Fermion and Photon gap-equations in Minkowski space within the Nakanishi Integral Representation method

The approach based on the Nakanishi integral representation of n-leg transition amplitudes is extended to the treatment of the self-energies of a fermion and an (IR-regulated) vector boson, in order to pave the way for constructing a comprehensive application of the technique to both gap- and Bethe-Salpeter equations. This step offers the possibility of embedding the characteristics of the non-perturbative regime at a more fundamental level.


Introduction
The description of bound states, fully taking into account the general principles of the relativistic quantum field theory (QFT) and the needed non perturbative regimes, is a longstanding and highly challenging problem. As it is well-known, the formal solution of the problem can be traced back to the birth of QFT, with the seminal paper by Salpeter and Bethe [1]. Starting from the analysis of the pole contributions to the Green's function relevant for the bound state under scrutiny (e.g., the four-points Green's function for investigating two-body bound states), they introduced an integral equation, known as the Bethe-Salpeter equation (BSE) for the bound-state amplitude, where the kernel is obtained from the two-particle irreducible diagrams, describing the dynamics inside the system. The systematic evaluation of the interaction kernel needs in turn the knowledge of other key ingredients: i) self-energies of both intermediate particles and quanta and ii) vertex functions (see also Ref. [2]). Unfortunately, those 2-and 3-point functions are quantities to be determined through the infinite tower of Dyson-Schwinger equations (DSEs) [3][4][5] (see for introductory reviews, e.g., Refs. [6][7][8][9][10][11], and references quoted therein) that govern the whole set of N -point functions. Therefore, in order to make feasible the construction of more and more realistic interaction kernels, model builders have to elaborate strategies for truncating the DSEs infinite tower, as much self-consistently as possible, while retaining the dynamical effects, at the greatest extent.
In the last decades, significant progresses have been done for implementing the aforementioned program, and a high degree of sophistication has been achieved, but mainly in Euclidean space (see Refs. [6][7][8][9][10][11][12] and also Refs. [13][14][15][16][17][18][19]). As is well known, the Euclidean space allows one to address the space-like observables, leaving aside quantities that live onto the light-cone or generically inside the causal region. Within the framework based on a Euclidean space and truncated DSEs 1 , the most widely investigated field is the continuum QCD, with an impressive wealth of applications in hadron physics, ranging from baryon and meson spectra to elastic electromagnetic (em) form factors and transition ones (see, e.g., [6][7][8][9][10][11][12]), but also applications to QED have been pursued at large extent (see, e.g. Refs. [20][21][22][23] and for a recent study in Minkowski space Ref. [24]). The interest in investigating the QED, in the whole dynamical range, may be surprising, given the extraordinary accuracy achieved in the comparison with the data by using perturbative tools (see, e.g., the case of the muon anomalous magnetic moment [25]). Indeed, a close study of the non-perturbative regime on the one side could be relevant for shedding light onto QED at very short distances (e.g., the Landau singularity, time by time considered as an academic issue or a problematic topic, could be a potential target, as well as the distinctive features of the QED critical coupling), and on the other side could represent a possible warm-up to go beyond linear theories, when the self-energies of particles and quanta modify each other in a very sizable way.
Our investigation will focus on to the study of QED in the non perturbative regime, and in order to help the reader to better appreciate the differences with other approaches it is useful to indicate, even in a simplistic way, the directions along which we will move in what follows. For this reason, let us immediately mention the two key ingredients we will adopt: i) the Minkowski space, where the physical processes take place and ii) the structure of the vertex function. The vertex will be composed by the well-known part introduced by Ball and Chiu in Ref. [26], that fulfills the Ward-Takahashi identities (WTIs) (both the differential form and the finite-difference one), plus a transverse contribution (see, e.g., Refs [20,22,24,[27][28][29][30] for a wide discussion), based on a minimal Ansatz proposed in Ref. [14]. Such a transverse term is able to restore the full multiplicative renormalization of both fermion and photon propagators (solutions of suitably truncated DSEs), and in turn gracefully implements a workable truncation scheme. As to this point, there is a very important remark about the fundamental role of the transverse part of the vertex, that is somewhat hided by the great relevance of WTI. A part of the general structure that can be deduced by the curl of the vertex, its dynamical content is not only related to the fermion self-energy (as in the case of the Ball-Chiu vertex [26]), but it needs the knowledge of the full off-shell scattering matrix. In principle, this should lead to consider the whole tower of DSEs, if one does not introduce a self-consistent truncation scheme.
