Exploring smoking-gun signals of the Schwinger mechanism in QCD

Exploring smoking-gun signals of the Schwinger mechanism in QCD A. C. Aguilar ,1M. N. Ferreira ,1and J. Papavassiliou 2 1University of Campinas—UNICAMP, Institute of Physics “Gleb Wataghin”, 13083-859 Campinas, São Paulo, Brazil 2Department of Theoretical Physics and IFIC, University of Valencia and CSIC, E-46100 Valencia, Spain (Received 22 November 2021; accepted 10 January 2022; published 27 January 2022) In Quantum Chromodynamics, the Schwinger mechanism endows the gluons with an effective mass through the dynamical formation of massless bound-state poles that are longitudinally coupled. The presence of these poles affects profoundly the infrared properties of the interaction vertices, inducing crucial modifications to their fundamental Ward identities. Within this general framework, we present a detailed derivation of the non-Abelian Ward identity obeyed by the pole-free part of the three-gluon vertex in the softgluon limit, and determine the smoking-gun displacement that the onset of the Schwinger mechanism produces to the standard result. Quite importantly, the quantity that describes this distinctive feature coincides formally with the bound-state wave function that controls the massless pole formation. Consequently, this signal may be computed in two independent ways: by solving an approximate version of the pertinent BetheSalpeter integral equation, or by appropriately combining the elements that enter in the aforementioned Ward identity. For the implementation of both methods we employ twoand three-point correlation functions obtained from recent lattice simulations, and a partial derivative of the ghost-gluon kernel, which is computed from the corresponding Schwinger-Dyson equation. Our analysis reveals an excellent coincidence between the results obtained through either method, providing a highly nontrivial self-consistency check for the entire approach. When compared to the null hypothesis, where the Schwinger mechanism is assumed to be inactive, the statistical significance of the resulting signal is estimated to be 3 standard deviations. DOI: 10.1103/PhysRevD.105.014030 I. INTRODUCTION The systematic study of the fundamental n-point correlation (Green’s) functions, such as propagators and vertices, forms an essential element in the ongoing quest for unraveling the nonperturbative properties and underlying dynamical mechanisms of Quantum Chromodynamics (QCD) [1]. In recent years, this challenging problem has been tackled by means of approaches formulated in the continuum, such as the Schwinger-Dyson equations (SDEs) [2–12] or the functional renormalization group [13–17], in conjunction with numerous gauge-fixed lattice simulations [18–27].This intense activity has delivered new insights on the nature and phenomenology of the strong interactions and has broadened our basic understanding of non-Abelian gauge theories [28–48]. In this context, the characteristic feature of infrared saturation displayed by the gluon propagator has attracted particular attention, being often linked with the emergence of a mass gap in the gauge sector of QCD [49–67]. This property has been explored both in large-volume simulations [18–24], and in various functional approaches [61–73], and is rather general, manifesting itself in the Landau gauge, away from it [74–80], and in the presence of dynamical quarks [81–85]. In general terms, the scalar form factor, Δðq2Þ, of the gluon propagator reaches a finite nonvanishing value in the deep infrared, and the gluon mass, m, is identified as Δ−1ð0Þ¼m2. One of the nonperturbative mechanisms put forth in order to explain this special behavior of the gluon propagator is based on a non-Abelian extension of the wellknown Schwinger mechanism [86,87]. According to the fundamental observation underlying this mechanism, if the self-energy develops a pole at zero momentum transfer (q2¼0), then the corresponding vector meson (gluon) acquires a mass, even if the gauge symmetry forbids a mass term at the level of the fundamental Lagrangian [86–89]. The precise implementation of this idea at the level of the SDE describing the momentum evolution of Δðq2Þrequires the inclusion of longitudinally coupled massless poles at the level of the fundamental interaction vertices of the theory [90–94]. These poles are produced as massless bound state excitations, whose formation is governed by a special set of Bethe-Salpeter equations (BSEs) [90–93].In addition, their presence is crucial for maintaining intact the Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 105, 014030 (2022) 2470-0010=2022=105(1)=014030(26) 014030-1 Published by the American Physical Society form of the Slavnov-Taylor identities (STIs) [95,96] satisfied by the corresponding vertices. Since the fully dressed vertices enter in the diagrammatic expansion of the gluon SDE, their massless poles end up triggering the Schwinger mechanism, enabling a completely dynamical generation of an effective gluon mass [64,90–93]. It is clearly important to further scrutinize the dynamical picture described above, and identify certain characteristic properties that would corroborate its validity and discriminate it from alternative dynamical scenarios. In the present work we explore a distinctive signal of the non-Abelian Schwinger mechanism, which is intimately connected with the three-gluon vertex, and has the advantage of being reliably calculable by means of well-established inputs, such as twoand three-point correlation functions obtained from large-volume lattice simulations. The pivotal ideas underlying this study may be summarized as follows. The massless poles are longitudinally coupled, and therefore drop out from “on-shell”observables [49,88,89,97,98], or from the transversely projected vertices employed in lattice simulations [85,99–103], where only the pole-free part of the corresponding vertex survives. Nonetheless, the imprint of the poles is invariably encoded into the pole-free part, as may be seen by considering the Ward identity (WI) that this latter part satisfies, namely the limit of the STI as the gluon momentum in the channel of the pole is taken to zero.1 Since the poles contribute nontrivially to the STIs, the corresponding WI involves the standard building blocks (e.g., propagators) and a residual contribution with a nontrivial momentum dependence, which is directly related to the Schwinger mechanism. As a result, in that kinematic limit, the relevant form factor of the pole-free part of the vertex is displaced with respect to the case where the Schwinger mechanism is absent. The above considerations become particularly relevant in the case of the three-gluon vertex, because the form factor of its pole-free part has been evaluated rather accurately in recent lattice simulations [104–106]. As a result, the displacement originating from the onset of the Schwinger mechanism, to be denoted by Cðr2Þ, may be calculated by appropriately combining this form factor with all other constituents that enter into the WI of the three-gluon vertex; all of them are available from lattice simulations, with the exception of a particular partial derivative, denoted by Wðr2Þ, related to the ghost-gluon kernel that appears in the STI [107,108]. The importance of the calculation put forth above becomes particularly transparent when an additional theoretical ingredient is taken into account. Specifically, as will become clear in the main body of the article, Cðr2Þ coincides exactly with the wave function amplitude of the massless bound state poles associated with the threegluon vertex [90–93]. Thus, the form of Cðr2Þis determined from an entirely different procedure, namely as the solution of the BSE mentioned earlier. This solution, in turn, serves as a benchmark of our analysis, in the sense that signals emerging from the WI treatment are expected to be qualitatively compatible with the Cðr2Þobtained from the BSE [90–93]. Our numerical analysis reveals that the Cðr2Þconstructed by putting together all the ingredients of the WI deviates markedly from zero, showing an impressive resemblance to the results obtained from the corresponding BSE. On average, the signal obtained is 3.1σaway from the null hypothesis value, Cðr2Þ¼0, which corresponds to the absence of the Schwinger mechanism. Moreover, for momenta r≥2GeV the deviation of the signal from Cðr2Þ¼0exceeds the 5σ, owing to a characteristic peak of Cðr2Þin the vicinity of 2 GeV, and to the fact that the error bars assigned to the lattice points get reduced as one moves away from the deep infrared region. Let us finally mention that the principal uncertainty associated with the WI determination originates from the computation of the function Wðr2Þ, which is not available from lattice simulations, and has been approximated by a truncated version of the SDE of the ghost-gluon kernel. As was explained in [108], the simulation of this function on the lattice is theoretically conceivable, but practically rather cumbersome. The article is organized as follows. In Sec. II we explain in an Abelian context how the presence of longitudinally coupled massless poles modifies the form of the WI satisfied by the pole-free part of a vertex. In Sec. III we derive the corresponding WI for the pole-free part of the three-gluon vertex, introducing the displacement function Cðr2Þ. Then, in Sec. IV we express Cðr2Þin terms of the three-gluon form factor, the gluon propagator and its derivative, the ghost dressing function, and the function Wðr2Þ. Next, in Sec. Vwe present the BSE determination of Cðr2Þ. In Sec. VI we use lattice inputs for the components of the WI in order to determine the form of Cðr2Þ, and compare it to the corresponding result obtained from the BSE. In Sec. VII we discuss issues related to Cðr2Þand its physical implications. Finally, in Sec. VIII we present our discussion and conclusions. In addition, certain topics have been relegated to three appendices: Appendix A contains technical details of the BSE treatment, in Appendix Bwe discuss the SDE-based determination of Wðr2Þ, while in Appendix Cwe collect the fits employed in our numerical analysis. 1The standard Takahashi identity of QED, qμΓμðq; p; pþqÞ¼S−1ðpþqÞ−S−1ðpÞ, is an Abelian STI; the corresponding WI, Γμð0;p;−pÞ¼∂S−1ðpÞ=∂pμ, is obtained from it by expanding around q¼0. AGUILAR, FERREIRA, and PAPAVASSILIOU PHYS. REV. D 105, 014030 (2022) 014030-2 II. WARD IDENTITIES IN THE PRESENCE OF MASSLESS POLES In this section we focus on the modifications induced to the form of the WIs when the vertices involved contain longitudinally coupled massless poles, which is one of the trademarks of the Schwinger mechanism at the level of the vertices. In general, the derivation of the WI from the corresponding Takahashi identity, or, in general, from a given STI, involves a Taylor expansion around the contracting momentum [67]. In the case of a function of a single variable, fððpþqÞ2Þ, such as a propagator, the Taylor expansion proceeds through the elementary formula (q→0) fððpþqÞ2Þ¼fðp2Þþqα∂fðp2Þ ∂pαþOðq2Þ ¼fðp2Þþ2ðq·pÞ∂fðp2Þ ∂p2þOðq2Þ:ð2:1Þ For a function fðq; r; pÞ,withqþrþp¼0, such as a three-particle vertex or kernel, the Taylor expansion around q¼0(and p¼−r)gives fðq; r; pÞ¼fð0;r;−rÞþqα∂fðq; r; pÞ ∂qαq¼0þOðq2Þ: ð2:2Þ Note that if the fðq; r; pÞ¼−fðq; p; rÞ, as happens in the case of the term associated with the massless pole in the q channel (see below), then fð0;r;−rÞ¼0. In order to fix the ideas, we employ a vertex with reduced tensorial structure, which obeys an Abelian STI. In particular, we consider one of the typical vertices of the background field method (BFM) [109–116], namely the BðqÞ¯ cðrÞcðpÞ vertex, where Bdenotes the background gluon2and ¯ c(c)the antighost (ghost) fields. Due to the residual invariance of the action under background gauge transformations, this vertex satisfies an Abelian STI that relates it to the inverse ghost propagator. Specifically, suppressing the gauge coupling g and the color factor fabc, and denoting the remainder of the vertex by ˜ Γαðq; r; pÞ,wehave[6,60] qα˜ Γαðq; r; pÞ¼D−1ðp2Þ−D−1ðr2Þ;ð2:3Þ where the ghost propagator is given by Dabðq2Þ¼ iδabDðq2Þ. Note that, at tree level, ˜ Γα 0ðq; r; pÞ¼ðr−pÞα. At this point we will assume that the Schwinger mechanism is inactive, such that the form factors comprising ˜ Γαðq; r; pÞdo not contain poles. In that case, one may carry out the Taylor expansion of both sides of Eq. (2.3) according to Eqs. (2.1) and (2.2). Specifically, the left-hand side (l.h.s) of (2.3) yields ½l:h:s¼qα˜ Γαð0;r;−rÞþOðq2Þ;ð2:4Þ while the right-hand side (r.h.s) is simply given by ½r:h:s¼qα∂D−1ðr2Þ ∂rαþOðq2Þ:ð2:5Þ Then, equating the coefficients of the terms linear in qαon both sides, one obtains the simple, QED-like relation [67] ˜ Γαð0;r;−rÞ¼∂D−1ðr2Þ ∂rα:ð2:6Þ Finally, at the level of the single form factor comprising ˜ Γαð0;r;−rÞ, namely ˜ Γαð0;r;−rÞ¼ ˜ Aðr2Þrα;ð2:7Þ we obtain directly from Eq. (2.6) [67] ˜ Aðr2Þ¼2∂D−1ðr2Þ ∂r2:ð2:8Þ Let us now turn the Schwinger mechanism on, and denote the resulting full vertex by e IΓαðq; r; pÞ. The vertex e IΓαðq; r; pÞ, diagrammatically represented in Fig. 1,is comprised by two distinct pieces, e IΓαðq; r; pÞ¼ ˜ Γαðq; r; pÞþ˜ Vαðq; r; pÞ;ð2:9Þ where ˜ Γαðq; r; pÞcontains all pole-free contributions, while the pole term ˜ Vαðq; r; pÞhas the general form [49,88,89,97,98], ˜ Vαðq; r; pÞ¼qα q2 ˜ Cðq; r; pÞ:ð2:10Þ We emphasize that the pole-free terms ˜ Γαðq; r; pÞare different when the Schwinger mechanism is turned on or off. In particular, the infrared finiteness of the gluon propagator affects the behavior of all other Green’s functions, due to its nontrivial interconnection with them imposed by the corresponding coupled SDEs. A typical qualitative example of the type of modifications that the emergence of a gluonic mass scale induces to one-loop contributions is the conversion of “unprotected”logarithms into “protected”ones, according to lnðq2=μ2Þ→ ln½ðq2þm2Þ=μ2[101,117]. 2Within the BFM, the gauge field Aa αis decomposed as Aa α¼Ba αþQa α, where Ba αis the background field and Qa αis the quantum (fluctuating) field. EXPLORING SMOKING-GUN SIGNALS OF THE SCHWINGER …PHYS. REV. D 105, 014030 (2022) 014030-3 Evidently, combining Eqs. (2.9) and (2.10) we get qαe IΓαðq; r; pÞ¼qα˜ Γαðq; r; pÞþ ˜ Cðq; r; pÞ;ð2:11Þ thus, the contraction by qαcancels the massless pole in q2. We next assume that the Becchi-Rouet-Stora-Tyutin (BRST) symmetry [118,119] of the theory remains intact as the Schwinger mechanism becomes operational. In this context, there are three important points that are worth emphasizing. First, a sharp distinction must be drawn between the standard BRST symmetry, understood throughout the present work, and the modified nonperturbative BRST symmetry exhibited by the refined GribovZwanziger action [68], as first demonstrated in [120]. Second, the nonperturbative quartet mechanism (see [121] and references therein) may be intrinsically related with the Schwinger mechanism, in the sense that it too relies on the dynamical formation of massless bound states (see additional comments in Sec. VIII). Third, motivated by arguments based mostly on the Kugo-Ojima formalism [122], a nonvanishing infrared-finite