Confinement within the use of Minkowski space integral representation
aa r X i v : . [ h e p - l a t ] N ov Confinement within the use of Minkowski integral representation
V. ˇSauli Department of Theoretical Physics, NPI Rez near Prague, Czech Academy of Sciences
Confinement defined as the absence of certain (colored in QCD) particle like excitations in S-matrix is investigated in the nonperturbative framework of Schwinger-Dyson equations solved inMinkowski space. We revise the method based on utilization of generalized spectral representationand improve existing techniques, such that the method turns to be particularly suited for strongcoupling quantum field theories with confinement. The method is applied to strong quenched QEDand to SU (3) Yang-Mills theory in 3+1 dimensions. Result for the gluonic spectral function has beenobtained, for which purpose the equation for the gluon propagator has been solved in the gaugeinvariant manner. The gluon propagator, instead of having a real pole, has an unusual infraredstructure that consists from two oppositely signed Cauchy resonances. PACS numbers:
I. INTRODUCTION
Confinement and chiral symmetry breaking are essential features of QCD, they govern the low energy processes aswell as they dominate late hadronization phase of high energy collisions. The confinement is phenomena that we donot observe the excitations of quark and gluon fields as a particle in isolation, while the chiral symmetry phenomena isassociated with explanation of mass hierarchy of hadrons. These phenomena are intuitively well understood, howeverthey are not yet satisfactorily encoded in a known framework which might be useful for calculations of hadronicobservable e.g. continuous production form factors. Presented paper is devoted to the first unavoidable step: to theaccurate evaluation of Green’s function in strongly coupled QFT in the entire region of momentum Minkowski space.QCD is gauge quantum field theory and therefore it requires a particular gauge fixing to be manageable in practice.Owing to a given gauge a particular confinement pictures have emerged in various studies. The Gribov-Zwanzigerpicture of confinement [1],[2] is likely established in Coulomb gauge [3], while not confirmed in the class of lineargauges (e.g. in Landau gauge), where the lattice simulation of ghost propagator does not show enhancement- therequired condition for the Gribov-Zwanziger scenario. Other scenario of confinement- the vortex condensation [4–9] with associated condensation of magnetic monopoles are detected on lattices in special gauges [10],[11],[12], e.g.in maximal Abelian gauge [13]. Thus a different manifestation and realizations of confinement exist in variousgauges, while they do not show up in other gauges. Actually, reflecting the difficulty of calculation performance, anoverwhelming majority of canonical Schwinger-Dyson equations (SDEs) studies exist in the Landau gauge- a singleexample ( ξ = 0) out of a larger class of linear gauges ξ = 0.Provided the structure of Green’s functions is known in the entire region of Minkowski space, the analytical prop-erties of correlator associated with confined modes could clarify the topic of confinement in a great extent. Followingthe intuition one can expect that the correlator, which does not show up any peak or cusp in the spectrum can beascribed to confined objects. While those peaks associated with particle like excitations and which stay pronouncedin the S-matrix, they belong to standard observable particle excitations: freely moving particles or familiar hadronicresonances.What might be an expected structure of confining modes is largely speculated in waste amount of literature andhere we concern only those, that does not contradict standard analyticity and Unitarity. In case of QCD, the spectraldecomposition or integral representation methods follow such a strategy [14–16]. The spectral decomposition of theLandau gauge lattice propagators have been questioned in papers [17, 18]. The resulting spectral function of latticegluon propagator was found to be a smooth function and the absence of the real pole in the propagator can be takenas a striking evidence of confinement.In order to shed a more light on this subject, we are going to concern several correlators and ask what is therelevant ”spectral” structure of associated presumably confined objects in given approximation. To achieve desiredanalytical properties (i.e. spectrality) we develop relatively powerful technique, which rely on two steps analyticalcontinuation of DSEs in between Minkowski and Euclidean space. The first step requires converting the SDEs systemin momentum space into the equation for spectral functions by method analytical continuation. Although this is veryconventional and certainly not new step when solving SDEs, [14–16, 19–24] it, up to the exception, did not provideconvergent system of equations for strongly coupled theories with confinement. To achieve the later, we follow thetrick introduced in [15] and perform the finite subtraction at the timelike scale of the momentum and perform thesearch of the solution, which is consistent with the spectrality constrain. It not only works, but it automaticallyensures