Another important ingredient, though more technical, is represented by the so-called Nakanishi integral representation (NIR) of a generic n-leg transition amplitude [31][32][33]. Indeed, such a tool has allowed one to undertake new efforts for developing methods for solving in Minkowski space both truncated DSEs (see, e.g. Refs. [24,[34][35][36][37][38][39]) and BSE (see, e.g., Refs. [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54], where systems with and without spin degrees of freedom are investigated). The main motivation for adopting the NIR, closely related to the Stieljes transform (see an application in Ref. [55]), is given by the possibility to express the n-leg transition amplitudes through their all-order perturbative form. The freedom needed for exploring a non perturbative regime is assured through the unknown Nakanishi weight functions (NWFs), that are real functions [33]. Such a freedom has shown all its relevance in the numerical studies of the bound states (by using both ladder and cross-ladder interaction kernels), that are the main instance where the realistic description of the non perturbative regime is necessary. Furthermore, the NWFs to be used for the self-energies do not depend upon the external momenta, greatly simplifying our formal elaboration, as shown in what follows. The advantage of the NIR is that the four-momentum dependence is made explicit, allowing direct algebraic manipulations, and eventually making affordable analytic integrations. This is an important virtue of the NIR approach, since it simplifies the treatment of the expected singularities. On the phenomenology side, when light-cone observables have to be evaluated, e.g. for describing the partonic structure of hadrons [56][57][58][59][60][61][62][63][64][65], the explicit dependence upon the momenta facilitates the needed projection onto the light-cone. However, in the NIR context, the dynamical assumptions are still much simpler than the one made in Euclidean calculations. For instance, in the above mentioned works when solving Minkowskian BSE for mesons, the constituent fermions are most of the time considered perturbative-like, i.e. omitting the running of the dressed quark mass (with the exception of Ref. [52] where it has been proposed to import the running mass of the quarks from the Euclidean lattice into the BSE framework).
Our present effort aims at formally developing a method based on NIR for solving a coupled system composed by the gap-equations for both fermion and gauge boson, directly in Minkowski space. The integral-equation system is obtained by implementing a self-consistent truncation scheme of DSEs, valid in the whole dynamical range of QED, and by adopting both dimensional regularization and momentum-subtraction procedure for the renormalization. Moreover, to pragmatically remove the well-known IR divergences, a tiny mass-regulator has been introduced for the gauge boson, (see, e.g., Ref. [66] for a more general discussion and Ref. [67] for a recent analysis). In general, we share the same spirit of works as: i) Ref. [24] where, within a quenched approximation, a spectral representation of the fermion propagator was adopted, in combination with its Källén-Lehman (KL) representation, and, importantly, a vertex function was constructed by exploiting the form suggested by the Gauge Technique [68] plus transverse terms, added for matching the perturbative expressions of the renormalized fermion self-energy; ii) Refs. [34,36,37] (see also Ref. [69] for a first study of the transverse vertex contribution), where a more direct link to the NIR technique (with different sets of approximations) can be found. Simplifying, the main difference with the previous works is a fully dressing of fermion and photon selfenergies, by introducing a vertex function composed by the standard Ball-Chiu component [26] and a minimal Ansatz for the purely transverse contribution [14], able to assure the multiplicative renormalizability of the whole approach.
The paper is organized as follows. In Sect. 2, the general formalism is introduced for fermion and photon propagators and self-energies, in terms of the KL representations and the NIR, respectively. In Sect. 3, the adopted vertex function is discussed. In Sect. 4, the gap-equations are introduced and the main result of our formal analysis, i.e. the coupled system of integral equations for determining the NWFs, of both electron and photon selfenergies, is illustrated. In Sect. 5 an initial application of the coupled system, based on its first iteration, is shown. In Sect. 6, the conclusions of our analysis of the truncated DSEs within the NIR framework are drawn and the perspectives of the future numerical studies are presented. Finally, it has to be emphasized that the Appendices have been written in a detailed form for making as simple as possible a check of the whole formalism, and therefore they have to be considered an essential part of the work.

