Elucidating the effect of intermediate resonances in the quark interaction kernel on the time-like electromagnetic pion form factor
?ngel S. Miramontes López, Hèlios Sanchis-Alepuz, Reinhard Alkofer
EElucidating the effect of intermediate resonances in the quark interaction kernel onthe time-like electromagnetic pion form factor ´Angel S. Miramontes L´opez ∗ Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´as de Hidalgo, Morelia, Michoac´an 58040, Mexico
H`elios Sanchis Alepuz † Silicon Austria Labs GmbH, Inffeldgasse 25F, 8010 Graz, Austria
Reinhard Alkofer ‡ Institute of Physics, University of Graz, NAWI Graz, Universit¨atsplatz 5, 8010 Graz, Austria (Dated: February 26, 2021)An exploratory study of the time-like pion electromagnetic form factor in a Poincar´e-covariantbound state formalism in the isospin symmetric limit is presented. Starting from a quark interactionkernel representing gluon-intermediated interactions for valence-type quarks, non-valence effects areincluded by introducing pions as explicit degrees of freedom. The two most important qualitativeaspects are, in view of the presented study, the opening of the dominant ρ -meson decay channel andthe presence of a multi-particle branch cut setting in when the two-pion threshold is crossed. Basedon a recent respective computation of the quark-photon vertex, the pion electromagnetic form factorfor space-like and time-like kinematics is calculated. The obtained results for its absolute value andits phase compare favorably to the available experimental data, and they are analyzed in detailby confronting them to the expectations based on an isospin-symmetric version of a vector-mesondominance model. PACS numbers: 11.10.St, 13.30.Eg, 13.40.Gp, 14.40.n
I. INTRODUCTION
Hadronic time-like form factors will be measured inupcoming experiments to an unprecedented precision.An understanding of these quantities which are display-ing pronounced structures originating from hadron reso-nances will contribute significantly to our knowledge onthe relation between the hadrons’ substructure and thehadron spectrum. On the one hand, based on respectiveexperimental and theoretical progress in the last decades,it is by now evident that hadron resonances can be, atleast in principle, described in terms of quarks and glu-ons. The latter, being the QCD degrees of freedom, areconsidered to be complete in the sense that they allow fora computation of every hadronic observable. Changingthe perspective in an attempt to understand the StrongInteraction starting from the low-energy regime, a pos-sible way of phrasing the phenomenon of confinementin QCD is the statement that all possible hadronic de-grees of freedom will form also a complete set of physicalstates. Therefore, the equivalence of descriptions of ob-servables in either quark and glue or hadronic degrees offreedom is a direct consequence of confinement and uni-tarity. This is the gist of a chain of arguments whichcan be sophisticated and applied to many different phe-nomena involving hadrons. The related picture is known ∗ [email protected] † [email protected]@gmail ‡ [email protected] under the name “quark-hadron duality”, and its conse-quences have been verified on the qualitative as well asthe semi-quantitative level, for a review see, e.g., Ref. [1].A verification of this duality is, beyond the trivial fact ofthe absence of coloured states, the clearest experimen-tal signature for confinement. To appreciate the scope ofsuch a scenario it is important to note that a perfect or-thogonality of the quark-glue degrees of freedom on theone hand and hadronic states on the other hand, andthus the perfect absence of “double-counting” in any ofthe two “languages”, is nothing else but another way toexpress confinement.Gaining insight into the interplay between formationof hadronic bound states, consisting of quarks and glu-ons, and the open decay channels of the respective reso-nance is an essential element of every study of time-likeform factors in kinematic regions close to a resonance.Here, attention should be paid to the fact that the hadronwhose form factor is investigated and the hadronic res-onance which is apparent in the form factor are bothto be described as composite objects of quarks and glu-ons. The same is true for the hadronic decay productsin a hadronic or semi-leptonic decay of the resonance.This makes evident that an approach to calculate a time-like form factor from QCD, or from a microscopic modelbased on QCD degrees of freedom, faces the challengingtask to treat all elements appearing in the calculation onthe same footing and to a sufficient degree of sophistica-tion if the result is intended to allow for conclusions onthe dynamics underlying such a form factor.Herein, we will report on an exploratory study of thetime-like pion electromagnetic form factor using func- a r X i v : . [ h e p - ph ] F e b tional methods. More precisely, we will employ a com-bination of Bethe-Salpeter and Dyson-Schwinger equa-tions (for recent reviews on this and related approachessee, e.g. , [2–5]). Although such an approach is capable ofallowing a first-principle calculation (see, e.g. , the com-putation of the glueball spectrum reported in ref. [6]),for the task at hand this is yet out of reach. To graspall essential features of the pions’ time-like form factor one needs to describe at least (i) the pion as bound stateof quark and antiquark thereby at the same time takinginto account its special role as would-be Goldstone bo-son of the dynamically broken chiral symmetry of QCD,(ii) the mixing, respectively, the interference of the ρ -meson, being described also as a quark-antiquark boundstate, with a virtual photon when this photon is in turncoupled to a quark-antiquark pair via the fully renormal-ized quark-photon vertex, and (iii) the dominant decaychannel of the ρ -meson, namely, ρ → ππ . The studypresented here is now in two aspects exploratory. Firstof all, the interaction between quarks and antiquarks ismodelled in such a way that the essential features, asimplied by QCD and phenomenology, are taken into ac-count but that it is on the other hand still manageable insuch an involved calculation. Second, in several places wewill make technical simplifications, especially when thesuch introduced error can for good reasons assumed tobe small and the reduction in the computing time neededis substantial. We thus aim here more for an understand-ing of how the different features of the form factor arisefrom the QCD degrees of freedom than for a quantitativeagreement with the experimental data.In the chosen model for the quark-antiquark inter-action, besides a gluon-mediated interaction also pi-ons will be included explicitly. The reason for this isas follows: if one were able to take into account thefully renormalized quark-gluon vertex exactly within thisapproach, hadronic