that obtained solution must be equivalent to the solution of the Euclidean space counterpartner DSE.In the next Sections II we explain how to get spectral function within the simple model: the quenched QED in3+1 dimensions. Using the L renormalization scheme [25] we also marry the phenomena of mass generation andconfinement with spectral representation of the electron propagator in classically massless case m = 0, keeping allrenormalization constant finite and equal to one.The method is also applied to the Background Field Method- Pinch Technique (BFM-PT) DSE in Landau gauge,which equation has been already studied in the paper [16] with limited numerical outputs. In the Section IV we revisenumeric used in [16] and offer much precise solution there. II. STRONG QUENCHED QED IN 3+1 DIMENSIONS
Strong coupling QED3+1 represents the truncation of DSEs system where the technique and associated amount ofcalculations is reduced to minimum. For classically massless fermions it requires the introduction of scale by hand.Using a hard cut-off it, the model was already studied in seventies [26], where the absence of the fermion pole wasnoticed due to the obscure grow of the electron dynamical mass function at large timelike q . Notably, the model canbe interpreted as an extreme case of the so called Walking Technicolor theory, where cut-off plays the role of the scalewhere asymptotic freedom start to emerge.In the chiral limit, the classical Lagrangian mass is taken zero m = 0 and if a nontrivial mass function is generated,we use to use say it is generated dynamically through quantum loops. Such effect is most easily studied within theuse of Euclidean metric, whilst to gain further information about confinement property of generated modes is lessstraightforward. The known critical behavior associated with Miransky scaling M (0) ≃ Λ exp ( α − π/ − / persistsin more sophisticated approximations [28] and is associated with chiral symmetry breaking- generation of the mass.However even in classically massive case, the system can be characterized by the critical coupling α c ≃ π/
3. Forcouplings satisfying inequality α < α c one gets non-confining spectral solution for the fermion propagator [22]. Suchspectral function has a delta function associated with free propagation delta function plus the continuum part, whichstart to be nonzero at the threshold (identical to pole mass). We argue such solution does not exist for super-criticalcouplings α > α c , for which case we provide another -confining- spectral solution for the first time. Avoiding hardcutoff is a net but necessary change which is required in our study, albeit in a very strict sense we are facing differenttheories since regularization plays it own role in quenched QED. For regularization we need to use more sophisticatedregularization/renormalization scheme such us dimensional regularization [29], BPHZ scheme [30] or an L-operationscheme [25], which introduce the mass scale in a slightly different manner.Conventional renormalization schemes (dimensional or BPHZ one, to name some out of the above list) togetherwith simultaneous use of the spectral representation for the fermion propagator S : S ( k ) = Z ∞ da pρ v ( a ) + ρ s ( a ) p − a + iǫ (2.1)always forces us to renormalize the mass. The reason is that the Euclidean space loop integral turns to be divergent,irrespective of absence or presence of mass term in the Lagrangian. Thus to cover m = 0 case, we adopt L- regular-ization scheme, which does not require subtraction and allows to match spectral representation with dynamical massgeneration in the chiral limit.Our convention for Minkowski metric tensor reads: g µν = diag (+1 , − , − , − E when we want to specify the Euclidean momentum, i.e. for instance q E = − q for some spacelike momentum q .The unrenormalized quenched ladder-rainbow approximated fermion DSE in the Landau gauge reads: S − = p − m o − Σ( p )Σ( p ) = ie Z d k (2 π ) γ µ S ( k ) γ ν P µν ( k − p )( k − p ) + iǫ , (2.2)where e is the fermion charge, P ( z ) is transverse projector. Two functions or distributions ρ v,s in the Eq. (2.1) areenough to complete two scalar functions S v,s or A, B alternatively. In our notation the are defined as S ( p ) = S v ( p ) p + S s ( p ) = 1 pA ( p ) − B ( p ) . (2.3) Dimensional renormalization scheme
Substituting the representation (2.1) into the DSE, swapping the order of integrations and making the integrationover the momentum one gets B ( p ) = m + 3 e (4 π ) Z ∞ daρ s ( a ) Z dt (cid:18) ln (cid:20) p t − a + iǫµ t (cid:21) + C (cid:19) = m ( µ t ) + 3 e (4 π ) Z ∞ daρ s ( a ) Z dtln (cid:20) p t − a + iǫµ t (cid:21) ,A = 1 (2.4)where µ t is t’Hooft renormalization scale of MS bare dimensional renormalization scheme, which has been used.In the next step we will add the zero of the following form0 = B ( ζ ) − m ( µ t ) − e (4 π ) Z ∞ daρ s ( a ) Z dtln (cid:20) ζt − a + iǫµ t (cid:21) (2.5)to the rhs. of (2.4), i.e. we subratct the equation with itself evaluated at the scale ζ . We thus get B ( p ) = B ( ζ ) + 3 e (4 π ) Z ∞ daρ s ( a ) Z dtln (cid:20) p t − a + iǫζt − a + iǫ (cid:21) . (2.6)Note also, the function M = B/A is renormalization scheme invariant here as well as in other gauges.
