D-deformed Wess-Zumino model and its renormalizability properties
aa r X i v : . [ h e p - t h ] A p r D-deformed Wess-Zumino model and itsrenormalizability properties
Marija Dimitrijevi´c and Voja Radovanovi´c
University of Belgrade, Faculty of PhysicsStudentski trg 12, 11000 Belgrade, Serbia
Abstract
Using the methods developed in earlier papers we analyze a new type of de-formation of the superspace. The twist we use to deform the N = 1 SUSY Hopfalgebra is non-hermitian and is given in terms of the covariant derivatives D α .A SUSY invariant deformation of the Wess-Zumino action is constructed andcompared with results already known in the literature. Finally, by calculatingdivergences of the two-point Green functions a preliminary analysis of renormal-izability properties of the constructed model is done. As expected, there is norenormalization of mass and no tadpole diagrams appear. Keywords: supersymmetry, non-hermitian twist, deformed Wess-Zumino model,renormalization eMail: dmarija,[email protected]
Introduction
The idea of noncommuting spacetime coordinates goes back to Heisenberg who sug-gested [1] that uncertainty relations between coordinates could resolve the ultraviolet(UV) divergences arising in quantum field theories. The issue was then investigated bySnyder in [2], but did not attract much interest at the time. However, in the last twodecades noncommutative geometry has found applications in many branches of physicssuch as quantum field theory and particle physics, solid state physics and many others.Comprehensive reviews on the subject can be found in references [3], [4], [5], [6], [7],[8],[9]. Having in mind problems which physics encounters at small scales (high energies),in recent years attempts were made to combine supersymmetry (SUSY) with non-commutative geometry. Different models were constructed, see for example [10], [11],[12], [13], [14], [15]. Some of these models emerge naturally as low energy limits ofstring theories in backgrounds with a constant Neveu-Schwarz two form and/or a con-stant Ramond-Ramond two form. In [13] the anticommutation relations between thefermionic coordinates were modified in the following way { θ α ⋆ , θ β } = C αβ , { ¯ θ ˙ α ⋆ , ¯ θ ˙ β } = { θ α ⋆ , ¯ θ ˙ α } = 0 , (1.1)where C αβ = C βα is a complex constant symmetric matrix. The analysis is done inEuclidean space, since the deformation (1.1) is hermitian only in Euclidean signaturewhere undotted and dotted spinors are not related by the usual complex conjugation.Note that the ⋆ -product used in (1.1) is also well defined in Minkowskian signaturealthough in that case it is not hermitian [14]. The chiral coordinates y m = x m + iθσ m ¯ θ commute in this setting, therefore the notion of chirality is preserved, i.e. the ⋆ -product of two chiral superfields is again a chiral superfield. On the other hand, theconstructed models break one half of the N = 1 SUSY so they are invariant only underthe so-called N = 1 / N = 1 / N = 1 SUSY transformations. Our choice of the twist is different from thatin [19]. We work in Minkowski space-time and choose a hermitian twist. As undottedand dotted spinors are related by the usual complex conjugation, we obtain { θ α ⋆ , θ β } = C αβ , { ¯ θ ˙ α ⋆ , ¯ θ ˙ β } = ¯ C ˙ α ˙ β , { θ α ⋆ , ¯ θ ˙ α } = 0 , (1.2)1ith ¯ C ˙ α ˙ β = ( C αβ ) ∗ . The deformed Wess-Zumino Lagrangian was formulated and an-alyzed; the action which follows is invariant under the twisted SUSY transformations.Superfields transform in the undeformed way, while the Leibniz rule for SUSY trans-formations (when they act on a product of superfields) is modified. Since the action isnon-local it is difficult (but not impossible) to discuss renormalizability properties ofthe model.Nevertheless, we are interested in renormalizability properties of theories withtwisted symmetries. It is important to understand whether deforming (by a twist)of symmetries spoils some of the renormalizability properties of SUSY invariant theo-ries. Therefore, in this paper we analyze a simpler model with the twist given by F = e C αβ D α ⊗ D β , (1.3)where C αβ = C βα ∈ C is a complex constant matrix and D α = ∂ α − iσ α ˙ αm ¯ θ ˙ α ∂ m arethe SUSY covariant derivatives.Following the method of [20] in the next section we introduce the deformationand the ⋆ -product which follows from it. Due to our choice of the twist (1.3), thecoproduct of SUSY transformations remains undeformed, leading to the undeformedLeibniz rule. Being interested in deformations of the Wess-Zumino model, we discusschiral fields and their products. The product of two chiral fields is not a chiral fieldand we have to use the projectors defined in [21] to separate chiral and antichiral parts.All possible invariants are listed in Section 4 and the deformed Wess-Zumino actionis constructed in Section 5. Using the background field method we then analyze two-point functions and their divergences. Finally, we give some comments and compareour results with the results already present in the literature. Some details of thecalculations are collected in appendices A and B. D -deformation of the Hopf algebra of SUSY transfor-mations The undeformed superspace is generated by the coordinates x , θ and ¯ θ which fulfill[ x m , x n ] = [ x m , θ α ] = [ x m , ¯ θ ˙ α ] = 0 , { θ α , θ β } = { ¯ θ ˙ α , ¯ θ ˙ β } = { θ α , ¯ θ ˙ α } = 0 , (2.4)with m = 0 , . . . α, β = 1 ,
