Infrared self-consistent solutions of bispinor QED 3 ∗ Tomasz Radoz˙ycki Faculty of Mathematics and Natural Sciences, College of Sciences, Cardinal Stefan Wyszyn´ski University, Woýcickiego 1/3, 01-938 Warsaw, Poland Quantum electrodynamics in three dimensions in the bispinor formulation is considered. It is shown that the Dyson-Schwinger equations for fermion and boson propagators may be self- consistently solved in the infrared domain if on uses the Salam’s vertex function. The parameters definingthebehavior of thepropagators are found numerically for different valuesof coupling constant andgauge parameter. For weak coupling theapproximated analytical solutions are obtained. The renormalized gauge boson propagator (transverse part) is shown in the infrared domain to be practically gauge independent. 3 PACSnumbers: 11.10.Kk,11.15.Tk 1 0 2 I. INTRODUCTION forthevertexfunction,whichischosenmostofteninthe form satisfying Ward-Takahashi identity. Naturally this n a identity fixes only longitudinalpartofthe vertexleaving QED in three space-time dimensions has become a J the transverse part a subject of further discussion and testing laboratory for certain nonperturbative aspects 5 improvements [15, 23–27]. of QFT like chiral symmetry breaking [1–10], bound 2 In our previous paper [28] we applied a method elabo- ] satpaptreosx[i1m1a–t1i3o]nosrchceomnfiesn.emDuenetto[8t,h1e4d–i1m8e]nasniodnfaolrityvaorfiothues ratedearlierinQED4 [29]toQED3 withtwo-component h fermions. It consists on the following five steps: couplingconstant(as√mass)itisasuperrenormalizable t - theorysoitisfreeofinfiniterenormalizationambiguities 1. a certain infrared form of two basic propagators, p e typical for four-dimensional theory. S(p)andDµν(k)(suggestedbyperturbativecalcu- h Itcanbeformulatedintwoinequivalentversions: with lationsorothermethods)asdependentonacouple [ two- and four-component fermions [19, 20]. The proper- of unknown parameters is assumed, ties of the theory are different in these two cases. To 1 v the investigation of both versions much attention has 2. the fermion propagator is represented in the spec- 5 been payed over the last twenty years. The work has tral form with one known spectral density ρ(M), 7 been especially concentrated on nonperturbative solu- whichisingeneralpossibleinthe infrareddomain, 1 tions of Dyson-Schwinger (DS) equations with different 6 3. Salam’s form of the vertex function [30], together approximations incorporated in the theory as quenched . with S(p) and Dµν(k), is substituted into the first 1 approximation, rainbow approximation, 1/N expansion 0 and various models of vertex function [1–4, 6, 10]. Par- two of the set of DS equations, 3 ticularly often the multi-flavor theory with the limit 1 4. from these two equations the set of self-consistent N has been used, since it avoids infrared prob- : → ∞ equations for parameters is derived, v lems, which usually become troublesome in lower num- i ber of dimensions. The other important point in this X 5. the obtained equations are solved numerically or analysis has been the unpleasant gauge dependence of r analytically. a the nonperturbative results which constitute the common problem of the approximated studies based on DS This method proved to be relatively effective both in equations [6, 9, 10, 16, 21, 22]. QED , where all parameters were correctly found with- 4 As it is well known, DS equations constitute an in- outthenecessityofinfiniterenormalization,andinspinor finite set of relations involving Green’s functions