Analytical solution for the correlator with Gribov propagators
aa r X i v : . [ h e p - t h ] O c t Analytical solution for the correlator with Gribov propagators
V. ˇSauli ∗ Department of Theoretical Physics, Institute of Nuclear Physics Rez near Prague, CAS, Czech Republic
Propagators approximated by a meromorphic functions with complex conjugated poles are widelyused to model infrared behavior of QCD Green’s functions. In this paper, analytical solutions fortwo point correlator made out of functions with complex conjugated poles or branch points havebeen obtained in the Minkowski space for the first time. As a special case the Gribov propagatorhas been considered as well. The result is different from the naive analytical continuation of thecorrelator primarily defined in the Euclidean space. It is free of ultraviolet divergences, and insteadof Lehmann it rather satisfies Gribov integral representation.
PACS numbers: 11.10.St, 11.15.Tk
I. INTRODUCTION
The explanation of hadron properties in terms of QCD degrees of freedom represents hard non-perturbative task,especially when the energy of a process does not comply with the asymptotic freedom and particle like description ofhadron constituents. Chiral symmetry breaking and confinement are the main phenomena beyond the applicabilityof perturbation theory.That confinement can be naturally encoded in analytical properties of QCD Green’s functions is an old-fashionableconjecture [1–9]. Quark and gluon propagators with complex conjugated singularities are longstanding outcomes ofmany Bethe-Salpeter and Schwinger-Dyson equations (SDEs) studies [10–16], noting here that a considered propa-gators, either calculated or phenomenologically used, usually exhibit not one but several (infinite numbers with zeromeasure i.e. the cuts is possible as well) complex conjugated poles. For instance the quark propagator considered asa series S q = X i r i p − m i + r i p − m ∗ i , (1.1)where m i are complex numbers and r i real residua, actually provides a good ingredient for calculations of pionobservables. Note that in practice, most of (and all cited above) non-perturbative studies are based on the useof Euclidean metric from the beginning and the calculations performed in Minkowski space are due to the knownobstacles very rare [17, 18]. Guiding by simple assumption, that physics in Minkowski space can be read from theanalytical continuation of the solutions via Euclidean theory, the lattice data has been checked against the form ofStjieltjes representation [19, 20] as well as the solutions of SDEs system [21–23] have been discussed in the context ofusual dispersion relation.Assumed structure of propagators describing the confinement of quarks and gluons, i.e. the form represented by(1.1), implies the lost of perturbative analyticity. In this paper we readdress some important issues of analyticalproperties of Green’s functions in QFT. For this purpose we consider 2-point correlation function of the followingform Π( p ) = i Z d l (2 π ) Γ G ( l )Γ G ( l − p ) . (1.2)Using for this purpose the Gribov propagator: G ( p ) = 1 p + µ p , (1.3)(here we ignore the spin structure) we will explicitly calculate the correlator (1.2) in Minkowski space and show thatthe correlator does not satisfy spectral representation but it reflects, and in fact it reproduces again, the Gribovform in its own continuous integral representation. Recall that the lost of reflection positivity is expected for theory ∗ Electronic address: [email protected] with confinement. Here this is the specific form of analyticity, which ensures non existence of spectral (Lehmann)representation at all.The choice of the function (1.3) is well motivated due to the fact that the Gribov propagator appears as the solutionof Gribov copy problem [1, 3]. Actually, it was shown in [24–26] that if QCD is properly quantized then the gluonpropagator in Landau gauge receives the Gribov form in non-perturbative manner. Notice trivially, it has the simplestnontrivial complex conjugated poles, which are located on the imaginary axis of p complex plane.The purpose of presented paper is twofold: the first point is to show that the evaluation