Superconformal Sum Rules and the Spectral Density Flow of the Composite Dilaton (ADD) Multiplet in $\mathcal{N}=1$ Theories

We discuss the signature of the anomalous breaking of the superconformal symmetry in $\mathcal{N}=1$ super Yang Mills theory, mediated by the Ferrara-Zumino hypercurrent ($\mathcal{J}$) with two vector ($\mathcal V$) supercurrents $(\mathcal{JVV})$ and its manifestation in the anomaly action, in the form of anomaly poles. This allows to investigate in a unified way both conformal and chiral anomalies. The analysis is performed in parallel to the Standard Model, for comparison. We investigate, in particular, massive deformations of the $\mathcal{N}=1$ theory and the spectral densities of the anomaly form factors which are extracted from the components of this correlator. In this extended framework it is shown that all the anomaly form factors are characterized by spectral densities which flow with the mass deformation. In particular, the continuum contributions from the two-particle cuts of the intermediate states turn into into poles in the zero mass limit, with a single sum rule satisfied by each component. Non anomalous form factors, instead, in the same anomalous correlators, are characterized by non-integrable spectral densities. These tend to uniform distributions as one moves towards the conformal point, with a clear dual behaviour. As in a previous analysis of the dilaton pole of the Standard Model, also in this case the poles can be interpreted as signaling the exchange of a composite dilaton/axion/dilatino (ADD) multiplet in the effective Lagrangian.[...]


Introduction
Dilaton fields are expected to play a very important role in the dynamics of the early universe and are present in almost any model which attempts to unify gravity with the ordinary gauge interactions (see for instance [1]). Important examples of these constructions are effective field theories derived from strings, describing their massless spectra, but also theories of gravity compactified on extra dimensional spaces, where the dilaton (graviscalar) emerges in 4 spacetime dimensions from the extra components of the higher dimensional metric (see for instance [2,3,4,5,6]). In these formulations, due to the geometrical origin of these fields, the dilaton is, in general, a fundamental (i.e. not a composite) field. Other extensions, also of significant interest, in which a fundamental dilaton induces a gauge connection for the abelian Weyl symmetry in a curved spacetime, have been considered (see the discussion in [7,8,9]). However, also in this case, the link of this fundamental particle to gravity renders it a crucial player in the physics of the early universe, and not a particle to be searched for at colliders. In fact, its interaction with ordinary matter should be suppressed by the Planck scale, except if one entails scenarios with large extra dimensions.
More recently, following an independent route, several extensions of the Standard Model with an effective dilaton have been considered. They conjecture the existence of a scale-invariant extension of the Higgs sector [10,11,12]. In this case the breaking of the underlying conformal dynamics, in combination with the spontaneous breaking of the electroweak symmetry [9], suggests, in fact, that the dilaton could emerge as a composite field, appearing as a Nambu-Goldstone mode of the broken conformal symmetry. A massless dilaton of this type could acquire a mass by some explicit potential and could mix with the Higgs of the Standard Model. By reasoning in terms of the conformal symmetry of the Standard Model, which should play a role at high energy, the dilaton would be the physical manifestation of the trace anomaly in the Standard Model, in analogy to the pion, which is interpolated by the U (1) A chiral current and the corresponding AV V (axial-vector/vector/vector) interaction in QCD. As in the AV V case, this composite state should be identified with the anomaly pole of the related anomaly correlator (the T V V diagram, with T the energy momentum tensor (EMT)), at least at the level of the 1-particle irreducible (1PI) anomaly effective action [11]. Considerations of this nature brings us to the conclusion that the effective massless Nambu-Goldstone modes which should appear as a result of the existence of global anomalies, should be looked for in specific perturbative form factors under special kinematical limits. For this reason they are easier to investigate in the on-shell anomaly effective action, with a single mass parameter which drives the conformal/superconformal deformation. This action has the advantage of being gauge invariant and easier to compute than its off-shell relative.

Anomalies and anomaly poles
To exploit at a full scale the analogy between chiral and conformal anomalies, one should turn to supersymmetry, where the correlation between poles and anomalies should be more direct. In fact, in an ordinary quantum field theory, the T V V diagram (and the corresponding anomaly action) is characterized, as we are going to show, by pole structures both in those form factors which multiply tensors that contribute to the trace anomaly and in those which don't. For this reason we turn our attention to the effective action of the superconformal (the Ferrara-Zumino, FZ) multiplet, where chiral and conformal anomalies share similar signatures, being part of the same multiplet. Therefore one would expect that supersymmetry should help in clarifying the significance of these singularites in the effective action. We are going to prove rigorously in perturbation theory that the anomaly of the FZ multiplet is associated with the exchange of three composite states in the 1PI superconformal anomaly action. These have been discussed in the past, in the context of the spontaneous breaking of the superconformal symmetry [13]. They are identified with the anomaly poles present in the effective action, extracted from a supersymmetric correlator containing the superconformal hypercurrent and two vector currents, and correspond to the dilaton, the dilatino and the axion. This exchange is identified by a direct analysis of the anomalous correlators in perturbation theory or by the study of the flow of their spectral densities under massive deformations. The flow describes a 1-parameter family of spectral densities -one family for each component of the correlator -which satisfy mass independent sum rules, and are, therefore, independent of the superpotential. This behaviour turns a dispersive cut of the spectral density ρ(s, m 2 ) into a pole (i.e. a δ(s) contribution) as the deformation parameter m goes to zero. Moreover, denoting with k 2 the momentum square of the anomaly vertex, each of the spectral densities induces on the corresponding form factor a 1/k 2 behaviour also at large k 2 , as a consequence of the sum rule. We also recall that the partnership between dilatons and axions is not new in the context of anomalies, and it has been studied in the past -for abelian gauge anomalies -in the case of the supersymmetric Stückelberg multiplet [14,15,16,17]. The three states associated to the three anomaly poles mentioned above are described -in the perturbative picture -by the exchange of two collinear particles. These are a fermion/antifermion pair in the axion case, a fermion/antifermion pair and a pair of scalar particles in the dilaton case, and a collinear scalar/fermion pair for the dilatino. The Konishi current will be shown to follow an identical pattern and allows the identification of extra states, one for each fermion flavour present in the theory. This pattern appears to be general in the context of anomalies, and unique in the case of supersymmetry. In fact, we are going to show that in a supersymmetric theory anomaly correlators have a single pole in each component of the anomaly multiplet, a single spectral flow and a single sum rule, proving the existence of a one-to-one correspondence between anomalies and poles in these correlators.
Our work is organized as follows. We first illustrate the motivations of our study by overviewing the analysis of the spectral densities performed in the investigations of the conformal anomaly in the T V V vertex at nonzero momentum transfers in QED [18] [19] and QCD [20]. The case of a general non-abelian theory, with the inclusion of scalars, is new and is discussed -in the massless case -in Section 3. This may serve to highlight some specific properties of these types of vertices which have not received sufficient attention in the past.
Then we turn to a perturbative study of the effective action in the case of supersymmetric N = 1 theories, focusing on the components of the FZ multiplet and on the corresponding anomalies. This is followed by a study of the spectral densities of the relevant diagrams which are responsible for the superconformal anomaly. We show the existence of a unique sum rule for each component of the multiplet, and of a spectral density flow driven by the mass perturbations. As the deformation (mass) parameter turns to zero, restoring the superconformal symmetry, the flow gets localized at zero invariant mass, signalling the exchange of a massless pole in the anomaly effective action. We will compare these results for supersymmetric theories with the ordinary non supersymmetric case, highlighting the differences between the two. In particular we show in detail how the cancellation of the extra poles of non anomalous form factors is realized in supersymmetric theories. Finally, we present the structure of the anomaly action as a combination of the pole contributions, plus the non anomalous (logarithmic) terms. In superspace, the first had been identified in the past by supersymmetric arguments [2]. We will conclude our analysis with some comments on the possible implications of our results for what concerns the physical manifestation of anomaly poles -for global anomalous symmetries -and their cancellations in the case of their superconformal gaugings.

