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SU Non-renormalizability of the noncommutative (2) gauge theory Maja Burić∗ and Voja Radovanović† Faculty of Physics, P.O. Box 368, 11001 Belgrade, Serbia and Montenegro 4 0 0 2 n a J Abstract 5 1 We analyze the divergent part of the one-loop effective action for the noncommuta- 1 tiveSU(2)gaugetheorycoupledtothefermionsinthefundamentalrepresentation. We v showthatthedivergencies inthe2-pointandthe3-pointfunctionsintheθ-linearorder 3 0 can be renormalized, while the divergence in the 4-point fermionic function cannot. 1 1 0 4 0 / h t - p e h : v i X r a ∗E-mail: majab@ff.bg.ac.yu †E-mail: rvoja@ff.bg.ac.yu 1 1 Introduction Although discussed for quite some time, the question of renormalizability of field theories on noncommutative R4 has not been settled in a satisfactory way yet. Noncom- mutativity of the coordinates, i.e., a relation of the type [xˆµ,xˆν]= iθµν, (1.1) puts the lower bound on the coordinate measurements, so one would expect that it also implies a natural ultraviolet cut-off and acts as a regulator. However, this idea has not been successfully implemented in models. UsuallyonerepresentsthenoncommutativefieldΦˆ(xˆ)byafunctionΦˆ(x)onR4 and encodes the noncommutativity in themultiplication rule(⋆-product). For example, for constant θµν, the field multiplication is given by the Moyal-Weyl product: φ(x)⋆χ(x)= e2i θµν∂x∂µ∂∂yνφ(x)χ(y)|y→x . (1.2) This product is nonlocal, and so is the field theory defined by it. The most extensive study of renormalizability of noncommutative (NC) field theo- rieswasdoneforthescalarfieldtheory[1]. Onequantizesperturbatively: thediagrams in the perturbation theory split into planar and nonplanar. The planar diagrams re- produce the behavior of the underlying commutative theory; on the other hand, UV divergencies in the nonplanar graphs get regulated by the effective cut-off (θp)−1. But for the exceptional values of the momenta (when the momentum flow into the loop is zero), these contributions become infinite. The reappearance of the UV divergencies in the infrared sector is related to nonlocality of the theory. Renormalizability of noncommutative Φ4 has been analyzed in the recent papers [2, 3, 4] from the point of view of the Wilson-Polchinski RG equation. The renormalization procedure was defined in some special cases; it was shown that this procedure is different from the usual planar renormalization. Noncommutative gauge theories, in particular U(1) and U(N), have been studied on the similar lines, too. The UV/IR mixing appears in the Feynman graphs in the same way as for the scalar field theory [5, 6], so the question of renormalizability has the same status. However, for the gauge theories there is another representation. As shownbySeibergandWitten [7],noncommutativeandcommutativegaugetheoriesare equivalent. Thisequivalenceisrealizedbyamappingrelatingtherepresentation(gauge fields, matter fields) of NC gauge symmetry to the fields carrying the representation of its commutative counterpart. The SW map is given as a series in powers of θµν. Classical action is also expanded in θ: in the zero-th order it reduces to the action of ordinarygaugetheory; additionaltermscanbetreatedascouplings. Nonlocality shows up in the infinite