On the Renormalization of Theories of a Scalar Chiral Superfield
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On the Renormalization of Theories of a Scalar Chiral Superfield
Oliver J. Rosten ∗ Dublin Institute for Advanced Studies,10 Burlington Road, Dublin 4, Ireland
Abstract
An exact renormalization group for theories of a scalar chiral superfield is formulated, directlyin four dimensional Euclidean space. By constructing a projector which isolates the superpotentialfrom the full Wilsonian effective action, it is shown that the nonperturbative nonrenormalizationtheorem follows, quite simply, from the flow equation. Next, it is argued that there do not exist anyphysically acceptable non-trivial fixed points. Finally, the Wess-Zumino model is considered, as alow energy effective theory. Following an evaluation of the one and two loop β -function coefficients,to illustrate the ease of use of the formalism, it is shown that the β -function in the massless casedoes not receive any nonperturbative power corrections. ∗ Electronic address: [email protected] ontents I. Introduction Acknowledgments II. The Flow Equation
III. The Wilsonian Effective Action IV. Diagrammatics V. The Nonrenormalization Theorem
VI. The Dual Action VII. Critical Fixed Points γ ⋆ ≥ γ ⋆ < VIII. The β -function β -Function from the Dual Action 41B. Perturbative Computations 431. The One-Loop Coefficient 432. The Two-Loop Coefficient 45C. Nonperturbative Considerations 46 IX. Conclusion Note Added . SUSY Conventions B. Classical Two-Point Vertices References I. INTRODUCTION
A crucial question that should be asked of any quantum field theory is whether or not it isrenormalizable. However, to definitively answer this question is often far from easy. A case inpoint is scalar field theory in d = 4 dimensions. Let us start by supposing that we introducean overall momentum cutoff, Λ , the ‘bare scale’. Now, without any further restrictions,there are an infinite number of different theories we could consider, corresponding to differentchoices of the bare interactions. At least within perturbation theory, one such choice appearsto be special: if we take just a mass term and a λϕ interaction then it is very well knownthat the theory is perturbatively renormalizable. In other words, if we send Λ → ∞ (a.k.a.taking the continuum limit), then all ultraviolet (UV) divergences can be absorbed into justthe two couplings and the anomalous dimension of the field. However, beyond perturbationtheory, this breaks down. For example, defining this λϕ theory on a lattice, it can be(essentially) proven that the only continuum limits are trivial [1].The resolution to this apparent paradox is that taking the limit Λ → ∞ within per-turbation theory amounts to a sleight of hand. Imagine integrating out degrees of freedombetween the bare scale and a much lower, effective scale, Λ. The point is that perturbationtheory done at the scale Λ is in fact only correct up to O (Λ / Λ ) terms. Formally, one cansend Λ → ∞ , after which all quantities can be written in ‘self-similar’ form [2, 3]: i.e.the results of all perturbative calculations can be expressed as functions of the renormal-ized couplings, m (Λ) and λ (Λ), and the anomalous dimension, η (Λ). Indeed, self-similarityis precisely a statement of renormalizability, since nothing has any explicit dependence onΛ / Λ . The sleight of hand has come about because the various perturbative series are not,by themselves, well defined: when one attempts to resum the hopefully asymptotic pertur-bative series using e.g. the Borel transform, it is found that there are poles on the positive3eal axis of the Borel plane, impeding this procedure. Whilst one can avoid these poles by deforming the contour of integration, there is anambiguity relating to whether the contour goes above, or below, each pole. To arrive atan unambiguous result one must include the Λ / Λ terms which were earlier thrown away.Doing so manifestly spoils self-similarity and hence renormalizability.If we define the β -function, as usual, according to β ≡ Λ dλd Λ , (1.1)and denote the one-loop β -function by β then it is apparent that the Λ / Λ contributionsare indeed nonperturbative: Λ / Λ ∼ e − / β λ . (1.2)So, it is quite possible that perturbative conclusions about renormalizability differ from thenonpertubative ones. Consequently, it is quite consistent that the perturbatively renormal-izable λϕ model does not strictly have an interacting continuum limit. But what about allthe other possible models we could have written down at the bare scale?At first sight, answering this question is nigh impossible: after all, we can hardly checkevery single such model to see whether, nonperturbatively, an interacting continuum limitexists. Fortunately, the question can be rephrased in a different way which, whilst stillhard to answer in general, is nevertheless much more amenable to solution. To do this, wemust adopt Wilson’s picture of renormalization, whereby nonperturbatively renormalizabletheories follow directly from critical fixed points of the renormalization group (RG) andthe ‘renormalized trajectories’ emanating from them [5]. The first point to make is thatcritical fixed points correspond to conformal field theories. These theories are thereforerenormalizable in the nonperturbative sense: since they are scale independent, they mustbe independent of Λ , which can thus be trivially sent to infinity.It is very simple to show, nonperturbatively, that scale dependent renormalizable the-ories follow by considering flows out of some critical fixed point along the relevant andmarginally relevant directions as defined at this fixed point [2]. A crucial feature of these Poles of this type can have different origins; those arising due to small/large loop momentum behaviourare known as renormalons—for a review see [4]. For theories which are perturbatively renormalizable butfor which an interacting continuum limit based around the Gaussian fixed point nevertheless does notexist, ultraviolet (UV) renormalons give rise to the poles along the positive real axis. Henceforth, we will take ‘relevant’ to include marginally relevant. d = 4 scalarfield theory: with respect to this fixed point, the only relevant direction is the mass; λ ismarginally irrelevant and all other directions are even more irrelevant still. However, if anon-trivial fixed point is found, then everything changes. If this fixed point were to haverelevant directions, then these could be used to construct a continuum limit. Now, supposethat there exist RG flows from this putative fixed which take us down towards the Gaussianfixed point. As we begin our journey into the infrared, at some point we pass the scalewe earlier denoted by Λ . We can, if we choose, still call the action at this scale the bareaction. But now it is determined by our choice of renormalized trajectory (this informationis encoded in the integration constants associated with the relevant directions). It is forthis reason that the bare action along a renormalized trajectory is sometimes referred to asthe ‘perfect action’ in the vicinity of the UV fixed point [6]. Continuing our journey, weultimately reach the vicinity of the Gaussian fixed point. Here, all interactions die away,with the exception of the mass, which is relevant at the Gaussian fixed point. However, ofthe other interactions, λ dies away by far the slowest (logarithmic decay, compared to powerlaw decay) and so, sufficiently close to the Gaussian fixed point, we are effectively back toa λϕ model. Indeed, this model is the good low energy effective theory; but note that,crucially, all other interactions would have to be retained if one wished to reconstruct theRG trajectory back into the UV.This scenario, whereby a low energy effective theory is the result of a flow down froma UV fixed point is often called asymptotic safety [7]. Recently, however, such a scenariowas ruled out for scalar field theory in d ≥ η ⋆ (we will use ⋆ to denotefixed point quantities). First, fixed points with η ⋆ ≥ d = 4, in thecase where η ⋆ = 0, Pohlmeyer’s theorem [10] implies that the only critical fixed point is theGaussian one.) For η ⋆ ≥
0, it was demonstrated that no non-trivial fixed points exist in d ≥
4. As for fixed points with η ⋆ <
0, it was shown that, should such fixed points exist,then they are necessarily non-unitary since the kinetic term lacks the standard p part. Thiscan be seen explicitly for the exotic Gaussian fixed points discovered by Wegner [11].The aim of this paper is to explore various aspects of the renormalizability of theoriesof a scalar chiral superfield in four dimensions. In line with the previous discussion, weavoided explicit mention of the Wess-Zumino model in the previous sentence. As before,this is because in this supersymmetric case1. it is very well known that the Gaussian fixed point does not support interacting renor-malized trajectories;2. there are no interacting continuum limits of the Wess-Zumino model.The latter fact can be deduced much more straightforwardly [12, 13] than in the case of d = 4 scalar field theory, on account of the nonrenormalization theorem [14] and Pohlmeyer’stheorem. Indeed, we can state in complete generality that there cannot be any non-trivialfixed point with a three-point superpotential coupling, λ , as we now discuss. (Henceforth,we exclusively use λ to denote this coupling.)The first point to make is that, to uncover fixed point behaviour, we should rescale todimensionless variables by dividing all quantities by Λ raised to the appropriate scalingdimension. This means that the superpotential does now renormalize, but only via thescaling dimension of the field. In particular, the three-point superpotential coupling, whichhas zero canonical dimension, acquires a scaling from the anomalous dimension of the field.Now, at a fixed point, all couplings must stop flowing, by definition. Therefore, if the fixedpoint action possesses a three-point superpotential term, the anomalous dimension mustvanish. But Pohlmeyer’s theorem implies that any critical fixed point (in integer dimension)with vanishing anomalous dimension must be the trivial one.Of course, this says nothing as to the existence, or otherwise, of non-trivial fixed pointswithout a three-point superpotential term. Moreover, such fixed points could potentiallyfurnish an asymptotic safety scenario for the Wess-Zumino model: since we are working in6imensionless variables, λ does scale and so can in principle be a relevant direction at a fixedpoint (this is no different from saying that the mass is relevant at the Gaussian fixed point,despite the fact that there are no quantum corrections along the trivial mass direction).However, if such fixed points are to exist, it was recently shown that they can only be usedto construct an asymptotic safety scenario for the Wess-Zumino model if the fixed point has1. negative anomalous dimension;2. at least one relevant direction coming from the K¨ahler potential.The proof of this is very simple, utilizing only the nonrenormalization theorem andPohlmeyer’s theorem [15].However, by adapting the methodology of [8], we will show that, should any fixed pointswith negative anomalous dimension exist, they necessarily correspond to non-unitary theo-ries. Consequently, an asymptotic safety scenario for the Wess-Zumino model is ruled out.Furthermore, it will be shown that there are no physically acceptable non-trivial fixed pointswith positive anomalous dimension, either (just because such a fixed point cannot possess atrajectory that flows towards the Wess-Zumino action does not mean that such a fixed pointcannot exist; a separate argument is required to show this). Thus, an asymptotic safetyscenario is ruled out for general theories of a scalar chiral superfield.In addition to this comprehensive study of the non-existence of useful fixed points, a newproof of the nonperturbative renormalization theorem will be provided. It is not as elegantas Seiberg’s beautiful argument [14] but it has the advantage of being less heuristic, as itfollows directly (and, it should be added, rather simply) from the flow equation.Finally, the β -function of the Wess-Zumino model—considered as a low energy effectivetheory—is studied. First, an explicit computation of the one and two-loop coefficients isprovided, to illustrate the ease of use our approach which, we note, is formulated directly in d = 4. Secondly, we adapt an analysis performed in QED [16] to show that the β -functionin the massless model (given the definition of the coupling implicit in the approach) is freeof nonperturbative power corrections and hence is expected to be (Borel) resummable.The formalism that will be employed throughout this paper is the Exact RenormalizationGroup (ERG), which is essentially the continuous version of Wilson’s RG [5, 17]. Centralto the approach is the effective cutoff, Λ, (introduced earlier) above which the modes of thetheory under examination are regularized. The physics at the effective scale is encapsulated7y the Wilsonian effective action, S Λ , whose evolution with Λ is given by the ERG equation.It is curious that, despite the success of the ERG in addressing nonperturbative problemsin Quantum Field Theory (QFT) (see [3, 18–23] for reviews) and despite the fact thatsome of the most penetrating insights into supersymmetric theories utilize the Wilsonianeffective action (including the nonrenormalization theorems [14] and the Seiberg-Wittensolution [24, 25]) applications of the ERG to supersymmetric theories are rather limited,both in number and in scope [26–37] (see also the note added at the end of the paper). Itis hoped, then, that the concrete results that this paper provides will lead to a developmentof this—surely fruitful—area.The rest of this paper is arranged as follows. In section II we will discuss generalizedERGs and adapt the formalism to theories of a scalar chiral superfield. Our subsequentanalysis is facilitated by the introduction, in section III, of a form for the Wilsonian effectiveaction in which all the superspace coordinates are Fourier transformed. This allows us todirectly develop a simple diagrammatic representation for the flow equation, which is done insection IV, and to prove the nonrenormalization theorem, which is the subject of section V.In section VI, a construction is introduced (the ‘dual action’ of [8]) which is necessary forthe analysis of the existence of critical fixed points (section VII) and aids the discussion onthe β -function of the Wess-Zumino model (section VIII). We conclude in section IX. Acknowledgments
I would like to thank IRCSET for financial support.
