Towards flavored bound states beyond rainbows and ladders
TTowards flavored bound states beyond rainbows and ladders
B. El-Bennich ∗ , E. Rojas ∗ , M. A. Paracha ∗ ,† and J. P. B. C. de Melo ∗ ∗ Laboratório de Física Teórica e Computacional, Universidade Cruzeiro do Sul, São Paulo 01506-000 SP,Brazil † Centre for Advanced Mathematics and Physics, National University of Science and Technology, Islamabad,Pakistan
Abstract.
We give a snapshot of recent progress in solving the Dyson-Schwinger equation with a beyond rainbow-ladderansatz for the dressed quark-gluon vertex which includes ghost contributions. We discuss the motivations for this approachwith regard to heavy-flavored bound states and form factors and briefly describe future steps to be taken.
Keywords:
Mesons, Quarks & Gluons, Nonperturbative QCD, Dyson-Schwinger equations, Bethe-Salpeter equations
PACS:
1. MOTIVATION
The challenge of understanding bound states in terms of the elementary fields of a given quantum field theory ispersistent and of particular interest in Quantum Chromodynamics (QCD). The question of how precisely the quarksand gluons form hadrons immediately leads into the domain of relativistic quantum fields whose key propertiescan only be understood with nonperturbative methods. Paramount among the challenges is the understanding ofconfinement and dynamical chiral symmetry breaking (DCSB), both of which are likely to be intimately related [1].Whereas certain simple two-body bound states and their resonances can be adequately described by potentialmodels, this is not the case in QCD. Did light quarks not exist, the picture of string-like potentials arising from aflux tube between two (infinitely) heavy quarks would be correct. Yet, in the real world, where light current-quarks areubiquitous, it is a feature of QCD that light-pair creation and annihilation effects are essentially nonperturbative andcannot be described by a quantum mechanical potential [1, 2]. Such potentials are a poor guide to understanding theGoldstone boson of QCD and must be necessarily fine-tuned. In particular, in typical applications to hadronic formfactors with “light" constituent quarks, the quark’s propagation, S ( k ) = ( γ · k − m q ) − , is scale independent and doesnot describe confinement. In applications of relativistic quark models it was noted that this can lead to significantmodel dependance at larger momentum transfers [3, 4, 5, 6, 7].Heavy-light mesons are of additional interest since they exhibit some features of light-quark confinement. The im-portant asymmetry in quark masses of flavor-nonsinglet Q ¯ q mesons leads to a disparate array of energy scales to bedealt with in solving the meson’s relativistic bound-state equation. Thus, heavy mesons provide an excellent opportu-nity to study additional aspects of nonperturbative QCD and can be used to test simultaneously all manifestations ofthe Standard Model, namely the interplay between electroweak and strong interactions. Some of the major advancesin heavy quark effective theory (HQET) [8] deal with factorization theorems allowing for a disentanglement of short-distance or hard physics, which includes electroweak interactions and perturbative QCD (pQCD) contributions, fromlong-distance or soft physics, dominated by nonperturbative hadronic effects. The systematic reorganization of weakand QCD interactions in HQET has been treated in various approaches; e.g., with QCD factorization (QCDF) [9],pQCD [10] and soft-collinear effective theory (SCET) [11].On the other hand, progress on nonperturbative matrix elements involving heavy-light states with flavor quantumnumbers, C = ± B = ±
1, has been slower: while factorization theorems provide the means to systematicallyintegrate out energy scales in the perturbative domain, valid in the infinitely heavy-quark limit, a reliable evaluation ofthe latter is notoriously difficult. Consider, for instance, the weak non-leptonic decay of a B meson: B → M M . If M is a heavy or light(er) meson and M a light meson [9], then the decay amplitude can be schematically written as, (cid:104) M M | O i | B (cid:105) = (cid:104) M | j | B (cid:105)(cid:104) M | j | (cid:105) (cid:20) + ∑ n r n α ns + O ( Λ QCD / m h ) (cid:21) , (1) a r X i v : . [ nu c l - t h ] N ov here j and j are the bilinear currents and m h is the heavy quark mass. The dimension-six effective four-quarkoperators, O i , result from integrating out the weak gauge bosons W ± in the operator product expansion. Multipliedby the appropriate Cabbibo-Kobayashi-Maskawa (CKM) matrix elements and Wilson coefficients, C i ( µ ) , whichencode perturbative