Mandelstam’s seminal work [1] established that the rainbow-ladder truncation of the gap and bound-state equations is an adequate approximation to describe dynamical chiral symmetry breaking (DCSB) [2–12] in quantum chromodynamics (QCD). Later on, within this same truncation scheme, Munczek and Nemirovsky succeeded in reproducing the masses of pseudoscalar and vector meson ground states [13]. More sophisticated models succeeded in the following decades that satisfy theoretical and phenomenological constraints [14–26]. Their popularity owes to a wide range of successful applications to mesons, baryons, hyperons, their excited states and parity partners [22–49].

In this work we propose a model that describes effectively a massive gluon interaction in the infrared region, where we are inspired by the functional structure of generalized quantum electrodynamics (GQED) proposed long ago by Podolsky [50,51]. Historically, this generalization aimed at remedying pathologies inherent to the Maxwell theory and consisted in introducing higher-order derivatives in the Lagrangian of electrodynamics, maintaining at the same time linearity of the equations of motion in the fields. In other words, the goal was to eliminate the infinities that arise in higher-order corrections of point charges and the associated coupling.

However, since this extension of the Lagrangian preserves gauge invariance in a consistent treatment [52,53], GQED has come to be viewed more as a prototype of a theory that contains massless as well as massive photons that do not break gauge invariance. This is because Podolsky’s extension of electrodynamics is the only possible linear, Lorentz and U(1) invariant generalization of the Maxwell theory [54] and a consistent quantization of GQED was shown to require a generalized Landau gauge condition [53], while the proper covariant quantization of GQED in this generalized gauge was obtained with functional methods in Ref. [55]. More recently, the Podolsky approach to QED was also reinterpreted as a natural way of providing a Pauli-Villars regularization in ordinary QED [56]. GQED introduces therefore in a consistent manner a mass parameter mP in the vector-boson propagator while preserving gauge invariance and acting as an effective ultraviolet cutoff in Landau gauge.

These features are clearly attractive for modeling the nonperturbative gluon interaction in an Abelianized truncation of QCD, given the compelling body of work that evidence an infrared-finite gluon propagator [57–79] and which can be related to an effective gluon mass. It turns out that in Landau gauge and in the leading truncation of the quark’s Dyson-Schwinger equation (DSE) we may interpret the Podolsky propagator as a nonperturbative model for the gluon propagator, at least in the low-momentum region, where its massive “dressing function” effectively drives the strength of the DCSB.

In Sec. II we introduce the DSE that describes the quark-gap equation with a Podolsky propagator in Landau gauge and obtain its solutions for different flavors on the spacelike real axis. The functional behavior of the quark’s mass and wave-renormalization function is reminiscent of that found with the Maris-Tandy [24] or Qin-Chang [26] models and the obvious question arises whether this interaction is useful for hadron phenomenology. We solve the Bethe-Salpeter equation (BSE) as usual in Euclidean space to find antiquark-quark bound states, which implies that the arguments of the quark propagators are complex-valued momenta. To obtain the quark propagators on the complex plane, we apply Cauchy’s integral theorem that requires DSE solutions on a contour defined by a parabola describing the complex-momentum distribution [80,81]. We note that the convergence of the DSE on such a contour is not generally guaranteed for a given interaction regardless of its convergence on the real axis. Nonetheless, using the Podolsky propagator with an appropriate parametrization the DSE converges rapidly on this contour and the BSE solutions reproduce the mass spectrum and weak decay constants of pseudoscalar mesons as we discuss in Sec. III. We finish with some concluding remarks about possible extensions and applications of the Podolsky propagator in Sec. IV.

The gap equation for a quark of flavor f is expressed by a DSE for the inverse propagator in Minkowski space as Sf-1(p)=Z2γ·p-Z4mf(μ)-Z1g2∫Λd4k(2π)4Gμν(q)λa2γμSf(k)λa2Γν(k,p),(1)where Z2(μ,Λ), Z4(μ,Λ) and Z1(μ,Λ) are the wave function, mass and vertex renormalization constants, respectively. Moreover, Γμa(k,p)=12λaΓμ(k,p) is the quark-gluon vertex and λa are the SU(3) color matrices in the fundamental representation, while Λ is a Poincaré-invariant regularization scale, chosen such that Λ≫μ. In GQED the vector-boson propagator in a covariant gauge with gauge-fixing parameter ξ and momentum, q=k-p, is given by Gμν=-iΔ(q2)Pμν(q),Pμν(q)=Δμν(q)-[gμν+(1-ξ)qμqνq2-mP2]1q2-mP2+(1-2ξ)qμqνq2(q2-mP2)+qμqν(q2-mP2)2,(2)with the standard gauge-boson propagator, Δμν(q)=[gμν-(1-ξ)qμqνq2]1q2.(3)

In general, the Dirac structure of the fermion propagator is fully defined by two covariants and associated scalar functions, the wave-function renormalization Ff(p) and the mass function Mf(p), so that Sf(p)=Ff(p)γ·p-Mf(p).(4)In order to determine the renormalization constants and to make quantitative matching with pQCD, one imposes the renormalization conditions, Ff(p2)|p2=μ2=1,Sf-1(p)|p2=μ2=γ·p-mf(μ),(5)where μ2≫ΛQCD2 and mf(μ) is the renormalized running quark mass; in particular, mf(μ) is nothing else but the dressed-quark mass function evaluated at one particular deep spacelike point, p2=μ2, namely mf(μ)=Mf(μ).

