On the determination of anomalies in supersymmetric theories

We develop an efficient technique to compute anomalies in supersymmetric theories by combining the so-called nonlocal regularization method and superspace techniques. To illustrate the method we apply it to a four dimensional toy model with potentially anomalous N=1 supersymmetry and prove explicitly that in this model all the candidate supersymmetry anomalies have vanishing coefficients at the one-loop level.


I. INTRODUCTION
Supersymmetric quantum field theories have many remarkable properties. In particular quantum corrections are usually better under control in such theories than in others due to nonrenormalization properties implied by supersymmetry. However, it is not clear from the outset whether the supersymmetry of a classical theory survives as a symmetry of the quantized theory, due to the lack of consistent regularization methods which manifestly preserve supersymmetry in perturbation theory. Nevertheless, supersymmetry 'miraculously' appears to be preserved in standard supersymmetric theories.
An indirect but powerful and regularization independent tool to investigate whether or not supersymmetry can be anomalous consists in an analysis of the supersymmetric analog of the Wess-Zumino consistency condition [1]. Nontrivial solutions to this consistency condition are candidate supersymmetry anomalies whereas the absence of such solutions indicates that supersymmetry is not anomalous.
The consistency condition for supersymmetry anomalies, in combination with the usual Wess-Zumino consistency condition in the case of supersymmetric gauge theories, has been studied already for various D = 4, N = 1 globally supersymmetric models, see e.g. [2,3], and recently also for minimal supergravity [4]. It turns out that whether or not candidate supersymmetry anomalies exist depends decisively on the way supersymmetry is represented on the fields, i.e. on the structure of the supersymmetry multiplets present in the model in question. For standard representations, such as multiplets that can be described in terms of unconstrained or chiral scalar superfields, one finds that candidate anomalies for supersymmetry itself do not exist. However, this does not exclude the existence of supersymmetrized versions of other candidate anomalies such as ABJ chiral anomalies in super Yang-Mills theories. Moreover, there are non-standard representations of supersymmetry ("non-QDS-representations" in the terminology of [3]) which do give rise to candidate anomalies for supersymmetry itself.
When the cohomological analysis alone is not sufficient to exclude candidate anomalies due to the existence of nontrivial solutions to the consistency condition (for supersymmetry or other symmetries), one has to check by an explicit calculation whether or not these candidate anomalies have vanishing coefficients. To that end one needs an appropriate regularization method. One of the main disadvantages of most of the regularization methods designed for supersymmetric theories is the lack of a consistent implementation of the superspace techniques [5,6] -one of the main tools in supersymmetry-at the regularized level [5]. This drawback, somewhat analogous to the dimensional regularization troubles when dealing with chiral theories, becomes then relevant in analyzing the presence of anomalies in the model under consideration. Indeed, naive manipulations in superspace may lead to inconsistencies or ambiguities when computing divergent expressions, making impossible to detect and calculate (unambiguously) such anomalies. It would thus be desirable to design a method in which superspace computations were unambiguously defined.
In this paper we develop a new efficient technique to investigate anomaly issues in supersymmetric theories. It combines naturally superspace techniques, which facilitate the perturbative calculations in supersymmetric theories considerably, with the so-called nonlocal regularization [7,8], which has already been successfully used to compute one [9] and higher loop anomalies [8] in other (nonsupersymmetric) theories. Among others, the method allows to check whether or not supersymmetry itself is anomalous. We illustrate this by applying the method to a four dimensional supersymmetric toy model whose supersymmetry is potentially anomalous, as cohomological results indicate [3].
The paper is organized as follows. First we describe our method in section II. To that end we briefly recall the basic concepts of nonlocal regularization, emphasizing its use to determine anomalies, and describe how superspace techniques are naturally implemented in it. In section III we introduce the toy model and present its candidate supersymmetry anomalies. In section IV we then apply our method to this toy model and prove the absence of supersymmetry anomalies at the one-loop level. Three appendices finally collect our conventions.