QCD effects above the renormalisation point µ , the sum of these operators forms the heavy-quark effective hamiltonian. Neglecting power corrections in α s and taking the limit m b → ∞ , the naive factorization isrecovered. Higher orders in α s break the factorization, yet in the limit m h (cid:29) Λ QCD , pQCD corrections beyond the naivefactorization can systematically be accounted for. In the case of B decays, the factorization, formally suppressed in Λ QCD / m b , can be broken by weak annihilation decay amplitudes [12, 13] and final-state interactions between daughterhadrons [14, 15, 16, 17, 18, 19].Moreover, since the charm quark is neither a light nor really a heavy quark, HQET may not be the adequate guide tocharm physics and Λ QCD / m c corrections are significant [20, 21]. Whilst effective Lagrangians based on approximateSU(4) flavor-spin symmetries are successful, see for example Refs. [22, 23, 24, 25], care should be taken when thissymmetry is applied to the Lagrangian’s effective couplings. As has been noted, SU(4) F relations underestimate, forexample, the D ρ D coupling by a factor of four to five [26, 27]. In this context, it is also noteworthy that SU(3) F andheavy-quark spin symmetry breaking effects are by no means insignificant and are manifest in the decay constantratios, f D s / f D , f B s / f B as well as f D ∗ / f D and f B ∗ / f B ; see, e.g., Section 4 in Ref. [21] and similar observations inlattice-QCD computations [28].With respect to the weak decay constants, our focus in Eq. (1) is on the hadronic matrix elements, (cid:104) M | j | B (cid:105) and (cid:104) M | j | (cid:105) . The latter represents the weak decay constant of M , which in the case of a pseudoscalar meson is given by, f M P µ = (cid:104) | ¯ q a γ µ γ q b | M (cid:105) = Z (cid:90) d k ( π ) tr CD (cid:104) γ γ µ S aq ( k + η P ) Γ a , bM ( k ; P ) S bq ( k − ˆ η P ) (cid:105) , (2)where a , b collect flavor and color indices, Z is the wave function renormalization constant, S q ( k ) are dressed quarkpropagators and Γ ( k ; P ) is the mesons’s Bethe-Salpeter amplitude (BSA). Note that in a Poincaré invariant treatment,the BSA and the weak decay constant — and any other hadronic matrix element — are independent of the momentumpartitioning parameters η + ˆ η =
1. In some cases, for instance for the pseudoscalar mesons, π , K , D , D s and B , thevalues of the weak decay constants are known experimentally [29]. Much effort has also been invested to determinethe heavy-light decay constants with lattice-regularized QCD, recently with unquenched fermions and increasinglybetter extrapolations to the continuum limit and physical pion masses [28, 30, 31, 32, 33, 34, 35]. Note that while the c -quark is treated as a propagating mode, the b -quark is usually implemented as a static fermion in lattice calculationsof f B and f B s .The first form factor in Eq. (1), (cid:104) M | j | B (cid:105) , describes the transition of a heavy to a light(er) meson via the weak V − A current and includes the propagation of the light spectator quark. Their precise evaluation is crucial in determiningbranching fractions and associated CP -violating observables of non-leptonic weak D and B decays [14, 16, 17] andoscillations [18, 19]. In semi-leptonic decays of heavy and light mesons they play a pivotal role in the determinationof the Standard Model parameters, more precisely its weak sector via the CKM matrix elements. For example, the K e decay, K → π e ν e , can be used to extract the matrix element | V us | ; D e to obtain | V cs | ; and the semi-leptonic decays of B and B c mesons, in particular B → D (cid:96) ν (cid:96) , B c → D ∗ s (cid:96) + (cid:96) − and B → π (cid:96) ν (cid:96) , inform the matrix elements V cb , V cs and V ub ,respectively [36, 37]. As an example, we consider the case of heavy H ( − + ) to light(er) P ( − + ) transitions mediatedby the weak HQET operators, ¯ q l γ µ ( − γ ) Q , where the hadronic matrix element is completely described by twoLorentz vectors, (cid:104) P ( p ) | ¯ q l γ µ ( − γ ) Q | H ( p ) (cid:105) = F + ( q ) P µ + F − ( q ) q µ , (3)with the total heavy-meson momentum, P µ = ( p + p ) µ , P = − M H , q µ = ( p − p ) µ , Q = c , b and q l = u , d , s .Applications of light-quark propagators solutions of QCD’s Dyson-Schwinger equations (DSE) in conjunction withthe heavy-quark expansion to the form factors, F + ( q ) and F − ( q ) , are in qualitative and quantitative agreement withheavy-quark symmetry [38, 39]. Yet, when the form factors in Eq. (3) are calculated both ways [38, 39], namelywith the fully dressed heavy-quark propagator and the propagator in the heavy-quark limit, it is possible to verify thevalidity of said limit: corrections are of the order of (cid:39) −