which version of QED3, where obtained results stay in general form the ‘inverted ladder’ type structure: the n-point agreement with other works. In the present paper we functions depend on n+1-point ones and so on up to would like to extend its application to four-component infinity. This set cannot be solved without the trunca- QED3, without Chern-Simons term, i.e. when gauge tionofsuchahierarchy. Butevensuchtruncation,which bosons (‘photons’) remain massless in spite of interac- turns the infinite set of equations into only a couple of tion. them, leads to the system which is far from being trivial The common effect of various simplifications of DS andrequiresfurther simplifications. Usually this trunca- equations is the gauge dependence of the results (even tion is accomplished by performing certain assumption ofthe physicalobservables). The sourcesofthis undesir- able behavior are the approximations made to the propagators and to the vertex. It seems therefore valuable to test various possible approaches with regard to that ∗Electronicaddress: t.radozycki@uksw.edu.pl particularfeature andmuch work has alreadybeen done 2 in this direction (see the references above). This point where we choose for the metric tensor: lies in the scope of interest of the present work too. This paper is organizedas follows. In the next section g00 = g11 = g22 =1. (6) − − wedefinethemodelitselfandgivetheresultingsetofDS The trace of the product of an odd number of gamma equations. In sectionIII the infraredGreen’s function in matrices equals zero as in four dimensions (this was not question are formulated up to several unknown parame- the case in the spinor representation, where it was pro- ters. In section IV we substitute these Green’s functions portional to the antisymmetric tensor εµνρ). Addition- into DS equations and obtain the set of relations for the ally we have the identity: introduced parameters. In the last section we present analytical and numerical results and some conclusions. γµγνγ = γν . (7) µ − Using the Lagrangian density (1), one can derive in II. FORMULATION OF THE MODEL. the standard way – for instance through Feynman path integral – the Dyson-Schwinger equations for propaga- The model is defined through following Lagrangian tors [31]. For boson propagatorwe obtain the relation density : 1 kµkα 1kµkα L(x)= Ψ(x)(iγµ∂µ−m0−e0γµAµ(x))Ψ(x) (1) Dµν(k)= k2 −gµα+ k2 − λ k2 δαν −ie20× 1 λ (cid:18) (cid:19)(cid:20) −4Fµν(x)Fµν(x)− 2 (∂µAµ(x))2 , Trγ d3p S(p)Γ (p,p k)S(p k)Dβν(k) , × α (2π)3 β − − where λ is the gauge parameter. The quantities m and Z (cid:21) 0 (8) e denote here the bare fermion mass and the bare cou- 0 pling constant respectively. As mentioned in the Intro- which may be given the graphical form shown in Fig- duction, the latter for D = 2+1 is a quantity with the ure 1. The propagator Dµν(k) on the right hand side is dimensionality of √mass. That means that in the quan- not integrated over three-momenta and therefore it can tumtheoryhighertermsofperturbationexpansionshave fully be represented through fermion functions (and the betterultravioletmomentumdependenceinloopintegra- vertex). tions and the model is superrenormalizable. Asalreadytold,inthepresentpaperwedealwithfour- componentfermionfield,choosingthefollowingrepresen- = + tation for gamma matrices used also in four-dimensional QED: FIG.1: Dyson-Schwingerequationforthegaugebosonprop- σ 0 iσ 0 γ0 = 3 , γ1 = 1 , agator Dµν(k). Light lines represent free propagators and 0 σ 0 iσ (cid:18) − 3 (cid:19) (cid:18) − 1 (cid:19) heavyonesdressedpropagators. Thefullcirclestandsforthe iσ 0 full fermion-boson vertex. γ2 = 2 . (2) 0 iσ 2 (cid:18) − (cid:19) The DS equation for the fermion propagator does not There are two other gamma matrices, which anticom- allow for such a separation, and has the form mute with all above, and which can serve for defining chiral transformations: 1 S(p)= 1+ie2γµ (9) p m 0 × γ3 = 0 11 , γ5 = 0 i11 . (3) 6 − 0(cid:20) 11 0 i11 0 d3k (cid:18) (cid:19) (cid:18)− (cid:19) × (2π)3S(p+k)Γν(p+k,p)S(p)Dµν(k) , The fermion mass term, that was chosen in (1) in the Z (cid:21) form m ΨΨ, breaks the chiral symmetry defined by any 0 of the matrices (3). There is, however, the possibility of ′ the other choice for the mass term [3]: m0ΨτΨ, where = + i 11 0 τ = γ3,γ5 = . (4) 2 0 11 FIG. 2: Dyson-Schwinger equation for the fermion propaga- (cid:18) − (cid:19) tor. As in Figure 1, heavy lines stand for full functions and (cid:2) (cid:3) This term does not break chiral symmetry, since matrix light for free ones. γ0τ commutes with both γ3 and γ5, but it does violate parity symmetry (contrary to the choice made in (1)). Its graphical representation is shown in Figure 2. For matrices (2) we have ordinary relations: While introducing the fermion self-energy Σ(p), it can be rewritten in a simpler manner γµ,γν =gµν , trγµ =0, Tr [γµγν]=4gµν , { } Tr [γµγνγργσ]=4(gµνgρσ gµρgνσ+gµσgνρ) , (5) (p m )S(p)=1+Σ(p)S(p), (10) 0 − 6 − 3 where The free fermion propagator has the usual form: Σ(p)S(p)= (11) 1 S(0)(p)= . (17) d3k p m =ie20γµ (2π)3S(p+k)Γν(p+k,p)S(p)Dµν(k). 6 − 0 Z Due to the masslessness of the gauge boson and the ab- Higher equations are not considered in our present sence of the mass gap for emission of soft photons, the approach. The vertex function Γµ is not taken from pole at mass m should, in the full propagator, turn 0 DS equations, but postulated in the form proposed by into a branch point at p2 = m2, where m is a phys- Salam [30] and used afterwards in the so called ‘gauge ical mass. This is expected also on the basis of gen- technique’[24,25,32,33]. Theeventualextensionofour eralconsiderationsonthe analyticalstructureoffermion method on higher Green’s functions will be considered propagator [17, 35] and of confinement, which prohibits elsewhere. S(p) from having a simple pole. The numerical results, performedin 1/N expansionwith bare vertex and in euclidean space, suggesting the existence of complex sin- III. INFRARED GREEN’S FUNCTIONS gularities rather than real may as well constitute the effect of the coarse approximations made while solving The free propagator of massless vector boson has the DS equations: the approximations that are inevitable in standard form: any nonperturbative approach. Such singularities may be a signal of confinement, but are not prerequisite. In 1 kµkν 1 kµkν D(0)µν(k)= gµν + . (12) the Schwinger Model, massless electrodynamics in two k2 − k2 − λ(k2)2 (cid:18) (cid:19) space-time dimensions, which exhibits confinement of fermions, the real singularity in the infrared domain in As is well known, the interaction with fermions does the fermion propagator has been found (it has the form notchangethelongitudinalpartofDµν(k). Thisisguar- of 1/( p2)5/4) [36, 37]. anteed by the following Ward-Takahashi (WT) identity − Our infrared assumption for S(p) is then: 1 kν k Dµν(k)=k D(0)µν(k)= . (13) µ µ −λ k2 1 S(p)= . (18) (p m)(1 p2/m2)β Consequently the nonperturbative propagator may be 6 − − then written as Wewillseelaterthatitwillturnouttobeself-consistent. Z kµkν 1 