluonvertex. The upshot of these considerations is the derivation oftheoretical expressions for the form factors Γ i , which will becontrasted with the new lattice results in the next section.Within the PT-BFM framework it is natural to cast the in-frared ﬁnite ∆ ( q ) as the sum of two distinct pieces [62] , ∆ − ( q ) = q J ( q ) + m ( q ) , (18)2here J ( q ) denotes the so-called “kinetic term”, while m ( q ) represents a momentum-dependent mass scale. Clearly, m (0) = ∆ − (0) is the saturation point of the gluon propagator.The emergence of m ( q ) hinges crucially on the structure ofI Γ αµν , entering in the SDE for ∆ ( q ). In particular, I Γ αµν must bedecomposed asI Γ αµν ( q , r , p ) = Γ αµν ( q , r , p ) + V αµν ( q , r , p ) , (19)where V αµν is comprised by longitudinally coupled masslesspoles , i.e. P αα (cid:48) ( q ) P µµ (cid:48) ( r ) P νν (cid:48) ( p ) V αµν ( q , r , p ) =

0, and Γ αµν denotesthe pole-free part of the vertex. By virtue of the above property, V αµν drops out from the Γ αµν in Eq. (3), and, consequently, thelattice projection of Eq. (5) depends only on Γ αµν . Γ αµν ( q , r , p ) is usually decomposed into a longitudinal and atransverse contribution [2, 3, 20, 22] Γ αµν ( q , r , p ) = Γ αµν L ( q , r , p ) + Γ αµν T ( q , r , p ) , (20)with q α Γ αµν T ( q , r , p ) = r µ Γ αµν T ( q , r , p ) = p ν Γ αµν T ( q , r , p ) = Γ αµν L ( q , r , p ) = (cid:88) i = X i ( q , r , p ) (cid:96) αµν i ( q , r , p ) , (21a) Γ αµν T ( q , r , p ) = (cid:88) i = Y i ( q , r , p ) t αµν i ( q , r , p ) , (21b)where the explicit form of the basis tensors (cid:96) i and t i is given inEqs. (3.4) and (3.5) of [20].Note that the tree-level expression for Γ αµν is recovered fromEq. (21a) by setting X = X = X =

1, and X i = Γ i interms of the X i and Y i . Speciﬁcally, in Euclidean space, weobtain Γ sym1 ( s ) = X ( s ) − s X ( s ) + s Y ( s ) − s Y ( s ) , Γ sym2 ( s ) = s X ( s ) − s Y ( s ) − s Y ( s ) , (22)where X i ( s ) ≡ X i ( s , s , s ) and Y i ( s ) ≡ Y i ( s , s , s ). More-over, one has Γ asym3 ( q ) = X ( q , q , − q X ( q , q , . (23)Past this point, we will determine the X i by resorting to aconstruction relying on the STIs satisﬁed by I Γ αµν , i.e. , p ν I Γ αµν ( q , r , p ) = F ( p )[ T µα ( r , p , q ) − T αµ ( q , p , r )] , (24)with T µα ( r , p , q ) : = ∆ − ( r ) P σµ ( r ) H σα ( r , p , q ) . (25) F ( p ) denotes the ghost dressing function, while H νµ ( q , p , r )is the ghost-gluon scattering kernel [2, 3, 63], whose tensorialdecomposition is given by [ A i ≡ A i ( q , p , r )] H νµ ( q , p , r ) = g νµ A + q µ q ν A + r µ r ν A + q µ r ν A + r µ q ν A . (26) J ( q ) ∝ ddq ... q = 0 + q ( c ) µ νµq ( c ) νµ ν ( c ) q Figure 1: SDE diagrams contributing to the derivative of the gluon propagatorat the origin. Blue (red) circles indicate fully dressed propagators (vertices).

