Exact Correlators from Conformal Ward Identities in Momentum Space and Perturbative Realizations
Claudio Corianò, Matteo Maria Maglio, Alessandro Tatullo, Dimosthenis Theofilopoulos
EExact Correlators from Conformal Ward Identities inMomentum Space and Perturbative Realizations
Open issues on conformal anomaly actions
Claudio Corianò ∗ INFN Lecce, Dipartimento di Matematica e Fisica "Ennio De Giorgi",Università del Salento and INFN-Lecce,Via Arnesano, 73100 Lecce, ItalyE-mail: [email protected]
Matteo Maria Maglio, Alessandro Tatullo, Dimosthenis Theofilopoulos
INFN Lecce, Dipartimento di Matematica e Fisica "Ennio De Giorgi",Università del Salento and INFN-Lecce,Via Arnesano, 73100 Lecce, ItalyE-mail: [email protected],[email protected],[email protected]
The general solution of the conformal Ward identities (CWI’s) in momentum space, and theirmatching to perturbation theory, allows to uncover some specific characteristics of the breakingof conformal symmetry, induced by the anomaly. It allows to compare perturbative features of the1-particle irreducible (1PI, nonlocal) anomaly action with the prediction of a similar (but exact)nonlocal action identified by the CWI’s. The two predictions can be exactly matched at the levelof 3-point functions. The analysis of the
T JJ and
T T T shows that both approaches - based eitheron 1PI or on the exact solutions of the CWI’s - predict massless (dynamical) scalar exchangesin 3-point functions as the signature of the conformal anomaly. In a local formulation such 1PIactions exhibit a ghost in the spectrum which may induce ghost condensation. We also discussalternative approaches, which take to Wess-Zumino forms of the action with an asymptotic dila-ton, which should be considered phenomenological alternatives to the exact nonlocal action. Ifderived by a Weyl gauging, they also include a ghost in the spectrum. The two formulations, non-local and of WZ type, can be unified under the assumption that they describe the same anomalyphenomenon at two separate (UV/IR) ends of the renormalization group flow, possibly separatedby a vacuum rearrangement at an intermediate scale. A similar analysis is presented for an N = Proceedings of the Corfu Summer Institute 2018 "School and Workshops on Elementary Particle Physicsand Gravity"1-27 September 2018Corfu, Greece ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ a r X i v : . [ h e p - ph ] M a y xact Correlators from Conformal Ward Identities Claudio Corianò
1. Introduction
The breaking of conformal symmetry by the conformal anomaly is a fascinating topic whichhas played a key role in d = v , but there is no specific theoretical reason, except phenomenological, why theelectroweak scale lays around 246 GeV. This is not the only open issue in the SM, as there are oth-ers which remain unsolved. For instance, there is no simple explanation of the fact that the numberof fermion families is three, unless one considers very special extensions. A rare example is the331 model [1], which is quite simple and unique, where the embedding of the third generation andthe constraints from anomaly cancelation select the number of families to be exactly three, at thecost of breaking universality. Given such shortcomings and the puzzle raised by the gauge hierarchy problem, the inclusionof conformal symmetry may provide an alternative approach for answering at least some of thesepuzzles. In a conformal extension motivated at a lower (TeV) scale, one can still envision the SMwith its current field content, preserving the fundamental nature of the Higgs field, but with anelectroweak scale generated by the vev of a second field ( Σ ) , whose role is to enforce a larger(conformal) symmetry in the classical Lagrangian.In fact, it is possible - and quite simple - at tree level at least, to reconcile conformal symmetry andthe Higgs mechanism by the introduction of an extra scalar field Σ ( x ) in such a way to restore thissymmetry. In this case the role of the Higgs remains the usual one, but the new scalar can mix withthe Higgs, giving an ordinary mass eigenstate which would correspond to the SM Higgs, and to adilaton. The real problem, in this scenario, is how to break the new symmetry in a simple way. Werecall at this point that the dilaton ( τ ( x )) is related to Σ via a nonlinear realization with Σ ∼ Λ exp ( τ ( x ) / Λ ) , (1.1)where Λ denotes the conformal symmetry breaking scale. While this is one possibility, in whichthe dilaton is generated by enlarging the degrees of freedom of the SM, it is not the only one. Adynamical solution is also possible, as we are going to elaborate, where the dilaton emerges fromthe conformal dynamics in a specific way, as an effective degree of freedom. The arguments thatwe bring forward towards a resolution of some conflicting issues related to this topic are not nec-essarily advocated around the TeV scale, but may also reach very large scales, being generic and2 xact Correlators from Conformal Ward Identities Claudio Corianò probably more of cosmological relevance than anything else.In absentia of new physics at the LHC, and with the success of the SM, a way to explain the nat-urality of the Higgs mass, according to ’t Hooft’ s principle, is to invoke a larger symmetry whichprotects the small masses present in the SM from the large quadratic divergences of the scalar sec-tor [2, 3].In order to touch ground with the ordinary S-matrix formalism, it is necessary to promote the anal-ysis of such conformal extensions to momentum space, where several non-perturbative tools, suchas the conformal Ward identities, the operator product expansion and the conformal algebra at op-eratorial level, may allow to progress towards the analysis of multi-point correlation functions in asystematic way. One of the advantages of a momentum space analysis is the possibility of identi-fying new effective degrees of freedom in such theories, and this brings us directly to investigatethe form of the anomaly action, which is usually addressed within perturbation theory.Given the obvious limitations of the perturbative approach, it is necessary to compare perturbativeand non-perturbative methods in order to shed light on the issue of the nonlocality of the confor-mal anomaly action, which plays a key role in this context. This provides the main motivation forturning to a non-perturbative discussion of 3-point functions in momentum space. In particular,the statement that anomaly poles, which are the perturbative signature of the anomaly, do not cor-respond to physical states, should be taken with extreme care and correctly interpreted in a widercontext, being a perturbative analysis probably insufficient to come to conclusions, given the com-plex nature of the phenomenon.We will argue that in the presence of conformal anomaly poles (and the same occurs for chiralanomaly poles) a certain theory rearranges its vacuum structure in such a way that the dynamicsof such nonlocal interactions will generate an ordinary asymptotic state. Such a state would be aquasi Nambu-Goldstone mode in a local effective action and would correspond to a dilaton.We believe that in this way we can reconcile the two main formulations of the conformal anomalyaction - the nonlocal and the local one - the latter incorporating an asymptotic dilaton, which oth-erwise appear to be unrelated. However, it is naturally expected that an intermediate potential willprovide a small mass for such Nambu-Goldstone mode. A similar hypothesis has been formulatedin the case of a Stückelberg field, which carries analogous properties, and that may be renderedmassive by a mechanism of misalignment as for an ordinary Peccei-Quinn axion. Our analysis isdriven by this analogy.The goal of this review is to clarify some of the issues raised when comparing perturbative andnon-perturbative approaches in theories affected by the conformal anomaly. For 3-point functions,the nonlocal structure of such action and its expression in terms of anomaly poles has been workedout directly from the solution of the conformal Ward identities (CWI’s) in momentum space. Forthis reason, the comparison between the two descriptions, whenever possible, plays an importantrole. However, only in momentum space such combined analysis connect CFT’s with their par-ticle interpretations and help in clarifying these aspects. We are going to briefly summarise ourarguments.
2. The conformal anomaly action
In a conformal theory in even spacetime dimensions, the conformal symmetry can be broken3 xact Correlators from Conformal Ward Identities
Claudio Corianò by the conformal anomaly. Anomalies are related to the field content of a given theory and thoughapparently very different in their conformal and chiral versions, they are unified by the same phe-nomenon: the emergence of an anomaly pole, i.e. of a massless exchange in those diagrams whichare held responsible for the origin of the anomaly, which are part of the 1PI (1-particle irreducible)effective action. They can be identified in a given anomaly vertex if we keep all the tensor struc-tures of the same vertex uncontracted. By taking a trace or a divergence of an anomaly vertex,the pole is washed out, while only the residue at the pole remains in the interaction. Given theperturbative nature of the 1PI action, and its simplicity, this behaviour may well be considered anartifact of perturbation theory, deprived of any specific meaning.One of the goals of our analysis will be to show that such effective interactions are present also ifwe move away from perturbation theory and discuss the same anomalous vertices using completelydifferent methods.In a CFT the CWI’s fix the 3-point functions almost completely, modulo few constants, and it is inthis case that the matching between the perturbative and non-perturbative approaches allows us tocome to conclusions in regards to the emergence of such massless interactions.
An anomaly action modifies the classical action by the anomaly contribution. For instance,in the presence of an axial vector current, generated by an external gauge boson B , the anomalycontribution triggers the transition of such off-shell current into two photons, or into two gluons bya fermion loop, in QED and QCD respectively. Since an intermediate one-loop AVV interaction isinvolved, it is part of a simple 1PI action, and a direct computation shows that the interaction thatensues is mediated by the exchange of a massless pole [4–7] which takes to a nonlocal action, aswe will discuss in section 6.However, on the other hand, it is sufficient to couple linearly an axion b ( x ) to the anomaly in orderto obtain an effective action which now includes a scale ( M ) and a dimension-5 operator ( b ( x ) / M ) F ˜ F (2.1)to account for the anomaly, which takes to a local action. Operatorial terms of this type can also beintroduced as counterterms in order to restore a gauge symmetry if an anomalous gauge boson isalso part of the spectrum and not an external source for an axial-vector current.As in the conformal case, where a similar coupling is present for the dilaton, in the local formu-lation the axion field b ( x ) shifts under the local gauge symmetry as a typical Nambu-Goldstonemode. In the case of an anomalous axial-vector coupling, this construction takes to StückelbergLagrangians [8, 9], if the axial-vector gauge field is part of the dynamics. The Stückelberg field,introduced to cancel a gauge anomaly, turns into a physical gauge invariant component after mixingwith the CP-odd phases of the Higgs sector. This is obtained by a mechanism of vacuum misalign-ment.There are some issues which need to be addressed in the analysis of a perturbative anomaly actionwhich is related to the choice of the regularization scheme, but it is clear that dimensional regular-ization (DR) plays a special role in this context. Other (mass-dependent) regularization schemes4 xact Correlators from Conformal Ward Identities Claudio Corianò do not preserve the conformal symmetry of the theory and as such are not optimal.The appearance of an anomaly pole in perturbation theory in mass-dependent corrections for anyvalue of a correcting mass parameter m i is a feature which is typical of DR and not of other schemes.Indeed, DR allows to separate the anomaly contribution from the explicit mass-dependent correc-tions of a diagram in a very natural way for any value of the external momentum ( p ) running inthe loop. This holds even for p < m , where m is the mass of the fermion in the loop, giving anon-vanishing contribution to the β function of a given theory, and hence to the anomaly for anyvalue of the external momentum.Wess-Zumino types of action, the other variant of the conformal anomaly actions, as clear fromthe discussion above and from Eq. (2.1), enlarge the number of degrees of freedom by introducingan axion or a dilaton in order to generate the same anomalies of the original theory. Obviously,a dilaton is not part of the original Lagrangian, though it is part of the anomaly action, and onehas to view such actions as effective actions which need to be related to UV descriptions of thesame phenomenon using renormalization group/effective field theory arguments. Connecting thetwo descriptions requires special care and while this is done by preserving the global symmetriesof a theory in its UV and IR phases, some aspects of this transition may not be easy to disentangle,as in the case of QCD versus the chiral Lagrangian. There are various ways to generate such actions, typically by the Noether method, where theyare obtained starting from a linear coupling of the dilaton to the anomaly (see for instance [10,11]).A second approach is to use field-enlarging transformations, where the dilaton is introduced as acompensator. A compensator is not a dynamical field, and for this reason usually, such actionsare modified by introducing by hand an (extra) kinetic term for the dilaton field. It is important toobserve that if we decided to generate such a term from the Weyl gauging [11, 12] of the EinsteinLagrangian, the procedure would generate a kinetic term which is ghost-like.Flipping the sign of this term is a standard procedure, which, however, needs to be motivated sincethe inclusion of a Goldstone mode by hand, while possible, should be compared against the de-scription of the same theory in the (UV) conformal phase. In the UV this degree of freedom iscompletely absent and the violation of the conformal symmetry (i.e. the anomaly) appears as apurely radiative effect. Clearly, the latter is a pure phenomenological approach which allows toderive a possible final expression of the Lagrangian in a broken conformal phase without address-ing the nature of the breaking itself. The closest example is the QCD chiral Lagrangian, where theglobal symmetries of the two theories in the UV and IR are matched, but the intermediate dynamicsis essentially non-perturbative.In order to shed some light on this, one possibility is to turn to exact methods, if these are available.Up to 3-point functions, CFT’s fix the structure of their correlators in an essentially unique way,except for few constants. It is then clear that the effective action built by combining the correlatorsof 2- and 3-point functions - which are solutions of the conformal constraints - naturally determinethe simplest expression of the anomaly action of a certain theory. This action is specifically nonlo-cal, but obviously, it is not unique. As we are going to discuss next, also in the case of a nonlocalconformal anomaly action, as well as in the chiral case, once this is rewritten in a local form, itmanifests a kinetic mixing between two scalar degrees of freedom. In the chiral case, the scalars5 xact Correlators from Conformal Ward Identities
Claudio Corianò are replaced by pseudoscalars. If we try to decouple the two states we need to define a scale M atwhich the decoupling takes place. It is then easy to realize that the new decoupled Lagrangian ischaracterized by a ghost in its spectrum, at least at the level of trilinear interactions.As emphasized in previous work, a simple analysis of this Lagrangian - in the case of a chiral theory- [13] shows that the resolution of the kinetic mixing requires the inclusion of two axions, with oneof them being a ghost. In the unmixed case, a standard Coleman-Weinberg analysis of the potentialfor the ghost term shows that the Lagrangian induces a ghost condensation, with the possibility ofa redefinition of the vacuum. We are going to briefly review such features. The analysis has beendone in the case of an external axial-vector coupling, although the result in the case of a dilatonpole we expect it to be quite similar.