of the loop with thepropagators (1.3,1.1) in Minkowski space is feasible and its analytical evaluation is certainly not too much demanding.As the second point, we mention possible physics related with.Of course, elaborating a correct analytical relations between relations in Minkowski and Euclidean spaces is alwaysuseful and obviously possible here. However let us stressed, that we start with Minkowskian definition and thusthe final result is not given by analytical continuation of the expression primarily defined within the use of a realEuclidean theory. As a consequence, we will show that the correlator with Gribov propagators produces purelyimaginary result, contradicting not only the use naive Wick rotation, but the usual strategy. Furthermore, theobtained result is ultraviolet finite, which is another striking outcomes of presented calculation performance.Regarding two point correlator matrix (1.2), it has played longstanding and historically unpredentecable role inphysics. Such two point function can stands for QED like V − V or A − A current correlators considered originally inQCD Sum Rules [27, 28] or it can represent colored gluon polarization function, quark selfenergy etc. depending onwhat we mean by the vertex Γ and the propagator(s) G . While the results do not support too much the old -fashionableconjecture of quark-hadron duality, the integral representation derived thorough this paper can be actually used inpractice for bound states and form factors calculations. Remind here its weak coupling perequisitor: the PerturbationTheory Integral Representation [29], which has found its own application in bound state calculations [30, 31] innonconfinig theory.We will consider not only Gribov propagator, but a wider class of the functions which have the complex conjugatedsingularities, including the poles as well as branch point in general. Before going ahead we simplify and concern theanalytical structure alone. However note, that using suitable ”trace projectors” and after some trivial algebra, anycorrelator matrix (1.2) can be cast into the sum of the product of matrices,tensors and the following scalar formfactors Π( p ) = iI ( p ) = i Z d l (2 π ) Γ( l, p ) G ( l ) G ( l − p ) , (1.4)where all the functions in Rel. (1.4) are Lorentz scalars. Thorough this paper, we neglect also the momentumdependence of the vertex function and simply take Γ = 1, implying thus, that all the analytical behaviour is solelydue to the Minkowski space measure and the form of functions G . Obviously, a more complete solution of the SDEswith nontrivial momentum dependence of the vertex would require analysis beyond the scope of presented study. Inwhat follow we present Minkowski space calculation for the correlator (1.4) made out of Gribov propagators in thenext Section (II).In order to see the effect of changing the pole position, we also consider the correlator made out of the propagatorswith shifted poles. For this purpose we consider the following superconvergent toy model propagator function G ( l ) = 1( l − a ) + b , (1.5)where a represents a real part of the pole location. As the last case we consider also the convolution of ”propagators”which have complex conjugated branch points. The later example with the function G defined as G ( l ) = 1 p ( l − a ) + b , (1.6)is considered in the Section IV. II. CORRELATOR WITH GRIBOV PROPAGATORS .The Gribov propagator (1.3) represents a simple rational function, whilst it has a usual perturbative ultravioletasymptotic, its infrared structure is drastically different from the free particle propagator. Instead of the real poleassociated with free particle modes, it has two simple imaginary poles associated with the confinement scale b . Itsreality and the absence of the real axis singularity implies that the function I = − i Π should be real again. Also,the direct integration in the Minkowski space is well established without a need of sometimes unavoidable analyticalcontinuation to the auxiliary Euclidean space. However, as the momentum integration is particularly easy within theuse of the Euclidean metric, the method of analytical continuation still remains a powerful