II. THE FLOW EQUATIONA. Generalized ERG Equations
Throughout this paper, we will work in d = 4 Euclidean space. We will generally usethe same symbol for four-vectors and their moduli, the meaning hopefully being clear fromthe context. In appendix A we review the approach of [38] to the problem of implementingEuclidean N = 1 superfields, and set our conventions. These conventions are such that onewill get the correct signs when doing spinor algebra by using the appropriate formulae ofWess and Bagger [39], but replacing the Minkowski metric by δ µν . Digging inside, however,8here are some differences—notably in the definition of σ µ —but these can largely be for-gotten about. Note that Hermitian conjugation is replaced by ‘Osterwalder and Schrader’conjugation, which we will denote by OSC (schematically, for what we will do, this makesno difference).Working, for the moment, in some generic QFT with fields ϕ , a generalized ERG followsfrom the fundamental requirement that the partition function is invariant under the flow [40,41]: − Λ ∂ Λ e − S Λ [ ϕ ] = Z x δδϕ ( x ) (cid:0) Ψ x [ ϕ ] e − S Λ [ ϕ ] (cid:1) , (2.1)this property being ensured by the total derivative on the right-hand side. The Λ derivativeis performed at constant ϕ . The functional, Ψ, parametrizes the continuum version of ageneral Kadanoff blocking [42]. To generate the family of flow equations to which Polchinski’sformulation [43] of the ERG belongs, we take:Ψ x = 12 ˙∆ ϕϕ ( x, y ) δ Σ Λ δϕ ( y ) , (2.2)where it is understood that we sum over all the elements of the set of fields ϕ . The ˙∆s arethe ERG kernels, which are generally different for each of the elements of ϕ . In momentumspace, each kernel incorporates a cutoff function, c ( p / Λ ), which dies off sufficiently fastas p / Λ → ∞ to implement ultraviolet regularization. The dot on top of the ∆ is definedaccording to ˙ X ≡ − Λ ∂ Λ X. Returning to (2.2), and henceforth dropping the various subscripted Λs, we takeΣ ≡ S − S, (2.3)where ˆ S is the ‘seed action’ [44–47], a nonuniversal input which controls the flow but of whichall physical quantities should be independent. Given the choice (2.2), and a choice of cutofffunction, the seed action encodes the residual blocking freedom. The only restrictions on theseed action are that it is infinitely differentiable and leads to convergent loop integrals [44,47]. The first requirement is that of ‘quasi-locality’ (mentioned in the introduction), whichmust apply to all ingredients of the flow equation. Quasi-locality ensures that each ERGstep is free of IR divergences or, equivalently, that blocking is performed only over a localpatch. The seed action has the same structure and symmetries as the Wilsonian effective9ction; however, we choose the former, whereas we solve for the latter. Our flow equationreads: − Λ ∂ Λ S = 12 δSδϕ · ˙∆ · δ Σ δϕ − δδϕ · ˙∆ · δ Σ δϕ (2.4)where, as ususal, we employ the shorthand A · B ≡ R d D x A ( x ) B ( x ). Similarly, A · ˙∆ · B ≡ R x,y A x ˙∆( x, y ) B y = R d D p / (2 π ) D A ( p ) ˙∆( p ) B ( − p ). The two terms on the right-hand sideof (2.4) are often referred to as the classical and quantum terms, respectively, for reasonsthat will become apparent when we discuss the diagrammatics.At this point, an example is useful. Suppose that we take ϕ to be a single scalar fieldand make the choice ∆( p ) = c ( p / Λ ) p . (2.5)We interpret ∆( p ) as a UV regularized or ‘effective’ propagator. Using this definition, wesplit the actions according to S [ ϕ ] = 12 ϕ · ∆ − · ϕ + S IΛ [ ϕ ] , ˆ S [ ϕ ] = 12 ϕ · ∆ − · ϕ + ˆ S IΛ [ ϕ ] . (2.6)These latter two expressions serve as a definition for what we mean by S I [ ϕ ] and ˆ S I [ ϕ ];clearly, they can be interpreted as the interaction parts of the Wilsonian effective action andseed action, respectively. Note that, just because we have not included a mass term in theeffective propagator, (2.5), does not necessarily mean that the theory is massless: a massterm could be included in, or generated by, S I [ ϕ ]. Thus (2.6) should be viewed simply as aconvenient way of splitting the actions.If we now substitute (2.5) and (2.6) into (2.4) we get, up to a discarded vacuum energyterm coming from the quantum term: − Λ ∂ Λ S I = 12 δS I δϕ · ˙∆ · δ Σ I δϕ − δδϕ · ˙∆ · δ Σ I δϕ − ϕ · ∆ − · ˙∆ · δ ˆ S I δϕ . (2.7)Note that all (non-vacuum) terms involving explicit ∆ − s, besides the final term whichdepends on the interaction part of the seed action, have cancelled amongst themselves; thisobservation will be important when we come to construct an ERG for theories of a scalarchiral superfield. If we were to set the interaction part of the seed action to zero—as we arequite at liberty to do—then the resulting equation is none other than Polchinski’s form ofthe ERG equation.Ideally, since universal results must be independent of the choice of seed action, we wouldlike to retain a general seed action for all calculations. Unfortunately, the methodology for10he work pertaining to the (non) existence of fixed points has only been figured out for thesimplest seed action ( ˆ S I [ ϕ ] = 0). For other calculations in this paper, however, we are ableto keep a general seed action and will do so. B. An ERG for Theories of Scalar Chiral Superfields
In this section, we construct an ERG for theories of a scalar chiral superfield. For mostof this paper, we will not consider any particular theory (i.e. bare action) but rather willtake the space of all possible (quasi-local) theories as our arena: in other words, we considerall (quasi-local) theories of a scalar chiral superfield, Φ, and its conjugate, Φ. This is thecorrect setting for asking the question as to whether or not there are any non-trivial fixedpoint theories. Only in section VIII will we look at a specific theory—the Wess-Zuminomodel.
1. General Formulation
In the case of theories of a scalar chiral superfield, we find it convenient to automaticallysatisfy the chirality constraint by taking the set of fields represented by ϕ to be ‘potentialsuperfields’ (see e.g. [48]), φ and φ , which are related to the scalar chiral superfield, Φ, andits conjugate, Φ, as follows: Φ = D φ, Φ = D φ. (2.8)In condensed notation, our flow equation reads: − Λ ∂ Λ S = 12 (cid:18) δSδφ · ˙∆ φφ · δ Σ δφ + δSδφ · ˙∆ φφ · D · δ Σ δφ − δδφ · ˙∆ φφ · δ Σ δφ − δδφ · ˙∆ φφ · D · δ Σ δφ (cid:19) + OSC , (2.9)where we have anticipated that it is convenient to extract a D from the φφ kernel. To bemore explicit about what the dots mean in (2.9), we expand e.g. δδφ · ˙∆ φφ · δδφ = Z d x d x ′ d θ d θ ′ δδφ ( x, θ, θ ′ ) ˙∆ φφ ( x, θ, θ ; x ′ , θ ′ , θ ′ ) δδφ ( x, θ, θ ′ ) . (2.10)Given the superspace operators, Q and Q [see (A13a) and (A13b)], supersymmetry of theflow equations follows straightforwardly, by considering the transformation δ ζ φ = ( ζ Q +11 Q ) φ , so long as we recognize that˙∆ XY ( x, θ, θ ; x ′ , θ ′ , θ ′ ) = ˙∆ XY ( x − x ′ , θ − θ ′ , θ − θ ′ ) , where X and Y can each be either the potential superfield or its conjugate.For what follows, including the development of a diagrammatic representation of theflow equation, it is useful to work in completely Fourier transformed superspace; i.e. wetransform the fermionic coordinates as well as the spatial ones. Focussing first on thespatial coordinates, we have the usual definitions: φ ( x, θ, θ ) = Z d p (2 π ) φ ( p, θ, θ ) e − ip · x , φ ( x, θ, θ ) = Z d p (2 π ) φ ( p, θ, θ ) e ip · x , (2.11)˙∆ XY ( x, θ, θ ; x ′ , θ ′ , θ ′ ) = Z d p (2 π ) ˙∆ XY ( p ; θ, θ, θ ′ , θ ′ ) e ip · ( x − x ′ ) . (2.12)The fermionic Fourier transforms are defined as follows: φ ( p, θ, θ ) = 4 Z d ρ e − iρ · θ φ ( p, ρ, ρ ) , φ ( p, ρ, ρ ) = 4 Z d θ e iρ · θ φ ( p, θ, θ ) , (2.13)where ρ · θ ≡ ρθ + ρθ . That we choose a factor of four to accompany both the Fouriertransform and its inverse is a matter of convention. Indeed, any choice of prefactors whoseproduct is sixteen would be consistent, as is apparent from (A14).When we completely Fourier transform the flow equation, equation (2.10) becomes: Z d p (2 π ) Z d ρ δδφ ( p, ρ, ρ ) ˙∆ φφ ( p ) δδφ ( p, ρ, ρ ) , (2.14)where we write ˙∆ φφ ( p, θ, θ, θ ′ , θ ′ ) = ˙∆ φφ ( p ) δ (4) ( θ − θ ′ ) . (2.15)For the terms in the flow equation involving explicit D s or D s we define D ( p, ρ, ρ, κ, κ ) ≡ Z d θ e − iρ · θ D ( p, θ, θ ) e − iκ · θ (2.16a)= 4 p (( ρ + κ )( ρ + κ )) − ρρ )(( ρ + κ ) pκ ) + 4( κκ )(( ρ + κ ) pρ ) − ( κκ )( ρρ )(( ρ + κ )( ρ + κ )) , (2.16b)and so arrive at the following building block of the flow equation: Z d p (2 π ) Z d ρ Z d κ δδφ ( − p, ρ, ρ ) ˙∆ φφ ( p ) D ( p, ρ, ρ, κ, κ ) δδφ ( p, κ, κ ) . (2.17)12ur aim now is to mimic the decomposition (2.6). To this end, we write S [ φ, φ ] = − φ · D · c − · D · φ − m φ · c − · D · φ − m φ · c − · D · φ + S I [ φ, φ ] , (2.18)where m is the bare mass. Actually, as a consequence of the nonrenormalization theorem,the mass is the same at all scales and so there is no need to call it the bare mass. However,we will shortly perform some rescalings, after which the superpotential will renormalize, viathe scaling dimension of the field. In this case, it will be useful to distinguish the bare massfrom the running mass.It is worth pointing out that, in contrast to the case of plain scalar field theory, we findit convenient to pull out the mass terms from S I . As we will see below, the reason for thisis because, unlike ∆ φφ (or the effective propagator in scalar field theory), ∆ φφ vanishes for m = 0.Note that, since we include a momentum dependent cutoff function in the two-point φφ vertex, this term contributes to both the superpotential and the K¨ahler potential, as can beseen by expanding c ( p / Λ ) = 1+ O ( p / Λ ). If we now make the following (very natural [49])choices for the momentum space integrated ERG kernels∆ φφ ( p ) = 116 c ( p ) p + m , (2.19a)∆ φφ ( p ) = ∆ φφ ( p ) = 164 m c ( p ) p ( p + m ) , (2.19b)then we once again find that the only place where the explicitly written two-point termsin (2.18) appear is in a term containing the seed action [cf. (2.7)]: − Λ ∂ Λ S I = (cid:16) φ · D + 4 m φ (cid:17) · c − · D · ˙∆ φφ · δ ˆ S I δφ + ˙∆ φφ · D · δ ˆ S I δφ ! + 12 (cid:18) δS I δφ · ˙∆ φφ · δ Σ I δφ + δS I δφ · ˙∆ φφ · D · δ Σ I δφ − δδφ · ˙∆ φφ · δ Σ I δφ − δδφ · ˙∆ φφ · D · δ Σ I δφ (cid:19) + OSC . (2.20)Setting ˆ S I = 0 yields the supersymmetric version of Polchinski’s equation. Deriving (2.20)is, however, somewhat more involved than in the case of scalar field theory, due to the factthat the two-point K¨ahler vertex is not invertible. Nevertheless, we do have at our disposalthe relationship D D D = − p D , (2.21)13nd it is this is which ensures that everything goes through.It is tempting to identify the integrated kernels as regularized propagators, but we mustbe careful doing so. In scalar field theory, it is both natural and convenient to make thisidentification. However, in the current case we cannot invert the kinetic term, and so it isnot immediately obvious that we can define a propagator.This situation is somewhat similar to what occurs in the manifestly gauge invariantERGs for QCD [50] and QED [51] where, again, the two-point vertex cannot be inverted.The central point is that ERG kernels exist, first and foremost, as ingredients of a perfectlywell defined ERG equation, and there is nothing to stop us from integrating them. Ifit so happens that one can additionally identify the integrated kernels with regularizedpropagators then all the better, but this occurs only in special cases and not for generalfield content. Nevertheless, even when this identification cannot be made, the integratedkernels have a structural similarity to regularized propagators and play an analogous role inERG diagrams to the role played by normal propagators in Feynman diagrams. With thisin mind, the phrase ‘effective propagator’ was coined [44].In the current scenario, things are somewhere between the case of scalar theory andmanifestly gauge invariant formulations. As emphasised by Weinberg [48] (chapter 30), thetheory is invariant under the ‘gauge’ transformations φ → φ + D ˙ α ω ˙ α , φ + φ → D α ω α , where ω and its conjugate are unconstrained superfields. This invariance comes aboutbecause the theory is built out of gauge invariant objects, Φ and Φ, in contrast to gaugetheories where the theory is built using the gauge variant connection. Now, in the context oftheories of a scalar chiral superfield, so long as one is only interested in correlation functionsof gauge invariant objects, then one can proceed without fixing the gauge by introducing newvariables of integration in the path integral. This involves separating out the zero mode ofthe two-point operator [48]. The resulting propagators are (modulo the UV regularization)precisely what we obtain for the integrated ERG kernels. Thus, with this understanding,we can interpret the integrated ERG kernels as regularized propagators.Returning to (2.20), it is worth adding that, reassuringly in this supersymmetric scenario,the vacuum terms vanish. 