3. Conformal symmetry and its classical breaking
Let’s come to a brief description of the conformal invariant extension of the SM with a funda-mental Higgs and a dilaton.A scale invariant extension of a given Lagrangian can be obtained if we promote all the dimension-ful constants to dynamical fields. It is natural to ask whether the new degree of freedom introducedto restore the conformal symmetry of the theory can be generated dynamically, emerging from theeffective interactions which can be held responsible for the generation of an anomaly. We illustratethis point in the case of a simple interacting scalar field theory incorporating the Higgs mechanism.At a second stage, we will derive the structure of the dilaton interaction at order 1 / Λ , where Λ is the scale characterizing the spontaneous breaking of the dilatation symmetry and discuss somepossible phenomenological constraints on Λ .The two equivalent forms of the scalar Higgs potential V ( H , H † ) = − µ H † H + λ ( H † H ) = λ (cid:18) H † H − µ λ (cid:19) − µ λ V ( H , H † ) = λ (cid:18) H † H − µ λ (cid:19) (3.1)generate two different scale-invariant extensions V ( H , H † , Σ ) = − µ Σ Λ H † H + λ ( H † H ) V ( H , H † , Σ ) = λ (cid:18) H † H − µ Σ λ Λ (cid:19) , (3.2)where H is the Higgs doublet, λ is its dimensionless coupling constant, while µ has the dimensionof a mass. The constant µ term present in V which in a non-scale invariant theory can be absorbedby a redefinition of the Lagrangian, is clearly insignificant in flat space and generates two differentscale-invariant potential, where only the second one is stable. L = ( ∂ φ ) − V ( φ ) = ( ∂ φ ) + µ φ − λ φ − µ λ , (3.3)6 xact Correlators from Conformal Ward Identities Claudio Corianò obeying the classical equation of motion (cid:3) φ = µ φ − λ φ . (3.4)This theory is not scale invariant due to the appearance of the mass term µ , as one can easily noticefrom the trace of the canonical ( c ) energy-momentum (EMT) tensor T µν c ( φ ) = ∂ µ φ ∂ ν φ − η µν (cid:20) ( ∂ φ ) + µ φ − λ φ − µ λ (cid:21) , T µ c µ ( φ ) = − ( ∂ φ ) − µ φ + λ φ + µ λ . (3.5)The EMT of a scalar field can be improved so that its trace is proportional only to the scale breakingparameter, i.e. the mass µ . This can be achieved by adding an extra contribution T µν I ( φ , χ ) whichis symmetric and conserved T µν I ( φ , χ ) = χ (cid:0) η µν (cid:3) φ − ∂ µ ∂ ν φ (cid:1) , (3.6)where the χ parameter is specifically choosen. The combination of the canonical plus the improve-ment EMT, T µν ≡ T µν c + T µν I has the off-shell trace T µ µ ( φ , χ ) = ( ∂ φ ) ( χ − ) − µ φ + λ φ + µ λ + χφ (cid:3) φ . (3.7)Using the equation of motion (3.4) and selecting χ = /
6, the trace relation given in the expressionabove becomes proportional to the scale-breaking term µ T µ µ ( φ , / ) = − µ φ + µ λ . (3.8)The scale-invariant extension of the Lagrangian given in Eq.(3.3) is obtained by promoting themass terms to dynamical fields using the replacement µ → µ Λ Σ ( x ) , (3.9)obtaining L = ( ∂ φ ) + ( ∂ Σ ) + µ Λ Σ φ − λ φ − µ λ Λ Σ , (3.10)where we have included a kinetic term for the dilaton Σ . The new Lagrangian is dilatation invariant,as shown from the trace of the EMT T µ µ ( φ , Σ , χ , χ (cid:48) ) = ( χ − ) ( ∂ φ ) + (cid:0) χ (cid:48) − (cid:1) ( ∂ Σ ) + χ φ (cid:3) φ + χ (cid:48) Σ (cid:3) Σ − µ Λ Σ φ + λ φ + λ µ Λ Σ , (3.11)an expression that vanishes upon using the equations of motion for the Σ and φ fields, (cid:3) φ = µ Λ Σ φ − λ φ , (cid:3) Σ = µ Λ Σ φ − λ µ Λ Σ , (3.12)7 xact Correlators from Conformal Ward Identities Claudio Corianò and after setting the value of the χ , χ (cid:48) parameters at the special value χ = χ (cid:48) = /
6, correspondingto conformally coupled scalars. The scalar potential V triggers the spontaneous breaking of thescale symmetry around a stable point of minimum. Expanding around the vacuum, parameterizedby the conformal scale Λ and the Higgs vev v respectively Σ = Λ + ρ , φ = v + h (3.13)one can describe the theory in the broken phase.For our present purposes, it is enough to expand the Lagrangian (3.10) around the vev for thedilaton field, as we are interested in the structure of the couplings of its fluctuation ρ L = ( ∂ φ ) + ( ∂ ρ ) + µ φ − λ φ − µ λ − ρ Λ (cid:18) − µ φ + µ λ (cid:19) + . . . , (3.14)where we have neglected terms of higher order in 1 / Λ . It is clear, from (3.8) and (3.14), that onecan write a dilaton Lagrangian at order 1 / Λ , as L ρ = ( ∂ ρ ) − ρ Λ T µ µ ( φ , / ) + . . . , (3.15)where we have used the equations of motion in order to re-express the trace of the energy momen-tum tensor.It is clear, from this simple analysis, that a dilaton, in general, does not couple to the anomaly, butonly to the sources of the explicit breaking of scale invariance, which are proportional to the massterms of the action. In V we parameterize the Higgs around the electroweak vev v as in Eq. (3.13),and indicate with Λ the vev of the dilaton field Σ = Λ + ρ , with φ + = φ = ρ and h ,which are given by with the potential V exhibiting a massless mode due to the existence of a flatdirection. The Higgs and the dilaton will mix according to the mass matrix (cid:32) ρ h (cid:33) = (cid:32) cos α sin α − sin α cos α (cid:33) (cid:32) ρ h (cid:33) (3.16)with cos α = (cid:112) + v / Λ sin α = (cid:112) + Λ / v . (3.17)We denote with ρ the massless dilaton generated by this potential, while h denotes the Higgsscalar whose mass is given by m h = λ v (cid:18) + v Λ (cid:19) with v = µ λ , (3.18)and with m h = λ v being the mass of the Standard Model Higgs. Notice that the Higgs mass, inthis case, is corrected by the new scale of the spontaneous breaking of the dilatation symmetry ( Λ ),which remains a free parameter.Obviously, the presence of a massless dilaton in the spectrum is troublesome, a problem whichsend us back to the issue of how to select a single vacuum state from the underlying vacuumdegeneracy. This can be lifted by the introduction of extra (explicit breaking) terms which give a8 xact Correlators from Conformal Ward Identities Claudio Corianò (a) (b) (c)
Figure 1:
The triangle diagram in the fermion case (a), the collinear fermion configuration responsible forthe anomaly (b) and a diagrammatic representation of the exchange via an intermediate state (dashed line)(c). small mass to the dilaton field. To remove such degeneracy, one can introduce, for instance, theterm L break = m ρ ρ + m ρ ρ Λ + . . . , (3.19)where m ρ represents the dilaton mass. The coupling of a dilaton to an anomaly is necessary, sincethe dilaton is the pseudo Nambu-Goldstone mode of the dilatation symmetry and the anomaly isa source of such breaking. Thus, this coupling has to be introduced by hand and the role of theconformal anomaly action is to account for it.
4. Phenomenology of a classical scale invariant extension of the Standard Model
In the case of the SM, the dilaton interaction takes the form discussed above L int = − Λ ρ T µµ SM . (4.1)where T is the EMT of the SM. As usual, it can be easily derived by embedding the SM Lagrangianin the background metric g µν S = S SM + S I = (cid:90) d x √− g L SM + ξ (cid:90) d x √− g R H † H , (4.2)where H is the Higgs doublet and R the scalar curvature of the same metric, and then defining T µν ( x ) = (cid:112) − g ( x ) δ [ S SM + S I ] δ g µν ( x ) , (4.3)or, in terms of the SM Lagrangian, as12 √− gT µν ≡ ∂ ( √− g L ) ∂ g µν − ∂∂ x σ ∂ ( √− g L ) ∂ ( ∂ σ g µν ) . (4.4)The complete expression of the energy-momentum tensor can be found in [14]. S I is responsible forgenerating a term of improvement ( I ) , which induces a mixing between the Higgs and the dilatonafter spontaneous symmetry breaking. As usual, we parameterize the vacuum H in the scalarsector in terms of the electroweak vev v as H = (cid:32) v √ (cid:33) (4.5)9 xact Correlators from Conformal Ward Identities Claudio Corianò ρ (a) W ± W ± W ± ρ (b) φ ± φ ± φ ± ρ (c) H ZHρ (d) ρ (e) W ± W ± ρ (f) φ ± φ ± ρ (g) W ± φ ± ρ (h) Z Hρ (i) W ± ρ (j) Figure 2:
Typical amplitudes of triangle and bubble topologies contributing to the ργγ , ργ Z and ρ ZZ interactions. They include fermion ( F ) , gauge bosons ( B ) and contributions from the term of improvement(I). Diagrams (a)-(g) contribute to all the three channels while (h)-(k) only in the ρ ZZ case. and we expand the Higgs doublet in terms of the physical Higgs boson H and the two Goldstonebosons φ + , φ as H = (cid:32) − i φ + √ ( v + H + i φ ) (cid:33) , (4.6)obtaining from the term of improvement of the stress-energy tensor the expression T I µν = − ξ (cid:20) ∂ µ ∂ ν − η µν (cid:3) (cid:21) H † H = − ξ (cid:20) ∂ µ ∂ ν − η µν (cid:3) (cid:21)(cid:18) H + φ + φ + φ − + v H (cid:19) , (4.7)which is responsible for a bilinear vertex V I , ρ H ( k ) = − i Λ ξ s w M W e k where s W is the Weinberg angle. The trace takes contribution from the massive fields, the fermionsand the electroweak gauge bosons, and from the conformal anomaly in the massless gauge bosonsector, through the β functions of the corresponding coupling constants.One can directly verify the the separation between the anomalous and the explicit mass-relatedterms in the expression of the correlators responsible of the conformal anomaly. It has been verifiedin QED and in the neutral sector of the SM [15, 16] by explicit computations. Such separation doesnot necessarily hold in other regularization schemes. The reason for such difference is related to thespecial role of DR compared to other regularization schemes, which explicitly break the conformalsymmetry by introducing a cutoff. This point has been discussed by Bardeen [2] in the contextof ’t Hooft’s naturalness principle applied to the Higgs mass. Also in this case there are strongarguments that convey a truly specials role to such regularization scheme compared to others. Inthe case of the TVV vertex, for instance, by taking a trace one can derive an anomalous Wardidentity of the form Γ αβ ( z , x , y ) ≡ η µν (cid:68) T µν ( z ) V α ( x ) V (cid:48) β ( y ) (cid:69) = δ A ( z ) δ A α ( x ) δ A β ( y ) + (cid:68) T µ µ ( z ) V α ( x ) V (cid:48) β ( y ) (cid:69) . (4.8)10 xact Correlators from Conformal Ward Identities Claudio Corianò where A ( z ) is the anomaly functional, while A α indicates the gauge fields coupled to the current V α . Γ αβ is a generic dilaton/gauge/gauge vertex, whose Feynman expansion takes a form depictedin Fig. 2. It is obtained from the TVV (cid:48) vertex by tracing the spacetime indices µν . A ( z ) is derivedfrom the renormalized expression of the vertex by tracing the gravitational counterterms in 4 − ε dimensions (see for instance [10, 11]) (cid:104) T µµ (cid:105) = A ( z ) , (4.9)which in a curved background is given by the metric functional A ( z ) = − (cid:20) bC + b (cid:48) (cid:18) E − (cid:3) R (cid:19) + c F (cid:21) , (4.10)where b , b (cid:48) and c are parameters. For the case of a single fermion in an abelian gauge theory theyare given by b = / π , b (cid:48) = − / π , and c = − e / π . C is the square of the Weyltensor and E is the Euler density given by C = C λ µνρ C λ µνρ = R λ µνρ R λ µνρ − R µν R µν + R E = ∗ R λ µνρ ∗ R λ µνρ = R λ µνρ R λ µνρ − R µν R µν + R . (4.12)In a flat metric background the expression of such functional reduces to the simple form A ( z ) = ∑ i β i g i F αβ i ( z ) F i αβ ( z ) , (4.13)where β i are clearly the mass-independent β functions of the gauge fields and g i the correspondingcoupling constants. Obviously, for a theory which which is quantum conformal invariant, the β i vanish. We refer to [17, 18] for more details concerning the phenomenology of such models ofdirect LHC relevance.It is possible, using such effective interaction which couples the dilaton to the anomaly, to addresssome phenomenological issues which are can be studied at the LHC.We show in Figs 3 and 4 some results of a phenomenological analysis of the production and decayof a dilaton ( ρ ) as a function of its mass and branching ratios, together with some comparisons withthe Higgs. In particular Fig. 5 shows the results of an analysis of the bounds on Λ , the conformalscale by a comparison with experimental data. In general it is possible to select a value of Λ inthe few TeV region in such a way that such additional interactions are in agreement with the SMresults. We refer to [18] for further details.
5. Wess Zumino actions with an asymptotic dilaton from a kinetic flip
The inclusion of an asymptotic dilaton, as done above, and the derivation of the correspondinganomaly action can be performed in various ways. One possibility is to use the procedure of Weylgauging (see [12]). It is possible to generate such an action by the application of this procedureto the anomaly counterterms. In this case one can also show that multiple traces of stress energy11 xact Correlators from Conformal Ward Identities
Claudio Corianò
100 200 300 400 500 60010 (cid:45) m Ρ (cid:72) GeV (cid:76) (cid:66) (cid:114) (cid:72) Ρ (cid:174) (cid:88) (cid:45) (cid:88) (cid:76) gg (cid:116) (cid:45) (cid:116) H H W (cid:177) W (cid:161) ZZ ΓΓ (cid:99) (cid:45) (cid:99) (cid:98) (cid:45) (cid:98) (a)
100 200 300 400 500 60010 (cid:45) m h (cid:72) GeV (cid:76) (cid:66) (cid:114) (cid:72) h (cid:174) (cid:88) (cid:45) (cid:88) (cid:76) gg (cid:116) (cid:45) (cid:116) W (cid:177) W (cid:161) ZZ ΓΓ (cid:99) (cid:45) (cid:99) (cid:98) (cid:45) (cid:98) (b) Figure 3:
The mass dependence of the branching ratios of the dilaton (a) and of the Higgs boson (b).