technical tool and we willuse it carefully in our case as well. Before starting doing so, it is convenient to make a little algebra and we rewritethe correlator Π( p ) in the following way:Π( p ) = i Z d l (2 π ) (cid:20) l + ib )( q + ib ) + 1( l − ib )( q − ib )+ 2( l + ib )( q − ib ) (cid:21) . (2.1)where each line in bracket is hermitean and q = l − p . Feynman parametrization represent a useful tricks, whichallows to evaluate perturbative loops in closed form. Using this we can write for the first line in (2.1) i Z d l (2 π ) Z dx (cid:20) l + p x (1 − x ) + ib ] + 1[ l + p x (1 − x ) − ib ] (cid:21) (2.2)where, we assume the shift of variable l as well as the interchange of the integrations order is granted by the symmetry(and we will add omitted factor 1 / l o integration we will use the usual contour,which is literally known as a Wick rotation (WR) for the first term, while we will use the contour which is mirrorsymmetric to the Wick rotation (MWR) to integrate the second term (for Wick rotation see the standard textbook([32]) . Cauchy lemma then allows to write Z d l E (2 π ) Z dx (cid:20) − − l E − p E x (1 − x ) + ib ] + 1[ − l E − p E x (1 − x ) − ib ] (cid:21) . (2.3)where − p → p E = p + p + p + p in our metric convention.Matching two terms in (2.3) together in the following manner Z d l E (2 π ) Z dx ib ( − l E − p E x (1 − x ))[ − l E − p E x (1 − x ) − ib ] [ − l E − p E x (1 − x ) + ib ] = Z d l E (2 π ) Z dx Z dy Γ(4)4 iby (1 − y )( − l E − p E x (1 − x ))[ l E + p E x (1 − x ) + ib (1 − y )] . (2.4)The result is obviously finite and one can integrate over the momentum l without use of any regulator. However itrequires the integration over two auxiliary variables x, y and there is a slightly easy way, which is to integrate eachterm individually. This step requires some usual (translation and other symmetries keeping) regularization, howeverthe infinite peaces cancel each other. Independently on the procedure, it leads to the result:1(4 π ) Z dx ln (cid:18) p E x (1 − x ) + ibp E x (1 − x ) − ib (cid:19) . (2.5)In order to get the analytical structure more explicit , it is advantageous to have all logs linearly dependent onintegral variable. An easy way is to exploit the substitution u = x − / u one can change the u-integral boundaries such that R / − / du → R / du . Then changingvariable u → ω such that u = p (1 / − b/ω ) one gets b (4 π ) Z ∞ b dωω q − bω ln (cid:18) p E + iωp E − iω (cid:19) = − ib (4 π ) Z ∞ b dωω q − bω tan − (cid:18) − ωp (cid:19) (2.6)The expression (2.6) defines function of p which has two symmetric cuts along imaginary axis going from i b to i ∞ and from − i b to − i ∞ .Note that here is no real cut associated with the particle threshold and the usual of dispersion relation betweenabsorptive and real part does not apply there. In other words: there is no Lehmann representation for the correlatormade out of Gribov propagators in Minkowski space. The correlator has no real part anywhere for the real Minkowskispace argument p . In formal analogy with perturbation theory which deals with the usual particle like propagators,an analogous integral representation to spectral ones can be written down, however here, they have Gribov form again(i.e. denominator of such representation has complex conjugated zeros). To write down such representation explicitlyone can use per-partes integration in ω variable getting the following: i (4 π ) π s gn ( p ) − Z ∞ b dω p q − bω p + ω , (2.7)which shows us how the spectral representation for particles turns to the continuous sum of Gribov propagators ofconfined objects. Anticipate here, the same arguments apply for the remaining term in Eq. (2.1) for which we aregoing to derive the appropriate result now. Matching together its denominators by using Feynman variable x we canwrite for the second line of Rel. (2.1) i Z d l (2 π ) Z l + p x (1 − x ) + ib (2 x − . (2.8)To arrive to the known Euclidean integral we use x − parameter dependent contours in the complex l o plane. For b (2 x −
1) positive (negative) we can use WR (MWR) and employ Cauchy lemma (assuming the same for the externalmomentum). This directly gives the following result:2 Z d l E (2 π ) Z dx [ − Θ(2 x −