Sum rules
As pointed out long ago in the literature on the chiral anomaly [21], its perturbative signature is in the appearance of a massless pole (an anomaly pole) in the spectrum of the AV V diagram. The pole is present, in perturbation theory, only in a specific kinematic configuration, namely at zero fermion mass and with on-shell vector lines. The intermediate state which is exchanged in the effective action, see Fig. (1), and which is mediated by the AV V diagram, is characterized by two massless and collinear fermions moving on the light cone. It is rather compelling to interpret the appearance of this intermediate configurationwithin the obvious limitations of the perturbative picture -as signalling the possible exchange of a bound state in the quantum effective action. In a phenomenological context, what gives a far broader significance to this kinematical mechanism is the appearance of a certain kinematic duality, which accompanies any perturbative anomaly. In this case it is better known as Q 2 -duality, relating the resonance and the asymptotic region of a certain correlator in a nontrivial way [22]. This property, in general, finds its justification in the existence of a sum rule for the spectral density ρ(s, m 2 ). Generically, it is given in the form with the constant f independent of any mass (or other) parameter which characterizes the thresholds or the strengths of the resonant states eventually present in the integration region (s > 0). It should be stressed that sum rules formulated for the study of the structure of the resonances, i.e. their strengths and masses, as in the QCD case, involve a parameterization of the resonant behaviour of ρ(s, m) at low s values, via a phenomenological approach, with the inclusion of the asymptotic behaviour of the correlator, amenable to perturbation theory, for larger s. For this significant interplay between the infrared (IR) and the ultraviolet (UV) regions, the term duality is indeed quite appropriate to qualify the implications of a given sum rule. It was pointed out, some time ago, that one specific tract of the chiral anomaly is in the existence of a sum rule for the AV V diagram [23], later extended to a similar study of the trace of the energy momentum tensor (trT ) for the trT V V in QED (with V a vector current), at zero momentum transfers [24,25]. Similar analysis were performed on the T T correlator in 2-dimensional gravity [26], which is affected by the trace anomaly. The analysis brought substantial evidence that the sum rule, combined with the original identification of the anomaly pole from the perturbative spectral density [21], were two important and related aspects of the anomaly phenomenon. We recall that the study of these types of correlators has a quite long history [27,28,29].
More recently, very general perturbative analysis of the T V V correlator (or graviton-gauge-gauge vertex), performed off-shell and at nonzero momentum transfer, have shown that the general features observed in the AV V and trT V V cases where preserved [18,19,20,30]. A specific feature of the spectral density of the chiral and conformal anomalies is that the pole is introduced in the spectrum in a specific kinematical limit, as a degeneracy of the two-particle cut when any second scale (for instance the fermion mass) is sent to zero. The property of the cut turning into a pole is peculiar for finite (non superconvergent) sum rules. It is related to a spectral density which is normalized by the sum rule just like an ordinary weighted distribution, and whose support is located at the edge of the allowed phase space (s = 0) as the conformal deformation turns to zero. This allows to single out a unique interpolating state out of all the possible exchanges permitted in the continuum, i.e. for s > 4m 2 , as the theory flows towards its conformal/superconformal point.

Sum rule and the UV/IR conspiracy of the anomaly
As we have just mentioned, the existence of a sum rule for the form factor responsible for a certain anomaly indicates a UV/IR connection manifested by the corresponding spectral density, but it is not exclusively related, obviously, to the anomaly phenomenon. In fact, non anomalous form factors, in some cases, as we are going to show next, share a similar behaviour. We would expect, though, that the breaking of a symmetry should manifest in the apperance of a massless state in the spectrum of the effective theory, and in this respect the saturation of the spectral density with a single resonance in an anomaly form factor acquires a special status. Elaborating on Eq. (1), one can show that the effect of the anomaly is, in general, related to the behaviour of the spectral density at any values of s, although, in some kinematical limits, it is the region around the light cone (s ∼ 0) which dominates the sum rule, and amounts to a resonant contribution. In fact, the combination of the scaling behaviour of the corresponding form factor F (Q 2 ) (equivalently of its density ρ) with the requirement of integrability of the spectral density, essentially fix f to be a constant and the sum rule (1) to be saturated by a single massless resonance. Obviously, a superconvergent sum rule, obtained for f = 0, would not share this behaviour. At the same time, the absence of subtractions in the dispersion relations guarantees the significance of the sum rule, being this independent of any ultraviolet cutoff.
It is quite straightforward to show that Eq. (1) is a constraint on asymptotic behaviour of the related form factor. The proof is obtained by observing that the dispersion relation for a form factor in the spacelike once we expand the denominator in Q 2 as 1 . and make use of Eq. (1), induces the following asymptotic behaviour on The F ∼ f /Q 2 behaviour at large Q 2 , with f independent of m, shows the pole dominance of F for Q 2 → ∞. The UV/IR conspiracy of the anomaly, discussed in [30,31,19], is in the reappearance of the pole contribution at very large value of the invariant Q 2 , even for a nonzero mass m. In fact, as we are going to show in the following sections, the spectral density has support around the s = 0 region (ρ(s) ∼ δ(s)), as in the massless (m = 0) case. This point is quite subtle, since the flows of the spectral densities with m show the decoupling of the anomaly pole for a nonzero mass. Here, the term decoupling will be used to refer to the non resonant behaviour of ρ. Therefore, the presence of a 1/Q 2 term in the anomaly form factors is a property of the entire flow which a) converges to a localized massless state (i.e. ρ(s) ∼ δ(s)) as m → 0, while b) the presence of a non vanishing sum rule guarantees the validity of the asymptotic constraint illustrated in Eq. (3). Notice that although for conformal deformations driven by a single mass parameter the independence of the asymptotic value f on m is a simple consequence of the scaling behaviour of F (Q 2 , m 2 ), it holds quite generally even for a completely off-shell kinematics [18]. In summary, in complete agreement with a previous analysis by Giannotti and Mottola [18], we are going to verify that for a generic supersymmetric N = 1 theory, the two basic features of the anomalous behaviour of a certain form factor responsible for chiral or conformal anomalies are: 1) the existence of a spectral flow which turns a dispersive cut into a pole as m goes to zero and 2) the existence of a sum rule which relates the asymptotic behaviour of the anomaly form factor to the strength of the pole resonance. In a supersymmetric theory this correspondence, as we are going to show, is unique, since the only poles present in the explicit expressions are those of the anomaly form factors. This feature is shared also by the AV V in non supersymmetric theories, where one can identify a single pole in the related form factor, a single sum rule and a single spectral density flow. In the T V V diagram, for a general field theory, instead, this feature is absent. The appearance of extra poles in the form factors of the traceless parts of this second correlators leaves unanswered the question about the physical meaning of these additional singularities [18,19,20,30]. On the other end, the effective massless states emerging from the anomaly sectors should be identified with the Nambu-Goldstone modes of the corresponding broken symmetries, which are such because of non conserved dilatation and chiral currents.
The reason for turning to supersymmetry should be obvious. One expects, in general, that the perturbative structure of the chiral and conformal [32] anomalies, in this case, should be similar. This should occur for a supersymmetric anomaly multiplet, where both the T V V and the AV V -like diagrams are components of the same anomalous correlator. At the same time, we expect to recover, for each single component, the properties found in the past, separately in the chiral and in the conformal cases [23,24,25], but, hopefully, without the extra poles which are present in the non anomalous tensor structures and form factors of T V V . We are going to prove, by an explicit computation, that this is indeed true. We also stress the fact that our analysis, in particular in the T V V case, is entirely performed at nonzero momentum transfer, working with the insertion of the uncontracted T , rather than with its trace, as done in [24,25]. The study is centered around the J VV correlator, in N = 1 and N = 4 theories, containing the hypercurrent J (the Ferrara-Zumino multiplet [33]) and two vector supercurrents V. We will show that the two requirements enunciated above are satisfied by every component of this multiplet. However, before moving to the supersymmetric case, we briefly review the results of the computation of the T V V in an ordinary non-abelian gauge theory. This may serve to illustrate the differences between an ordinary gauge theory and its supersymmetric version, which has triggered the current analysis.