II. NONLOCAL REGULARIZATION OF SUPERSYMMETRIC THEORIES
There exist many ways in the literature to algebraically compute (one-loop) anomalies. All of them are essentially based in testing the response of the -suitably regulated-partition function of the model under the (infinitesimal version of the) symmetry transformation under study. Departures from unity of the jacobian arising upon this change which can not be absorved by suitable counterterms reflect then the presence of anomalies in the model.
The so-called "nonlocal regularization" method, recently introduced in [7,8], fits perfectly well in this philosophy. Indeed, this approach proceeds by constructing from the original action S(Φ A ) and symmetry transformations δΦ A of the model a regulated action S Λ (Φ A ), invariant under a "regulated" version of the original symmetry, δ Λ Φ A , where Λ stands for a cut-off or regulating parameter. Such invariant action, exponentiated afterwards in the path integral, generates then a modified set of Feynman rules and propagators that yield finite Feynman integrals for finite values of the cut-off at all loop levels, and thus a finite partition function.
For our purposes, there are two main advantages of this approach relative to other "standard" regularization methods. First of all, the nonlocally regularized action S Λ (Φ A ) can just be seen as a "smooth" deformation of the original one such that its main features (dimensionality, field content, symmetries...) remain unaltered. Therefore, when dealing with supersymmetric theories, in particular, superspace computations at regulated level can be performed in exactly the same way as in the original theory. Second, and on top of that, the invariance of S Λ under δ Λ directly relates potential one-loop anomalies to the finite part of the functional trace -now completely regulated-of the jacobian matrix, namely 1 where (−1) A ≡ (−1) |ΦA| stands for the Grassmann parity of the field Φ A . In view of these facts, nonlocal regularization appears thus as an excellent candidate to implement our programme.
In what follows, we briefly summarize the construction of the nonlocal action S Λ and of its symmetries δ Λ , as well as the specific form of the anomaly (2.1), along the lines of refs. [7,8], implementing afterwards the standard superspace techniques in this framework.

A. Basics on Nonlocal Regularization
Consider a theory defined by a classical action S(Φ A ), which admits a sensible perturbative decomposition into free and interacting parts Introduce now a field independent operator (T −1 ) A B such that a second order derivative 'regulator' R A B arises through the combination and construct from this object the so-called smearing operator, ε A B , and shadow kinetic operator, To each original field Φ A it is now associated an auxiliary, or 'shadow', field Ψ A with the same statistics. Both sets of fields are then coupled by means of the auxiliary actioñ (2.5) with A(Ψ), the kinetic term for the auxiliary fields, constructed with the help of (2.4) as and where the "smeared" fieldsΦ A appearing in the free part of the auxiliary action (2.5) are defined, using (2.3), bŷ Φ A ≡ (ε −1 ) A B Φ B . The perturbative theory described by (2.5), when only external Φ lines are considered, is then seen to describe the same theory as the original action (2.2). However, the special form of propagators and couplings in (2.5) lead the loops formed with shadow propagators to isolate the divergent parts of the original diagrams. As a consequence, dropping out these loop contributions, i.e., the quantum fluctuations of the shadow fields by hand, regularizes the theory. Such ad hoc procedure may however be simply implemented by putting the auxiliary fields Ψ classically on-shell. The classical shadow field equations of motion should then be solved, in general, in a perturbative fashion and its solution Ψ 0 (Φ) substituted in the auxiliary action (2.5). The result of this process is the nonlocalized action to be used in regularized perturbative computations Moreover, as mentioned above, the nonlocalization procedure just presented has the merit of preserving at tree level a distorted version of any of the original continuous symmetries of the theory. Indeed, assume the original action (2.2) be invariant under the infinitesimal transformation Then, the auxiliary action (2.5) is seen to be invariant under the auxiliary infinitesimal transformations while the nonlocally regulated action S Λ (Φ) (2.7) becomes invariant under with Ψ 0 (Φ) the solution of (2.6). In this way, an extensive use of the chain rule allows to determine a closed form for the anomaly (2.1) in terms of propagators and vertices of the original theory as and with the regulated identity (δ Λ ) A B defined by in terms of the functional hessian of the original interaction in (2.2) The proof of these statements is straightforward and can be found in the original references [7,8], to which we refer the reader for further details.