The mass function Mf(p2) and the renormalization wave function Ff can be projected out from the DSE (1) and rewritten in Euclidean space one obtains¹

See Appendix for details of the calculation in arbitrary covariant gauge.

We note that the Podolsky mass, mP≈0.77  GeV, which serves as an infrared mass scale in our model, agrees with other gluon mass scales [14,15,82]. In particular, in Landau gauge, the integral expressions for Mf(pE) and Ff(pE) in Eqs. (6) and (7) can also be obtained by taking the projections and traces of the DSE (1) using a modified perturbative gluon propagator with a mass term: Dμν(q)≔mP2qE2+mP2(gμν-qEμqEνqE2)1qE2=mP2qE2+mP2Δμνξ=0(qE2).(9)Indeed, for ξ=0 and with Δ(q2)=1 the Podolsky propagator (2) reduces to Eq. (9) and is equivalent to multiplying by a factor mP2/qE2 the massive gluon propagator introduced in Ref. [14]. The latter is motivated by soft breaking of Becchi-Rouet-Stora-Tyutin (BRST) symmetry and described by a gluon-mass term in the Lagrangian corresponding to a particular case of the Curci-Ferrari model [83]. Alternatively, here the vector-boson mass mP originates in higher-order derivatives which preserve gauge symmetry in a generalized Lagrangian of the Abelian field theory. The coupling αeff, on the other hand, cannot be directly compared with the running strong coupling at this scale. This is because we merely employ the rainbow truncation of the DSE in which αeff partially accounts for the lack of DCSB from a fully dressed quark-gluon vertex [84–100].

In this context, we remind that one can infer qualitative and analytic properties of the interaction kernel from the mass and wave-renormalization functions via an inversion process of the DSE [87,88] which allows for the comparison of different models. We also stress that it is possible to modify the interaction (8) to include the perturbative ∼1/q2 running at larger momenta. However, our aim here merely consists in verifying the simple expression’s (8) capacity to yield the adequate DCSB observed in hadron phenomenology.

We close this section with the graphs of Mu(k2) and Fu(k2) functions on the complex plane, i.e., their solutions calculated on a parabola of complex momenta p2 defined by the arguments of the quark propagators in the BSE (11), k±2=(k±η±P)2=k2-η±2MP2±2iη±MP|k|zk,(10)where P2=-MP2 is the external meson mass of the pseudoscalar meson, zk=k·P/|k||P|, -1≤zk≤+1, is an angle and henceforth the Euclidean metric is implicit: k2=kE2. The real and imaginary parts of Mu(k2) and Fu(k2) are plotted in Figs. 2 and 3, respectively. The behavior of the real part of Mu(k2) is smooth, monotonically decreasing in both, real and imaginary k±2 directions, whereas the real part of Fu(k2) tends towards its perturbative limit both on and off the real axis. The imaginary part of both functions is characterized by complex-conjugate extrema near the origin of the parabola. For details of the DSE solutions on the complex plane we refer to Refs. [30,80,81]. Within the complex momentum range considered herein, the Podolsky model results in complex DSE solutions that differ somewhat in magnitude but not much in their analytical behavior from the ones obtained with the gluon model in Ref. [26]. However, to really compare the analytic properties of these models, a more detailed study on the complex plane is necessary that allows to trace poles and branch cuts outside the parabolic contour presented in Figs. 2 and 3.

The homogeneous BSE for a q¯q bound state with relative momentum p and total momentum P can be written as Γmnfg(p,P)=∫Λd4kE(2π)4Kmnkl(p,k,P)[Sf(k+)Γfg(k,P)Sg(k-)]lk,(11)where m, n, k, l collect Dirac and color indices, f, g are flavor indices and k+=k+η+P, k-=k-η-P; η++η-=1. Since we work within the rainbow-ladder truncation, the BSE kernel is given by Kmnkl(p,k,P)=-4παeff(λa2γμ)knPμν(q)(λa2γν)ml,(12)which satisfies the axial-vector Ward-Green-Takahashi identity [23] and as a consequence ensures a massless pion in the chiral limit. As such, Eqs. (11) and (12) define an eigenvalue problem with physical solutions at the on-shell points, P2=-MP2. As in the DSE (1), the vertex renormalization is absorbed in αeff.