The T V V and AV V vertices in an ordinary gauge theory
We consider a non-abelian gauge theory containing massless scalars, fermions and gauge fields, with fermions and scalars assigned to the representations R f and R s respectively, and define a correlator with a symmetric EMT (T ) and two vector (gauge) currents (V ). The correlator can be interpreted, in a weak gravitational background, as describing the one graviton-two gauge fields vertex, which is affected by the trace anomaly. The analysis in QED is contained in [18,19], while generalization to QCD and to the Standard Model can be found in [20,34]. The general tensor structure of this type of vertex has been given in QED by Giannotti and Mottola [18] in terms of a non minimal basis of 13 form factors (t 1 , t 2 , . . . , t 13 ) in the course of their studies on the 1PI conformal anomaly effective action for gravity. Here we present the tensor expansion of the one-loop T V V vertex with on-shell vector lines in a non-abelian gauge theory. The details can be found in [20]. The on-shell expansion of the T V V correlator in a non-abelian gauge theory is expressed in terms of just 3 independent form factors [20] Γ µναβ where the tensor structures are defined by with Here k = p + q is the incoming momentum in the EMT line, while p α and q β are the two outgoing momenta from the two vector currents. For massless fields running in the loops, of these 3 tensor structures only φ 1 is traceful, contributing to the trace anomaly, the remaining ones being traceless. Fermions, scalars and gauge fields give contributions which are separately gauge invariant. Both f 1 , the anomaly form factor, and the form factor f 2 of the traceless tensor φ 2 are found to be finite, while f 3 , the form factor of the traceless φ 3 , gets renormalized.
In the most general paremeterization of the vertex -assuming nonzero virtualities of all the external lines and an internal mass (m) for the field in the loops -t µναβ 1 is still the only traceful and anomalous tensor structure. t 2 is also traceful, but describes the explicit breaking of the conformal symmetry (its form factor is proportional to m and therefore it is not anomalous) and t 13 is the only tensor structure affected by renormalization. The remaining form factors, corresponding to the contributions (t 1 , t 2 , t 3 , . . . , t 12 ) are finite. In the on-shell and massless case, for a Dirac fermion (f ) in the representation R f running in the loops, the form factors are given by Analogous results hold for a conformally coupled complex scalar (s) in the representation R s f (s) while for a gauge field (A) in the adjoint representation one obtains A discussion of the scalar integrals is given in Appendix A. It is a common lore to denote with B 0 and C 0 the scalar 2-and 3-point functions. Note that in the expression of C 0 (k 2 , m 2 ), the scalar triangle integral, the first entry is the only nonzero external invariant, while m is the mass of the virtual particles. Also note that f 1 and f 2 are both finite, while f 3 needs renormalization, as we have just mentioned. Each contribution is separately gauge invariant and it is characterized by an anomaly pole in the corresponding form factor f 1 , described by a spectral density which is a Dirac delta (∼ δ(k 2 )). However, additional poles are present also in f 2 , which multiplies a traceless structure, and are not directly linked to the conformal anomaly. These extra poles have demised, so far, any interpretation, but they seem to be on the same ground as the anomaly poles of the correlator. It is then clear that both f 1 and f 2 , in this case, should be treated on the same footing since, as we are going to show, they are both characterized by spectral densities satisfying the conditions enunciated in the previous sections. We will see, however, that supersymmetry gives a surprisingly simple answer on this issue, since the extra, non anomalous poles in the supersymmetric case are simply not present. The existence of extra poles is a characteristic of the T V V correlator in ordinary gauge theories, but not of the AV V one, where A is an axial-vector current. We recall that for an axial anomaly, the usual Rosenberg parameterization in terms of six form factors (A 1 , . . . , A 6 ), and the use of the Ward identities and on-shellness conditions on the vector lines, reduce the anomaly amplitude ∆ λµν to the simple form [31] with k denoting the incoming momentum of the axial-vector line (of Lorentz index λ), and with p and q denoting the outgoing momenta of the (µ, ν) vector lines. Note that in this case the transversality condition for the vector currents removes the second combination of form factors, leaving only a nonzero A 6 , which is given by with τ (k 2 , m 2 ) = 1 − 4m 2 /k 2 . In the massless limit, the spectral density of this form factor, for k 2 > 0, is proportional to a Dirac δ-function, since the logarithmic term vanishes, and is accompanied by a sum rule. In any case, the spectral density of the A 4 + A 6 form factor is not integrable, and the link between the chiral anomaly and the corresponding pole is again unique. We will illustrate this point in a later section. We now turn to discuss the structure of the correlator which is responsible for the superconformal anomaly, proceeding with a perturbative analysis of its components and of its related spectral densities.

Theoretical framework
In this section we review the definition and some basic properties of the Ferrara-Zumino supercurrent multiplet, which from now on we will denote also as the hypercurrent, in order to distinguish it from its fermionic component, usually called the supercurrent. We consider a N = 1 supersymmetric Yang-Mills theory with a chiral supermultiplet in the matter sector.
In the superfield formalism the action is given by where the supersymmetric field strength W A and gauge vector field V are contracted with the hermitian generators T a of the gauge group to which the chiral superfield Φ belongs. In particular V = 2gV a T a , and with Tr T a T b = T (R)δ ab . In order to clarify our conventions we give the component expansion of the chiral superfield Φ and of the superfields W a A and V a in the Wess-Zumino gauge where φ i is a complex scalar and χ i its superpartner, a left-handed Weyl fermion, A a µ and λ a are the gauge vector field and the gaugino respectively, F a µν is the gauge field strength while F i and D a correspond to the F -and D-terms. Moreover, we have defined σ µν = (i/4)(σ µσν − σ νσµ ). Using the component expansions introduced in Eq. (14) and (15) we obtain the supersymmetric lagrangian in the component formalism, which we report for convenience where the gauge covariant derivatives on the matter fields and on the gaugino are defined respectively as with t abc the structure constants of the adjoint representation, and the scalar potential is given by For the derivatives of the superpotential we have been used the following definitions where the symbol | on the right indicates that the quantity is evaluated at θ =θ = 0. Notice that in the above equations the F -and D-terms have been removed exploiting their equations of motion. Having defined the model, we can introduce the Ferrara-Zumino hypercurrent where ∇ A is the gauge-covariant derivative in the superfield formalism whose action on chiral superfields is given by The conservation equation for the hypercurrent J AȦ is where γ is the anomalous dimension of the chiral superfield. The first two terms in Eq. (23) describe the quantum anomaly of the hypercurrent, while the last is of classical origin and it is entirely given by the superpotential. In particular, for a classical scale invariant theory, in which W is cubic in the superfields or identically zero, this term identically vanishes. If, on the other hand, the superpotential is quadratic the conservation equation of the hypercurrent acquires a non-zero contribution even at classical level. This describes the explicit breaking of the conformal symmetry.
We can now project the hypercurrent J AȦ defined in Eq.(21) onto its components. The lowest component is given by the R µ current, the θ term is associated with the supercurrent S µ A , while the θθ component contains the energy-momentum tensor T µν . In the N = 1 super Yang-Mills theory described by the Lagrangian in Eq. (17), these three currents are defined as where L is given in Eq. (17) and S µ I and T µν I are the terms of improvement in d = 4 of the supercurrent and of the EMT respectively. As in the non-supersymmetric case, these terms are necessary only for a scalar field and, therefore, receive contributions only from the chiral multiplet. They are explicitly given by T µν The terms of improvement are automatically conserved and guarantee, for W(Φ) = 0, upon using the equations of motion, the vanishing of the classical trace of T µν and of the classical gamma-trace of the supercurrent S µ A .
The anomaly equations in the component formalism, which can be projected out from Eq. (23), are The first and the last equations are respectively extracted from the imaginary and the real part of the θ component of Eq. (23), while the gamma-trace of the supercurrent comes from the lowest component.