30% are encountered in b → c transitions and can beas large as a factor of 2 in c → d transitions, as verified in a vast array of light- and heavy-meson observables [39]. i.e., models which implement SU(4) F flavor symmetry in their Lagrangian approaches yet break these symmetries with the empirically knownhadron masses. oreover, the following ratios of transition form factors serve as a measure of SU(3) F breaking: F B → K + ( ) F B → π + ( ) = . , A B → K ∗ ( ) A B → ρ ( ) = . . (4)In Eq. (4), A ( q ) are the appropriate form factors in H ( − + ) to V ( −− ) transitions [39]. The flavor breaking is ofsimilar order as for the decay constant ratios, f D s / f D and f B s / f B , discussed in detail in Ref. [21].For a summary of heavy-to-light transition form factor data from lattice-regularized QCD, see the review inRef. [40]; we merely stress that contemporary lattice results are obtained for large squared-momentum transfer, i.e., q (cid:38)
16 GeV in the case of B → π transitions and q (cid:38) or q in B → D transitions [41, 42, 43]. Values atlow q must necessarily be extrapolated by means of appropriate parametrizations [44]. The rather strong quantitativedifferences between several predictions for the B → π form factor are emphasized in table 1 of Ref. [45] from whichit is plain that model dependence is still the major obstacle to a precision calculation of even the simplest transitionform factors.In order to significantly improve on these form factor predictions in nonperturbative continuum QCD approaches,much progress has to be made in the hadronic description of heavy-light bound states. Within the framework of theDSEs and Bethe-Salpeter equations (BSE), recent efforts come to the conclusion that the so-called rainbow-laddertruncation fails to adequately reproduce basic static observables, such as the weak decay constant via Eq. (2) [46].Other nonperturbative quantities are light-front distribution amplitudes which play an important role in QCDF analysesof hard exclusive processes. In particular, light-front projections of the pseudoscalar’s BSA, namely the pseudoscalarand pseudotensor projections, are identified as twist-three two-particle distribution amplitudes for which estimationswith QCD sum rules exist [47, 48, 49]. The pseudoscalar projection has recently been derived [50] and the samemethod may be applied to the pseudotensor projection. These projections can then be extended to D and B mesonsprovided a reliable BSA exists which allows for a faithful reproduction of experimental data on the respective weakdecay constants. It is the aim of this contribution to sketch the path to a successful computation of the heavy-lightmeson’s BSA.
2. TWO, THREE, FOUR ... HOW MANY POINTS IN A MESON?
In the continuum formulation, QCD’s two-point Green functions are described by DSE, which provide the adequatenonperturbative approach. Likewise, mesons are quark-antiquark bound states which appear as poles in the 2-quark,2-antiquark Green’s function, G ( ) = (cid:104) | q q ¯ q ¯ q | (cid:105) These poles are found from studies of the inhomogeneouspseudoscalar and axialvector BSE [1, 51, 52, 53], as will be discussed shortly.The Dyson or gap equation determines how quark propagation is influenced by interaction with the gauge fields.For a given quark flavor, the solutions of the quark DSE, S − ( p ) = Z ( i γ · p + m bm ) + Z g (cid:90) Λ k ∆ µν ( q ) λ a γ µ S ( k ) Γ a ν ( − p , k , q ) , (5)where (cid:82) Λ k ≡ (cid:82) Λ d k / ( π ) represents a Poincaré invariant regularization of the integral with the regularization massscale, Λ , and Z , ( µ , Λ ) are the vertex and quark wave-function renormalization constants. The (infinitely many)nonperturbative interactions alter the current-quark bare mass, m bm ( Λ ) , which receives corrections from the self-energy given by the second term in Eq. (5), where the integral is over the dressed gluon propagator, ∆ µν ( q ) , the dressedquark-gluon vertex, Γ a ν ( − p , k , q ) , and λ a are the usual SU(3) color matrices of the fundamental representation. Thegluon propagator is purely transversal in Landau gauge, which offers advantages in phenomenological interactionansätze [1, 54]: ∆ ab µν ( q ) = δ ab (cid:18) g µν − q µ q ν q (cid:19) ∆ ( q ) . (6)The quark-gluon vertex is given by Γ a µ ( p , p , p ) = g λ a Γ µ ( p , p , p ) with the convention: p + p + p = S ( p ) = (cid:2) i γ · p A ( p ) + B ( p ) (cid:3) − with the renor-malization condition, Z ( p ) = / A ( p ) | p = µ = µ (cid:29) Λ . The mass function, M ( p ) = B ( p , µ ) / A ( p , µ ) , is independent of the renormalization point µ . In order to make quantitative matching withQCD, another renormalization condition, S − ( p ) (cid:12)(cid:12) p = µ = i γ · p + m ( µ ) , (7)is imposed, where m ( µ ) is the renormalized running quark mass.Before discussing Bethe-Salpeter amplitudes, we turn our attention to the quark-gluon vertex Γ µ ( p , p , p ) ,which is one of QCD’s three-point functions and satisfies its own BSE. In perturbation theory, that is for momenta p = p = p (cid:38) µ , quark dressing effects are suppressed and Γ µ ( p , p , p ) → γ µ . However, since the tremendousimpact of DCSB on Z ( p ) and M ( p ) is nowadays well established, it is natural to accept that this also be true for thecorresponding three-point functions.In applications to hadron physics, practical models for the fermion-gauge boson vertex ought to satisfy funda-mental symmetries of QCD. General ansätze to the nonperturbative vertex impose constraints of quantum field the-ory; as just mentioned, one insists that the vertex must reduce to the bare vertex γ µ in the large-momentum limit(when dressed propagators can be replaced by bare propagators); it must have the same transformation propertiesas the bare vertex under charge conjugation C , parity transformation P and time reversal T ; it must ensure gaugecovariance and invariance; and one demands that the vertex must be free of kinematic singularities. Finally, thefull nonperturbative vertex can always be decomposed into a longitudinal and a transverse part, Γ µ ( p , p , p ) = Γ L µ ( p , p , p ) + Γ T µ ( p , p , p ) [55]. Clearly, gauge invariance is not satisfied for a bare vertex since it does not satisfythe Ward-Green-Takahashi identity (WGTI), i γ · p (cid:54) = − i γ · p A ( p ) + B ( p ) − i γ · p A ( p ) − B ( p ) . (8)Models that are largely consistent with the field theoretical constraints just mentioned have also been used to representthe dressed quark-gluon vertex, the most prominent amongst which is the Ball-Chiu ansatz for the longitudinalvertex [55]. However, while employed in studies of hadronic observables, the Ball-Chiu vertex satisfies a WGTIwhereas the true quark-gluon vertex satisfies a Slavnov-Taylor identity (STI). The form of the latter, see Eq. (11),makes it plausible that within certain approximations a solution of p µ i Γ µ ( p , p , p ) = B ( p ) (cid:2) S − ( − p ) − S − ( p ) (cid:3) (9)can provide a reasonable approximation to the correct vertex.In view of the scarce information on the quark-gluon vertex from first principle calculations, the strategy to combinedifferent nonperturbative approaches to QCD was explored in Ref. [56]. Therein, lattice-QCD data for the dressed-quark functions, A ( p ) and B ( p ) [57, 58], as well as for the gluon and ghost propagators, ∆ ( q ) and F ( q ) [59, 60],were employed to numerically extract a momentum-dependent effective function ˜ X ( q ) from the quark gap equationvia an inversion procedure. In order to apply this inversion, one defines a "ghost-improved" Ball-Chiu vertex [61, 62],˜ Γ BC µ ( p , p , p ) = ˜ X ( p ) F ( p ) Γ BC µ ( p , p , p ) , (10)which can be derived from the constraints of the STI, p µ i Γ µ ( p , p , p ) = F ( p ) (cid:104) S − ( − p ) H ( p , p , p ) − H ( p , p , p ) S − ( p ) (cid:105) , (11)where F ( q ) is the ghost-dressing function and the quark-ghost scattering kernel is parameterized in terms of thematrix-valued function, H ( p , p , p ) , and its conjugate, H ( p , p , p ) [63]. The decomposition of these two functionsin terms of Lorentz covariants requires eight form factors, H ( p , p , p ) = X I D + i X γ · p + i X γ · p + i X σ αβ p α p β , (12) H ( p , p , p ) = X I D − i X γ · p − i X γ · p + i X σ αβ p α p β . (13)Perturbative expressions for the form factors X i have been computed to one-loop order [63] and yield X = + O ( g ) and X i = O ( g ) , i = , ,
3. Thus, X is the dominant form factor at large momenta and using the approximations X , , (cid:39) X = X = ˜ X ( q ) the dressed quark-gluon vertex reduces to the expression in Eq. (10) which satisfiesthe identity (9) with B ( q ) = ˜ X ( q ) F ( q ) .The quark-gluon vertex built from the resulting ˜ X is enhanced in the infrared region and recovers the perturbativebehavior as one approaches larger momenta. The two different inversion methods employed in Ref. [56], linear .01 0.1 1 10 q² [GeV²] X ( q ²) MEM maximum errorMEM root-mean-sqare errorMEM best fit (for a eff ) q² [GeV²] a e ff ( q ²) M g = 400 MeVM g = 500 MeVM g = 600 MeV FIGURE 1.