The decompositions given in Eqs. (18) and (19) prompt theseparation of the above STI into two “partial” STIs, obtainedby implementing the matching Γ ↔ J and V ↔ m [62, 64].Based on this hypothesis, one may extend the BC constructionof [2] to the case of infrared ﬁnite gluon propagator, expressingthe X i in terms of the J , the F , and the A i . In particular, weobtain X ( s ) = Z sym1 F ( s ) J ( s ) R sym1 ( s ) , X ( s ) = Z sym1 F ( s ) (cid:104) J (cid:48) ( s ) R sym2 ( s ) + J ( s ) R sym3 ( s ) (cid:105) , (27)and X ( q , q , = Z asym1 F ( q ) J ( q ) R asym1 ( q ) , (28) X ( q , q , = Z asym1 F (0) (cid:104) J (cid:48) ( q ) R asym2 ( q ) + J ( q ) R asym3 ( q ) (cid:105) , where the R sym j and R asym j are linear combinations of the A i andtheir derivatives, whereas Z sym1 and Z asym1 are, respectively, the ﬁnite renormalization constants [65] of the ghost-gluon kernelin the symmetric and asymmetric MOM schemes, deﬁned inEq. (15).Note that this procedure leaves the transverse vertex formfactors Y i undetermined; nonetheless, as we will see in the nextsection, their qualitative features may be deduced from the cor-responding SDE governing the vertex Γ .The ingredients comprising Eqs. (27) and (28) are obtainedas follows. R sym j and R asym j may be computed using the SDEresults for the form factors A i presented in [21]. The valuesof Z sym1 and Z asym1 have been estimated by means of a one-loopcalculation in [65], while F ( q ) is accurately known both fromlattice simulations and functional studies. Finally, the gluonkinetic term J ( q ) requires a more elaborate treatment, which isoutlined below.To determine J ( q ), we ﬁrst compute m ( q ) from its owndynamical equation (see, e.g. , [22]); the result is shown in theinset of Fig. 2. Then, we subtract the m ( q ) from the latticedata for the gluon propagator [30], by employing Eq. (18), i.e. , J ( q ) = [ ∆ − ( q ) − m ( q )] / q . While this procedure is com-pletely stable for a wide range of momenta, it becomes lessreliable as q →

0, due the fact that J ( q ) diverges logarithmi-cally at the origin, e.g. , J ( q ) (cid:39) q → a ln( q /µ ) + b , (29)as a direct consequence of the the nonperturbative masslessnessof the ghost [9].It turns out that the behavior of J ( q ) near the origin may becomputed from the SDE of the gluon propagator, by recogniz-ing that, in the limit q →

0, di ﬀ erentiation with respect to q J ( q ), e.g. , d ∆ − ( q ) / dq = q → J ( q ) + . . . , (30)where the ellipses denote infrared ﬁnite terms.The direct di ﬀ erentiation of the diagrams contributing to theSDE of ∆ ( q ) [see Fig. 1] leads to major technical simpliﬁca-tions, yielding ﬁnally the value of a ≈ . c i contribute to the value of a . Speciﬁ-cally, diagram ( c ) furnishes the primary diverge, owing to themasslessness of the ghost propagators, while ( c ) and ( c ) con-tribute secondary divergences, due to fully-dressed three-gluonvertices attached to the their external leg (Lorentz index ν ).Thus, the combined treatment furnishes J ( q ) for the entirerange of q , as shown in Fig. 2. q [GeV] -1-0.500.51 J ( q ) q [GeV] m ( q ) [ G e V ] Figure 2: Gluon kinetic term, J ( q ), and gluon mass, m ( q ) (inset). Whencombined according to Eq. (18), they reproduce accurately the lattice dataof [30] for ∆ ( q ). Finally, putting together all ingredients described above, weobtain from Eqs. (27) and (28) the SDE-derived result for the X i in the two kinematic conﬁgurations of interest. In partic-ular, in the asymmetric limit we obtain the X ( q , q ,

0) and q X ( q , q ,

0) shown in Fig. 3.As will become apparent in the next section, X divergeslogarithmically, inheriting directly from Eqs. (27) and (28) thecorresponding logarithmic divergence of the J ( q ), given byEq. (29). Instead, while X is dominated by the divergent J (cid:48) ( q ), the combinations s X ( s ) and q X ( q , q ,

0) appearingin Eqs. (22) and (23) saturate to ﬁnite constants in the infrared.