B. Implementation of superspace techniques
The nonlocal regularization procedure outlined above applies of course to all kinds of perturbative models, including supersymmetric ones. Now, it is well-known that in supersymmetric theories perturbative calculations can often be considerably simplified by means of superspace techniques due to the cancellation of terms caused by supersymmetry. It is therefore natural to look for a way to implement these techniques in the nonlocal regularization procedure. An obvious idea is to replace ordinary fields by superfields. However one faces immediately the following related difficulties: how should one define functional derivatives w.r.t. arbitrary (constrained) superfields and integrations over their 'superspace coordinates' ? These two problems appear to make the simple substitution 'fields → superfields' impossible except in very special cases where one deals only with particular superfields such as unconstrained or chiral ones. Thus in general we cannot simply take the Φ's of the previous subsections to be superfields.
Fortunately this is not necessary at all since superspace techniques are of course not restricted to true superfields 2 . In fact, we will show now that they apply also to "constituents" of superfields such as provided a, b, c are elementary fields. Namely then we can define functional derivatives 3 w.r.t. ϕ simply through Summation over their indices in de Witt's condensed notation includes then simply an integration d 6z ≡ d 4 xd 2θ . Alternatively we can (and will) use instead of ϕ the quantity which is antichiral in the sense that where the standard covariant derivatives are defined as (2.14) However Φ is not in general a superfield (see appendix B) i.e. (2.13) does not reflect the transformation properties of Φ. The functional derivative w.r.t. Φ is then defined by means of (2.11) according to This results in (2.15) can indeed be found in many textbooks on supersymmetry for functional derivatives w.r.t. to antichiral superfields -we just extend it to constituents of superfields satisfying (2.13). Due to the presence of the antichiral 2 See appendix B for a discussion of the concept of superfield. 3 For definiteness all formulae are written for left-derivatives in this subsection. projector 1 2 D 2 in (2.15), summation over the indices of these constituents does not involve the integration d 8 z but again only an integration d 6z . Analogous formulae hold of course for functional right-derivatives and chiral quantities.
We conclude that we can use quantities like (2.10) or (2.12) in nonlocal regularization instead of ordinary fields. This remains true even if it is impossible to combine all the elementary fields in such quantities -the remaining elementary fields may be treated as usual, i.e. one can use quantities (2.10) or (2.12) and ordinary fields simultaneously if necessary. The only thing one has to keep in mind when dealing with such constituents is that operators such as (2.14) or the usual generators of supersymmetry transformations do not have the same interpretation on constituents of superfields as on superfields themselves: in particular the operators (2.16) do not represent the supersymmetry transformations anymore on all of the constituent fields.

A. Multiplet and supersymmetry transformations
The four dimensional toy model we are going to use contains only a supersymmetry multiplet considered in section 7 of [3]. This multiplet consists of complex Weyl spinors χ, ψ and η, a complex vector field V , and two complex scalar fields A and F . On these fields the abstract supersymmetry algebra Table 1 The assignment of the dimensions (dim) to the fields in table 1 follows from the choice dim(χ)=1/2, which will be the power counting dimension of χ, and from the standard convention dim(D α )= dim(Dα)=1/2, dim(∂ a )=1. Supersymmetry transformations, δ susy , of the fields in table 1 are then defined according to the relation where the parameters ǫ α are constant anticommuting spinors. The supersymmetry multiplet and transformation laws of table 1 can also be formulated in superspace (c.f. appendix B) which will be useful within the computation of the anomaly coefficients. However, for the reasons we have just explained, we will apply a somewhat unconventional approach involving not only true superfields but also special constituents of them, which will be introduced and discussed in the following.