DSE in L- renormalization scheme
The momentum integral stays divergent in dimensional regularization prescription and the singular pole term1 /ǫ d = 1 / ( d −
4) in the constant C is absorbed into the renormalized mass, such that m ( µ t ) = m − e (4 π ) Z daρ s ( a )[1 /ǫ d + γ E + 4 π ] (2.7)In order to maintain renormalizability, mass term is inevitably presented, otherwise spectral representation couldnot be used in this scheme.Recall, the derivation of the equation (2.4) is crucially based on the change of ordering of the several integrations. Inorder to include m = 0 case as well, one unavoidably needs to keep the trace of selfenergy finite. For this purpose it ismore convenient to use regularization called L − operation ([25]), which is based on further exploatation of Feynmanparamaterization, which does not require subtraction of infinities at least at one loop level. Applying this in our case,we get B ( p ) = m + 3 e (4 π ) Z ∞ daρ s ( a ) Z dtln (cid:20) p t − a + iǫµ F (cid:21) ,A = 1 , (2.8)The second term on the right side represents regularized self-energy in L operation scheme and since the expressionis finite, the main outcomes is that to take m = 0 is formally possible.Subtracting the Eq. (2.8) at some timelike scale ζ , we get the Eq. (2.6) again. In this tricky way, we are ableto include massless case as well, however stress here, that what make difference between dynamical mass generationin classically massless and in classically massive theory, is the property of solution. In our case, that the solution of(2.6) should provide the solution of Eq. (2.8) as well. It also means there must exist unique parameter µ F , whichcomplies with the existence of assumed spectral representation (2.1) in case of chiral limit.Since the quenched QED in 3+1D is not an asymptotically free theory, there are other subtleties. Thus a hardcutoff was introduced in order to regularize momentum space integral in the Euclidean space theory [26]. It makesnonperturbative solutions obtained within the spectral technique herein, very hardly comparable with the Euclideansolution in details. Quenched LRA QED in 3+1 dimensions split into different models according to what regularizationmethod is used. Let us anticipate at this place, that we do not see meaningfull numerical solution in the chiral limitin quenched QED. All observed solutions are in fact indistinguishable from the explicitly massive case. However, asknown from QCD studies, we expect that the entire dynamically mass generation can be studied within proposedmethod in asymptotically free theories. The method of solution
For the spacelike p and the negative ζ we can in principle drop out Feynman iǫ and solve the equation in thespacelike domain of Minkowski space. Note plainly, that for this purpose one should know the spectral function ρ s in advance. There is no working method (at least known to the author), which would allow to extract spectralfunction ρ v,s by solving DSE in the spacelike region alone. However it turns that the Eq. (2.6) is quite easily solvablenumerically.In order to determine the function ρ it is advantageous to consider the Eq. (2.4) at timelike scale, where the runningmass B is complex valued function. The analytical continuation of the DSE (2.6) for p > ℜ B ( p ) = ℜ B ( ζ ) + 3 e (4 π ) Z ∞ daρ s ( a ) Z dtln (cid:12)(cid:12)(cid:12)(cid:12) p t − aζt − a (cid:12)(cid:12)(cid:12)(cid:12) , ℑ B ( p ) = − e π Z ∞ daρ s ( a ) (cid:20) (1 − ap ) θ ( p − a ) − (1 − aζ ) θ ( ζ − a ) (cid:21) , (2.9)withe θ ( x ) is usual Heaviside step function, which for x > θ ( x ) = 1 and θ ( x ) is zero otherwise.Keeping the equation for the dynamical mass function B in hand, one can reconstruct the propagator S . Comparingthe real and the imaginary parts of the Eq. (2.6) and the Eq.