degrees of freedom will effectivelyemerge and thus be included in the interaction betweenquarks and antiquarks, respectively, they will back-feedon the quarks’ dynamics. Due to the pions’ Goldstoneboson nature, and especially due to the implied smallpion mass, the pions are the most important low-energydegrees of freedom within the Strong Interaction, as, e.g. ,also elucidated by chiral perturbation theory. And as inorder to describe the physics of decays non-valence ef-fects need to be taken into account, it is some minimalrequirement for the investigation reported here to includepions as the most important non-valence-type interactionmediator in the sub-GeV region.The interaction model herein is also chosen in view ofa possible generalisation to the study of baryon form fac-tors, and hereby especially the nucleons’ time-like form There are several investigations of the space-like pion electro-magnetic form factor in the Dyson-Schwinger–Bethe-Salpeter ap-proach, early examples include [7–9]. A calculation of this formfactor spanning the entire domain of space-like momentum trans-fers is described in ref. [10]. factor. In this respect one can build on existing calcula-tions of space-like form factors from bound state ampli-tudes, see, e.g. , refs. [3, 11–15] for some recent respectivework. A thorough understanding of the proton time-likeform factor at very low Q is a very timely subject asthe upcoming PANDA experiment possesses the uniquepossibility to measure the proton’s electromagnetic formfactors in the so-called unphysical region through the pro-cess ¯ pp → l + l − π , l = e, µ [16]. At large Q the questionof the onset of the convergence scale between the space-like and the time-like form factors arises.However, also the pions’ time-like electromagnetic formfactor will be studied further by upcoming experiments,among other reasons because recently the consistency ofthe available data sets has been questioned [17]. Basedon the long-known fact that the τ radiative decay al-lows to extract the pion form factor [18] and that a verylarge number of τ -leptons are produced at B-meson fac-tories further high-precision data in sub-GeV region willbecome available. Besides earlier lattice QCD calculations of the pions’space-like electromagnetic form factor [20–26] recentlycalculations of the time-like pion form factor have be-come available [27–30]. These results typically show agood agreement with the experimental data. In thosecalculations, the extraction of the time-like form factoremploys a parameterisation based on the vector mesondominance (VMD) picture to extract from the latticedata at discrete energies the time-like pion form factor.The exploratory calculation presented herein is done inthe isospin symmetric limit. One of the effects of isospinbreaking clearly visible in the time-like pion form factoris ρ - ω mixing, see, e.g. , the review [31]. To this end wewill employ the VMD-based fit given in [31] and modifyit such that an expected form of the pion form factorwithout this mixing effect is extracted. Furthermore,we will present a simplified but numerically quite accu-rate VMD parameterisation of the time-like form factorwhich then serves as a basis for a detailed analysis of ourresults. Here, the focus is more on a comparison of ourresults with the form expected on the basis of the VMDparameterisation than on numerical agreement. This paper is organized as follows: In Sect. II we reviewsome facts about the electromagnetic pion form factor,employ the VMD-based fit given in [31] to remove the ρ - ω mixing effects, and provide an expected form of the pionform factor. In Sect. III we present our approach basedon Bethe-Salpeter and Dyson-Schwinger equations. Our In ref. [32] another method has been used to remove the ρ - ω mixing effects from the data. A precise representation of the time-like form factor requires pa-rameterisations including excited ρ -mesons, respective examplescan be found in refs. [33, 34]. Two-photon effects, on the otherhand, can very likely be safely neglected, for a correspondingstudy of the form factor at large momentum transfer see [35]. results are presented and analyzed in Sect. IV. In Sect. Vwe present conclusions and an outlook. Some technicaldetails are deferred to two appendices. II. THE TIME-LIKE ELECTROMAGNETICPION FORM FACTOR (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) γ (cid:63) (cid:64)(cid:64)(cid:82)(cid:64)(cid:64) e − (cid:0)(cid:0)(cid:9)(cid:0)(cid:0) e + (cid:26)(cid:25)(cid:27)(cid:24) F π π + π − (cid:0)(cid:0) (cid:0)(cid:0)(cid:64)(cid:64) (cid:64)(cid:64) FIG. 1. Electron-positron pair annihilating to a virtual pho-ton with time-like momentum which then decays to a pionpair.
The pion, being a composite object, does not have apoint-like interaction with the electromagnetic field, andthe related substructure, the pion being a pseudoscalar,is related to one form factor. Considering, for example,the scattering of an electron off a pion π + one can de-scribe the leptonic part of the interaction quite preciselyin lowest-order perturbation theory, i.e. , one considersthe process in which the electron emits a virtual photon,and the latter couples to the pion. Defining the formfactor F π ( s ) via the relation (cid:104) π + ( p ) | j µe.m. | π + ( p ) (cid:105) = e ( p + p ) µ F π ( q ) , (1)where q µ = p µ − p µ is the virtual photon momentum,and e is the elementary electric charge. The S -matrixelement for electron-pion scattering is then proportionalto the form factor, i M eπ → eπ = e ( p + p ) µ F π ( q ) D µν photon ( q )( ie ¯ u ( k , s ) γ ν u ( k , s )) , (2)where D µν photon ( q ) is the photon propagator, and u ( k i , s i )is the electron spinor, see, e.g. , Sect. 8.4 of [36] for moredetails. The kinematics of this scattering process is suchthat the photon momentum is space-like, and withoutloss of generality one can assume the form factor F π ( s )to be real. Charge conservation requires that for a realphoton one has F π (0) = 1.Turning to the process of electron-positron annihila-tion into a pion pair the corresponding S -matrix elementis again proportional to the form factor, see , e.g. , Sect.8.5 of [36], i M e + e − → π + π − = e ( p + p ) µ F π ( q ) D µν photon ( q )( − ie ¯ v ( k , s ) γ ν u ( k , s )) , (3)with appropriately redefined momenta. Especially, thevirtual photon momentum is now time-like, cf. Fig. 1, and one measures in such an annihilation process to apion pair the form factor for time-like momenta. Abovethe two-pion production threshold, i.e. , in the physicalregion, the corresponding cut in the amplitude (3) neces-sitates to treat the time-like pion form factor as a com-plex quantity, it fulfils the dispersion relation F π ( q ) = 1 + q π (cid:90) m π ds I m F π ( s ) s ( s − q − i(cid:15) ) . (4)Especially, it is expected that the phase of the pion formfactor varies strongly