The fundamental ('defining') superfield of the multiplet of table 1 is with where we used the identity (A1), table 1 and the notation θ∂θ = θ α ∂ ααθα , θ 2 = θ α θ α andθ 2 =θαθα. The split of G α into the constituents H α and K will be useful later on, in particular since the latter are 'antichiral' in the sense that 4 whereas G α itself satisfies the 'constraint' It is important to realize and keep in mind that H α is not a superfield since it does not satisfy the first identity (B1). Rather, its supersymmetry transformations are given by In contrast, K is a true superfield and thus satisfies (B1), We remark that the supersymmetry multiplet of table 1 can be truncated (consistently with the supersymmetry algebra) in two ways, by setting to zero either all the fields χ, V, η or all the fields A, ψ, F . One would then be left with standard antichiral supersymmetry multiplets given by (A,ψ, F ) and (χ, V, η) respectively, corresponding to K and H α respectively. Hence, the supersymmetry multiplet of table 1 may be regarded as a nontrivial merger of these two multiplets. Alternatively, one can regard it itself as the truncation of a full complex vector multiplet corresponding to an unconstrained complex scalar superfield.

B. Action
Using the techniques of [3] one can prove that the most general real action for the supersymmetry multiplet of table 1 which is a) polynomial in the elementary fields and their derivatives, b) constructible out of field monomials of dimension ≤ 4 (with dimensions as in table 1), c) Poincaré invariant and d) invariant (up to surface terms) under the supersymmetry transformations D α andDα given in table 1, can be written, up to surface terms, in terms of superspace integrals in the form 12) 14) where G α and K are the superfields given in (3.3) and (3.4); µ 2 , a 3 , m, b 1 , b 2 are complex parameters and a 1 , a 2 , b 3 are real parameters. The action is spelled out explicitly in appendix C. Some special features of this general action merit now special consideration. First of all, the terms in (3.12-3.15) corresponding to the parameters µ 2 , m, b 2 give rise to a superpotential (µ 2 K −mK 2 −b 2 K 3 ) for the antichiral multiplet (A,ψ, F ) since one has where ∼ = denotes equality up to a total derivative. Expression (3.16) together with the kinetic term corresponding to the parameter a 2 constitute thus nothing but the familiar action of a Wess-Zumino model for the fields A, ψ, F making up the (anti)chiral superfields K,K. The other terms in the action involve also the fields χ, V, η and in particular couple them to A, ψ, F . For simplicity we will later not work with the above general action but restrict ourselves to the simpler action 3.17) i.e. we will set to zero the Wess-Zumino superpotential (3.16) as well as the coefficients a 3 and b 3 . Furthermore we will assume since otherwise (3.17) does not give well-defined propagators for all the fields. a 1 = 0 is imposed since otherwise the kinetic terms of (3.17) reduce to those of the Wess-Zumino model for A, ψ, F and the remaining fields would not propagate. a 1 + a 2 = 0 warrants that (3.17) has no gauge invariance.

C. Candidate anomalies
By standard arguments, analogous to those used in [1] and applied to the vertex functional (effective action), one concludes from the (classical) supersymmetry algebra (3.1) that at lowest order inh supersymmetry anomalies must satisfy the consistency conditions where the contributions ∆ α and ∆α to such anomaly are local functionals of the fields. Furthermore one can assume for any local functional Γ 0 of the fields since otherwise the anomaly can be removed through a local counterterm, at least up to terms of higher order inh. The consistency condition (3.19) and the non-triviality condition (3.20) are most efficiently formulated and analysed using cohomological techniques. To that end one introduces a 'BRST'-operator s corresponding to the algebra (3.1) where ξ α are constant commuting supersymmetry ghosts and C a are constant anticommuting translation ghosts (D α andDα vanish on the ghosts). s is nilpotent and allows to reformulate (3.19) and (3.20) through with ∆ = ξ α ∆ α +ξα∆α .
In (3.19) and (3.21) it is understoood that the operators (D α resp. s) act on the integrands of the ∆'s and Γ 0 and, in general, equalities need to hold only on-shell (up to surface terms). For the model in question two complex solutions of (3.21) have been given in section 7 of [3]: whereψ ′ is the combinationψ ′α =ψα + 2i∂ αα χ α . (3.23) The explicit form of ∆ 2 is given in appendix C. We note that both ∆ 1 and ∆ 2 give in fact rise to two independent real solutions of (3.21), given by their real and imaginary part respectively.