The perturbative expansion in the component formalism
In this section we will present the one-loop perturbative analysis of the one-particle irreducible correlators, built with a single current insertion contributing -at leading order in the gauge coupling constant -to the anomaly equations previously discussed. We define the three correlation functions, Γ (R) , Γ (S) and Γ (T ) as with k = p + q and where we have factorized, for the sake of simplicity, the Kronecker delta on the adjoint indices. These correlation functions have been computed at one-loop order in the dimensional reduction scheme (DRed) using the Feynman rules listed in Appendix E. We recall that in this scheme the tensor and scalar loop integrals are computed in the analytically continued spacetime while the sigma algebra is restricted to four dimensions. In order to provide more details, we will present the results for the matter chiral and gauge vector multiplets separately, for on-shell external gauge lines. The chiral contribution will be discussed first, and the result will be given with the inclusion of the corresponding mass corrections. Notice that the matter chiral superfield belongs to a certain representation R of the gauge group. If the representation is complex, for instance the fundamental of SU (N ), then the superfield must be accompanied by another superfield (eventually with the same mass) belonging to the complex-conjugate representation R. In this case, the generatorT a ofR are related to those of R by the equationT a = −(T a ) T = −(T a ) * . For simplicity, in the following we will consider just the case of a single chiral superfield in a real representation of the gauge group. The extension to a complex representation amounts just to a factor of 2 in front of all the expressions which are generated for the chiral multiplet. Indeed these terms are all proportional to T (R), which is equal to T (R). The one-particle irreducible correlation functions of the Ferrara-Zumino multiplet are ultraviolet (UV) divergent, as one can see from a direct computation, and we need a suitable renormalization procedure in order to get finite results. In particular we have explicitly checked that, at one-loop order, among the three correlators defined in Eq. (32), only those with S µ A and T µν require a UV counterterm. The renormalization of the correlation functions is guaranteed by replacing the bare operators in Eq. (25) and Eq. (26) with their renormalized counterparts. This introduces the renormalized parameters and wave-function renormalization constants which are fixed by some conditions that specify the renormalization scheme. In particular, for the correlation functions we are interested in, the bare S µ A and T µν current become where the suffix R denotes renormalized quantities. Z A and Z λ are the wave-function renormalization constants of the gauge and gaugino field respectively, while the ellipses stand for all the remaining operators.
In the previous equations we have explicitly shown only the contributions from which, at one-loop order, we can extract the counterterms needed to renormalize our correlation functions. All the other terms, not shown, play a role at higher perturbative orders.
Expanding the wave-function renormalization constants at one-loop as Z = 1 + δZ we obtain the vertices of the counterterms with p and q outgoing momenta and where the two tensor structures C µναβ and D µναβ (p, q) are given in Appendix E. The δZ counterterms can be defined, for instance, by requiring a unit residue of the full two-point functions on the physical particle poles. This implies that where the one-loop corrections to the gauge and gaugino two-point functions are defined as with Using the previous expressions we can easily compute the wave-function renormalization constants and therefore obtain the one-loop counterterms needed to renormalize our correlators. In the following we will always present results for the renormalized correlation functions. It is interesting to observe that, accordingly to Eq. (34), the one-loop counterterm to the supercurrent correlation function is identically zero for the vector gauge multiplet, due to a cancellation between δZ A and δZ λ . Therefore we expect a finite result for the vector supermultiplet contribution to the Γ µα (S) . Indeed this is the case as we will show below. The correctness of our computations is secured by the check of some Ward identities. These arise from gauge invariance, from the conservation of the energy-momentum tensor and of the supercurrent. In particular, for the three point correlators defined above, we have from the conservation of the vector current, and for the conservation of the supercurrent and of the EMT respectively, whereΓ (AA) andΓ (λλ) are the renormalized self-energies. Their derivation follows closely the analysis presented in [20]. Notice that, for on-shell gauge and gaugino external lines, the two identities in Eq. (42) simplify considerably because their righthand sides vanish identically. 6 The supercorrelator in the on-shell and massless case In this section we discuss the explicit results of the computation of supercorrelator when the components of the external vector supercurrents are on-shell and the superpotential of the chiral multiplet is absent. We will consider first the contributions due to the exchange of the chiral multiplet, followed by a subsection in which we address the exchange of a virtual vector multiplet.

The chiral multiplet contribution
We start from the chiral multiplet, presenting the result of the computation for massless fields and on shell gauge and gaugino external lines.
-Three-point function of the R µ current The diagrams defining the one-loop expansion of the Γ (R) correlator are shown in Fig. (2). They consist of triangle and bubble topologies with fermions, since the scalars do not contribute. The explicit result for a massless chiral multiplet with on-shell external gauge bosons is given by Figure 2: The one-loop perturbative expansion of the RV V correlator with a massless chiral multiplet running in the loops.
There is no much surprise, obviously, for the anomalous structure of Eq. (43) which is characterized by a pole 1/k 2 term, since in the on-shell case and for massless fermions (which are the only fields contributing to the RV V at this perturbative order), we recover the usual structure of the AV V diagram.
-Three-point function of the S µ A current The perturbative expansion of the Γ µα (S) AḂ correlation function is depicted in Fig. (3). For simplicity we will remove, from now on, the spinorial indices from the corresponding expressions. The explicit result for a massless chiral supermultiplet with on-shell external gauge and gaugino lines is then given by where the form factor Φ 2 (k 2 , 0) is defined as Figure 4: The one-loop perturbative expansion of the T V V correlator with a massless chiral multiplet running in the loops. The last diagram, being a massless tadpole, is identically zero in dimensional regularization. and the two tensor structures are The B 0 function appearing in Eq.(46) is a two-point scalar integral defined in Appendix A. Notice that the form factor multiplying the second tensor structure s 2 is ultraviolet finite, due to the renormalization procedure, but has an infrared singularity inherited by the counterterms in Eq. (34).
It is important to observe that the only pole contribution comes from the anomalous structure s µα 1 , which shows that the origin of the anomaly has to be attributed to a unique fermionic pole (σ ρ k ρ /k 2 ) in the correlator, in the form factor multiplying s µα 1 . It is easy to show that Eq. (45) satisfies the vector current and EMT conservation equations. Moreover, the anomalous equation reads as where only the first tensor structure contributes to the σ-trace of the correlator. This result is clearly in agreement with Eq.(30) after Fourier transform (F.T .), owing to Notice also that -Three-point function of the energy-momentum tensor T µν The diagrams appearing in the perturbative expansions of the Γ (T ) are depicted in Fig.(4). They consist of triangle and bubble topologies. There is also a tadpole-like contribution, Fig.(4j), which is non-zero only in the massive case. The explicit expression of the Γ (T ) correlator for a massless chiral supermultiplet and on-shell gauge lines is given by where the Φ 2 is defined in Eq.(46) and where φ µναβ 1 , φ µναβ 3 and u αβ are given in Eqs. (5) and (6). As in the previous cases we have explicitly checked all the Ward identities originating from gauge invariance and conservation of the energy-momentum tensor. As one can easily verify by inspection, only the first one of the two tensor structures is traceful and contributes to the anomaly equation of the Γ (T ) correlator The comparison of Eq.(54) to Eq. (31) is evident if one recognizes that For completeness we give also the inverse Fourier transform of t µναβ 2S (p, q) which is obtained from where T µν gauge is the pure gauge part of the energy-momentum tensor. Notice that t 2S is nothing else than the tree-level vertex with two onshell gauge fields on the external lines.
As in the previous subsection, concerning the supersymmetric current S µ A , also in the case of this correlator there is only one structure containing a pole term, which appears in the only form factor (which multiplies t 1S ) with a nonvanishing trace. Differently from the non supersymmetric case, such as in QED and QCD, with fermions or scalars running in the loops, as shown in Eqs.

The vector multiplet contribution
Finally, we come to a discussion of the perturbative results for the vector (gauge) multiplet to the three anomalous correlation functions presented in the previous sections. Notice that due to the quantization of the gauge field, gauge fixing and ghost terms both must be taken into account, increasing the complexity of the computation. This technical problem is completely circumvented with on-shell gauge boson and gaugino, which is the case analyzed in this work. Concerning the diagrammatic expansion, the topologies of the various contributions defining the three correlators is analogous to those illustrated in massless chiral case. The explicit results are given by where The tensor expansion of the correlators is the same as in the previous cases. The only differences are in the form factors. In particular, the first in each of them is the only one responsible for the anomaly and is multiplied, respect to the chiral case, by a factor −3 and by a different group factor. The result reproduces exactly the anomaly Eqs (29,30,31). Concerning the ultraviolet divergences of these correlators, the explicit computation shows that the vector multiplet contribution to Γ µν (S) is indeed finite at one-loop order before any renormalization. This confirms a result obtained in the analysis of the renormalization properties of these correlators presented in a previous section, where it was shown the vanishing of the counterterm of Γ µα (S) for the vector multiplet. Also for the vector multiplet, the result is similar, since the only anomaly poles present in the three correlators (57), (58) and (59) are those belonging to anomalous structures. We conclude that in all the cases discussed so far, the signature of an anomaly, in a superconformal theory, are anomaly poles.

The supercorrelator in the on-shell and massive case
We now extend our previous analysis to the case of a massive chiral multiplet. This will turn out to be extremely useful in order to discuss the general behaviour of the spectral densities away from the conformal point.
The diagrammatic expansion of the three correlators for a massive chiral multiplet in the loops get enlarged by a bigger set of contributions characterized by mass insertions on the S µ A and T µν vertices and on the propagators of the Weyl fermsions. A sample of them are shown in Fig. (5). An explicit computation, A a α (p) (c) Figure 5: A sample of diagrams, for a massive chiral multiplet, mass insertions in the fermion propagators.
in this case, gives with The expressions above show that the only modification introduced by the mass corrections is in the form factors, while the tensor structure remains unchanged.
As we have previously discussed, if the superpotential is quadratic in the chiral superfield, the conservation equation of the hypercurrent is non homogeneous. Its four-divergence equals a classical (non-anomalous) contribution due to the explicit breaking of the conformal symmetry. Therefore, in this case, the anomaly equations (44),(48), and (54) must be modified in order to account for the mass dependence. The new conservation equations for a massive chiral supermultiplet become It is interesting to observe that supersymmetry prevents the appearance of new structures in the conservation equations, at least for these correlation functions, being the explicit classical breaking terms just a correction to the anomaly coefficient. This is not the case for non-supersymmetric theories [18,19].