kµkν Dµν(k)= 3 gµν + , (14) The values of the exponent β and of mass renormal- d(k2) − k2 − λ(k2)2 izationconstantδm=m m willbe establishedby the (cid:18) (cid:19) − 0 requirementofconsistency. The presenceofβ in denom- withcertainunknownfunctiond(k2). Intheinfrareddo- inatorimprovesthe ultravioletbehaviorofloopintegrals mainweassumethisfunctiontohaveaTaylorexpansion (we assume that 0<β <1, to be verified a posteriori). starting from k2, since for bispinor fermions the boson For our further purposes S(p) has to be written in the field remains massless [16] and we expect d(0)=0. This spectral form is not the case for massive two-component fermions, for which the topological boson mass is generated. 1 1 Taking the first two terms of this expansion, we have S(p)= dMρ(M) , (19) p m − p M Z (cid:18)6 − 6 − (cid:19) k2 d(k2)=k2 1+ , (15) κ2 with the one spectral density : (cid:18) (cid:19) whereκ2,togetherwiththerenormalizationconstantZ , 3 sin(πβ) 1 will be determined from consistency conditions. From ρ(M)= π (M m)(M2/m2 1)β the unitarity we expect the value of Z3 to satisfy the re- − − [Θ(M m) Θ( M m)], (20) striction 0< Z 1 [34]. This expectation was actually 3 × − − − − ≤ confirmedinourpreviousworksonspinorQED [28]and 3 QED [29]. sufficient to define the infrared behavior of the propaga- 4 With the above assumptions the inverse of Dµν may tor. ΘisheretheHeavisidestepfunction. Theparticular be written as form of ρ(M) was given in our previous works [28, 29]. Using ρ(M), one can write the vertex function, which D−1(k)µν = (16) we need to put into DS equations (8) and (9). As men- Z−1 1+ k2 ( k2gµν +kµkν) λkµkν . tionedinthe Introductionweuse the Salam’svertex[30] 3 κ2 − − (with slight and obvious modification resulting from the (cid:18) (cid:19) 4 form(19)ofS(p),whichinourcasecontainstwoterms): M in (24) with the use of (20), similarly as it was done in [28], we obtain: S(p+k) Γµ(p+k,p)S(p) e2Γ(β+1/2) 1 1 Πµν(k)= 0 (27) = dMρ(M) γµ π3/2Γ(β+1)m × p+ k m p m " Z 6 6 − 6 − 1 x(1 x) 1 γµ 1 . (21) × −k2gµν +kµkν dx(1 k2/m2x−(1 x))β+1/2 . −6p+6k−M 6p−M# (cid:0) (cid:1)Z0 − − ItautomaticallyguaranteesthecompliancewiththeWT The integral over Feynman parameter x leads to the identity: hypergeometric (Gauss) function kµS(p+k)Γµ(p+k,p)S(p)=S(p) S(p+k). (22) 1 x(1 x) − dx − = (1 yx(1 x))β+1/2 Z0 − − 1 IV. SELF-CONSISTENT INFRARED F (2,β+1/2;5/2;y/4), (28) 2 1 EQUATIONS 6 and the DS equation (23) may be given the form A. Gauge boson propagator D−1(k)µν = λkµkν +( k2gµν +kµkν) (29) − − × Inserting (19) and (21) into the right hand side of (8) e2Γ(β+1/2) 1+ 0 F (2,β+1/2;5/2;k2/4m2) . we obtain × 6π3/2Γ(β+1)m 2 1 (cid:20) (cid:21) Afterthesubstitutionoftheexpression(16)fortheleft D−1(k)µν =−k2gµν +kµkν −λkµkν +ie20Trγµ× hand side and cancellation of the tensor structures, we d3p 1 1 areleftwiththescalarequation,forwhichwerequirethe dMρ(M) γν × (2π)3 p m+iε p k m+iε adjustment the first two terms of the Taylor expansion Z Z (cid:16)6 − 6 −6 − in k2. In that way we get two equations for Z and κ2: 1 1 3 γν , (23) −6p−M +iε 6p−6k−M +iε(cid:17) Z−1 = 1+ e20Γ(β+1/2) , (30) where we have rewritten this equation for inverse propa- 3 6π3/2mΓ(β+1) gator,whichiseasiertohandle,becauseinthiscaseDµν Z−1 1 = e20Γ(β+3/2) , (31) decouples from other functions. 3 κ2 30π3/2m3Γ(β+1) The vacuumpolarizationtensormaybe definedasthe spectral integral whereweexpandedtheGaussfunctionforsmallmomen- tum according to the formula: Πµν(k)= dMρ(M)(Πµν(k) Πµν(k)) . (24) m − M ab Z 2F1(a,b;c;z) 1+ z+ (z2), ≈ c O where Πµν(k) denotes the usual perturbative tensor: m and Γ is the Euler function. These are two of the set of Πµmν(k)= (25) four equations for parameters of the model, to be solved d3p 1 1 in section V. ie2Trγµ γν . 