2. To x m we refer as to bosonic and to θ α and ¯ θ ˙ α werefer as to fermionic coordinates. Also, x = x m x m = − ( x ) + ( x ) + ( x ) + ( x ) .A general superfield F ( x, θ, ¯ θ ) can be expanded in powers of θ and ¯ θF ( x, θ, ¯ θ ) = f ( x ) + θφ ( x ) + ¯ θ ¯ χ ( x ) + θθm ( x ) + ¯ θ ¯ θn ( x ) + θσ m ¯ θv m + θθ ¯ θ ¯ λ ( x ) + ¯ θ ¯ θθϕ ( x ) + θθ ¯ θ ¯ θd ( x ) . (2.5) We only consider the N = 1 SUSY in this paper. The generalization to N = 2 SUSY and highercan be obtained by following the same steps as in Section 2. δ ξ F = ( ξQ + ¯ ξ ¯ Q ) F, (2.6)where ξ and ¯ ξ are constant anticommuting parameters and Q and ¯ Q are the SUSYgenerators Q α = ∂ α − iσ mα ˙ α ¯ θ ˙ α ∂ m , (2.7)¯ Q ˙ α = ¯ ∂ ˙ α − iθ α σ mα ˙ β ε ˙ β ˙ α ∂ m . (2.8)As in [22], [20], we introduce a deformation of the Hopf algebra of infinitesimalSUSY transformations by choosing the twist F in the following way F = e C αβ D α ⊗ D β , (2.9)with the complex constant matrix C αβ = C βα ∈ C . Note that this twist is nothermitian, F ∗ = F . The usual complex conjugation is denoted by ” ∗ ”. It can beshown [23] that (2.9) satisfies all the requirements for a twist [24]. The Hopf algebraof SUSY transformation does not change since { Q α , D β } = { ¯ Q ˙ α , D β } = 0 (2.10)and it is given by • algebra { Q α , Q β } = { ¯ Q ˙ α , ¯ Q ˙ β } = 0 , { Q α , ¯ Q ˙ β } = 2 iσ mα ˙ β ∂ m , [ ∂ m , ∂ n ] = [ ∂ m , Q α ] = [ ∂ m , ¯ Q ˙ α ] = 0 . (2.11) • coproduct ∆ Q α = Q α ⊗ ⊗ Q α , ∆ ¯ Q ˙ α = ¯ Q ˙ α ⊗ ⊗ ¯ Q ˙ α , ∆ ∂ m = ∂ m ⊗ ⊗ ∂ m . (2.12) • counit and antipode ε ( Q α ) = ε ( ¯ Q ˙ α ) = ε ( ∂ m ) = 0 ,S ( Q α ) = − Q α , S ( ¯ Q ˙ α ) = − ¯ Q ˙ α , S ( ∂ m ) = − ∂ m . (2.13)This means that the full supersymmetry is preserved.Strictly speaking, the twist (2.9) does not belong to the universal enveloping algebraof the Lie algebra of infinitesimal SUSY transformations. Therefore, to be mathemat-ically correct we should enlarge the algebra (2.11) by introducing the relations for theoperators D α as well. Note that the same happened in [20], where the twist was givenby F = e C αβ ∂ α ⊗ ∂ β + ¯ C ˙ α ˙ β ¯ ∂ ˙ α ⊗ ¯ ∂ ˙ β , (2.14)3ith the complex constant matrix C αβ = C βα and C αβ and ¯ C ˙ α ˙ β were related bythe usual complex conjugation. There we had to enlarge the algebra by adding therelations for the fermionic derivatives ∂ α and ¯ ∂ ˙ α .The inverse of the twist (2.9) F − = e − C αβ D α ⊗ D β , (2.15)defines the ⋆ -product. For two arbitrary superfields F and G the ⋆ -product reads F ⋆ G = µ ⋆ { F ⊗ G } = µ {F − F ⊗ G } = µ { e − C αβ D α ⊗ D β F ⊗ G } = F · G −
12 ( − | F | C αβ ( D α F ) · ( D β G ) − C αβ C γδ ( D α D γ F ) · ( D β D δ G ) , (2.16)where | F | = 1 if F is odd (fermionic) and | F | = 0 if F is even (bosonic). Thesecond line of (2.16) is the definition of the µ ⋆ multiplication. No higher powers of C αβ appear since the derivatives D α are Grassmanian. The ⋆ -product (2.16) is associative ,noncommutative and in the zeroth order in the deformation parameter C αβ it reducesto the usual pointwise multiplication. One should also note that it is not hermitian,( F ⋆ G ) ∗ = G ∗ ⋆ F ∗ . (2.18)The ⋆ -product (2.16) leads to { θ α ⋆ , θ β } = C αβ , { ¯ θ ˙ α ⋆ , ¯ θ ˙ β } = { θ α ⋆ , ¯ θ ˙ α } = 0 , [ x m ⋆ , x n ] = − C αβ ( σ mn ε ) αβ ¯ θ ¯ θ, [ x m ⋆ , θ α ] = − iC αβ σ mβ ˙ β ¯ θ ˙ β , [ x m ⋆ , ¯ θ ˙ α ] = 0 . (2.19)The chiral coordinates y m also do not commute[ y m ⋆ , y n ] = − θ ¯ θC αβ ( σ mn ε ) αβ . (2.20)Other (anti)commutation relations follow in a similar way.Relations (2.19) enable us to define the deformed superspace. It is generated bythe usual bosonic and fermionic coordinates (2.4) while the deformation is containedin the new product (2.16). From (2.19) it follows that both fermionic and bosonic part The associativity of the ⋆ -product follows from the cocycle condition [24] which the twist F hasto fulfill F (∆ ⊗ id ) F = F ( id ⊗ ∆) F , (2.17)where F = F ⊗ F = 1 ⊗ F . It can be shown that the twist (2.9) indeed fulfills this condition,see for details [23].