gluon propagator has often been associated with a soft breaking of the standard BRST symmetry (see, e.g., [3,62]); nonetheless, this conclusion depends on the precise mechanism responsible for this special behavior. In particular, the Schwinger mechanism, implemented via longitudinally coupled massless poles [49,50,61,64,89,90,98,123], appears to be compatible with an unbroken BRST symmetry [67,124], at least at the level of the STIs satisfied by the elementary vertices. In particular, the STIs retain their standard form, but are now realized through the nontrivial participation of the massless pole terms. Accordingly, the full e IΓαðq; r; pÞsatisfies, as before, precisely (2.3), namely qαe IΓαðq; r; pÞ¼D−1ðp2Þ−D−1ðr2Þ;ð2:12Þ where Dðq2Þis the ghost propagator in the presence of the Schwinger mechanism. For the same reasons described above for the case of ˜ Γαðq; r; pÞ,Dðq2Þalso differs from the corresponding quantity when the Schwinger mechanism is not operational. Then, using Eq. (2.11), we obtain for the pole-free part qα˜ Γαðq;r;pÞ¼½D−1ðp2Þ−D−1ðr2Þ− ˜ Cðq;r;pÞ:ð2:13Þ The WI obeyed by ˜ Γαðq; r; pÞmay be derived again by means of a Taylor expansion, since, after the contraction by qα, all terms appearing in the STI of Eq. (2.13) contain no poles, as q→0. In particular, qα˜ Γαð0;r;−rÞ¼ ˜ Cð0;r;−rÞ þqα∂D−1ðr2Þ ∂rα−∂ ˜ Cðq; r; pÞ ∂qαq¼0 þOðq2Þ:ð2:14Þ The comparison between Eqs. (2.14) and (2.5) reveals that the only zeroth-order contribution, namely ˜ Cð0;r;−rÞ, must vanish, ˜ Cð0;r;−rÞ¼0:ð2:15Þ Note that the result of Eq. (2.15) may be independently obtained from the property ˜ Cðq; r; pÞ¼− ˜ Cðq; p; rÞ, which follows directly from the general ghost-antighost symmetry of the BðqÞ¯ cðrÞcðpÞvertex. Then, the matching of the terms linear in qyields the WI ˜ Γαð0;r;−rÞ¼∂D−1ðr2Þ ∂rα−∂ ˜ Cðq; r; pÞ ∂qαq¼0 |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} WI displacement ;ð2:16Þ which, when compared to that of Eq. (2.6),is“displaced” by the partial derivative of the form factor associated with the pole term. In order to determine the displaced analog of Eq. (2.8), we set ∂ ˜ Cðq;r;pÞ ∂qαq¼0¼2rα ˜ Cðr2Þ; ˜ Cðr2Þ≔∂ ˜ Cðq;r;pÞ ∂p2q¼0 ; ð2:17Þ and obtain immediately from Eqs. (2.7) and (2.16) (a) (b) (c) FIG. 1. (a) Compact diagrammatic representation of the vertices e IΓαðq; r; pÞ(Bgluon) and IΓαðq; r; pÞ(Qgluon); (b) the pole-free parts, ˜ Γαðq; r; pÞand Γαðq; r; pÞ; (c) the pole parts, ˜ Vαðq; r; pÞand Vαðq; r; pÞ. The black circle denotes the “transition amplitude,” describing the mixing of the gluon with a massless excitation; its detailed diagrammatic content may be found in [91]. AGUILAR, FERREIRA, and PAPAVASSILIOU PHYS. REV. D 105, 014030 (2022) 014030-4 ˜ Aðr2Þ¼2∂D−1ðr2Þ ∂r2− ˜ Cðr2Þ:ð2:18Þ Note that the displacement of the WI exemplified above becomes especially relevant within the framework that combines the pinch technique (PT) [6,50,55,125] with the BFM, known as “PT-BFM scheme”[60,126]. In particular, the action of terms such as ˜ Cðr2Þis instrumental for the evasion of a powerful nonperturbative cancellation that operates at the level of the gluon SDE [127], which would otherwise enforce the result Δ−1ð0Þ¼0. In fact, the contribution of the ghost loop to the nonvanishing Δ−1ð0Þ, to be denoted by Δ−1 gh ð0Þ,isgivenby[67] Δ−1 gh ð0Þ∼Zd4kk2D2ðk2Þ ˜ Cðk2Þ:ð2:19Þ Let us finally point out that the displacement associated with the conventional ghost-gluon vertex IΓαðq; r; pÞ(see Fig. 1), to be denoted by Cðr2Þ[see Eq. (5.7)], is related to ˜ Cðr2Þby the simple relation Cðr2Þ¼Fð0Þ ˜ Cðr2Þ;ð2:20Þ where we have introduced the ghost dressing function, Fðq2Þ, related to the ghost propagator by Fðq2Þ¼ q2Dðq2Þ. The demonstration of Eq. (2.20) relies on the “background-quantum identity”that relates ˜ IΓαðq; r; pÞand IΓαðq; r; pÞ[6,128]; details will be presented elsewhere. III. THREE-GLUON VERTEX AND ITS WARD IDENTITY DISPLACEMENT In this section we consider the case of the three-gluon vertex in the conventional Landau gauge. If this vertex develops longitudinally coupled massless poles, its polefree part satisfies a displaced WI, whose derivation is the focal point of this section. Before commencing, we introduce the gluon propagator, Δab μν ðqÞ¼−iδabΔμνðqÞ; in the Landau gauge that we employ in this work, it is given by the completely transverse form ΔμνðqÞ¼Δðq2ÞPμνðqÞ;P μνðqÞ≔gμν −qμqν=q2: ð3:1Þ Furthermore, we define the two tensorial structures Pμ μ0ðrÞPν ν0ð−rÞ≔Tμν μ0ν0ðrÞ;λμναðrÞ≔2rαPμνðrÞ;ð3:2Þ and the tree-level three-gluon vertex, Γαμν 0ðq; r; pÞ,as Γαμν 0ðq; r; pÞ¼ðq−rÞνgαμ þðr−pÞαgμν þðp−qÞμgνα; ð3:3Þ where the gauge coupling gand the color factor fabc were suppressed. In order for the Schwinger mechanism to be activated, the full three-gluon vertex, to be denoted by IΓabc αμνðq; r; pÞ¼ gfabcIΓαμνðq; r; pÞ, is written as (see Fig. 2) IΓαμνðq; r; pÞ¼Γαμνðq; r; pÞþVαμνðq; r; pÞ;ð3:4Þ where Γαμνðq; r; pÞis the pole-free component, while Vαμνðq; r; pÞcontains longitudinally coupled poles, i.e., it assumes the general form Vαμνðq; r; pÞ¼qα q2Cμνðq; r; pÞþrμ r2Aανðq; r; pÞ þpν p2Bαμðq; r; pÞ:ð3:5Þ For the particular kinematic limit that we will eventually consider in the present work (q→0), we only require the tensorial decomposition of the term Cμνðq; r; pÞin Eq. (3.5), given by Cμνðq; r; pÞ¼C1gμν þC2rμrνþC3pμpν þC4rμpνþC5pμrν;ð3:6Þ where Ci≔Ciðq; r; pÞ. Due to its special form given by Eq. (3.5),Vαμνðq; r; pÞ satisfies the crucial condition (a) (b) (c) FIG. 2. (a) Compact diagrammatic representation of the vertices e IΓαμνðq; r; pÞ(Bgluon) and IΓαμνðq; r; pÞ(Qgluon); (b) the pole-free parts, ˜ Γαμνðq; r; pÞand Γαμνðq; r; pÞ; (c) the pole parts, ˜ Vαμνðq; r; pÞand Vαμνðq; r; pÞ. EXPLORING SMOKING-GUN SIGNALS OF THE SCHWINGER …PHYS. REV. D 105, 014030 (2022) 014030-5 Pα α0ðqÞPμ μ0ðrÞPν ν0ðpÞVαμνðq; r; pÞ¼0;ð3:7Þ and, consequently, it drops out from the typical lattice observables involving the transversely projected threegluon vertex [see Eqs. (4.4) and (A2)]. The full vertex IΓαμνðq; r; pÞsatisfies the STI qαIΓαμνðq; r; pÞ¼Fðq2Þ½Δ−1ðp2ÞPσ νðpÞHσμðp; q; rÞ −Δ−1ðr2ÞPσ μðrÞHσνðr; q; pÞ;ð3:8Þ analogous expressions are obtained when contracting by rμ or pν. Note that the ghost-gluon kernel, Habc νμ ðq; p; rÞ¼ −gfabcHνμðq; p; rÞ, defined in Fig. 12, enters in the STI nontrivially; for the nonperturbative structure of its relevant form factors, see [129]. In addition, we point out that the Hσμðp; q; rÞand Hσνðr; q; pÞalso contain massless poles in the rμand pνchannels, respectively, which are completely eliminated by the transverse projections in Eq. (3.12). It is clear from Eqs. (3.4) and (3.6) that Pμ μ0ðrÞPν ν0ðpÞ½qαIΓαμνðq; r; pÞ ¼Pμ μ0ðrÞPν ν0ðpÞ½qαΓαμνðq; r; pÞþCμνðq; r; pÞ;ð3:9Þ while, from the STI of Eq. (3.8) Pμ μ0ðrÞPν ν0ðpÞ½qαIΓαμνðq; r; pÞ ¼Pμ μ0ðrÞPν ν0ðpÞFðq2ÞRνμðp; q; rÞ;ð3:10Þ where Rνμðp; q; rÞ≔Δ−1ðp2ÞHνμðp; q; rÞ−Δ−1ðr2ÞHμνðr; q; pÞ: ð3:11Þ Then, equating the right-hand sides of Eqs. (3.9) and (3.10) we obtain qα½Pμ μ0ðrÞPν ν0ðpÞΓαμνðq; r; pÞ ¼Pμ μ0ðrÞPν ν0ðpÞ½Fðq2ÞRνμðp; q; rÞ−Cμνðq; r; pÞ: ð3:12Þ Due to the presence of the projectors Pμ μ0ðrÞPν ν0ðpÞ,itis clear from Eq. (3.6) that only the terms C1gμν and C5pμrν contribute to Cμνðq; r; pÞ. Note, however, that since Pμ μ0ðrÞPν ν0ðpÞC5pμrν¼Pμ μ0ðrÞPν ν0ðpÞC5qμqν,thistermis subleading, i.e., of order Oðq2Þ, when the limit q→0 is taken. We next proceed with the implementation of the limit q→0. In particular, as was done in the previous section, we carry out the Taylor expansion of both sides of Eq. (3.12) around q¼0, and collect terms linear in q. The computation of the l.h.s. of Eq. (3.12) is immediate: using Eq. (3.2),wehave ½l:h:s¼qαTμν