0 (2π)3 p m+iε p k m+iε Z 6 − 6 −6 − It may be evaluated in the standard way with the use B. Fermion propagator of identities (5), and for instance by performing Wicks rotation and introducing Feynman parameters. Passing Togetothertwoequationsforparameters,weput(19), back to Minkowski space, we have (21)and(14)togetherwith(15)intotheDSequation(9). We obtain, after some simplifications Πµν(k)= (26) m 1 (p m )S(p)= (32) e2 x(1 x) 6 − 0 0( k2gµν +kµkν) dx − . 1+[Σ(p)S(p)] +[Σ(p)S(p)] +[Σ(p)S(p)] , π − (m2 k2x(1 x))1/2 A B C Z0 − − where we divided fermion self-energy into pieces com- For k2 < 4m2 the x integral is well defined. The ing from different tensor structures in Dµν(k): gµν and transversalityofΠµν(k)isafavorableconsequenceofus- kµkν fromthetransversepartandagainfromthegauge- m ing the vertex function in the form (21), satisfying the dependent longitudinal part. After small rearrangement WT identity (22). Performing the spectral integral over they are: 5 [Σ(p)S(p)] = (33) A d3k 1 1 iZ κ2e2 dMρ(M) γµ γ (m M) 3 0 (2π)3 p+ k m+iε µ(k2+iε)(k2 κ2+iε)(p m+iε) − → Z Z (cid:20) 6 6 − − 6 − (cid:21) [Σ(p)S(p)] = (34) B d3k 1 iZ κ2e2 dMρ(M) k (m M) 3 0 (2π)3 6 p+ k m+iε)(k2+iε)2(k2 κ2+iε) − → Z Z (cid:20) 6 6 − − (cid:21) [Σ(p)S(p)] = (35) C ie2 d3k 1 0 dMρ(M) k (m M) . λ (2π)3 6 (p+ k m+iε)(k2+iε)2 − → Z Z (cid:20) 6 6 − (cid:21) A comment should be made here. To avoid technical the integrals, we will come back to real values of κ. The difficulty while performing Wick’s rotation in the above procedure was discussed in [29]. With this trick each momentumintegrals,weanalyticallycontinuedthevalue of the above momentum integrals can be performed in of κ to imaginaryvalues on the upper half plane of com- anordinarywayknownfromperturbationtheory. Itcan plex κ. Now the deformation of the integration contour easilybeseenthatallintegralsarefinitewithouttheneed as is requiredby passing into euclidean space,is not dis- of any regularization. Omitting details of this standard turbedbytheinappropriatelocationofpolessinceallsin- calculation, we find (in Minkowski space) gularitieshavethe ‘Feynman’position. Afterperforming d3k 1 1 I = ie2 γµ γ = (36) A 0 (2π)3 p+ k m+iε µ(k2+iε)(k2 κ2+iε) Z 6 6 − − 1 e2 1 1 0 dx(3m p(1 x)) , 8πκ2 −6 − (m2x p2x(1 x))1/2 − (m2x p2x(1 x)+κ2(1 x))1/2 Z0 (cid:20) − − − − − (cid:21) 1 d3k 1 e2 p(p+m) I = ie2 k = 0 dx 1 x6 6 (37) B 0 (2π)3 6 (p+ k m+iε)(k2+iε)2(k2 κ2+iε) 8πκ2 − κ2 × Z 6 6 − − Z (cid:20)(cid:18) (cid:19) 0 1 1 p(p+m) x(1 x) + 6 6 − , × (m2x p2x(1 x))1/2 − (m2x p2x(1 x)+κ2(1 x))1/2 2 (m2x p2x(1 x))3/2 (cid:18) − − − − − (cid:19) − − (cid:21) 1 d3k 1 e2 1 I = ie2 k = 0 dx C 0 (2π)3 6 (p+ k m+iε)(k2+iε)2 −8π (m2x p2x(1 x))1/2 Z 6 6 − Z0 (cid:20) − − p(p+m) x(1 x) +6 6 − . (38) 2 (m2x p2x(1 x))3/2 − − (cid:21) Now the contributions to Σ(p)S(p) may be written as All the abovespectral integrals canbeen found similarly as in [28], therefore we omit the technicalities. [Σ(p)S(p)] = (39) A In the infrared domain, when p2 m2, the left hand 1 Z3κ2 dMρ(M) IA (m M) , sideoftheDSequation(32),aftersu→bstituting(18),con- p m+iε − → Z (cid:20) 6 − (cid:21) tainstwotypesofsingularterms(theonlytermsthatare [Σ(p)S(p)] = (40) B important): Z κ2 dMρ(M)[I (m M)] , 3 B − → Z 1 m2βδm(p+m) m2β [Σ(p)S(p)] = dMρ(M)[I (m M)] . (41) 6 + . (42) C λ C − → − (m2 p2)β+1 (m2 p2)β Z − − 6 Therefore, to