(2.1) one can write down the following complementaryequation: ρ s ( s ) = − π ℑ B ( s ) R D ( s ) + ℜ B ( s ) I D ( s ) R D ( s ) + I D ( s ) (2.10)where s = p > R D and I D stand for the square of the real and the imaginarypart of the function sA ( s ) − B ( s ), i.e. R D ( s ) = s [ ℜ A ( s )] − s ( ℑ A ( s )) − [ ℜ B ( s )] + ( ℑ B ( s )) ,I D ( s ) = 2 s ℜ A ( s ) ℑ A ( s ) + 2 ℜ B ( s ) ℑ B ( s ) , (2.11)where we keep A non constant for more general purpose (( A = 1) in the approximation employed here).To get the solution, we start with some ad hoc initial guess for the constant ℑ B ( ζ ) and trial spectral function ρ s ( s )and substitute it into the Eq. (2.9). Then three equations (2.9) and (2.10) have been solved numerically by methodof iterations. Very importantly, the method works equally well in case of QCD quark propagator [15, 23].In this way the obtained function ρ s is still not what we are looking for. The system is ill constrained by ourrandom choice of the complex phase φ = arg ℑ B ( ζ ) / ℜ B ( ζ ) and only thanks to high nonlinearity it provides precisenumerical solution with the artbitrarily high numerical accuracy. Recall, there must exist a single value ℑ B ( ζ ) for afixed mass ℜ ( ζ ) at fixed scale ζ . To get rid of the problem we fix ℜ B ( ζ ) and repeat iteration procedure describedabove for a new value ℑ B ( ζ ) and look at the quality of equality: L ( s ) = R ( s ) = 14 ℜ T rS ( s ) = ℜ S s ( s ) (2.12)where L ( s ) = P. Z ∞ da ρ s ( a ) p − aR ( s ) = −ℑ B ( s ) I D ( s ) + ℜ B ( s ) R D ( s ) R D ( s ) + I D ( s ) (2.13)for a given choice of the phase φ ( ζ ). Similar constrains can be written for the function S v (see Appendix), howeverin our approximation S v is completely, albeit non-linearly, determined by the function S s itself.In the Fig. (1) solid line represents result for the propagator which corresponds to the ratio ℑ B (0 . / ℜ B (0 .
1) =0 . ± . B ( ζ ) is our choice, while the imaginary part ℑ B ( ζ = 1) was the subject of thenumerical scan. In order to visualize our numerical search we plot two lines corresponding to L and R as defined inby (2.13). Lines are labeled by two numbers representing values ℜ B ( ζ ) , ℑ B ( ζ ). More they can be distinguished, morethe equality Eq. (2.12) is incompleted.The absence of the real pole in the propagator is more then obvious and instead of physical threshold the propagatorhas the zero branch point. The imaginary part of propagators for super-critical coupling does not correspond to thedecay width of particle mode, but rather it tells us how much fast is the creation-reabsorption process of associatedquantum field. Confinement is there due to the abrupt generation of absorptive part of selfenergy in infrared domainof the timelike momenta. It is worthwhile to mention that the generation of zero anomalous threshold is the old s/ ξ -2-1.5-1-0.500.51 s R e S s ( s ) (1; 0.1)(1; 0.2)(1,0.325)(1; 0.4)(1; 0.6)(1; 1) FIG. 1: sS ( s ) function constructed from L and R functions. Each type of line corresponds to a given choice of the phase, theare labeled by real and imaginary part of B ( ζ ) /ζ . The exception is the best matching case, where “L” is dotted , while “R”solution is depicted by the solid line. conjecture of Schwinger [27] made for 1+1 QED, which has emerged several years before QCD has been accepted asa correct theory of hadrons.In the Fig. 2 we show the spectral function of the fermion propagator. In non-confining quantum field theory,the Osterwalder-Schrader axiom of reflection positivity [31] is equivalent to the positive definiteness of the norm inHilbert space of the corresponding Quantum Field Theory. Violation of reflection positivity is often regarded as amanifestation of confinement. Obviously the property of reflection positivity is not lost in our case, however thefermion turns to be a short living excitation according to suggestion made (albeit for the photon at that time) by J.Schwinger half century ago. Violation of reflection positivity turns out to be a weak