100 1000500200 300150 7000.010.11101001000 m Ρ (cid:72) GeV (cid:76) Σ gg (cid:174) Ρ (cid:72) pb (cid:76) (cid:76) (cid:61) T e V (cid:76) (cid:61) T e V (cid:76) (cid:61) T e V (a)
100 1000500200 300150 7000.10.51.05.010.050.0100.0 m Ρ (cid:72) GeV (cid:76) Σ V B F Ρ (cid:72) f b (cid:76) (cid:76) (cid:61) T e V (cid:76) (cid:61) T e V (cid:76) (cid:61) T e V (b) Figure 4:
The mass dependence of the dilaton cross-section via gluon fusion (a) and vector boson fusion (b)for three different choices of the conformal scale, Λ = , ,
10 TeV respectively. tensors are functionally dependent on the first 4 ones in d =
4, the first 6 ones in d = d =
4, beforestressing one crucial aspect of this approach, i.e. the presence of a ghost in the procedure. Themethod amounts to a field-enlarging transformation, with the removal of the ghost by flipping thesign of the kinetic term. In the next sections, once we turn to the analysis of 1PI anomaly actions,which are not constructed by this formal procedure, but are directly computed either in perturbationtheory or by solving the CWI’s in momentum space, we will point out a similar feature.The metric tensor g µν ( x ) , the vierbein V a ρ ( x ) and the fields Φ change under Weyl scalingsaccording to g (cid:48) µν ( x ) = e σ ( x ) g µν ( x ) , V (cid:48) a ρ ( x ) = e σ ( x ) V a ρ ( x ) , Φ (cid:48) ( x ) = e d Φ σ ( x ) Φ ( x ) , (5.1)with σ ( x ) being a dimensionless function parameterizing a local Weyl transformation. Here d Φ is12 xact Correlators from Conformal Ward Identities Claudio Corianò
300 400 500 600 700 800 900 10000.1110100 m Ρ (cid:72) GeV (cid:76) Σ gg (cid:174) Ρ (cid:66) (cid:114) (cid:72) Ρ (cid:174) ZZ (cid:174) (cid:123) Ν (cid:76)(cid:72) f b (cid:76) CMSSM (cid:76)(cid:61) (cid:76)(cid:61) (cid:76)(cid:61)
10 TeV (a)
200 300 400 500 6000.010.050.100.501.005.00 m Ρ (cid:72) GeV (cid:76) Μ WW CMS (cid:76)(cid:61) (cid:76)(cid:61) (cid:76)(cid:61)
10 TeV SM (b)
300 400 500 600 700 800 9000.11101001000 m Ρ (cid:72) GeV (cid:76) Σ gg (cid:174) Ρ (cid:66) (cid:114) (cid:72) Ρ (cid:174) Τ (cid:45) Τ (cid:76)(cid:72) f b (cid:76) CMS SM (cid:76)(cid:61) (cid:76)(cid:61) (cid:76)(cid:61)
10 TeV (c)
260 280 300 320 34010205010020050010002000 m Ρ (cid:72) GeV (cid:76) Σ gg (cid:174) Ρ (cid:66) (cid:114) (cid:72) Ρ (cid:174) HH (cid:174) Τ (cid:76)(cid:72) f b (cid:76) CMS SM (cid:76)(cid:61) (cid:76)(cid:61) (cid:76)(cid:61)
10 TeV (d)
Figure 5:
The mass bounds on the dilaton from heavy scalar decays to (a) ZZ , (b) W ± W ∓ , (c) ¯ ττ and (d) to H H for three different choices of conformal scale, Λ = , ,
10 TeV respectively. the scaling dimension of the generic field Φ , the latin suffix a in V a ρ denotes the flat local index,while the Greek indices are the curved indices of the spacetime manifold. A way to build a Weylinvariant theory containing the fields in (5.1) consists in making the metric tensor, the vierbein andthe fields Φ , Weyl invariant through the substitutions V a ρ ( x ) → ˆ V a ρ ( x ) ≡ e − τ ( x ) Λ V a ρ ( x ) , Φ ( x ) → ˆ Φ ( x ) ≡ e − d Φ τ ( x ) Λ Φ (5.2)and take the form of field-enlarging transformations.Under a Weyl scaling (5.1), the dilaton τ is required to shift as a Goldstone mode τ (cid:48) ( x ) = τ ( x ) + Λ σ ( x ) . (5.3)The Weyl invariant terms may take the form of any scalar contraction of ˆ R µνρσ , ˆ R µν and ˆ R and can be classified by their mass dimension. Typical examples are J n ∼ Λ ( n − ) (cid:90) d x (cid:112) ˆ g ˆ R n , (5.4)and so forth, with the case n = Γ [ ˆ g ] ≡ Γ [ g , τ ] , extended with the inclusion of τ ( x ) . Γ [ ˆ g ] ∼ ∑ n J n [ ˆ g ] . (5.5)13 xact Correlators from Conformal Ward Identities Claudio Corianò
The leading contribution to Γ is the kinetic term for the dilaton, which can be obtained in twoways. The first method is to consider the Weyl-gauged Einstein-Hilbert term (cid:90) d d x (cid:112) ˆ g ˆ R = (cid:90) d d x √ g e ( − d ) τ Λ (cid:20) R − ( d − ) (cid:3) τ Λ + ( d − ) ( d − ) ∂ λ τ ∂ λ τ Λ (cid:21) = (cid:90) d d x √ g e ( − d ) τ Λ (cid:20) R − ( d − ) ( d − ) ∂ λ τ ∂ λ τ Λ (cid:21) , (5.6)with the inclusion of an appropriate normalization S ( ) τ = − Λ d − ( d − ) ( d − ) (cid:90) d d x (cid:112) ˆ g ˆ R , (5.7) which reverses the sign in front of the Einstein term. Indeed, the extraction of a conformal factor( ˜ σ ) from the Einstein-Hilbert term from a fiducial metric ¯ g µν ( g µν = ¯ g µν e ˜ σ ) generates a kineticterm for ( ˜ σ ) which is ghost-like.An alternative method consists in writing down the usual conformal invariant action for ascalar field χ in a curved background S ( ) χ = (cid:90) d d x √ g (cid:18) g µν ∂ µ χ ∂ ν χ − d − d − R χ (cid:19) . (5.8)By the field redefinition χ ≡ Λ d − e − ( d − ) τ Λ Eq. (5.8) becomes S ( ) τ = Λ d − (cid:90) d d x √ g e − ( d − ) τ Λ (cid:18) ( d − ) Λ g µν ∂ µ τ ∂ ν τ − d − d − R (cid:19) , (5.9)which, for d =
4, reduces to the familiar form S ( ) τ = (cid:90) d x √ g e − τ Λ (cid:18) g µν ∂ µ τ ∂ ν τ − Λ R (cid:19) (5.10)and coincides with the previous expression (5.7), obtained from the formal Weyl invariant con-struction.In four dimensions we can build the following possible subleading contributions (in 1 / Λ ) tothe effective action which, when gauged, can contribute to the fourth order dilaton action S ( ) τ = (cid:90) d x √ g (cid:18) α R µνρσ R µνρσ + β R µν R µν + γ R + δ (cid:3) R (cid:19) . (5.11)The fourth term ( ∼ (cid:3) R ) is just a total divergence, whereas two of the remaining three terms canbe traded for the squared Weyl tensor F and the Euler density G . As √ g F is Weyl invariant and G is a topological term, neither of them contributes, when gauged according to (5.2), so that the onlynon vanishing four-derivative term in the dilaton effective action in four dimensions is S ( ) τ = γ (cid:90) d x (cid:112) ˆ g ˆ R = γ (cid:90) d x √ g (cid:20) R − (cid:18) (cid:3) τ Λ − ∂ λ τ ∂ λ τ Λ (cid:19)(cid:21) , (5.12)14 xact Correlators from Conformal Ward Identities Claudio Corianò with γ a dimensionless constant. If we also include a possible cosmological constant term, S ( ) τ , weget the final form of the dilaton effective action in d = S τ = S ( ) τ + S ( ) τ + S ( ) τ + · · · = (cid:90) d x (cid:112) ˆ g (cid:26) α − Λ d − ( d − ) ( d − ) ˆ R + γ ˆ R (cid:27) + . . . , (5.13)where the ellipsis refer to additional operators which are suppressed in 1 / Λ . In flat space ( g µν → δ µν ), (5.12) becomes S τ = (cid:90) d x (cid:20) e − τ Λ α + e − τ Λ ∂ λ τ ∂ λ τ + γ (cid:18) (cid:3) τ Λ − ∂ λ τ ∂ λ τ Λ (cid:19)(cid:21) + . . . (5.14)This approach can be extended to the anomaly counterterms as well, by considering the renormal-ized action Γ ren [ g , τ ] . This takes to a Wess-Zumino (WZ) action which consistently accounts forthe anomaly, at the price of including an extra dynamical degree of freedom and a kinetic flip. Thelatter is obtained by the relation Γ WZ [ g , τ ] = Γ ren [ g , τ ] − ˆ Γ ren [ g , τ ] (5.15)and takes the WZ form Γ WZ [ g , τ ] = (cid:90) d x √ g (cid:26) β a (cid:20) τ Λ (cid:18) F − (cid:3) R (cid:19) + Λ (cid:18) R ∂ λ τ ∂ λ τ + ( (cid:3) τ ) (cid:19) − Λ ∂ λ τ ∂ λ τ (cid:3) τ + Λ (cid:16) ∂ λ τ ∂ λ τ (cid:17) (cid:21) + β b (cid:20) τ Λ G − Λ (cid:18) R αβ − R g αβ (cid:19) ∂ α τ ∂ β τ − Λ ∂ λ τ ∂ λ τ (cid:3) τ + Λ (cid:16) ∂ λ τ ∂ λ τ (cid:17) (cid:21)(cid:27) . (5.16)This action is local and contains an asymptotic dilaton, but does not appear to be equivalent to thenonlocal action (the Riegert action) S NL anom [ g ] = (cid:90) d x √− g x (cid:16) E − (cid:3) R (cid:17) x (cid:90) dx (cid:48) √− g x (cid:48) D ( x , x (cid:48) ) (cid:20) b (cid:48) (cid:0) E − (cid:3) R (cid:1) + bC (cid:21) x (cid:48) (5.17)where D ( x , x (cid:48) ) = ( ∆ − ) xx (cid:48) and ∆ ≡ ∇ µ (cid:18) ∇ µ ∇ ν + R µν − Rg µν (cid:19) ∇ ν = (cid:3) + R µν ∇ µ ∇ ν − R (cid:3) + ( ∇ µ R ) ∇ µ (5.18)is the unique fourth order scalar kinetic operator that is conformally covariant √− g ∆ = √− ¯ g ¯ ∆ (5.19)under the local conformal reparameterization g µν = e σ ¯ g µν , for an arbitrary rescaling σ ( x ) [19,20].Such second form of the anomaly action, without an asymptotic dilaton field, is what one rediscov-ers both from a perturbative analysis and from the solution of the CWI’s for 3-point functions, aswe will be discussing below. We are going to emphasize one key feature of such two actions.15 xact Correlators from Conformal Ward Identities Claudio Corianò
6. Turning to general issues: The perturbative structure of PI anomaly actions forchiral and conformal anomalies The type of actions discussed above, which enlarge the spectrum of the SM by the inclu-sion of an asymptotic dilaton field, are the most popular ones, and are justified within an ordi-nary phenomenological approach. Obviously, they do not introduce a dilaton dynamically, but areviable anyhow, as far as the issue of the explicit breaking of the dilaton mass is accepted on apurely phenomenological basis. An alternative approach consists in starting from a given classi-cally conformal invariant theory and investigate the quantum corrections which are responsible forthe generation of the conformal anomaly. In this case, the basic procedure for chiral and conformalanomalies are quite similar, although for a chiral anomaly the approach is far simpler, since onlya linear coupling of a Nambu-Goldstone mode to the anomaly is sufficient in order to generatethe anomaly contribution. This type of approach is at the core of the Stückelberg mechanism forthe cancelation of the gauge anomalies generated by a certain anomalous gauge interaction ( B µ ) .The method allows to obtain a physical axion only in the presence of a non-perturbative periodicpotential, which can be easily justified under the assumption that at a phase transition the instantonsector can generate it. We refer to a recent review for a discussion of such a mechanism [9]. For a chiral anomaly, the main features of the 1PI effective actions have been discussed indetail in [6, 7]. They are based on a perturbative representation of the anomaly vertex which isequivalent to the original description in terms of 6 form factors found on textbooks, but formulatedin terms of longitudinal and transverse components, as we are going to illustrate.We recall that the
AVV amplitude with off-shell external lines is parameterized in the form ∆ λ µν = V ( k , k ) ε [ k , µ , ν , λ ] + V ( k , k ) ε [ k , µ , ν , λ ] + V ( k , k ) ε [ k , k , µ , λ ] k ν + V ( k , k ) ε [ k , k , µ , λ ] k ν + V ( k , k ) ε [ k , k , ν , λ ] k µ + V ( k , k ) ε [ k , k , ν , λ ] k µ (6.1)where ε [ k , µ , ν , λ ] ≡ ε αµνλ k α , and so on, with k and k the momenta of the two vector lines.The four invariant amplitudes V i for i ≥ V ( k , k ) = − V ( k , k ) = − π I ( k , k ) , (6.2) V ( k , k ) = − V ( k , k ) = π [ I ( k , k ) − I ( k , k )] , (6.3)where the general massive I st integral is defined by I st ( k , k ) = (cid:90) dw (cid:90) − w dzw s z t (cid:2) z ( − z ) k + w ( − w ) k + wz ( k k ) − m (cid:3) − , (6.4)Both A and A are instead represented by formally divergent integrals, which can be renderedfinite only by imposing the Ward identities on the two vector lines, giving V ( k , k ) = k · k V ( k , k ) + k V ( k , k ) , (6.5) V ( k , k ) = k V ( k , k ) + k · k V ( k , k ) , (6.6)16 xact Correlators from Conformal Ward Identities Claudio Corianò which allow to re-express the formally divergent amplitudes in terms of the convergent ones. TheBose symmetry on the two vector vertices with indices µ and ν is fulfilled thanks to the relations V ( k , k ) = − V ( k , k ) (6.7) V ( k , k ) = − V ( k , k ) . (6.8)Coming to the second parameterization of the three-point correlator function, this is the one pre-sented in [22]. One of the features of this parameterization is the presence of a longitudinal con-tribution (i.e. of an anomaly pole), apparently for generic virtualities of the external momentaof the two vector lines. For on-shell photons there is a single form factor with a 1 / k behaviour ( k = k + k ) in the only longitudinal structure of the vertex. In the general off-shell case severalstructures are affected by other poles in their transverse components as well, raising some doubtsabout the significance of the longitudinal pole, for general kinematics. The various contributionsand parameterizations can be related one to the other by the Schoutens relations, as discussed in [6].This second parameterization plays an important role in the description of the anomalous magneticmoment of the muon [22]. For this reason we start by recalling the structure of such L/T parame-terization, which separates the longitudinal (L) from the transverse (T) components of the anomalyvertex, which is given by W λ µν = π (cid:104) W L λ µν − W T λ µν (cid:105) , (6.9)where the longitudinal component W L λ µν = w L k λ ε [ µ , ν , k , k ] (6.10)(with w L = − i / k ) describes the anomaly pole, while the transverse contributions take the form W T λ µν ( k , k ) = w (+) T (cid:0) k , k , k (cid:1) t (+) λ µν ( k , k ) + w ( − ) T (cid:0) k , k , k (cid:1) t ( − ) λ µν ( k , k )+ (cid:101) w ( − ) T (cid:0) k , k , k (cid:1) (cid:101) t ( − ) λ µν ( k , k ) , (6.11)with the transverse tensors given by t (+) λ µν ( k , k ) = k ν ε [ µ , λ , k , k ] − k µ ε [ ν , λ , k , k ] − ( k · k ) ε [ µ , ν , λ , ( k − k )]+ k + k − k k k λ ε [ µ , ν , k , k ] , t ( − ) λ µν ( k , k ) = (cid:20) ( k − k ) λ − k − k k k λ (cid:21) ε [ µ , ν , k , k ] (cid:101) t ( − ) λ µν ( k , k ) = k ν ε [ µ , λ , k , k ] + k µ ε [ ν , λ , k , k ] − ( k · k ) ε [ µ , ν , λ , k ] . (6.12)taking to a parameterization of the vertex in the form Γ ( ) = Γ ( ) pole + ˜ Γ ( ) (6.13)with the pole part, coupled to the external axial vector field B µ given by Γ ( ) pole = − π (cid:90) d x d y ∂ · B ( x ) (cid:3) − x , y F ( y ) ∧ F ( y ) (6.14)17 xact Correlators from Conformal Ward Identities Claudio Corianò and the rest ( ˜ Γ ( ) ) given by a complicated nonlocal expression which contributes homogeneouslyto the Ward identify of the anomaly graph.All the terms in this parameterization are linked together, and the isolation of a single contribu-tion from the rest is only possible for specific kinematics. Nevertheless, around the light-cone ( k → ) the anomaly can be attributed to the pole-like behaviour of the longitudinal part. Noticethat the absence of the pole structure in Rosenberg’s formulation of the AVV [21] is due to theredundancy of the latter. Poles can be removed and reinserted in a given parameterization by usingthe Schoutens relations, but there is no way one can remove the anomaly pole of W L consistently.One can naturally try to to unconver the meaning of such interactions. As we are going to see,at least from the analysis of 3-point functions, the effective action shows the emergence of someinstabilities, signalling the possibility that the vacuum will be restructured in the presence of suchinteractions. This feature is probably shared by the effective actions of both chiral and conformalanomalies, which get unified in the context of supersymmetric theories. We are going to illustratethis phenomenon in the chiral case, where it has been worked out in some detail. As an example we consider a gauge boson B coupled to a single chiral fermion which generatesa 1PI effective action of the form L = ¯ ψ ( i (cid:54) ∂ + g (cid:54) B γ ) ψ − F B + (cid:104) ∆ BBB
BBB (cid:105) + c ∂ B (cid:3) F B ˜ F B + . . . (6.15)where the ellipsis refer to additional transverse terms identified in the trilinear BBB vertex, which isanomalous. We have isolated the longitudinal contribution related to the anomaly pole and we willfocus in this contribution. It is easy to show that the 1 / (cid:3) term can be generated by the introductionof two pseudoscalar fields a and b which allow to remove the nonlocal contribution of the action L = ψ ( i (cid:54) ∂ + g (cid:54) B γ ) ψ − F B + (cid:104) ∆ BBB
BBB (cid:105) + c F B ∧ F B ( a + b )+ (cid:0) ∂ µ b − M B µ (cid:1) − (cid:0) ∂ µ a − M B µ (cid:1) , (6.16)where both a and b shift under the gauge symmetry δ b = M θ B ( x ) δ B µ = ∂ µ θ B ( x ) (6.17)and where θ B ( x ) parameterizes a gauge transformation. The inclusion of two pseudoscalars whichacquire Stückelberg mass terms, given by the second line of the equation above, is a specific fea-ture of this reformulation, with a Stückelberg mass M . The equivalence between (6.15) and (6.16)can be proven directly from the functional integral, integrating out both a and b , which gives twogaussian integrations [23]. Notice that b has a positive kinetic term and a is ghost-like. A similarbehaviour obviously holds also in the case of an external axial-vector current coupled to a corre-sponding classical gauge field if the fermion spectrum is anomalous. In this case the appearance ofa ghost in the effective action indicates the onset of an instability. Notice also that the inclusion ofa Stückelberg mass term indicates that we need to introduce a suitable scale in order to be able to18 xact Correlators from Conformal Ward Identities Claudio Corianò define a local action.There is a third equivalent formulation of the same action (6.16) which can be defined with theinclusion of a kinetic mixing between the two pseudoscalars. This has been given for QED (with asingle fermion) coupled to an external axial-vector field B µ [7] and takes the form L = ∂ µ η∂ µ χ − χ∂ B + e π η F ˜ F , (6.18)where F is the field strength of the photon A µ while B µ takes the role of a source. It is quitestraightforward to relate (6.16) and (6.18). This can be obtained by the field redefinitions η = ( a + b ) M , χ = M ( a − b ) , (6.19)showing that indeed a mixing term is equivalent to the presence of either an anomaly pole or to twopseudoscalars in the spectrum of the theory, one of them being a ghost, and the inclusion of a scaleat which to define their decoupling, which is the Stückelberg scale. Notice that in Eq. (6.19) χ isgauge invariant while η is not.