avoid lengthy expressions we will not give V. SOLUTIONS AND CONCLUSIONS theresultsfor(39),(40),and(41)intheirfullcomplexity, because it is sufficient for our goal to only pick out from The four equations we have obtained, i.e (30), (31), them the identical singular terms. We find: (49) and (50) are sufficient to determine all parameters. e2Z 1 1 Werewritethemwiththeuseofrenormalizedquantities: [Σ(p)S(p)]A ≈ 08π3 m2 I1(m2,κ2)− m3 × fermion mass m = m0 + δm, gauge coupling constant (cid:26)(cid:20) (cid:21) e = Z1/2e and gauge parameter λ = Z λ. Besides, it p(p+m) p(p+m) 1 3 0 R 3 ×(1 6 p62/m2)β+1 + 6 6 2 I2(m2,κ2)− m3 isusefultointroduceadimensionlessparameterζ = 4πe2m. − (cid:20) (cid:18) (cid:19) Afterexecutingtheparametricintegralsin and and 1 3 3 1 I I , (43) performing the reverse analytical continuation in κ, we −4βm (1 p2/m2)β get (cid:21) − (cid:27) e2Z 1 p(p+m) [Σ(p)S(p)]B ≈ 08π3 m3 (1 6 p62/m2)β+1 (44) δm = ζ 1 1 4m2−κ2 ln(4m2/κ2+1) ,(52) (cid:20) − m 2 λ − − 4m2 p(p+m) 1 β 1 1 (cid:20) R (cid:21) 6 6 − , ζ 3 1 4m2 κ2 − 2m3 β − 4βm (1 p2/m2)β 1= 2 2 − (cid:18) (cid:19) − (cid:21) 2 4β λ − − 4m2+κ2 (cid:20) (cid:18) R (cid:19) e2 p(p+m) 1 κ2 [Σ(p)S(p)]C ≈−8π0λ 6 6 m3 (1 p2/m2)β+1 −2m2 ln(4m2/κ2+1) , (53) (cid:20) − (cid:21) p(p+m) β 1 1 1 2ζ Γ(β+1/2) + 6 6 − + ,(45) Z = 1 , (54) 2m3 β 4βm (1 p2/m2)β 3 − 3 √πΓ(β+1) (cid:18) (cid:19) − (cid:21) where refers to diverging terms, when p2 m2. The κ2 15√πΓ(β+1) ≈ → = . (55) functions 1 and 2 have the following form: m2 2ζ Γ(β+3/2) I I 1 x+2 Thissetofequationsforδm/m,β, Z3 andκ2/m2 may (m2,κ2)= dx , (46) I1 (m2x2+κ2(1 x))1/2 Z0 − 1 Κ x2(x+2) (m2,κ2)= dx . (47) I2 (m2x2+κ2(1 x))3/2 30 Z0 − If the solution is to be self-consistent close to the 20 fermion mass shell, the divergent terms on both sides of DS equation (10) must be identical. Equating them, and making use of the fact that up to finite terms we 10 have p(p+m) 2m2 6 6 , (48) (1 p2/m2)β ≈ (1 p2/m2)β Ζ − − 0.1 0.2 0.3 0.4 p(p+m) m(p+m) m2 6 6 6 , (1 p2/m2)β+1 ≈ (1 p2/m2)β+1 − (1 p2/m2)β Β − − − wederivethefollowingtworelationsforunknownparam- eters δm and β: δm=e20Z3 m (m2,κ2)+ 1 , (49) 0.2 1 8π − I λZ (cid:20) 3(cid:21) e2Z 3 1 1= 0 3 κ2 3(m2,κ2)+ 2 . 0.1 8π − I 4βm λZ − (cid:20) (cid:18) 3 (cid:19)(cid:21) (50) The function (m2,κ2) is defined as follows Ζ I3 0.1 0.2 0.3 0.4 1 = (m2,κ2) m2 (m2,κ2) = I3 κ2 I1 − I2 FIG. 3: The dependence of κ in units of m (upper plot) and 1 (cid:0) (cid:3) (1 x)(x+2) of power β (lower plot) on the parameter ζ. The dashed dx − . (51) line corresponds to approximated solutions defined by equa- (m2x2+κ2(1 x))3/2 Z0 − tions (56)-(59). The gauge parameter is chosen as λR =0.3. 7 be solved numerically for certain chosen values of renor- Now, from (55) it becomes obvious that for small ζ the malizedgaugeparameterλ andtheresultsplotasfunc- left hand side must be large. This is in agreement with R tions of parameter ζ. Let us consider the case of weak our expectations concerning the subsequentterms in the coupling, when ζ is small. Since the applicability of our Taylorexpansion(15). Butforκ2 m2 thefirsttermin ≫ method requires 0 < β < 1, then from equation (55) squarebracketsonthe righthand side of(53) dominates we deduce that κ2/m2 should be large. It may be jus- overallother,whichmaybeeasilyverified. Theexpected tified by the elementary estimation given below. First positivity of β requires then considering only gauges for we rewritefollowingexpressioninterms ofbeta function which 0 < λ < 1/2. This is not surprising, since β is R B(x,y) and its