number of interactions. This realization gives another framework to address the issue of renormalizability: it in particular makes sense in the limit of small noncommutativity, θ → 0. One of the features of ’θ-expanded’ approach is that arbitrary gauge groups and tensor products can be represented, so noncommutative generalizations of the standard model have been constructed [8]. In this paper, we study the renormalizability of NC SU(2) in the θ-expanded approach. Renormalizability of the θ-expanded NC gauge theories has been addressed in papers[9,10,11,12]forthegaugegroupU(1)withorwithoutfermionicmatterandfor itssupersymmetricextension. HerewediscusstheSU(2)theorycoupledtofermionsin the fundamental representation. Although the SU(2) gauge theory technically differs 2 from the abelian U(1) in many details, the conclusion concerning renormalizability is the same. If fermions are massive, theory is not renormalizable. For massless fermions the theory is ”almost renormalizable”, meaning that there is only one divergent term in the effective action which cannot be absorbed by the SW field redefinition scheme. The method which we used to calculate the divergent contributions is the background field method. As it was thoroughly explained in [12], we skip the technical points here, referring also to the standard literature [13]. The calculations for SU(2) are quite involved in comparison with U(1), so in this paper we discuss only the θ-linear order and find the divergent parts of the 2-point, 3-point and fermionic 4-point functions. Previously,theresultsforU(1)intheθ-linearorderweregivenin[9,10]; forthe2-point functions they were extended in [12] to the θ2-order. The plan of the paper is the following. In the second section we describe the classical action for noncommutative SU(2). In the third section all necessary steps for the perturbative quantization are done. The results for the divergencies of the 2-point, 3-point and fermionic 4-point functions are given in the sections 4 and 5. The results which are obtained and some further issues concerning renormalizability are discussed in the last section. 2 The model Thegeneralconstruction ofgaugetheories on noncommutative spaceandtheirrelation to theSWmap were introducedin [14,15, 16]; we willrepeatonly afew relevant steps. The noncommutative space is an algebra generated by a set of noncommuting coordinates xˆµ; in general they obey relations, R(xˆ) = 0; for example (1.1). Physical fields ψˆ(xˆ) are functions of the coordinates. We want to describe the gauge theory: let α(x) = αa(x)Ta be a (commutative) gauge parameter and Ta – generators of a Lie group. The field ψˆ(xˆ) transforms covariantly under the infinitesimal gauge transformation Λˆ (xˆ) if δˆ ψˆ(xˆ) = iΛˆ (xˆ)ψˆ(xˆ) . The gauge degrees of freedom are inner, so α α α the coordinates xˆµ are invariant under gauge transformations: δˆ xˆµ = 0. However, α one can define the ’covariant coordinates’ Xˆµ introducing the gauge potentials Aˆ (xˆ): µ Xˆµ = xˆµ + θµνAˆ (xˆ). They have the transformation property δˆ Xˆµ = i Λˆ ,Xˆµ , ν α α h i when the vector potential transforms appropriately. In order that the infinitesimal gauge transformations close, one impose δˆαδˆβ −δˆβδˆα = δˆα×β, (2.3) where α×β = −i[α,β] denotes the composition