criterion for confinement in themodel presented herein.We do not show the evolution with the coupling however within decreasing coupling the shape gradually rises elbowsand the on shell pole rises for subcritical value of couplings. For small couplings then one needs to determine the poleposition and its residuum as done for theory in even dimensions [22],[21] as well as for theory with odd number ofspacetime dimensions [32]. We also do not go beyond quenched approximation due to the Abelian character of theinteraction, however we can expect that when the photon polarization function is taken into account, the quantitativechanges can be quite dramatic.At last but not at least, as we have already anticipated, we do not get any truly conformal solution. The renormal-ization scale makes the theory effectively massive and in order to eliminate scale µ F completely one would need thecondition R ρ ( a ) = 0 is fullfiled. The later we have not achieved herein. III. YANG-MILLS THEORY, CONFINEMENT OF GLUONS
The analytical structure of the gluon propagator on the entire domain of momenta is largely unknown. Theultraviolet behavior of the gluon propagator should be governed by perturbation theory, while at low q it is partiallyknown only at the spacelike domain of momenta, where it is accessible by lattice theory and other Euclidean spacenonperturbative methods.For our purpose we will use a simple truncation of the PT-BFM [33] gluon Dyson-Schwinger equation which hasbeen already studied in [34, 35] in the Euclidean space and in the paper [16] in the Minkowski space. Remind that ξ ρ FIG. 2: Spectral function of strongly interacting fermion in units of renormalized mass ζ . the Background Field method (BFM) fields and Pinch Technique (PT) are originally perturbative constructs in whichGreen’s function satisfy Ward identities rather then Slavnov-Taylor identities. Guiding by the principles of PinchTechnique , the gauge invariant truncation of Yang-Mills DSEs was constructed, providing nonperturbative gaugeinvariant Green’s functions (in any covariant gauge[36]). Apart of lattice development, there exists a certain progress[44, 46–50] in developments of SDEs in Yang-Mills theory in various gauges. However, since the most of studies areperformed in Landau gauge we have chosen this gauge as well.Here, using the method of subtractions in the timelike scale we revise numeric used in [16] and offer much precisesolution. We show that instead of two sharp poles as naively suggested in [16], we get the solution characterized bya smooth spectral function with few nodes and several modes of opposite signs.To get infrared finite gluon propagator, the gluon propagator must loose its perturbative 1 /q pole through theSchwinger mechanism in Yang-Mills theory [51]. It underlies on the assumption of the form of transverse piece of thegluon vertex in a way it leaves polarization tensor gauge invariant (transverse to its momentum). In this paper wesimply use the derived equation in [34], where Schwinger mechanism is employed through the simple Anstaz for theunproper (two leg dressed) three gluon PT-BFM dressed vertex d ( k )˜Γ ναβ ( k, q ) d ( k + q ) = Z dωρ ( ω ) 1 k − ω + iǫ Γ Lναβ k + q ) − ω + iǫ + d ( k )˜Γ Tναβ d ( k + q ) , (3.1)where Γ Lναβ satisfies tree level WTI and d is scalar function related to the all order PT-BFM gluon propagator whichin Landau gauge reads G µν = (cid:20) − g µν + k µ k ν k (cid:21) d ( k ) (3.2)and satisfies generalized Lehman representation d ( k ) = Z ∞ dω ρ ( ω ) k − ω + iǫ (3.3)and ˜Γ T is the rest of the three gluon unproper vertex which is not specified by gauge invariance. In the expression(3.3) we assume the branch point is located in the beginning of complex plane, albeit not generated by poles, it is inits usual “non-anomalous “ position.The essential feature of the vertex Γ ναβ is that apart the structure dictated by WTI it also includes 