Left panel: the effective quark-gluon vertex function, ˜ X MEM0 , from a nonlinear inversion based on MEM; see Ref. [56]for details. Note that the functional form of ˜ X MEM0 differs from that in Ref. [56] where ˜ X MEM0 ( q → ) →
1. Here it is clearlyenhanced in the infrared due to the requirement that q ˜ X ( q ) be finite in the MEM inversion. However, considering the case ofmaximal correlation of the MEM fit parameters, represented by the maximum error (yellow) band, the solution for ˜ X in Ref. [56]is compatible with the present one. Owing to the lack of lattice-QCD data on A ( p ) and B ( p ) , the inversion procedure is simplynot constrained below 0 .
14 GeV. Nonetheless, either solution yields the mass functions in the left-hand plot of Fig. 2 since the DSEkernel is vanishing for q (cid:46) . X thus becomes irrelevant in this momentum range. Right panel:the effective charge defined in Eq. (10) with ˜ X ≡ ˜ X MEM0 . regularization and the maximum entropy method (MEM), produce ˜ X form factors compatible with each other forthe range of momenta where lattice-simulation data is available, i.e. in the domain (cid:39) . − . − X ( q ) becomesless reliable owing to the lack of lattice-data constraints. Similarly, the steep increase of ˜ X ( q ) below (cid:39) . is tobe taken with caution. Nonetheless, the functional form of ˜ X above q (cid:38) . bears strong similarities with thatof the ghost-dressing function [59, 60], F ( q ) , which a posteriori justifies the prescription F ( q ) → F ( q ) employedin Ref. [61]. We thus, in analogy with Ref. [64], define an effective charge via the combination, α eff ( q ) = α s ˜ X ( q ) F ( q ) ∆ ( q ) (cid:2) q + m g ( q ) (cid:3) ; m g ( q ) = M g q + M g , (14)plotted in Fig. 1, where α s ( . ) = .
295 and M g typically of the order 500 −
600 MeV. p² [GeV²] M ( p ²)[ G e V ] chiralu quark, m curr = 6 MeVs quark, m curr = 82 MeVc quark, m curr = 1.29 GeVb quark, m curr = 4.19 GeV MEM p² [GeV²] M ( p ²)[ G e V ] chiralu quark, m curr = 6 MeVs quark, m curr = 82 MeVc quark, m curr = 1.29 GeVb quark, m curr = 4.19 GeV (D w) = 0.8 FIGURE 2.
Flavor dependence of the solutions for the mass function M ( p ) = B ( p ) / A ( p ) ; left panel: using the effectiveinteraction model of Eq. (14); right panel: DSE with interaction model and given parameter set of Ref. [65]. Re q [GeV ] -4-2024 I m q [ G e V ] Re q [GeV ] -4-2024 I m q [ G e V ] ImA <-0.250-0.250< ImA< -0.125 -0.125< ImA <-0.025ImA = 00.025< ImA < 0.1250.125< ImA < 0.2500.250< ImA
Re q [GeV ] -4-2024 I m q [ G e V ] Re q [GeV ] -4-2024 I m q [ G e V ] Im B < -0.26 -0.26< Im B < -0.17 -0.0072< Im B< -0.00026 Im B = 00.00026 FIGURE 3. Contour levels of the real and imaginary parts of the DSE solutions, A ( p ) and B ( p ) , for complex momenta usingthe interaction model of Ref. [65]. The flavor dependence of the solutions for the mass function is depicted in the left panel of Fig. 2. For the lightquarks, u and d , the DCSB leads to M ( ) (cid:39) 220 MeV in accordance with lattice results [57, 58]. Yet, in comparisonwith the best available phenomenological interaction model [65] whose functional behavior accords qualitatively withresults of modern DSE and lattice studies, the effect of the DCSB is much weaker: as seen in the right panel ofFig. 2, the model of Ref. [65] yields M ( ) (cid:39) 600 MeV. Although the consequences of DCSB are less marked for theheavy quarks’ mass functions, which remain almost constant over a large momentum domain, we do note a difference ∆ M b ( ) (cid:39) 600 MeV between both models. It is thus expected that the interaction defined by Eq. (14) is too weakfor applications to hadron phenomenology. This shortcoming may be remedied by the inclusion of the transversequark-gluon vertex component which describes the quark’s anomalous chromomagnetic moment [66].Mass functions are not physical observables and to test the validity and efficacy of an interaction model