of two transformations. In principle, for a given set of relations R = 0, the noncommutative coordinates xˆµ can be represented by the coordinates xµ on a commutative manifold, and the multiplication by some ⋆-product. As already mentioned, in the case of the constant noncommutativity (1.1) (which we here consider), the ⋆-product is the Moyal product (1.2); it is possible to construct ⋆-products for the other cases, too. Expanding the gauge parameter and physical fields in θ, the condition (2.3) becomes an equation (Seiberg-Witten) for the coefficients in the expansion. A solution of the SW equation is 1 Λˆ(x) = α(x)+ θµν{∂ α(x),A (x)}+... µ ν 4 1 Aˆ (x) = A (x)− θµν{A (x),∂ A (x)+F (x)}+... (2.4) ρ ρ µ ν ρ νρ 4 3 1 i ψˆ(x) = ψ(x)− θµνA (x)∂ ψ(x)+ θµνA (x)A (x)ψ(x)+... ; µ ν µ ν 2 4 theterms ofthesecondorderinθ aregiven in[16]. Inthelastformulaα(x), A (x)and µ ψ(x) denote the commutative gauge parameter, the vector potential and the fermionic field: A = AaTa, D ψ =∂ ψ−iA ψ , (2.5) µ µ µ µ µ F = ∂ A −∂ A −i[A ,A ] (2.6) µν µ ν ν µ µ ν which correspond to the noncommutative quantities. As terms linear in θ in (2.4) contain anticommutators of Ta, it is clear that NC gauge fields Λˆ(x), Aˆ (x) do not µ close into the Lie algebra; they take values in the enveloping algebra of the gauge group. The other property of the solution (2.4) is that it is not unique. It can be shown [9, 17] that, if A(n), ψ(n) is a solution of the SW equation (2.3) up to the order n µ in θ, then by adding any gauge covariant expression (of appropriate dimension) with exactly n factors of θ to A(n), ψ(n) one obtains another solution. This is similar to the µ relation between the solutions of inhomogeneous and the corresponding homogeneous linear equation; in a way, it expresses the nonlocality of the theory. As pointed out in [9], this nonuniqueness can be used to subtract the divergent terms in the effective action and to regularize the theory – one can think of such a procedure as a sort of ’dressing’ of the ’bare’ fields, that is, the choice of the physical ones. Wenowproceedtotheaction. For NCSU(2)Yang-Mills theorytheclassicalaction is given with S = d4xψˆ¯⋆(iγµDˆ −m)ψˆ− 1 d4xTr(Fˆ ⋆Fˆµν) , (2.7) µ µν Z 4 Z where the noncommutative field strength Fˆ is µν Fˆ = ∂ Aˆ −∂ Aˆ −i(Aˆ ⋆Aˆ −Aˆ ⋆Aˆ ) , (2.8) µν µ ν ν µ µ ν ν µ and the covariant derivative Dˆ ψˆ= ∂ ψˆ−iAˆ ⋆ψˆ . (2.9) µ µ µ The commutative counterpart ψ(x) of the matter field ψˆ(x) is in the fundamental representation of SU(2), Ta = σa. Inserting the expansion (2.4) into (2.7), we get the 2 action in theθ-linear order [16]: S = S +S +S , (2.10) 0 1,A 1,ψ 1 S = d4x ψ¯(iγµD −m)ψ− FµνaFa , (2.11) 0 Z (cid:18) µ 4 µν(cid:19) S = 0 , 1,A 1 1 S = θρσ d4x −iψ¯γµF D ψ+ ψ¯F (−iγµD +m)ψ (2.12) 1,ψ µρ σ ρσ µ 2 Z (cid:18) 2 (cid:19) 1 m = d4x − θρσ∆µναψ¯F γβ(i∂ +A )ψ+ θµνψ¯F ψ . Z (cid:18) 8 ρσβ µν α α 4 µν (cid:19) αβγ In order to simplify the notation we introduced the symbol ∆ defined as σρµ ∆αβγ = δαδβδγ −δαδβδγ +(cyclic αβγ) = −ǫαβγλǫ . (2.13) σρµ σ ρ µ ρ σ µ σρµλ 4 The bosonic θ-linear term S vanishes (unlike in the U(1) case) because it is pro- 1,A portional to the symmetric coefficients dabc = Tr({Ta,Tb}Tc), and for SU(2) these coefficients are zero for all irreducible representations. In the