4. Presentation and analysis of the results

The lattice evaluation of the form factors Γ sym1 , and Γ asym3 pro-ceeds through the direct simulation of the projections T i and ofthe gluon propagator ∆ [Eqs. (8), (13) and (4)], and subsequentuse of Eqs. (16) and (17), respectively. This is accomplished byexploiting lattice gauge ﬁeld conﬁgurations obtained from sim-ulations with the Wilson action on a 48 lattice at β = q [GeV] -1-0.500.51 q i - X i ( q , q , ) i=1i=3 Figure 3: The form factor X ( q , q ,

0) (black continuous) which composes thetree-level tensor structure of the full three-gluon vertex and the dimensionlesscombination q X ( q , q ,

0) (red dashed). conﬁgurations) and 5.6 (980 conﬁgurations), and on a 52 lat-tice at β = lat-tice [16, 19], making thereby apparent that di ﬀ erent discretiza-tions of the QCD action provide practically the same results forthe three-gluon form factors. In the case of Γ sym1 , for the sakeof both comparison and implementation of the renormalizationcondition at µ = ﬀ ect of dynamical quarks.The implications of this approximation has been recently as-sessed in [21], where only minor quantitative but no qualitativee ﬀ ects have been detected.The new lattice results are shown in Figs. 4 and 5. It shouldbe stressed that the results for Γ sym1 and Γ asym3 are considerablyimproved with respect to previous analyses [16, 19], capitaliz-ing on a better statistical sample and the careful treatment ofdiscretization artifacts, especially for propagators [66]. In ad-dition, to the best of our knowledge, results for the form factor Γ sym2 are presented for the ﬁrst time in this letter.As a very apparent and distinctive feature, Γ sym1 and Γ asym3 clearly display the infrared suppression previously re-ported [16, 19, 21], accompanied by the characteristic logarith-mic divergence near the origin. Instead, Γ sym2 appears to saturateat a small negative constant at low momenta. As we explain be-low, these behaviors are well understood within the context of For Eq. (16) to work properly , the bare quantities evaluated both at s and µ must be computed from conﬁgurations simulated at the same β , such thatthe cut-o ﬀ dependence, assumed to be multiplicative, cancels out in the ratios.Therefore, as the accessible momenta to β = Γ sym1 also applies for Γ sym2 . s [GeV] -1-0.500.511.5 G s y m ( s ) Latt [ b =5.8, L=48]Latt [ b =5.6, L=48]Latt [ b =5.6, L=52]Latt [ b =5.9-6.0]Latt [ b =3.9, L=64]0 1 2 s [GeV] -1-0.500.51 G s y m ( s ) Latt [ b =5.8, L=48]Latt [ b =5.6, L=48]Latt [ b =5.6, L=52]Latt [ b =3.9, L=64] Figure 4: Results for Γ sym1 ( s ) (upper panel) and Γ sym2 ( s ) (lower) obtainedfrom three simulations with the Wilson action in a 48 lattice at β = β = lattice at β = lattice at β = Γ sym1 covering a range of larger momenta (turquoise),obtained from several β ’s and volumes and previously published in [67, 68],have been also used to ﬁx the subtraction point at µ = the SDE analysis of the previous section.Quite interestingly, the transverse form factors Y i of thefull vertex do not contribute to the projection Γ asym3 ( q ) in theasymmetric limit, which is thus completely determined throughEq. (23) by X ( q , q ,