in a two-pion resonance region. Be-low the inelastic threshold, i.e. , for s < m π , the time-like pion form factor is, via Watson’s final state the-orem, related to the isovector P-wave scattering phaseshift δ , ( s ): I m F π ( q ) = 12 i (cid:16) − e − iδ , ( s ) (cid:17) F π ( s + i(cid:15) )= sin( δ , ( s )) e − iδ , ( s ) F π ( s + i(cid:15) ) . (5)As depicted in Fig. 1, the pion form factor contains,for time-like as well as for space-like photon virtualities,all kind of interaction processes turning a photon into apion pair. As strong-interaction processes dominate thecorresponding amplitude, and as gluons do not coupledirectly to photons, one decisive element of the pion formfactor is the amplitude describing how the photon couplesto a quark, taking hereby all possible contributing QCDprocesses into account. This amplitude is exactly the fullquark-photon vertex. As we will see in the following, thiscorrelation function carries information about the virtualphoton’s hadronic substructure. And as will be describedin detail in the next section, the other quantities neededthen for calculating the pion form factor are the fullyrenormalized quark propagator and the pion bound stateamplitude, the latter describing how an antiquark and aquark form a pionic bound state.While our calculation is based on QCD degrees of free-dom it is capable of providing an understanding why inthe resonance region a vector-meson dominance (VMD)picture provides very good results for the time-like pionform factor. In addition, it will elucidate to which extenta VMD picture might be applicable in other kinematicregions.In the VMD picture, the hadronic contribution to thephoton propagator is given by mixing with electricallyneutral vector mesons. Restricting to light-quark mesons,the corresponding vector meson is the ρ , i.e. , the un-charged member of the isotriplet of vector mesons. In awould-be isospin symmetric world this would be the onlyvector meson below one GeV with which a virtual pho-ton mixes because the isosinglet ω will not mix with thephoton due to G -parity.In the real world, isospin symmetry is broken, and oneof the many effects of isospin breaking is ρ - ω mixing, see, e.g. , the review [31]. This mixing and the resulting inter-ference of states lead to quite some pronounced structurein the time-like pion form factor around m ω . Indeed, itis by now well understood that the sharp dip in the ex-perimental data stems from a combination of ρ - ω mixingand interference effects between the decays ρ → ππ and ω → ππ , the latter being isospin breaking, see, e.g. , Ref.[31]. Moreover, it has been estimated that this combi-nation of mixing and interference effects decreases theheight of the bump of the order of up to 10% (see [32]and references therein).As the here presented exploratory calculation is per-formed in the isospin limit and thus ρ - ω mixing is ne-glected we estimate its effect by comparing the VMD-based fit to the pion form factor given in ref. [31] witha plot of the same expression but the mixing matrix ele-ment put to zero, Π ρω = 0, see Fig. 2. As expected theresulting curve is much smoother than the one includingthe ρ - ω interference effect. Around the ω mass the de-viation of the two curves can be as large as almost tenpercent, however, this effect is limited to a small inter-val, and the two curves are practically indistinguishableoutside this small interval. Therefore we expect our cal-culation to reproduce all qualitative features of the curverepresenting the case with ρ - ω mixing switched off, andto be in a reasonable quantitative agreement with it. − . − . − . − . − . − . − . − . FIG. 2. Absolute value of the pion form factor in the time-like( Q <
0) domain from the VMD-based fit given in ref. [31](full line) in comparison to the experimental data [17]. Thedashed line is based on the same expression but the mixingmatrix element put to zero, Π ρω = 0. It is instructive to analyze the momentum behaviourof a simplified version of the fit given in ref. [31]. Inorder to be in agreement with the notation in the follow-ing presentation we introduce the photon virtuality withthe convention that Q < ρ -meson is used. It displays Isospin breaking and thus the effect of ρ - ω mixing on the pionform factor is currently under investigation, the correspondingresults will be published elsewhere. the two-pion cut,Γ ρ ( Q ) ∝ (cid:0) − Q − m π (cid:1) / Θ( − Q − m π ) , (6)which is important for a qualitatively correct analyticstructure of the pion form factor. However, especiallyclose to the maximum of the pion form factor, the pionmass is quantitatively negligible, and for vanishing pionmass the momentum dependent width assumes for time-like Q < ρ ( Q ) = ¯Γ ρ | Q | /m ρ , (7)where ¯Γ ρ = Γ ρ ( − m ρ ). This then leads to the simplifiedbut still quite accurate form for the fit F π ( Q + i(cid:15) ) = 1 − g ρππ g ρ Q ( Q + m ρ )( Q + m ρ ) + Q ¯Γ /m ρ + i g ρππ g ρ Q ¯Γ /m ρ ( Q + m ρ ) + Q ¯Γ /m ρ , (8)where (cid:15) → + has been introduced to fix the sign of theimaginary part. Hereby, the coupling constant g ρππ isdetermined from Γ ρππ = ¯Γ = 149 MeV to be g ρππ ≈ g ρ is a parameter reflecting the strength of the γ - ρ -mixing, which is then described by the effective La-grangian L ργ = − e m ρ g ρ ρ µ A µ . From the partial width Γ ρe + e − = 7 keV one infers g ρ ≈ III. DYSON-SCHWINGER ANDBETHE-SALPETER FORMALISM
We determine the necessary input for the calculation ofthe pion electromagnetic form factor using a combinationof Bethe-Salpeter (BSE) and Dyson-Schwinger equations(DSE). To make this presentation self-contained we sum-marise in this section the most relevant aspects of the In contrast to a constant width approximation the pole is in thisparameterization and thus in the employed fit not located at Q = − m ρ + im ρ ¯Γ ρ but at Q = ( − m ρ + im ρ ¯Γ ρ ) / (1 + ¯Γ ρ /m ρ ), i.e. , real and imaginary part of the pole position are decreasedby 3.7% if the same values for the ρ ’s mass and width are used. approach, for more details see, e.g. , the recent reviews[2–5] as well as references therein. All expressions in thefollowing are understood to be formulated in Euclideanmomentum space, i.e. , after a Wick rotation.In the DSE/BSE formalism, the fully-dressed quark-photon vertex Γ µ , which describes the interaction bet-ween quarks and photons in a quantum field theory, canbe obtained as the solution of an inhomogeneous BSE(Γ µ ) aα,bβ ( p, Q ) = Z ( γ µ ) ab t αβ (9)+ (cid:90) q K rρ,sσaα,bβ ( Q, p, q ) S rρ,e(cid:15) ( k ) × (cid:0) Γ i,µ (cid:1) e(cid:15),nν ( Q, q ) S nν,sσ ( k ) . Here, Q is the photon momentum, p is the relative mo-mentum between quark and antiquark, q is an internalrelative momentum which is integrated over, the internalquark and antiquark momenta are defined as k = q + Q/ k = q − Q/
2, respectively, such that Q = k − k and q = ( k + k ) /