4f the superspace are deformed. This is different from [20] where only the fermioniccoordinates were deformed.The deformed infinitesimal SUSY transformation is defined as δ ⋆ξ F = ( ξQ + ¯ ξ ¯ Q ) F. (2.21)Since the coproduct (2.12) is undeformed, the usual (undeformed) Leibniz rule follows.Then the ⋆ -product of two superfields is again a superfield. Its transformation law isgiven by δ ⋆ξ ( F ⋆ G ) = ( ξQ + ¯ ξ ¯ Q )( F ⋆ G )= ( δ ⋆ξ F ) ⋆ G + F ⋆ ( δ ⋆ξ G ) . (2.22) Since we are interested in possible deformations of the usual Wess-Zumino action, wenow analyze chiral fields and their ⋆ -products.A chiral field Φ fulfills ¯ D ˙ α Φ = 0, where ¯ D ˙ α = − ¯ ∂ ˙ α − iθ α σ mα ˙ α ∂ m and ¯ D ˙ α is relatedto D α by the usual complex conjugation. In terms of the component fields the chiralsuperfield Φ is given byΦ( x, θ, ¯ θ ) = A ( x ) + √ θ α ψ α ( x ) + θθH ( x ) + iθσ l ¯ θ ( ∂ l A ( x )) − i √ θθ ( ∂ m ψ α ( x )) σ mα ˙ α ¯ θ ˙ α + 14 θθ ¯ θ ¯ θ ( (cid:3) A ( x )) . (3.23)The ⋆ -product of two chiral fields readsΦ ⋆ Φ = Φ · Φ − C αβ C γδ D α D γ Φ D β D δ Φ= Φ · Φ − C ( D Φ)( D Φ)= A − C H + 2 √ Aθ α ψ α − i √ C H ¯ θ ˙ α ¯ σ m ˙ αα ( ∂ m ψ α ) + θθ (cid:16) AH − ψψ (cid:17) + C ¯ θ ¯ θ (cid:16) − H (cid:3) A + 12 ( ∂ m ψ ) σ m ¯ σ l ( ∂ l ψ )) (cid:17) + iθσ m ¯ θ (cid:16) ∂ m ( A ) + C H∂ m H (cid:17) + i √ θθ ¯ θ ˙ α ¯ σ m ˙ αα ( ∂ m ( ψ α A ))+ √
22 ¯ θ ¯ θC ( − Hθ (cid:3) ψ + θσ m ¯ σ n ∂ n ψ∂ m H )+ 14 θθ ¯ θ ¯ θ ( (cid:3) A − C (cid:3) H ) , (3.24) In this paper “usual“ always refers to undeformed, that is to the case C αβ = 0. C = C αβ C γδ ε αγ ε βδ . Because of the ¯ θ , ¯ θ ¯ θ and the θ ¯ θ ¯ θ terms (3.24) is not achiral field. Following the method developed in [20] we decompose the ⋆ -products ofchiral fields into their irreducible components by using the projectors defined in [21].The antichiral, chiral and transversal projectors are defined as follows P = 116 D ¯ D (cid:3) , (3.25) P = 116 ¯ D D (cid:3) , (3.26) P T = − D ¯ D D (cid:3) . (3.27)The chiral part of (3.24) is undeformed and it is given by (for details we refer to[20]) P (Φ ⋆ Φ) = ΦΦ= A + 2 √ Aθ α ψ α + θθ (cid:16) AH − ψψ (cid:17) + iθσ m ¯ θ (cid:16) ∂ m ( A ) (cid:17) + i √ θθ ¯ θ ˙ α ¯ σ m ˙ αα ( ∂ m ( ψ α A ))+ 14 θθ ¯ θ ¯ θ (cid:3) A . (3.28)The antichiral part reads P (Φ ⋆ Φ) = − C H − i √ C H ¯ θ ¯ σ m ∂ m ψ + C ¯ θ ¯ θ (cid:16) − H (cid:3) A + 12 ( ∂ m ψ ) σ m ¯ σ l ( ∂ l ψ )) (cid:17) + iθσ m ¯ θC H∂ m H + √
22 ¯ θ ¯ θC ( − Hθ (cid:3) ψ + θσ m ¯ σ n ∂ n ψ∂ m H ) − θθ ¯ θ ¯ θC (cid:3) H . (3.29)In this case there is no transverse part of Φ ⋆ Φ, P T (Φ ⋆ Φ) = 0 . (3.30)Next, we calculate the ⋆ -product of three chiral fields. The following identity applies(Φ ⋆ Φ) ⋆ Φ = (Φ · Φ + P (Φ ⋆ Φ)) ⋆ Φ= ΦΦΦ − C D (ΦΦ) D Φ + P (Φ ⋆ Φ)Φ , (3.31)with − C D (ΦΦ) D Φ = C h − AH + 12 H ( ψψ ) − i √ AH ¯ θ ¯ σ n ( ∂ n ψ ) − i √ θ ¯ σ m ∂ m ( Aψ )) H + i √
22 (¯ θ ¯ σ n ∂ n ψ )( ψψ )6¯ θ ¯ θ (cid:16) − AH (cid:3) A − H (cid:3) A + 12 ψψ (cid:3) A + ∂ m ( Aψ ) σ m ¯ σ n ( ∂ n ψ ) (cid:17) + iθσ m ¯ θ∂ m (cid:16) AH − Hψψ (cid:17) + √
22 (¯ θ ¯ θ ) (cid:16) − AHθ (cid:3) ψ + 12 ( θ (cid:3) ψ )( ψψ ) − Hθ (cid:3) ( Aψ )+ θσ n ¯ σ m ∂ m ( Aψ )( ∂ n H ) + 12 θσ l ¯ σ m ( ∂ m ψ ) ∂ l (2 AH − ψψ ) (cid:17) + 14 θθ ¯ θ ¯ θ (cid:3) (cid:16) − AH + 12 ( ψψ ) H (cid:17)i (3.32)and P (Φ ⋆ Φ) ⋆ Φ = P (Φ ⋆ Φ) · Φ= C h − AH − √ θψH − θθH − i √ θ ¯ σ m ( ∂ m ψ )) AH +(¯ θ ¯ θ )( − HA (cid:3) A + 12 A ( ∂ l ψ ) σ l ¯ σ m ( ∂ m ψ ))+ i ( θσ l ¯ θ )( 12 A ( ∂ l H ) − H ( ∂ l A ) + ( ψσ l ¯ σ m ( ∂ m ψ )) H )+ i √ (cid:16)
14 ( θθ )(¯ θ ¯ σ l ψ )( ∂ l H ) −
54 ( θθ )(¯ θ ¯ σ m ( ∂ m ψ )) H (cid:17) + √
22 ¯ θ ¯ θ (cid:16) θσ m ¯ σ n ∂ m ( H∂ n ψ ) A − θψ )( H (cid:3) A −
12 ( ∂ m ψ ) σ m ¯ σ n ( ∂ n ψ )) − ( H∂ l A ) θσ l ¯ σ m ( ∂ m ψ ) (cid:17) + θθ ¯ θ ¯ θ (cid:16) − A (cid:3) H − H (cid:3) A − ψσ m ¯ σ n ∂ m ( H∂ n ψ )+ H ( ∂ m ψ ) σ m ¯ σ n ( ∂ n ψ ) + 14 ( ∂ m A )( ∂ m H ) (cid:17)i . (3.33)It is easy to see that P (Φ ⋆ Φ) ⋆ Φ = ΦΦΦ − C D (ΦΦ) D Φ . (3.34)The projections are given by P ( P (Φ ⋆ Φ) ⋆ Φ) = − C D (ΦΦ) D Φ P ( P (Φ ⋆ Φ) ⋆ Φ) = ΦΦΦ . (3.35) Let us now examine the transformation laws under the deformed SUSY transformations(2.21) of terms which could be relevant for the construction of a SUSY invariant action.7here are two quadratic (in the number of fields) invariants , I and I : I = P (Φ ⋆ Φ) (cid:12)(cid:12)(cid:12) θθ = 2 AH − ψψ, (4.36) I = P (Φ ⋆ Φ) (cid:12)(cid:12)(cid:12) ¯ θ ¯ θ = − C (cid:16) H (cid:3) A −