14 . Rescalings One of the applications of our flow equation will be to analyse the existence of fixedpoints. Fixed point behaviour is most easily seen by rescaling to dimensionless variables, bydividing by Λ to the appropriate scaling dimension (by this it is meant, of course, the fullscaling dimension, and not the canonical dimension). As it turns out, there is a subtletyrelated to scaling out the anomalous dimension from φ (and φ ), so we will consider thisrescaling first, in isolation. Thus, we make the following transformation: φ → φ √ Z, φ → φ √ Z (2.22)where Z is the field strength renormalization, from which we define the anomalous dimension: γ ≡ Λ d ln Zd Λ . (2.23)The problem with this transformation is that it produces an annoying factor of 1 /Z on theright-hand side of the flow equation. However, we can remove this factor by utilizing theimmense freedom inherent in the ERG, encapsulated by (2.1), to shift the kernels ˙∆ XY → Z ˙∆ XY . For orientation, the resulting flow equation is therefore not obtainable from thePolchinski equation by a simple rescaling of the fields: it is a cousin, rather than a descendent.In the case of scalar field theory, such a flow equation (with ˆ S I = 0) was first consideredin [52]; the version with more general seed action has been considered in [47, 53].With this change to the flow equation, (2.9) becomes: − Λ ∂ Λ S + γ (cid:18) φ · δSδφ + φ · δSδφ (cid:19) = 12 (cid:18) δSδφ · ˙∆ φφ · δ Σ δφ + δSδφ · ˙∆ φφ · D · δ Σ δφ − δδφ · ˙∆ φφ · δ Σ δφ − δδφ · ˙∆ φφ · D · δ Σ δφ (cid:19) + OSC . (2.24)Note that, as a consequence of our rescalings, the superpotential does now renormalize, butonly through the field strength renormalization.We now complete the rescalings started with (2.22). To this end, we define the ‘RG-time’ t ≡ ln µ/ Λ , (2.25)where µ is an arbitrary mass scale, and also scale out the various canonical dimensions: p i → p i Λ , ρ i → ρ i √ Λ . (2.26)15n these units, fixed point solutions satisfy the condition ∂ t S ⋆ [ φ, φ ] = 0 . (2.27)This follows because, if all variable are measured in terms of Λ, independence of Λ impliesscale independence. (Subscript ⋆ s will be used to denote fixed-point quantities.)With these rescalings, the flow equation in the massless case reads (cid:20) ∂ t + γ (cid:18) φ · δδφ + φ · δδφ (cid:19) + ∆ D − (cid:21) S = 116 (cid:18) δSδφ · c ′ · δ Σ δφ − δδφ · c ′ · δ Σ δφ (cid:19) + OSC , (2.28)where, with p now being dimensionless, c ′ ( p ) ≡ ∂∂p c ( p ) , and the ‘superderivative counting operator’, ∆ D , (utterly unrelated to the effective propa-gator, ∆) is given by∆ D ≡ (cid:20) − Z d p (2 π ) d ρ φ ( p, ρ, ρ ) (cid:18) p µ ∂ ′ ∂p µ + 12 ρ α ∂∂ρ α + 12 ρ ˙ α ∂∂ρ ˙ α (cid:19) δδφ ( p, ρ, ρ )+ Z d p (2 π ) d ρ φ ( p, ρ, ρ ) (cid:18) p µ ∂ ′ ∂p µ + 12 ρ α ∂∂ρ α + 12 ρ ˙ α ∂∂ρ ˙ α (cid:19) δδφ ( p, ρ, ρ ) (cid:21) . (2.29)The prime on the momentum derivative, viz. ∂ ′ /∂p µ , means that the derivative is not allowedto strike the momentum conserving δ -function which belongs to each vertex. The flowequation (2.28) generalizes the dimensionless flow equation of scalar field theory [3, 52, 54],in an obvious way.Finally, in anticipation of our study of the β -function of the Wess-Zumino model (sec-tion VIII), it is convenient to return to the flow equation (2.24). Studying the β -functionis a different problem from looking for the complete spectrum of fixed point theories andthere is nothing to be gained by scaling the out the various canonical dimensions. However,in this context, it is worth rescaling the fields by the three-point, superpotential coupling, λ : φ → φ/λ (and similarly for φ ). By doing so, the perturbative expansion in λ coincideswith the one in ~ , and this is the natural way to do perturbation theory (of course, there isno absolute need to perform this rescaling, but it does make life somewhat easier if we doso).We absorb the change on the left-hand side of the flow equation resulting from thisrescaling into the term involving the anomalous dimension, γ . With this latter rescaling,16he perturbative expansion of the action, should we choose to perform one, reads: S ∼ ∞ X i =0 λ i − S i , (2.30)where S is the classical action, and the S ≥ are the quantum corrections.The flow equation in the current scenario reads: − Λ ∂ Λ S + ˜ γ (cid:18) φ · δSδφ + φ · δSδφ (cid:19) = 12 (cid:18) δSδφ · ˙∆ φφ · δ Σ λ δφ + δSδφ · ˙∆ φφ · D · δ Σ λ δφ − δδφ · ˙∆ φφ · δ Σ λ δφ − δδφ · ˙∆ φφ · D · δ Σ λ δφ (cid:19) + OSC (2.31)where Σ λ = λ ( S − S ) . (2.32) III. THE WILSONIAN EFFECTIVE ACTION
As emphasised throughout this paper, most of the time we will consider a general (quasi-local) theory of scalar chiral superfields. This means that, apriori, the superpotential andthe K¨ahler potential possess all possible interactions. Expanding the action in powers ofthe fields, the superpotential possesses a two-point vertex with coupling f (2) , a three-pointvertex with coupling f (3) , and so forth. (Note that we can choose to exclude one-pointvertices in the superpotential through a classical renormalization condition: there are noquantum corrections as a consequence of the nonrenormalization theorem.) With this inmind, we write the superpotential as f [ φ ] = ∞ X n =2 f ( n ) n ! Z d x d θ δ (2) ( θ ) " n Y j =1 Φ( x, θ, θ ) = − ∞ X n =2 f ( n ) n ! Z d x d θ φ ( x, θ, θ ) " n Y j =2 D ( x, θ, θ ) φ ( x, θ, θ ) . (3.1)We will take Weinberg’s definition [48] of the K¨ahler potential: it is a real scalar functionof Φ and Φ, where we allow terms in which superderivatives act on these fields. (Sometimes Note that, in contrast to some other works [44, 47], we have pulled a λ out of the seed action, as well asthe Wilsonian effective action. n Φs and m Φs. Actually, we chooseto write the K¨ahler potential in terms of the potential superfield, φ , and its conjugate.Suppressing superspace coordinates, the vertex for the n , m contribution is K ( n,m ) , wherethis object is understood to be a differential operator, containing at least n D s and m D s: K [ φ, φ ] = − ∞ X n + m ≥ − n − m n ! m ! " n Y j =0 Z d x j d θ ′ j φ ( x ′ j , θ ′ j , θ ′ j ) m Y k =0 Z d x k d θ k φ ( x k , θ k , θ k ) K ( n,m ) ( x ′ , . . . , x ′ n , θ ′ , . . . , θ ′ n , θ ′ , . . . , θ ′ n ; x , . . . , x m , θ , . . . , θ m , θ , . . . , θ m ) . (3.2)In this expression, the operator K ( n,m ) acts to its left (to save space!). Notice that we useprimed coordinates for φ s and unprimed coordinates for φ s. The factor of 4 − n − m is insertedfor later convenience. Although not manifest in the way we have written things, every verteximplements locality in the fermionic coordinates. For small numbers of fields, we will oftenuse a notation where the fields are indicated, explicitly e.g. K φφ ≡ K (1 , .The fermionic Fourier transforms of the vertices follow from substituting (2.13) and itsconjugate into (3.2) and (3.1). Let us start by considering the completely Fourier trans-formed superpotential. To cast this in a neat form, we use a trick. In the second lineof (3.1), we pretend that there are n different θ s. Each field (and each D ) is taken todepend on one of these θ i and its conjugate. With this in mind, we rewrite Z d θ = Z d θ · · · Z d θ n δ (4) ( θ − θ ) · · · δ (4) ( θ n − − θ n )= 16 n − Z d θ · · · Z d θ n Z d ω · · · Z d ω n − n e iω · ( θ − θ ) · · · e iω n − n · ( θ n − − θ n ) , where we have used the representation of the Fermionic δ -function, (A14). Using (2.16a), itis now a simple matter to check that f [ φ ] = − ∞ X n =2 n − f ( n ) n ! " n Y j =1 Z d p j (2 π ) d ρ j φ ( p j , ρ j , ρ j ) ˆ δ ( p + · · · + p n ) " n Y i =2 Z d ω i − i D ( p i , ω i − i − ω i i +1 , ω i − i − ω i i +1 , ρ i , ρ i ) δ (4) ( ω − ρ ) , (3.3)where ω n n +1 ≡ δ ( p ) ≡ (2 π ) δ (4) ( p ) . β -function coefficients, it will be useful to write this as f [ φ ] = − ∞ X n =2 n ! " n Y j =1 Z d p j (2 π ) d ρ j φ ( p j , ρ j , ρ j ) ˆ δ ( p + · · · + p n ) F ( n ) ( p , ρ , ρ ; . . . ; p n , ρ n , ρ n ) . (3.4)We will find a similar structure to (3.3) when we completely Fourier transform the K¨ahlerpotential. To get a feeling for this, let us start by looking at the classical two-point vertices.In position superspace, the two-point, classical contribution to the φφ vertex is given by − Z d x d x ′ d θ c − ( x − x ′ ) D φ ( x, θ, θ ) D φ ( x ′ , θ, θ ) , (3.5)where we recall that, in momentum space, c ( p / Λ ) is a smooth ultraviolet cutoff function[see (2.12) for the definition of the Fourier transform], which regularizes the theory abovethe scale Λ. Since the only dependence of D and D on position coordinates occurs viaspacetime derivatives, in Fourier transformed superspace we have: K φφ ( − p, ρ, ρ ; p, κ, κ ) = − c − ( p / Λ ) Z d θ h D ( − p, θ, θ ) e iρ · θ ih D ( p, θ, θ ) e − iκ · θ i , (3.6)where the subscript ‘0’ on the vertex indicates that we are considering only the classicalcontribution, (3.5). Notice that, if we were to integrate by parts in superspace, so as totransfer the D from the φ to the φ , then we should remember to change the argument − p to + p . Applying (2.16a) and (A14), it is straightforward to show that (3.6) can be rewrittenin the intuitive form: K φφ ( − p, ρ, ρ ; p, κ, κ ) = − c − ( p / Λ ) Z d ω D ( − p, ω, ω, ρ, ρ ) D ( p, ω, ω, κ, κ ) , (3.7a)= − c − ( p / Λ ) Z d ω D ( p, ρ, ρ, ω, ω ) D ( − p, κ, κ, ω, ω ) , (3.7b)where the last line, which will be useful later, follows from inspection of (2.16b). Contractingtwo such vertices into one another gives Z d ω K φφ ( − p, ρ, ρ ; p, ω, ω ) K φφ ( − p, ω, ω ; p, κ, κ ) = +16 p c − ( p / Λ ) K φφ ( − p, ρ, ρ ; p, κ, κ ) , (3.8)which is a manifestation of the superspace relationship (2.21).As mentioned earlier, since we include a cutoff function in the mass term, the classical,two-point mass vertices contribute to both the superpotential and the K¨ahler potential. In19osition space we have the contribution to the action −