0) and X ( q , q , Γ asym3 ( q ), given by the black continuous curve in Fig. 5; ev-idently, the coincidence with the lattice data is rather notable.This ﬁne agreement may be ultimately attributed to the accuratedetermination of J ( q ) over the full range of momenta, follow-ing the considerations outlined in section 3.We next focus on the features of the Γ i in the deep infrared,contrasting the common behavior of Γ sym1 and Γ asym3 to that of Γ sym2 . The upshot of this comparison is that, while the formerquantities display the infrared divergence known from previousstudies, the latter saturates at a ﬁnite constant.To that end, we turn to the SDE for the three-gluon ver- q [GeV] -0.500.511.5 G a s y m ( q ) Latt [ b =5.8, L=48]Latt [ b =5.6, L=48]Latt [ b =5.6, L=52]Latt [ b =5.9-6.0]SDE Figure 5: Results for Γ asym3 ( q ) obtained from the same lattice simulations withthe Wilson action quoted in the caption of Fig. 4 (same color code). The blacksolid line corresponds to the SDE-based computation which, in the asymmetriclimit, determines entirely the transversally projected three-gluon vertex. tex, shown in Fig. 6, and study the transverse form factors Y i , which are not accessible through the STI-based construc-tion of the previous section. In particular, a detailed analysisin the symmetric limit reveals that, as s → Y ( s ) ∼ c / s and Y ( s ) ∼ d / s , for constants c ≈ − .

07 and d ≈ − .

20, and,consequently, the combinations s Y and s Y appearing inEq. (22) approach constant values at the origin.The approximation of the Y i ( s ) through the SDE of Fig. 6proceeds as follows. First, the projectors that extract the scalarform factors Y i from the tensor structure of the full vertex weredetermined algebraically. Then, it was veriﬁed that the dia-gram ( d ) and its permutations do not contribute to the Y i aslong as the four-gluon vertex entering there is kept at tree level.The ghost-gluon and three-gluon vertices appearing in diagrams( d ) and ( d ) are then approximated by retaining only form fac-tors that possess a nonvanishing tree-level value.Speciﬁcally, for the three-gluon vertex, we keep only theform factors X , X and X [see Eq. (21)], whereas the ghost-gluon vertex is approximated to Γ µ ( q , p , r ) = q µ B ( q , p , r ),where q , r and p , denote the momenta of the anti-ghost, ghostand gluon, respectively. Furthermore, to simplify the numeri-cal treatment, these form factors are all considered as functionsof the single momentum scale, s , and evaluated in their cor-responding totally symmetric limits. Finally, for the ghost andgluon propagators we use ﬁts to lattice data, and for the form = − ++ ( d ) ( d ) ( d ) Figure 6: The SDE of the three-gluon vertex at the one-loop dressed level.Blue (red) circles indicate fully dressed propagators (vertices). .1 1 s [GeV] -1-0.500.51 G , s y m ( s ) G G a SDE G = G G = G G = G Figure 7: [upper panel] Γ sym1 ( s ) (orange) and Γ sym2 ( q ) (green) plotted interms of the momentum in logarithmic scale. The solid magenta and blue linesshow the asymptotic infrared behavior for the two form factors according, re-spectively, to Eqs. (32) and (33), the former supplemented by an intercept ﬁttedas explained in the text. [lower panel] The logarithmic derivatives of Γ sym1 , ( s )and Γ asym3 ( q ) computed from the ﬁt of f ( x ) = α ln x + β [ x = q , s ] to latticedata (solid circles), with errors representing the statistical deviation from theseﬁts, compared to their SDE estimates in Eqs. (32) and (33) (solid lines). Thegrey bands for Γ sym1 ( s ) and Γ asym3 ( q ) display the uncertainty obtained fromthe SDE value of a , by propagating in it a systematic error of 5 %. factors X i ( s ) and B ( s ) we use the results of Refs [20, 63].Thus, one reaches the conclusion that the only term that fur-nishes logarithmically divergent contributions to the Γ i throughEqs. (22) and (23) is X , while all others provide numerical con-stants, i.e. , X ( s ) (cid:39) s → Z sym1 F ( s ) J ( s ) (cid:39) s → Z sym1 F (0) (cid:104) a ln( s /µ ) + b (cid:105) , s X ( s ) (cid:39) s → − Z sym1 F ( s ) s J (cid:48) ( s ) (cid:39) s → − Z sym1 F (0) a , s Y ( s ) (cid:39) s → c , s Y ( s ) (cid:39) s → d , (31)Then, from Eqs. (22), (23) and (31) follows that the leading infrared contributions of Γ sym1 ( s ) and Γ asym3 ( q ) are given by Γ sym1 ( s ) (cid:39) s → Z sym1 F (0) a ln( s /µ ) ≈ . s /µ ) , (32) Γ asym3 ( q ) (cid:39) q → Z asym1 F (0) a ln( q /µ ) ≈ . q /µ ) . Evidently, both form factors diverge logarithmically to −∞ atthe origin, as captured by the corresponding Figs. 4 and 5, re-spectively.Instead, in the same limit, Γ sym2 ( s ) saturates to a negativeconstant near zero, Γ sym2 ( s ) (cid:39) s → − (cid:34) Z sym1 F (0) a + c + d (cid:35) ≈ − . . (33)In obtaining the numerical values quoted above, the ﬁnite renor-malization constants Z sym1 and Z asym1 were evaluated perturba-tively at µ = F (0) = µ ). In addition, as mentionedabove, a ≈ . c ≈ − .