μ0ν0ðrÞΓαμνð0;r;−rÞþOðq2Þ:ð3:13Þ The computation of the r.h.s. of Eq. (3.12) is considerably more complicated. We start by noticing that, to lowest order in q, only the term C1ðq; r; pÞgμν survives. In addition, since it is clear from Eq. (3.11) that Rνμð−r; 0;rÞ¼0, the vanishing of the zeroth order contribution imposes the condition C1ð0;r;−rÞ¼0;ð3:14Þ in exact analogy to Eq. (2.15). Thus, the r.h.s. of Eq. (3.12) becomes ½r:h:s¼qαTμν μ0ν0ðrÞFð0Þ∂Rνμðp; q; rÞ ∂qαq¼0 −qαPμ0ν0ðrÞ∂C1ðq; r; pÞ ∂qαq¼0þOðq2Þ:ð3:15Þ In order to compute the first partial derivative in Eq. (3.15), we exploit the fact that, in the Landau gauge, the ghost-gluon kernel may be cast in the form [91,107] Hνμðp; q; rÞ¼˜ Z1gνμ þqρKνμρðp; q; rÞ;H μνðr; q; pÞ¼˜ Z1gμν þqρKμνρðr; q; pÞ;ð3:16Þ where the kernels Kdo not contain poles as q→0. Moreover, ˜ Z1is the finite constant renormalizing Hνμðp; q; rÞin the “asymmetric”momentum subtraction (MOM) scheme, employed in the lattice simulation of [85,101]; its numerical value, estimated in [130],is ˜ Z1≈0.95. AGUILAR, FERREIRA, and PAPAVASSILIOU PHYS. REV. D 105, 014030 (2022) 014030-6 Then, to lowest order in q, ∂Hνμðp; q; rÞ ∂qαq¼0¼Kνμαð−r; 0;rÞ; ∂Hμνðr; q; pÞ ∂qαq¼0¼Kμναðr; 0;−rÞ:ð3:17Þ Consider next the tensor decomposition of Kμναðr; 0;−rÞ [107], Kμναðr; 0;−rÞ¼Kðr2Þgμνrαþ;ð3:18Þ where the ellipses denote terms proportional to gναrμ,gμαrν, and rαrμrν, which get annihilated by contraction with Tμν μ0ν0ðrÞ. Clearly, Tμν μ0ν0ðrÞKνμαð−r;0;rÞ¼ −Tμν μ0ν0ðrÞKμναðr;0;−rÞ. Then, it is straightforward to demonstrate that Tμν μ0ν0ðrÞ∂Rνμðp; q; rÞ ∂qαq¼0 ¼λμ0ν0αðrÞf˜ Z1½Δ−1ðr2Þ0−Kðr2ÞΔ−1ðr2Þg;ð3:19Þ where the “prime”denotes differentiation with respect to r2. As for the second partial derivative in Eq. (3.15), applying the chain rule we have ∂C1ðq;r;pÞ ∂qαq¼0¼2rαCðr2Þ;Cðr2Þ≔∂C1ðq;r;pÞ ∂p2q¼0 ; ð3:20Þ such that Pμ0ν0ðrÞ∂C1ðq; r; pÞ ∂qαq¼0¼λμ0ν0αðrÞCðr2Þ;ð3:21Þ and, therefore, Eq. (3.15) becomes ½r:h:s¼qαλμ0ν0αðrÞ½Fð0Þf˜ Z1½Δ−1ðr2Þ0−Kðr2ÞΔ−1ðr2Þg−Cðr2ÞþOðq2Þ:ð3:22Þ The final step is to equate the terms linear in qthat appear in Eqs. (3.13) and (3.22), to obtain the WI Tμν μ0ν0ðrÞΓαμνð0;r;−rÞ¼λμ0ν0αðrÞ½Fð0Þf˜ Z1½Δ−1ðr2Þ0−Kðr2ÞΔ−1ðr2Þg−Cðr2Þ:ð3:23Þ Thus, the inclusion of the term Vαμνðq; r; pÞin the vertex of Eq. (3.4) leads ultimately to the displacement of the WI satisfied by the pole-free part Γαμνðq; r; pÞ, by an amount given by the special function Cðr2Þ. Evidently, if Cðr2Þ¼0 one recovers the WI in the absence of the Schwinger mechanism. We end this section with some remarks related to the PT-BFM scheme. Note that if the gauge field carrying the momentum qis a background gluon instead of a quantum one (see Fig. 2), then the corresponding three-gluon vertex, ˜ Γαμνðq; r; pÞ, satisfies a simplified version of the WI in Eq. (3.23), where F→1, ˜ Z1→1, and Kðr2Þ→0, i.e., Tμν μ0ν0ðrÞ ˜ Γαμνð0;r;−rÞ¼λμ0ν0αðrÞ½½Δ−1ðr2Þ0− ˜ Cðr2Þ: ð3:24Þ As has been demonstrated in [67], the contribution of the gluon loops to the nonvanishing Δ−1ð0Þ, to be denoted by Δ−1 gl ð0Þ, is controlled by ˜ Cðr2Þ, Δ−1 gl ð0Þ∼Zd4kk2Δ2ðk2Þ½1−6παsCAYðk2Þ ˜ Cðk2Þ;ð3:25Þ where αs≔g2=4π,CAis the Casimir eigenvalue of the adjoint representation [Nfor SUðNÞ], and Yðk2Þrepresents a particular one-loop correction (see, e.g., Fig. 3 in [67]). Evidently, Eq. (3.25) is the exact analog of Eq. (2.19). The total mass, identified with Δ−1ð0Þ, is obtained by summing up Eqs. (3.25) and (2.19). Finally, the relation between ˜ Cðr2Þand Cðr2Þis given by Cðr2Þ¼Fð0Þ ˜ Cðr2Þ;ð3:26Þ in exact analogy to Eq. (2.20). IV. DISPLACEMENT FUNCTION IN TERMS OF LATTICE QUANTITIES In this section we establish a crucial connection between the l.h.s. of Eq. (3.23) and the results of recent lattice simulations. This, in turn, will allow us to relate the EXPLORING SMOKING-GUN SIGNALS OF THE SCHWINGER …PHYS. REV. D 105, 014030 (2022) 014030-7 characteristic ingredient of the Schwinger mechanism, namely Cðr2Þ, to quantities obtained directly from lattice QCD. The advantage of such a connection is that the lattice is intrinsically “blind”to particular field theoretic constructs (such as the Schwinger mechanism), furnishing results obtained through the model-independent functional averaging over gauge-field configurations. We start our analysis by considering the pole-free part Γαμνðq; r; pÞof the three-gluon vertex, in the kinematic limit of interest, q→0. Given that only a single momentum (r) is available, the general tensorial decomposition of Γαμνð0;r;−rÞis given by Γαμνð0;r;−rÞ¼2A1ðr2Þrαgμν þA2ðr2Þ ×ðrμgνα þrνgμαÞþA3ðr2Þrαrμrν;ð4:1Þ where the form factors Aiðr2Þmay diverge at most logarithmically as r→0, but do not contain stronger singularities. At tree level, we have that Γαμν 0ð0;r;−rÞ¼2rαgμν −ðrμgνα þrνgμαÞ;ð4:2Þ corresponding to Að0Þ 1ðr2Þ¼1,Að0Þ 2ðr2Þ¼−1, and Að0Þ 3ðr2Þ¼0. It is then elementary to derive from Eq. (4.1) that Tμν μ0ν0ðrÞΓαμνð0;r;−rÞ¼A1ðr2Þλμ0ν0αðrÞ:ð4:3Þ We next establish a connection between the form factor A1ðr2Þand the projection of the three-gluon vertex studied in the lattice simulations of [85,99,100,102,103,106,131–136]. Specifically, after appropriate amputation of the external legs, the lattice quantity Lsgðr2Þis given by Lsgðr2Þ¼Γαμν 0ðq; r; pÞPαα0ðqÞPμμ0ðrÞPνν0ðpÞIΓα0μ0ν0ðq; r; pÞ Γαμν 0ðq; r; pÞPαα0ðqÞPμμ0ðrÞPνν0ðpÞΓα0μ0ν0 0ðq; r; pÞq→0 p→−r :ð4:4Þ Now, by virtue of Eq. (3.7), it is clear that the term Vα0μ0ν0ðq; r; pÞassociated with the poles drops out from Eq. (4.4) in its entirety, amounting effectively to the replacement IΓα0μ0ν0ðq; r; pÞ→Γα0μ0ν0ðq; r; pÞ. Then, the numerator, N, and denominator, D, of the fraction on the r.h.s. of Eq. (4.4), after employing Eqs. (4.1) and (4.2), become N¼4ðd−1Þ½r2−ðr·qÞ2=q2A1ðr2Þ; D¼4ðd−1Þ½r2−ðr·qÞ2=q2:ð4:5Þ Evidently, the path-dependent contribution contained in the square bracket drops out when forming the ratio N=D, and Eq. (4.4) yields the important relation Lsgðr2Þ¼A1ðr2Þ:ð4:6Þ Combining Eqs. (4.3) and (4.6), we get Tμν μ0ν0ðrÞΓαμνð0;r;−rÞ¼Lsgðr2Þλμ0ν0αðrÞ:ð4:7Þ At this point, after substitution of Eq. (4.7) into Eq. (3.23), we arrive at Cðr2Þ¼Fð0Þf˜ Z1½Δ−1ðr2Þ0−Kðr2ÞΔ−1ðr2Þg−Lsgðr2Þ: ð4:8Þ The final step consists in passing the result of Eq. (4.8) from Minkowski to Euclidean space, following the standard conversion rules. Specifically, we set r2¼−r20 E, with r2 E>0the positive square of an Euclidean four-vector, and use ΔEðr2 EÞ¼−Δð−r2 EÞ;F Eðr2 EÞ¼Fð−r2 EÞ; LE sgðr2 EÞ¼Lsgð−r2 EÞ;CEðr2 EÞ¼−Cð−r2 EÞ:ð4:9Þ In what follows we suppress the indices “E”to avoid notational clutter. Then, Eq. (4.8) is converted to Cðr2Þ¼Lsgðr2ÞþFð0ÞfKðr2ÞΔ−1ðr2Þ− ˜ Z1½Δ−1ðr2Þ0g; ð4:10Þ which is one of the central results of this article. Finally, it is convenient to introduce the dimensionless function Wðr2Þ, defined as Kðr2Þ¼− Wðr2Þ r2;ð4:11Þ thus casting Eq. (4.12) into the form Cðr2Þ¼Lsgðr2Þ−Fð0ÞWðr2Þ r2Δ−1ðr2Þþ˜ Z1½Δ−1ðr2Þ0; ð4:12Þ AGUILAR, FERREIRA, and PAPAVASSILIOU PHYS. REV. D 105, 014030 (2022) 014030-8 which will be employed in the numerical evaluation given in Sec. VI. V. DYNAMICAL DETERMINATION OF THE DISPLACEMENT FUNCTION In this section we elaborate on the determination of Cðr2Þfrom the BSEs that describe the dynamical formation of massless colored bound states; the analysis is based on the derivations given in [90,93], adapted to the present context. Note that this procedure determines also the analog of ˜ Cðr2Þ, introduced in Sec. II, for the case of the conventional ghost-gluon vertex, to be denoted by Cðr2Þ. The starting point of this study is the BS version of the SDEs that govern the momentum evolution of the threegluon vertex, IΓαμνðq; r; pÞ, and of the conventional ghostgluon vertex, IΓαðq; r; pÞ, shown in Fig. 3. In