1) + Θ(1 − x )][ l E + p E x (1 − x ) − ib (2 x − . (2.9)Integrating over the Euclidean momentum we arrive into the finite expression24 π − Z / dx + Z / dx ! ln (cid:0) − p E x (1 − x ) + ib (2 x − (cid:1) , (2.10)noting that separate integration over the first or over the second step function in Eq. (2.9) would require theintroduction of some regulator method (like in the previous case, the regulator can be avoided by summing theseterms before the integration, for which purpose one can use Feynman trick once again. We are not showing thesetrivial details).In addition we offer several integral representations, which can be in principle useful in future. To arrive in, wemake the substitution x = (1 + z ) / x = (1 − z ) / π ) Z dz ln − p E (1 − z ) / − ibz − p E (1 − z ) / ibz = 2 i (4 π ) Z dz tan − bzp E (1 − z ) . (2.11)As aforementioned , it cannot be cast into the form of spectral representation. The reason is obvious , the correlatorhas branch cut along the imaginary axis of p instead of the real one. Actually, using the substitution z → ω one getsfor (2.11) the following representation i (4 π ) " − π s gn ( p ) + Z ∞ dω ( p + ω ) p ω ( b + p b + ω / (2.12)or alternatively 1(4 π ) " − iπ s gn ( p ) − Z ∞−∞ dω ( ip − ω ) | ω | ( b + p b + ω / . (2.13) -100 -50 0 50 100 p -0,0500,050,10,150,20,250,30,350,40,450,50,550,60,650,70,750,80,850,9 - i ( π ) Π ( p ) sum1s1 FIG. 1: Gribov correlator in units where b = 1. To conclude, the correlator Π is given by (quarter of) the sum of two contributions (2.7) and (2.12) and satisfiescontinuous ”Gribov” integral representationΠ( p ) = i Z ∞ dω ρ G ( ω ) p p + ω ρ G ( ω )(4 π ) = 14 ωb + p b + ω / − r − bω Θ( ω − b ) (2.14)Obviously, likewise the Lehmann representation reflects the analyticity of Feynman propagator G − = p − m + iǫ ,the correlator made out two Gribov propagators copiously reproduces the analytical structure of Green’s functionsinvolved.The last undone integration can be performed as well, a bit lengthy expression is not presented here. Numericalvalues for the function Π is shown against the Minkowski variable p in Fig. (1). We plot separately the resultstemming from the first line for comparison. It is shown that the terms which separately allows either WR or MWRgives relatively smooth contribution in the infrared. While In contrast, there is infrared discontinuity in the originof the complex p plane, which stems from the second line, i.e. from the term which involves poles at both sides-upper and low semi-half- of complex plane. Recall, this discontinuity is lowered when the pole of Gribov propagatoris shifted away from the imaginary axes. This analytical behaviour is worthwhile to study and we do this in the nextSection.At last but not at least, we comment the ultraviolet finiteness of the result. The result we obtained is finite here,however we should stress that ultraviolet finiteness is not a general property of the correlator made out of propagatorswith complex conjugated poles. Usual ”log” divergence arises for the convolution of two different Gribov propagators.This divergence is proportional to the difference of the pole positions ≃ b − b and if necessary it can removed bysome sort of symmetry keeping regularization. Recall here, the Gribov form of QCD propagators is assumed to be agood approximation in the infrared, while it is likely less useful for the study of ultraviolet properties. However, asthe selfconsistent and serious Minkowski space study of QCD with Gribov propagators is lacking, an outcomes canbe challenging. III. CORRELATORS WITH GENERALIZED GRIBOV PROPAGATORSA. Shifting the branch point