Using the methods of [3] and extending them to the on-shell problem 5 one can prove that, up to trivial solutions of the form sΓ 0 and surface terms, the functionals (3.22) and their complex conjugates are indeed the only inequivalent solutions to (3.21) in our model which have the correct Lorentz transformation properties and are polynomials in all the fields and their derivatives with dim(∆) ≤ 4 (using dim(ξ) = −1/2).
Furthermore ∆ 1 and ∆ 2 are cohomologically nontrivial, i.e. there is no local functional Γ 0 of the fields such that sΓ 0 equals ∆ 1 or ∆ 2 on-shell modulo a surface term. This can be verified straightforwardly by an explicit inspection of all the relevant candidates for Γ 0 . In fact there are only finitely many such candidates as only functionals need to be considered which have the same dimension as the respective ∆ (1 resp. 4) and which are Lorentz-invariant, thanks to the properties of s. Without going into details we remark that the presence of candidate supersymmetry anomalies in our model is due to the fact that the representation of the supersymmetry algebra given in table 1 of section III A does not have "QDSstructure" in the terminology of [3], in contrast to more standard representations of supersymmetry. Furthermore we note that the non-QDS-property itself can be traced back to the 'constraint' (3.8).
Finally we add two comments concerning the consistency condition for supersymmetry anomalies in general and its solutions ∆ 1 and ∆ 2 : a) In superspace notation ∆ 1 and ∆ 2 read with G α as in (3.3) andΨ ′ being the antichiral superfield whose lowest component field isψ ′ (3.23) The presence of θ 2 in the integrands in (3.24) indicates that ∆ 1 and ∆ 2 cannot be written as superspace integrals d 8 z (or d 6z ) over true (antichiral) superfields. This shows that in general it would be misleading to formulate the consistency conditions (3.19) resp. (3.21) in terms of the operators ∇ α defined in (2.16) instead of the D α (recall that the ∇'s represent the supersymmetry transformations only on true superfields).
b) The dimensions of ∆ 1 and ∆ 2 indicate that they would play different roles if they would occur in the (anomalous) jacobian of supersymmetry transformations: ∆ 1 has dimension 1 and thus would eventually arise as a divergent contribution to that jacobian, in contrast to ∆ 2 which has canonical dimension 4 and is interpreted as a genuine potential anomaly.

IV. COMPUTATION OF THE ANOMALY COEFFICIENTS
Let us finally pass to investigate the actual presence of the candidate anomalies (3.22) in our toy model by applying expression (2.8) of the nonlocally regularized form of the anomaly to it. For the sake of simplicity, to illustrate the procedure and results we restrict ourselves to the simple version (3.17) of the general action (3.11).