1 /q pole termwhich gives rise to infrared finite solution. For this purpose the following form d ( k )˜Γ ναβT ( k, q ) d ( k + q ) = Z dωρ ( ω ) 1 k − ω + iǫ Γ Tναβ k + q ) − ω + iǫ Γ ναβT ( k, q ) = c [(2 k + q ) ν + q ν q ( − k.q − q )] g αβ + [ c + c q (( k + q ) + k ))]( q β g να − q α g νβ ) , (3.4)has been proposed in [34]. This vertex is transverse in respect to q ( q. Γ = 0) and it respects Bose symmetry to twoquantum legs interchange.After the renormalization, it leads to the following form of SDE in Euclidean space: d − E ( q E ) = q E ( K + bg Z q E / dz s − zq E d E ( z ) ) + γbg Z q E / dzz s − zq E d E ( z ) + d − E (0) , (3.5)where the second line appears due to the Ansatz for the gluon vertex (3.4) and K is the renormalization constant.Thus the strength of the dynamical mass generation is triggered through the adopted coupling constants c , c , c which is fully equivalent to the introduction of (in principle arbitrary) constant γ and infrared value d − E (0) (forcompleteness recall that in the paper [34] d (0) has been calculated since one of c i was fixed by hand).Analytical continuation q E = − q = − s → q , s = q > s reads d − ( s ) = s ( K + bg Z s/ dz r − zs d ( z ) ) + γbg Z s/ dzz r − zs d ( z ) + d − (0) + iǫ (3.6)where we use standard convention d E ( q E ) = − d ( s ) for s = q < d − (0) should take a negative value to prevent tachyonic solution.Subtracting the equation once again at the timelike fixed scale ζ one gets: d − ( s ) = s ( K + bg Z s/ dz r − zs d ( z ) ) − ( s → ζ )+ γbg Z s/ dzz r − zs d ( z ) − ( s → ζ ) − d − ( ζ ) . (3.7)Now, the function d is assumed to be complex for all s >
0, while it stays real for negative s .Like in previous study of quenched QED, we should mention that the DSE equation (3.7) has infinity many stablesolutions that do not match the Eq. (3.8) and the numerical scan of the phase of the function d ( ζ ) must be performed.Thus we arbitrarily fix the real part Red − ( ζ ) at some fixed scale ζ and then search for the value of ℑ d − ( ζ ) tillassumed integral representation, e.g. the relation ℜ d ( k ) = − π P. Z dω ℑ d ( ω ) k − ω , (3.8)will hold.Like in previous study of quenched QED, we should mention that the DSE equation (3.7) has infinity many stablesolutions that do not match the Eq. (3.8) and the numerical scan of the phase of the function d ( ζ ) must be performed.Recall, that while detailed values of renormalization constant K and d − (0) are not crucial for the subject ofconfinement, they should match with other quantities calculated in given specific scheme [45]. Notably, the gluonicgap Eq. (3.5) is derivable in various symmetry preserving renormalization scheme, to name a few: BPHZ MOMscheme, the dimensional renormalization as well as it can be obtained within the L operational scheme [25]. All theseschemes necessarily lead to the Eq. (3.7). Contrary to this, an unintegrated form of the Eq. (3.5) does not necessaryensure the same result when one uses regularization technique, which does not respect symmetry of the theory andfake unphysical solutions can appear.The solution of the gluon propagator is shown in the figures 3 and 4, where only the solution fitting the Eq. (3.8) isshown. The shape of the gluon propagator at the timelike region is obviously something that we are not experienced,the real part is represented by two broad peaks which have simultaneously opposite signs.To get presented solution we set ℜ ζd ( ζ ) = 1 . g = 15 . γ = 1 /
20. Wegot ℑ ζd ( ζ ) = 0 .
140 after the scan. Comparison of gluonic SDE with RGE improved SDE is shown as well (interestedreader can find the meaning in [16] ). Contrary to achievements in [16], now the access to numerical solution in thewhole complex plain is an easy and straightforward and the reader can find our numerical codes at the author’s webpage.At ultraviolet the studied gluon propagator vanishes as 1 / ( sln γ )( s ) with γ ≃ .