12 ( ∂ m ψ ) σ m ¯ σ n ( ∂ n ψ ) (cid:17) . (4.37)Their transformation laws are given by δ ⋆ξ I = 2 i √ ξ ¯ σ m ∂ m ( Aψ ) , (4.38) δ ⋆ξ I = √ C ξσ m ¯ σ n ∂ m ( H ( ∂ n ψ )) . (4.39)Looking at cubic terms we see that there are more candidates for possible invariants.The first two are I and I : I = P ( P (Φ ⋆ Φ) ⋆ Φ) (cid:12)(cid:12)(cid:12) θθ = 3( A H − Aψψ ) , (4.40) I = P ( P (Φ ⋆ Φ) ⋆ Φ) (cid:12)(cid:12)(cid:12) ¯ θ ¯ θ = C (cid:16) − AH (cid:3) A − H (cid:3) A + 12 ψψ (cid:3) A + ∂ m ( Aψ ) σ m ¯ σ n ( ∂ n ψ ) (cid:17) . (4.41)One can check that they indeed transform as total derivatives. Two more candidatesare given by I = P ( P (Φ ⋆ Φ) ⋆ Φ) (cid:12)(cid:12)(cid:12) ¯ θ ¯ θ = C (cid:16) − AH (cid:3) A + 12 A ( ∂ l ψ ) σ l ¯ σ m ( ∂ m ψ ) (cid:17) , (4.42) I = P ( P (Φ ⋆ Φ) ⋆ Φ) (cid:12)(cid:12)(cid:12) θθ = − C H . (4.43)However they do not transform as total derivatives δ ⋆ξ I = C ξ α (cid:16) − H ( A (cid:3) ψ α + ψ α (cid:3) A ) + 2( σ m ¯ σ l ) βα ( ∂ l ψ β )( ∂ m H ) A + ψ α ( ∂ l ψ ) σ l ¯ σ m ( ∂ m ψ ) (cid:17) (4.44) = ∂ m ( . . . ) ,δ ⋆ξ I = − i √ C ¯ ξ ¯ σ m ( ∂ m ψ ) H (4.45) = ∂ m ( . . . ) . Inclusion of these terms will not lead to a SUSY invariant action. The last candidatefor a cubic invariant is I : I = P ( P (Φ ⋆ Φ) ⋆ Φ) (cid:12)(cid:12)(cid:12) θθ ¯ θ ¯ θ = − C (cid:16) A (cid:3) H + 5 H (cid:3) A (4.46) − H ( ∂ m ψ ) σ m ¯ σ l ( ∂ l ψ ) + 2 ψσ m ¯ σ l ∂ m ( H ( ∂ l ψ )) (cid:17) . Strictly speaking, terms I and I are invariant only under the integral R d x , that is when includedin an action. Since the construction of an invariant action is our aim, we continue with this abuse ofnotation and call ”invariant“ all terms that under SUSY transformations transform as total derivatives. P ( P (Φ ⋆ Φ) ⋆ Φ) (cid:12)(cid:12)(cid:12) θθ ¯ θ ¯ θ and P ( P (Φ ⋆ Φ) ⋆ Φ) (cid:12)(cid:12)(cid:12) θθ ¯ θ ¯ θ are equal up to a total derivative term and therefore lead to the sameequations of motion. Since I is the highest component of a superfield, under (2.21) ittransforms as a total derivative and can be included in a SUSY invariant action. In order to write the SUSY invariant action we collect all invariant terms and obtainthe following Lagrangian L = Φ + ⋆ Φ (cid:12)(cid:12)(cid:12) θθ ¯ θ ¯ θ + h m (cid:16) P (Φ ⋆ Φ) (cid:12)(cid:12)(cid:12) θθ + aP (Φ ⋆ Φ) (cid:12)(cid:12)(cid:12) ¯ θ ¯ θ (cid:17) + λ (cid:16) P ( P (Φ ⋆ Φ) ⋆ Φ) (cid:12)(cid:12)(cid:12) θθ + bP ( P (Φ ⋆ Φ) ⋆ Φ) (cid:12)(cid:12)(cid:12) ¯ θ ¯ θ +2 c ( P + P )( P (Φ ⋆ Φ) ⋆ Φ) (cid:12)(cid:12)(cid:12) θθ ¯ θ ¯ θ (cid:17) + c.c. i , (5.47)with m , λ , a , b and c real constant parameters. Terms P ( P (Φ ⋆ Φ) ⋆ Φ) (cid:12)(cid:12)(cid:12) θθ ¯ θ ¯ θ and P ( P (Φ ⋆ Φ) ⋆ Φ) (cid:12)(cid:12)(cid:12) θθ ¯ θ ¯ θ are equal up to a total derivative term and are therefore includedwith the same coefficient. The action in component fields which follows from (5.47)reads S = Z d x n A ∗ (cid:3) A + i∂ m ¯ ψ ¯ σ m ψ + H ∗ H + m ( AH − ψψ ) + m ( A ∗ H ∗ −
12 ¯ ψ ¯ ψ )+ λ ( A H − Aψψ ) + λ (( A ∗ ) H ∗ − A ∗ ¯ ψ ¯ ψ )+ h C (cid:16) ma ( 12 ψ (cid:3) ψ − H (cid:3) A ) + λa ( − AH (cid:3) A − H ( (cid:3) A )+ 12 ψψ ( (cid:3) A ) + Aψ (cid:3) ψ ) + λa ( − H (cid:3) A + 32 H ( ∂ m ψ ) σ m ¯ σ l ( ∂ l ψ )) (cid:17) + c.c. io . (5.48)The coefficients a , b and c are related to a , a and a : a/ a , b/ a and c = a .Note that (5.48) is the full action, i.e. no higher order terms in the deformationparameter C αβ appear.Varying the action (5.48) with respect to the fields H and H ∗ we obtain the equa-tions of motion for these fields H ∗ = − mA − λA + ma C ( (cid:3) A ) + λa C (cid:16) A (cid:3) A + 12 ( (cid:3) A ) (cid:17) − λa C (cid:16) − H ( (cid:3) A ) + ( ∂ m ψ ) σ m ¯ σ l ( ∂ l ψ ) (cid:17) , (5.49)9 = − mA ∗ − λ ( A ∗ ) + ma ¯ C ( (cid:3) A ∗ ) + λa ¯ C (cid:16) A ∗ (cid:3) A ∗ + 12 ( (cid:3) A ∗ ) (cid:17) − λa ¯ C (cid:16) − H ∗ ( (cid:3) A ∗ ) + ( ∂ m ¯ ψ )¯ σ m σ l ( ∂ l ¯ ψ ) (cid:17) . (5.50)Unlike in the undeformed theory, equations (5.49) and (5.50) are nonlinear in H and H ∗ . Nevertheless they can be solved H ∗ = (cid:16) − λa ) C ¯ C ( (cid:3) A ∗ )( (cid:3) A ) (cid:17) − n − mA ∗ − λ ( A ∗ ) + ma ¯ C ( (cid:3) A ∗ ) + λa ¯ C (cid:16) A ∗ (cid:3) A ∗ + 12 ( (cid:3) A ∗ ) (cid:17) − λa ¯ C ( (cid:3) A ∗ )( mA + λA ) − λa ¯ C ( ∂ m ¯ ψ )¯ σ m σ l ( ∂ l ¯ ψ )+3 λa ¯ C ( (cid:3) A ∗ ) h ma C ( (cid:3) A ) + λa C (cid:16) A (cid:3) A + 12 ( (cid:3) A ) (cid:17) − λa C ( ∂ m ψ ) σ m ¯ σ l ( ∂ l ψ ) io , (5.51)and similarly for H . These solutions we can expand up to second order in the defor-mation parameter and insert in the action (5.48). The action then becomes S = S + S , (5.52)with S = Z d x n A ∗ (cid:3) A + i ( ∂ m ¯ ψ )¯ σ m ψ − m ψψ + ¯ ψ ¯ ψ ) − λ ( A ∗ ¯ ψ ¯ ψ + Aψψ ) − m A ∗ A − mλA ( A ∗ ) − mλA ∗ A − λ A ( A ∗ ) o , (5.53) S = Z d x n C ma (cid:16) ψ ( (cid:3) ψ ) + ( (cid:3) A )( mA ∗ + λ ( A ∗ ) ) (cid:17) + C λa (cid:16) ψψ ( (cid:3) A ) + Aψ ( (cid:3) ψ ) + ( mA ∗ + λ ( A ∗ ) )( A ( (cid:3) A ) + 12 ( (cid:3) A )) (cid:17) − C λa ( mA ∗ + λ ( A ∗ ) ) (cid:16) ( (cid:3) A )( mA ∗ + λ ( A ∗ ) ) + ( ∂ m ψ ) σ m ¯ σ l ( ∂ l ψ ) (cid:17) + ¯ C ma (cid:16)