it must standthe comparison with experimental data. To this end, bound state equations must be solved and while the propagatorsatisfies the gap equation, the vertices are determined by an inhomogeneous BSE. Consider, for instance, the exactinhomogeneous axialvector BSE [67] which is valid when the quark-gluon vertex is fully dressed, i.e. for an ansatzbeyond the rainbow-ladder truncation: Γ f g µ ( k ; P ) = Z γ γ µ − g (cid:90) Λ q D αβ ( k − q ) λ a γ α S f ( q + ) Γ f g µ ( q ; P ) S g ( q − ) λ a Γ g β ( q − , k − )+ g (cid:90) Λ q D αβ ( k − q ) λ a γ α S f ( q + ) λ a Λ f g µβ ( k , q ; P ) , (15)where P is the total meson momentum, q ± = q ± P / , k ± = k ± P / f , g denote the flavor indices of a light-lightor heavy-light bound state. The 4-point Schwinger function Λ f g µβ is entirely defined via the quark self energy. Thiscomes about that a WGTI can be derived for the Bethe-Salpeter kernel whose solution provides a symmetry-preservingclosed system of gap and vertex equations [67, 68]. The pseudoscalar vertex, Γ f g ( k ; P ) , satisfies an analogous equationto Eq. (15) and it is a well known feature of QCD that both the axialvector and pseudoscalar vertices exhibit poleshenever P = − m M n , where m M n is the mass of the meson M or any of its radial excitations [69, 70] : Γ µ ( k ; P ) (cid:12)(cid:12) P + m Mn (cid:39) = f M n P µ P + m M n Γ M n ( k ; P ) + Γ reg . µ ( k ; P ) , (16)Here, Γ M n ( k ; P ) is the pseudoscalar bound state’s BSA. The solutions of the BSE for P = − m M n in Euclideanmomentum space requires the knowledge of the quark propagator at complex momenta whose squares lie inside aparabola and which in the past presented a considerable numerical challenge at large quark masses, m q > m c . Improvednumerical methods which facilitate the treatment of the quark’s DSE are now available [71] and in Fig. 3 we presentthe real and imaginary parts of the complex solutions A ( p ) and B ( p ) , where P (cid:39) − . Solutions for heavierquarks are currently being investigated and in conjunction with Eq. (15) first results for the heavy meson’s BSA usingEq. (10) will soon be available. 3. EPILOGUE We have summarized recent progress towards computing the DSE and BSE for heavy-light systems beyond therainbow-ladder truncation based on an ansatz for the quark-gluon vertex which correlates the tensor structure of theBall-Chiu vertex with a nonperturbative vertex function. The functional form of the latter is extracted from lattice-regulated QCD data on the quark’s dressed propagator via an inversion of the DSE [56]. When the vertex modelis re-inserted in the quark’s DSE, its solutions yield mass functions which are qualitatively comparable with thoseobtained with a recent interaction model [66] but whose magnitude of DCSB is considerably smaller. Computationsof the BSA for heavy-light systems which make use of the exact form of the BSE, valid for the fully dressed quark-gluon vertex, and either interaction models are underway. The first test any beyond the rainbow-ladder ansatz for thequark-gluon vertex must pass is the numerical value one obtains for one of the most elementary observable, i.e. theweak decay constant which is more sensitive to the BSA normalization. We shall report results of its computation for D ( s ) and B ( s ) mesons in a future communication. This will be the first in a series of steps to obtain their form factorsand parton distribution amplitudes that are our original motivation (see Section 1) and for which the well knownrainbow-ladder ansätze [69, 70] are not phenomenologically valuable. ACKNOWLEDGMENTS This work is supported by the São Paulo Research Foundation, Fundação de Amparo à Pesquisa do Estado de SãoPaulo (FAPESP), and the federal agency, Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).We acknowledge valuable communication with Orlando Oliveira and Tobias Frederico. B. 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