12! 4 m Z d x d x ′ d θ c − ( x − x ′ ) (cid:16) φD φ + φD φ (cid:17) , (3.9)where we have pulled out a factor of 1 / S φφ ( − p, ρ, ρ ; p, κ, κ ) = − m c − ( p / Λ ) D ( p, ρ, ρ, κ, κ ) , (3.10a) S φφ ( p, ρ, ρ ; − p, κ, κ ) = − m c − ( p / Λ ) D ( − p, ρ, ρ, κ, κ ) . (3.10b)For completeness, we give the explicit expression for the completely Fourier transformedclassical, two-point vertices in appendix B.Now we want to deal with general contributions to the K¨ahler potential. Noting that,if superfields carry positive momenta into the vertices, then anti-superfields carry positivemomenta out of the vertices, we define the momentum space vertices (suppressing fermioniccoordinates) via: K ( n,m ) ( − p ′ , . . . , − p ′ n , . . . ; p , . . . p m , . . . ) ˆ δ − n X j =1 p ′ j + m X k =1 p k ! = n Y i =1 Z d x ′ i ! m Y j =1 Z d x j ! K ( n,m ) ( x ′ , . . . , x ′ n , . . . ; x , . . . , x m , . . . )exp i n X k =1 p ′ k · x ′ k − i m X l =1 p l · x l ! (3.11)so that all momenta flow into the vertex coefficient functions.The idea now is to break up K ( n,m ) into n pieces, denoted by K ′ ( n,m ) i ( − p ′ i , θ ′ i , θ ′ i ), as-sociated with the φ ( p ′ i , θ ′ i , θ ′ i ) and m pieces, denoted by K ( n,m ) j ( p j , θ j , θ j ), associated withthe φ ( p j , θ j , θ j ). These objects might, individually, possess loose spinor indices which arecontracted together in some way but, for brevity, we will not explicitly indicate this: K [ φ, φ ] = − ∞ X n + m ≥ − n − m n ! m ! " n Y j =0 Z d p ′ j (2 π ) d θ ′ j K ′ ( n,m ) j ( p ′ j , θ ′ j , θ ′ j ) φ ( p ′ j , θ ′ j , θ ′ j ) m Y k =0 Z d p k (2 π ) d θ k K ( n,m ) k ( p k , θ k , θ k ) φ ( p k , θ k , θ k ) δ (4) ( θ ′ − θ ′ ) · · · δ (4) ( θ m − − θ m )ˆ δ − n X j =1 p ′ j + m X k =1 p k ! . (3.12)20ith this in mind, we can obtain a particularly useful form for the completely Fouriertransformed vertices by generalizing (2.16a): K ( n,m ) j ( p, ρ, ρ, κ, κ ) ≡ Z d θ j e − iρ · θ j K ( n,m ) j ( p, θ j , θ j ) e − iκ · θ j . (3.13)Using the same trick we used for completely Fourier transforming the superpotential, wefind: K [ φ, φ ] = − ∞ X n + m ≥ n ! m ! n Y j =1 Z d p ′ j (2 π ) d ρ ′ j d ω ′ j j +1 φ ( p ′ j , ρ ′ j , ρ ′ j ) K ′ ( n,m ) j ( − p ′ j , ω ′ j j +1 − ω ′ j − j , ω ′ j j +1 − ω ′ j − j , ρ ′ j , ρ ′ j ) m Y i =1 Z d p i (2 π ) d ρ i d ω i − i φ ( p i , ρ i , ρ i ) K ( n,m ) i ( p i , ω i − i − ω i i +1 , ω i − i − ω i i +1 , ρ i , ρ i )ˆ δ ( − p ′ − · · · − p ′ n + p + · · · p m ) δ (4) ( ω − ω ′ n n +1 ) , (3.14)where ω ′ , ω m m +1 ≡ − n − m has disappeared.We conclude this section with some remarks on the form of the K φφ vertex, which we willrequire later. The vertex must possess at least one D and at least one D . The observationwe will require is that general two-point vertices can be taken to have only additional powersof momenta and no further superderivatives. To see this, we start by noting that, as usual, { D α , D β } = { D ˙ α , D ˙ β } = 0 , (3.15a) { D α , D ˙ α } = − i∂ α ˙ α . (3.15b)Since space-time derivatives can thus be written in terms of superderivatives, a generaltwo-point vertex goes like D · · · D , (3.16)where the ellipsis stands for an arbitrary string of superderivatives (with epsilon tensorsincluded, as appropriate) and we have used integration by parts in superspace to arrangefor all superderivatives to strike one of the fields. If the ellipsis represents unity, then ourassertion is clearly satisfied. Otherwise, we must have either D · · · D ˙ α D ˙ β D , or D · · · D α D ˙ α D . ǫ ˙ α ˙ β D · · · D D , or p α ˙ α D · · · D . Iterating the procedure until the ellipses have been removed, we see that a general two-pointvertex can be written as a string of D s and D s, up to powers of momentum. However, wecan use the relationship (2.21) to reduce these strings to a single D and a single D , up topowers of momentum, thereby proving the original assertion. IV. DIAGRAMMATICS
One of the advantages of Fourier transforming all superspace coordinates is that thevertices are converted from differential operators to functions. These vertices can thus begiven a straightforward diagrammatic interpretation. The diagrammatics for the action ismost simply introduced by considering the two-point vertex, S φφ : S φφ ( − p, ρ, ρ ; p, κ, κ ) ≡ κ, κ − pρ, ρ p S . (4.1)The arrows on the lines emanating from the vertex indicate whether the correspondingfields are potential superfields or potential anti-superfields. We could instead have simplytagged each line with a φ or φ , as appropriate. However, we have avoided doing this toemphasise that the diagrammatics involves only the vertex coefficient functions, the fieldsand symmetry factors having been stripped off. To represent higher point vertices, we simplyadd more legs, as appropriate. Usually, we will drop all coordinate labels, and arrows, forbrevity.The diagrammatic form of the various flow equations follows by direct substitution of thediagrammatic form of the action and identifying terms with the same field content. Takingthe flow equation (2.31), for definiteness, the result is shown in figure 1, where { f } is a setof any n f φ s and/or φ s. Note that, since all fields have been stripped off, we can write theΛ-derivative as a total, rather than partial, derivative.22 − Λ dd Λ + 12 ˜ γn f (cid:19) (cid:20) S (cid:21) { f } = 12 • Σ λ S − Σ λ • { f } FIG. 1: The diagrammatic form of the flow equation for vertices of the Wilsonian effective action.
The lobe on the left-hand side is the Wilsonian effective action vertex corresponding tothe fields, { f } . On the right-hand side of the flow equation, we identify X • Y ≡ ˙∆ XY ,where both X and Y can be either φ or φ . Since the kernels are always internal lines, wesum over all realizations of X and Y and integrate over the associated fermionic coordinates.The kernels attach to vertex coefficient functions which can, in principle, have any numberof additional legs. The rule for determining how many legs each of these vertices has—equivalently, the rule for decorating the diagrams on the right-hand side—is that the n f available legs are distributed in all possible, independent ways. For much greater detail onthe diagrammatics, see [47, 55, 56].In view of their suggestive structure, the two diagrams on the right-hand side of the flowequation are often called the classical and quantum terms, respectively. However, it shouldbe noted that whilst the classical term does look like a tree diagram, the vertices have reallyabsorbed quantum fluctuations from the bare scale all the way down to the effective scale.For what follows, it will be useful to consider the effect of the quantum term, in themassless case. Since the massless effective propagator ties together a φ and a φ , only theK¨ahler potential survives being operated on by the quantum term. Now, bearing in mindthe representation (3.14), suppose that it is K ′ ( n,m )1 and K ( n,m )1 that are tied together bythe kernel, which we take to carry momentum, k . There is now a straightforward argumentthat we can take K ′ ( n,m )1 and K ( n,m )1 to go as D and D , up to some function of k . Thepoint is that, when two legs are tied together by an internal line, we can integrate by partsin superspace. This means that K ′ ( n,m )1 and K ( n,m )1 combine to produce D · · · D , where the ellipsis is some string of superderivatives. Now, if this string comprises just D s or D s, then our assertion is immediately verified, on account of (2.21). Suppose instead thatthe string contains superderivatives with loose spinor indices, which might be contracted23 lsewhere in the diagram (this option was not available in the two-point case discussedearlier). On account of the relationships D ˙ α D α D ∼ ∂ α ˙ α D , D α D β D ∼ ǫ αβ D D , D D α D = 0 , (4.2)it is clear that our assertion is true, in complete generality. V. THE NONRENORMALIZATION THEOREMA. Projectors
To prove the nonrenormalization theorem, we will construct a projector which, whenacting on the Wilsonian effective action, picks out just the superpotential: P f ( y ) G ( φ, φ ) ≡ (5.1) (cid:20) − y Z d ρ δδφ (0 , ρ , ρ ) + y Z d ρ d ρ ( ρ ρ ) δδφ (0 , ρ , ρ ) δδφ (0 , ρ , ρ ) − · · · (cid:21) G | φ,φ =0 . This projector is inspired by Hasenfratz & Hasenfratz [57] who constructed a similar pro-jector in scalar field theory, with a view to projecting out the local potential.To see how this works, let us first consider its action on the superpotential, as givenby (3.3). To this end, we note from (2.16b) and (A15) that( ρρ ) D (0 , ω, ω, ρ, ρ ) = − δ (4) ( ω ) δ (4) ( ρ ) . (5.2)Therefore, P f ( y ) f [ φ ] = +4 ∞ X n =2 n − f ( n ) y n n ! " n Y j =1 d ρ j δ (4) ( ρ ) · · · δ (4) ( ρ n )ˆ δ (0) Z d ω · · · d ω n − n δ (4) ( ρ − ω ) δ (4) ( ω − ω ) · · · δ (4) ( ω n − n − − ω n − n ) δ (4) ( ω n − n )= 4 ∞ X n =2 n − f ( n ) y n n ! ˆ δ (0) ≡ − f ( y )ˆ δ (0) , (5.3)where the ill-defined ˆ δ (0) can always be regularized at intermediate stages by working in afinite-sized box.In (5.1), it is crucial that the number of ( ρρ ) factors is one less than the number of func-tional derivatives. Had we included an extra such factor in each term, the projector wouldhave yielded zero. Let us now analyse the effect of the projector on the K¨ahler potential,24oting that each K ( n,m ) j possesses some combination of superderivatives, in addition to thenecessary D , arranged in some order: K ( n,m ) j (0 , ω, ω, ρ j , ρ j ) = − δ (2) ( ω ) δ (2) ( ρ j ) Z d θ e − i ( ωθ ) · · · e − i ( ρ j θ ) , (5.4)where the ellipsis represents some combination of superderivatives (including overallconstants)—beyond the D which is always present, and whose effects have been takeninto account. Note that all superderivatives are evaluated at zero momentum, since we haveset the first argument of K ( n,m ) j equal to zero. Therefore,( ρ j ρ j ) K ( n,m ) j (0 , ω, ω, ρ j , ρ j ) ∝ δ (4) ( ω ) δ (4) ( ρ j ) , (5.5)as we now explain in more detail. First, observe that the ( ρ j ρ j ) converts the δ (2) ( ρ j ) of (5.4)into δ (4) ( ρ ). Consequently, the e − i ( ρ j θ ) can be replaced with unity. Now, the only combinationof superderivatives evaluated at zero momenta acting on unity which yields a non-zero answeris the trivial combination of no superderivatives. This means that the θ -integral can beperformed, yielding the right-hand side of (5.5). Similarly,( ρ ′ j ρ ′ j ) K ′ ( n,m ) j (0 , ω, ω, ρ ′ j , ρ ′ j ) ∝ δ (4) ( ω ) δ (4) ( ρ ′ j ) . (5.6)Thus we find that: P f ( y ) K [ φ, φ ] ∝ ∞ X m =2 y m m ! Z d ρ d ω K (0 ,m )1 (0 , ω , ω , ρ , ρ ) δ (4) ( ω )ˆ δ (0) = 0 , (5.7)as follows from (5.4). Consequently, acting on the entire action, our projector does indeedpick out just the superpotential.Before moving on, it is worth noting that a particularly effective and powerful approxima-tion scheme within the ERG is the derivative expansion (see [3] for a review of the literature,and [2] for the key ideas), whereby the action is expanded in powers of derivatives. Withthis in mind, it is tempting to mimic this in the supersymmetric case and thus construct a‘superderivative expansion’. We write the K¨ahler potential as K Λ [Φ , Φ] ∼ Z d x d θ V Λ (Φ , Φ) + . . . , (5.8)where V Λ (Φ , Φ) depends on Φ , Φ, but not superderivatives thereof, and the ellipsis indicatesterms with extra superderivatives. 25e can pick V out of the full K¨ahler potential by using the projector P ( y, y ) ≡ P f ( y ) P f ( y ) (5.9)where, of course, we set φ, φ = 0 after the derivatives from both operators have acted.Now, a serious health warning should be given. Suppose that we are interested in search-ing for pure K¨ahler fixed points using the superderivative expansion. Unfortunately, if wework to lowest order then, as can be straightforwardly checked, the fixed point equation for V is in fact linear and, as a consequence, leaves the anomalous dimension entirely undeter-mined. Moreover, the reparemtrization invariance of the flow equation is catastrophicallybroken. Indeed, as recognized by Wegner [41] and very nicely put by Morris [2], the ERGequation at a fixed point can be thought of as a non-linear eigenvalue equation for theanomalous dimension. So, the lowest order in the superderivative expansion looks to beuseless for finding fixed points. Of course, we can always go to higher orders by appropri-ately generalizing (5.9) and, indeed, the resulting coupled equations do become non-linear.Nevertheless, reparametrization invariance is still broken, and so a unique determination ofthe anomalous dimension at a putative non-trivial fixed point is not possible within thisapproach. However, this is not something new for Polchinski-style flow equations [58] andso it might be profitable to develop this idea further. B. Proof of the Nonrenormalization Theorem