7. The Coleman-Weinberg potential and ghost condensation at trilinear level
We have clarified that a Lagrangian containing a pole counterterm shows some nontrivial fea-tures. In particular, the presence of the nonlocal ∂ B (cid:3) − F ˜ F interaction induced by an anomalypole, rewritten in a local version, allows to proceed with some further perturbative analysis whichsheds some light on the character of the effective potential of the ghost field a defined in (6.16).We can use the Coleman-Weinberg approach, which indicates the presence of an instability in theaction, obtained after integration over all the remaining fields of the model. The instability is sig-nalled by the presence of a ghost condensate at 1-loop level. To illustrate this formal result wefollow closely the analysis of [13].Consider the gauge-fixed version of the Lagrangian in (6.16) given by L = ¯ ψ ( i (cid:54) ∂ + e (cid:54) B γ ) ψ − F B − ( ∂ µ B µ ) α + e π M F B ∧ F B ( a + b )+ ( ∂ µ b − M B µ ) − ( ∂ µ a − M B µ ) (7.1)where α is the gauge parameter. We shift the ghost field, separating the classical ghost background(still denoted as a ( x )) , from its quantum fluctuating part on which we will integrate, A ( x ) a ( x ) −→ a ( x ) + A ( x ) . (7.2)Dropping the linear terms in the quantum fluctuation field A ( x ) and taking just the quadratic partof all the quantum fields we get the quadratic Lagrangian L quad = ¯ ψ i (cid:54) ∂ ψ − F B − ( ∂ µ B µ ) α + e π M aF B ∧ F B + ( ∂ b ) − M ∂ µ bB µ − ( ∂ µ A ) + M B µ ∂ µ A , (7.3)19 xact Correlators from Conformal Ward Identities Claudio Corianò from which we can determine, after an integration by parts, the contribution which is quadratic inthe anomalous gauge field B (cid:90) d xB µ (cid:104) g µν (cid:3) − (cid:18) − α (cid:19) ∂ µ ∂ ν + e π M ∂ α a ε µαρν ∂ ρ (cid:105) B ν . (7.4)The one loop effective action in the background of the ghost a is obtained by integration overall the quantum fields in the form e i Γ [ a ] = (cid:90) [ DA ][ D ψ ][ D ¯ ψ ][ DB ][ Db ] × exp { i (cid:90) d x (cid:104) ¯ ψ i (cid:54) ∂ ψ − F B − ( ∂ µ B µ ) α + ( ∂ b ) − M ∂ µ bB µ − ( ∂ µ A ) + M B µ ∂ µ A + e π M F B ∧ F B a (cid:105)(cid:111) . (7.5)The integration over the quantum fluctuations of the ghost field A gives (cid:90) [ DA ] exp (cid:104) i (cid:90) d x (cid:16) A (cid:3) A − M ∂ µ B µ A (cid:17)(cid:105) ∝ exp (cid:104) − (cid:90) d xd y (cid:16) M ∂ µ B µ ( x ) D F ( x − y ) M ∂ ν B ν ( y ) (cid:17)(cid:105) (7.6)where D F ( x − y ) = (cid:90) d p ( π ) − ie ip ( x − y ) p − i ε (7.7)is the propagator for the quantum fluctuations of the ghost field. The integration over the axion b induces some cancelations of various terms giving (cid:90) [ Db ] exp (cid:104) i (cid:90) d x (cid:16) − b (cid:3) b + M ∂ µ B µ b (cid:17)(cid:105) ∝ exp (cid:104) − (cid:90) d xd yM ∂ µ B µ ( x ) D F ( x − y ) M ∂ ν B ν ( y ) (cid:105) , (7.8)where we have introduced the propagator of the axion field D F ( x − y ) = (cid:90) d p ( π ) ie ip ( x − y ) p + i ε . (7.9)Notice that D F ( x − y ) + D F ( x − y ) vanishes in the limit ε →
0, thus the integration in A and b eliminates the terms (7.6) and (7.8) from the action. Therefore we are just left with the expression e i Γ [ a ] ∝ (cid:90) [ DB ] exp (cid:104) − i (cid:90) d x ( F B ) − (cid:90) d x ( ∂ µ B µ ) α + i e π M (cid:90) d xaF B ∧ F B (cid:105) . (7.10)Defining l ≡ e π M and φ α ≡ ∂ α a , (7.11)20 xact Correlators from Conformal Ward Identities Claudio Corianò the effective action of the classical background ghost field is then given by i Γ [ a ] = −
12 Tr log (cid:18) (cid:3) g µ ν − (cid:18) − α (cid:19) ∂ µ ∂ ν + l ε µαρ ν φ α ∂ ρ (cid:19) (7.12)where the trace Tr, as usual, must be taken in the functional sense.To perform the calculation of (7.12) we use the heat kernel method and define the functionaldeterminant in (7.12) using a ζ function regularization. We take φ α to be constant. We havelog det Q = − lim s → dds µ s Γ ( s ) (cid:90) + ∞ dt t s − Tr ( e − tQ ) (7.13)with the functional trace performed in the plane wave basisTre − tQ = (cid:90) d x tr < x | e − tQ | x > = (cid:90) d x tr (cid:90) d k ( π ) e − ikx e − tQ e ikx , (7.14)and with tr denoting the trace on the Lorentz indices. Further manipulations give ( e − ikx e − tQ e ikx ) µ τ = g µ τ e tk exp (cid:0) − itl ε τ ν αρ φ α k ρ (cid:1) + ( − α ) k µ k ν k ( − e tk ) , (7.15)where we have used the relation e − tk µ k ν = g µν + k µ k ν k ( e − tk − ) k µ k τ e − itl ε τναρ φ α k ρ = k µ k ν . (7.16)We need to consider in (7.15) just the φ -dependent part. As usual, the Coleman-Weinberg potentialis gauge-dependent. In this case the dependence on the gauge-fixing parameter α can be assimilatedto the constant terms. The functional trace receives contributions only from the terms with n even,and after some manipulations we obtain tr ( e − ikx e − tQ e ikx ) = − e tk cosh tl (cid:113) k φ − ( k · φ ) + const . (7.17)Inserting (7.17) into (7.14) we get, apart from a constant factor of infinite volume, the expressionof the trace Tr e − tQ ∼ − (cid:90) d k ( π ) e tk cosh tl (cid:113) k φ − ( k · φ ) , (7.18)giving an effective potential for the background φ α of the form V [ φ ] = − lim s → dds µ s Γ ( s ) (cid:90) + ∞ dt t s − (cid:90) d k ( π ) e tk cosh tl (cid:113) k φ − ( k · φ ) . (7.19)We can obtain the leading contribution of this effective potential by expanding the integrand in l , i.e. in 1 / M . After performing the expansion and restoring the infinite space-time volume weobtain the effective action S = (cid:90) d xdt (cid:26)(cid:18) − − l π (cid:19) ( ∂ a ) + l π ( ∂ a ) (cid:27) (7.20)21 xact Correlators from Conformal Ward Identities Claudio Corianò which obviously can be rewritten as S = (cid:90) d xdt (cid:26) − ( ∂ a ) + l π ( ∂ a ) (cid:27) . (7.21)Notice that the polynomial in the integrand P ( φ ) = − φ + l π φ (7.22)has a minimum at ¯ φ ∼ l (cid:114) π > . (7.23)To investigate the character of the minimum and of the fluctuations around this minimum, we selecta time-like frame, where the background takes the form¯ a = ¯ φ t . (7.24)If we now consider small fluctuactions around this configuration of minimum, denoted as π a = ¯ φ t + π (7.25)and expanding (7.21) we obtain the action S = (cid:90) d xdt (cid:40) ˙ π + (cid:114) π l ˙ π + l π ˙ π + l π ˙ π | ∇ π | − (cid:114) π l ˙ π | ∇ π | + · · · (cid:41) . (7.26)This action has the same form as in [24] (see formula (4.2)). As in this previous analysis, we do notget the term | ∇ π | since its coefficient is proportional to P (cid:48) ( ¯ φ ) =
0. Clearly, the Lorentz symmetryis broken, at least at 1-loop level, and is signalling an instability of the local model (6.15) generatedin the infrared region. Notice, in fact, that in the Coleman-Weinberg approach we are closing thegauge boson loop and we are taking the long wavelength limit of the external background ghostfield. Finally, one should also notice that the dependence of the effective potential on M , in thisapproach, is recovered at higher orders. For instance, additional contributions, suppressed by 1 / M ,are obtained by the insertion of the self-energy of the anomalous gauge boson on the lowest ordercontribution (the gauge boson loop). These features of actions containing Wess-Zumino terms havebeen studied in the past with similar results [25] [26].There are some conclusions that one can draw from this simple analysis. There are some indicationsthat the presence of an anomaly pole in a chiral theory, taken face value, generates a vacuuminstability which leads to ghost condensation, and a similar feature is expected from the analysisof a conformal anomaly pole. It is natural to think that the vacuum gets redefined if the one loopanalysis can be trusted.