The structure of the superfield (3.3) and the previous considerations immediately suggest to work with its '(anti)chiral' constituents (3.4) and use as basis to express matrix-like operators where latin indices express compactly antichiral (a) and chiral (ā) components. In terms of these (anti)chiral components, t he action (3.17) reads then S = d 8 z ia 1 (H α + θ α K)∂ αα (Hα +θαK) + a 2 KK As pointed out in subsection II B (and in many textbooks), the constrained character of these (anti)chiral components requires some reinterpretation of their superspace integration and functional differentiation rules. First of all, the functional derivative rules for (anti)chiral fields (2.15), now reading where δ a b encodes, according to the compact notation we are using, a discrete identity as well as the 8-dimensional delta function δ 8 (z −z ′ ) in superspace, express nothing but the fact that (anti)chiral fields and operators obtained from functional differentiation with respect to them naturally live in six dimensional superspace. This fact is conveniently expressed by introducing the projector in the space of antichiral-chiral superfields (4.2) and an analogous relation for the chiral sector. '(Anti)chiral' kernels will thus be typically expressed, in compact notation, as so that super matrix multiplication will then yield, according to (4.2) The nonlocal regularization of the model (4.1) requires now the identification of the basic quantities involved in the computation, namely the jacobian (2.8) of the original transformation, the hessian of the interaction (2.9) and the regulating objects related to the kinetic operator (2.2). The jacobian of the original transformation (3.2) adopts in the above basis, according to eqs. (3.9), (3.10), the form with its antichiral and chiral sectors given by In an analogous way, the hessian of the interaction term in (4.1) results in I A B = (P q IP q ) A B , with the 'naive' hessian I A B expressed as Finally, the kinetic operator is found to be F A B = (P q F P q ) A B , with the 'naive' kinetic term F A B given by Introducing then as operator T −1 the free propagator of the model in superspace up to (−✷) −1 , namely ( a suitable regulator, diagonal and quadratic in space-time derivatives, arises In this way the corresponding smearing and shadow kinetic operators (2.3), (2.4), adapted to the chiral case, result in with ε 2 andσ defined as The form of the candidate anomalies (3.22) -involving only either products of antichiral fields H α , K, or of chiral fieldsHα,K, but no crossed terms-indicates that the evaluation of their coefficients by means of the supertrace (2.8) can now be considerably simplified by considering for instance only the antichiral sector, i.e. by neglecting the fieldsHα andK, and by further restricting the computation to only linear and trilinear terms in H α , K, namely to the first and third order interaction terms 7 . The coefficients coming from the chiral sector contributions can then be automatically determined by complex conjugation. Therefore, from now on we are going to concentrate our attention in the termsÃ 3, (4.4) where the subscript 'anti' indicates that all terms involvingHα andK are neglected. Our main task shall now consist in determining the diagonal entries of the matrix involved in expression (4.4). First of all, the nth power of the matrix O A C I C B reads, under the above restrictions Its diagonal blocks -the relevant ones taking into account the block diagonal form of the jacobian (4.3)-can be easily found by using the commutation relation resulting in in terms of the quantities S α , G α defined as where all the operators are understood to act on everything on their right. Terms indicated by dots in the above matrices turn out to be irrelevant for the present computation. Afterwards, straightforward matrix multiplication yields where the expressions for the antichiral sector operators are found to be, upon use of the commutation relation ǫ β ∇ β , θ α = ǫ α , whereas the chiral sector operator is directly given by The general expression ofÃ n (4.4) is thus (4.9) where the extra minus sign comes from taking the discrete trace over the fermionic fields, while the symbols Tr and Tr stand respectively for the functional traces in the antichiral and chiral superspaces, namely (4.10) Upon substitution of expressions (4.7) and (4.8), both traces in (4.9) are then seen to share similar structures. However, there is the fundamental difference that such functional traces are taken in different superspaces, according to (4.10). Therefore, in order to compare both expressions, some mechanism should be found to relate supertraces of antichiral expressions to those of chiral ones. Fortunately, it is not difficult to verify, as shown in appendix D, that for chiral operatorsĀ, namely those verifyingDαĀ = 0, the following relation holds Using this result as well as the commutation relation (4.5) and the cyclic property of the regulated trace, the antichiral sector contribution Tr [−(A n ) α α + A n ] to (4.9) can be rewritten in chiral form as with the operator B n given by after substitution of S α by its explicit expression (4.6). In this way, B n is seen to 'almost' coincide with C n (4.8) when reading it from the right to the left. This similarity may conveniently be exploted by using the property that the trace of an operator and of its transpose coincide. Combining further this fact with the cyclic property of regulated traces, the following relations are seen to hold so that the contribution coming from the antichiral sector, Tr[B n ] (4.12), is seen to exactly cancel that coming from the chiral sector, Tr[C n ], for all n. The present computation leads thus to the vanishing ofÃ n (4.9) for all n and with it, of the potential anomalies of our model. Therefore, we conclude that the latter, potentially present on cohomological grounds, actually do not show up in the model we have analyzed at the one loop level. We have also checked that this remains valid for supersymmetric actions which differ from (3.17) and arise from (3.11) by turning on other (combinations of) coefficients such as a 3 , m or b 3 . However, we have not performed the computation for the most general action (3.11), as the main purpose of considering the toy model was the illustration of the method outlined in section II.