We will now prove the nonrenormalization theorem for the massless theory (the massivecase can be done in exactly the same way). To this end, we apply the projector, P f ( y ), termby term to the flow equation (2.28). The effect on the left-hand side is obvious. On theright-hand side, the most awkward term to deal with is the quantum one, so we treat thisfirst. However, there are a number of simplifications we can make. First, it does not makeany difference to the following analysis whether we take the Wilsonian effective action or seedaction contribution to Σ, so we just take the former. Secondly, since we are dealing with themassless theory, only the K¨ahler potential yields surviving contributions to the quantum It is interesting to note that, in scalar field theory, reparametrization invariance can be maintained withinthe derivative expansion by using the 1PI flow equation, with a particular form of cutoff [54]. However,there is a price to pay: with this choice of cutoff function, the derivative expansion does not converge [59]! P f ( y ), the only surviving contributions arethose where all external fields are φ . Consequently, we wind up with contributions from thevertices K (1 ,m ) which we split according to (3.14): P f ( y ) δδφ · c ′ · δKδφ ∝ X m y m − ( m − Z d k (2 π ) c ′ ( k ) Z d κ d ρ d ω d ζK ′ (1 ,m )1 ( − k, ω, ω, κ, κ ) K (1 ,m )1 (0 , ζ − ω, ζ − ω, ρ, ρ ) K (1 ,m )2 ( k, ζ , ζ, κ, κ ) . (5.10)But we know from the discussion around (4.2) that, since the K ′ (1 ,m )1 and the K (1 ,m )2 are tiedtogether by a loop integral, we can take them to go as a D and a D , respectively, up tosome function of k . Thus, using (3.7b) we have that Z d κ K ′ (1 ,m )1 ( − k, ω, ω, κ, κ ) K (1 ,m )2 ( k, ζ , ζ, κ, κ ) ∝ K φφ ( k, ω, ω ; − k, ζ , ζ ) . Furthermore, we have from (5.4) that Z d ρ K (1 ,m )1 (0 , ζ − ω, ζ − ω, ρ , ρ ) ∝ Aδ (4) ( ζ − ω ) + Bδ (2) ( ζ − ω ) , for some A and B . Therefore, the fermionic integrals in (5.10) produce Z d ω d ζ (cid:2) Aδ (4) ( ζ − ω ) + Bδ (2) ( ζ − ω ) (cid:3) K φφ ( k, ω, ω ; − k, ζ , ζ ) = 0 , as can be easily checked by using (B1).The classical terms are easy to project on to with P f ( y ). First we note that, because weare working in the massless case, the effective propagator must link a φ to a φ , and so atleast one of the vertices must be K¨ahler in order to end up with a contribution possessingexternal fields of all one type. It is simple to check that the classical terms do not yield anycontributions to the superpotential, and so the nonrenormalization theorem is satisfied.Note that, at a heuristic level, we can see that the nonrenormalization theorem must betrue, just by counting superderivatives. Ignoring one-point vertices, for the moment, everyvertex must possess at least one D or D . Furthermore, every n -point vertex must havea combined number of D s and D s which is at least n −
1. Now, diagrams generated bythe classical term of the flow equation have two vertices with, say, n and m legs, each ofwhich has had one field differentiated. Therefore, the diagram has a combined number ofat least n + m − D s and D s, which is at least equal to the number of external fields. Tostand any chance of generating contributions to the superpotential, we must remove enough27f the D s and D s, such that the remaining combined number is n + m −
3, without endingup with any positive powers of momenta. The only way to perform this removal is viathe relationship (2.21), but this generates two powers of momentum. Quasi-locality of thevertices means that this cannot be cancelled by negative powers of momenta in the vertices.Since the flow equation involves the differentiated effective propagators, rather than theeffective propagators themselves, no negative powers of momenta appear on the internallines. Consequently, the classical term in the flow equation cannot generate contributionsto the superpotential.The diagrams generated by the quantum term in the flow equation have n legs and acombined number of at least n + 1 D s and D s. Again, we see that it is impossible togenerate contributions to the superpotential.Were we to include one-point vertices, the discussion for the quantum term remains thesame, since vertices contributing to such diagrams must have at least two legs (correspondingto the two ends of the ERG kernel). As for the classical term, diagrams involving a one-pointvertex vanish. A one-point vertex carries zero momentum and, since it necessarily belongsto the superpotential, carries a δ -function in its external fermionic coordinates. Clearly, twoone-point vertices yield zero upon mutual attachment. If a one-point vertex attaches to anyother vertex, then we can always integrate by parts in superspace to ensure that a D or D is explicitly associated with the attachment [in the massive case, these superderivativescould also occur as part of the internal line, as in (2.17)]. From (A17a) and (A17b)—andremembering to set the momentum to zero—it is clear that such an attachment yields zero.Since we have rescaled the fields, a flow of the superpotential is induced. For the flowequation (2.31), where we recall that we have rescaled using first Z and then λ (Λ), theclassical action comes with an overall 1 /λ . Using the flow equation, together with thenonrenormalization theorem, it follows that˜ γ = − β λ , (5.11)where the β -function is defined according to (1.1).Alternatively, using the flow equation (2.24) or (2.28), where the rescaling by λ is notperformed, we find the more familiar relationship γ = 2 β λ . (5.12)28 I. THE DUAL ACTION
In [8], the key object in the demonstration of the triviality of scalar field theory in d ≥ ϕ , and the effective propagator by ∆, the dual action is defined according to −D [ ϕ ] = ln (cid:26) exp (cid:18) δδϕ · ∆ · δδϕ (cid:19) e − S I [ ϕ ] (cid:27) . In the case that the flow equation is strictly the Polchinski equation, there is a simplerelationship between the dual action vertices, D ( n ) , and the correlation functions, G : G ( p , . . . , p n ) = −D ( n ) ( p , . . . , p n ) n Y i =1 p i , n > ,G ( p ) = 1 p (cid:20) − D (2) ( p ) 1 p (cid:21) . However, for other flow equations, these relationships no longer hold. In section II B 2, we in-troduced a supersymmetric ERG which has a convenient form after scaling the field strengthrenormalization out of the field. Let us consider the analogue of this equation in the non-supersymmetric case. Now it turns out that the above relationship between the two-pointdual action vertex and the two-point correlation function only holds for small momentum. Moreover, if we were to take a non-trivial seed action, then the simple relationships betweenthe dual action vertices and the correlation functions are greatly complicated. Mindful ofthese points, we stick to the terminology ‘dual action’, rather than conflating it with thecorrelation functions.Returning to the supersymmetric case, the dual action is defined according to − D m [ φ, φ ] = ln n e Y m [ δ/δφ,δ/δφ ] e − S I [ φ,φ ] o , (6.1)where Y m [ δ/δφ, δ/δφ ] ≡ δδφ · ∆ φφ · δδφ + 12 δδφ · ∆ φφ · D · δδφ + 12 δδφ · ∆ φφ · D · δδφ (6.2)and the subscript m reminds us that we are working with the massive theory, implying thepresence of the second and third terms on the right-hand side. In the massless case, we This point was not made in versions one through five of [8]. It does not affect any of the results, however,on the one hand because this is only really important for the interpretation of the dual action and on theother because the various analyses are anyway performed at small momenta. D [ φ, φ ] = lim m → D m [ φ, φ ] . The construction (6.2) has in mind the flow equation (2.24), and so the fields in (6.1)have been rescaled. Note that, if we also rescale the superspace coordinates to arrive atflow equation (2.28) (or its massive counterpart), then the form of the dual action stays thesame. However, if we work with the flow equation (2.31), we must introduce − D m,λ [ φ, φ ] = ln n e Y m,λ [ δ/δφ,δ/δφ ] e − S I [ φ,φ ] o , (6.3)with Y m,λ [ δ/δφ, δ/δφ ] ≡ Y m [ λδ/δφ, λδ/δφ ] . (6.4)Let us now compute the flow of the dual action, using (2.24): − (cid:20) Λ ∂ Λ + γ (cid:18) φ · δSδφ + φ · δSδφ (cid:19)(cid:21) D m = γ (cid:16) φ · D · c − · D · φ + 2 m φ · c − · D · φ + 2 m φ · c − · D · φ (cid:17) + " e D m (cid:16) φ · D + 4 m φ (cid:17) · e Y m c − · D · ˙∆ φφ · δ ˆ S I δφ + ˙∆ φφ · D · δ ˆ S I δφ ! + OSC . (6.5)Notice that the seed action contributions are restricted to just one term (and its conjugate).Although other seed action terms are generated, they cancel amongst themselves—eitherdirectly, or courtesy of the relationship δδφ · ˙∆ φφ · D D p · δ ˆ S I δφ e − S I = − δδφ · ˙∆ φφ · δ ˆ S I δφ e − S I . (6.6)This follows because, in order to give a non-vanishing contribution, ˆ S I must possess atleast one D (recall that the 1 /p is nullified by the derivative of the cutoff function in˙∆ φφ and so if the δ/δφ strikes a one-point superpotential vertex, the entire term just van-ishes). Integrating by parts in superspace, we can always ensure that this D —with nofurther superderivatives—is associated with the leg hit by the functional derivative. Thenwe use (2.21), remembering that the D left over belongs to the vertex.As an aside, it is well worth mentioning that, in the past, cancellations of the seed actionwere demonstrated using elaborate (though increasing sophisticated) diagrammatics [45,47, 50, 51, 56, 60, 61]. However, as recognized in [8], by employing the dual action, thesecancellations can instead be done with a few lines of algebra, as has been done here.30or what follows, we set ˆ S I = 0. As mentioned earlier, we would ideally like to keep theseed action general, but then it is not known how to proceed for the calculations we will do.If we now introduce the vertices of the dual action, which we will denote by D ( i,j ) m , then itis clear that the flow for those with i + j = 2 is particularly simple (as a direct consequenceof taking ˆ S I = 0) and yields: D ( i,j ) m ( − p ′ , . . . , − p ′ i , p , . . . , p j ) = Z − ( i + j ) / A ( − p ′ , . . . , − p ′ i , p , . . . , p j ) , i + j = 2 (6.7)where A is independent of Λ.In the following analysis concerning the existence or otherwise of non-trivial fixed points,we will draw conclusions using the dual action in two ways. On the one hand, we will drawsome conclusions without having to dig around inside the dual action. In fact, we have anexample of this already in (6.5). On the other hand, certain conclusions will be drawn byre-expressing the dual action. To be specific, we will expand the exponential e Y m [ δ/δφ,δ/δφ ] = ∞ X i =0 i ! (cid:0) Y m [ δ/δφ, δ/δφ ] (cid:1) i and then allow the functional derivatives to act on e − S I before we perform the sum over i .Interchanging the order of these two operations is potentially dangerous. Part of the (admit-tedly heuristic) justification for this procedure is that S I is the full nonperturbative solutionto the flow equation. Thus, even though we have performed this interchange of operations,the resulting series certainly contains more than just standard perturbation theory (standardperturbation theory would correspond to additionally replacing the full Wilsonian effectiveaction with only its perturbative contributions). Moreover, we will perform a partial resum-mation of this series and assume that the resulting series can, in principle, be (re)summed togive the full nonperturbative answer. Rather than performing these manipulations directlyat the algebraic level, we prefer to use the diagrammatic representation of the dual action.From (6.1), the dual action comprises all connected diagrams built out of vertices ofthe interaction part of the Wilsonian effective action and effective propagators (it is thelogarithm which, as usual, ensures connectedness). A selection of terms contributing to D (2) [or D (2) m ], by which we mean all D ( i,j ) with i + j = 2, is shown in figure 2.31 (2) = S I + 12 S I − S I S I − S I S I + · · · FIG. 2: The first few terms that contribute to D (2) . Momentum arguments have been suppressed.Each of the lobes represents a vertex of the interaction part of the Wilsonian effective action. VII. CRITICAL FIXED POINTS