particular, we replace (inside the loops) the tree-level vertices (with incoming momentum q) by their fully dressed counterparts, modifying the corresponding multiparticle kernels Kij accordingly, to avoid overcounting (see e.g., Fig. 7 of [90]). The main advantage of this conversion is that various vertex renormalization constants, which otherwise would appear explicitly multiplying the corresponding diagrams, are naturally absorbed by the additional dressed vertices. Note that, in order to simplify the pertinent set of SDEs, we omit from our analysis the fully dressed four-gluon vertices (with incoming momentum q), whose impact is expected to be subleading [12,137]. In what follows, we will introduce a longitudinally coupled massless pole also in the ghost-gluon vertex IΓαðq; r; pÞ, casting it into a form analogous to Eq. (2.9), and diagrammatically represented in Fig. 1, where the incoming gluon is Qa α. In particular, we set IΓαðq; r; pÞ¼Γαðq; r; pÞþVαðq; r; pÞ;ð5:1Þ where Γαðr; p; qÞdenotes the pole-free component, while Vαðq; r; pÞ¼qα q2Cðq; r; pÞ;ð5:2Þ describes the pole multiplied by the associated form factor. Then, the BSEs of Fig. 3may be written schematically as IΓαμνðq; r; pÞ¼Γαμν 0ðq; r; pÞ−ig2CA 2Zk IΓαβγðq; k; −sÞΔβρðkÞΔγσðsÞKμνσρ 11 ðr; p; s; −kÞ þig2CAZk IΓαðq; k; −sÞDðk2ÞDðs2ÞKμν 12ðr; p; s; −kÞ; IΓαðq; r; pÞ¼Γα 0ðq; r; pÞ−ig2CA 2Zk IΓαβγðq; k; −sÞΔβρðkÞΔγσðsÞKσρ 21ðr; p; s; −kÞ −ig2CA 2Zk IΓαðq; k; −sÞDðk2ÞDðs2ÞK22ðr; p; s; −kÞ;ð5:3Þ (a) (b) (c) (d) FIG. 3. The coupled system of inhomogeneous BSEs for the three-gluon and ghost-gluon vertices, IΓαμνðq; r; pÞand IΓαðq; r; pÞ, respectively. Circles represent full propagators or vertices, while the orange ellipses correspond to four-point scattering kernels, Kij. The omitted diagrams contain five-point scattering kernels, associated with fully dressed four-gluon vertices with incoming momentum q, which are neglected in our truncation scheme. EXPLORING SMOKING-GUN SIGNALS OF THE SCHWINGER …PHYS. REV. D 105, 014030 (2022) 014030-9 scaling ghost dressing function diverges at the origin, and so do the threeand four-gluonvertex functions. It is important to emphasize that, despite their clear discrepancies in the deep infrared, the scaling and decoupling solutions practically coincide for momenta higher than a few hundred MeV (see, e.g., [143]). Let us also point out that the Schwinger mechanism has been recently implemented in the context of the scaling solutions [94]. In this case, the generation of the mass scale does not affect the deep infrared regime of the corresponding Green’s functions, but accounts for their considerable support in the intermediate range of momenta. Finally, one may hypothesize on the properties of Cðr2Þif scaling solutions were used for its determination. Given the observations made above, it is reasonable to expect that its qualitative behavior would remain unaffected for momenta larger than a few hundred MeV, while differences would manifest themselves in the deep infrared. In fact, it is perfectly plausible that Cðr2Þmight vanish at the origin, exactly as Cðr2Þdoes. VIII. DISCUSSION AND CONCLUSIONS In the present work we have investigated in detail a characteristic feature that is intimately linked with the onset of the Schwinger mechanism in QCD, and the ensuing emergence of an effective gluon mass scale. The action of this mechanism relies on the inclusion of massless longitudinal poles in the fundamental vertices of the theory, which participate nontrivially in the realization of the corresponding STIs. This, in turn, causes a distinct displacement to the WI satisfied by the pole-free part of the vertices involved, quantified by the function Cðr2Þ, which is formally identical to the bound-state wave function that governs the dynamical formation of the massless poles. We have computed Cðr2Þfor the case of the three-gluon vertex in two distinct ways: by solving the corresponding BSE and by appropriately combining the ingredients appearing in the non-Abelian WI. In both cases we have relied predominantly on results obtained from lattice simulations, with the exception of the special function Wðr2Þ, which was determined from an SDE. The results found for Cðr2Þ are clearly nonvanishing and in excellent mutual agreement, providing additional support to the details of the general dynamical picture put forth in a series of articles. In particular, the dual role played by Cðr2Þis especially noteworthy, hinting towards deeper connections that have yet to be unraveled. It is important to stress that the computation of the null hypothesis, presented in Sec. VI, proceeds by assuming that all inputs of Eq. (6.1) retain their known form; in particular, salient features, such as the saturation of the gluon propagator and the ghost dressing function, persist unaltered. In that sense, this specific implementation probes the compatibility between the lattice results and the absence of a displacement in the non-Abelian WI of the three-gluon vertex. Seen from this point of view, one might state that this particular possibility is excluded at the level of 3.1σ. As mentioned in Appendix A, the value of αsthat corresponds to the eigenvalue of the system is αs¼0.63, which is considerably different from the value αs¼0.27 found within the “asymmetric”MOM scheme that we employ. The discrepancy may be interpreted as a truncation artifact, given that the corresponding BSE kernels has been approximated by their one-particle exchange diagrams, as depicted in Fig. 4. In addition, the expressions employed for the fully dressed vertices comprising these kernels contain a certain amount of uncertainty. Quite interestingly, a preliminary numerical exploration indicates that minor modifications of the kernel affect the value of αsconsiderably, without practically modifying the form of the solution found for C⋆ðr2Þand C⋆ðr2Þ. This observation suggests that, while the decrease of αstowards its MOM value may require a more refined knowledge of the corresponding BSE kernels, the obtained solutions should be considered as fairly reliable. The proximity between the Wðr2Þ, computed in Appendix B, and the W⋆ðr2Þobtained from Eq. (6.4), suggest that minor modifications of the inputs used for the SDE of Eq. (B5) might lead to an even better coincidence. In this context, it is interesting to point out that the determination of the transverse form factors Yi[see item (iii) in Appendix B] is subject to a considerable uncertainty, originating from the approximations implemented to the complicated SDE satisfied by the three-gluon vertex (Fig. 6 in [106]). Given the relevance of Wðr2Þfor the systematic scrutiny of the Schwinger mechanism, as exposed in the present work, it may be worthwhile revisiting this particular computation. As mentioned below Eq. (3.8), the Schwinger mechanism induces poles also in the ghost-gluon kernel, Hμνðr; q; pÞ. This may be understood qualitatively by considering the diagrams (d1) and (d2) in Fig. 12: the fully dressed ghost-gluon and three-gluon vertices (with Lorentz index νand incoming momentum p) contain poles, which are transmitted to the form factors of Hμνðr; q; pÞ associated with the tensorial structures pνpμand pνrμ. It would be important to compute in detail the pole structure of Hμνðr; q; pÞ, especially in view of the STI rμHμνðr; q; pÞ¼Γνðp; r; qÞ, which links nontrivially the form factors of Hμνðp; r; qÞand Γμðp; r; qÞ, in general, and the corresponding pole terms, in particular. Specifically, one may explore how accurately the appropriate combination of pole terms coming from Hμνðp; r; qÞwill reproduce AGUILAR, FERREIRA, and PAPAVASSILIOU