2. Latin letters represent Dirac indices,and Greek letters represent flavour indices. The isospinstructure of the vertex is given by t αβ = diag ( / , − / ).The Dirac structure of the vertex can be expanded in abasis consisting of twelve elements [37], and all of themare considered in our calculation.Similarly, mesons as bound states of two quarks are de-scribed in this framework by Bethe-Salpeter amplitudesΓ which are obtained as solutions of a homogeneous BSE,(Γ) aα,bβ ( p, P ) = (cid:90) q K rρ,sσaα,bβ ( P, p, q ) × S rρ,e(cid:15) ( k ) (Γ) e(cid:15),nν ( q, P ) S nν,sσ ( k ) , (10)where for clarity we have here used P for the total mesonmomentum (instead of Q as above). For pions, the Diracpart of the Bethe-Salpeter amplitude Γ can be expandedin a tensorial basis with four elements.In the equations above, the interaction kernel K de-scribes the interaction between quark and antiquark, and S is the fully-dressed quark propagator. We will discussin detail the interaction kernels below. The quark propa-gator S ( p ) is obtained as the solution of the quark DSE,(11) S − = S − − Z f (cid:90) q γ µ S ( q )Γ qglν ( q, k ) D µν ( k ) , with S − the renormalised bare propagator, S − ( p ) = Z ( i / p + Z m m ) , (12)and Z f , Z and Z m are renormalisation constants, m is the (renormalisation-point dependent) current quarkmass, Γ qgl is the full quark-gluon vertex and D µν is thefull gluon propagator which, in the Landau gauge, isparametrised as D µν ( k ) = (cid:18) δ µν − k µ k ν k (cid:19) Z ( k ) k , (13)with Z ( p ) being the gluon dressing function. For sim-plicity, we have suppressed the color indices. A. Interaction Kernels
The interaction kernel K in eqs. (9) and (10) encodesall possible interactions processes between a quark andan antiquark. In a diagrammatic representation, it con-tains a sum of infinitely many terms. In practical calcu-lations, the expansion of the interaction kernel must betruncated to a sum of a finite number of terms, chosensuch that the relevant dynamics and global symmetriesare correctly implemented. Chiral symmetry and its dy-namical breaking ensures that pions are massless boundstates in the chiral limit as a consequence of Goldstone’stheorem. On the other hand, U(1) vector symmetry en-sures charge conservation in electromagnetic processes.Chiral symmetry will be correctly implemented in theDSE/BSE formalism only if the kernel fulfills the axial-vector Ward-Takahashi identity (Ax-WTI) (14) i Σ ar ( p + ) γ rb t iαβ + iγ ar Σ rb ( p − ) t iαβ = (cid:90) q K rρ,sσaα,bβ ( Q, p, q ) (cid:2) i t iρν γ rn S nν,sσ ( q − )+ i S rρ,e(cid:15) ( q + ) γ es t i(cid:15)σ (cid:3) , with Σ the quark self-energy and v ± = v ± Q/
2. Simi-larly, vector symmetry will be correctly implemented ifthe kernel satisfies the vector Ward-Takahashi identity(V-WTI) (15) i Σ ab ( p + ) t iαβ − i Σ ab ( p − ) t iαβ = (cid:90) q K rρ,sσaα,bβ ( Q, p, q ) (cid:2) i t iρν S rν,sσ ( q − ) − i S rρ,s(cid:15) ( q + ) t i(cid:15)σ (cid:3) . In DSE/BSE studies the most widely used truncationis the so-called rainbow-ladder (RL) truncation, wherebythe BSE kernel consists of a vector-vector gluon ex-change, namely (omitting again color indices) (16) K rρ,sσaα,bβ ( Q, p, q ) = α (cid:0) k (cid:1) γ µar γ νsb D µν ( k ) δ αρ δ σβ , with k = p − q the gluon momentum. In order to pre-seve the Ax-WTI and V-WTI the kernel (16) is used incombination with a truncated quark DSE, defined by thereplacement Z f γ µ Z ( k )Γ qgl ν ( q, p ) → Z γ µ πα ( k ) γ ν (17)such that α ( k ) provides an effective coupling that de-scribes the strength of the quark-antiquark interaction.To parametrise this effective interaction we use theMaris-Tandy model [38, 39] (18) α ( q ) = πη (cid:18) q Λ (cid:19) e − η q + 2 πγ m (1 − e − q / Λ t )ln[ e − q / Λ QCD ) ] , FIG. 3. Diagrams relevant for the calculation of the pion form factor in the truncation employed herein, as determined by thegauging method. The impulse approximation implies considering the first diagram and the corresponding permutation only. where the second term on the right-hand side repro-duces the one-loop QCD behavior of the quark propa-gator in the ultraviolet, and the Gaussian term providesenough interaction strength for dynamical chiral symme-try breaking to take place. The model parameters Λ and η are determined as explained in the next section. Thescale Λ t = 1 GeV is introduced for technical reasons andhas no impact on the results. For the anomalous dimen-sion we use γ m = 12 / (11 N C − N f ) = 12 /
25 with N f = 4flavours and N c = 3 colours. For the QCD scale we useΛ QCD = 0 .
234 GeV.In the RL truncation, bound states cannot develop adecay width, correspondingly their masses are real num-bers. Note that bound states, determined as solutionsof BSEs, appear as poles in Green’s functions with thecorresponding quantum numbers. In the RL approxima-tion these poles occur for real (and in the conventionemployed here, negative) momentum-squared values incertain kinematic configurations. In particular, the pho-ton being described by a vector field, electrically neutralvector mesons appear as poles of the quark-photon ver-tex. In the RL approximation these poles are located atnegative and real values of Q (for which M = − Q with M being the mass of the vector meson). Such poles inthe quark-photon vertex also manifest as poles in the cal-culation of time-like form factors, in contradiction withphenomenology. Generally speaking, any physical phe-nomenon that is triggered by the presence of virtual in-termediate particles, as, e.g. , decays, will be absent fromany calculation using the RL truncation only.It is possible to improve the RL truncation in thisrespect by re-introducing the presence of intermediateparticles explicitly. The simplest implementation of suchan idea was introduced in [40, 41] , where, based on therole of the pion as lightest hadron, explicit pion-quark in-teractions were introduced in the truncated quark DSEand in the BSE kernel K , with the pion-quark interac-tion vertex given by the pion Bethe-Salpeter amplitudeΓ, calculated consistently via a truncated BSE. The cor- Note that effects of intermediate virtual states like the decay ofhadrons are in principle present in the full quark-gluon vertexwhich, however, is drastically simplified in the RL truncation. See, however, ref. [42] for considering pion loop contributions tothe electromagnetic pion radius in the DSE/BSE approach. responding additional BSE kernels are given in AppendixA and shown in Fig. 8, and the technical difficulties aris-ing for time-like momenta when those kernels are used inBSEs have been described in detail in [37]. This type ofkernels enable the possibility of intermediate virtual de-cays in the BSE interaction kernel to occur. As a conse-quence, certain BSE solutions signal a finite decay widthand thus represent (a) hadron resonance(s).