functional integration we will treat (A , ψ) as a multiplet so we want both µ fields to be real (or to be complex). Therefore we write the Dirac spinor ψ in terms of the Majorana spinors ψ . For the charge-conjugated spinor ψC = Cψ¯T Majorana 1,2 spinorsaregivenbyψ = 1(ψ±ψC); viceversa, ψ = ψ +iψ 1. Toexpresstheaction 1,2 2 1 2 in terms of Majorana spinors, one has to use explicitly the form of Pauli matrices (i.e., the representation of the group generators). As σ is antisymmetric and σ , σ are 2 1 3 symmetric matrices, the gauge field A2 couples to Majorana spinors differently from µ A1, A3. The action reads: µ µ σ σ S = d4x ψ¯ (i/∂ −m+A/2 2)ψ +ψ¯ (i/∂ −m+A/2 2)ψ 0 1 1 2 2 Z (cid:16) 2 2 σ σ σ σ 1 + iψ¯ (A/1 1 +A/3 3)ψ −iψ¯ (A/1 1 +A3 3)ψ − FµνaFa , (2.14) 1 2 2 2 2 2 2 1 4 µν (cid:17) 1 σ σ −→ ←− σ σ S = − θρσ∆µνα d4x iψ¯ γβ((F1 1 +F3 3)∂ − ∂ (F1 1 +F3 3))ψ 1 16 ρσβ Z (cid:16) 1 µν 2 µν 2 α α µν 2 µν 2 1 σ σ −→ ←− σ σ + iψ¯ γβ((F1 1 +F3 1)∂ − ∂ (F1 1 +F3 3))ψ 2 µν 2 µν 2 α α µν 2 µν 2 2 σ −→ ←− σ −→ ←− − ψ¯ γβ 2(F2 ∂ − ∂ F2 )ψ +ψ¯ γβ 2(F2 ∂ − ∂ F2 )ψ 1 2 µν α α µν 2 2 2 µν α α µν 1 i + Fa Aa(ψ¯ γβψ −ψ¯ γβψ ) . (2.15) 2 µν α 1 2 2 1 (cid:17) This will be the initial point for the quantization. 3 One-loop effective action Background field method is one of the standard methods to obtain divergent and finite quantum contributions to the classical action [13]. In the first step, one expands fields around their classical configuration, i.e. splits the fields into the background (classical) part and the quantum correction: A → A +A , ψ → ψ +Ψ . (3.16) µ µ µ 1,2 1,2 1,2 Quantum fields are denoted here by A , Ψ . The functional integration over the µ 1,2 quantum fields in the generating functional is then performed; the effective action, Γ, is the Legendre transformation of the generating functional. In the saddle-point approximation, the integration gives: i Γ[A ,ψ ,ψ ] = S[A ,ψ ,ψ ]+ Sdet logS(2)[A ,ψ ,ψ ] . (3.17) µ 1 2 µ 1 2 µ 1 2 2 Sdet denotes the functional superdeterminant and S(2) is the second functional derivative of the classical action. For polynomial interactions, the second derivative can be obtained from the quadratic part of the action; it is an expression of the type: Ae ν d4x(Afµ Ψ¯1 Ψ¯2)BΨ1 , (3.18) Z Ψ 2   1We encounter some of the identities which Majorana spinors satisfy: φ¯χ = χ¯φ ; φ¯γ χ = −χ¯γ φ; µ µ φ¯σµνχ=−χ¯σµνφ; φ¯γ5χ=χ¯γ5φ; φ¯γµγ5χ=χ¯γµγ5φ. 5 where we wrote B instead of S(2) as, in fact, we include the gauge fixing term, too. In our case the gauge fixing term is 1 S = − d4x(D Aµa)2 , (3.19) GF µ 2Z D is the background covariant derivative, D Aνα = ∂ Aνa+ǫabcAbAνc . To calculate µ µ µ µ the one-loop correction i i Γ(1) = logSdetB = STr logB 2 2 perturbatively, one usually expands logB. In correspondence with the notation for the fields, B is a 3×3 block matrix B B B 11 12 13 B = B21 B22 B23 . B B B 31 32 33   ThesubmatricesB ,B ,B andB areGrassmann-oddwhiletherestareGrassmann- 12 13 21 31 even; the supertrace is defined by STrB = TrB −TrB −TrB . B depends on the 11 22 33 classical fields. One should keep in mind that Aa (for a= 1,2,3) is a triplet, while ψ µ 1 and ψ are dublets of the SU(2) group. 2 In the absence of the interaction, B is the inverse propagator; in general case, one can separate the kinetic part: 1g δab(cid:3) 0 0 2 αβ B =  0 i/∂ 0 +M . 