8. The 1PI for the
T JJ in QED and QCD
The perturbative character of chiral and conformal anomalies is encoded in both cases in suchsingularities, that we will describe jointly in the case of a supersymmetric N = xact Correlators from Conformal Ward Identities Claudio Corianò = (a)k p q (b)p + ll − q l q p k + exch. (c)l l − q k p q + + exch. Figure 6:
The complete
T JJ one-loop vertex (a) given by the sum of the 1PI contributions with triangle (b)and pinched topologies (c). i t µναβ i ( p , q ) (cid:0) k g µν − k µ k ν (cid:1) u αβ ( p . q ) (cid:0) k g µν − k µ k ν (cid:1) w αβ ( p . q ) (cid:0) p g µν − p µ p ν (cid:1) u αβ ( p . q ) (cid:0) p g µν − p µ p ν (cid:1) w αβ ( p . q ) (cid:0) q g µν − q µ q ν (cid:1) u αβ ( p . q ) (cid:0) q g µν − q µ q ν (cid:1) w αβ ( p . q ) [ p · q g µν − ( q µ p ν + p µ q ν )] u αβ ( p . q ) [ p · q g µν − ( q µ p ν + p µ q ν )] w αβ ( p . q ) (cid:0) p · q p α − p q α (cid:1) (cid:2) p β ( q µ p ν + p µ q ν ) − p · q ( g βν p µ + g β µ p ν ) (cid:3) (cid:0) p · q q β − q p β (cid:1) (cid:2) q α ( q µ p ν + p µ q ν ) − p · q ( g αν q µ + g αµ q ν ) (cid:3) (cid:0) p · q p α − p q α (cid:1) (cid:2) q β q µ q ν − q ( g βν q µ + g β µ q ν ) (cid:3) (cid:0) p · q q β − q p β (cid:1) (cid:2) p α p µ p ν − p ( g αν p µ + g αµ p ν ) (cid:3) (cid:0) p µ q ν + p ν q µ (cid:1) g αβ + p · q (cid:0) g αν g β µ + g αµ g βν (cid:1) − g µν u αβ − (cid:0) g βν p µ + g β µ p ν (cid:1) q α − (cid:0) g αν q µ + g αµ q ν (cid:1) p β Table 1:
The basis of 13 fourth rank tensors satisfying the vector current conservation on the external lineswith momenta p and q . appear in a single anomaly multiplet. Before coming to that point, it is convenient to summarizesome basic findings concerning the structure of the 1PI anomaly action in the conformal case,which allow to extend the considerations presented before in the AVV diagram to the new case.Once we move to the analysis of the conformal anomaly, the diagrammatic expansion induces at 1-loop level a trace contribution, identified at lowest order in the gravitational coupling in correlatorswith a single insertion of a stress-energy tensor. The T JJ vertex, in QED, describes the coupling ofa graviton to two photons and provides the simplest realization of this phenomenon. The correlatoris shown in Fig. 7 for QED.The anatomy of this vertex is due to Giannotti and Mottola [7], who have classified its possibletensor structures in terms of 13 form factors. On this basis, which is built by imposing on the
T JJ vertex all the Ward identities derived from diffeomorphism invariance, gauge invariance and Bosesymmetry, the original 43 tensor structures can be reduced to this smaller number (see also the23 xact Correlators from Conformal Ward Identities
Claudio Corianò = (a)k p q (b)p + ll − q l q p k + exch. (c)l l − q k p q + + exch. Figure 7:
The complete
T JJ one-loop vertex (a) given by the sum of the 1PI contributions with triangle (b)and pinched topologies (c). discussion in [15]. We report them in Table 1. The vertex can be written as Γ µ ν µ µ ( p , p ) = ∑ i = F i ( s ; s , s , ) t µ ν µ µ i ( p , p ) , (8.1)where the invariant amplitudes F i are functions of the kinematic invariants s = p = ( p + p ) , s = p , s = p , and the t µ ν µ µ i define the basis of the independent tensor structures.The set of the 13 tensors t i is linearly independent for generic k , p , q different from zero.Five of the 13 are Bose symmetric, t µναβ i ( p , q ) = t µνβα i ( q , p ) , i = , , , , , (8.2)while the remaining eight tensors are Bose symmetric pairwise t µναβ ( p , q ) = t µνβα ( q , p ) , (8.3) t µναβ ( p , q ) = t µνβα ( q , p ) , (8.4) t µναβ ( p , q ) = t µνβα ( q , p ) , (8.5) t µναβ ( p , q ) = t µνβα ( q , p ) . (8.6)In the set are present two tensor structures u αβ ( p , q ) ≡ ( p · q ) g αβ − q α p β , (8.7) w αβ ( p , q ) ≡ p q g αβ + ( p · q ) p α q β − q p α p β − p q α q β , (8.8)which appear in t and t respectively. Each of them satisfies the Bose symmetry requirement, u αβ ( p , q ) = u βα ( q , p ) , (8.9) w αβ ( p , q ) = w βα ( q , p ) , (8.10)and vector current conservation, p α u αβ ( p , q ) = = q β u αβ ( p , q ) , (8.11) p α w αβ ( p , q ) = = q β w αβ ( p , q ) . (8.12)24 xact Correlators from Conformal Ward Identities Claudio Corianò
They are obtained from the variation of gauge invariant quantities F µν F µν and ( ∂ µ F µλ )( ∂ ν F νλ ) u αβ ( p , q ) = − (cid:90) d x (cid:90) d y e ip · x + iq · y δ { F µν F µν ( ) } δ A α ( x ) A β ( y ) , (8.13) w αβ ( p , q ) = (cid:90) d x (cid:90) d y e ip · x + iq · y δ { ∂ µ F µλ ∂ ν F νλ ( ) } δ A α ( x ) A β ( y ) . (8.14)All the t i ’s are transverse in their photon indices q α t µναβ i = p β t µναβ i = . (8.15) t . . . t are traceless, t and t have trace parts in d =
4. With this decomposition, the two vectorWard identities on the photon lines are automatically satisfied by all the amplitudes, as well as theBose symmetry.Diffeomorphism invariance, instead, is automatically satisfied (separately) by the two tensor struc-tures t and t , which are completely transverse, while it has to be imposed on the second set( t . . . t ).In this way it is possible to extract from the 9 traceless tensor structures a (completely) transverseand traceless set of 5 amplitudes, two of them related by the bosonic symmetry.To summarize, from the original 13 tensor structures t i , split into a set of two transverse and tracecomponents and a remaining set of 11 partially transverse but traceless ones (in d = A i ’s introduced in a completely independent reconstruction method [27], based on the so-lutions of the CWI’s, with no reference to the perturbative expansion. We will re-investigate thisdifferent decomposition in a following section once we turn to the analysis of the CWI’s of 3-pointfunctions for the same TJJ vertex and for the T T T .The F i ’s are functions of the kinematical invariants s = k = ( p + q ) , s = p , s = q and of theinternal mass m . Explicit expressions of these form factors are given in [15]. Also in the massivecase, as already pointed out, in DR one finds a neat separation between the anomaly contributionand the mass dependent corrections.In the massless case only few form factors survive and one gets F ( s , , , ) = − e π s , (8.16) F ( s , , , ) = F ( s , , , ) = − e π s , (8.17) F ( s , , , ) = − F ( s , , , ) , (8.18) F , R ( s , , , ) = − e π (cid:20)
12 log (cid:18) − s µ (cid:19) − (cid:21) , (8.19)where F R is the only renormalized form factor ( F ) of the entire amplitude.The anomaly is entirely given by F , which indeed shows the presence of an anomaly pole . Furtherdetails on the organization of the effective action mediated by the trace anomaly can be found in25 xact Correlators from Conformal Ward Identities Claudio Corianò [15]. Perturbative investigations of this correlator have shown that the pole contribution is describedin the 1PI effective action by the term S pole = − e π (cid:90) d xd y (cid:0) (cid:3) h ( x ) − ∂ µ ∂ ν h µν ( x ) (cid:1) (cid:3) − xy F αβ ( x ) F αβ ( y ) , (8.20)where R ( ) = (cid:3) h ( x ) − ∂ µ ∂ ν h µν ( x ) (8.21)is the linearized curvature and the background metric has been expanded as at first order in itsfluctuations h µν . A similar phenomenon, in perturbation theory, occurs in each gauge invariant sector of thenon-abelian
TVV correlator, as shown in the case of QCD.In fact, coming to QCD, the on-shell vertex (the two gluons are taken on-shell), which is the sumof the quark and pure gauge contributions, can be decomposed by using three appropriate tensorstructures φ µναβ i , given in [4]. The anomaly pole appears in the expansion of quark ( Γ µναβ q ( p , q ) )and gluon ( Γ µναβ g ( p , q ) ) subsets of diagrams Γ µναβ ( p , q ) = Γ µναβ g ( p , q ) + Γ µναβ q ( p , q ) = ∑ i = Φ i ( s , , ) δ ab φ µναβ i ( p , q ) , (8.22)with form factors defined as Φ i ( s , , ) = Φ i , g ( s , , ) + n f ∑ j = Φ i , q ( s , , , m j ) , (8.23)where the sum runs over the n f quark flavors. They are given by Φ ( s , , ) = − g π s ( n f − C A ) + g π n f ∑ i = m i (cid:26) s − s C ( s , , , m i ) (cid:20) − m i s (cid:21)(cid:27) , (8.24) Φ ( s , , ) = − g π s ( n f − C A ) − g π n f ∑ i = m i (cid:26) s + s D ( s , , , m i ) + s C ( s , , , m i ) (cid:20) + m i s (cid:21) (cid:27) , (8.25) Φ ( s , , ) = g π ( n f − C A ) − g C A π (cid:20) B MS ( s , ) − B MS ( , ) + s C ( s , , , ) (cid:21) + g π n f ∑ i = (cid:26) B MS ( s , m i ) + m i (cid:20) s + s D ( s , , , m i ) + C ( s , , , m i ) (cid:20) + m i s (cid:21) (cid:21)(cid:27) , (8.26)with C A = N C . The scalar integrals B MS , D and C are defined in [4]. Notice the appearance in thetotal amplitude of the 1 / s pole in Φ , which is present both in the quark and in the gluon sectors,and which saturates the contribution to the trace anomaly in the massless limit. In this case theentire trace anomaly is just proportional to this component, which becomes Φ ( s , , ) = − g π s ( n f − C A ) . (8.27)26 xact Correlators from Conformal Ward Identities Claudio Corianò
Further elaborations show that the effective action takes the form S pole = − c (cid:90) d x d y R ( ) ( x ) (cid:3) − ( x , y ) F a αβ F a αβ = g π (cid:18) − C A + n f (cid:19) (cid:90) d x d y R ( ) ( x ) (cid:3) − ( x , y ) F a αβ F a αβ (8.28)and is in agreement with the same action derived from the nonlocal gravitational action proposedby Riegert long ago. Here R ( ) denotes the linearized expression of the Ricci scalar R ( ) x ≡ ∂ x µ ∂ x ν h µν − (cid:3) h , h = η µν h µν (8.29)and the constant c is related to the non-abelian β function as c = − β ( g ) g . (8.30)The presence of such effective scalar interactions mediated by the anomaly diagrams show thatonly one component of an entire anomaly vertex is responsible for the generation of the anomaly,which, obviously, cannot be isolated from the entire vertex. It is not an artificial component intro-duced by a specific decomposition of the same vertex, but it is simply the signature of the samevertex.We are going to see conclusively, at least from the point of perturbation theory, that such interac-tions are generated by a region in the loop integration where the two intermediate particle emerg-ing from the stress-energy tensor T (or from the axial- vector current J A , in the chiral case), movecollinearly before reaching the final state, made of two gluons, two photons or, more generally, twostress-energy tensors, and describe a pairing. This pairing is what is identified Fig. 1 and it is de-scribed by a spectral density whose support is on a single point in phase space, being proportionalto δ ( s ) , with s being the invariant mass of the lines on which T or J A are inserted.A similar pairing has been noticed in the case of topological insulators [28] and of Weyl semimet-als [29], which are materials in which chiral and conformal anomalies play a significant role [30].
9. Non-perturbative solutions of the TJJ and TTT from the CWI’s
The same
T JJ vertex andt its anomaly can be completely determined, up to few constants,in a completely independent way just by solving the CWI’s of a general CFT in d =
4. The goalof this section will be to show how the results coming from the perturbative analysis and the per-turbative approach can be merged completely, bringing to conclusive evidence that such masslessinteractions are genuinely present in any anomaly vertex. A similar result, in fact, will be shownto hold also in the
T T T case. The result is also in agreement with the prediction coming from anonlocal version of the conformal anomaly action given in Eq. (5.17), recently discussed in [31],which accounts for the same anomaly structure which we are going to discuss here.