E.g. , thedescription of the ρ -meson as a finite-width resonance isthen mostly due to the intermediate process ρ → ππ (see[43] for a treatment in the here discussed approach as wellas [44, 45] and references therein for respective calcula-tions based on DSEs and BSEs), the partial decay widthto the latter process representing more than 99% of thetotal ρ decay width. Additionally, and thereby complet-ing the physical effects of intermediate virtual ππ -states,in this truncation the quark-photon vertex develops amulti-particle branch cut along the negative real Q axis,starting as expected at the two-pion production thresh-old [37]. Clearly, including these two effects of the inter-mediate virtual ππ -states is especially important whenit comes to the calculation of the pion form factor fortime-like momenta in the sub-GeV kinematic region.We wish to stress here that, for computational fea-sibility, for the pion vertices in (A1)–(A4) we used theleading γ component of the pion Bethe-Salpeter ampli-tude in the chiral limit, given by B/f π with B one of thequark’s dressing functions, see Eq. (A5). On the otherhand, the pion amplitudes used in the form factor calcu-lation of Eq. (19) are considered in full, including theirleading and sub-leading contributions. In that regard,our calculations contains two types of treatments of pi-ons .Even though the “pionic” kernels (A1)–(A4) are phe-nomenologically justified and, as we will see in the nextsection, constitute a first step in the correct direction, itmust be noted that they have not been (yet) rigorouslyderived from QCD. Lacking a solid quantum-field the-oretical basis the use of these kernels comes with someshortcomings, especially it implies that the Ax-WTI andV-WTI are not fulfilled simultaneously. Indeed, one canchoose to preserve either the Ax-WTI (and hence chiralsymmetry) or the V-WTI (and hence charge conserva-tion), but not both [37, 40]. It turns out, however, thatthe respective violation of either of them induce typi-cally only small errors in physical observables, as we willdemonstrate for some quantities in the next section. B. Form factor calculation
Meson form factors are extracted from a current J µ encoding the coupling of a meson to an external electro-magnetic current. In the BSE framework, the currentis calculated by means of the coupling of an externalphoton to each of the constituents of the bound state,as specified by a procedure known as gauging and devel-oped in [46–50]. The conserved current J µ that describesthe coupling of a single photon with a quark-antiquark,a three-quark or other multi-quark system is given by, J µ = ¯Ψ f G ( Γ µ − K µ ) G Ψ i , (19)with Ψ i,f the incoming and outgoing Bethe-Salpeter am-plitudes of the meson, the baryon or some other multi-quark state, and G represents the appropriate productof dressed quark propagators. This equation is shown di-agrammatically for the quark-antiquark / meson case inFig. 3. The term Γ µ represents the impulse approxima-tion diagrams where the photon couples to the valencequarks only Γ µ = (cid:0) S − ⊗ S − (cid:1) µ = Γ µ ⊗ S − + S − ⊗ Γ µ , (20)with Γ µ the quark-photon vertex. The term K µ describesthe interaction of the photon with the Bethe-Salpeter ker-nel, which in our truncation includes the coupling of thephoton to the quark-pion vertex and to the propagat-ing pions. Including both terms in (19) is necessary inorder to implement current conservation precisely. How-ever, the new vertices appearing in the term K µ repre-sent an enormous computational challenge. Given theexploratory purpose of the present calculation, we thusdecided to omit the coupling of the photon to the quark-pion vertex and to the propagating pions, and to considerthe impulse diagram only (first diagram in Fig. 3). Aswe will show in the next section, the thereby impliedviolation of charge conservation is at the level of approx-imately one percent. IV. RESULTS
Following the formalism sketched above, we have cal-culated the pion electromagnetic form factor in the time-like Q < Q > η and Λ of our inter-action model (18) as well as the value of the current quarkmass are fixed. It is customary in studies using the RLtruncation to adjust those parameters such that the pion decay constant agrees with the experimental value . Inref. [37] the quark-photon vertex has been calculated forthe first time with the above discussed interaction kernelstaken into account and also used herein. The parameters,including the isospin symmetric light quark current mass m q , were adjusted such that the pion mass and decayconstant as well as the ρ -meson mass have been correctlyreproduced in the employed approximation. Here, onlythe gluon- and pion-exchange kernels need to be usedfor fixing the parameters because the pion decay kernels(A3) and (A4) do not contribute to the pion BSE.For the present exploratory calculation, we choose toadjust the parameters in a slightly different and simplermanner, especially as we aim at a qualitative understand-ing of the physical mechanisms involved in determiningthe shape of the pion form factor, and not so much atachieving an accurate quantitative agreement with ex-periment. First, we set initially η = 1 .
5. Second, al-though we assume (confirmed by our calculation) thatthe pion-decay kernels will not only move the ρ -mesonpole into the complex plane but also shift down its realvalue, we nevertheless adjust the parameter Λ such thatwe obtain a ρ -meson mass close to the phenomenologicalvalue already in the calculations with gluon- and pion-exchange kernels only. Third, we require to reproducea quite accurate value for the pion decay constant. Fi-nally, we adjust m q to obtain the correct value for thepion mass as well. In this way we chose the model pa-rameters to be η = 1 .
5, Λ = 0 .
78 GeV and m q = 6 . µ = 19 GeV. As a rudimen-tary test of model dependence we additionally performthe calculations for η = 1 . m q unchanged). The results for the pion mass and decayconstant as well as for the ρ -meson and ω -meson masses,without the pion decay kernels being taken into account,are shown in Tab. I. TABLE I. The pion mass m π , the pion decay constant f π ,the ρ -meson and ω -meson masses m ρ and m ω for the twodifferent parameterizations of the model used herein and forthe case with rainbow-ladder and pion-exchange kernels butwithout decay kernels are shown. The light quark mass hasbeen set to m q = 0 . ρ -meson pole position defined as M pole = M ρ − iM ρ Γ ρ , as discussed in the text. All values for dimensionfulquantities are given in GeV.Λ = 0 . m π f π m ρ m ω M ρ Γ ρ η = 1 . η = 1 . As mentioned in the previous section, the full QCDquark-photon vertex possesses poles reflecting the masses Note that, for this observable, the result is quite independentof the value of η around η = 1 . and widths of the electrically neutral vector meson res-onances. Correspondingly, and as discussed in detail inrefs. [37, 43], a solution for the quark-photon vertex inthe DSE/BSE framework allows to extract the ρ -mesonmass and width via the position of the poles of the ver-tex dressing functions. For the RL truncation as well asfor the RL plus pion exchange approximation this poleis located on the real negative Q axis indicating thatthe ρ mesons were stable for those truncations. Includ-ing the s - and u -channel decay kernels (A3) and (A4),the pole of the dressing functions moves into the com-plex plane and can be extracted from the data on thereal axis via a Pad´e fit. Parametrising the pole positionas M pole = M ρ − iM ρ Γ ρ , we extract the correspond-ing results for the ρ -meson mass and width, M ρ andΓ ρ for this truncation (see Tab. I) in reasonable agree-ment with the experimental values. Hereby, it has tobe noted that underestimating the ρ -meson width doesnot come unexpected because taking into account onlythe leading γ component of the pion Bethe-Salpeter am-plitude in the kernels (A1)–(A4) misses some strengthstherein. Whether considering in addition also the sub-leading pion amplitudes will provide a much better resultfor the ρ -meson width can only be answered by perform-ing the corresponding calculation. This computation isthen, however, an order of magnitude more expensivethan the present exploratory calculation.It is interesting to note that the pion exchange kernelslift the degeneracy in between the isovector ρ - and theisosinglet ω -meson present at the level of RL calculationsthe interaction kernels of which are flavour blind and thusflavour U(2) (resp., flavour U( N f )) symmetric. The pionexchange kernels lead to a splitting such that m ω − m ρ =10 MeV which compares favorably with the experimentalsplitting of 7 - 8 MeV.We turn now to the calculation of the pion form factor.As already indicated in the previous section, thereare, besides restricting to the leading pion amplitude inthe kernels(A1)–(A4), two major approximations that wemust perform in order to keep the calculation technicallymanageable. First, using the decay kernels as describedin this work entails that one must choose whether theaxial-vector or the vector WTI are preserved while theother one is violated. Following [37] we choose to preservethe vector identity. A violation of the axial-vector WTI ismanifested, among others, in the pion not being masslessin the chiral limit, and therefore the value of the currentmass for which the pion becomes massless allows for aquantification of the violation of the axial-vector WTI.In Fig. 4 we therefore show the evolution of the pionmass with varying m q in the employed truncation. First,the relation is linear as expected from the Gell-Mann–Oakes–Renner relation which is a direct consequence ofthe dynamical breaking of chiral symmetry. Second, thepion does not become massless in the chiral limit butfor a value of the current mass m (0) q ( µ = 19 GeV) =3 MeV. On the one hand, this explains the relativelylarge value of m q ( µ = 19 GeV) = 6.8 MeV we needed to obtain the correct pion mass: The related explicitlychiral-symmetry-breaking term is m q − m (0) q = 3.8 MeV,and thus much closer to what is expected from the knownparameters of QCD. Second, as the masses of the vectormesons depend linearly on the current mass the inducederror on m ρ and m ω is of the order of m (0) q = 3 MeV, andthus it is as small or even smaller than other uncertaintiesin our calculation of the vector meson masses. . . . . . .
002 0 .
003 0 .
004 0 .
005 0 .
006 0 . FIG. 4. The pion mass squared m π vs. the current mass m q ( µ = 19 GeV) for the employed truncation. Second, in the calculation of the form factor we usethe impulse approximation which, in this context, im-plies neglecting the second and third diagrams in Fig. 3.The consequence of discarding diagrams is the violationof charge conservation or, equivalently, a deviation from F π (0) = 1. As can be seen in the inset in Fig. 5, thiseffect is of the order of ∼
1% only. . . . .
81 0 0 . . FIG. 5. Pion form factor in the spacelike Q > η = 1 . η = 1 . As can be also seen from Fig. 5, the results for thepion form factor in the space-like Q > η parame-ter of the model. Even more remarkable is the fact thatour calculation shows a very good agreement with ex- − . − . − . − . − . − . FIG. 6. Absolute value of the pion form factor in the time-like Q < η = 1 . η = 1 . perimental data in the space-like domain even though,as evident from the above discussion, we aimed at in-cluding all physical effects which are important in thetime-like regime. An interpretation of this result in viewof the dispersion relation (4) provides an indication thatthe imaginary part in the time-like region is preciselyenough reproduced to provide very good results for thespace-like form factor.We show our results for the pion form factor for time-like ( Q <
0) virtualities in Figs. 6 and 7. As discussed inthe previous section, as a consequence of the decay ker-nels in our truncation, the pion form factor develops abranch cut along the real negative axis starting from thetwo-pion threshold Q = − m π , induced by the corre-sponding cut in the quark-photon vertex [37]. Hence, inthat region the (complex) form factor is defined from itsanalytic continuation as F ( Q + i(cid:15) ). In the numerical cal-culations we have typically chosen (cid:15) = 0 . afterverifying that this value is small enough to not disturbthe presented results. In Fig. 6 we present the absolutevalue for the model parameter η = 1 . η = 1 .