0 0 i/∂   We are to expand logB around identity I = diag(g δab,1,1). To achieve this, we µν multiply B by C [18]: 2 0 0 C = 0 −i/∂ 0  , 0 0 −i/∂   and then for the one-loop correction we obtain i i Γ(1) = STr log(BC)+ STr logC−1 2 2 i i i = STr log(I +(cid:3)−1MC)+ STr logC−1+ STr log(cid:3) . (3.20) 2 2 2 The second and the third terms, being independent on the fields, can be included in the infinite renormalization. Note that now the propagator for all fields is (cid:3)−1. The operator (cid:3)−1MC defines the rules in the perturbation expansion. To get the structure of the expansion more clearly, we decompose MC into the sum MC = N +N +N +T +T +T (3.21) 0 1 2 1 2 3 with respect to the number of fields – indices denote their number in a given term. N , N and N originate from the commutative theory, while T , T and T are the 0 1 2 1 2 3 noncommutative interactions linear in θ. One can read N ...T from the action 0 3 (2.14-2.15), after the separation of the part quadratic in the quantum fields. 6 In the θ0 order we get 0 0 0 N0 = 0 im/∂ 0  . (3.22) 0 0 im/∂   N is lengthy. If we write it as 1 N u u 12 13 N1 = u21 u22 u23 , (3.23) u −u u 31 23 22   its matrix elements are given by −→ ←− Nfe = 2ǫefb(∂ Ab −∂ Ab)+ǫefbg (Aαb∂ − ∂ Aαb) µν µ ν ν µ µν α α u = (−ψ¯ σ1 −iψ¯ σ2 −ψ¯ σ3 )T γµ/∂ 12 2 2 1 2 2 2 u = (ψ¯ σ1 −iψ¯ σ2 ψ¯ σ3 )T γµ/∂ 13 1 2 2 2 1 2 u = 2γν(iσ1ψ σ2ψ iσ3ψ ) 21 2 2 2 1 2 2 u = 2γν(−iσ1ψ σ2ψ −iσ3ψ ) 31 2 1 2 2 2 1 σ u = −i/A2 2/∂ 22 2 σ σ u = (A/1 1 +A/3 3)∂/ . 23 2 2 The remaining, N , is 2 M 0 0 N2 =  0 0 0 , (3.24) 0 0 0   with Mfe =−g δǫfAaAαa+g AeAfα+2AeAf −2AfAe . µν µν α µν α µ ν µ ν We will introduce T , T and T symbolically; the full expressions are given in the 1 2 3 Appendix. They are of the form 0 p p 12 13 T1 = p21 p22 p23 , (3.25) p −p p 31 23 22   Q q q R r r 12 13 12 13 T2 =q21 q22 q23 , T3 = r21 0 r23. (3.26) q −q q r −r 0 31 23 22 31 23     As already stressed, the representation of SU(2) is notreal (symmetric), and the interaction of the gauge fields with the Majorana spinors cannot be written in a manifestly covariant way. But all given operators have the same structure which is the conse- quence of the fact that the action was originally in Dirac spinors. Note that for the massless fermions N = 0; N , T and T drastically simplify, as well. 0 1 1 2 7 4 Divergencies, 2-point functions Introducing the decomposition (3.21), for the one-loop correction (3.20) we get i Γ(1) = STr log(I +(cid:3)−1MC) (4.27) 2 ∞ i (−)n+1 = STr((cid:3)−1N +(cid:3)−1N +(cid:3)−1N +(cid:3)−1T +(cid:3)−1T +(cid:3)−1T )n . 0 1 2 1 2 3 2 n nX=1 Our notation allows to extract the contributions from different terms in the expansion (4.27) easily. Parts of the effective action which give the 2-point functions have two classical fields. This means that the sum of indices in the monomials which we are interested in is equal to 2. Since there is an operator with the index 0, in principle, infinitely many terms contribute to the n-point functions. However, as we are calcu- lating the divergencies only, we will need just finite number of terms: (cid:3)−1N behaves 0 as p−1 in the momentum space, and the integrals become convergent for ((cid:3)−1N )k of 0 a high enough degree. For the 2-point function in the zero-th order, power counting gives that the traces whichcontributeare: STr((cid:3)−1N (cid:3)−1N ),STr((cid:3)−1N (cid:3)−1N )andSTr((cid:3)−1N ((cid:3)−1N )2). 