The flavor chiral symmetries and the Konishi anomaly
If the superpotential W(Φ) is absent, the action in Eq. (12) is also invariant under a phase rotation of the chiral superfield alone. This transformation, differently from the R transformation, does not affect the θ,θ coordinates. If the theory contains N f flavor chiral superfields Φ f , then we can construct N f chiral currents associated to the each of the independent U (1) flavor rotations. In the superfield formalism these are given by where J f is the Konishi operator defined as In the component formalism the chiral currents are extracted from the θθ component of the Konishi operator and are given by Differently from the R current, which belongs to a supermultiplet together with the supercurrent and the energy-momentum tensor, the U (1) chiral currents discussed here are the only non-trivially conserved components of the Konishi operator.
As for non-supersymmetric theories, these U (1) chiral symmetries suffer from an anomaly whose equation in the superfield formalism is given bȳ or, equivalently, in components as The one-loop perturbative computation for the three-point function, Γ µαβ (J f ) , with a J f µ current insertion and two on-shell gauge fields on the external lines can be easily recovered from the previous computations. Indeed, due to its chiral nature, the J f current is quite similar to the R current. Taking into account the fact that the scalar part of J f µ in Eq.(70) does not contribute to the one-loop correlator, the result for Γ µαβ (J f ) is obtained from Eq.(43) with a multiplicative factor −3, or from the vector multiplet contribution of Eq.(57) with a different group theoretical factor. Therefore, for massless chiral multiplets we have which manifests, also in this case, an anomaly pole. We conclude this section by giving the expression of the correlator responsible for the Konishi anomaly in the massive case with Φ 1 (k 2 , m 2 ) given in Eq. (64), in full analogy with the result for the correlator of the R current.
In the next section we investigate the sum rule and the spectral density flows associated with these correlators, showing the universality of their behaviour.

Mass deformations and the spectral densities flow
In this and in the following section, we turn to a detailed discussion of the dispersive structure of the form factors of the correlators computed above, since their spectral densities carry significant information on the anomaly. As before, we will be setting the momenta p, q on-shell, and choose the incoming momentum k to be either spacelike, timelike or null. Being interested in the analysis of the spectral density of the anomalous form factor Φ 1 (k 2 , m 2 ), it is convenient first to describe the analytic properties of the three-point scalar integral C 0 (k 2 , m 2 ) which enters in the definition of Φ 1 , as clear from Eq. (64). We start by introducing the spectral density ρ(k 2 ), which is the discontinuity of C 0 along the cut (k 2 > 4m 2 ), as with the usual i prescription ( > 0) To determine the discontinuity above the two-particle cut we can proceed in two different ways. We can use the unitarity cutting rules and therefore compute the integral where τ (k 2 , m 2 ) = 1 − 4m 2 /k 2 . The integral has been computed by sitting in the rest frame of the offshell line of momentum k. Alternatively, we can exploit directly the analytic continuation of the explicit expression of the C 0 (k 2 , m 2 ) integral in the various regions. This is given by From the two branches encountered with the ±i prescriptions, the discontinuity is then present only for k 2 > 4m 2 , as expected from unitarity arguments, and the result for the discontinuity, obtained using the definition in Eq. (76), clearly agrees with Eq. (77), computed instead by the cutting rules. The dispersive representation of C 0 (k 2 , m 2 ) in this case is written as which, for k 2 < 0 gives the identity with ρ(s, m 2 ) given by Eqs. (75) and (77). The identity in Eq. (80) allows to reconstruct the scalar integral C 0 (k 2 , m 2 ) from its dispersive part.
Having determined the spectral function of the scalar integral C 0 (k 2 , m 2 ), we can extract the spectral density associated with the anomaly form factors in Eqs. (61), (62), (63) and (74), which is given by and which can be computed as Using the principal value prescription we obtain where we have defined and This gives, together with the discontinuity of C 0 (k 2 , m 2 ) which we have computed previously in Eq. (77), The discontinuity of the anomalous form factor χ(k 2 , m 2 ) is then given by The total discontinuity of χ(k 2 , m 2 ), as seen from the result above, is characterized just by a single cut for k 2 > 4m 2 , since the δ(k 2 ) (massless resonance) contributions cancel between the first and the second term of Eq. (82). This result proves the decoupling of the anomaly pole at k 2 = 0 in the massive case due to the disappearance of the resonant state. The function describing the anomaly form factor, χ(k 2 , m 2 ), then admits a dispersive representation over a single branch cut corresponding to the ordinary threshold at k 2 = 4m 2 , with ρ χ (s, m 2 ) = 1 2i Disc χ(s, m 2 ) = 2πm 2 s 2 log From the spectral function given above and from the corresponding integral representation one can extract a new nontrivial integral relation which is the analogue of Eq. (80). As we have anticipated above, a crucial feature of these spectral densities is the existence of a sum rule. In this case it is given by At this point, to show the convergence of the family of spectral densities to a resonant behaviour, it is convenient to extract a discrete sequence of functions, parameterized by an integer n and let n go to infinity.
in a distributional sense. We have shown in Fig. (6), on the left, the sequel of spectral densities which characterize the flow as we turn the mass parameter to zero. The area under each curve is fixed by the sum rule and is a characteristic of the entire flow. Clearly, the ρ (n) are normalized distributions for each given value of m. They describe, for each invariant mass value s, the absolute weight of the intermediate state -of that specific invariant mass -to a given anomaly form factor. Notice that the function χ(s, m 2 ) is a universal function, since it provides a full description of the flow for the anomaly form factors of all the components of the multiplet.
One can see from the same figure how the density gets more and more peaked towards the lower end of the region of the interval 4m 2 n ≤ s < ∞ as m 2 n tends to zero. In physical terms this means that the branch cut is replaced by a single massless anomaly pole. In Fig. (6), on the right, we have included a 3D plot of ρ χ (s, m 2 ) in the (s, m 2 ) plane, giving a visual perspective on the entire flow.

The analytic structure of Φ 2
Here we discuss the spectral representation of the second of the form factors appearing in the same Γ T and Γ S correlators, which is proportional to the renormalized function Φ 2 which needs a subtraction for its integrability, due to the UV singularities of Φ 2 . Clearly, in this case, Φ 2 does not admit a dispersive representation, due to its logarithmic divergence at large k 2 , and, as we are going to show, it is characterized just by an ordinary cut for k 2 > 4m 2 , as in the previous case. We are now going to briefly illustrate this point. As for C 0 (k 2 , m 2 ) also in this case we give the three branches of B 0 (k 2 , m 2 ) in the k 2 < 0, 0 < k 2 < 4m 2 and k 2 > 4m 2 regions The discontinuity of the two-point scalar integral B 0 (k 2 , m 2 ) is then easily computed and it is given by From the previous equation and from Eq. (77) we extract the discontinuity of Φ 2 in the form This shows that both Φ 1 /k 2 and Φ 2 are characterized by a single 2-particle cut for a nonzero mass deformation. It is important to observe that the spectral density of Φ 2 tends to a uniform distribution in the massless limit. It is obvious, from this analysis, that the spectral density ρ Φ 2 of Φ 2 , which characterizes all the non anomalous form factors of the correlators that we have investigated, does not satisfy an unsubtracted dispersion relation. There is however a sort of duality between the spectral densities of the two form factors, since while ρ χ becomes more and more localized at k 2 = 0 as m → 0, the opposite is true for the spectral density of the non anomallous form factor ρ Φ 2 , as clear from Fig. 8. In this case, as m goes to zero, the flow singles out -in the form factor which is relevant for the anomalya single massless state, while all the continuum region carries uniform weight in ρ Φ 2 . π (s, m 2 ) versus s. As m 2 decreases they turn to a unit step function θ(s).