PHYS. REV. D 105, 014030 (2022) 014030-16 the corresponding term contained in Γμðp; r; qÞ. We hope to undertake such a study in the near future. The generation of massless poles in the ghost correlation functions and kernels, such as Γνðp; r; qÞand Hμνðr; q; pÞ, may herald a deeper connection with the nonperturbative BRST quartet mechanism mentioned earlier. In particular, the dynamical realization of the quartet mechanism put forth in [121] relies on the formation of massless ghostgluon bound states, which would manifest themselves in the ghost-gluon four-point kernel that enters in the corresponding BSE (see Fig. 1 in [121]). Evidently this issue deserves detailed scrutiny, both from the point of view of the decoupling as well as the scaling solutions. Such a study could be particularly revealing, establishing the compatibility and harmonious interplay between two key nonperturbative mechanism operating in the gauge sector of QCD. The scale ambiguity associated with the BSE amplitudes Cðr2Þand Cðr2Þresults from considering only the leading order terms of the BSEs in an expansion around q¼0, which furnishes homogeneous linear equations. In general kinematics, however, the presence of inhomogeneous terms in the BSEs resolves this ambiguity. As such, the scales of Cðr2Þand Cðr2Þcan be fixed by taking the q¼0limit of the solution of the corresponding inhomogeneous BSEs, treated beyond leading order in q. In the context of conventional bound states, this procedure is well understood [144] and explicit scale-setting equations have been derived, which are sometimes referred to as “canonical normalization condition”[28,145]. It is our intention to pursue this point in an upcoming study, and settle dynamically the scale of the corresponding solutions. ACKNOWLEDGMENTS The work of A. C. A. is supported by the Brazilian CNPq Grants No. 307854/2019-1 and No. 464898/2014-5 (INCTFNA). A. C. A. and M. N. F. also acknowledge financial support from the FAPESP Projects No. 2017/05685-2 and No. 2020/12795-1, respectively. J. P. is supported by the Spanish AEI-MICINN Grant No. PID2020–113334GBI00/AEI/10.13039/501100011033, and the grant Prometeo/ 2019/087 of the Generalitat Valenciana. APPENDIX A: TECHNICAL DETAILS ON THE BSE SYSTEM In this appendix we present details related to the numerical treatment of the BSE system formed by Cðr2Þ and Cðr2Þ, shown in Fig. 4. For the Bose symmetric three-gluon vertex appearing in the diagrams of Fig. 4we employ the tensor basis of BallChiu [146,147], Γαμνðq; r; pÞ¼X 10 i¼1 Xiðq; r; pÞlαμν iþX 4 i¼1 Yiðq; r; pÞtαμν i; ðA1Þ where the explicit form of the basis tensors lαμν iand tαμν iis given in Eqs. (3.4) and (3.6) of [147]. At tree level, Xð0Þ 1¼Xð0Þ 4¼Xð0Þ 7¼1, while all other Xð0Þ iand Yð0Þ ivanish. It is convenient to introduce the transversely projected vertex, ¯ Γαμνðq; r; pÞ, defined as ¯ Γαμνðq; r; pÞ≔Pα0 αðqÞPμ0 μðrÞPν0 νðpÞΓα0μ0ν0ðq; r; pÞ:ðA2Þ Similarly, the tensorial decomposition of the vertex Γμðq; r; pÞis given by Γμðq; r; pÞ¼rμB1ðr; p; qÞþqμB2ðr; p; qÞ;ðA3Þ at tree level, Bð0Þ 1¼1and Bð0Þ 2¼0. The ghost-antighost symmetry of Γμðq; r; pÞin the Landau gauge implies that B1ðr; p; qÞ¼B1ðp; r; qÞ. Next, we pass to Euclidean space and employ spherical coordinates. For convenience, we define the variables x≔r2,y≔k2, and u≔ðr−kÞ2¼xþy−2ffiffiffiffiffi xy pcϕ, with ϕdenoting the angle between the momenta kand r, while sϕ≔sin ϕ,cϕ≔cos ϕ. Furthermore, we parametrize scalar form factors, such as B1ðq; r; pÞ, in terms of the squares of their first two arguments and the angle between them, e.g., B1ð−r; k; r −kÞ→B1ðx; y; π−ϕÞ. Then, the final set of BSEs reads CðxÞ¼λZ∞ 0 dyy ffiffiffiffiffi xy pΔ2ðyÞCðyÞZπ 0 dϕs2 ϕcϕΔðuÞNðx;y;uÞ−2Z∞ 0 dy ffiffiffi y x rF2ðyÞCðyÞZπ 0 dϕs4 ϕcϕ FðuÞ uB2 1ðu;y;χÞ; CðxÞ¼3λZ∞ 0 dyy ffiffiffiffiffi xy pΔ2ðyÞCðyÞZπ 0 dϕs4 ϕcϕ FðuÞ uB2 1ðx;u;θÞþZ∞ 0 dy ffiffiffiffiffi xy pF2ðyÞCðyÞZπ 0 dϕs4 ϕcϕ ΔðuÞ uB2 1ðx;y;π−ϕÞ: ðA4Þ In the above equation, EXPLORING SMOKING-GUN SIGNALS OF THE SCHWINGER …PHYS. REV. D 105, 014030 (2022) 014030-17 λ≔αsCA 12π2;ðA5Þ where αs≔g2=4π. The angles χand θare given by χ¼cos−1ffiffiffi x pcos ϕ−ffiffiffi y p ffiffiffi u p; θ¼cos−1ffiffiffi y pcos ϕ−ffiffiffi x p ffiffiffi u p:ðA6Þ Finally, Nðx; y; uÞ≔− 1 x½¯ Γαβγð−r; k; r −kÞ¯ Γαβγð−r; k; r −kÞE; ðA7Þ where the subscript “E”indicates that Eq. (A7) is to be converted to Euclidean coordinates. The Nðx; y; uÞof Eq. (A7) can be written in terms of the form factors Xiand Yiof Eq. (A1); note that the Xiwith i¼2, 5, 8, 10 drop out, because they are annihilated by the transverse projection in Eq. (A2). Then, we obtain Nðx; y; uÞ¼ s2 ϕ 2u2x2yfuT1½T1ððu−x−yÞ2þ2xyÞþ4xyðT2þT3Þþ4T4ðu−x−yÞ þxT2½T2ððu−xþyÞ2þ2uyÞÞþ4uyT3−4T4ðu−xþyÞ þyT3½T3ððuþx−yÞ2þ2uxÞ−4T4ðuþx−yÞþ4T2 4ðuþxþyÞg;ðA8Þ where T1¼−uðX1−X4−X7þxyY1Þ−xðX1−X4þX7−2yX3Þ−yðX1þX4−X7Þ; T2¼uðX1−X4−X7þ2yX6−xyY2ÞþxðX1−X4þX7Þ−yðX1þX4−X7Þ; T3¼uðX1−X4−X7þ2xX9−xyY3Þ−xðX1−X4þX7ÞþyðX1þX4−X7Þ; T4¼1 2fu2ðX1−X4−X7Þ−ðx−yÞ½xðX1−X4þX7Þ−yðX1þX4−X7Þþ2u½xðX7−yY4ÞþyX4g;ðA9Þ and we suppress the functional dependence Xi≡Xiðx; y; π−ϕÞand Yj≡Yjðx; y; π−ϕÞ. For the numerical evaluation of Eq. (A4), in addition to the lattice propagators given in Appendix C,weneedB1, together with all Xiand Yithat comprise Nðx; y; uÞ, in general kinematics. These form factors are obtained as follows: (i) For the B1we employ recent results (see Fig. 6 of [130]), obtained from an SDE analysis that uses as inputs lattice data that have been cured from volume and discretization artifacts. (ii) The Xiare obtained from the nonperturbative generalization of the Ball-Chiu solution; the relevant formulas are given in Eq. (3.11) of [147], and involve the ghost dressing function, the kinetic term of the gluon propagator, and certain components of the ghost-gluon kernel. Note that the inputs have been calibrated to exactly reproduce the lattice projection Lsgðr2Þ, through the relation Lsgðr2Þ¼X1ðr2;r 2;πÞ−r2X3ðr2;r 2;πÞ:ðA10Þ In this indirect way, the error bars assigned to the lattice calculation of Lsgðr2Þ, encompassed by the functions L sgðr2Þof Eq. (C13), find their way into our BSE determination of Cðr2Þand Cðr2Þ, giving rise to the errors band indicated in Fig. 6. (iii) For the transverse components Yi, which cannot be deduced from the fundamental STIs, we resort to a SDE determination, along the lines of the analysis presented in [106]; see, in particular, Fig. 6therein. (iv) The results for X1ðq2;r 2;θÞ,qrX3ðq2;r 2;θÞ, q2r2Y1ðq2;r 2;θÞ, and qrY4ðq2;r 2;θÞare shown in Fig. 11, for the special case θ¼2π=3;qand r denote now the magnitudes of the corresponding Euclidean momenta. (v) By virtue of the Bose symmetry of the three-gluon vertex, the remaining form factors of the three-gluon vertex entering in Eq. (A7) can be obtained from those shown in Fig. 11 by appropriate permutations of their arguments, as explained in [147]. AGUILAR, FERREIRA, and PAPAVASSILIOU PHYS. REV. D 105, 014030 (2022) 014030-18 Employing the ingredients described above, we solve the coupled system of BSEs of Eq. (A4) numerically, obtaining the Cðr2Þand Cðr2Þshown in Fig. 5, together with the corresponding error estimates. Since Eq. (A4) does not have an inhomogeneous term and is linear in Cðr2Þand Cðr2Þ, it corresponds to an eigenvalue problem. The resulting eigenvalues correspond to αs¼0.63 ∓0.05, with signs opposite to those of the L sgðr2Þerror bands, i.e., using Xicorresponding to a higher Lsgðr2Þleads to a smaller αs. In these results, the overall constant was determined by matching the BSE prediction for Cðr2Þto the result obtained from the WI, as explained in Sec. VI. APPENDIX B: COMPUTATION OF Wðr2Þ The function Wðr2Þis a central ingredient for our analysis, whose determination proceeds through the study of the corresponding SDE. In this appendix we present the technical details related to this calculation, and discuss the validity of our approximations. 