07 and d ≈ − .

20. The errors have been estimated and displayed inEqs. (32) and (33), for illustrative purposes, through the prop-agation in them of an uncertainty of 5 % in the determinationof a , c and d . Note that the numerical di ﬀ erence in the loga-rithmic slopes of Γ sym1 and Γ sym3 in Eq. (32) is entirely due to thedi ﬀ erence between Z sym1 and Z asym1 .The asymptotic behaviors of Γ sym1 ( s ) and Γ sym2 ( s ), given inEqs. (32) and (33), are next compared to the lattice data; the re-sults of this comparison are shown in the upper panel of Fig. 7.Speciﬁcally, we introduce the function f ( x ) = α ln x + β , whichrepresents a straight line on a logarithmic plot ( x = s , q ).Then, in the case of Γ sym1 ( s ), the slope α is ﬁxed at the valuepredicted by Eq. (32), namely α = .

11, while the value of itsintercept β (not predicted by our calculation) is adjusted suchthat one gets the best ﬁt to lattice data below s = Γ sym2 ( s ), one simply ﬁxes α and β at their theoretical values α = β = − .

006 (no ﬁtting), thus obtaining the blue line.A second comparison involves the logarithmic slopes of Γ sym1 , ( s ) and Γ asym3 ( q ). In particularly, we now ﬁt the latticedata below 0.5 GeV with f ( x ), treating both α and β as free pa-rameters, determined by a least-squares ﬁt, including statisticalerrors. The resulting values for α , together with the associatederrors, are then compared with the theoretical predictions, asshown in the lower panel of Fig. 7. In all cases, the agreementis excellent, indicating a consistent picture from both SDE andlattice computations.

5. Conclusions

In this work we have explored crucial nonperturbative as-pects of the quenched three-gluon vertex through the combina-tion of new lattice data obtained from large-volume simulationsand a detailed SDE-based analysis within the PT-BFM frame-work.To begin with, we have acquired a clearer view of the in-frared logarithmic divergences associated with the form factors Γ sym1 and Γ asym3 by reducing considerably the statistical errors ofthe lattice simulation. Thus, the presence of these characteristicdivergences, already identiﬁed in earlier studies (see e.g. , [16]),6s further supported by the present data. In addition, lattice re-sults for the form factor Γ sym2 are reported here for the ﬁrst time,strongly supporting its ﬁniteness at the origin.The new lattice results o ﬀ er an invaluable opportunity to fur-ther scrutinize key dynamical mechanisms from new angles andperspectives. In particular, the nonperturbative features of theLandau-gauge two-point sector of QCD, especially the infraredﬁniteness of the gluon propagator and the ghost dressing func-tion, are instrumental for obtaining infrared divergent Γ sym1 and Γ asym3 , and, and the same time, a ﬁnite Γ sym2 . The observed agree-ment between lattice and SDE results clearly corroborates thephysical picture put forth, and bolsters up the conﬁdence in thepredictivity of continuous functional methods in general. Acknowledgments

The work of A. C. A. is supported by the CNPq grant307854 / / / / / /

087 of the GeneralitatValenciana. F. D. S. and J. R. Q. are supported the SpanishMICINN grant PID2019-107844-GB-C2, and regional Andalu-sian project P18-FR-5057.

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