As a first application of the dual action formalism, we will investigate the existence ofcritical fixed points. This analysis mimics that of [8], but with a few small modifications. Tothis end, we set the mass to zero (as we must, since we are looking for critical fixed points)and work with dimensionless variables. Thus our flow equation for the dual action becomes: (cid:20) ∂ t − γ (cid:18) φ · δδφ + φ · δδφ (cid:19) + ∆ D − (cid:21) D [ φ, φ ] = γφ · D · c − · D · φ (7.1)Noting that in rescaled variables c = c ( p ) is independent of Λ, it is apparent from thedefinition (6.1) that, if m = 0, then (2.27) implies ∂ t D ⋆ [ φ, φ ] = 0 . (7.2)Now, let us solve (7.1) for the two-point dual action vertex, D φφ⋆ . To this end, we recall fromthe end of section III that we can write D φφ⋆ ( − p, ρ, ρ, p, κ, κ ) = z ( p ) Z d ω D ( − p, ω, ω, ρ, ρ ) D ( p, ω, ω, κ, κ ) , (7.3)and so we have: (cid:18) − γ ⋆ + 2 p ddp (cid:19) z ( p ) = γ ⋆ c − ( p ) . (7.4)This equation has solution z ( p ) = p γ ⋆ / (cid:20) b ( γ ⋆ ) − γ ⋆ Z dp c − ( p ) p γ ⋆ / (cid:21) , (7.5)where 1 /b ( γ ⋆ ) is the (finite) integration constant and is a functional of the cutoff function.In the case where γ ⋆ = 0, b is defined by the form of z ( p ) taken if we perform the indefiniteintegral by Taylor expanding the cutoff function. For γ ⋆ = 0, we make a choice such that32he leading behaviour in the first case coincides with the behaviour in the second case, as γ ⋆ →
0. Thus, for small momentum, we have z ( p ) = b p γ ⋆ / − (1 + subleading) , γ ⋆ = 0 , b − , γ ⋆ = 0 . (7.6)Note that the subleading terms are cutoff dependent, not just with regards to their prefactors,but also to their structure. For example, if γ ⋆ = 2 and c ′ (0) = 0, then the subleading piecehas a nonpolynomial component p ln p , but this is absent altogether if c ′ (0) = 0. However,the real point to make here is that, so long as γ ⋆ <
2, the subleading term in the bracketsis always subleading compared to bp γ ⋆ / . We will now exclusively take γ ⋆ < I ( i,j ) . At the two-point level we have that D φφ is built up from I φφ according to the geometric series D φφ ( − p, ρ, ρ ; p, κ, κ ) = I φφ ( − p, ρ, ρ ; p, κ, κ ) − Z d ω I φφ ( − p, ρ, ρ ; p, ω, ω )∆ φφ ( p ) I φφ ( − p, ω, ω ; p, κ, κ ) + · · · (7.7)Noting that our aim now is to sum the series (7.7), we can schematically write (7.7) as: D φφ = I φφ φφ I φφ . However, this is no more than a mnemonic for (7.7), due to the fermionic integrals thatmust be performed. To perform these integrals, we recall from the end of section III thatan arbitrary two-point vertex can be written as a single D and single D , up to powers ofmomentum. Applying (2.21), we see that we can remove all of these D s and D s, with theexception of those on the lines which are external with respect to D φφ , at the expense of afactor of − p for each ∆ φφ ( p ). Up to the minus sign, this cancels the 1 / p coming fromeach effective propagator, in each case leaving behind a c ( p ).Denoting what is left after we strip off the external D and D from I φφ by ˜ z , we nowreally can write z ( p ) = ˜ z ( p )1 − c ( p )˜ z ( p ) , (7.8)33hich can be inverted to yield: ˜ z ( p ) = z ( p )1 + c ( p ) z ( p ) . (7.9)The final ingredient that we will need is the dressed effective propagator, defined accordingto ˜∆ φφ ( p ) ≡ ∆ φφ ( p )1 − c ( p )˜ z ( p ) . (7.10)At a fixed point with γ ⋆ < φφ⋆ ( p ) ∼ p − γ ⋆ / , (7.11)which is exactly what we expect at a critical fixed point.Now, the dressed effective propagator can be used to resum sets of loop diagrams con-tributing to ˜ z , such that all internal lines become dressed, as indicated in figure 3. ˜ z = S I + 12 S I − S I S I + · · · FIG. 3: Resummation of diagrams contributing to ˜ z : the thick lines represent dressed effectivepropagators, (7.10), and the stops at the ends of the external lines indicate that the external D and D have been stripped off from each diagram. A. γ ⋆ ≥ In the case where γ ⋆ = 0, we know from Pohlmeyer’s theorem that the only criticalfixed point is the Gaussian one, So let us now consider critical fixed points with γ ⋆ > z ⋆ ( p ) = − bp − γ ⋆ / + 1 + · · · . (7.12)34econdly, we recognize that, by considering the diagrammatic expression for ˜ z ,˜ z ⋆ ( p ) = constant + f ( p ) , (7.13)where lim p → f ( p ) = 0. This follows from power counting, so long as we assume that theWilsonian effective action vertices are Taylor expandable for small momenta—this being oneof our requirements for physical acceptability. Given I internal lines and V vertices, thereare L = I − V + 1 loops. If we temporarily ignore the superderivatives associated with eachof the internal legs of the vertices, then the degree of IR divergence is D ′ ≥ I − V + 1) − − γ ⋆ / I, where we understand D ′ > I ≥ V − . Consequently [for 4 ≥ − γ ⋆ / D ′ ≥ − V + 2 + (cid:18) V − (cid:19) γ ⋆ . However, now we must take account of the internal superderivatives in each diagram.This is easy to do. Let us denote the corrected degree of divergence by D . Since we areinterested in the smallest possible value of D , we need only consider diagrams built out ofthree-point vertices: taking vertices with more legs either leaves D unchanged, if pairs ofthese legs are tied together, or increases it if the legs attach to other vertices. Similarly, wecan consider the minimal number of superderivatives, amounting to one pair per leg. Now,from (2.16b), we see that the i th leg—either internal or external—in some diagram carries6 − P i Grassmann numbers, where P i is the number of powers of momentum taken on thegiven leg. However, from (3.14), each vertex—being three-point—contains an integral overa pair of dummy coordinates. Thus, the total number of Grassmann numbers is V X i =1 (6 − P i ) − V. Next we notice that, from (B1), a diagram with an external D and an external D , in whichthe external momentum has been set to zero [cf. (7.13)], has 8 external Grassmann numbers.35hus, the total number of internal Grassmann numbers is V X i =1 (6 − P i ) − V + 1) . However, since the external momentum is set to zero, we can set P V − and P V —thesebeing the P i we choose to associate with the external legs—to zero. Thus leaves V − X i =1 (6 − P i ) − V + 1) + 12 . Now, each internal line contains a fermionic integral, each one of which counts − V − X i =1 (6 − P i ) − V + 4 = 4 I = 6 V − ⇒ V − X i =1 (6 − P i ) = 14 V − . Consequently, the total number of powers of internal momenta is P ≡ V − X i =1 P i = 2( V − , (7.14)yielding a corrected degree of divergence D ≥ V + (cid:18) V − (cid:19) γ ⋆ . Given that we are considering γ ⋆ >
0, it is obvious that this is always positive, and so all ofour diagrams are IR safe, confirming (7.13).It is therefore apparent that, for γ ⋆ >
0, equations (7.12) and (7.13) are inconsistent andso we conclude that there are no non-trivial fixed points with γ ⋆ > b = 0 is not acceptable, since this would mean that z ( p ) is singular]. Pohlmeyer’s theorem,of course, rules out non-trivial fixed points with γ ⋆ = 0, meaning that, at this stage of theanalysis, if any non-trivial fixed points are to exist, then they must have negative anomalousdimension.Note that if we were to consider diagrams possessing vertices belonging to the superpo-tential, then the degree of divergence is lowered, since superpotential vertices lack (at least)one D or one D compared to K¨ahler potential vertices. Although this observation is ofno use here, since we know that there cannot be a superpotential at a critical fixed pointwith γ ⋆ > β -function of the Wess-Zumino model.36 . γ ⋆ < Again, this analysis is based on that in [8], but with some minor modifications. The vitalproperty of fixed points with negative anomalous dimension, which we will now exploit, isthat lim p → ˜ z ⋆ ( p ) = 1 , (7.15)completely independently of the shape of the cutoff function. Note that for fixed points withpositive anomalous dimension, the right-hand side of (7.15) instead diverges.The next step is to further resum the diagrams in figure 3. We cannot do anything withthe first two diagrams. However, the third can be resummed such that the vertices arereplaced with I (4) s. Actually, as discussed in [8], this double counts certain contributionsbut, crucially, these diagrams are also built entirely out of I ( n ) s. Thus, we arrive at theexpression in figure 4. ˜ z = S I + S I − S I I + · · · − II + · · · FIG. 4: Further resummation of diagrams contributing to ˜ z . The brackets contain terms in whichboth fields decorate the same vertex. The second ellipsis represents diagrams built out of I ( i + j> vertices. After we take the limit p →
0, we will denote the first contribution on the right-handside of figure 4 by w , and the rest by W so that, at a fixed point we have1 = w + W ⋆ . (7.16)Note that w is a finite number. If w < full actionhas a kinetic term of the right sign. In this case, w is a free parameter corresponding to thenormalization of the field, with w = 0 being canonical normalization.Let us now suppose that, at a fixed point, W ⋆ > W ⋆ <
0, though such fixed points are already ruled out by our requirement relating to37nitarity). From (7.16), we know that W ⋆ is independent of the shape of the cutoff function.Now, the only contribution to the cutoff function which is independent of its shape—i.e.universal—is c (0) = 1. Heuristically, then, we expect that any surviving contributions to W ⋆ come from when the loop momenta are precisely equal to zero. To be more precise aboutthis we note that, since every contribution to W ⋆ contains at least one loop integral, we canwrite W ⋆ = Z d D k (2 π ) D c ( k ) F [ c ]( k ) . Given that δW ⋆ /δc = 0 and noting that W ⋆ is the finite number 1 − w , it appears that wecould take F [ c ]( k ) = c − ( k ) g ( k ), where δg/δc = 0 and the integral over g gives 1 − w .However, since δg/δc = 0, we can always choose c such that F diverges arbitrarily stronglyin the UV. In [8] it was shown that such behaviour of F is inconsistent. Consequently,taking F [ c ]( k ) = c − ( k ) g ( k ) is ruled out and that the only option, besides W ⋆ = 0, is that F [ c ]( k ) has net contributions only when both k , and also all momenta internal to F , arezero. It is tempting to say that such contributions must have zero support but this does notfollow immediately, as it is quite possible that individual terms contributing to ˜ z diverge as p →
0. However, as we will now argue, the resummations we have performed in figure 4guarantee that there are no such contributions to W ⋆ .The contributions to figure 4 which might have support for vanishing loop momenta arethose containing I ( n ) vertices, since some of these terms look like they might possess IRdivergences for p →
0. Now, to show that this does not occur, we need the momentumdependencies of the I ( n ) ⋆ . Let us begin by noting that (7.1) gives us some useful informationabout the D ( n ) ⋆ . In particular, a dual action vertex with i φ s and j φ s has a total number ofsuperderivatives r ij = 4 + ( i + j ) γ ⋆ , (7.17)where we recall that momenta can always be written in terms of superderivatives. Now, sinceeach φ or φ necessarily comes with a pair of superderivatives, we can define the number of‘extra’ superderivatives by s , where s ij = 4 + ( i + j )( γ ⋆ − . (7.18)However, we are not interested in s i,j , per se, but rather the corresponding quantity for the I ( n ) ⋆ , which we will denote by ˜ s i,j . To go from s ij to ˜ s ij , we strip off the leg decorations from38he D ( i + j> and, to this end, define D ′ ( i + j> via D ( i + j> ( p , . . . , p n ) = D ′ ( i + j> ( p , . . . , p n ) Q i + jk =1 [1 − c ( p k )˜ z ( p k )] , (7.19)where we have suppressed the fermionic coordinates. Notice that D ′ (3) = I (3) but, beyondthe three point level, there are additional contributions. However, one of the contributionsto D ′ ( i + j> is always I ( i + j> and so, from (7.18) and (7.19), it is apparent that˜ s ij = s ′ i,j = 4 − ( i + j )( γ ⋆ + 2) . When considering two-point diagrams built out of I ( i + j> s, we know from the discussionsat the end of sections III and IV that all extra superderivatives can be converted into powersof momenta. Indeed, each vertex effectively comes with˜ s ij − ( i + j )( γ ⋆ / − ( γ ⋆ / R ′ , of a diagram contributing to lim p → ˜ z ( p ) builtout of V I ( i + j> ⋆ vertices and I dressed effective propagators. Totting up the dependenciesfrom the loop integrals, the dressed effective propagators and the vertices, we have: R ′ = 4( I − V + 1) − I (2 − γ ⋆ ) − ( I + 1)( γ ⋆ + 2) + 2 V = 2(1 − V ) − γ ⋆ . Now, just as we did at the end of section VII A, we must correct this, to take account ofthe D s and D s associated with each of the internal legs (recall that the external ones havebeen stripped off). It is straightforward to check that, once again, the correction is givenby (7.14) and so we find that R = − γ ⋆ . Therefore, the diagrams just analysed do indeed go like p − γ ⋆ / and so, for γ ⋆ <