Constraining the flow: scaling behaviour and sum rules
The large momentum behaviour of the anomaly form factors, beside the sum rule, can be studied directly also from the explicit expressions of these. For this goal, we are going to investigate the behaviour of both Φ 1 and Φ 2 in the two opposite limits k 2 → 0 and k 2 → −∞, which cover the light-cone as well as the deep euclidean regions of the correlators. For k 2 approaching zero we have while for a large and negative k 2 we find Because these form factors are characterized by only two mass scales, namely m 2 and k 2 , performing the k 2 → −∞ limit is equivalent to taking the massless limit. Indeed it is easy to show that the leading order terms in Eq. (101) reproduce the massless chiral contributions described in the previous sections. Notice also the presence of an infrared singularity, for m 2 → 0, in the Φ 2 form factor (see Eq. (101)). This is due to the B 0 (0, 0) scalar integral appearing in Eq. (46). The argument can be formally stated as follows. The anomaly form factor χ = Φ 1 /k 2 satisfies the relation under rescaling with a constant λ being a homogeneous function. A similar property of homogeneity holds for the spectral density itself which under a partial rescaling, involving only the mass parameter m, with m 2 → m 2 /λ and λ large (which is the same as m → 0) has the resonant behaviour At this point, using Eq. (103) a large rescaling of the invariant mass s gives showing that the asymptotic behaviour of ρ χ under a rescaling of s with λ identifies its support on the s = 0 region. Notice that Eq. (105) should be interpreted as a light-cone dominance (s → 0) of the asymptotic limit of the correlator as λ goes to infinity. On the other hand, the vanishing of the massive form factors in the k 2 → 0 region, and the consequent disappearance of the 1/k 2 pole in the anomalous correlators, may be understood as a consequence of decoupling of the massive states. Scaling relations (a), combined with the sum rule (b) and the resonant behaviour of the densities for m going to zero (c), provide some important constraints on the structure of the flow, although they are not exclusive to anomalous form factors. We recall that as a consequence of the scaling relation, one has the constraint Similar conditions are satisfied by the related spectral density (ρ χ ) The combination of scaling behaviour and of the sum rule, together with the vanishing of ρ χ (s, m 2 ) at the threshold (i.e. at s = 4m 2 ), induces further constraints on its functional form, for instance In the previous equation, and in the following ones, f is a nonzero constant which normalizes the sum rule of the spectral density. For ρ χ introduced in the previous section f = 1.
Eq. (108) can be generalized to give an infinite set of ordinary and superconvergent sum rules Additional constraints come from the scaling relation expanded to second order, Using the information that the density has only a branch cut for nonzero m, integrating over the cut Eq.
(110) we get At this point, the sign of the dispersive integrals above can be determined by exploiting the derivative form of the sum rule which is satisfied because of the convergence condition of the integral of ρ χ . Differentiated respect to m 2 the sum rule above gives which relates the integral of the mixed derivatives to the spectral density at the threshold. If the spectral density is properly normalized with a positive constant f in the sum rule, then it will be always positive along the entire cut and, in particular, at threshold t. Notice that as m goes to zero, the density is saturated by the pole behaviour, and it is then clear that it implies the local positivity relation being the integral dominated just by the region around the threshold s ∼ 4m 2 . Clearly this implies that having used Eq. (111). Also in this case, in the m → 0 limit, the inequality becomes a local condition which has to be satisfied by the flow. Notice that in the presence of multiple thresholds at specific masses m n the density jumps at every threshold by a positive or a negative amount. The jump is proportional to the contribution of the new threshold to the β function of the theory. This point can be easily illustrated by reintroducing the prefactor contribution of each massless state in front of the corresponding density. For this purpose we define the contributions of each field to the β function of a theory at 1-loop, which for a Dirac fermion and a complex scalar in the representation R f and R s respectively, and for a spin 1 in the adjoint are with with T (R f ), T (A), T (R s ) being the Dynkin indices of the respective representations. Real scalars and Weyl fermions contribute with an additional factor of 1/2 respect to complex scalars and Dirac fermions. We recall that in a SU (N ) N = 1 theory, the vector multiplet contributes with −11/3 T (A) and 2/3 T (A) for the gauge field and the gaugino respectively, while the chiral supermultiplet gives 2/3 T (R) and 1/3 T (R) for the Weyl fermion and the complex scalar. We use the notation to refer to the total spectral density of a certain theory, with intermediate thresholds at increasing mass values {m 2 n } ≡ (m 2 1 , m 2 2 , . . . , m 2 I ) with (m 1 < m 2 < . . . < m I ), where I counts the total number of degrees of freedom. The corresponding anomaly form factor will be given by (120) Notice that if Q 2 4m 2 n , for a certain mass threshold n, then we can set Q 2 = 4m 2 n λ, with 1/λ = 4m 2 n /Q 2 1. Due to scaling, the n th threshold will then contribute to the total form factor with the amount which in the 1/λ 1 limit will give Eq. (122) reduces to the anomaly pole contribution times the contribution of the state (n) to the expression of the total β function. As Q 2 grows larger than any intermediate scale, the total spectral density ρ in the dispersive integral is asymptotically given by the expression where we have used Eq. (104). Notice that ρ(s, {m 2 n }) satisfies a total sum rule to which contribute all the intermediate thresholds for In supersymmetric theories this function is the only one which developes a resonant behaviour at the conformal point and satisfies a sum rule, as we have pointed out. The sum of the densities stripped of the gauge factors, integrated over the thresholds simply counts the number of degrees of freedom (I).
Notice that the analysis of this section related to Eqs. (106-115) remains valid also for any form factor which is characterized by a finite (non superconvergent) sum rule. The asymptotic analysis discussed in Eqs. (120-125), can be also easily extended to cases unrelated to the anomaly, with coefficients c (n) replaced by some new coefficients, not related to the β function.

Comparing supersymmetric and non supersymmetric cases: sum rules and extra poles in the Standard Model
In this section and in the following one, we compare the structure of the spectral densities between supersymmetric and non supersymmetric theories in the presence of mass terms, looking for the additional sum rules not directly related to the anomalies, which may be present in the T V V and AV V correlators. We anticipate that these are found in the T V V in the non suspersymmetric case in all the gauge invariant sectors of the Standard Model. We start our analysis with the conformal anomaly action of QCD, described by the EMT-gluon-gluon vertex and then move to the EMT-γγ vertex in the complete electroweak theory. Obviously, the spectral densitites develope anomaly poles in the limit in which all the second scales of the vertices turn to zero. By this we mean fermion masses, the W mass and external virtualities. Moreover, we identify the explicit form of the sum rules satisfied in perturbation theory.

The extra pole of QCD
For definiteness we focus our attention on a specific gauge theory, QCD. We write the whole amplitude Γ µναβ (p, q) of the T V V diagram in QCD as having separated the quark (Γ q ) and the gluons/ghosts (Γ g ) contributions. We have omitted the colour indices for simplicity, being the correlator diagonal in colour space. As described before in Section 3 for the massless case, also in the massive case the amplitude Γ is expressed in terms of 3 tensor structures. In the M S scheme these are given by [20] Γ µναβ q/g (p, q) = For on-shell and transverse gluons, only 3 invariant amplitudes contribute, which for the quark loop case are given by where the on-shell scalar integrals D(k 2 , m 2 ), C 0 (k 2 , m 2 ) and B M S 0 (k 2 , m 2 ) are given in Appendix A. Here we concentrate on the two form factors which are unaffected by renormalization, namely Φ 1,2q . Both admit convergent dispersive integrals of the form in terms of spectral densities ρ 1,2q (s, m 2 ). From the explicit expressions of these two form factors, the corresponding spectral densities are obtained using the relations where A(s) is defined in Eq.(85) and we have used the general relation with δ (n) (x) the n-th derivative of the delta function. The contribution proportional to δ (s) in Eq.(132) can be rewritten in the form giving for the spectral densities ρ 1q (s, m 2 ) = g 2 12π Both functions are characterized by a two particle cut starting at 4m 2 , with m the quark mass. Notice also that in this case there is a cancellation of the localized contributions related to the δ(s), showing that for nonzero mass there are no pole terms in the dispersive integral. The crucial difference, respect to the supersymmetric case discussed above, is that now we have two independent sum rules one for each form factor, as it can be verified by a direct integration. We can normalize both densities as in order to describe the two respective flows, which are homogeneuos, since both densities carry the same physical dimension and both converge to a δ(s) as the quark mass m is sent to zero H Figure 9: Amplitude with the graviton -Higgs mixing vertex generated by the term of improvement. The blob represents the SM Higgs -VV' vertex at one-loop.
Indeed at m = 0, Φ 1,2q are just given by pole terms, while Φ 3q is logarithmic in momentum It is then clear, from this comparative analysis, that the supersymmetric and the non supersymmetric anomaly correlators can be easily differentiated in regards to their spectral behaviour. In the non supersymmetric case the spectral analysis of the T V V correlator shows the appearance of two flows, one of them anomalous, the other not. A similar pattern is found in the gluon sector, which obviously is not affected by the mass term. In this case the on-shell and transverse condition on the external gluons brings to three very simple form factors whose expressions are The M S renormalized scalar integrals can be found in Appendix A. Also in this case, it is clear that the simple poles in Φ 1 g and Φ 2 g , the two form factors which are not affected by the renormalization, are accounted for by two spectral densities which are proportional to δ(s). The anomaly pole in Φ 1 g is accompanied by a second pole in the non anomalous form factor Φ 2 g . Notice that Φ 3g is affected by renormalization, and as such it is not considered relevant in the spectral analysis.

T V V and the two spectral flows of the electroweak theory
The point illustrated above can be extended to the entire electroweak theory by looking at some typical diagrams which manifest a trace anomaly. The simplest case is the T V V in the full electroweak theory, where V , in this case, denotes on-shell photons. At one loop level it is given by the vertex Γ µναβ and expanded onto two terms Γ µναβ (p, q) = Σ µναβ (p, q) + ∆ µναβ (p, q) , where Σ µναβ (p, q) is a full irreducible contribution derived from the set of diagrams given in the appendices and depicted in Figs. (11), corresponding to topologies of triangles, bubbles and tadpoles. In this case Σ µναβ (p, q) is given by the expression [35,34,11] Σ µναβ (p, q) = Σ µναβ corresponding to the exchange of fermions (Σ F ), gauge bosons (Σ B ) and to a term of improvement (Σ I ). The latter is generated by an EMT of the form and is responsible for a bilinear mixing between the EMT and the Higgs field. The term ∆ µναβ (p, q) in Eq.(143) comes from the insertion of the EMT of improvement given above with the Standard Model Hγγ vertex. The relevant diagram is reported in Fig. (9). The inclusion of this term is necessary, from a careful analysis of the Ward identities, as shown in [34]. They full irreducible contributions are expanded as with s = k 2 = (p + q) 2 , φ µναβ i (p, q) given in Eq. (5) and while the ∆ term reads as ∆ µναβ (p, q) = ∆ µναβ I (p, q) This is built by combining the tree level vertex for EMT/Higgs mixing, coming from the improved EMT, and the Standard Model Hγγ correlator at one-loop. For convenience, we have included in Appendix B the explicit expression of these form factors, from which we extract the corresponding spectral densities and sum rules. The spectral densities of the fermion contributions, related to Σ F have structure similar to those computed above in Eq. (135), as one can easily deduce from the explicit expression of the form factor given in Eq.(171), with ρ Φ 1F ∼ ρ 1q (s) and ρ Φ 2F ∼ ρ 2q (s). Therefore we have two sum rules and two spectral flows also in this case, following the pattern discussed before for the spectral densities in Eq. (135). A similar analysis on the two form factors Φ B in the gauge boson sector gives while ρ φ 2B has the same functional form of ρ φ 2F , modulo an overall factor, with m, the fermion mass, replaced by the W mass M W . Notice that both ρ φ 1B and ρ φ 2B , as well as ρ φ 1F and ρ φ 2F are deprived of resonant contributions, being the diagrams massive.
Coming to the form factors in Σ I , whose explicit expressions are given in Eq.(175), one immediately realizes that the spectral density of Φ 1I shares the same functional form of ρ χ , extracted from Eq. (90), and there is clearly a sum rule associated to it. Also in this case, this result is accompanied by the 1/k 2 behaviour of the corresponding form factor, due to the anomaly. Finally, for the case of ψ 1I , one can also show that the spectral density finds support only above the two particle cuts. The cuts are linked to 2m and 2M W . In this case there is no sum rule and the contribution is not affected by an anomaly pole, as expected, being the virtual loop connected with the Hγγ vertex (see Fig. 9). The explicit expression of this density is given in Appendix B.

The non-transverse AV V correlator
Before closing the analysis on the spectral densitites of non supersymmetric theories, we pause for few comments on the structure of the AV V diagram, which, as we are going to show, is affected by a single flow even if we do not impose the transversality condition on the two photons. We consider once more the anomaly vertex as parameterized in Eq. (10), and consider the second form factor A 4+6 ≡ A 4 + A 6 , which contributes to the anomaly loop for non transverse (but on-shell) photons. The expression of A 6 , the anomalous form factor, has been given in Eq. (11), while A 4 is given by and A 4+6 takes the form Its discontinuity is given by Notice that in this case there is no sum rule satisfied by this spectral density, being non-integrable along the cut. Coming to the spectral density for the anomaly coefficient A 6 , this is proportional to the density of χ(s, m 2 ) given in Eq. (90) and shares the same behaviour found for ρ χ (s, m 2 ), as expected. This analysis shows that in the AV V case one encounters a single sum rule and a single massive flow which degenerates into a δ(s) behaviour, as in the supersymmetric case. This condition remains valid also for non-transverse vector currents. It is then clear that the crucial difference between the non supersymmetric and the supersymmetric case is carried by the T V V diagram, due to the extra sum rule discussed above.

Cancellations in the supersymmetric case
In order to further clarify how the cancellation of the extra poles occurs for the supersymmetric T V V , we consider the non-anomalous form factor f 2 in a general theory (given in Eqs. (7,8,9)), with N f Weyl fermions, N s complex scalars and N A gauge fields. We work, for simplicity, in the massless limit. In this case the non anomalous form factor f 2 , which is affected by pole terms, after combining scalar, fermions and gauge contributions can be written in the form where the fermions give a negative contribution with respect to scalar and gauge fields. If we turn to a N = 1 Yang-Mills gauge theory, which is the theory that we are addressing, we need to consider in the anomaly diagrams the virtual exchanges both of a chiral and of a vector supermultiplet. In the first case the multiplet is built out of one Weyl fermion and one complex scalar, therefore in Eq.(155) we have Here we present the quantum effective action obtained from the three-point correlation functions discussed previously. We consider the massless case for the chiral supermultiplet and on-shell external gauge bosons and gauginos. The anomalous part is given by the three terms S anom = S axion + S dilatino + S dilaton (156) which are A a α (p) T µν (k) We show in Figs In the case of the SV F correlator, the intermediate state is a collinear scalar/fermion pair, interpreted as a dilatino. In the T V V case, the collinear exchange is a linear combination of a fermion/antifermion and scalar/scalar pairs. The non-anomalous contribution is associated with the extra term S 0 which is given by whereΦ 2 (z − x) andṼ (z − x) are the Fourier transforms of Φ 2 (k 2 , 0) and V (k 2 ) respectively. Their contributions in position space correspond to nonlocal logarithmic terms. The relation between anomaly poles, spectral density flows and sum rules appear to be a significant feature of supersymmetric theories affected by anomalies. It is then clear that supersymmetric anomaly-free theories should be free of such contributions in the anomaly effective action. In this respect, it natural to turn to the N = 4 theory, which is free of anomalies, in order to verify and validate this reasoning. Indeed the β function of the gauge coupling constant in this theory has been shown to vanish up to three loops [36,37,38], and there are several arguments about its vanishing to all perturbative orders. As a consequence, the anomaly coefficient in the trace of the energy-momentum tensor, being proportional to the β function, must vanish identically and the same occurs for the other anomalous component, related to the R and to the S currents in the Ferrara-Zumino supermultiplet. We recall that in the N = 4 theory the spectrum contains a gauge field A µ , four complex fermions λ i (i = 1, 2, 3, 4) and six real scalars φ ij = −φ ji (i, j = 1, 2, 3, 4). All fields are in the adjoint representation of the gauge group.
From the point of view of the N = 1 SYM, this theory can be interpreted as describing a vector and three massless chiral supermultiplets, all in the adjoint representation. Therefore the T V V correlator in N = 4 can be easily computed from the general expressions in Eqs (51) and (59) which give One can immediately observe from the expression above the vanishing of the anomalous form factor proportional to the tracefull tensor structure t µναβ 1S . The partial contributions to the same form factor, which can be computed using Eqs. (51) and (59) for the various components, are all affected by pole terms, but they add up to give a form factor whose residue at the pole is proportional to the β function of the N = 4 theory. It is then clear that the vanishing of the conformal anomaly, via a vanishing β function, is equivalent to the cancellation of the anomaly pole for the entire multiplet. Notice also that the only surviving contribution in Eq. (161), proportional to the traceless tensor structure t µναβ 2S , is finite. This is due to the various cancellations between the UV singular terms from V (k 2 ) and Φ 2 (k 2 , 0) which give a finite correlator without the necessity of any regularization.
We recall that the cancellation of infinities and the renormalization procedure, as we have already seen in the N = 1 case, involves only the form factor of tensor t µναβ 2S , which gets renormalized with a counterterm proportional to that of the two-point function AA , and hence to the gauge coupling. For this reason the finiteness of the second form factor and then of the entire T V V in N = 4 is directly connected to the vanishing of the anomalous term, because its non-renormalization naturally requires that the β function has to vanish.

Conclusions and Perspectives
Our analysis and results show the consistency of a conjecture about the perturbative structure of the anomalies in supersymmetric theories, formulated by us in a previous work [35]. We have presented additional evidence that anomaly poles are the signature of the anomalies in the perturbative anomaly action of these theories, extending former studies [18,20,30]. For global anomalies it is expected that the massless states identified by the pole contributions can be promoted to new composite degrees of freedom by some non perturbative dynamics, as for the chiral anomaly and the pion. In the QCD case [20], for instance, the breaking of classical scale invariance -in this perturbative pictureshould manifest in the emergence of a dilaton, if gluons were asymptotic states. We have noticed, though, that the T V V vertex in QCD, as we have shown, has one extra pole and and one extra flow related to a non anomalous form factor which is both IR and UV safe, which should be interpreted as an extra interpolating state. A similar perturbative pattern emerges in the Standard Model [35,11], as clear from the analysis of the conformal anomaly in the electroweak sector [35]. However, by turning to supersymmetry, we have shown that here the connection between anomalies, poles and sum rules for anomaly vertices are one to one. The 1/k 2 feature of the anomaly form factors has been investigated in connection with the scaling properties of their spectral densities and with the finite (non zero) sum rule which it satisfies, in agreement with a previous analysis by Giannotti and Mottola [18]. We have seen that the anomalous behaviour emerges from the s ∼ 0 region of the spectral density of a given form factor and covers, therefore, the entire light-cone surface. The resonant behaviour at s = 0 is present, as we have shown, also at very high momentum. In supersymmetry we have focused our attention on the perturbative correlators which are responsible for the generation of the superconformal anomaly, and shown that the Ferrara-Zumino multiplet, as well as the Konishi currents, allow to identify some composite states in the effective action, interpolating between the currents and the on-shell final states. They correspond to a dilaton, a dilatino and an axion, plus a number of pseudoscalar states, one for each fermion flavour. The description of these effective degrees of freedom not as anomaly poles but as asymptotic states of the S-matrix remains, obviously, an open issue, which goes beyond the simple perturbative picture discussed here, as demonstrated by the complex pattern of chiral dynamics in QCD. Following this pattern, it is then natural to ask if global anomalies are always connected to the generation of effective degrees of freedom, and hence to compositeness, as indicated by the poles of the effective action. These results are valid for all the anomalies characterized by a single flow, in particular for all the chiral currents affected by global anomalies. From this perspective, also the Peccei-Quinn current should induce as an interpolating state a composite axion rather an elementary one, being our argument generic to anomalous global currents. There are also some drastic implications of our analysis, at least in the supersymmetric case, whenever the symmetries of the hypercurrent are gauged, which concern the way anomalies should cancel when a theory affected by a superconformal anomaly is coupled to gravity. We have seen that the anomaly is entirely given by the β/k 2 term, in terms of the β function of the theory, and it appears obvious that the coupling to gravity has necessarily to provide an extra massless sector in order to remove such contribution. This could only take place if the gravitational sector can contribute by an equal and opposite amount to the pole residue, at the cost, otherwise, of being left with an inconsistency in the total theory. It is important to remark, as we have already pointed out, that the cancellation of the pole may not be an identical cancellation of the anomaly vertex. We have in fact explicitly shown that in a N = 4 theory, for instance, by setting the β function to zero, one indeed is canceling the pole contributions, and hence the anomaly, but not the entire anomaly vertex, as clear from Eq. (161). This situation is new compared to the case of anomalous abelian symetries, where anomaly cancellation by charge assignments on the massless matter spectrum forces the entire AV V / AAA vertices to vanish.

A Appendix. Scalar integrals
One-, two-and three-point functions are denoted respectively as A 0 , B 0 and C 0 with , . (162) Moreover, for equal internal masses and for p 2 = q 2 = 0 we have used the more compact notation In the spacelike region (k 2 < 0), using two regulators for the ultraviolet and infrared singularities (n = 4 − U V = 4 + IR ), where n denotes the spacetime dimensions, the relevant 2-point functions appearing in the computation are with τ (k 2 , m 2 ) = 1 − 4m 2 /k 2 , while for k 2 null we obtain In the QCD computations we have also used the following finite two-point scalar integrals and we have renormalized all the divergent B 0 functions in the M S scheme in which the 1/ε U V divergences have been subtracted. The massless scalar 3-point function, for k 2 < 0, is given by while the massive C 0 (k 2 , m 2 ) is given in Eq. (78).

D Appendix. Polology
A pole in a correlation function may correspond either to the exchange of a fundamental particle in the defining Lagrangian, as shown in Fig. (12), or to the exchange of a composite particle, as in Fig. (13). We summarize the proof of the relation between the existence of poles in the S-matrix and the nature of the intermediate exchange for the specific case of the 3-point functions that we investigate. We consider a generic correlator G as a function of the momenta of the external lines. In coordinate space it is given by with the operatros A denoting either fields appearing in the Lagrangian or even composite local operators. We specialize to a simpler case with just three operators, say O(z), A 1 (x 1 ) and A 2 (x 2 ). This is the situation encountered in our studies on chiral and conformal anomalies. Then we move to momentum space and consider the correlator G(k, p 1 , p 2 ) = d 4 zd 4 x 1 d 4 x 2 e −ikz−ip 1 x 1 −ip 2 x 2 0|T {O(z)A 1 (x 1 )A 2 (x 2 )} |0 as a function of the virtuality of O, namely k 2 = (−p 1 − p 2 ) 2 . Notice that the virtualities of the external momenta p 2 1 and p 2 2 are not fixed by any on-shellness condition and can be arbitrary. We isolate the operator O from the T product and retain only the term in which O appears to the far left where the ellipsis stand for the other time ordering products which we have ignored. They do not contribute with any pole structure to the correlator. Now we insert a complete set of intermediate states between the operator O and the other ones, isolating only single particle states with a specific mass m. We discard the other single particle states with different masses (they will contribute with poles but at other kinematical positions) and multi particle states (which appear as branch cuts). We obtain where | p, σ is a single particle state with mass m (p 2 = m 2 ) and with quantum numbers collectively identified by σ. We extract the z and x 1 dependences from the matrix elements appearing in the previous equation, and introduce the new integration variable y = x 1 −x 2 in place of x 2 . Finally we insert the integral representation of the step function θ(t) given by where is an infinitesimal and positive constant. We have G(k, p 1 , p 2 ) = i 2π σ d 4 z d 4 x 1 d 4 y d 3 p dω ω + i e −ikz−i(p 1 +p 2 )x 1 −ip 2 y × e −iω(z 0 −x 0 1 −max{0,y 0 }) e ipz−ipx 1 0|O(0)| p, σ p, σ|T {A 1 (0)A 2 (y)} |0 + . . . , where the integration over z and x 1 is straightforward and gives only delta functions G(k, p 1 , p 2 ) = i 2π σ d 4 y d 3 p dω ω + i e −ip 2 y+iω max{0,y 0 } 0|O(0)| p, σ p, σ|T {A 1 (0)A 2 (y)} |0 ×(2π) 8 δ (3) ( k − p) δ(k 0 − p 2 + m 2 + ω)δ (3) ( p 1 + p 2 + p) δ(p 0 1 + p 0 2 + p 2 + m 2 − ω) + . . . . (188) The integrations over the momenta p and ω are now trivial due to the delta functions and lead to G(k, p 1 , p 2 ) = (2π) 4 δ (4) (k + p 1 + p 2 )i (2π) 3 k 2 + m 2 − k 0 + i × σ d 4 y e i √ k 2 +m 2 −k 0 max{0,y 0 } e −ip 2 y 0|O(0)| k, σ k, σ|T {A 1 (0)A 2 (y)} |0 .
The appearance of the pole in the limit k 0 → k 2 + m 2 in the correlation function is now explicitly manifest and originates from the massless pole in ω, which comes, in turn, from the integral parameterization of the step function. In order to make the pole structure more clear we notice that near the pole while the exponential function under integration goes to unity. This allows us to define the matrix elements (2π) 4 δ (4) (k + p 1 + p 2 ) M (k,σ)|0 (k, p 1 , With these definitions and simplifications the pole behaviour of the correlator is now explicit and reads as G(k, p 1 , p 2 )

E Appendix. Feynman rules
We report the Feynman rules used for the massless computation. All momenta are incoming.