V. CONCLUSION
The purpose of this paper is to show that implementation of superspace techniques in the framework of nonlocal regularization constitutes a suitable and efficient tool to analyze anomaly issues. To outline and illustrate the method, we have applied it to a toy model whose supersymmetry, by cohomological arguments, is potentially anomalous, but turns out to be actually nonanomalous at the one-loop level. As a byproduct, the result of the computation gives further evidence that the remarkable quantum stability of supersymmetry even extends to models which admit nontrivial solutions of the consistency condition for supersymmetry anomalies. Finally, although not proven, our construction also points out to nonlocal regularization as a possible candidate for a supersymmetric invariant regularization method.

Contraction of spinor indices:
ψχ := ψ α χ α ,ψχ :=ψαχα Lorentz vector indices in spinor notation: The grading (Grassmann parity) |X| of a field or an operator X is determined by the number of its spinor indices and its ghost number (gh), The grading of the fields φ i determines their statistics, Complex conjugation of a field or operator X is denoted byX. Complex conjugation of products of fields and operators is defined by In particular this implies ∂/∂φ = (−) |φ| ∂/∂φ and thus the minus sign in front of ∂/∂θ in (2.16) and (2.14).

APPENDIX B: SUPERFIELDS AND CONSTITUENTS
In this appendix, we briefly review the construction of superfields out of ordinary fields for given supersymmetry transformations of the latter according to the conventions used in this paper. As usual we implement the supersymmetry transformations on superfields through the operators ∇ α ,∇α (2.16). Then, given a (linear) representation D α ,Dα of the supersymmetry algebra (3.1) on ordinary fields φ i such as in table 1 of section III A, superfields are defined as functions Σ of the θ α ,θα, φ i and of the derivatives of the φ i , Σ = Σ(θ,θ, φ, ∂φ, . . .), satisfying where D α andDα act nontrivially only on the φ i and their derivatives and anticommute with all the θ's andθ's. The operators ∇ α ,∇α (2.16) provide then a representation of the supersymmetry algebra (3.1) with (P a , Q α ,Qα) ≡ (−∂ a , −∇ α , −∇α). Note that ∇ α Σ is not a superfield since itsDα transformation is not given by∇α∇ α Σ, but rather byDα Instead, and in contrast to the ∇'s, the standard 'covariant derivatives D α ,Dα (2.14) map superfields to superfields because they anticommute both with the D's and with the ∇'s. Having characterized superfields abstractly by (B1), we can now construct them explicitly: any superfield, i.e. any solution of (B1) can be written in the form Σ = exp(θD +θD) f (φ, ∂φ, . . .) , where f (φ, ∂φ, . . .) is a function of the (ordinary) fields and their derivatives and we used the summation conventions θD = θ α D α andθD =θαDα. The proof of this statement is straightforward using that (i) (B2) satisfies (B1) for any f (φ, ∂φ, . . .), as can be easily checked directly, (ii) any nonvanishing superfield has a nonvanishing θ-independent part which is required by (B1). The assertion is now proved as follows: given a nonvanishing solution Σ of (B1) with θ-independent part f (φ, ∂φ, . . .) we consider Σ ′ = Σ − exp(θD +θD)f (φ, ∂φ, . . .). The latter is a superfield due to (i) and must vanish due to (ii) since by construction it has no θ-independent part.
with F ab = ∂ a V b − ∂ b V a and ✷ = ∂ a ∂ a = 1 2 ∂ αα ∂α α . The integrand of the candidate anomaly ∆ 2 in (3.22) reads explicitly APPENDIX D: PROOF OF RELATION (4.11) In the perturbative computation of the anomaly coefficients performed in section IV, relation (4.11) has been seen to be crucial in checking their vanishing. In this appendix, we prove that relation.