An independent analysis of the features described above requires a momentum space approachin the solution of the CWI’s. These have been discussed in several papers [32] [33] and several27 xact Correlators from Conformal Ward Identities
Claudio Corianò comparisons against the free field theory realizations of the same theories have been discussedin [34–36]. A systematic discussion of how to move to momentum space from coordinate space inthe analysis of scalar and tensor correlators in CFT’s can be found in [35]. Here we just recall thatthe dilatation and the special conformal transformations, in momentum space take the forms (cid:32) n ∑ j = ∆ j − ( n − ) d − n − ∑ j = p α j ∂∂ p α j (cid:33) Φ ( p , . . . p n − , ¯ p n ) = . (9.1) n − ∑ j = (cid:32) p κ j ∂ ∂ p α j ∂ p α j + ( ∆ j − d ) ∂∂ p κ j − p α j ∂ ∂ p κ j ∂ p α j (cid:33) Φ ( p , . . . p n − , ¯ p n ) = . (9.2)The latter is the momentum space version of the differential constraint n ∑ j = (cid:32) − x j ∂∂ x κ j + x κ j x α j ∂∂ x α j + ∆ j x κ j (cid:33) Φ ( x , x , . . . , x n ) = . (9.3)Once we move to momentum space, one needs to select the independent momenta, which in ourconventions will be the first n −
1, with the n − th as a dependent one ¯ p n = − ( p + . . . p n − ) . The simplest case, investigated in the conformal approach is the Φ = (cid:104) OOO (cid:105) , correspondingto the scalar 3-point function. Here we will sketch the derivation of the equations in this simplercase. We refer to [32] for more details.In the case of a scalar correlator all the anomalous conformal WI’s can be re-expressed in scalarform by taking as independent momenta the magnitude p i = (cid:113) p i as the three independent vari-ables, the dilatation equation becomes (cid:32) ∆ − d − ∑ i = p i ∂∂ p i (cid:33) Φ ( p , p , ¯ p ) = . (9.4)The relation above is derived using the chain rule ∂ Φ ∂ p µ i = p µ i p i ∂ Φ ∂ p i − ¯ p µ p ∂ Φ ∂ p , (9.5)where ∆ is the sum of of the scaling dimensions of each operator in the 3-point function. It isa straightforward but lengthy computation to show that the special (non anomalous) conformaltransformation in d dimensions takes the form, for the scalar component K scalar κ Φ = K κ scalar = ∑ i = p κ i K i (9.7) K i ≡ ∂ ∂ p i ∂ p i + d + − ∆ i p i ∂∂ p i (9.8)28 xact Correlators from Conformal Ward Identities Claudio Corianò with the expression (9.7) which can be split into the two independent equations ∂ Φ ∂ p i ∂ p i + p i ∂ Φ ∂ p i ( d + − ∆ ) − ∂ Φ ∂ p ∂ p − p ∂ Φ ∂ p ( d + − ∆ ) = i = , . (9.9)The expressions of K κ as given in (9.8) has been first defined in [33]. Similar results have beenobtained in [32] using a direct change of variables that reduces the special CWI to a hypergeometricsystem of equations.In the scalar case, defining K i j ≡ K i − K j (9.10)Eqs. (9.9) take the homogeneous form K κ Φ = K κ Φ = . (9.11)The solutions of such equations and their reduction to hypergeometric forms is obtained by theansatz Φ ( p , p , p ) = p ∆ − d x a y b F ( x , y ) (9.12)with x = p p and y = p p . Here we are taking p as "pivot" in the expansion, but we couldequivalently choose any of the 3 momentum invariants. Φ is required to be homogenous of degree ∆ − d under a scale transformation, according to (9.4), and in (9.12) this is taken into account bythe factor p ∆ − d . The use of the scale invariant variables x and y leads to the hypergeometric formof the solution. One obtains K φ = p ∆ − d − x a y b (cid:18) x ( − x ) ∂∂ x ∂ x + ( Ax + γ ) ∂∂ x − xy ∂ ∂ x ∂ y − y ∂ ∂ y ∂ y + Dy ∂∂ y + ( E + Gx ) (cid:19) × F ( x , y ) = A = D = ∆ + ∆ − − a − b − d γ ( a ) = a + d − ∆ + G = a ( d + a − ∆ ) E = − ( a + b + d − ∆ − ∆ − ∆ )( a + b + d − ∆ − ∆ + ∆ ) . (9.14)Similar constraints are obtained from the equation K Φ =
0, with the obvious exchanges ( a , b , x , y ) → ( b , a , y , x ) K φ = p ∆ − d − x a y b (cid:18) y ( − y ) ∂∂ y ∂ y + ( A (cid:48) y + γ (cid:48) ) ∂∂ y − xy ∂ ∂ x ∂ y − x ∂ ∂ x ∂ x + D (cid:48) x ∂∂ x + ( E (cid:48) + G (cid:48) y ) (cid:19) × F ( x , y ) = A (cid:48) = D (cid:48) = A γ (cid:48) ( b ) = b + d − ∆ + G (cid:48) = b ( d + b − ∆ ) E (cid:48) = E . (9.16)29 xact Correlators from Conformal Ward Identities Claudio Corianò
The exponents a and b , as shown in [35], are exactly those that allow to remove the 1 / x and 1 / y terms in (9.15), which implies that a = ≡ a or a = ∆ − d ≡ a . (9.17)From the equation K Φ = b by setting G (cid:48) / y =
0, thereby fixingthe two remaining indices b = ≡ b or b = ∆ − d ≡ b . (9.18)The four independent solutions of the CWI’s will all be characterised by the same 4 pairs of indices ( a i , b j ) ( i , j = , ) . Setting α ( a , b ) = a + b + d − ( ∆ + ∆ − ∆ ) β ( a , b ) = a + b + d − ( ∆ + ∆ + ∆ ) (9.19)then E = E (cid:48) = − α ( a , b ) β ( a , b ) A = D = A (cid:48) = D (cid:48) = − ( α ( a , b ) + β ( a , b ) + ) , (9.20)the solutions take the form F ( α ( a , b ) , β ( a , b ) ; γ ( a ) , γ (cid:48) ( b ) ; x , y ) = ∞ ∑ i = ∞ ∑ j = ( α ( a , b ) , i + j ) ( β ( a , b ) , i + j )( γ ( a ) , i ) ( γ (cid:48) ( b ) , j ) x i i ! y j j ! (9.21)where ( α , i ) = Γ ( α + i ) / Γ ( α ) is the Pochammer symbol. We will refer to α . . . γ (cid:48) as to the first, . . . ,fourth parameters of F .The 4 independent solutions are then all of the form x a y b F , where the hypergeometric functionswill take some specific values for its parameters, with a and b fixed by (9.17) and (9.18). Specifi-cally we have Φ ( p , p , p ) = p ∆ − d ∑ a , b c ( a , b ,(cid:126) ∆ ) x a y b F ( α ( a , b ) , β ( a , b ) ; γ ( a ) , γ (cid:48) ( b ) ; x , y ) (9.22)where the sum runs over the four values a i , b i i = , c ( a , b ,(cid:126) ∆ ) , with (cid:126) ∆ = ( ∆ , ∆ , ∆ ) . Eq. (9.22) is a very compact way to write down the solution, which can be mademore explicit. For this reason it is convenient to define α ≡ α ( a , b ) = d − ∆ + ∆ − ∆ , β ≡ β ( b ) = d − ∆ + ∆ + ∆ , γ ≡ γ ( a ) = d + − ∆ , γ (cid:48) ≡ γ ( b ) = d + − ∆ (9.23)as the 4 basic (fixed) hypergeometric parameters. All the remaining solutions are defined by shiftswith respect to these. The 4 independent solutions can be re-expressed in terms of the parametersabove as S ( α , β ; γ , γ (cid:48) ; x , y ) ≡ F ( α , β ; γ , γ (cid:48) ; x , y ) = ∞ ∑ i = ∞ ∑ j = ( α , i + j ) ( β , i + j )( γ , i ) ( γ (cid:48) , j ) x i i ! y j j ! (9.24)30 xact Correlators from Conformal Ward Identities Claudio Corianò and S ( α , β ; γ , γ (cid:48) ; x , y ) = x − γ F ( α − γ + , β − γ +
1; 2 − γ , γ (cid:48) ; x , y ) , S ( α , β ; γ , γ (cid:48) ; x , y ) = y − γ (cid:48) F ( α − γ (cid:48) + , β − γ (cid:48) + γ , − γ (cid:48) ; x , y ) , S ( α , β ; γ , γ (cid:48) ; x , y ) = x − γ y − γ (cid:48) F ( α − γ − γ (cid:48) + , β − γ − γ (cid:48) +
2; 2 − γ , − γ (cid:48) ; x , y ) . Notice that in the scalar case, one is allowed to impose the complete symmetry of the correlatorunder the exchange of the 3 external momenta and scaling dimensions, as discussed in [32]. Thisreduces the four constants to just one.We are going first to extend this analysis to the case of the A − A form factors of the T JJ . In-terestingly, this approach can be generalized to 4-point functions, showing that at large energy andmomentum transfers, scatterings at fixed angle are characterized by similar solutions, with somegeneralizations, taking to Lauricella’s functions [37].
T JJ and the matching to perturbation theory
The solution in the case of the
T JJ is far more involved compared to the scalar case and itrequires a completely new approach. A way to solve the CWI’s for tensor correlators has beenformulated in [27]. In order to establish a link between the perturbative approach of the previoussections and the non-perturbative one based on [27], we start by stating the special CWI satisfiedby this correlator ∑ j = (cid:34) ( ∆ j − d ) ∂∂ p κ j − p α j ∂∂ p α j ∂∂ p κ j + ( p j ) κ ∂∂ p α j ∂∂ p j α (cid:35) (cid:104) T µ ν ( p ) J µ ( p ) J µ ( ¯ p ) (cid:105) + (cid:32) δ κ ( µ ∂∂ p α − δ κα δ λ ( µ ∂∂ p λ (cid:33) (cid:104) T ν ) α ( p ) J µ ( p ) J µ ( ¯ p ) (cid:105) + (cid:32) δ κµ ∂∂ p α − δ κα δ λ µ ∂∂ p λ (cid:33) (cid:104) T µ ν ( p ) J α ( p ) J µ ( ¯ p ) (cid:105) = . (9.25)The first line in the equation above corresponds to the scalar action K κ , while the second andthe third lines are the contributions coming from the spin parts. The correlator is decomposed intoits transverse traceless ( (cid:104) t µ ν j µ j µ (cid:105) ) and longitudinal (local) parts in the form (cid:104) T µ ν J µ J µ (cid:105) = (cid:104) t µ ν j µ j µ (cid:105) + (cid:104) T µ ν J µ j µ loc (cid:105) + (cid:104) T µ ν j µ loc J µ (cid:105) + (cid:104) t µ ν loc J µ J µ (cid:105)− (cid:104) T µ ν j µ loc j µ loc (cid:105) − (cid:104) t µ ν loc j µ loc J µ (cid:105) − (cid:104) t µ ν loc J µ j µ loc (cid:105) + (cid:104) t µ ν loc j µ loc j µ loc (cid:105) . (9.26)with T µν = t µν + t µν loc (9.27)with the longitudinal/trace terms given by t µν loc ( p ) = p µ p Q ν + p ν p Q µ − p µ p ν p Q + π µν d − ( T − Qp ) = Σ µναβ T αβ Q µ = p ν T µν , T = δ µν T µν , Q = p ν p µ T µν . (9.28)31 xact Correlators from Conformal Ward Identities Claudio Corianò
The decompositon above allows to separate the equations into primary and secondary constraints,the secondary ones involving equations with the inclusion of two-point functions.At the core of the decomposition there is the transverse traceless sector, which is parameterized bya minimal set of form factors, as proposed in [27].It is possible to show that these amplitudes are in a one-to-one correspondence with the formfactors A j ( j = , . . . ) introduced in the parameterization of the T JJ correlator presented in [27].In that work the full 3-point function is parameterized in terms of transverse (with respect to all theexternal momenta) traceless components plus extra terms identified via longitudinal Ward identitiesof the
T JJ (the so-called local or semi local ) characterised by pinched topologies (cid:104)(cid:104) T µ ν J µ J µ (cid:105)(cid:105) = (cid:104)(cid:104) t µ ν j µ j µ (cid:105)(cid:105) + local terms . (9.29)Here we have switched to a symmetric notation for the external momenta, with ( p , p , p ) ≡ ( k , p , q ) , and with the transverse traceless parts expanded in terms of a set of the form factors A j mentioned above (cid:104) t µ ν ( p ) j µ ( p ) j µ ( p ) (cid:105) = Π µ ν α β π µ α π µ α (cid:16) A p α p β p α p α + A δ α α p α p β + A δ α α p β p α + A ( p ↔ p ) δ α α p β p α + A δ α α δ α β (cid:17) . (9.30)The equation above provides the basic decomposition of the T JJ in terms of a minimal set of formfactors which can be mapped into the set of the F (cid:48) i s presented in the previous sections. We are goingto show that the 1 / k behaviour found in the anomaly form factor ( F ) is in agreement with theexplicit expressions obtained from the solutions of the CWI’s, which fix the A i . But before comingto this point, we briefly comment on the form of the equations obtained.In this expression Π µ ν α β is a transverse and traceless projector built out of momentum p , while π µ α and π µ α denote transverse projectors with respect to the momenta p and p .Coming to the explicit form of the A j , they are extracted from the solution of the scalar CWI’s0 = ˜ C = K A = ˜ C = K A − A = ˜ C = K A = ˜ C = K A ( p ↔ p ) + A = ˜ C = K A + A = ˜ C = K A = ˜ C = K A − A = ˜ C = K A + A = ˜ C = K A ( p ↔ p ) = ˜ C = K A + A ( p ↔ p ) , (9.31)which can be obtained in two ways. The approach of [27, 38, 39] expresses such solutions in termsof 3-K integrals, i.e. integrals of three Bessel functions, and can be related to specific combinationsof hypergeometric functions F , also known as Appell functions. An equivalent method can beformulated which allows to work out the explicit form of the solutions using properties of thefunctions F [35]. The two approaches have been combined, more recently, in the analysis of scalar4-point functions, with the introduction of 4K integrals [37]. F and the A bases We have taken into consideration two separate bases in the analysis of the
T JJ . In the F basisonly one form factors needs to be renormalized, which is F , by dimensional counting. We can32 xact Correlators from Conformal Ward Identities Claudio Corianò show that the renormalization of F is responsible for the emergence of a 1 / k pole in the insertionof the T operator. Also, the singularities of F are mirrored by the singular behaviour of the A i form factors which contain such form factor, following the map shown below in Eq. (9.33).Let’s see how these points can be proven, following the discussion of [34].The T JJ correlator in QED is conformal in d dimension, with finite form factors which arenot affected by the conformal anomaly, being dimensionally regulated. We can use the F -basis ofthe 13 form factors F i introduced before and tensor structures t i to parameterize them.Notice that the separation of these 13 structures into trace-free and trace parts is valid only in d = t , t , t and t , which remain traceless in d dimensions.conctractions with the metric tensor are, at this point, performed in d dimensions with a metric g µν ( d ) . The 4-dimensional metric, instead, will be denoted as g µν ( ) .For example, a contraction of t and t in d- dimensions will give g µν ( d ) t µναβ = ( d − ) k u αβ ( p , q ) g µν ( d ) t µναβ = ( d − ) k w αβ ( p , q ) , (9.32)and similarly for all the other structures, except for those mentioned above, which are trace-free inany dimensions.Using the completeness of the F -basis we can identify the mapping between the form factors ofsuch basis and those of the A -basis which parameterize the transverse-traceless contributions inthe reconstruction method of [27]. They are conveniently expressed in terms of the momenta ( k , p , q ) ≡ ( p , p , p ) in the form A = ( F − F − F ) − p F − p F A = ( p − p − p )( F − F − F ) − p p ( F − F + F ) − F A = p ( p − p − p ) F − p p F − F A = ( p − p − p ) F , (9.33)which are transverse and traceless, with A , A and A symmetric. T JJ anomaly pole from renormalization
Starting from d -dimension and using the F -basis, we require that this correlator has no trace(i.e. be anomaly free) in d dimensions. The anomaly will emerge in dimensional regularization aswe take the d → F = ( d − ) p ( d − ) (cid:2) F − p F − p F − p · p F (cid:3) (9.34)and F = ( d − ) p ( d − ) (cid:2) p F + p F + p · p F (cid:3) . (9.35)33 xact Correlators from Conformal Ward Identities Claudio Corianò
Both equations are crucial in order to understand the way the renormalization procedure works forsuch correlator. From Eq. (9.35) it is clear that by sending d → F vanishes, F = ε ( d − ) p (cid:2) p F + p F + p · p F (cid:3) → , (9.36)since all the form factors F , F and F are finite for dimensional reasons, and therefore F is indeedzero in this limit, since the right-hand side of (9.35) has no poles in ε ≡ d − d = F − basis with 4 independentcombinations of form factors from the original seven, given in (9.33), which are sufficient to de-scribe the complete transverse traceless sector of the theory, plus an additional form factor F .Therefore, by taking the d → F − set contains only one single tensor structure of nonzerotrace and associated form factor, which should account for the anomaly in d =
4. This result is ob-viously confirmed in perturbation theory in QED [15].As already mentioned, F is the only form factor that needs to be renormalized in the F -set andit is characterized by the appearance of a single pole in 1 / ε in dimensional regularization. Thefact that such singularity will be at all orders of the form 1 / ε and not higher is a crucial ingredientin the entire construction, and is due to conformal symmetry. Such assumption is consistent withthe analysis in conformal field theory since the only available counterterm to regulate the theory isgiven by 1 ε (cid:90) d x √ gF µν F µν (9.37)which renormalizes the 2-point function (cid:104) JJ (cid:105) and henceforth F . An explicit computations in QEDgives the result F = G ( p , p , p ) − [ Π ( p ) + Π ( p )] (9.38)with G being a lengthy expression which remains finite as d →
4. Therefore, the origin of the sin-gularity is traced back to the scalar form factor Π ( p ) of the photon 2-point function. Concerningthe fact that the singularity in Π ( p ) stops at 1 / ε , we just recall that the structure of the two-pointfunction of two conserved vector currents of scaling dimensions η and η is given by [32] G αβ V ( p ) = δ η η c V π d / η − d / Γ ( d / − η ) Γ ( η ) (cid:32) η αβ − p α p β p (cid:33) ( p ) η − d / , (9.39)with c V being an arbitrary constant. It requires the two currents to share the same dimensionsand manifests only a single pole in 1 / ε . In dimensional regularization, in fact, the divergence canbe regulated with d → d − ε . Expanding the product Γ ( d / − η ) ( p ) η − d / , which appears in thetwo-point function, in a Laurent series around d / − η = − n (integer) gives the single pole in 1 / ε behaviour [32] Γ ( d / − η ) ( p ) η − d / = ( − ) n n ! (cid:18) − ε + ψ ( n + ) + O ( ε ) (cid:19) ( p ) n + ε , (9.40)where ψ ( z ) is the logarithmic derivative of the Gamma function, and ε takes into account thedivergence of the two-point correlator for particular values of the scale dimension η and of thespace-time dimension d . Therefore, the divergence in F is then given by a single pole in ε is ofthe form 34 xact Correlators from Conformal Ward Identities Claudio Corianò F = d − F + F f (9.41)In QED, for instance, one finds by an explicit computation that ¯ F = − e / ( π ) at one-loop and¯ F f is finite [15] and gets renormalized into F R only in its photon self-energy contributions [15] ( s = p , s = p , s = p ) F , R ( s ; s , s , ) = − [ Π R ( s , ) + Π R ( s , )] + G ( s , s , s ) (9.42)with Π R ( s , ) = − e π (cid:20) − log (cid:18) − s µ (cid:19)(cid:21) , (9.43)denoting the renormalized scalar form factor of the JJ correlator at one-loop and with G implicitlydefined in (9.38).Inserting (9.41) into (9.34) we obtain F = ( d − ) p ( d − ) (cid:18) d − F + F f − p F − p F − p · p F (cid:19) , (9.44)which in the d → F = ¯ F p (9.45)and specifically, in QED F = − e π s , (9.46) ( s ≡ k ) showing that the anomaly pole in F is indeed generated by the renormalization of thesingle divergent form factor F . In the case of QED, the relation between the prefactor in frontof the 1 / s pole and its relation to the QED β -function has been extensively discussed in [7, 15],to which we refer for further details. In performing the limit we have used the finiteness of theremaining form factors.The analysis presented above proves the consistency of the conjecture concerning the origin ofthe anomaly pole which is attributed to the renormalization of the T JJ correlator as we reach thephysical dimensions. At the same time, the agreement with perturbation theory, in QED, holdsonly at 1-loop, where the theory is conformal. But this is sufficient to establish a link between ageneral CFT result and a specific perturbative realization of a given correlator in free field theory.The consistency between the results obtained in the F basis and the transverse-traceless one of the A i can be easily figured out from a cursory look at (9.33). One can show that the singularities ofthe A i are in direct correspondence with the presence of F in their relations to the F basis. Forinstance A , A and A need to be renormalized, while A does not.35 xact Correlators from Conformal Ward Identities Claudio Corianò p p p l p l + p lT µ ⌫ T µ ⌫ T µ ⌫ p p p l + p lT µ ⌫ T µ ⌫ T µ ⌫ p p p l + p lT µ ⌫ T µ ⌫ T µ ⌫ p p p l p l + p lT µ ⌫ T µ ⌫ T µ ⌫ p p p l + p lT µ ⌫ T µ ⌫ T µ ⌫ p p p l + p lT µ ⌫ T µ ⌫ T µ ⌫ p p p l p l + p lT µ ⌫ T µ ⌫ T µ ⌫ p p p l + p lT µ ⌫ T µ ⌫ T µ ⌫ p p p l + p lT µ ⌫ T µ ⌫ T µ ⌫ Figure 8:
Typical one-loop scalar diagrams for the three-graviton vertex.