12 ¯ ψ ( (cid:3) ¯ ψ ) + ( (cid:3) A ∗ )( mA + λA ) (cid:17) (5.54)+ ¯ C λa (cid:16)
12 ¯ ψ ¯ ψ ( (cid:3) A ∗ ) + A ∗ ¯ ψ ( (cid:3) ¯ ψ ) + ( mA + λA )( A ∗ ( (cid:3) A ∗ ) + 12 ( (cid:3) ( A ∗ ) )) (cid:17) −
32 ¯ C λa ( mA + λA ) (cid:16) ( (cid:3) A ∗ )( mA + λA ) + ( ∂ m ¯ ψ )¯ σ m σ l ( ∂ l ¯ ψ ) (cid:17)o . Renormalizability properties: Two-point Green func-tions
In this section we investigate some renormalizability properties of our model. Usingthe background field method [25] and the dimensional reduction [26] the divergentpart of the effective action up to second order in fields is calculated. Note that wework with the action (5.48) and not with (5.52).To start with, we rewrite the deformed action (5.48) introducing the real fields S , P , E and G as A = S + iP √ , H = E + iG √ ψ M = (cid:18) ψ α ¯ ψ ˙ α (cid:19) . The deformation parameter C αβ can bewritten in the following way C αβ = K ab ( σ ab ε ) αβ , ¯ C ˙ α ˙ β = K ∗ ab ( ε ¯ σ ab ) ˙ α ˙ β . (6.56)Since K ab is a self dual tensor we write it as K ab = κ ab + i ǫ abcd κ cd , (6.57)where κ ab is a real antisymmetric tensor. In this way we obtain C + ¯ C = 4 κ ab κ ab (6.58) C − ¯ C = 2 iǫ abcd κ ab κ cd . (6.59)In order to simplify our calculation we will assume that C − ¯ C = 0. This choice canbe obtained by setting κ i = 0.With all this and introducing g = λ √ the action (5.48) becomes S = S + S with S = Z d x n S (cid:3) S + 12 P (cid:3) P −
12 ( i ¯ ψγ m ∂ m ψ + m ¯ ψψ ) + 12 ( E + G )+ m ( SE − P G ) − gS ¯ ψψ + gP ¯ ψγ ψ + g ( ES − EP − SP G ) o , (6.61) The method of dimensional regularization has a draw-back that it might not preserve the super-symmetry. Therefore one uses a modification of it, the so-called dimensional reduction. The index M on the Majorana spinors will be omitted in the following formulas. In the notation of [21] the matrix γ and the Lorentz generators Σ mn are defined as γ = γ γ γ γ , Σ mn = 14 [ γ m , γ n ] . (6.60) = C Z d x n ma ( 12 ¯ ψ (cid:3) ψ − E (cid:3) S + G (cid:3) P )+ ga ( P G (cid:3) S − SE (cid:3) S + P E (cid:3) P + SG (cid:3) P −
12 ( S (cid:3) E − P (cid:3) E − SP (cid:3) G ) + 12 ¯ ψψ (cid:3) S −
12 ¯ ψγ ψ (cid:3) P + ¯ ψ (cid:3) ψS − ¯ ψγ (cid:3) ψP )+ 32 ga ( − E (cid:3) S + G (cid:3) S + 2 EG (cid:3) P − E∂ m ¯ ψ∂ m ψ + G∂ m ¯ ψγ ∂ m ψ − ψ Σ mn ∂ n ψ∂ m E + 2 ¯ ψ Σ mn γ ∂ n ψ∂ m G ) o . (6.62)We split the fields into their classical and quantum parts, for example E → E + E .The action quadratic in quantum fields is S (2) = 12 ( ¯Ψ S P E G ) M Ψ SPEG , (6.63)where Ψ , ¯Ψ , S , P , E , G are quantum fields. The one loop effective action is thenΓ = i h (cid:3) − m ) − M C i , (6.64)with C = − i / ∂ + m − m
00 0 1 0 m − m (cid:3)
00 0 m (cid:3) . (6.65)The matrix M C can be decomposed into three parts
M C = N + T + V. (6.66)The zeroth order (in the deformation parameter) term is given by N = 2 g ( − S + γ P )( − i / ∂ + m ) − ψ γ ψ mψ mγ ψ − ¯ ψ ( − i / ∂ + m ) ( E − mS ) − ( G + mP ) − mE + S (cid:3) − mG − P (cid:3) ¯ ψγ ( − i / ∂ + m ) − G + mP − E − mS mG − P (cid:3) − mE − S (cid:3) S − P − mS − mP − P − S mP − mS . (6.67) The second order term (in the deformation parameter) which contains no fields is T = ma C (cid:3) ( − i / ∂ + m ) 0 0 0 00 m ←− (cid:3) −←− (cid:3) −→ (cid:3)
00 0 m ←− (cid:3) ←− (cid:3) −→ (cid:3) − (cid:3) m (cid:3)
00 0 (cid:3) m (cid:3) . (6.68)12he matrix V is second order in the deformation parameter and contains classical fieldslinearly. Its matrix elements are given in Appendix A.The one-loop divergent part of the effective action we calculate up to second order in g , second order in fields (two-point functions) and up to second order in the deformationparameter C αβ . Therefore the effective action is given byΓ = i h (cid:3) − m ) − ( N + T + V ) i = i h STr(( (cid:3) − m ) − ( N + T + V )) −
12 STr(( (cid:3) − m ) − N ( (cid:3) − m ) − N ) − STr(( (cid:3) − m ) − N ( (cid:3) − m ) − ( T + V ))+STr((( (cid:3) − m ) − N ) ( (cid:3) − m ) − T ) i . (6.69)The calculation of divergent parts of supertraces is tedious but straightforward and herewe give only the results. The details are given in Appendix B. Denoting K = (cid:3) − m ,we have STr( K − ( N + T + V )) = 0 , (6.70)STr( K − N K − N ) = ig π ǫ Z d x h S (cid:3) S + P (cid:3) P − ¯ ψi / ∂ψ + E + G i , (6.71)STr h K − N K − T i = 0 , (6.72)STr( K − N K − V ) = − g C i π ǫ Z d x h a m ( − P (cid:3) G − ¯ ψ (cid:3) ψ + 2 S (cid:3) E )+ a (( (cid:3) S ) + ( (cid:3) P ) + 4 m S (cid:3) S − ¯ ψi / ∂ (cid:3) ψ − m ¯ ψi / ∂ψ + 4 m P (cid:3) P +4 m E + E (cid:3) E + G (cid:3) G + 4 m G ) i , (6.73)STr( K − N K − N K − T ) = 2 iC a m g π ǫ Z d x h S (cid:3) S + P (cid:3) P − ¯ ψi / ∂ψ + E + G i . (6.74)In (6.73) terms ( (cid:3) S ) and ( (cid:3) P ) appear. Since these terms do not have classicalcounterparts we take a = 0. Then the divergent part of the one loop effective action(6.69) is given byΓ = g π ǫ Z d x h