5. This behavior should lead tosuperconvergent relation sum rule [37],[38],[39]: I = Z ∞ ρ ( s ) ds = 0 , (3.9)which turns to be satisfied within reasonable accuracy I ≃ . ρ ( s ) = sγ ( s − s o κ ) + ( s Γ) (3.10)was suggested as an artificial mathematical model for the spectral function of the photon in the Schwinger model[27]. Interestingly, the spectral gluon function can be quite accurately parametrized as a difference of two near shortliving excitations. Comparing to Schwinger model, there are unavoidable further but subtle changes since SU (3)gluodynamics should comply with asymptotic freedom, e.g. with the spectral sum rule (3.9), thus instead of using(3.10), it turns that more suited fit can be constructed from the sum of several Cauchy distributions ρ ( s ) = Σ i [ R ( s/ζ ) λ ( s − s o ) + (Γ) ] i , (3.11)where the exponent λ = 0 for the positive modes of the function ρ (negative modes in figures), while for the negativemode the exponent λ = 3 / q ). Since we are comparingtwo theoretically rather different objects, anyone of aforementioned lattice data fits is suited for purpose of our visualcomparison. The differences are obviously large, quantitatively comparable to difference between gluon propagatorscalculated in the Feynman and the Landau gauge [44]. IV. SHORT REMARK ON THE QUARK PROPAGATOR
Following similar method described in the Sect. 2 the ladder-rainbow approximation (LRA) for the quark SDEhas been already solved in the paper [15]. The obtained quark propagator has an anomalous threshold located atbeginning of complex plane of square of momenta. Also the confinement manifest itself as missing real pole in thequark propagator. On the other hand a simple phenomenological kernel used in [15] did not follow from QCD directly,and it is also failing in detailed description of anomalous correction of electromagnetic form factors.To this point, let us mention that the LRA within a naive use of the lattice landau gauge gluon propagator does nothave enough strength to provide correct amount of chiral symmetry breaking. It is thus very likely that the kernel ofLRA must mimic also the contribution from neglected crossed boxes as well as there should be involved changes dueto the quark-gluon vertex (effectively but purely presented in the LRA).In our case the scale invariant quantity sd ( s ) is several times suppressed when comparing to lattice landau gaugegluon propagator. Not surprisingly the LRA does not provide a known slope of the quark dynamical mass function,neither it can provide correct pion observable. The best understanding of this inefficiency is obvious from the modernversion of Goldberger-Treiman-like relation [52],[53], which reads f π Γ A (0 , k ) = B ( k ) (4.1)where Γ A is the leading piece of the pion’s Bethe-Salpeter amplitude and B is the quark dynamical mass times thequark renormalization function. ζ -30-20-100102030 s d ( s ) Re s d(s)Im s d(s)Re sd(s) rgDSE
FIG. 3: DSE solution for gluon propagator plotted at the timelike domain of momenta (in units of ζ ). The real part of thesolution for the so called Renormgroup improved SDE (see [16] for the meaning) is added for comparison. ]00.511.52 s d ( s ) DSE m=0.8 GeVlattice DSE m=0.36 GeV
FIG. 4: Solutions of gluonic DSE compared to the lattice fit in the Landau gauge. Dot-dashed line states for rescaled solutionwhich corresponds to 0.36 GeV position of the first peak in the gluon propagator.
V. CONCLUSION
We have applied the method of subtractions of SDEs at the timelike scale of momenta and get the confined solutionin Yang-Mills theory for the gluon propagator as well as we have illustrated the method in case of fermion propagatorin quenched QED. In both strong coupling models the solutions for two point correlator were obtained in the entiredomain of Minkowski space momenta. The method should be and in fact it has already been useful for evaluation ofhadronic form factors.
Appendix A: Appendix A
The second constrains which should be checked beyond ladder-rainbow approximation can be derived by makingthe trace of p -projected fermion propagator, where L v ( s ) = R v ( s ) = 14 p ℜ 6 pT rS ( s ) = ℜ S v ( s ) . (A1)with the left and right hand sides defined as L v ( s ) = P. Z ∞ da ρ v ( a ) s − aR v ( s ) = −ℜ A ( s ) R D ( s ) + ℑ A ( s ) I D ( s ) R D ( s ) + I D ( s ) (A2)where symbol P. stands for principal value integration. Appendix B: Appendix B
We provide the details of the numerical fit in this Appendix. The fit was chosen as following ρ ( s ) = Σ i =1 , R i ( s − s o,i ) + ( w i ) + Σ i =2 , R i ( s/ζ ) / ( s − s o,i ) + ( w i ) , (B1)with the numerical values of parameters listed in the following table: i R/ζ s o /ζ w/ζ Proposed fit is compared to the obtained numerical results in the Fig. 5, note that the curve labeled as “Re fit”was obtained obtained with the spectral function given by (B1) and used in the Eq. (3.3). The large “widths” as wellas the small value of residuum of the fourth term makes this term already negligible. [1] V. Gribov, Nucl. Phys. B139, 1 (1978). ζ -30-20-100102030 s d ( s ) Re dseIm dseFitRe fit
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