6, withthe remaining model parameters kept constant, as dis-cussed above. As a manifestation of the fact that the ρ -meson pole in the quark-photon vertex moves into thecomplex plane when the decay kernels are included inthe calculation, the pion form factor develops a bumpon the real and negative Q axis with an approximatelycorrect height and width. Therefore our calculation over-comes a major deficiency of the RL truncation, withoutor even with the pion exchange term, for which the form factor diverges instead at the Q -value corresponding tothe ρ -meson mass in those truncations, see e.g. [51]. Thisconstitutes already one main result of the here presentedinvestigation.We note, however, that, contrary to the results forspace-like regime, the position and height of the bumpof the form factor depends strongly on the value of the η parameter shape of the form factor. Of course, thisreflects the different positions of the ρ -meson pole, cf. Tab. I. Nevertheless, there are features that appear tobe independent of η , most prominently that the heightof the bump is underestimated. As expected from thediscussion in Sec. II the form factor behaves smoothlyafter the bump, in contrast to the sharp dip in the ex-perimental data. Even though an unambiguous analysisof the origin of such discrepancies can only result fromthe inclusion of all relevant physical mechanisms in ourcalculations, it is evident from the analysis performed inSec. II that one of the main missing elements is the ρ - ω mixing due to isospin breaking, which is completelyabsent in the present study.Particularly sensitive to the deficiencies of our trunca-tion is the phase of the form factor, shown in Fig. 7. Eventhough our data shows the expected behaviour near theresonance value, it severely underestimates the experi-mental data, particularly in the elastic region. This isa manifestation of the absence of some hadronic effectsin our approximation scheme, which only includes thosestemming from the resonance complex pole in the quark-photon vertex and the ρ → ππ induced branch cut. In0 . . . . . . . . . − FIG. 7. Phase of the pion form factor in the time-like Q < η = 1 . η = 1 . addition to the isospin breaking effects discussed above,and which would be more relevant in the region above theresonance, the impulse approximation used in our calcu-lation of the form factor entails that effects coming fromthe coupling of the photon to the intermediate hadrons,via its coupling to the exchanged pion or to the quark-pion vertex (see Fig. 3), are missing. It has been shown[52] that considering only impulse-like diagrams leads toa very small pion-pion scattering amplitude in the elasticregion in the isospin I = 1 channel. This, due to unitar-ity, implies a very small value of the imaginary part ofthe pion form factor in the elastic region (which is, infact, what we observe) which translates into a very smallphase, as seen in Fig. 7 (and as could be inferred fromWatson’s theorem).Last but not least, we are comparing Pad´e fits, resp.rational fits of the order (3,3), for the real part and forthe imaginary part of the form factor to the expression(8). Trying first, Re F π ( Q ) − F π (0) ≈ − a + a Q + a ( Q ) + a ( Q ) b + b Q + b ( Q ) + b ( Q ) Im F π ( Q ) ≈ c + c Q + c ( Q ) + c ( Q ) d + d Q + d ( Q ) + d ( Q ) , (21)we obtain tiny values for the coefficients a , a , b , c , c and d . Note that this confirms the structure expected TABLE II. The coefficients of the rational fits to the pionform factor as discussed in the text. All values for dimension-ful quantities are given in GeV. η =1.5 η =1.6 Eq. (8) a a b b b c c d d d from the VMD form (8). We repeated the fits for Re F π ( Q ) − F π (0) ≈ − a Q + a ( Q ) b + b Q + b ( Q ) Im F π ( Q ) ≈ c Q + c ( Q ) d + d Q + d ( Q ) . (22)The coefficients resulting from these fits as well as theones resulting from expression (8) are given in table II.From this we conclude that the expression based on theVMD is an astonishingly good representation of our re-sults. Therefore, our investigation makes it plausible thatthe VMD picture can be derived from QCD. At least, the1results of the here presented microscopic approach give astrong hint into this direction. V. CONCLUSIONS AND OUTLOOK
In this work we have presented an exploratory study ofthe pion form factor in the DSE/BSE approach. Our fo-cus has been to explore how the interplay between hadronstructure (as described by form factors) and the hadronspectrum (as described by resonance masses and widths)can be realised in the microscopic approach presentedherein. In particular, we focused on the effect of inter-mediate pions in the BSE interaction kernel, the inclu-sion of which is sufficient to describe the ρ meson as aresonance [43]. As elucidated by the detailed analysis inthe last section our calculation represents a verificationof the vector meson dominance picture and provides anexplanation how at the quark level vector meson dom-inance becomes effective. A more complete calculationthan the one presented here might then actually providea derivation of vector meson dominance from QCD.Despite the fairly drastic approximations used in thispreliminary study, the agreement with experiment is re-markable. On the space-like side our calculations agreewith experimental data at the quantitative level. Fortime-like momentum transfers the agreement is mostlyqualitative and consistent with the fact that in our ap-proximation scheme time-like physics is dominated by thelowest-lying ρ -meson resonance only. The absolute valueof the calculated form factor features a bump at approx-imately the correct Q region as caused by a resonancepole. Our result lacks, however, other features such asthose caused by the isospin-breaking ρ - ω mixing and in-terference. The phase of the form factor also shows defi-ciencies caused by the employed impulse approximation.However, the overall qualitatively correct behaviour is,nevertheless, very encouraging for future studies on time-like phenomenology with BSE methods as it shows thatthe necessary computational techniques are getting moreand more under control, and that within a functional-method-based bound-state approach to QCD direct cal-culations in the time-like regime are becoming feasible.Among the different physical mechanisms absent in ourcalculation, the most relevant one appears to be isospinbreaking by the light quarks’ masses and electric charges,and the different phenomena associated with it. The pre-sented exploratory calculation paved the way to includein a bound-state approach formulated in QCD degrees offreedom the effects of isospin violation, and hereby mostprominently ρ - ω mixing, on the time-like pion electro-magnetic form factor. Thus including isospin violationin a BSE approach is the topic of ongoing work. A fur-ther related topic is the study of the form factor for thecoupling of a photon to three pions. On the one hand,this process is of special theoretical interest because therelated form factor is at the soft point completely de-termined by the Abelian chiral anomaly. On the other hand, data of the COMPASS experiment are currentlyanalysed [54], and therefore experimental data for thisform factor in the time-like region will become available.Combing previous studies in the DSE/BSE approach forthe space-like γπππ form factor [55–58] with the tech-niques of the here presented calculation will thus enablea respective investigation of this form factor.Another aspect we want to investigate is how to realisethe idea of decay kernels like (A3) and (A4) in a baryonbound state equation. This is a necessary step in orderto tackle time-like nucleon form factors in the DSE/BSEapproach, which is one of our major goals due to theincreased effort and interest from the experimental sidein highly precise measurements over a wide kinematicaldomain of the nucleon form factors. ACKNOWLEDGEMENTS
This work was partially supported by the the AustrianScience Fund (FWF) under project number P29216-N36.A.S. Miramontes acknowledges CONACyT for financialsupport.The numerical computations have been performed atthe high-performance compute cluster of the Universityof Graz.