14 ( S (cid:3) S + P (cid:3) P + ¯ ψi / ∂ψ + E + G )+ 34 a C m (2 P (cid:3) G + ¯ ψ (cid:3) ψ − S (cid:3) E )13 C a m ( S (cid:3) S + P (cid:3) P − ¯ ψi / ∂ψ + E + G ) i . (6.75)Let us now discuss the one-loop renormalizability properties of our model. Tocancel the divergences we have to add to the classical Lagrangian the counterterms L B = L + L − Γ . (6.76)In this way we obtain the bare Lagrangian L B . It is important to note that theterm I in the classical action (5.48) produces divergences proportional to I (compare(6.73) and (6.62)), so both of them are necessary in order to absorb the divergences inthe effective action. From the form of the bare Lagrangian we see that all fields arerenormalized in the same way: S = √ ZS, P = √ ZP, ψ = √ Zψ, E = √ ZE, G = √ ZG, (6.77)with Z = 1 − g π ǫ (1 − a m C ) . (6.78)The tadpole contributions add up to zero as in the commutative case. Also, δm = 0, i.e.there are no δm ¯ ψψ and δm ( SE + P G ) counterterms. It is obvious that the deformationparameter has to be renormalized too, C = (cid:16) − a g π a ǫ (cid:17) C . (6.79)The present analysis is not complete and we plan to consider the vertex correctionsin a forthcoming publication. From the vertex corrections we should draw conclusionsabout the renormalization of the coupling constant g and about the renormalizabilityof the full model.Finally, let us make a comment concerning the non-renormalization theorem. From(6.75) we see that the divergent part of the effective action consists of the usual term(Φ + Φ) (cid:12)(cid:12)(cid:12) θθ ¯ θ ¯ θ and a new (compared to the undeformed case) term P (Φ ⋆ Φ) (cid:12)(cid:12)(cid:12) ¯ θ ¯ θ . Bothterms are expressible as integrals over the whole superspace. In particular, for the newterm we have P (Φ ⋆ Φ) (cid:12)(cid:12)(cid:12) ¯ θ ¯ θ = Z d x d ¯ θ d θ θθP (Φ ⋆ Φ)= − C Z d x d ¯ θ d θ θθ ( D Φ)( D Φ)= 18 C Z d x d ¯ θ d θ Φ( D Φ) . (6.80)We see that at the level of two-point Green functions there is no need to deform thenonrenormalization theorem. This conclusion is different from [27].14 Conclusions
In order to see how a deformation by twist of the usual Wess-Zumino model affectsits renormalizability properties, we considered a special example of the twist (2.9).Compared with the undeformed SUSY Hopf algebra, the twisted SUSY Hopf algebrais unchanged. In particular, the twisted coproduct is undeformed, which leads to theundeformed Leibniz rule (2.22). However, the notion of chirality is lost and we haveto apply the method of projectors introduced in [20]. By including all constructedinvariants, we formulate a deformation of the usual Wess-Zumino action (5.48). Finally,we discuss some preliminary renormalizability properties of the model. As expected,there are no tadpole diagrams and no mass renormalization counterterms. All fields arerenormalized in the same way, which is another property of SUSY invariant theories.As the renormalization of the coupling constant g is concerned, at present we cannotsay if it is renormalized and how. However, we see that the freedom in choosing terms inthe action is partially fixed by demanding the cancellation of divergences. That requestleads to a = 0 and additionally we see that both a and a terms are necessary.Let us remark that the twist (2.9) leads to the ⋆ -product (2.16) which has alreadybeen discussed in [14]. In that paper the deformed Wess-Zumino Lagrangian hasbeen constructed in two different ways. The difference was present in the interactionterms. Namely, one can take the term Φ ⋆ (cid:12)(cid:12)(cid:12) θθ which (since Φ ⋆ is not chiral) breaks1 / I (4.43) and was not included in our deformedmodel (5.48) since it is not SUSY invariant. Adding its complex conjugate breaks thefull supersymmetry. The other possibility which was considered in [14] was to takethe term Φ ⋆ (cid:12)(cid:12)(cid:12) θθ ¯ θ ¯ θ as an interaction term. Is is equal to our I (4.46). Since it is thehighest component of the superfield Φ ⋆ , it transforms as a total derivative and theaction is invariant under the full supersymmetry. However, its commutative limit iszero and it is not a deformation of the usual interaction term. The commutative limitis obtained in [14] by adding the term (Φ + ) ⋆ which is undeformed and its complexconjugate reproduces the proper