10. The 3-graviton vertex from the CWI’s
As we move to the
T T T vertex, the analysis gets more involved but it remains substantiallyunchanged. Similarly to the
T JJ , the longitudinal/transverse-traceless decomposition takes theform (cid:104) T µ ν T µ ν T µ ν (cid:105) = (cid:104) t µ ν t µ ν t µ ν (cid:105) + (cid:104) t µ ν loc t µ ν t µ ν (cid:105) + (cid:104) t µ ν t µ ν loc t µ ν (cid:105) + (cid:104) t µ ν t µ ν t µ ν loc (cid:105) + (cid:104) t µ ν loc t µ ν loc t µ ν (cid:105) + (cid:104) t µ ν loc t µ ν loc t µ ν (cid:105) + (cid:104) t µ ν t µ ν loc t µ ν loc (cid:105) + (cid:104) t µ ν loc t µ ν loc t µ ν loc (cid:105) (10.1)with the (cid:104) t µ ν t µ ν t µ ν (cid:105) parameterized by five form factors ( A i ) in their transverse traceless sector.All the primary CWI’s can be solved in terms of the general 3 K integral I α { β β β } ( p , p , p ) = (cid:90) ∞ dx x α ∏ j = p β j j K β j ( p j x ) , (10.2)where K ν is a Bessel K function. This integral depends on four parameters, namely the power α ofthe integration variable x , and the three Bessel function indices β j . α = d − + N , β j = ∆ j − d + k j , j = , , . (10.3)Here we assume that we concentrate on some particular 3-point function and the conformal dimen-sions ∆ j , j = , , J N { k j } = I d − + N { ∆ j − d + k j } , (10.4)with { k j } = { k , k , k } , the solutions are expressed in the form [27, 39] A = α J { } , (10.5) A = α J { } + α J { } , (10.6) A = α J { } + α J { } + α J { } , (10.7) A = α J { } − α J { } + α J { } , (10.8) A = α J { } + α (cid:0) J { } + J { } + J { } (cid:1) + α J { } (10.9)36 xact Correlators from Conformal Ward Identities Claudio Corianò in terms of 3K integrals. One important issue is how to relate these explicit solutions to free-fieldtheory.
We can show that such solutions can be perfectly matched in perturbation theory by introduc-ing three independent sectors, a scalar, a fermion and spin one [36].Also in this case, as for the
T JJ , one performs a parallel analysis of the decomposition of the cor-relator in free field theory using each such three sectors. In d = (cid:104) T µ ν ( p ) T µ ν ( p ) T µ ν ( p ) (cid:105) = ∑ I = F , G , S n I (cid:104) T µ ν ( p ) T µ ν ( p ) T µ ν ( p ) (cid:105) I . (10.10)For the triangle V and contact topologies W , the latter take the form (cid:104) t µ ν ( p ) t µ ν ( p ) t µ ν ( p ) (cid:105) I = Π µ ν α β ( p ) Π µ ν α β ( p ) Π µ ν α β ( p ) × (cid:20) − V α β α β α β I ( p , p , p ) + ∑ i = W α β α β α β I , i ( p , p , p ) (cid:21) (10.11)where we have included all the three sectors ( I ) . Both the longitudinal and the transverse sectorsare characterized by divergences in the forms of single poles in 1 / ε ( ε = ( − d ) /
2) which needsto be removed by renormalization. The singular contributions of the A i in DR take the form A Div = π ε (cid:2) n G − n F − n S (cid:3) (10.12a) A Div = π ε (cid:2) ( s + s ) (cid:0) n F + n S + n G (cid:1) + s ( n F + n G + n S ) (cid:3) (10.12b) A Div = π ε (cid:2) ( s + s ) (cid:0) n F + n G + n S (cid:1) + s ( n F + n G + n S ) (cid:3) (10.12c) A Div = π ε (cid:26) n F (cid:0) s − s ( s + s ) + s − s s + s (cid:1) + (cid:2) n G (cid:0) s + s ( s + s ) + s + s s + s (cid:1) + n S (cid:0) s − s ( s + s ) + s − s s + s (cid:1) (cid:3)(cid:27) (10.12d)and are renormalized by the addition of two counterterms in the defining Lagrangian. In perturba-tion theory the one loop counterterm Lagrangian is S count = − ε ∑ I = F , S , G n I (cid:90) d d x √− g (cid:18) β a ( I ) C + β b ( I ) E (cid:19) (10.13)corresponding to the Weyl tensor squared and the Euler density, omitting the extra R operatorwhich is responsible for the (cid:3) R term in (4.13), having choosen the local part of anomaly ( ∼ β c (cid:3) R ) xact Correlators from Conformal Ward Identities Claudio Corianò vanishing ( β c = (cid:104) T µ ν ( p ) T µ ν ( p ) T µ ν ( p ) (cid:105) count == − ε ∑ I = F , S , G n I (cid:18) β a ( I ) V µ ν µ ν µ ν C ( p , p , p ) + β b ( I ) V µ ν µ ν µ ν E ( p , p , p ) (cid:19) (10.14)and can be separated in their transverse-traceless components and their longitudinal ones. The A i ’sare renormalized by the first, while the second renormalize the longitudinal components of the T T T . The approach has been detailed in [40]. The method consists in taking the transverse trace-less projectors (with (open indices) as d -dimensional tensors and in expanding their d-dependenceas ( d − )+
4, generating new tensors which are transverse traceless only in d =
4, plus a remainder.All the index contractions are performed with a d-dimensional metric tensor, while the inclusionof the counterterms generated by (10.13) allows to remove the 1 / ε terms. At the final stage theexpression of the renormalized vertex is given only by the 4-dimensional component of such a finaltensor, which can be traced as an ordinary 4- d tensor, thereby generating an anomaly.An extensive analysis shows that the entire 4- d solution of the CWI’s A i can be reconctructedjust by a superposition of the three perturbative sectors mentioned above. In [33] the general 3Ksolution is uniquely parameterized in terms of some constants α , α and c T . If we choose, forinstance α = π ( n S − n F ) , α = π n F , c T = π / ( n S + n F ) , c g = d =
3, we can match perturbative and non-perturbative results in the same dimension. Itis then clear that, given the rather complex structure of the 3K integrals, the use of perturbationtheory and the matching allow to re-express such integrals in a very simple form, using only thescalar 2- and 3-point functions of the ordinary Feynman expansion as a basis. This implies thatthe general hypergeometric solution in d = d = d =
3, for instance, can be worked out only atnumerical level, since the procedure developed for the renormalization of such integrals is ratherinvolved. The general expressions of the A i in d =
4, valid non-perturbatively, can be found in [36].It is interesting to observe how the structure of the anomalous contributions of the correlator appearin this formulation. They are associated to terms which develop a nonvanishing single, double andtriple trace, with massless exchanges in the three separate legs of the correlator, as shown in Fig. 9,which generalizes the behaviour of Fig. 1. Such anomalous ( (cid:104) .. (cid:105) a ) terms are given by38 xact Correlators from Conformal Ward Identities Claudio Corianò
Figure 9:
The anomaly contribution in the TTT correlator due to massless exchanges on each separate leg,represented by dashed lines. (cid:104) T µ ν ( p ) T µ ν ( p ) T µ ν ( p ) (cid:105) a = ˆ π µ ν ( p ) p (cid:104) T ( p ) T µ ν ( p ) T µ ν ( p ) (cid:105) a + ˆ π µ ν ( p ) p (cid:104) T µ ν ( p ) T ( p ) T µ ν ( p ) (cid:105) a + ˆ π µ ν ( p ) p (cid:104) T µ ν ( p ) T µ ν ( p ) T ( p ) (cid:105) a − ˆ π µ ν ( p ) ˆ π µ ν ( p ) p p (cid:104) T ( p ) T ( p ) T µ ν ( p ) (cid:105) a − ˆ π µ ν ( p ) ˆ π µ ν ( p ) p p (cid:104) T µ ν ( p ) T ( p ) T ( p ) (cid:105) a − ˆ π µ ν ( p ) ˆ π µ ν ( ¯ p ) p p (cid:104) T ( p ) T µ ν ( p ) T ( p ) (cid:105) a + ˆ π µ ν ( p ) ˆ π µ ν ( p ) ˆ π µ ν ( ¯ p ) p p p (cid:104) T ( p ) T ( p ) T ( ¯ p ) a . (10.16)As shown in [31], they can be reobtained from the nonlocal anomaly action (5.17) .
11. Moving to supersymmetry: Anomalies and sum rules for the anomalysupermultiplet
The kinematical features characterizing chiral and conformal anomalies become standard oncewe turn to consider a supersymmetric context. Supersymmetry, indeed, provides a unification of thephenomenon discussed above, identified in perturbation theory, as shown in the case of an N = m . As in theprevious cases, we will adopt dimensional regularization.39 xact Correlators from Conformal Ward Identities Claudio Corianò R µ ( k ) A bβ ( q ) A aα ( p ) (a) R µ ( k ) A bβ ( q ) A aα ( p ) (b) A bβ ( q ) A aα ( p ) R µ ( k ) (c) S µA ( k ) ¯ λ b ˙ B ( q ) A aα ( p ) (d) S µA ( k ) ¯ λ b ˙ B ( q ) A aα ( p ) (e) S µA ( k ) ¯ λ b ˙ B ( q ) A aα ( p ) (f) T µν ( k ) A bβ ( q ) A aα ( p ) (g) T µν ( k ) A bβ ( q ) A aα ( p ) (h) T µν ( k ) A bβ ( q ) A aα ( p ) (i) Figure 10:
Typical contributions to the one-loop perturbative expansion of the (cid:104)
RVV (cid:105) (cid:104)
SV F (cid:105) (cid:104)
T JJ (cid:105) dia-grams
The anomaly form factors are characterized by a unique function ( χ ) for which one can write downa spectral representation in terms of a spectral density ρ ( s , m ) supported by a single branch cut χ ( k , m ) = π (cid:90) ∞ m ρ χ ( s , m ) s − k ds (11.1)corresponding to the ordinary threshold at k = m , with ρ χ ( s , m ) = i Disc χ ( s , m ) = π m s log (cid:32) + (cid:112) τ ( s , m ) − (cid:112) τ ( s , m ) (cid:33) θ ( s − m ) . (11.2)A crucial feature of these spectral densities is the existence of a sum rule, given by1 π (cid:90) ∞ m ds ρ χ ( s , m ) = , (11.3)where the right hand side has been normalized to 1. In general, it equals the anomaly. Generically,it is given in the form 1 π (cid:90) ∞ ρ ( s , m ) ds = f , (11.4)with the constant f independent of any mass (or other) parameter which characterizes the thresholdsor the strengths of the resonant states eventually present in the integration region ( s > ) .It is quite straightforward to show that Eq. (11.4) is a constraint on the asymptotic behaviour ofthe related form factor. The proof is obtained by observing that the dispersion relation for a formfactor in the spacelike region ( Q = − k > F ( Q , m ) = π (cid:90) ∞ ds ρ ( s , m ) s + Q , (11.5)40 xact Correlators from Conformal Ward Identities Claudio Corianò once we expand the denominator in Q as s + Q = Q − Q s Q + . . . and make use of Eq. (11.4),induces the following asymptotic behaviour on F ( Q , m ) lim Q → ∞ Q F ( Q , m ) = f . (11.6)The F ∼ f / Q behaviour at large Q , with f independent of m , shows the pole dominance of F for Q → ∞ . Indeed, the resonant (pole) behaviour of such spectral density is obtained in the m → m → ρ χ ( s , m ) = lim m → π m s log (cid:32) + (cid:112) τ ( s , m ) − (cid:112) τ ( s , m ) (cid:33) θ ( s − m ) = πδ ( s ) , (11.7)where ρ converges to a delta function. Clearly, it is the region around the light cone ( s ∼
0) whichdominates the sum rule and it amounts to a resonant contribution.Therefore, the presence of a 1 / Q term in the anomaly form factors is a property of the entireflow which converges to a localized massless state (i.e. ρ ( s ) ∼ δ ( s ) ) as m →
0, while the presenceof a non vanishing sum rule guarantees the validity of the asymptotic constraint illustrated in Eq.(11.6). Notice that although for conformal deformations driven by a single mass parameter theindependence of the asymptotic value f on m is a simple consequence of the scaling behaviour of F ( Q , m ) , it holds quite generally even for a completely off-shell kinematics [7]. N = theory To illustrate this result, let’s consider the Lagrangian of an N = L = − F a µν F a µν + i λ a σ µ D ab µ ¯ λ b + ( D µ i j φ j ) † ( D ik µ φ k ) + i χ j σ µ D µ † i j ¯ χ i −√ g (cid:16) ¯ λ a ¯ χ i T ai j φ j + φ † i T ai j λ a χ j (cid:17) − V ( φ , φ † ) − ( χ i χ j W i j ( φ ) + h . c . ) , (11.8)where the gauge covariant derivatives on the matter fields and on the gaugino are defined respec-tively as D µ i j = δ i j ∂ µ + igA a µ T ai j , D ac µ = δ ac ∂ µ − gt abc A b µ , (11.9)with t abc the structure constants of the adjoint representation, and the scalar potential is given by V ( φ , φ † ) = W † i ( φ † ) W i ( φ ) + g (cid:16) φ † i T ai j φ j (cid:17) . (11.10)For the derivatives of the superpotential we have been used the following definitions W i ( φ ) = ∂ W ( Φ ) ∂ Φ i (cid:12)(cid:12)(cid:12)(cid:12) , W i j ( φ ) = ∂ W ( Φ ) ∂ Φ i ∂ Φ j (cid:12)(cid:12)(cid:12)(cid:12) , (11.11)where the symbol | on the right indicates that the quantity is evaluated at θ = ¯ θ = J A ˙ A = Tr (cid:2) ¯ W ˙ A e V W A e − V (cid:3) −
13 ¯ Φ (cid:20) ← ¯ ∇ ˙ A e V ∇ A − e V ¯ D ˙ A ∇ A + ← ¯ ∇ ˙ A ← D A e V (cid:21) Φ , (11.12)41 xact Correlators from Conformal Ward Identities Claudio Corianò s Ρ Χ (cid:144) Π Figure 11:
Representatives of the family of spectral densities ρ χ ( n ) π ( s ) plotted versus s in units of m . Thefamily "flows" towards the s = δ ( s ) function as m goes to zero. where ∇ A is the gauge-covariant derivative in the superfield formalism whose action on chiralsuperfields is given by ∇ A Φ = e − V D A (cid:0) e V Φ (cid:1) , ¯ ∇ ˙ A ¯ Φ = e V ¯ D ˙ A (cid:0) e − V ¯ Φ (cid:1) . (11.13)The conservation equation for the hypercurrent J A ˙ A is¯ D ˙ A J A ˙ A = D A (cid:20) − g π ( T ( A ) − T ( R )) Tr W − γ ¯ D ( ¯ Φ e V Φ ) + (cid:18) W ( Φ ) − Φ ∂ W ( Φ ) ∂ Φ (cid:19)(cid:21) , (11.14)where γ is the anomalous dimension of the chiral superfield.The first two terms in Eq. (11.14) describe the quantum anomaly of the hypercurrent, while the lastis of classical origin and it is entirely given by the superpotential. In particular, for a classical scaleinvariant theory, in which W is cubic in the superfields or identically zero, this term identicallyvanishes. If, on the other hand, the superpotential is quadratic, the conservation equation of thehypercurrent acquires a non-zero contribution even at classical level. This describes the explicitbreaking of the conformal symmetry. By projecting the hypercurrent J A ˙ A defined in Eq.(11.12)onto its components we get the explicit expressions of the three anomaly equations.The lowest component is given by the R µ current, the θ term is associated with the supercurrent S µ A , while the θ ¯ θ component contains the energy-momentum tensor T µν . In the N = xact Correlators from Conformal Ward Identities Claudio Corianò A a α ( p ) A b β ( q ) R µ ( k ) A a α ( p )¯ λ b ˙ B ( q ) S µA ( k ) A a α ( p ) A b β ( q ) T µν ( k ) + A a α ( p ) A b β ( q ) T µν ( k ) Figure 12:
The collinear diagrams corresponding to the exchange of a composite axion (top right), a dilatino(top left) and the two sectors of an intermediate dilaton (bottom). Dashed lines denote intermediate scalars.