Appendix A: Interaction kernels
The BSE kernel representing the exchange of an ex-plicit pionic degrees of freedom as defined in [40, 41] reads K ( t ) utrs ( q, p ; P ) = C jπ ] ru (cid:18) p + q − P p − q (cid:19) [ Z γ ] ts D π ( p − q )+ C jπ ] ru (cid:18) p + q − P q − p (cid:19) [ Z γ ] ts D π ( p − q )+ C Z γ ] ru [Γ jπ ] ts (cid:18) p + q + P p − q (cid:19) D π ( p − q )+ C Z γ ] ru [Γ jπ ] ts (cid:18) p + q + P q − p (cid:19) D π ( p − q ) , (A1)in combination with the following truncation of the quarkDSE S − ( p ) = S − ( p ) RL − (cid:90) q (cid:34) Z γ S ( q )Γ π (cid:18) p + q , q − p (cid:19) + Z γ S ( q )Γ π (cid:18) p + q , p − q (cid:19) (cid:35) D π ( k )2 , (A2)with S − ( p ) RL being the right-hand-side of the quarkDSE in the RL truncation with the gluon-mediated in-teraction as described in Sect. III. In Eqs. (A1) and (A2)the pion propagator is taken as D π ( k ) = ( k + m π ) − .2 FIG. 8. Truncations used herein for the BSE interaction kernel K (upper diagram) and the quark DSE one (lower diagram).In the upper diagram, the terms on the right-hand side correspond to the rainbow-ladder, pion exchange, and s- and u-channelpion decay contributions to the truncation, respectively. The s- and u-channel pion decay terms do not contribute to the quarkDSE. The factor 3 / C in (A1) shouldbe obtained. When done in the quark-photon vertex,the flavour factor leads to C = +3 /
2. However, suchvalue of C , violates the Ax-WTI while preserving the V-WTI. Since the V-WTI is related to charge conservationin electromagnetic form factors, we use C = +3 / π is taken to be the pion Bethe-Salpeter ampli-tude.The pions in the kernel can also appear in the s- andu- channels [40]. We used here a version of the kernelsslightly different to the one in [40] in order to be consis-tent with the construction for the t-channel, where oneof the pion vertices is kept bare and the kernel is thensymmetrised. They read K ( s ) heda ( q, p, r ; P ) = C D π (cid:18) P + r (cid:19) D π (cid:18) P − r (cid:19) (cid:104) [ Z γ ] dc S cb (cid:16) p − r (cid:17) [ Z γ ] ba × [Γ jπ ] hg (cid:18) q − P − r r − P (cid:19) S gf (cid:16) q − r (cid:17) [Γ jπ ] fe (cid:18) q + P − r − P + r (cid:19) + [Γ jπ ] dc (cid:18) p + P − r P + r (cid:19) S cb (cid:16) p − r (cid:17) [Γ jπ ] ba (cid:18) p − P − r P − r (cid:19) × [ Z γ ] hg S gf (cid:16) q − r (cid:17) [ Z γ ] fe (cid:105) , (A3)and K ( u ) heda ( q, p, r ; P ) = C D π (cid:18) P + r (cid:19) D π (cid:18) P − r (cid:19) (cid:104) [ Z γ ] dc S cb (cid:16) p + r (cid:17) [ Z γ ] ba × [Γ jπ ] hg (cid:18) q − P − r r − P (cid:19) S gf (cid:16) q − r (cid:17) [Γ jπ ] fe (cid:18) q + P − r − P + r (cid:19) + [Γ jπ ] dc (cid:18) p + P r P − r (cid:19) S cb (cid:16) p + r (cid:17) [Γ jπ ] ba (cid:18) p − P r P + r (cid:19) × [ Z γ ] hg S gf (cid:16) q − r (cid:17) [ Z γ ] fe (cid:105) , (A4)where now r is an additional integration momentum inthe BSE (cf. Eqs.(9) or (10)). The resulting truncationof the BSE kernel and the quark DSE is shown in Fig. 8.The inclusion of the two kernels given in equations(A3) and (A4) generates a highly non-trivial analyticstructure of the integrand of the BSE, induced by the in-termediate pions going potentially on-shell as well as by singularities in the quark propagators, see ref. [37] for de-tails where also the techniques for finding viable contourdeformations for performing the numerical integrationsin a mathematically correct way are described.The pion Bethe-Salpeter amplitude possesses four ten-sor components. Hereby, only one is generically smallsuch that for a precise calculation of pion properties it3would be necessary to take into account three of them,namely the leading pseudoscalar term and two sublead-ing ones related to the axialvector structure. However,using in the above described kernels three amplitudes forevery pion Bethe-Salpeter amplitude in these expressionsis numerically by an order of magnitude more expensiveand far beyond the scope of the present study. Havingrestricted to the leading component of the pion ampli-tude in the kernels (A1) and (A4) one can further exploitthat this leading amplitude may be well approximated us-ing the chiral limit value of the quark dressing function B ( p ) and normalize the amplitude by dividing throughthe pion decay constant f π ,Γ iπ ( p ; P ) = τ i γ B ( p ) f π . (A5)For light quarks the difference between calculated leadingorder amplitude and this approximation is at the levelof a few percent, see, e.g. , [53] and references therein.Therefore we use this simplified form. Appendix B: Poles of the quark propagator
In the truncations of the quark DSE used in this paper,the quark propagator features pairs of complex conjugate poles in the complex plane (see, e.g. [59]). In orderto facilitate the use of the quark propagators and easilyidentify the analytic structures generated by those polesin the form factor and vertex calculations, it is usefulto parametrise the quark propagator simply as a sum ofpoles (see e.g. [60]) S ( p ) = − i / pσ v ( p ) + σ s ( p ) ,σ v ( p ) = n (cid:88) i (cid:20) α i p + m i + α ∗ i p + m ∗ i (cid:21) ,σ s ( p ) = n (cid:88) i (cid:20) β i p + m i + β ∗ i p + m ∗ i (cid:21) , (B1)where the parameters m i , α i , β i can be obtained by fit-ting the corresponding quark DSE solution along the p real axis or, alternatively, on a parabola in the complexplane that does not enclose the poles.In the numerical solution of the quark DSE in the com-plex plane we tested fits with one real and one pair ofcomplex conjugated poles, with two pairs of complex con-jugated poles, and with three pairs of complex conjugatedpoles. 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