commutative limit. We have seen that the actionwith only the I term is not renormalizable.Renormalizability of the deformed Wess-Zumino models with the term Φ ⋆ ∝ H was studied, see for example [16]. To make these models renormalizable one has toadd additional terms to the original action. The main advantage of our model is theabsence of this problem. By including all possible invariants from the beginning wesee that no new terms are needed to cancel the divergences that appear. However, ourresults are not complete since we calculated here only the divergences in the two-pointfunctions. In the forthcoming paper we will consider the vertex contributions and thenwe will be able to tell if our present conclusions still hold.15 Matrix elements of V The matrix elements of V are given by V = gC (cid:16) a ( (cid:3) S + 2 S (cid:3) − γ (cid:3) P − P γ (cid:3) )+3 a ( E (cid:3) + ( ∂ m E ) ∂ m − ( ∂ m G ) γ ∂ m − Gγ (cid:3) − mn ∂ m E∂ n + 2Σ mn γ ∂ m G∂ n ) (cid:17) ( − i / ∂ + m ) , (1.81) V = gC h a (cid:16) ψ (cid:3) + 2 (cid:3) ψ (cid:17) − a m (cid:16) (cid:3) ψ + ∂ m ψ∂ m − mn ∂ n ψ∂ m (cid:17)i , (1.82) V = gC h a (cid:16) − γ ψ (cid:3) − γ (cid:3) ψ (cid:17) +3 a m (cid:16) − γ (cid:3) ψ − γ ∂ m ψ∂ m + 2Σ mn γ ∂ n ψ∂ m (cid:17)i , (1.83) V = gC h − ma (cid:16) ψ (cid:3) + 2 (cid:3) ψ (cid:17) + 3 a (cid:16) (cid:3) ψ + ∂ m ψ∂ m − mn ∂ n ψ∂ m (cid:17) (cid:3) i , (1.84) V = gC h ma (cid:16) − γ ψ (cid:3) − γ (cid:3) ψ (cid:17) +3 a m (cid:16) − γ (cid:3) ψ − γ ∂ m ψ∂ m + 2Σ mn γ ∂ n ψ∂ m (cid:17) (cid:3) i , (1.85) V = gC h a (cid:16) ←− (cid:3) ¯ ψ + 2 ¯ ψ (cid:3) (cid:17) ( − i / ∂ + m ) , (1.86) V = gC h − a (2 E (cid:3) + (cid:3) E ) + ma ( (cid:3) S + S (cid:3) + ←− (cid:3) S ) + 3 ma ←− (cid:3) E i , (1.87) V = gC h a ( (cid:3) G + G (cid:3) + ←− (cid:3) G )+ ma ( (cid:3) P + P (cid:3) + ←− (cid:3) P ) + 3 ma ←− (cid:3) G i , (1.88) V = gC h ma (2 E (cid:3) + (cid:3) E ) − a ( (cid:3) S + S (cid:3) + ←− (cid:3) S ) (cid:3) − a ←− (cid:3) E (cid:3) i , (1.89) V = gC h ma ( (cid:3) G + G (cid:3) + ←− (cid:3) G )+ a ( (cid:3) P + P (cid:3) + ←− (cid:3) P ) (cid:3) + 3 a ←− (cid:3) G (cid:3) i , (1.90) V = gC h a (cid:16) − ←− (cid:3) ¯ ψγ − ψγ (cid:3) (cid:17) ( − i / ∂ + m ) i , (1.91) V = gC h a ( (cid:3) G + G (cid:3) + ←− (cid:3) G ) − a ( (cid:3) P + P (cid:3) + ←− (cid:3) P ) − ma ←− (cid:3) G i , (1.92) V = gC h a (2 E (cid:3) + (cid:3) E ) + ma ( (cid:3) S + S (cid:3) + ←− (cid:3) S ) + 3 ma ←− (cid:3) E i , (1.93) V = gC h − a m ( (cid:3) G + G (cid:3) + ←− (cid:3) G )+ a ( (cid:3) P + P (cid:3) + ←− (cid:3) P ) (cid:3) + 3 a ←− (cid:3) G (cid:3) i , (1.94) V = gC h ma (2 E (cid:3) + (cid:3) E ) + a ( (cid:3) S + S (cid:3) + ←− (cid:3) S ) (cid:3) + 3 a ←− (cid:3) E (cid:3) i , (1.95)16 = gC h a (cid:16) − ∂ m ¯ ψ∂ m − ∂ m ¯ ψ Σ mn ∂ n (cid:17) ( − i / ∂ + m ) i , (1.96) V = gC h − a ( (cid:3) S + S (cid:3) + ←− (cid:3) S ) + 3 a ( m (cid:3) S − E (cid:3) ) i , (1.97) V = gC h a ( (cid:3) P + P (cid:3) + ←− (cid:3) P ) + 3 a ( m (cid:3) P + G (cid:3) ) i , (1.98) V = gC h a gm ( (cid:3) S + S (cid:3) + ←− (cid:3) S ) + 3 a ( − (cid:3) S (cid:3) + mE (cid:3) ) i , (1.99) V = gC h ma ( (cid:3) P + P (cid:3) + ←− (cid:3) P ) + 3 a (( (cid:3) P ) (cid:3) + mG (cid:3) ) i , (1.100) V = gC h a (cid:16) ∂ m ¯ ψγ ∂ m − ∂ m ¯ ψ Σ mn ∂ n (cid:17) ( − i / ∂ + m ) i , (1.101) V = gC h a ( (cid:3) P + P (cid:3) + ←− (cid:3) P ) + 3 a ( − m (cid:3) P + G (cid:3) ) i , (1.102) V = gC h a ( (cid:3) S + S (cid:3) + ←− (cid:3) S ) + 3 ma ( m (cid:3) S + E (cid:3) ) i , (1.103) V = − gC h ma ( (cid:3) P + P (cid:3) + ←− (cid:3) P ) − a ( − mG (cid:3) + ( (cid:3) P ) (cid:3) ) i , (1.104) V = gC h ma ( (cid:3) S + S (cid:3) + ←− (cid:3) S ) + 3 a ( mE (cid:3) + ( (cid:3) S )) (cid:3) i . (1.105) B Calculation of Supertraces
Here we calculate the divergent parts of two supertraces: STr( K − N K − V ) andSTr( K − N K − N K − T ). The following general formulas for the divergent parts oftraces are used Tr( K − f K − g ) = i π ǫ Z d x f g, (2.106)Tr( ∂ n K − f K − g ) = i π ǫ Z d x ∂ n f g, (2.107)Tr( ∂ n K − f ∂ m K − g ) = − i π ǫ Z d x (2.108) (cid:16) ∂ m ∂ n f g + 112 η mn (cid:3) f g − η mn m f g (cid:17) , Tr( K − f ∂ a K − g∂ b K − h ) = i π ǫ η ab Z d x f gh, (2.109)Tr( K − f ) = i π ǫ m Z d x f, (2.110)Tr( ∂ a K − f ) = 0 , (2.111)Tr( (cid:3) K − f ) = im π ǫ Z d x f, (2.112)Tr( (cid:3) K − f ) = im π ǫ Z d x f. (2.113)17 STr( K − N K − V )Using the definition of Supertrace we obtainSTr( K − N K − V ) = − X i Tr( K − N i K − V i )+ X i Tr( K − N i K − V i ) + . . . + X i Tr( K − N i K − V i ) . (2.114)The terms in (2.114) areTr h K − N K − V i = g C i π ǫ Z d x h a (( (cid:3) S ) + ( (cid:3) P ) − m S (cid:3) S − m