1. SDE, inputs, and solution The starting point of our determination of Wðr2Þis the SDE for the ghost-gluon scattering kernel, Hμνðr; q; pÞ, shown in Fig. 12, which is truncated at the one-loop dressed level, retaining only diagrams (d1) and (d2). In the Landau gauge it is immediate to factor out of these two diagrams the ghost momentum q, in order to obtain Kμναðr; q; pÞ,in accordance with Eq. (3.16). Finally, recalling Eq. (3.18), Wðr2Þis obtained by isolating the gμνrαform factor of Kμναðr; 0;−rÞand using Eq. (4.11). Since a detailed derivation and renormalization of the Wðr2Þequation has been carried out in Sec. 6 of Ref. [108], here we only collect the main results. Specifically, we obtain the general expression Wðr2Þ¼W1ðr2ÞþW2ðr2Þ;ðB1Þ FIG. 11. The three-gluon vertex form factors used in the BSE system: X1ðq2;r 2;2π=3Þ(top left), qrX3ðq2;r 2;2π=3Þ(top right), q2r2Y1ðq2;r 2;2π=3Þ(bottom left), and qrY4ðq2;r 2;2π=3Þ(bottom right). EXPLORING SMOKING-GUN SIGNALS OF THE SCHWINGER …PHYS. REV. D 105, 014030 (2022) 014030-19 where W1and W2denote the contributions from diagrams (d1) and (d2) of Fig. 12, respectively, given by W1ðr2Þ¼−ig2CA ˜ Z1 6Zk Δðk2ÞDðk2ÞDðt2ÞB1ðt; k; −rÞB1ðk; 0;−kÞðr·kÞfðr; kÞ; W2ðr2Þ¼−ig2CA ˜ Z1 6Zk Δðk2ÞΔðt2ÞDðt2ÞB1ðt; 0;−tÞrα¯ Γμ μαð−r; k; tÞ;ðB2Þ where fðr; kÞ≔1−ðk·rÞ2 k2r2;ðB3Þ and ¯ Γis defined in Eq. (A2). Note the appearance in Eq. (B5) of ˜ Z1, defined below Eq. (3.16), which implements the renormalization of the SDE in the asymmetric MOM scheme [108]. To express Eq. (B1) in Euclidean space, we use spherical coordinates and the kinematic variables x,yand udefined above Eq. (A4). Then, using the Ball-Chiu tensor basis of Eq. (A1) for the three-gluon vertex, we obtain WðxÞ¼W1ðxÞþW2ðxÞ;ðB4Þ with W1ðxÞ¼λ ˜ Z1Z∞ 0 dy ffiffiffiffiffi xy pΔðyÞFðyÞB1ðy; 0;0ÞZπ 0 dϕcϕs4 ϕ FðuÞ uB1ðu; y; χÞ; W2ðxÞ¼−2λ ˜ Z1Z∞ 0 dy y ffiffiffiffiffi xy pΔðyÞZπ 0 dϕs4 ϕΔðuÞB1ðu; 0;0ÞFðuÞ u2Kðx; y; uÞ;ðB5Þ where λis defined in Eq. (A5),χin Eq. (A6), and we define the kernel Kðx; y; uÞas Kðx; y; uÞ≔ffiffiffiffiffi xy pðc2 ϕþ2ÞX1þðy−cϕffiffiffiffiffi xy pÞcϕX4þðx−cϕffiffiffiffiffi xy pÞcϕX7−3cϕxyX3 −cϕyuX6−cϕxuX9þ1 2cϕxyuð3Y1þY2þY3Þ−uffiffiffiffiffi xy pY4;ðB6Þ where, again, Xi≡Xiðx; y; π−ϕÞand Yj≡Yjðx; y; π−ϕÞ. Using the Bose symmetry relations involving permutations of arguments of the Xiand Yi, given by Eqs. (3.7) to (3.10) of [147], it is possible to show that Kðx; y; uÞis symmetric under the exchange of x↔y. Then, for the gluon propagator Δðr2Þand the ghost dressing function Fðr2Þwe use the fits presented in Appendix C, while for the vertex form factors, B1,Xi, and Yiwe use the same inputs employed for the solution of the BSE of Eq. (A4). Note that, as mentioned in Appendix C, all inputs are renormalized within the “asymmetric”MOM scheme, at the renormalization point μ¼4.3GeV, for which αs¼0.27. With these ingredients, we obtain for Wðr2Þthe blue continuous line shown in the right panel of Fig. 13. The blue FIG. 12. The SDE satisfied by the ghost-gluon scattering kernel, Hμνðr; q; pÞ. AGUILAR, FERREIRA, and PAPAVASSILIOU PHYS. REV. D 105, 014030 (2022) 014030-20 band around it corresponds to an error propagated from the uncertainty in the lattice Lsgðr2Þ, through the same procedure explained in item (ii) of Sec. A. Given that Wðr2Þis one of the main ingredients in the analysis of Eq. (4.12), it is important to consider in more detail the uncertainty in our SDE determination of this function. It turns out that the contribution W1ðr2Þof Eq. (B5) is negligible in comparison to W2ðr2Þ(see Fig. 7 of [108]), except for r<0.5GeV, where W2ðr2Þ decreases significantly. Furthermore, diagram (d3) of Fig. 12, with the four-particle correlation function Γμσ nested in it, is known to affect the ghost-gluon vertex only by 2% [148]; thus, its omission is expected to have an insignificant effect on Wðr2Þ. Therefore, the main uncertainty originates from the term W2ðr2Þof Eq. (B5), and is related to our incomplete knowledge of the form factors Xiand Yifor general kinematics. In this regard, an examination of the integrand of W2ðr2Þ in Eq. (B5) shows that this contribution is dominated by the projection Lsgðr2Þof the full three-gluon vertex that it contains. In turn, this observation suggests that the SDE determination of Wðr2Þshould be fairly accurate provided that the ansatz employed for the general kinematics threegluon vertex reproduces in the soft-gluon limit the Lsgðr2Þ obtained on the lattice [106]. 2. A closer look at the SDE kernel To elucidate this last point, denote by Iðx; y; ϕÞthe integrand of W2ðxÞin Eq. (B5), i.e., Iðx; y; ϕÞ≔s4 ϕyffiffiffiffiffi xy pΔðyÞΔðuÞFðuÞ u2B1ðu; 0;0ÞKðx; y; uÞ: ðB7Þ Then, since FðuÞand, especially, ΔðuÞare decreasing functions of u, the term ΔðuÞFðuÞ=u2in the second line of Eq. (B7) causes the Iðx; y; ϕÞto decrease rapidly at large u. Hence, W2ðxÞshould be dominated by the small uregion of its integrand. Next, recalling that u¼xþy−2ffiffiffiffiffi xy pcϕ, we note that u¼0occurs when y¼xand ϕ¼0simultaneously. Also, we emphasize that, in spite of the presence of the factor u2 in the denominator, Eq. (B7) is finite at u¼0, due to the vanishing of s4 ϕwhen ϕ¼0. In the left panel of Fig. 13 we plot Iðx; y; ϕÞfor x¼1GeV2and general yand ϕ, using the general kinematics Xiand Yiof Fig. 11 in the evaluation of Kðx; y; uÞ. There we confirm that Iðx; y; ϕÞis largest around y¼x¼1GeV2and ϕ¼0, decaying rapidly to zero at large u. Other values of xlead to similar surfaces, with pronounced peaks at y¼x. Due to the sharply peaked structure of Iðx; y; uÞ, one expects that the value of the integral defining W2ðxÞin Eq. (B5) should depend mainly on the maximum value of Iðx; y; uÞ. To determine the value of this maximum, we expand Eq. (B7) around y¼x, and finally around ϕ¼0.5 To this end, first note that lim ϕ→0 y→xs4 ϕ u2¼1 x2:ðB8Þ The limit of the other terms in Eq. (B7) as u→0is straightforward, leading to FIG. 13. Left: integrand of W2ðxÞ,Iðx; y; ϕÞ, given by Eq. (B7), for x¼1GeV2. Note that its maximum occurs when yis first set to y¼x¼1GeV and then ϕ¼0, corresponding to u¼0.Iðx; y; ϕÞis intensely peaked around the maximum, dropping rapidly to zero away from it. Right: comparison of Wðr2Þ, obtained with Eq. (B4) (blue solid line), to Wsgðr2Þ, obtained with the replacement (B12) into Eq. (B4) (purple dot-dashed line). The bands correspond to propagated error from the lattice Lsgðr2Þof [106]. 5The limit of Iðx; y; ϕÞas u→0is path dependent, vanishing if ϕ¼0is set first; however, we are interested in its maximum, which occurs when y¼xis set first, and ϕ¼0after, as is clear from Fig. 13. EXPLORING SMOKING-GUN SIGNALS OF THE SCHWINGER …PHYS. REV. D 105, 014030 (2022) 014030-21 lim ϕ→0 y→x Iðx; y; uÞ¼3Δð0ÞFð0ÞB1ð0;0;0ÞxΔðxÞ ×½X1ðx; x; πÞ−xX3ðx; x; πÞ;ðB9Þ which, after using Eq. (A10) becomes lim ϕ→0 y→x Iðx;y;uÞ¼3Δð0ÞFð0ÞB1ð0;0;0ÞxΔðxÞLsgðxÞ:ðB10Þ Given that the Xiemployed reproduce LsgðxÞwhen the combination in Eq. (A10) is formed, the above considerations indicate that our result for Wðr2Þshould be rather accurate. Essentially W2ðxÞappears dominated by the “slice”that corresponds to Lsgðr2Þ, with little or no effect from all other kinematic configurations. 