1 1 0 2 0 1 Performing the Fourier transformation and the dimensional regularization we obtain i STr((cid:3)−1N (cid:3)−1N ) = d4x −24Aa(cid:3)Aµa−24(∂ Aµa)2+6iψ¯/∂ψ , 1 1 (4π)2ǫ Z (cid:16) µ µ (cid:17) STr((cid:3)−1N (cid:3)−1N ) = 0 , 0 2 i STr((cid:3)−1N ((cid:3)−1N )2) = d4x12mψ¯ψ . 0 1 (4π)2ǫ Z Adding, we get the standard result of the commutative theory i i Γ = − STr((cid:3)−1N (cid:3)−1N )+ STr((cid:3)−1N ((cid:3)−1N )2) 2A,2ψ 1 1 0 1 4 2 1 3i = −6Aa(cid:3)Aµa−6(∂ Aµa)2+ ψ¯/∂ψ−6mψ¯ψ . (4.28) (4π)2ǫ µ µ 2 (cid:16) (cid:17) The contribution of the ghost action (3.19) should be added, too: 1 1 Γ = d4xFa Fµνa . (4.29) 2,gh (4π)2ǫ 3 Z µν Now we consider the θ-linear order. Potentially divergent terms are STr((cid:3)−1N (cid:3)−1T ), STr(((cid:3)−1N (cid:3)−1N +(cid:3)−1N (cid:3)−1N )(cid:3)−1T ), STr((cid:3)−1N (cid:3)−1T ) 1 1 0 1 1 0 1 0 2 and STr(((cid:3)−1N )2(cid:3)−1N +(cid:3)−1N (cid:3)−1N (cid:3)−1N +(cid:3)−1N ((cid:3)−1N )2)(cid:3)−1T ). 0 1 0 1 0 1 0 1 The dimensional regularization gives the following: i i 1 1 STr((cid:3)−1N (cid:3)−1T )= θµν ǫ ψ¯γ γσ(cid:3)∂ρψ+ mψ¯σ ∂ ∂αψ− mψ¯σ (cid:3)ψ , 1 1 (4π)2ǫ 4 µνρσ 5 2 να µ 4 µν (cid:16) (cid:17) STr(((cid:3)−1N (cid:3)−1N +(cid:3)−1N (cid:3)−1N )(cid:3)−1T ) = 0 , 0 1 1 0 1 STr((cid:3)−1N (cid:3)−1T ) =0 , 0 2 and STr (((cid:3)−1N )2(cid:3)−1N +(cid:3)−1N (cid:3)−1N (cid:3)−1N +(cid:3)−1N ((cid:3)−1N )2)(cid:3)−1T 0 1 0 1 0 1 0 1 (cid:16) (cid:17) i 3 = θµν −im2ǫ ψ¯γ γβ∂αψ+m3ψ¯ψσ ψ . (4π)2ǫ 4 µναβ 5 νµ (cid:16) (cid:17) 8 All together, the divergent part of the 2-point function in the linear order reads: 1 1 i 1 1 Γ′ = θµν ǫ ψ¯γ γσ(cid:3)∂ρψ+ mψ¯σ ∂ ∂αψ− mψ¯σ (cid:3)ψ 2ψ (4π)2ǫ 2 4 µνρσ 5 2 να µ 4 µν (cid:16) (cid:17) 1 3 + θµν −im2ǫ ψ¯γ γβ∂αψ+m3ψ¯σ ψ . (4.30) (4π)2ǫ 8 µναβ 5 νµ (cid:16) (cid:17) This is a nice result: comparing (4.30) with the θ-linear correction for the 2-point functionsinNCQED[12],weseethatintheSU(2)casealso,onlyfermionicpropagator gets a correction; the correction is, up to a factor, the same as the one for U(1). This has further consequences. In NC QED we argued that the massive terms obstruct renormalization, as only for the case m = 0 one can redefine fields in such a way that thedivergenttermsdisappear. TheanalysiscanberepeatedforSU(2)withoutchange. Hence, we come to the conclusion: the NC gauge theories with massive fermions are not renormalizable. For this reason in the calculations of 3-point and 4-point functions we focus to the massless case. One might add that the calculations become so cumbersome that otherwise they would hardly be doable. 5 3-point and 4-point functions The background field method is a gauge covariant method and therefore it gives the covariantresults. Ontheotherhand,theseparationinto2-point,3-pointetc. functions ′ breaks the gauge covariance: for instance, Γ given by the formula (4.30) is not 2ψ covariant; it is a part of the covariant expression (written assuming that m = 0) which is, up to the order of the covariant derivatives, equal to: 1 i Γ′ = θµνǫ ψ¯γ γσD2Dρψ, (5.31) 2 (4π)2ǫ 8 µνρσ 5 ′ Writing Γ in theform (5.31), we includedthe