0, do indeedvanish for p → W ⋆ come from the diagrams enclosed by thebrackets in figure 4. These are most certainly IR safe for p →
0, since the external momentumnever flows around any of the loops and so the diagrams do have zero support for vanishing39oop momenta. Thus, there are no fixed points with W ⋆ = 0. Therefore, the only fixed pointswith negative anomalous dimension are those for which w = 1. But, these fixed points lacka standard kinetic term and so correspond to non-unitary theories, upon continuation toMinkowski space. VIII. THE β -FUNCTION Having made a statement about the space of all possible theories of a scalar chiralsuperfield—namely that there are no physically acceptable non-trivial fixed-points, we willnow return to more familiar territory. To be specific, we will consider the β -function for theWess-Zumino model, considered as a low energy effective theory.As is well known, if one chooses a particular set of renormalization schemes, then theone and two-loop coefficients come out the same. For other renormalization schemes—particularly those which involve masses— one gets different numbers which are no less correct(see [44] for a detailed discussion of this point). It is thus sensible to refer to the one andtwo loop β -function coefficients as pseudo-universal. In what follows, we will consider themassless Wess-Zumino model in order that we can make a meaningful comparison betweenour one and two loop results and the standard pseudo-universal answers. Encouragingly, weget the correct answers.It is important to point out that our calculations of the one and two-loop β -function aredone with general seed action (indeed, this is a nice example of a case where we know howto proceed without setting ˆ S I = 0). Whilst independence of these pseudo-universal numberson the seed action is expected, we actually find more than this: even nonperturbatively, the β -function turns out to have no explicit dependence on the seed action. In some sense, thisis quite surprising since the β -function, beyond two loops, is not even pseudo-universal.That we do see this unexpected degree of universality seems to be a feature of the structureof the ERG equation. Indeed, the equation has basically the same shape irrespective ofwhether one is considering scalar field theory, QED, QCD, or the case currently in question.Indeed, the same degree of universality has been found in these other theories [16, 60, 62].In order to compute the β -function, we must specify the renormalization conditions.Now, as a consequence of the nonrenormalization theorem, we know that λ is related to theanomalous dimension and the renormalization condition for γ is just that the kinetic term40s canonically normalized: K = − λ φ · D D · φ + · · · , (8.1)where the ellipsis denotes contribution of higher dimension operators to the K¨ahler potential.Note that the renormalization condition implies that the φ · D D · φ contribution to theinteraction part of the K¨ahler potential is zero. This is just the statement that, by 1 /λ ,we mean precisely the coefficient in front of the complete − φ · D D · φ part of the action.Furthermore, the three-point superpotential coupling, f (3) , is 1 /λ .When evaluating the β -function perturbatively in a theory which is perturbatively renor-malizable, but which may be nonrenormalizable beyond perturbation theory, there is a veryuseful trick we can use [47, 51, 53]. Namely, we recognize that, as discussed in the introduc-tion, the Wess-Zumino model is self-similar at the perturbative level . In the current variables,where the canonical dimensions have not been scaled out, this means that all dependenceon Λ can either be deduced by na¨ıve power counting or occurs through λ (Λ), equivalently γ (Λ). We will exploit this below.Beyond perturbation theory, self-similarity is destroyed, and we must allow for explicitoccurrences of the bare scale, Λ . Nevertheless, we can still formulate an equation for the β -function. However, the above considerations will, at least in principle, affect its evaluation.Actually, as we will see, the β -function is in fact free of nonperturbative power correctionsof the form Λ / Λ , just as in the manifestly gauge invariant approach to QED [16], given thedefinition of the coupling implicit in the approach [16]. A. The β -Function from the Dual Action To derive an expression for the β -function, we consider the dual action appropriate tothe case where we have rescaled the field by both √ Z and λ —see (6.3). For the followinganalysis, we will no longer take the interaction part of the seed action to be zero and so, inthe massless case, we have: (cid:18) Λ dd Λ + 4 βλ + ˜ γ (cid:19) z λ ( p ) = (cid:18) βλ + ˜ γλ (cid:19) c − ( p / Λ ) + seed action term , (8.2) When this analysis was first performed in QED, it was speculated whether resummability of the β -functionin the Wess-Zumino model might imply resummability of the dual action vertices (though this terminologyhad not yet been coined). However, there is no reason to expect this to be true. z λ is defined as what is left after the external D and D have been stripped off D φφλ .To compute the β -function, we must employ the renormalization condition (8.1), andso we are interested in considering (8.2) at p = 0. Now, at first sight we might worryabout strong IR divergences caused by one-particle reducible (1PR) diagrams; however, the1 /p s in the offending diagrams are compensated by factors of p arising from use of (2.21).We might also worry about weaker, logarithmic IR divergences occurring in loop integrals.These are most certainly present, but cancel out, as we will discuss in detail below. Atintermediate stages of computation, it is perhaps best to suppose that, term by term, weare looking at both the O ( p ) and O ( p ) × nonpolynomial contributions. Notice that thisrestriction kills the seed action term. To see this, consider the seed action term which,up to factors of λ , can be read off from (6.5) with m = 0. Now, by (6.6) it is apparentthat the explicitly written D D can be removed, yielding a factor of p . Thus, the seedaction term contributes at O ( p ) and O ( p ) × nonpolynomial and so can be removed fromour considerations. As claimed earlier, we have demonstrated that the β -function has noexplicit dependence on the seed action (there is, of course, implicit dependence buried in thevertices), which is true nonperturbatively since we have not yet performed a perturbativeexpansion of the vertices.Recalling (7.8), we introduce the 1PI contribution ˜ z λ , appropriate to the flow equa-tion (2.31), with z λ ( p ) = ˜ z λ ( p )1 − λ c ( p / Λ )˜ z λ ( p ) . (8.3)Utilizing (5.11), it is now straightforward to derive the following expression for the β -function: 2 β λ + O ( p ) = − Λ dd Λ ˜ z λ ( p )1 + 2 λ ˜ z λ ( p ) . (8.4)This can be rewritten in the compact form,Λ dd Λ ln (cid:20) λ (cid:18) λ ˜ z λ ( p ) (cid:19)(cid:21) = O ( p ) , or in the form convenient for computation,2 β λ + O ( p ) = − Λ ∂ Λ ˜ z λ ( p )1 + 2 λ ˜ z λ ( p ) + 3 λ / ∂ λ ˜ z λ ( p ) , (8.5)where the partial derivative with respect to Λ is performed at constant λ .42 . Perturbative Computations
1. The One-Loop Coefficient
To perform perturbative calculations, we recall (2.30) S ∼ ∞ X i =0 λ i − S i and also employ: ˜ z λ ( p ) ∼ ∞ X i =0 λ i − ˜ z λi ( p ) , (8.6) β ∼ ∞ X i =1 λ i +1 β i . (8.7)Noting that the one-loop, two-point vertex K φφ does not contribute to the β -function, asa consequence of the renormalization condition (8.1), we have:2 β O ( p ) = −
12 Λ dd Λ p − p , (8.8)where the zeros inside the vertices denote contributions to the classical action, S , and werecall that the stops on the ends of the external lines indicate that the external D and D have been removed.Let us consider the second diagram, taking the internal momentum to be k . Havingalready extracted the external D and D we suppose for the minute that the vertices donot contribute further powers of momenta. Temporarily neglecting the fermionic coordinatesand overall factors the diagram goes like (cid:20) Λ dd Λ Z d k (2 π ) c ( k / Λ ) k ( k − p ) (cid:21) p , (8.9)where we have explicitly indicated the fact that we wish to take the O ( p ) component, after performing the Λ-derivative [we have taken the liberty of setting p = 0 in c (( k − p ) / Λ )].Henceforth, throughout this section, we will use the shorthand c k ≡ c ( k / Λ ) . perturbative self-similarity, there are no hidden couplings/ dimensionful quantities buried in the vertices. Consequently, the only place where we cangenerate a scale is in the IR, as a consequence of the IR divergences present before the Λ derivative is taken as p →
0. In other words, the surviving contributions to (8.9) are of theform: Λ dd Λ ln p / Λ + O ( p ) . With this point in mind, we immediately see that the first diagram of (8.8) must vanish:there is no IR scale in this diagram.Let us now include the fermionic coordinates in our analysis of the second diagram in (8.8).We will begin by supposing that both vertices belong to the superpotential. For trans-parency, let us reinstate the external D and D . The diagram now translates to12 Λ dd Λ Z d k (2 π ) Z d ρ Z d ρ (cid:20) c k k ( k − p ) F (3)0 (0 , ρ, ρ ; − k, ρ , ρ ; k, ρ , ρ ) F (3)0 (0 , κ, κ ; k, ρ , ρ ; − k, ρ , ρ ) (cid:21) , (8.10)where we have used (3.4), have set p = 0 in the vertex coefficient functions, and recall thatsubscript zeros refer to classical quantities. Now, by the previous arguments, we cannot takeany powers of k from the vertices, if we want the diagram to survive. With this in mind, wenote that F (3) (0 , ρ, ρ ; 0 , ω , ω ; 0 , ω , ω ) = 4 Z d θ (cid:2) D (0 , θ, θ ) e iρ · θ (cid:3) (cid:2) D (0 , θ, θ ) e iω · θ (cid:3) e iω · θ = 16( ρρ )( ω ω )( ω ω )(( ρ + ω + ω )( ρ + ω + ω )) , (8.11)where we have used the renormalization condition which implies that f (3)0 = 1. Therefore,(8.10) becomes 12 [( ρρ )( ρρ )( κκ )( κκ )] Λ dd Λ Z d k (2 π ) c k k ( k − p ) , (8.12)where the contribution in square brackets turns out to be precisely the O ( p ) contributionto the external D and D . At this point we note that, were we to have taken either or44oth of the three-point vertices from the K¨ahler potential, then the resulting diagram wouldnot contribute to β : having arranged the superderivatives such that there are an external D and D , the diagram would either be too high an order in p , or would be killed by theΛ-derivative, due to additional powers of internal momenta. Combining (8.8) and (8.12)with the fermionic coordinates stripped off yields:2 β O ( p ) = 12 Λ dd Λ Z d k (2 π ) c k k ( k + p ) . (8.13)All that remains to be done is to compute the integral, which does not involve anyfermionic coordinates. There are several ways to do this. The most efficient involves usingdimensional regularization, not as a means of regularizing the integral in the UV, but as atrick for extracting the part which survives differentiation with respect to Λ. We empha-sise that using dimensional regularization in this way, and at this stage, is entirely valid,does not spoil our superspace implementation, and works to any number of loops (or evennonperturbatively). The key point is that it is simply a trick for evaluating a finite bosonicquantity. Clearly, given that the trick is known to work, the answer to (8.13) should notdepend on the history of how this equation was obtained. For the details of this elegantmethod, see [46, 51]; see [47] for an alternative technique formulate directly in d = 4. It isreassuring that we get the usual result: β = 32 1(4 π ) . (8.14)
2. The Two-Loop Coefficient