integral representation: a strongly gauge dependent quantity. For our numerical calculations we have then chosenthe values of the gauge 1 parameters from that range. √πΓ(β+1) 1 1 = B(β+1,1/2)= t−1/2(1 t)βdt, 2Γ(β+3/2) 2 2 − In figure 3 we show the dependence of parameters κ Z 0 and β on ζ for exemplary value of λ =0.3 (solid lines). R Infigure 4 we similarlyplotthe dependence oftwo other and next use the inequalities valid for 0<β <1: parameters: fermion mass renormalization constant δm and gauge field renormalization constant Z . It is nice 1 1 3 1 t−1/2(1 t)βdt < 1 t−1/2dt=1, to observe that we have 0<δm t−1/2(1 t)dt= . 2 − 2 − 3 In the weak coupling regime,assuming that we do not Z Z 0 0 choose the gauge parameter λ approaching zero (but R stillto be less than1/2),the equationsmaybe giventhe ∆m 0.6 Κ 30 0.4 20 0.2 10 Ζ 0.1 0.2 0.3 0.4 Z3 0.1 0.2 0.3 0.4 Ζ 1 Β 0.9 0.8 0.8 0.5 0.7 Ζ 0.2 0.1 0.2 0.3 0.4 Ζ 0.1 0.2 0.3 0.4 FIG.4: Thedependenceofmassrenormalization δminunits of m (upper plot) and charge renormalization constant Z3 (lowerplot)ontheparameterζ. Thedashedlinecorresponds FIG. 5: The comparison of the behavior of κ (upper plot) toapproximatedsolutionsdefinedbyequations(56)-(59). On and parameter β (lower plot) for different values of gauge the upper plot the dashed line is almost identical with the parameter: dotted line – λR = 0.1, dashed line – λR = 0.2, solidoneandthereforeitisnotvisible. Thegaugeparameter solid line – λR =0.3, mixed line – λR =0.5 (Yennie gauge). is chosen as λR =0.3. Theplotsofκforvariousgaugesfollowalmostthesamecurve. 8 The terms m2/κ2 in the first two equations arise from ∆m the expansion of and for large κ2, but in fact they 1 3 I I may be omitted, since they are of order ζ, as results 0.8 from the last equation. As already told, the solutions of thesesimplifiedequationsaredrawninfigures3and4as dashedlines. Onthe plots forκ and δm these curvesare 0.5 not visible since they are almost identical with the full solutions, but also on the other two they do not deviate from the ‘exact’ results too much. 0.2 From (57) we see that the exponent β in this approximation may be written as Ζ 0.1 0.2 0.3 0.4 Z3 3ζ 1 β = 2 , (60) 8 λ − 1 (cid:18) R (cid:19) 0.9 which means that λR = 1/2 is a kind of Yennie gauge. 0.8 It is interesting to observe, if and how the values of the parameters depend on the gauge. This is shown in figures 5 and 6 which are performed for the following 0.7 values from the range [0,1/2]: λ = 0.1, 0.2, 0.3, 0.5. R Ζ The Landaugaugecannotbe used because it wouldlead 0.1 0.2 0.3 0.4 to negative value of β . Theparticularstressdeservestheobservationthatthe FIG. 6: The comparison of the behavior of fermion mass gaugedependence ofthe parameterκ is extremely weak. renormalization δm (upper plot) and gauge field renormalization constant Z3 (lower plot) for different vales of gauge This means that the renormalizedgauge boson propaga- parameter: dotted line – λR = 0.1, dashed line – λR = 0.2, tor (strictly speaking its transverse part) is practically solid line – λR =0.3, mixed line – λR =0.5 (Yenniegauge). gauge independent, as it should be. Please note that inother approximationschemesapplied to DS equations in QED one obtains the gauge dependent value of the 3 following approximated form: polarization scalar [16]. δm ζ 1 6m2 ThedependenceofZ3onλRisrelativelyweakforsmall = , (56) ζ too. 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Infrared self-consistent solutions of bispinor QED3

Similar Infrared self-consistent solutions of bispinor QED3