In the previous section we have derived Gribov integral representation, which arises when two Gribov propagatorsconvolute in 3+1 momentum space. Obvious question arises: what would happen to the correlator Π when oneconsider more general structure of the propagator, e.g. shifted pole, branch points, etc. As an example we willconsider the correlator Π( p ) with propagators of the following form: G ( l ) = 1( l − a ) + b . (3.1)Again, without a detailed calculation one can see obvious properties of the correlator. Like in previous case, thefunction Π( p ) should be a purely imaginary function for all p . Further obvious is its finiteness, actually it shouldbe finite since its Euclidean counterpartner is.To arrive into the analytical expression here, we change a bit the calculation procedure and start with the Feynmanparametrization from the beginning. For this purpose let’s write1( p − a ) + b = 1 p − a + ib p − a − ib = Z dx p − a + ib − ibx ] . (3.2)The product of two such propagators in (1.4) can be written as Z dx dx l − a + ib − ibx ] l − p ) − a + ib − ibx ] = Z dydx dx y (1 − y )Γ(4)[ l y + ( l − p ) (1 − y ) − a + ib − ib ( x y + x (1 − y ))] (3.3)Lorentz invariance of the measure as well as the kernel here, both imply that the finite shift of the integral variableleaves the result invariant. We assume, this symmetry dictated property is a must of the theory, with possibleanomalies exceptions which we are not going to discuss here. Shifting momentum l one gets for the correlator:Π( p ) = i Z d l (2 π ) Z dydx dx y (1 − y )Γ(4)[ l + p (1 − y ) y − Ω] , Ω = a − ib + 2 ib ( x y + x (1 − y )) . (3.4)The position of the pole now depends on the parameters and we interchange ordering of integrations and integrateover the four-momentum as a first. Like in the previous case we use MWR for ℑ Ω < l , we will use the usual WR. In both cases the inner part ofintended curve is thus free of singularities and the use of Cauchy lemma switches to the Euclidean metric (performingsimilar for the external momentum). Doing this explicitly, one gets the following prescription for the integral i Z d l (2 π ) f ( l, p ) → − Z d l E (2 π ) Θ( ℑ Ω) f ( l E , p E ) + Z d l E (2 π ) Θ( −ℑ Ω) f ( l E , p E ) , (3.5)wherein the subscript E implies the arguments of f E uses an Euclidean metric l E = l + l + l + l . In accordancewith causality, we assume a > l one gets Π( p ) = Z dydx dx y (1 − y ) [ − Θ( ℑ Ω) + Θ( −ℑ Ω)](4 π ) [ p y (1 − y ) − Ω] , (3.6)where p is Minkowski momentum again, and the function Π( p >
0) is an analytical continuation of Π( p < x → z such that z = 1 − x y + x (1 − y )), one getsΠ( p ) = Z dydx Z − x (1 − y )1 − y + x (1 − y )) dz (1 − y ) [ − Θ( z ) + Θ( − z )](4 π ) [ p y (1 − y ) − a + ibz ] . (3.7)For the purpose of the integration over the variable z let’s distinguish three cases. The first case we define, is suchthat upper and down z − integral boundaries are negative. This allows to consider only the mirror Wick rotation.Integrating over the variable z leads to the following formula: − ib Z dydx (1 − y )(4 π ) Σ i =1 , ( − i [Θ(2 x (1 − y ) − p y (1 − y ) − a + ibz i , (3.8)where z = 1 − x (1 − y ) and z = 1 − y + x (1 − y )).The second case is defined by inequalities z > z <
1. Both step functions are relevant and they just splitthe z integration domain to two integrals with boundaries z , , z respectively. Integrating over the variable z is straightforward and the result for the second case reads − ib Z dy Z dx (1 − y )Θ( z )Θ( − z )(4 π ) (cid:20) p y (1 − y ) − a − Σ i =1 , p y (1 − y ) − a + ibz i (cid:21) , (3.9)with two variables z i defined previously.The third and the ultimate case corresponds with the condition z > z >
0, for which one gets − ib Z dy Z dx (1 − y )Θ( z )(4 π ) Σ i =1 , ( − i p y (1 − y ) − a + ibz i . (3.10)The scalar polarization correlator Π is then equal to the sum of expressions (3.8), (3.9) and (3.10).Further analytical integration is possible as well, for instance the integration over the variable x gives for (3.9)1(2 ib ) π ) "Z / dy ln R − ibyR + 2 iby + Z / ln R + ib (2 y − R + ib (1 − y ) + Z / ln R + ibR − ib + i b π ) "Z / dy − yR + Z / dy R , (3.11)where we have defined R = p y (1 − y ) − a for purpose of brevity.Using the identity ln R − ibyR + 2 iby = 2 i tan − byp y (1 − y ) − a (3.12)for the first and similarly for other terms in (3.11), one immediately sees that the result for (3.9) is purely imaginary.Further, summing (3.8) and (3.10) together and integrating over x gives after some trivial algebra the followingformula: 1( − ib ) π ) Z / dy [ − ln( R + 2 ib ) + ln( R + ib ) − ln( R + ib (1 − y )) + ln( R ) + c.c. ] , (3.13)where c.c. stands for complex conjugated