S − m P )+3 a ( − m P (cid:3) G − m SE + m S (cid:3) E − m P G ) i , (2.115)Tr h K − N K − V + K − N K − V i = ig C π ǫ Z d x h − a ( G (cid:3) G + 4 m G )+2 m a ( P (cid:3) P + 4 m P ) + 12 a m P G i , (2.116)Tr h K − N K − V + K − N K − V i = i π ǫ mg C Z d x h a m ( P (cid:3) P + 4 m P ) + 6 a m P G i , (2.117)Tr h K − V K − N + K − V K − N i = 3 g C a m i π ǫ Z d x ¯ ψ (cid:3) ψ, (2.118)Tr h K − N K − V + K − N K − V i = g C i π ǫ Z d x h − a (2 m E + 12 E (cid:3) E ) − m a ( S (cid:3) S + 4 m S ) − a m SE i , (2.119)Tr h K − N K − V + K − N K − V i = − g m C i π ǫ Z d x h a (4 m S + S (cid:3) S ) + 6 a m SE i . (2.120)Tr h K − N K − V + K − N K − V + K − N K − V + K − N K − V i = 3 ig C π ǫ Z d x h − a ( m S (cid:3) S + 2 m S )18 m a (6 m SE + E (cid:3) S ) i , (2.121)Tr h K − N K − V + K − N K − V + K − N K − V + K − N K − V i = 3 i π ǫ g C Z d x h − a ( m P (cid:3) P + 2 m P ) − m a (6 m P G − G (cid:3) P ) i , (2.122)Tr h K − N K − V − K − N K − V + K − N K − V − K − N K − V i = g C i π ǫ Z d x h a (2 im ¯ ψ / ∂ψ + i ¯ ψ / ∂ (cid:3) ψ )+ 32 a m ¯ ψ (cid:3) ψ i . (2.123)Adding all the terms (2.115)-(2.120) we obtainSTr( K − N K − V ) = − g C i π ǫ Z d x h a m ( − P (cid:3) G − ¯ ψ (cid:3) ψ + 2 S (cid:3) E )+ a (( (cid:3) S ) + ( (cid:3) P ) + 4 m S (cid:3) S − ¯ ψi / ∂ (cid:3) ψ − m ¯ ψi / ∂ψ + 4 m P (cid:3) P +4 m E + E (cid:3) E + G (cid:3) G + 4 m G ) i . (2.124) • STr( K − N K − N K − T )Again, from the definition of Supertrace it followsSTr( K − N K − N K − T ) = − Tr( K − N i K − N ij K − T j )+Tr( K − N i K − N ij K − T j )+ . . . +Tr( K − N i K − N ij K − T j ) . (2.125)The divergences appearing in (2.125) areTr( K − N K − N (cid:3) ( − i / ∂ + m ))= 4 g m i π ǫ Z d x h − S (cid:3) S − P (cid:3) P + 40 m S + 8 m P i , (2.126)Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x h − mE + 4 m SE − m S i , (2.127)Tr( K − N K − N K − (cid:3) ) 19 4 g i π ǫ Z d x h − mG + 2 mGP + 3 m P i , (2.128)Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x h m SE − m S i , (2.129)Tr( K − N K − N K − (cid:3) )= − g i π ǫ Z d x h m P + m P G i , (2.130)Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x h m P G + mG − m P i , (2.131)Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x h m ES + 3 m S + mE i , (2.132)Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x h m P − m GP i , (2.133)Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x h m ES + 3 m S i , (2.134) m Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x h m S − m ES i , (2.135) m Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x h m P + m P G i , (2.136) m Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x h m ES + 3 m S i , (2.137) m Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x h − m GP + 3 m P i , (2.138)Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x ( E + mS ) , (2.139)Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x h mG − m P i , (2.140) m Tr( K − N K − N K − (cid:3) ) 20 4 g i π ǫ Z d x h m ES + 3 m S i , (2.141)Tr( K − N K − N K − (cid:3) )= − g i π ǫ Z d x m P , (2.142)Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x m S , (2.143)Tr( K − N K − N K − (cid:3) )= 4 g i π ǫ Z d x ( E − mS ) , (2.144)Tr( K − N K − N K − (cid:3) )= 4 g im π ǫ Z d x h i ψ / ∂ψ − m ¯ ψψ i , (2.145)Tr( K − N K − N K − (cid:3) )= − g im π ǫ Z d x h i ψ / ∂ψ + m ¯ ψψ i , (2.146)Tr( K − N K − N K − (cid:3) )= − mg i π ǫ Z d x h i ψ / ∂ψ + m ¯ ψψ i , (2.147)Tr( K − N K − N K − (cid:3) )= 4 g m i π ǫ Z d x h − i ψ / ∂ψ + m ¯ ψψ i , (2.148)Tr( K − N K − N K − ( − i / ∂ + m ) (cid:3) ) + Tr( K − N K − N K − ( − i / ∂ + m ) (cid:3) )= 8 g i π ǫ Z d x i ¯ ψ / ∂ψ, (2.149)Tr( K − N K − N (cid:3) K − (cid:3) )= 4 g i π ǫ Z d x h m SE − m S i , (2.150)Tr( K − N K − N (cid:3) K − (cid:3) )= 4 g i π ǫ Z d x h m P G − m P i , (2.151)Tr( K − N K − N (cid:3) K − (cid:3) )= − g i π ǫ Z d x m S , (2.152)Tr( K − N K − N (cid:3) K − (cid:3) )= 12 g i π ǫ Z d x m P , (2.153)Tr( K − N K − N (cid:3) K − (cid:3) ) 21 − g i π ǫ Z d x m P , (2.154)Tr( K − N K − N (cid:3) K − (cid:3) )= 12 g i π ǫ Z d x m S , (2.155)Tr( K − N K − N (cid:3) K − (cid:3) )= 12 g i π ǫ Z d x h m P G + m P i , (2.156)Tr( K − N K − N (cid:3) K − (cid:3) )= 12 g i π ǫ Z d x h m SE + m S i . (2.157)Summing the terms (2.126)-(2.157) we obtainSTr( K − N K − N K − T ) = 2 ia C m g π ǫ (2.158) Z d x h S (cid:3) S + P (cid:3) P − ¯ ψi / ∂ψ + E + G i . Acknowledgments
The work of M.D. and V.R. is supported by the project 141036 of the SerbianMinistry of Science. M.D. also thanks INFN Gruppo collegato di Alessandria for theirfinancial support during one year stay in Alessandria, Italy where a part of this workwas completed.
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