Yang-Mills theory described by the Lagrangian in Eq. (11.8), these three currents are defined as R µ = ¯ λ a ¯ σ µ λ a + (cid:16) − ¯ χ i ¯ σ µ χ i + i φ † i D µ i j φ j − i ( D µ i j φ j ) † φ i (cid:17) , (11.15) S µ A = i ( σ νρ σ µ ¯ λ a ) A F a νρ − √ ( σ ν ¯ σ µ χ i ) A ( D ν i j φ j ) † − i √ ( σ µ ¯ χ i ) W † i ( φ † ) − ig ( φ † i T ai j φ j )( σ µ ¯ λ a ) A + S µ I A , (11.16) T µν = − F a µρ F a ν ρ + i (cid:20) ¯ λ a ¯ σ µ ( δ ac → ∂ ν − gt abc A b ν ) λ c + ¯ λ a ¯ σ µ ( − δ ac ← ∂ ν − gt abc A b ν ) λ c + ( µ ↔ ν ) (cid:21) + ( D µ i j φ j ) † ( D ν ik φ k ) + ( D ν i j φ j ) † ( D µ ik φ k ) + i (cid:20) ¯ χ i ¯ σ µ ( δ i j → ∂ ν + igT ai j A a ν ) χ j + ¯ χ i ¯ σ µ ( − δ i j ← ∂ ν + igT ai j A a ν ) χ j + ( µ ↔ ν ) (cid:21) − η µν L + T µν I , (11.17)where L is given in Eq.(11.8) and S µ I and T µν I are the terms of improvement in d = S µ I A = √ i (cid:104) σ µν ∂ ν ( χ i φ † i ) (cid:105) A , (11.18) T µν I = (cid:0) η µν ∂ − ∂ µ ∂ ν (cid:1) φ † i φ i . (11.19)The terms of improvement are automatically conserved and guarantee, for W ( Φ ) =
0, uponusing the equations of motion, the vanishing of the classical trace of T µν and of the classicalgamma-trace of the supercurrent S µ A . The anomaly equations in the component formalism, whichcan be projected out from Eq. (11.14), are ∂ µ R µ = g π (cid:18) T ( A ) − T ( R ) (cid:19) F a µν ˜ F a µν , (11.20)¯ σ µ S µ A = − i g π (cid:18) T ( A ) − T ( R ) (cid:19) (cid:0) ¯ λ a ¯ σ µν (cid:1) A F a µν , (11.21) η µν T µν = − g π (cid:18) T ( A ) − T ( R ) (cid:19) F a µν F a µν . (11.22)43 xact Correlators from Conformal Ward Identities Claudio Corianò
The first and the last equations are respectively extracted from the imaginary and the real part ofthe θ component of Eq.(11.14), while the gamma-trace of the supercurrent comes from the lowestcomponent.We define the three correlation functions, Γ ( R ) , Γ ( S ) and Γ ( T ) as δ ab Γ µαβ ( R ) ( p , q ) ≡ (cid:104) R µ ( k ) A a α ( p ) A b β ( q ) (cid:105) (cid:104) RVV (cid:105) , δ ab Γ µα ( S ) A ˙ B ( p , q ) ≡ (cid:104) S µ A ( k ) A a α ( p ) ¯ λ b ˙ B ( q ) (cid:105) (cid:104) SV F (cid:105) , δ ab Γ µναβ ( T ) ( p , q ) ≡ (cid:104) T µν ( k ) A a α ( p ) A b β ( q ) (cid:105) (cid:104) TVV (cid:105) , (11.23)with k = p + q and where we have factorized, for the sake of simplicity, the Kronecker delta on theadjoint indices. These correlation functions have been computed at one-loop order in the dimen-sional reduction scheme (DRed) using the Γ µαβ ( R ) ( p , q ) = − i g T ( R ) π k µ k ε [ p , q , α , β ] , (11.24)The correlator in Eq.(11.24) satisfies the vector current conservation constraints and the anomalousequation of Eq.(11.20) ik µ Γ µαβ ( R ) ( p , q ) = g T ( R ) π ε [ p , q , α , β ] . (11.25)Therefore, in the on-shell case and for massless fermions we recover the usual structure of the (cid:104) AVV (cid:105) diagram. In general we obtain Γ µαβ ( R ) ( p , q ) = i g T ( A ) π k µ k ε [ p , q , α , β ] , (11.26) Γ µα ( S ) ( p , q ) = i g T ( A ) π k s µα + i g T ( A ) π V ( k ) s µα , (11.27) Γ µναβ ( T ) ( p , q ) = g T ( A ) π k t µναβ ( p , q ) + g T ( A ) π V ( k ) t µναβ ( p , q ) , (11.28)where V ( k ) = − + B ( , ) − B ( k , ) − k C ( k , ) . (11.29)and with B ( k , ) and C ( k , ) denoting the scalar 2- and 3-point functions computing in themassless case. Notice that the two vector lines are kept on-shell for simplicity. The structure of thecorrelators is then given by Γ µαβ ( R ) ( p , q ) = i g T ( R ) π Φ ( k , m ) k µ k ε [ p , q , α , β ] , (11.30) Γ µα ( S ) ( p , q ) = i g T ( R ) π k Φ ( k , m ) s µα + i g T ( R ) π Φ ( k , m ) s µα , (11.31) Γ µναβ ( T ) ( p , q ) = g T ( R ) π k Φ ( k , m ) t µναβ S ( p , q ) + g T ( R ) π Φ ( k , m ) t µναβ S ( p , q ) , (11.32)44 xact Correlators from Conformal Ward Identities Claudio Corianò with Φ ( k , m ) = − − m C ( k , m ) , Φ ( k , m ) = − B ( , m ) + B ( k , m ) + m C ( k , m ) , (11.33)and with the anomalous broken Ward identities taking the form ik µ Γ µαβ ( R ) ( p , q ) = − g T ( R ) π Φ ( k , m ) ε [ p , q , α , β ] , (11.34)¯ σ µ Γ µα ( S ) ( p , q ) = − g T ( R ) π Φ ( k , m ) ¯ σ αβ p β , (11.35) η µν Γ µναβ ( T ) ( p , q ) = g T ( R ) π Φ ( k , m ) u αβ ( p , q ) . (11.36)A similar result holds also for the Konishi current J f µ = ¯ χ f ¯ σ µ χ f + i φ f † ( D µ φ f ) − i ( D µ φ f ) † φ f (11.37) Γ µαβ ( J f ) ( p , q ) = − i g T ( R f ) π Φ ( k , m ) k µ k ε [ p , q , α , β ] , (11.38)with Φ ( k , m ) given in Eq. (11.33), in full analogy with the result for the correlator of the R current.Defining χ ( k , m ) ≡ Φ ( k , m ) / k , (11.39)the discontinuity of the anomalous form factor χ ( k , m ) is then given byDisc χ ( k , m ) = χ ( k + i ε , m ) − χ ( k − i ε , m ) = − Disc (cid:18) k (cid:19) − m Disc (cid:18) C ( k , m ) k (cid:19) (11.40)giving Disc χ ( k , m ) = i π m ( k ) log 1 + (cid:112) τ ( k , m ) − (cid:112) τ ( k , m ) θ ( k − m ) . (11.41)The total discontinuity of χ ( k , m ) , as seen from the result above, is characterized just by a singlecut for k > m , since the δ ( k ) (massless resonance) contributions cancel between the first andthe second term of Eq. (11.40). This result proves the decoupling of the anomaly pole at k = S anom = S axion + S dilatino + S dilaton (11.42)45 xact Correlators from Conformal Ward Identities Claudio Corianò
UVIR (a)
Figure 13:
The UV/IR RG flow with a possible intermediate non-perturbative potential for the generationof ultralight masses for axions and dilatons where S axion = − g π (cid:18) T ( A ) − T ( R ) (cid:19) (cid:90) d z d x ∂ µ B µ ( z ) (cid:3) zx F αβ ( x ) ˜ F αβ ( x ) S dilatino = g π (cid:18) T ( A ) − T ( R ) (cid:19) (cid:90) d z d x (cid:20) ∂ ν Ψ µ ( z ) σ µν σ ρ ← ∂ ρ (cid:3) zx ¯ σ αβ ¯ λ ( x ) F αβ ( x ) + h . c . (cid:21) S dilaton = − g π (cid:18) T ( A ) − T ( R ) (cid:19) (cid:90) d z d x (cid:0) (cid:3) h ( z ) − ∂ µ ∂ ν h µν ( z ) (cid:1) (cid:3) zx F αβ ( x ) F αβ ( x ) . (11.43)The complete symmetry of the massless exchange present in each channel is evident.
12. Comments
There are obvious questions that one can ask, by looking at these results. One of them concernsthe possible physical meaning of such massless exchanges, which have motivated our analysis. Thepresence of ghosts in each anomaly channel seems to indicate that a mechanism of ghost conden-sation could take place, which causes a redefinition of the vacuum. On the other hand, there arelimitations to our analysis, the first being that it relies on the computation of a simple Coleman-Weinberg potential. The second one is its limitation to 3-point functions, i.e. at trilinear level,although at this level all the features of the anomaly functional are reproduced by the candidateaction, at least in d =
4, except for non anomalous contributions which require higher point func-tions.The physical excitations that emerge from such a vacuum rearrangement would be ultralight andshould play a role in cosmology, especially in the context of dark matter and/or dark energy stud-ies, for being Nambu-Goldstone modes which are dynamically generated by the superconformalanomaly. Obviously, such a speculative hypothesis requires further investigations in order to pro-vide solid predictions. We stress once again that this picture allows to reconcile two differentapproaches in the analysis of anomaly actions, the nonlocal one, based on a variational solution of46 xact Correlators from Conformal Ward Identities
Claudio Corianò the anomalous Ward identities, and the local one, based on the inclusion of a Nambu-Goldstonemode to account for the broken symmetry.In a supersymmetric context, this extension is realized by the inclusion of a supermultiplet, withan axion, an axino and a dilaton in the spectrum of the 1PI anomaly action. Lagrangians with suchfield content have been discussed in the past [42–44]. In general, one expects a mechanism of vac-uum misalignment to take place at a large scale in the generation of ultralight particles [9, 45] andit is conceivable that a similar mechanism could be induced also by the conformal anomaly, dueto the presence of a topological contribution. This scenario would then be summarized as in Fig.13. The UV and IR descriptions at the two upper and lower ends of this figure would correspondto the two versions of the anomaly action that we have analyzed, with an intermediate dynamicalpotential generated non-perturbatively at an intermediate scale, and connected by an RG flow. Suchpotential would be responsible for generating a mass for the dilatons. For instance, an ultralightaxion/dilaton pair would be of remarkable cosmological significance and would define a new pos-sibility for gravitational physics in the far infrared, although other scenarios a‘t this stage cannot beexcluded. In the case of Stuckelberg models, for instance, such potential is generated by a vacuummisalignment and can be attributed to instanton effects at a certain phase transition [9], which canbe tiny in its size. This and other challenging aspects of such class of models are left for futureinvestigations.
Acknowledgements
We would like to thank Antonio Costantini, Luigi Delle Rose, Raffaele Fazio, Carlo Marzoand Mirko Serino for contributing to these investigations. We thank Paul Frampton for discussions.C.C. warmly thanks all the Organizers and in particular George Zoupanos for the hospitality at theCorfu Summer Institute. This work is supported by INFN under Iniziativa Specifica QFT-HEP.
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