The general Poincaré-invariant form of the Bethe-Salpeter amplitudes (BSA), i.e., the solutions of Eq. (11), for the pseudoscalar channel JP=0- in a nonorthogonal base with respect to the Dirac trace, Aα(p,P)=γ5{iID,γ·P,γ·p(p·P),σμνpμPν}, is given by Γ(p,P)=γ5[iIDE(p,P)+γ·PF(p,P)+γ·pp·PG(p,P)+σμνpμPνH(p,P)],(13)where we suppress color, Dirac and flavor indices for the sake of readability and σμν=i/2[γμ,γν]. The functions Ffα(p,P)={E(p,P),F(p,P),G(p,P),H(p,P)} are Lorentz-invariant scalar amplitudes. For sake of completeness, we note that all BSA are normalized canonically as 2Pμ=∫Λd4kE(2π)4TrCD[Γ¯(k,-P)∂S(k+)∂PμΓ(k,P)S(k-)+Γ¯(k,-P)S(k+)Γ(k,P)∂S(k-)∂Pμ],(14)where we omit a third term that stems from the derivative of the kernel, ∂Kmnkl(p,k,P)/∂Pμ, since it does not contribute in the rainbow-ladder truncation of Eq. (12).²

We verify the values obtained with Eq. (14) with the equivalent normalization condition [101,102]: (dlnλn/dP2)-1=tr∫kΛ3Γ¯(k,-P)S(k+)Γ(k,P)S(k-).

The potential of this effective interaction is illustrated for the pseudoscalar q¯q channel in Table I. We work in the isospin-symmetric limit mu=md and set the light quark’s mass scale at 19 GeV with the pion mass; analogously we fix the strange- and charm-quark masses with the kaon and the ηc. The resulting weak decay constant of the pion is within 2% of the experimentally extracted value and that of the kaon is within 5% of its reference value. The charmonium’s decay constant is about 15% larger than a calculation using lattice QCD. We also compare the weak decay constants obtained with Eq. (15) with those making use of the Gell-Mann-Oaks-Renner (GMOR) relation described in detail, for example, in Ref. [30] and find very good agreement.

This is a first step to establish the model’s potential to correctly describe light-meson and quarkonia ground states and as an additional check we calculate the D and Ds meson’s masses and decay constants. As observed in Table I and previously in Ref. [30], their masses are overestimated by 12% for the D and 8% for the Ds with respect to experimental values. This is a consequence of the rainbow-ladder truncation which neglects the dramatically different impact of vertex dressing for heavy and light quarks and can be strongly improved by inclusion of this effect [36]. Likewise, the weak decay constants are also 22% larger than results reported by the FLAG collaboration [105]. Nonetheless, the Podolsky propagator facilitates the numerical treatment of the DSE with a large external Euclidean mass (using η+=0.8 and η-=0.2) on the complex plane in comparison with the model of Ref. [26] and both iterative treatments of the DSE and BSE mesons converge rapidly in case of the D and Ds.

We solve for the first time the DSE with the Podolsky propagator in rainbow truncation and find a mass gap for a typical hadronic scale of the Podolsky mass. Based on this observation, we propose a novel interaction model within the rainbow-ladder truncation of the DSE and BSE kernels, based on the Podolsky propagator which preserves gauge invariance in perturbative GQED. However, in our approach we merely interpret this massive propagator as a nonperturbative ansatz for the dressed gluon propagator. The associated mass scale we find is reminiscent of earlier DSE studies of the gluon. The Podolsky propagator has the same integrability properties as the perturbative propagator, a feature which is the practical motivation for our model, and when employed in the appropriate DSE we find well-behaved solutions on the complex plane even for larger timelike momenta. Employing these complex solutions for the quark propagators in the BSE, we fix the quark masses and interaction parameters with the masses of the π, K, ηc and find weak decay constants that agree very well with experimental reference values or results from lattice-QCD simulations. The D mesons are also obtained within this framework, where their masses are somewhat overestimated and the mass difference between the D and Ds is too small, a consequence of the too simplistic truncation for heavy-light systems.

The Podolsky propagator along with the ansatz in Eq. (8) ought to be used in future calculations of radial excitations and other JPC channels. Since we are interested in static properties of the mesons, we here omit the perturbative term of the interaction commonly included in other models. As a consequence of this simple form, we established that the angular integration of the DSE, Eqs. (A11) and (A12), can be performed analytically. Likewise, the simple pole structure of the Podolsky propagator allows, in given cases, for analytical calculations as with any perturbative propagator. These features are attractive enough to raise one’s curiosity about possible solutions of the DSE and BSE in Minkowski space, which is of practical importance with regard to the study of parton distribution functions defined on the light front. Amongst other applications, one may explore this interaction to obtain more sophisticated quark propagators and wave functions at finite density to understand the pion’s properties in a dense nuclear medium [106].