term. Recall, as was discussed in the beginning, the total result must becompletely imaginary. Thus since Rel. (3.9) already is, while the Rel. (3.13) is purely real, the later must be exactlyzero for all p and the total contribution is given solely by the expression (3.11).All integrals in (3.11) can be further integrated analytically, providing the following final result:Π = i b (4 π ) s ln ( a + s/ + b a + b − s ln ( a − s/ a + 12 b tan − ba − s q a − ss tan − r s a − s − b (4 π ) ( √ Ds tan − − ib + s √ D − √ Ds tan − − ib √ D − c.c. + √ D s tan − ib √ D − √ D s tan − ib − s √ D − c.c. − r D s tan − r sD − c.c. ) . (3.14)where we have used the following abbreviations D = s (4 a − s ) + 4 b ; D = s (4 a − s ) + 4 b + 4 ibs ; D = 4 a + 4 ib − s ; s = p . (3.15)The inverse tangent function of complex argument is defined through the multi-valuable complex logarithm. As onecan see, apart of complex singularities, there is a real branch point presented as well. This branch point has coincidedwith the origin of the complex plane in the previous case, where a = 0 case was considered only. B. Π composed from propagators with branch points Quark and gluon propagators can obtained via solution of SDEs and the result can be in principle approximatedas a series (1.1). It is well known that the interaction is reflected in the analytical properties of amplitudes. Forinstance, in a relatively simple theory like QED, the effect of dressing electron propagator by photon self-exchangeentails that a simple pole structure of the electron propagator turns to the branch point singularity. In QCD one canexpect the similar, if not stronger effect and it is plausible that the other considerable expansions are much fasterconvergent , especially when their first terms catch the main properties of the exact solution. As such a furtherreasonable candidate, we consider propagator with square root non-analyticity. Guiding by simplicity as well as itultraviolet asymptotic of a free propagator, let us calculate the correlator with the propagator (1.6). In this specialcase, the argument of the correlator reads 1 p (( l − p ) − a ) + b p ( l − a ) + b . (3.16)As the first step we start with the root decomposition of square-root arguments, then using Feynman trick gives us1( p − a + ib ) / p − a − ib ) / = Z dx x / (1 − x ) / (1 / p − a + ib − ibx . (3.17)Depending on the value of variable x , the denominator in rhs. of Eq. (3.17) has positive imaginary part for a small x and positive b . It turns to be negative for a larger value of x . Using the variable x and x for each propagator in(3.16) and further using the variable y to match propagators together one gets for Rel. (3.16) the following expression Z dx dx dy p x (1 − x ) x (1 − x ) Γ(2)Γ (1 / l y + ( l − p ) (1 − y ) − Ω , (3.18)where Ω is defined by (3.4) in the previous Section. After the standard shift one gets for the correlatorΠ( p ) = i Z d l (2 π ) Z dx dx dy p x (1 − x ) x (1 − x ) Γ(2)Γ (1 / [ l − ∆] ∆ = − p (1 − y ) y + Ω . (3.19)Now we will integrate over the momenta as in the previous case. It means that for case when ℑ ∆ < l o MWR is used.The conditions ℑ ∆ < ℑ ∆ > p ) = Z π ) Z dx dx dy p x (1 − x ) x (1 − x ) (cid:18) ǫ − ln(∆) − γ + O ( ǫ ) (cid:19) [Θ( −ℑ Ω) − Θ( ℑ Ω)] . (3.20)Note that only the ℑ parts of the expression in the large bracket in Eq. (3.20) can survive at the end, the result willnot depend on the renormalization scale at all and as mentioned it is finite at all. The remaining 3d integral over theFeynman variables can be performed numerically. If one wishes, the vanishment of the real part can regarded as testof numerical precision. Actually, numerically we get ℜ Π / ℑ Π < − (here we did not find the analytical expressionover the Feynman parameters due to the presence of square-root function). The resulting correlator is plotted in Fig.(2). Obviously, one can see the evidence for a real branch point at the momentum p = 4 a . -100 -50 0 50 100 p -6 -4 | I m Π | a=1; b=2 a=2; b=1a=4, b=4a=4 ; b=1 FIG. 2: Scalar crrelator made out of the two propagators with branch point singularity. Momentum is scaled by the values of a, b , which are shown explicitly.