General formalism
In this Section, we summarize the general formalism that will be used in our investigation of QED in Minkowski space (see the review in Ref. [6] for the Euclidean version). We introduce first the expression of the self-energy (2-leg transition amplitude in the Nakanishi language [33], that emphasizes the set of external momenta) in terms of NIR, for both fermion and photon. Then, the KL representations of the corresponding propagators are given. The main goal of this initial step is the relations between KL weights and NWFs (see Refs. [36,37], for an analogous approach, but with renormalization constants Z 1 = Z 2 = 1 and with a bare vertex function or the Ball-Chiu one, respectively).
The suitable renormalization scheme we adopt is the momentum subtraction one, applied on the mass-shell (MOM), as discussed in what follows. It should be anticipated that both electron and photon self-energies can be nicely renormalized by applying such a scheme, given the benefit from the presence of the transverse component of the vertex function.

The renormalized propagator of a fermion
By adapting the notations in Ref. [6], one can write the following relations involving the renormalized propagator of a fermion and the regularized self-energy.
The renormalized fermion propagator is given by with ζ the renormalization point and Σ R (ζ; p) the renormalized self-energy. From Lorentz invariance, one can write with A R (ζ; p) and B R (ζ; p) suitable scalar functions. In terms of the expression in Eq.
(2.2), the renormalized propagator reads Noteworthy, by requesting that the renormalized propagator for p 2 → ζ 2 has a pole at the mass m(ζ) = m phys 2 and the same residue of the free propagator, one finds the constraints to be fulfilled by the two scalar functions, at the renormalizzation point. Needless to say, those constraints are crucial for establishing the relations between the regularized self-energy and the two renormalization constants δm and Z 2 (ζ, Λ) (cf what follows). As a matter of fact, from the well-known general approach illustrated, e.g., in Ref. [66] (or adopting Eq.
These two equations define the standard on-shell QED renormalisation scheme. Bringing in mind that the natural outcome of our formal elaboration will be a system of integral equations, needed for determining A R and B R , we adopt the following renormalization conditions defining the RI'/MOM scheme It is worth noticing the following remarks about this choice: i) it preserves the pole at the physical mass of the fermion; ii) it allows a numerical simplification, avoiding to implement boundary conditions where there is an interplay between A R and B R ; and last but not least iii) it is exploited in the literature devoted to the non perturbative studies of QFT both on the lattice (see the discussions on the RI'/MOM scheme e.g., in Refs. [70,71]) as well as in continuous approaches (see, e.g., Refs. [6,72]). Since at the present stage of the novel approach we are exploring, the two boundary conditions in Eq. (2.5) turn out to simplify the determination of the two renormalization constants, we will leave the study of QED in the standard renormalisation scheme, Eq. (2.4), for further investigation. The propagator S R can be expressed in terms of the regularized quantity, Σ(ζ, Λ; p), where Λ stands for a Poincaré invariant regulator, e.g. Λ = 1/ within a dimensional regularization framework with d = 4 − . To make the mathematical notation less heavy, in what follows it is understood that the relations involving renormalized quantities hold only in the limit Λ → ∞. Hence, one writes where Z 2 (ζ, Λ) is the renormalization factor affecting the fermionic field and δm = m(ζ) − m 0 , with m 0 the bare mass. The analogous form of Eq. (2.2), for the regularized self-energy reads (it is useful to include the renormalization constant Z 2 in the definition) with A Z (ζ, Λ; p) and B Z (ζ, Λ; p) suitable scalar functions. In particular, comparing Eq. (2.1) and Eq. (2.6), one obtains Indeed, those relations amount to the outcomes of the subtraction scheme for the renormalization of each scalar function. Moreover, by taking into account Eq. (2.5), one has 9) and therefore in the limit Λ → ∞: Pursuing our goal of establishing a formal framework where one can get actual solutions of the gap equation, and eventually describe the renormalized propagator, we usefully introduce the NIR for the fermionic self-energy. This can be achieved by starting from the approach proposed for a scalar case by Nakanishi (see Ref. [33]), for summing up the infinite contributions to a given n-leg amplitude, and generalizing in two respects. One is the transition from scalars to fermions, and the second one, more important, from a perturbative to a non perturbative regime. Those steps have been explored for the BSEs in Refs. [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. For the fermion self-energy, it is necessary to introduce two NWFs, since one has to deal with two scalar functions. Hence, the regularized self-energy can be written in terms of the following scalar functions with s th the multiparticle threshold and ρ A(B) the NWFs. It should be recalled that the NWFs are real functions, and do not depend upon the external momenta. This last remark will be useful for simplifying the formal elaboration aiming to get the suitable integral equations for ρ A(B) .
It is easily seen that NWFs with a constant behavior for s → ∞ generate a logarithmic divergence.
It should be pointed out that a possible constant behavior of the NWFs ρ A(B) for s → ∞ would be regularized by the quadratic dependence upon s in the denominator, allowing to safely take Λ → ∞.
Dealing with the gap-equations, it is fruitful to use the KL representation of the renormalized propagators, and therefore one has to establish the relation between KL weights and NWFs of the corresponding self-energy (see also Ref. [34] for the scalar case and Refs. [36,37] for QED 3+1 ). Recalling the following KL representation where R S is the fermion propagator residue, controlled by the choice of the renormalisation scheme (here RI'/MOM). Using Σ R (ζ, p) from Eq. (2.2), one gets By evaluating the needed traces, one can obtain the following relations If one assumes that both KL weights and NWFs match the hypotheses for applying the Sokhotski-Plemelj formula, that reads Let us recall that ρ A(B) (s = ζ 2 , ζ) = 0 and values ω ≥ s th are relevant in what follows. By using Eqs. (2.20), (2.13) and (2.14), the real and the imaginary parts of A R (ζ; ω) become with the notation ρ A indicating the principal value in Eq. (2.20). Analogous expressions hold for B R (ζ; ω). Hence, one can formally gets the following relations between KL weights and NWFs for ω > ω th = s th where It has to be pointed out that the knowledge of the KL weights σ S(V ) (ω, ζ) for ω > ω th is enough for determining the fermion propagator for all the possible values of p 2 .

The renormalized propagator of a photon
In the Landau gauge, the free propagator of the photon reads where T µν (q) is the standard transverse projector with its useful properties, and ζ p is a IR-regulator, (see, e.g., Ref. [66]). For the sake of light notation, the dependence upon ζ p will be understood in the renormalized quantities. Hence, the renormalized photon propagator reads , (2.27) where Π R (ζ; q) can be called the photon self-energy, fulfilling the following condition, able to lead to the correct residue at photon pole Notice that the photon propagator would present a problematic pole if there exists a critcal value, q sing , such that 1 + Π R (ζ; q sing ) = 0. The relation between D µν R (ζ; q) and both the regularized self-energy and the renormal- . (2.29) Comparing the denominators in Eqs. (2.27) and (2.29) one has for Λ → ∞ By imposing the condition in Eq. (2.28), one gets the following normalization and writes for Λ → ∞ It is also useful to recall that the renormalized propagator fulfills the well-known integral equation, given by where Π µν R (ζ, q 2 ) is the renormalized vacuum polarization tensor, defined by This quantity is involved in the gap-equation for the photon (cf Sect. 4).
The KL representation of D R µν (ζ, q) reads and has to be compared with the following expression obtained from Eq. (2.27) .

The renormalized vertex function
The amputated three-leg transition amplitude, or vertex function, is the basic ingredient for any dynamical approach that aims at determining the self-energies of particle and quanta, involved in a given theory. Unfortunately, the fully dressed vertex function can be formally obtained only through the proper DSE where, in turn, the four-leg transition amplitude (i.e. the fully off-shell fermion-antifermion scattering kernel in the case of QED) is present. This fact makes clear the structure of the infinite tower of DSEs, where each n-leg transition amplitude fulfills an integral equation containing the (n+1)-leg transition amplitude. In spite of this, by using general principles, one can devise an overall form of the vertex, in terms of the Dirac structures allowed by both the Lorentz covariance, the parity conservation and time reversal (see, e.g., Refs [26,73]), when QED is investigated. Following well-known steps, one decomposes the vertex into two parts: i) the standard component introduced in the early eighties by Ball and Chiu [26], in order to fulfill WTIs and to avoid any kinematical singularity, and ii) a contribution purely transverse, i.e. containing the possible Dirac structures orthogonal to the momentum transfer q = p f − p i (see Fig. 1. for the pictorial representation and the kinematics). As a matter of fact, one writes the renormalized vertex (or the regularized one, with the proper modification in the notations) as follows The actual expression of Γ µ R;BC [26], is given by where While Γ µ R;BC is elaborated starting from WTIs and the crucial request of avoiding kinematical singularities, the transverse part Γ µ R;T has to fulfill the constraint imposed by the curl of the current, q µ Γ ν R − q ν Γ µ R [27] (see also the analysis in Ref. [14]), and it can be expressed in terms of eight Dirac structures, T µ i , such that q · T i = 0 (see Ref. [26] for the complete list) and eight scalar functions, In general the functions F i (p f , p i , ζ) cannot be written only in terms of A R and B R [14], but the whole set of functions has to cooperate for assuring another fundamental property: the multiplicative renormalizability of both self-energies (see Eqs. (2.6) and (2.29)) and vertex, viz with the constraint Z 1 (ζ, Λ) = Z 2 (ζ, Λ). It is fundamental to notice that, given the DSEs, the multiplicative renormalizability of both two-leg and three-leg functions are intimately related. This has been elucidated by a vast literature, in different frameworks. In particular, in Refs [20, 22-24, 28, 30] (and references quoted therein) a close analysis, ranging from a first perturbative study to non perturbative ones, was carried out, pointing to the role played by leading logarithms in determining the aforementioned property, through an unavoidable cooperation between the scalar functions present in Γ µ R;BC and Γ µ R;T . Differently, in Refs. [20,29,74], within a quenched approximation, the requirement of multiplicative renormalization is implemented by looking for solutions of the fermion gap-equation with a power-law behavior (it easily turned out that such a feature lead to multiplicative renormalization of the fermion self-energy).
In our unquenched approach, we take into account the transverse vertex, retaining only some contributions, as it will be explained in what follows. Indeed, this is a distinctive feature of our work, in comparison with approaches sharing the same spirit, i.e. exploiting spectral representations of both propagators and self-energies (see Refs. [24,34,[36][37][38][39]). In particular we consider the following two Dirac structures, of the eight identified in Ref. [73], with σ νρ = i[γ ν , γ ρ ]/2 and 0123 = +1. Notice that there is an overall different sign with respect to Ref. [26].
It is worth mentioning that in the fermion massless case (relevant for studying the dynamical generation of the mass) only the Dirac structures with i = 2, 3, 5, 8 contribute [22], and moreover, in the same limiting case, T µ 3 and T µ 8 allow one to implement the gauge covariance of the fermion propagator [75]. Finally, as pointed out in Ref. [14] the contribution T µ 8 is able to generate an anomalous magnetic moment term, within a perturbative framework [13]. The adopted expressions of F 3 and F 8 are the ones given in [14] (see also [29]), where a very detailed formal analysis of Γ µ T was carried out (adopting a Euclidean metric) and general expression were found. In particular, due to the curl of the current, it turns out that F 3 and F 8 can be minimally chosen as linear combinations of A and B (see subsect. 2.1). In this way, one has a workable Ansatz for Γ µ R;T , that has the virtue of closing the equations involving the fermion and photon self-energies. The actual F 3 and F 8 are given by (see also Ref. [29]) that match the expected perturbative behavior for p 2 f >> p 2 i (see the discussion in Ref. [29]). Hence, one can write In conclusion, we use Γ µ R given by the sum of the Ball-Chiu vertex [26] and one of the minimal Ansatzes for Γ µ R;T , proposed in Ref. [14]. In particular, by using Eq. (3.9) one can i) fulfill the multiplicative renormalizability, ii) establish a non-perturbative framework, where a closed coupled system of integral equations allows one to investigate the self-energies of both fermion and photon.

Coupled gap equations
This Section, in particular subsections 4.1 and 4.2, contains the main outcomes of our formal elaboration that aims to get a mathematical tool for determining the fermion and photon NWFs and eventually yield the fermion and photon self-energies. In order to accomplish such a task, it is necessary to proceed by writing down the DSEs for the self-energies (see, e.g., Ref. [6] for a general introduction), and insert the results obtained in Sect. 2 (see details in Appendices A and B).
The DSE for the regularized fermion self-energy, defined in Eq. (2.7), is given by where it is important to emphasize that the dependence upon Λ means that the rhs can have singular contributions (indeed this is the case). But such terms become finite after introducing a suitable regularization procedure, that in our case it turns out to be the dimensional one with d = 4 − and Λ = 1/ (see the details in Appendix A). Then, the renormalized self-energy fulfills The two scalar functions describing Σ R (ζ; p) (cf Eq. (2.2)) can be obtained by evaluating the suitable traces, i.e.
In the next subsect. 4.1 the results of the traces will be presented and the relation with the NWFs established.
In the Landau gauge we are adopting (recall that the polarization tensor is transverse in this gauge), one can start from the following expression of the regularized polarization tensor (cf Appendix B) in terms of the renormalized quantities Notice that T µν is a symmetric tensor, and therefore also the rightmost term it has to be. One can convince himself by recalling that one has at disposal only one four-vector, q µ for constructing antisymmetric contributions. It is understood that Π µν , microscopically described by the second line in Eq. (4.4), must satisfy the transversity property, i.e q µ Π µν = 0, Π µν q ν = 0. Hence, one has to verify that Since, we are adopting a vertex that automatically fulfills the WTI, the second line in Eq. (4.4) can be easily demonstrated by using the WTI itself and the dimensional regularization, in order to make formally allowed a shift in the integrand. As a matter of fact, one gets.
The equality in the first line of Eq. (4.5) is more involved, but for assuring that the microscopic calculation of Π µν be proportional to T µν one should recover the structure with the needed relations A = −B. This is guaranteed by Eq. (4.6), that follow from the fulfillment of WTI.
In order to single out the photon self-energy, Π(ζ, Λ; q), one can proceed by saturating the polarization tensor with any combination of g µν and q µ q ν /q 2 , but it is extremely useful to take full advantage and guidance from the analyses carried out in perturbative regime, (see, e.g. in Ref. [6], Ref. [22] and cf sect. 3.6). Hence, one can saturate both sides in Eq. (4.4) with the tensor P µν given by This tensor has been introduced in previous works (see Refs. [22,76,77]) in order to project T µν on its q µ q ν part, avoiding to deal with quadratic singularities proportional to g µν present in Π µν . We emphasize that such a projector is adopted for convenience reasons. As a matter of fact, apparent quadratic singularities are met in the following elaboration, but the choice of the vertex presented in Sect. 3 (cf also Appendix B) ensures their cancellations. These apparent singularities are easily bypassed by exploiting P µν , without carrying out a lengthy algebra (see also Refs. [76,77]). As a final remark, it should be pointed out that the formal manipulation shown in Appendix B needs a dimensional regularization of some terms and therefore one should substitute 4 with a generic dimension d in the expression of P µν .

The fermion gap equation and the NWFs
As it is shown in details in Appendix A, one can exploit the NIR of A R (ζ; p) and B R (ζ; p), Eq. (4.3), and the KL representations of both fermion and photon propagators, Eqs. (2.15) and (2.38) respectively, for obtaining the following relations and and In Eqs. (4.12) and (4.13), we havē is present. The one in F A + generates a severe divergent behavior in k (see Eqs. (4.12) and (4.13)) that cannot be regularized by subtraction, since the corresponding term in T A(B) , being evaluated at p 2 = ζ 2 , yields A R (ζ; ζ) = 0, by definition. A simple power counting in k 2 reveals that in T A and T B , only the combination proportional to F A + − (k 2 − p 2 )F A − allows one to mitigate the divergent behavior due to A R (ζ; p) in F A + , leading to a logarithmic divergence that can be regularized by the subtraction in Eqs.(4.10) and (4.11) (see details in Appendix A). In fact, one has This cancellation highlights the intrinsic limitation of the BC vertex, since it is necessary to go beyond such a contribution for restoring the multiplicative renormalizability. This has been known from a long time (see, e.g. Ref. [28]), but it is relatively more recent the suggestion that the constraints coming from the curl of the vertex allow one to elaborate transverse contributions suitable for assuring the multiplicative renormalizability (see, e.g., Ref. [14]). In Appendix A it is explicitly shown how non-multiplicatively renormalizable contributions, from the BC term, Eq. (3.3), and the transverse ones, Eq. (3.9), cancel each other.
Once the explicit expressions of the relations in Eqs (4.10) and (4.11), are obtained as in Eq (A.61) and (A.49), respectively, by using a spacelike external momentum p in order to avoid unnecessary formal complexities (recall that the NWFs are real functions that do not depend upon the external momenta as one can also assess a posteriori), one can extract the integral equations fulfilled by the corresponding NWFs ρ A and ρ B , after assuming that the uniqueness theorem by Nakanishi [33] can be applicable to the non perturbative regime.
In particular comparing Eq (A.61) and the lhs of Eq. (4.10), one gets the desired .
As to ρ B , after comparing Eq (A.49) and the lhs of Eq. (4.11), one extracts

The photon gap equation and the NWF
In Appendix B, the details are given for obtaining the integral equation fulfilled by the NWF ρ γ (cf Eq. (2.37)), exploiting both the integral equation that determines the renormalized photon self-energy, Eq. (4.9) and the uniqueness theorem [33]. One can write where (4.21) Then, following the formal steps in Appendix B, where a spacelike q 2 has been adopted for a straightforward elaboration without loss of generality (as in the fermionic case), one gets and . (4.24)

Some remarks
Concluding this Section, that presents our formal results, some remarks are in order. The coupled systems for determining ρ A , ρ B and ρ γ is composed by the set of Eqs. ( and (4.22) are derived for spacelike external momenta to avoid complications coming from singularities. Beside the above feature, the uniqueness theorem allows one to finalize the formal steps. Interestingly, derivatives of Dirac delta distribution naturally appear in our derivations. In summary, i) the NWFs are real functions that do not depend upon the external momentum, as a posteriori can be checked by a direct inspection of the coupled system; ii) once the NWFs are numerically evaluated, the scalar functions A R (ζ; p), B R (ζ; p) and Π R (ζ; q) are known for any value of any momenta; iii) the presence of the derivative of the delta-function is not an issue from the numerical point of view, as already observed when the NIR approach has been applied to the numerical solution of BSE (see, e.g., Refs. [46,48]).

A first application
After establishing the formal results, i.e. the system of integral equations that the NWFs ρ A , ρ B and ρ γ have to fulfill, it is important to test the consistency. Following the same spirit of the first applications of the NIR approach to the two-scalar system, where the honorable Wick-Cutokski model had to stem from the formal elaboration (see, e.g., Refs. [78,79]), we have performed the first iteration of the coupled system, as an initial step. Once the analytical expressions of ρ B and ρ (1) γ have been obtained, we have carried out the evaluations of i) the KL weights for both fermion and photon propagators, ii) the fermionic running mass and iii) the charge renormalization function, and eventually compared with one-loop results (see, e.g., Ref. [66], but noting the different renormalization scheme).
The numerical investigation, aimed at establishing the validity of our approach by assessing the convergence of the iterative method, will be presented elsewhere. We stress that, generally speaking, the result of the iterative procedure may differ significantly from the first iteration one (see for instance this study of the QCD ghost propagator [80]), and that we perform here a basic test of consistency. In Appendix C all the details for obtaining the aforementioned first iteration, Eqs. (C.13), (C.17) and (C.19) respectively, are illustrated with also some of their features, while in Appendix D, the full expressions of the coupled system is summarized for the convenience of the interested reader.
In Fig. 2, the first-order Källén-Lehman weights for the fermionic propagator, Eq.  with and Qualitatively σ V and σ S are quite similar, though width and tail of σ S are slightly larger than the σ V ones. The common features are i) the negative values (in Ref. [24] the scalar weight is also negative), that is a consequence of our choice of the renormalisation scheme, and ii) the sharp peaks for ζ p → 0. The latter are very close to the threshold and clearly depend upon the IR-regulator ζ p , while the tails are unaffected, as expected. To complete the discussion it is useful to remind that the residue at the pole of the renormalized fermion propagator, R S , is not equal to 1, as expected in the RI'/MOM scheme [70,71]. Table 1 summarises the different values of the residues associated with our curves in Fig. 2. Another simple check is the formal comparison between the expression of ρ with α em = e 2 R /4π. Moreover from the definition in Eq. (2.14) one writes After imposing B R (ζ, ζ) = 0, perturbative computations yield (see e.g. [66]) Hence, for p 2 > m 2 (ζ) the logarithm becomes complex, and adopting the same analytic continuation as in Ref. [66] (i.e. ln(−ρ) = ln ρ − iπ), eventually one has -21 - that coincides with the result one gets from Eqs. (5.5) and (5.3). As a final remark, one should point out that in the same limit A R (ζ, p) vanishes both in our case (see Eq. (C.16)) as well as in Ref. [66]. Figure 3 shows the Källén-Lehman weight for the photon propagator, Eq. (2.42), obtained from ρ The independence from the IR-regulator ζ p , as shown in the expression of ρ (1) γ , is the standard feature of the one-loop calculation, and only the higher-order contributions will make apparent such a dependence. Differently from the fermion case, the KL weight of the photon is positive, as expected. This bosonic result points to the highly non trivial interplay of the two scalar functions A R and B R , in order to obtain positive KL weights for the fermionic source. We have also calculated the running mass (cf Eq. (2.4) for the value at the renormalization point)   In Fig. 4, the running mass is shown for values of the four-momentum below the threshold, s th = m 2 (ζ), adopting a tiny ζ p up to ζ p /m(ζ) = 10 −4 , while in Fig. 5 both real and imaginary terms, generated for p 2 ≥ s th , are presented. Notice that the positive sign of the imaginary part is a consequence of the first-order calculation (cf Eq. (5.7) and the vanishining of A R for ζ p → 0).
In Figs. 6 and 7, the running charge defined in Eq. (5.10) is shown for values below and above the threshold, that in this case holds s p th = 4m 2 (ζ). The comparison with the results of Ref. [37] (where m{M (ζ, p)} has a negative sign) can be be performed only at the qualitative level, since the quantitative one is too early for our numerical efforts. In any case, one can recognize quite similar pattern, in particular, for the case of small values of the coupling constant α em . It should be pointed out that the first-order self-energies (both   Figure 6. The running charge G(ζ; q 2 ), Eq. (5.10), below the threshold vs q 2 , with s p th = 4m 2 (ζ).
the fermion and photon ones) depend linearly upon the coupling constant and therefore, the values of the running mass and the relative running charge are not affected by such a dependence.

Conclusions and Perspectives
This work belongs to the set of the early attempts (not too much numerous) to explore the non perturbative regime of QED 3+1 directly in Minkowski space, by exploiting the framework based on the so-called Nakanishi integral representation for describing the selfenergies of both fermion and photon. The originality of this work, elaborated within the RI'/MOM scheme, lies in the choice of the fermion-photon vertex, able to fulfill constraints coming from both the Ward-Takashi identity and the multiplicative renormazibility, that calls for purely transverse contributions. We have shown that despite the apparent complexity, it is possible to derive a well-defined system of equations for the Nakanishi weight functions, that we recall are real functions.
In addition, we have presented an intial check based on the evaluation of the first iteration of the coupled system. In particular, we have initiated the comparison with known results of i) the Källén-Lehmann weights for both fermion and photon, ii) the running mass and iii) the charge renormalization function, and put in evidence features and limitations of the simple first iteration. Beyond this, we have also verified that numerical stability remains under control, encouraging toward a more vast numerical investigation.
It has to be pointed out that the present results readily calls for three natural extensions on a short-time scale. First, complete numerical studies should be performed, allowing one to assess the convergence of the whole approach and to move the comparison to a quantitative level, e.g. with the results in Refs. [24,37]. Second, the expected residue equal to one at the mass pole should be implemented at the level of the NWFs, i.e. going from the RI'/MOM scheme to the standard on-shell renormalisation scheme. Eventually, one could include the Bethe-Salpeter equation within the present system, since all its ingredients are already present.
On a longer time-scale, the fourth desirable extension would be to move from QED to QCD. An educated reader might object that many ingredients we used are not available or available in a much more complicated way for QCD. For instance, there is no formal proof that the propagators of confined particles should have Källén-Lehmann-like representation (positive defined). Nonetheless, lattice-QCD computations seem to be consistent with a spectral representation (although not a positive one) [81]. Furthermore, the Ward-Takahashi identities must be replaced by the Slavnov-Taylor ones, forcing deep modifications of the quark-gluon vertex function, playing an important role in realizing a dynamical breakdown of chiral symmetry. Also for this issue, progresses have been recently done in that direction, with the definition of the non-abelian generalisation of the Ball-Chiu vertex [19]. Therefore, despite the technical difficulties to jump from QED to QCD, we believe that such a possibility should deserve a careful investivation.

Acknowledgments
We gratefully thank Massimo Testa for very stimulating discussions, and Si-xue Qin for illustrating in detail the issues related to the transverse vertex. C.M. acknowledges discussions with Jose Rodriguez-Quintero ands the warm hospitality of INFN Sezione di Roma, where his INFN PostDoc fellowship for the NINPHA project has been granted.

A DSE for the fermion self-energy
In this Appendix, the formal elaboration for obtaining the equation that determines the renormalized self-energy Σ R (ζ; p) is given in details.
The starting point is the integral equation fulfilled by the regularized self-energy Σ(ζ, Λ; p) (see Itzykson and Zuber [66], p. 275, for the adopted notations, and also Fig. 8), that reads By introducing in Eq. (A.1) the following relation between regularized and renormalized quantities with Σ Z (ζ, Λ; p) = Z 2 Σ(ζ, Λ; p). From Eq.(2.10), one gets the following integral equation for the renormalized self-energy The scalar functions A R (ζ; p) and B R (ζ; p) can be obtained by evaluating the following traces where the subscript T on a four-vector means with q = p − k (notice that the photon is outcoming), so that k β T = p β T . Moreover, from Eq. (3.4) one has For the purely transverse component Γ α R;T (ζ; k, p), Eq. (3.9), one has where (p − k) β T = 0 has been used.Summing the two contributions to Γ β R;T , one gets:  where and (A. 17) In Eq. (A. 16), the underbraces emphasize the contributions generated by each term present in the vertex (cf Eqs. (3.3) and (3.9)). This is motivated by the needed cooperation for eliminating the contribution produced by A R (ζ; p) present in F A+ . Such a contribution generates a singular integral in T A (ζ, Λ; p 2 ) that cannot be canceled by an analogous term in T A (ζ, Λ; p 2 = ζ 2 ), since A R (ζ; p = ζ) = 0. Also in F A− there is A R (ζ; p), but in a combination with A R (ζ; k), such that it does not plague the further calculation (see below). Notice that also in T B (ζ, Λ; p 2 ) the same issue will be met. In conclusion, all the terms in the vertex function play an essential role for properly restoring the multiplicative renormalizability of the self-energy, as expressed in Eq. (A.4). This result is expected from the perturbative analysis (see, e.g., [22]), but it is gratifying to be achieved within a non perturbative approach. The aforementioned cancellation of A R (ζ; p) in T A (ζ, Λ; p 2 ) can be attained by usefully recasting Eq. (A.16) as follows Then, the problematic A R (ζ; p) is canceled in the combination obtained from the contributions produced by T 3 and T 8 . It must be noticed in Eq. (A.18) that for getting the result one produces the term that in principle can generate a singular integral. Indeed, an other fortunate cancellation takes place by exploiting the 4D angular integration and the difference between T A (ζ, Λ; p 2 ) and T A (ζ, Λ; ζ 2 ) (cf in subsec. (A.4)). Remarkably the factor of 4 is essential for obtaining the finite result. For evaluating the trace in T B (see Eq. (A.12)), one gets and Finally, collecting all the results, one has the following expressions for T A (ζ, Λ; p) and T B (ζ, Λ; p) and By using Eq. (2.13) for A R (ζ; k) and (3.4) for F A − (k, p, ζ) and F B (k, p, ζ), one can write T A and T B , as follows ds σ S (s , ζ, s th ) p 2 I 2 (p, ω, s , s) − I 4 (p, ω, s , s) with ω, s , s ≥ 0.
It has to point out that I 0 , I 1 , and I 5 are divergent integrals for d = 4, and only after applying i) the dimensional regularization and ii) the subtraction of the corresponding integrals evaluated at p 2 = ζ 2 , one gets finite results for A R and B R , as it will be shown in the subsections A.3 and A.4, respectively.

A.2 Analytic Integrals
The evaluation of the analytic integrals in T A (ζ, Λ; p) and T B (ζ, Λ; p) represents the most lengthy part of the formal elaboration. It is helpful to recall that our goal is to achieve a form of both A R and B R suitable for exploiting the uniqueness of the NWFs ρ A and ρ B , as suggested by a theorem demonstrated by Nakanishi for a generic n-leg transition amplitude [33].
To proceed in a simple way, it is very useful to consider spacelike values for the external momentum p. This choice, as it becomes immediately clear, simplifies a lot the formal elaboration, and it is not restrictive, since the NWFs do not depend upon the values of the external momentum, but noteworthy they are used for obtaining the scalar functions A R and B R at any value of p 2 .
For the finite integral I 2 one gets The last step can be easily carried out without any concern, given the aforementioned choice of p 2 < 0.
The second finite integral, I 3 , becomes Finally, I 4 can be evaluated by using the result in Eq. (A.36), i.e.
In the above 4D integration on q the subscript Λ has been removed since the regularization is not necessary.

A.3 The B R contribution to the fermion self-energy
Let us start the evaluation of the contribution B R , since it contains only one divergent integral, i.e. I 0 , Eq.(A.28). To get a finite value for the contribution from I 0 , it is compulsory to exploit both the dimensional regularization, that legitimates the variable shift in the integrand, and the subtraction of the corresponding term in T B (ζ, Λ; p)| p 2 =ζ 2 , as shown in Eq. (A.5). With this in mind, we simplify the formal elaboration removing the dependence upon Λ in what follows, and directly using the 4D integration. B R (ζ; p) in Eq. (A.5) (cf also Eq. (A.27)) reads  The actual evaluation of the differences is briefly sketched. The first one, D 0 , can be obtained from Eq. (A.28) and recalling the need of the regularization, viz where χ 0 (p, q) = 1 44) and the formal manipulations are allowed by the dimensional regularization. Eventually, the last line has been introduced after applying an integration by parts for preparing the application of the uniqueness theorem [33] to the NWF ρ B . Notice that ξω+(1−ξ)s +η ≥ 0 and the exchange of the integration on η and on y has been assumed to be allowed. From Eq. (A.35), one gets (A.45) To evaluate D 3 , one can introduce the following difference with A > 0 and B > 0 where an integration by part of the cubic term has been applied for obtaining the last line and δ indicates the derivative with respect to y.

A.4 The A R contribution
In order to evaluate A R , (recall p 2 < 0, but without loss of generality on the final result for the NWFs) one has to face with the divergent behavior of I 1 , Eq. (A.29), and I 5 , Eq. (A.33). The strategy is exactly the same we have applied to I 0 in the subsec. A.3, combining the dimensional regularization for shifting the integration variable and then exploiting the subtraction.
Recalling that one has first to apply the dimensional regularization to I 5 , from Eq. (A.33) and using Eq. (A.46), and introducing one has  with

B DSE for the photon self-energy
This Appendix is devoted to obtain the integral equation determining the renormalized photon self-energy. Eq. (2.32). The initial step is given by the DSE for the regularized polarization tensor, Eq. (4.4), that we rewrite here for convenience, (the kinematical quantities are shown in Fig. 9) where Eq. (A.2) has been used for the renormalized quantities. From Eq. (B.1) and the properties (4.7), it follows that the renormalized photon self-energy, Eq. (2.32) reads where we have used: i) the KL representation of the fermion propagator, Eq. (2.15), ii) the definitions ofσ V (S) in Eq. (4.14). From the vertex contributions Eqs. (3.3), (3.9) and the relation γ 5 µανρ γ α (k − q) ν k ρ = γ 5 µανρ γ α k ν q ρ , one can define the relevant trace: Performing the traces (recall that 0123 = 1), one has for the first trace where k p = k −q has been used for getting a more compact expression and the contributions from the transverse vertex, i.e. T 3 and T 8 have been emphasized. After introducing one can obtain: The remaining traces are given by Saturating the tensor Sp µν with P µν one gets for T P (recall that g µν P µν = 0) and (cf Eq. (A.19), with p → k − q ) As in the case of the fermion self-energy (cf Appendix A), inserting the expressions of F A + , F A − and F B in terms of the NWFS (cf Eqs. (3.4)) one can write ds σ V (s , ζ, s th )σ V (s, ζ, s th ) 2 I 6 (q, s, s ) where the integrals I i , are defined as follows (B.17) with s, s , ω ≥ 0. Recall that the external momentum q 2 is chosen spacelike, for the sake of simplicity in the formal elaboration. Notice that I 6 presents an apparent quadratically divergence, as expected. Therefore, we exploit dimensional regularization, like the integral I 0 (cf Eq. (A.43)), and obtain that has a logarithmic divergence, harmless once we subtract the corresponding integral evaluated at ζ 2 p (see Eq. (B.2)).

C First iteration
This Appendix is devoted to present a first analytic result obtained by iterating one time the coupled system we have obtained.
The inputs are given by the zeroth-order NWFs ρ A , ρ B and ρ γ , i.e. with y > [m(ζ) + ζ p ] 2 . To complete our analysis, let us consider Θ y − s th ρ A (y, ζ) for ζ p → 0. In particular, one remains with the following limit  with ξ ± given in Eq. (C.9) after changing t → ξ. Differently from ρ is logarithmically divergent, (see Eq. (2.36)). The first-order photon self-energy, Eq. (2.37), is given by Notice that the imaginary part of Π R (ζ, q 2 ), when q 2 > 4m 2 (ζ) coincides with the result that can be found in Ref [66].

D Formulas Summary
For the sake of a quick focus on the main formal results have been obtained in the paper, in this Appendix we list the initial expressions useful for for a numerical calculations of KL weights in terms of NWFs.