At the two-loop level, although there are many diagrams which could, in principle, con-tribute to the β -function, only two give non-vanishing contributions:2 β O ( p ) = 12 Λ dd Λ p + 12 p Λ dd Λ p , (8.15)where the second term on the right-hand side comes from the second term in the denominatorof (8.5) (the third term in the denominator does not contribute until three loops).45s with β , only vertices belonging to the superpotential produce surviving contributionsand these can be cast in the form:2 β O ( p ) = Λ dd Λ Z d k (2 π ) Z d l (2 π ) (cid:20) c k c l − k c l k ( k − p ) ( l − k ) l − c k k ( k − p ) c l l ( l − p ) (cid:21) (8.16)Notice that a relative sign is introduced between the two terms, as compared with (8.15).This comes about as the result of employing (2.21) along the internal lines carrying theouter loop momentum [taking the outer loop momentum of the first diagram to be k , thisalso explains why the first term in (8.16) ∼ /k , rather than 1 /k ]. The relative factorof 1 / /
2. An evaluation of the integrals is given, directly in d = 4 in [47]. For detailsof the alternative method employing dimensional regularization, see [46]. Either way, theexpected answer is obtained: β = −
32 1(4 π ) . (8.17) C. Nonperturbative Considerations
We will now argue, along the lines of [16], that even in the case where there is anadditional physical scale present, violating self-similarity, the β -function does not receivenonperturbative corrections. First of all, let us recall from (1.2) that we can re-express anysuch terms using λ , according to ΛΛ ∼ e − / β λ (Λ) + . . . , where the prefactor contains the Λ dependence.Let us now return to the expression for the β -function, (8.5), before any perturbativeexpansion has been performed. Quite irrespective of whether we now perform a perturbativeexpansion and whether there are additional scales floating around, it is still the case thatthere are nonpolynomial contributions to ˜ z which blow up as p →
0. Moreover, since theleft-hand side of (8.5) is safe in the p → β λ + O ( p ) = F ( λ ) G ( λ , ln p / Λ ) F ( λ ) G ( λ , ln p / Λ ) = F ( λ ) F ( λ ) , (8.18)46here F , F and G are unknown functions.To begin with, let us reconsider perturbation theory. Let us suppose that, at order λ i ,the strongest IR divergence carried by ˜ z ( p ), at O ( p × nonpolynomial), goes like λ i ln j p / Λ . (8.19)In the numerator of (8.5), the Λ-derivative (which we recall is performed at constant λ )reduces this divergence to one of the form λ i ln j − p / Λ (8.20)whereas, in the denominator, a contribution of the form λ i +1) ln m p / Λ (8.21)is produced. At first sight, we have found that terms of the form (8.19) provide a divergentcontribution to the denominator which does not seem to exist in the numerator. Of course,there is no real problem here: all we need to do is consider diagrams with an extra loop. Insuch diagrams there are contributions of the form (8.19) but with i → i + 1 and j → j + 1.Terms like this in the numerator are, after differentiation with respect to Λ, of precisely theright form to cancel denominator contributions of the type (8.21).But now consider a contribution of the type λ i e − a/λ ln j p / Λ , (8.22)where again we assume that, for our choice of i , there is no stronger IR divergence. In thenumerator of (8.5) this contributes terms of the form λ i e − a/λ ln j − p / Λ (8.23)and in the denominator it yields terms of the form λ i e − a/λ ln j p / Λ + . . . , (8.24)where the ellipsis denotes terms higher order in λ . (The explicitly written term comesfrom the last piece of the denominator.) Crucially, (8.23) and (8.24) are the same order in λ . Since, by assumption, there are no terms in ˜ z ( p ) which are of order λ i e − a/λ butwhich have a stronger IR divergence than (8.22), there is no way that the denominator47ontribution (8.24) can ever be cancelled. From (8.18), we therefore conclude that terms ofthe type (8.22) must be absent from (8.18), unless j = 0. But it is easy to see that j = 0terms can appear only in G ( λ , ln p / Λ ) and not in F ( λ ) or F ( λ ): for if this conditionwere violated, then we would necessarily produce contributions of the form (8.22), when weexpand out F ( λ ) G ( λ , ln p / Λ ). In conclusion, the only contributions to the β -functionof the form (8.22) that are allowed—namely those with j = 0—cancel out!It is now straightforward to generalize this argument to show that only the perturbativecontributions to the β -function survive. First, we note that the above argument is notaffected if we consider terms which include e − b/g , e − c/g etc., or products of such terms.Secondly, we can allow additional functions of g to come along for the ride, so long as theydo not spoil the requirement that the ERG trajectory sinks into the Gaussian fixed point asΛ → β -function to be free ofpower corrections, since it is quite consistent to pick up terms like m Λ e − a/λ , because the mass now regularizes terms which previously diverged as p →
0. [Actually, withthe presence of more than one type of two-point vertex, even relationships like (7.8) need tobe rederived.] This observation could be important when inverting the relationship betweenthe dual action and the Wilsonian effective action: − S I [ φ, φ ] = ln n e −Y m [ δ/δφ,δ/δφ ] e −D m [ φ,φ ] o . (8.25)The point is that, since the dual action vertices are IR divergent, we presumably must take m = 0, at least at intermediate stages, in order to make sense of (8.25). Whilst it is truethat once S I has been computed, we should be able to safely send m →
0, it is quiteconceivable that contributions to the Wilsonian effective action of the form m / Λ × Λ /m are generated. Such terms are, of course, perfectly well defined in the m → IX. CONCLUSION
In this paper, we have given a comprehensive treatment of the renormalization of theoriesof a scalar chiral superfield. Central to our approach is the notion of ‘theory space’ rather48han a specific model. Theory space refers to the space of all possible (quasi-local) effectiveactions and it is in this space that we must search for non-trivial fixed points, the existenceof which is necessary if we are to construct interesting theories that are nonperturbativelyrenormalizable in d = 4. Unfortunately, just as in the case of scalar field theory, we arguedin section VII that there cannot be any non-trivial fixed points satisfying the conditionsof quasi-locality and unitarity (upon continuation to Minkowski space). Consequently, itis not possible to write down a physically acceptable non-trivial bare action for which thebare scale can be removed. Thus we conclude that the Wess-Zumino model suffers from theproblem of triviality.Of course, this does not stop one from treating the Wess-Zumino model as an effectivetheory and we did precisely that in section VIII, where the β -function was discussed. Itis heartening to see that, given familiarity with the formalism, it is no harder to computethe perturbative β -function within the ERG than within other formalisms. Moreover, theapproach has the added benefit of being defined nonperturbatively and we exploited this toshow that the β -function of the massless Wess-Zumino model (corresponding to the naturaldefinition of the coupling within the approach) is in fact free of nonperturbative powercorrections.The other main result of the paper is the proof of the nonrenormalization theorem,directly from the ERG, performed in section V.Besides these three results, it should be emphasised that the framework developed in thispaper is interesting in its own right. After all, it is formulated directly in d = 4, has UVregularization built in, is manifestly supersymmetric and is defined nonperturbatively. It ishoped that these features will encourage further research into this subject. Note Added
Since this paper was originally posted, there have been a number of interesting papersapplying the ERG to supersymmetric theories [64–67].49 ppendix A: SUSY Conventions
To define the N = 1 superfield formalism in four dimensional Euclidean space, we followLukierski and Nowicki [38] (see also [68] for a digestible summary). The lowest dimensionalfaithful spinor representation of SO(4) is described by two independent SU(2) spinors, whichwe will denote θ α ; , θ ; α . (A1)Note that, compared to [38], we have taken the indices to be upper, rather than lower, sothat our formulae map directly on to those of Wess and Bagger [39]. Furthermore, whencomparing to [38], the reader should be warned: some of the semicolons of [38] are inthe wrong place, some are either implicit or actually missing and the odd one has beenaccidentally replaced by a subscript j , which looks remarkably similar.The convention for complex conjugation is as follows:( θ α ; ) ∗ = θ ˙ α ; , ( θ ; α ) ∗ = θ ; ˙ α . (A2)Consequently, the lowest dimension Hermitean Euclidean superspace is S = ( x µ , θ α ; , θ ; α , θ ˙ α ; , θ ; ˙ α ) , (A3)which corresponds to N = 2 supersymmetry [69]. To obtain N = 1 superspace, we restrictourselves to non-Hermitean ‘Grassmann-analytic’ chiral superspaces: S − = ( x µ , θ α ; , θ ; ˙ α ) , S + = ( x µ , θ ; α , θ ˙ α ; ) . (A4)(The reader should be warned that the labelling of S ± is not consistent throughout [38].)Although we have lost Hermitean self-conjugacy for S + and S − , it is replaced by ‘Osterwalderand Schrader’ (OS) self-conjugacy [70], which involves Hermitean conjugation, followed bytime ( x ) reversal. Under this operation, θ α ; OS ←→ θ ; ˙ α , θ ; α OS ←→ θ ˙ α ; . (A5)Euclidean superfields which are OS-conjugate become Hermitean after continuation toMinkowski space and imposition of the Majorana condition. Focussing on S + , the σ matricesare chosen such that they are OS self-conjugate: σ µ ˙ α ;; α = ( σ j , i ) ˙ α ;; α . (A6)50f we now make the following identifications, where a ‘bar’ denotes OS-conjugation: θ ˙ α ; ≡ θ ˙ α , θ ; α ≡ θ α , σ µ ˙ α ;; α ≡ σ µ ˙ αα , (A7)then our spinor algebra conventions can be read off from those of Wess and Bagger, so longas we replace the Minkowski metric with δ µν and do not look inside σ µ .For completeness, we give the various formulae that were used to obtain the results inthis paper. Indices are raised and lowered with the epsilon tensors ǫ αβ , ǫ αβ , ǫ ˙ α ˙ β and ǫ ˙ α ˙ β with ǫ = ǫ = 1 etc. Defining σ µ ˙ αα ≡ ǫ ˙ α ˙ β ǫ αβ σ µ ˙ ββ (A8)we find ( σ µ σ ν + σ ν σ µ ) βα = − δ µν δ βα , (A9a)( σ µ σ ν + σ ν σ µ ) ˙ α ˙ β = − δ µν δ ˙ α ˙ β , (A9b)with the completeness relations: Tr σ µ σ ν = − δ µν , (A10a) σ µα ˙ α σ ˙ ββµ = − δ βα δ ˙ β ˙ α . (A10b)The spinor summation conventions are: ψχ = ψ α χ α = − ψ α χ α = χ α ψ α = χψ, (A11a) ψχ = ψ ˙ α χ ˙ α = − ψ ˙ α χ ˙ α = χ ˙ α ψ ˙ α = χψ, (A11b)where we will often enclose spinor products in round brackets, for clarity. We define( ρpθ ) ≡ ρ α σ µα ˙ α θ ˙ α p µ . (A12)It should be noted, to avoid possible confusion, that Lukierski et al. use what, in our notation,amounts to an ‘upper-lower’ convention of type (A11a) for both θ ; α and θ ˙ α ; . Consequently,whilst our superspace operators Q and Q , D and D take the same form as in Wess andBagger Q α = ∂∂θ α − iσ µα ˙ α θ ˙ α ∂ µ , (A13a) Q ˙ α = − ∂∂θ ˙ α − iθ α σ µα ˙ α ∂ µ , (A13b) D α = ∂∂θ α + iσ µα ˙ α θ ˙ α ∂ µ , (A13c) D ˙ α = − ∂∂θ ˙ α − iθ α σ µα ˙ α ∂ µ , (A13d)51hey differ from those in [38].When Fourier transforming the fermionic coordinates, the starting point is to recognizethat 16 Z d ρ e iρ · ( ω − θ ) = δ (4) ( ω − θ ) , (A14)where δ (4) ( θ ) = ( θθ )( θθ ) (A15)and Z d θ θθ = 1 , Z d θ θθ = 1 . (A16)Some useful formulae are: D ( − p, θ, θ ) e iρ · θ = (cid:2) − ( ρρ ) − i ( ρpθ ) + p ( θθ ) (cid:3) e iρ · θ , (A17a) D ( p, θ, θ ) e − iρ · θ = (cid:2) − ( ρρ ) + 2 i ( θpρ ) + p ( θθ ) (cid:3) e − iρ · θ . (A17b) Appendix B: Classical Two-Point Vertices
The completely Fourier transformed classical, two-point contribution to the K φφ vertexis given by: K φφ ( − p, ρ, ρ ; p, κ, κ ) = − c − ( p / Λ ) (cid:8)(cid:2) ( ρρ )( ρρ ) + 4( ρpρ ) − p (cid:3) (cid:2) ( κκ )( κκ ) + 4( κpκ ) − p (cid:3) +8( κpρ )( ρρ )( κκ ) + 16 p ( ρρ )( ρκ ) − p ( ρρ )( κκ ) + 16 p ( κκ )( ρκ ) + 32 p ( ρpκ ) (cid:9) , (B1)whilst the classical mass terms are given by: S φφ ( − p, ρ, ρ ; p, κ, κ ) = − m c − ( p / Λ ) × (cid:26) −
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