3. A special ansatz for the three-gluon vertex To confirm this hypothesis explicitly, we compute Wðr2Þ using a simpler ansatz for the three-gluon vertex, which also reproduces the limit given in Eq. (B10). Specifically, we substitute the full three-gluon vertex appearing in the second line of Eq. (B2) by ¯ Γμ μαð−r; k; tÞ→¯ Γ0 μ μαð−r; k; tÞ¯ Lsgðr2;k 2Þ; ¯ Lsgðr2;k 2Þ¼1 2½Lsgðr2ÞþLsgðk2Þ;ðB11Þ where ¯ Γαμν 0ðq; r; pÞis the tree-level equivalent of the ¯ Γαμνðq; r; pÞ, defined in Eq. (3.3). This ansatz amounts to substituting into Eq. (B6) X1¼X4¼X7¼¯ Lsgðr2;k 2Þ, with all other Xiand Yiset to zero. With this approximation, W2ðxÞis still given by Eq. (B5), but with Kðx; y; uÞ replaced by Kðx;y; uÞ→Ksgðx; y; uÞ¼½ ffiffiffiffiffi xy pðc2 ϕþ2Þþucϕ¯ Lsgðx; yÞ: ðB12Þ Then, substituting Eq. (B12) into Eq. (B7) it is straightforward to show that the limit in Eq. (B10) is exactly reproduced.6 The Wðr2Þthat is obtained through the use of Eq. (B12), denoted by Wsgðr2Þ, is shown as the purple dot-dashed curve in the right panel of Fig. 13, where it is compared to the result obtained from Eq. (B4) using the general kinematics Xiand Yi(blue solid line). The purple band around Wsgðr2Þ corresponds to propagated statistical errors in the Lsgðr2Þ of [106], obtained by implementing in Eq. (B11) the substitution Lsgðr2Þ→L sgðr2Þ[see Eq. (C13)]. In the right panel of Fig. 13 we see that the two approximations for Wðr2Þagree within the error bands, except for a small region around 3.5 GeV. This result indicates that the error in the lattice Lsgðr2Þis more important than the detailed general kinematics structure of the full three-gluon vertex, provided the limit in Eq. (B10) is respected. APPENDIX C: FITS FOR LATTICE INPUTS For the gluon and ghost propagators, as well as the threegluon vertex projection Lsgðr2Þ, we employ fits to lattice data [22,103,106,130], appropriately extrapolated to the continuum limit. The fitting functions used incorporate a number of features expected on physical grounds, particularly their asymptotic behaviors for small and large momenta. In particular: (i) In the UV, they reduce to the one-loop resummed behaviors dictated by renormalization-group arguments, namely lim r2→∞ Δ−1ðr2Þ¼r2Lδ UVðr2Þ; lim r2→∞ F−1ðr2Þ¼Lγ UVðr2Þ; lim r2→∞ Lsgðr2Þ¼Lδ−γ UV ðr2Þ;ðC1Þ where we have defined LUVðr2Þ¼ωln ðr2=Λ2Þ, with ω¼11CAαs=ð12πÞ. The anomalous dimensions are given by δ¼13=22 and γ¼9=44. (ii) Lsgðr2Þand the derivative of the gluon propagator diverge logarithmicaly at the origin, i.e., lim r2→0 Lsgðr2Þ¼llnr2 μ2; lim r2→0½Δ−1ðr2Þ0¼dlnr2 μ2;ðC2Þ with land ddimensionless constants. (iii) The Cðr2Þobtained from the BSE is finite at the origin [79,90,91,93]. On the other hand, the Lsgð0Þ and ½Δ−1ð0Þ0appearing in Eq. (4.12) diverge as given by Eq. (C2). Moreover, it can be shown that Wðr2Þhas the asymptotic behavior, lim r2→0 Wðr2Þ¼˜ Z1wΔð0Þr2lnr2 μ2;ðC3Þ with wa dimensionless constant, such that the combination Wðr2Þ=r2in Eq. (4.12) is also logarithmicaly divergent at the origin. Consequently, consistency of Eqs. (4.12),(C2), and (C3) with 6Any combination of the form ¯ Lsgðx; yÞ¼bLsgðxÞþ ð1−bÞLsgðyÞinstead of Eq. (B11) preserves Eq. (B10). The particular form used in (B11) has the advantage of preserving the symmetry of Kðx; y; uÞunder the exchange of x↔y. We have explicitly checked that the extreme cases b¼0and b¼1lead to results that are nearly identical to those obtained with b¼1 2, shown in Fig. 13. AGUILAR, FERREIRA, and PAPAVASSILIOU PHYS. REV. D 105, 014030 (2022) 014030-22 the BSE prediction for Cð0Þrequires that all these logarithmic divergences cancel. Specifically, we demand that l−Fð0Þ˜ Z1ðwþdÞ¼0:ðC4Þ (iv) We adopt the asymmetric MOM renormalization scheme [106–108,108], which imposes that Δ−1ðμ2Þ¼μ2;Fðμ2Þ¼1;L sgðμ2Þ¼1;ðC5Þ and we take the renormalization point to be μ¼4.3GeV. The fits for the lattice ingredients are all required to reduce exactly to the above values at μ. In order to incorporate all the above features, the fitting functions have rather elaborate forms. Starting with Fðr2Þ, an accurate fit to the lattice data is obtained with F−1ðr2Þ¼Aγðr2ÞþRðr2Þ;ðC6Þ where Aðr2Þ, Aðr2Þ≔1þωlnr2þη2ðr2Þ μ2þη2ðr2Þ;ðC7Þ with η2ðr2Þ¼ η2 1 1þr2=η2 2 ;ðC8Þ while Rðr2Þis a combination of rational functions, Rðr2Þ¼ b0þb2 1r2 1þðr2=b2 2Þþðr2=b2 3Þ2 −b0þb2 1μ2 1þðμ2=b2 2Þþðr2=b2 3Þ2:ðC9Þ Note that Rðr2Þvanishes quickly at infinity, and that Rðμ2Þ¼0and Aðμ2Þ¼1, enforcing the renormalization condition in Eq. (C5). Moreover, while Aðr2Þsaturates to a constant at the origin, in the UV it recovers the perturbative logarithm, since η2ðr2Þ→0at large r2, such that lim r2→∞ Aðr2Þ¼LUVðr2Þ;ðC10Þ where the function LUVðr2Þwas defined in Eq. (C1) with Λ2¼μ2e−1=ω. Turning to Δðr2Þ, a form that satisfies all the required conditions is Δ−1ðr2Þ¼r2d 1þðr2=κ2Þlnr2 μ2þAδðr2Þþν2Rðr2Þ; ðC11Þ where the unprotected logarithm in the first term in brackets describes the IR divergence of ½Δ−1ð0Þ0and drops out in the UV, while Aðr2Þand Rðr2Þare given by Eqs. (C7) and (C9). The parameter ν¼1GeV serves only to make the dimensionality of Rconsistent with that of Δ−1ðr2Þ, without changing the dimensions of the σiparameters in Eq. (C9). As for Lsgðr2Þwe use the fitting form Lsgðr2Þ¼ l 1þðr2=κ2Þlnr2 μ2þAδ−γðr2ÞþRðr2Þ;ðC12Þ with Aðr2Þand Rðr2Þgiven by Eqs. (C7) and (C9), respectively. Note that although in Eqs. (C6),(C11), and (C12) we use the same names for the parameters κ2,η2 i,b0, and b2 i, for economy, they are allowed to assume different values for each of the functions Fðr2Þ,Δðr2Þ, and Lsgðr2Þ. Next, the coefficient lin Eq. (C2), characterizing the rate of divergence of Lsgð0Þ, has been determined from lattice results to be l¼0.11 [106], and is held fixed during the fitting procedure. In contrast, the rate of divergence dof ½Δ−1ðr2Þ0is not accurately determined from the lattice, since the derivative is sensitive to the larger lattice noise in Δðr2Þin the deep IR. Moreover, the coefficient win FIG. 14. Left: lattice data of [22,130] (points), and the fit of Eq. (C11) (blue solid line), for the gluon propagator, Δðr2Þ. Center: derivative ½Δ−1ðr2Þ0, obtained through differentiation of Eq. (C11). Right: lattice data (points) of [103], and the fit of Eq. (C6) (blue solid), for the ghost dressing function, Fðr2Þ. EXPLORING SMOKING-GUN SIGNALS OF THE SCHWINGER …PHYS. REV. D 105, 014030 (2022) 014030-23 Eq. (C3) depends on the ingredients, including Δðr2Þ, used in the SDE evaluation of Wðr2Þthrough Eqs. (B4) and (B5). As such, in order to enforce Eq. (C4),wand dhave to be varied simultaneously, until the cancellation of the divergences has been reached to acceptable precision. The fitting parameters resulting for Δðr2Þ,Fðr2Þ, and Lsgðr2Þare given in Table II and its caption. The resulting curves for Δðr2Þand Fðr2Þare compared to the lattice data of [22,130] in Fig. 14, where we also show ½Δ−1ðr2Þ0. The lattice data of [106] and corresponding fit for Lsgðr2Þare shown as the points and black continuous curve, respectively, in the right panel of Fig. 6. Comparing the curve of Lsgðr2Þin Fig. 6to that of ½Δ−1ðr2Þ0in Fig. 14, we see that ½Δ−1ðr2Þ0is responsible for reproducing the overall shape of Lsgðr2Þin the WI of Eq. (4.12), with the other ingredients providing minor quantitative modulations. Now, it is clear from Figs. 6and 14 that the lattice quantity with the largest error in the present analysis is Lsgðr2Þ. In order to propagate the error of Lsgðr2Þto other quantities that depend on it, we make a band around Lsgðr2Þ, delimited by L sgðr2Þ¼Lsgðr2Þ δ1 1þðr2=δ2 2Þ2;ðC13Þ with parameters δ1¼0.08 and δ2 2¼5GeV2. Lastly, for the value of αsin the asymmetric MOM scheme, which appears in the SDE for Wðr2Þgiven by Eq. (B4), we use the value αs¼0.27 determined by lattice simulations [101,149]. For the Fð0Þappearing in Eq. (4.12) we obtain from Eq. 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