parts of the 3-point, 4-point and 5-point 2 functions. When one calculates the 3-point functions, from the dimensional analysis it is easy to see that, apart from the terms residual from (5.31), basically only two terms contribute (up to the partial integration and various combinations of indices). They are the leading terms in the covariant expressions θψ¯γF(Dψ) and θψ¯γ(DF)ψ , (5.32) (γ stands for the products of the γ-matrices). However, as we stressed already, the calculation i.e. the organization of terms becomes increasingly difficult, so in order to find the 3-point functions we use a trick. We calculate the coefficients of the terms (5.32) in the 4-point functions. When we are doing this, we can assume that the background spinor field is constant, so the covariant derivative D ψ reduces to A ψ; µ µ in this case N ...T also simplify. The gauge covariance enables us to recover the 1 3 result for the 3-point functions uniquely at the end of the calculation. Divergent parts of the3-point functionsare intheterms STr(((cid:3)−1N )2(cid:3)−1T )and 1 1 STr((cid:3)−1N (cid:3)−1T ); the corresponding traces in the 4-point functions are 1 2 STr((cid:3)−1N (cid:3)−1T ) and STr(((cid:3)−1N )2(cid:3)−1T ) . The divergent part of the first trace 1 3 1 2 is i STr((cid:3)−1N (cid:3)−1T )= − θµν 6AaFa ψ¯γαψ−3AaFa ψ¯γαψ , (5.33) 1 3 (4π)2ǫ ν µα α µν (cid:16) (cid:17) 9 while for the second one we get i 33 STr(((cid:3)−1N )2(cid:3)−1T ) = θµν Aa(∂ Aa −∂ Aa)ψ¯γαψ 1 2 (4π)2ǫ 24 µ ν α α ν (cid:16) 5 1 − Aa(∂ Aa −∂ Aa)ψ¯γαψ+ Aaα(∂ Aa −∂ Aa)ψ¯γ ψ 16 α ν µ µ ν 2 ν α α ν µ 5 σ + ǫabc(− ǫ Aaα(∂ Abβ −∂βAb)ψ¯ c γ γρψ 6 ναβρ µ µ 2 5 25 σ − ǫ Aa(∂αAcβ −∂βAcα)ψ¯ b γ γρψ 24 ναβρ µ 2 5 1 σ − ǫ Ac(∂βAaα −∂αAaβ)ψ¯ b γ γσψ 16 νµαβ σ 2 5 1 σ − ǫ Abα∂ Aaσψ¯ c γ γβψ) . (5.34) νµαβ σ 5 8 2 (cid:17) When we add these expressions and try to ’covariantize’ the result, we obtain a piece which does not match: it is precisely the part of the 2-point function (5.31). The rest gives the 3-point function in its covariant form: 1 1 111i Γ′ = − θµν − ψ¯γαF D ψ 3 (4π)2ǫ 2 6 να µ (cid:16) 43i 3i − ψ¯γαF D ψ− ψ¯γα(D F )ψ µν α α µν 4 4 + 2iψ¯γ F Dαψ+iψ¯γ (DαF )ψ µ να µ να 5 1 + ǫ ψ¯γ γρ(D Fαβ)ψ− ǫ ψ¯γ γσ(D Fαβ)ψ ναβρ 5 µ µναβ 5 σ 8 16 1 + ǫ (2ψ¯γ γβFραD ψ+ψ¯γ γβ(D Fρα)ψ) . (5.35) µναβ 5 ρ 5 ρ 8 (cid:17) 4-fermionic vertex has a very important role in the discussion of renormalizability. The corresponding 4-point function is relatively easy to find: to this end one can put Aa = 0 in N ...T . The divergent part comes from STr(((cid:3)−1N )2(cid:3)−1T ) and µ 1 3 1 2 STr(((cid:3)−1N )3(cid:3)−1T ); the final result is 1 1 1 9 Γ′ = θµνǫ ψ¯γ γσψψ¯γρψ . (5.36) 4ψ (4π)2ǫ 32 µνρσ 5 (5.31), (5.35) and (5.36) are the main results of our calculation. 6 Discussion As we mentioned in the introduction, there is no a priori criterion which would fix the nonuniqueness in the SW map. The redefinition of fields allowed by it changes the action; the terms which appear are of the forms: ∆S(n) = d4x(D Fµν)A(n) , (6.37) A Z ν µ ∆S(n) = d4x ψ¯i/DΨ(n) +Ψ¯(n)i/Dψ , (6.38) ψ Z (cid:16) (cid:17) writtenforthecaseofmasslessfermions. A(n) andΨ(n) aregaugecovariantexpressions µ of the n-th order in θ. The important thing in (6.37-6.38) is that, besides Fµν and ψ, they contain at least one derivative. 10