IV. CONCLUSION
Correlators defined as a convolution of two Green’s functions with various analytical structure, which admits con-finement, have been studied in the momentum Minkowski space. We have restricted to the choice of real propagatorswith complex conjugated singularities. The Gribov propagator, which plays important role in SU (3) Yang-Mills the-ory was considered and studied in great details. It was shown that Feynman parametrization allows the analyticalintegration over the momentum in all studied cases, providing unique and ultraviolet finite results in the Minkowskispace. Contrary to calculations performed in the Euclidean space [7–9], the Minkowski space correlator with Gribovpropagators remains finite and does not require renormalization. Neither of correlators satisfies Khallen-Lehmannrepresentation and in the case of Gribov propagator the integral representation copiously reproduces the Gribov formin its continuous version. In other cases, the analytical structure is more complicated and there is no easy way toclassify all branch points and related cuts. The correlator made out of generalized Gribov propagators exhibits thereal quasithreshold as well. The later strikes itself as a sharp cusp in the graph of Π( p ) and is located at the usualpoint p T = 4 a ( p T = 0 for Gribov).The ramifications of the Minkowski space calculations can only be explored in conjunction with quantum Chro-modynamics in strong coupling region. In this respect many steps remain to be finished and even the Gribov formis not an ultimate approximation of QCD propagators. Meanwhile, there are the first attempts [18, 33] to solveSchwinger-Dyson and Bethe-Salpeter equations directly in the momentum Minkowski space, the presented studyshow the pertinent existence of discontinuities (cuts) when passing the real axis of momenta in Minkowski space. Itobviously can make the direct Minkowskian momentum space integration of SDEs difficult, cumbersome, if not evenimpossible in some cases. In this respect, the integral representation derived here can be very useful when looking forhadronic observables in the framework of SDEs. [1] V. N. Gribov, Nucl. Phys. B 139 , 1 (1978).[2] M. Stingl, Phys. Rev.
D 34 , 3863 (1986).[3] D. Zwanziger, Nucl. Phys.
B 323 , 513 (1989).[4] P. MARIS and H.A. HOLTIES, Int. J. Mod. Phys.
A 07 , 5369 (1992).[5] D. Zwanziger, Nucl. Phys.
B 399 , 477 (1993).[6] M. Stingl, Z. Phys.
A 353 ,116 (1996). [7] D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel and H. Verschelde, Phys. Rev. D 78 , 065047 (2008).[8] D. Dudal and M. S. Guimaraes; Phys. Rev.
D 83 , 045013.[9] L. Baulieu, D. Dudal, M. S. Guimaraes, M. Q. Huber, S. P. Sorella, N. Vandersickel, and D. Zwanziger, Phys. Rev.
D 82 ,025021 (2010).[10] M. S. Bhagwat, M. A. Pichowsky, and P. C. Tandy, Phys. Rev.
D 67 , 054019 (2003).[11] S. M. Dorkin, L. P. Kaptari, B. Kampfer, arXiv:1412.3345.[12] S. M. Dorkin, L. P. Kaptari, T. Hilger, B. Kampfer, Phys. Rev.
C 89 , 034005 (2014).[13] T. Hilger, C. Popovici, M. Gomez-Rocha and A. Krassnig, arXiv:1409.3205.[14] C. S. Fischer, S. Kubrak, R. Williams, arXiv:1406.4370.[15] S.J. Stainsby, R.T. Cahill, Phys. Lett.
A146 , 467 (1990).[16] C. D. Roberts, Prog. Part. Nucl. Phys. ,50 (2008).[17] M. Praszalowicz and A. Rostworowski, Phys. Rev. D , 074003 (2001).[18] V. Sauli, submitted to FBS, arXiv:1505.03778.[19] D. Dudal, O. Oliveira and P. J. Silva, Phys. Rev. D 89
G 39 (2012).[22] V. Sauli, Few-Body Systems 45 .[23] V.Sauli, J.Adam, P. Bicudo, Phys.Rev. D75 (2007).[24] D. E. Kharzeev, Color confinement from fluctuating topology, contribution to the Gribov-85 Memorial volume,arXiv:1509.00465.[25] A. Pereira , A non-perturbative BRST symmetry for the Gribov-Zwanziger action, talk at ACHT2015, Graz 2015.[26] M. A. L. Capri et all, Phys. Rev. D , 065019 (2016).[27] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147 , 385 (1979).[28] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys.