Physics Letters B 781 (2018) 283–289

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Physics Letters B

www.elsevier.com/locate/physletb

Renormalization, conformal ward identities and the origin of a conformal pole ∗ Claudio Corianò , Matteo Maria Maglio

Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento and INFN Lecce, Via Arnesano, 73100 Lecce, Italy a r t i c l e i n f o a b s t r a c t

Article history: We investigate the emergence of a conformal anomaly pole in conformal field theories in the case of Received 5 February 2018 the TJJ correlator. We show how it comes to be generated in dimensional , using a Received in revised form 30 March 2018 basis of 13 form factors (the F -basis), where only one of them requires renormalization (F13), extending Accepted 1 April 2018 previous studies. We then combine recent results on the structure of the non-perturbative solutions Available online 5 April 2018 of the conformal Ward identities (CWI’s) for the TJJ in momentum space, expressed in terms of a Editor: A. Ringwald minimal set of 4 form factors (A-basis), with the properties of the F -basis, and show how the singular behaviour of the corresponding form factors in both basis can be related. The result proves the centrality of such massless effective interactions induced by the anomaly, which have recently found realization in solid state, in the theory of topological insulators and of Weyl semimetals. This pattern is confirmed in massless abelian and nonabelian theories (QED and QCD) investigated at one-loop. © 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction vector current in an AV V (axial-vector/vector/vector) correlator or that of a stress energy tensor (T ) in a TJJ vertex. Chiral and conformal anomalies are central in quantum field Previous studies in perturbation theory, away from the con- theory, due to the appearance in anomaly vertices of non-con- formal limit, by the inclusion, for instance, of a mass in served chiral or dilatation currents. Conditions of the loop, have shown that the form factors which appear in the cancellations—for gauge anomalies—and/or the identification of trace part of the TJJcorrelator are characterized by spectral den- possible global anomalies, play a key role in determining the parti- sities which satisfy mass-independent conformal [1] and, in the cle spectra of the corresponding theories, constraining their quan- supersymmetric case, superconformal [5]sum rules, related to the tum numbers. anomaly coefficients. In the massless fermion limit their spectral In general, most of the analysis has always been associated with densities converge to δ-functions, manifesting the exchange of an the investigation of the Ward identities (WI) of a given anoma- anomaly pole. This beautiful behaviour, obviously, is not just a co- lous correlator, in the form of conservation—for chiral—or trace incidence and suggests of something very special taking place in and conservation WI’s for conformal anomalies. These operations the conformal/chiral anomaly actions. reduce the number of free uncontracted indices of an anomalous The existence of chiral anomaly poles has been discussed in diagram and mix their defining tensor components and form fac- the literature since the work of Dolgov and Zakharov [6], while tors, providing less information with respect to that which is ob- conformal anomaly poles have been shown to be part of the TJJ tainable from the study of a full (uncontracted) vertex. vertex in QED [1,4], QCD and the electroweak sector of the Stan- It has been shown that in an uncontracted anomaly vertex of dard Model [7,8]only more recently. In the case of the Standard either chiral, conformal [1–4]or superconformal type [5], the ori- Model it has been argued that an effective dilaton-like interaction gin of an anomaly has to be attributed to the appearance of spe- could be mediated by the trace anomaly, due to such massless ex- cific form factors in its tensor structure, which are proportional to changes, which could be of phenomenological interest at the LHC 1/k2 in the massless limit. Such anomaly poles define massless ex- [9,10]. It is then natural to interpret such intermediate states as changes in momentum space and are the direct signature of the the signature of (anomaly) broken scale invariance of the Higgs anomaly. In all such cases k denotes the momentum of an axial- sector, if the zero mass limit of the Higgs sector is taken [9]. It is an open question, in the supersymmetric context, for instance, * Corresponding author. if the three anomaly poles of the superconformal currents super- E-mail address: claudio .coriano @le .infn .it (C. Corianò). multiplet, which interpolate with an axion–dilaton–dilatino (ADD) https://doi.org/10.1016/j.physletb.2018.04.003 0370-2693/© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 284 C. Corianò, M.M. Maglio / Physics Letters B 781 (2018) 283–289 composite multiplet, are an indication of the possible existence of a distinction between an anomaly pole and the remaining (non a broken conformal phase in N = 1 supersymmetric theories in anomalous) poles present in its (several) tensor structures, requires which supersymmetry is nonlinearly realized [5]. an in-depth study of the corresponding . If some Studies of such interactions in the context of both chiral and parameterizations obscure the pole behaviour, as in Rosenberg’s conformal anomaly diagrams have always been performed at the formulation, in others, such as the L/T one, the pole is present for perturbative level in the past, with the obvious limitations of the any momenta of the vertex. case. These studies show the presence of some universal features One possible way to resolve such a dispute is to go beyond of these interactions, confirming that anomaly poles are ubiquitous perturbation theory, if possible, using exact results if these are in the presence of anomalous interactions. The chiral and confor- available. Such is the case of conformal field theories (CFT’s) where mal anomaly coefficients are then proportional to the residues of the presence of extra conformal Ward identities (CWI’s)—with re- the corresponding correlators evaluated at the anomaly pole (times spect to Poincare’ invariance—allow to specify, at least for some a tensor structure which is the anomaly functional). correlators, their momentum dependence. In the case of a global U (1)B anomaly, with an external Bμ The goal of the present work is to illustrate how a pole emerges gauge field and field strength F B μν , coupled to anomalous (axial- from the renormalization of a single form factor (F13) in a specific vector, A) current, the anomaly can indeed be written in the (non minimal) basis of the TJJvertex. The result holds in general generic form for any TJJ vertex in CFT. We show how to combine such infor-    mation with a recent analysis of the solutions of CWI’s based on 1 4 4 ˜ a (minimal) basis of form factors (A1, ...A4) fixed by conformal an d xd y∂ · B(x) (x, y)F B F B (y) (1.1)  symmetry. Our results rely on recent solutions of the conformal equations presented in [16,17]and prove that the emergence of a 3 for a chiral anomaly of U (1)B type, corresponding to a vertex specific pole in the correlator is not the result of redundancy or an with three axial vector currents (AAA) and anomaly coefficient an. artefact of the parameterization. It should rather be thought of as A similar behaviour is expected from a (ag ), a conclusive manifestation of an anomaly, and it is not strictly as- generated at an axial-vector/graviton/graviton (ATT) vertex sociated to a specific configuration of the external momenta of a    vertex, but holds also off-shell. 1 a d4xd4 y∂ · B(x) (x, y)R R˜ (y) (1.2) In this work we will concentrate on the physical implications g  of our analysis, leaving aside the rather technical parts that will with Rμναβ being the Riemann tensor, when coupling an axial- be presented in a companion paper and in work in preparation. vector current mediated by an external pseudoscalar gauge field We are going to show how the combination of perturbative results Bμ to two stress–energy tensors. A third example is provided by in massless QED and of non-perturbative information derived from the TJJ vertex, that we are going to discuss below, which is af- the solution of the conformal Ward identities, allows us to trace fected by a conformal anomaly and manifests a similar interaction. back how an anomaly pole appears in a simpler correlator such as Understanding the key role played by these effective interac- the TJJ. tions at any energy scale, and in the presence of radiative ef- Our conclusions will be that the anomaly pole of the TJJ is fects which may corrupt their massless behaviour, is crucial for a crucial part of this anomalous interaction. We believe that sim- a more complete comprehension of their dynamics. In particular, ilar conclusions can be drawn in all cases in the corresponding their emergence in the anomaly effective action calls for a more anomaly actions. physical re-interpretation of the irreversibility of RG-flows from the UV to the IR, in theories with conformal anomalies, which should 1.2. The perturbative analysis be described, on physical grounds, also in terms of such effective interactions. The appearance of the β function at the numerator In momentum space, the emergence of these poles can be at- of an anomaly pole, and its dependence on the number of mass- tributed to a specific configuration of the loop integral in the Feyn- less degrees of freedom along the flow, is clearly an indication that man expansion of the correlator, with the exchange of a (fermion/ such possibility should not be excluded. gluon) collinear pair [1,3,4]. Anomaly poles trigger virtual inter- actions which redefine the of a theory and, in a simple 1.1. Poles in special and general kinematics perturbative analysis, cannot be immediately recognized as asymp- totic states of an effective S-matrix. Rather they can be thought of There are reasons why these contributions have been over- as effective intermediate exchanges mediated by an anomaly. The looked in the past, and they have to do with the proliferation solution of the conformal constraints that we are going to present of tensor structures of such vertices, as exemplified in the case in this work indicates that such viewpoint and limitations are a of the AV V diagram, for which at least two most valuable rep- consequence of perturbation theory. In this respect, the nonpertur- resentations exist. Most notably, these are the Rosenberg repre- bative approach provided by the solutions of the CWI’s shows that sentation [11], which is expressed in terms of 6 form factors, such pole-like contributions are generically present in the off-shell that reduce to 4 by applications of the vector Ward identities, anomaly vertex. and the longitudinal/transverse (L/T) parameterization [12], used This suggests that theories affected by anomaly poles may un- in the analysis of the anomalous magnetic moment of the muon. dergo a non-perturbative redefinition of their vacuum in such a In the latter case a complete 2-loop computation has shown the way that such interactions may describe, in a non-perturbative non-renormalizability of the entire vertex [13][14], not just of its phase of such theories, the exchange of composite (asymptotic) longitudinal part, as one would expect from the Adler–Bardeen states in the infrared, with specific quantum numbers. This tran- theorem [15]. sition requires a mechanism of dynamical breaking of a conformal The issue of whether poles are genuine or artificially introduced symmetry, of which the anomaly is probably just one component. by a certain ad hoc parameterization of a given vertex has gen- The most interesting case were such behaviour has been con- erated wide disagreement over the years, and it has also been a jectured [5]is in supersymmetric theories, where the superconfor- source of confusion. In fact, in general, an anomalous correlator has mal anomaly manifests itself with the appearance of 3 anomaly extra poles beside the anomaly poles. Therefore in order to make poles. These affect vertices containing one superconformal and C. Corianò, M.M. Maglio / Physics Letters B 781 (2018) 283–289 285

Fig. 1. The complete TJJone-loop vertex (a) given by the sum of the 1PI contributions with triangle (b) and pinched topologies (c). two (super)vector currents, which cover both the AVV and the TJJ Table 1 cases, plus a third anomaly vertex with the insertion of a super- The basis of 13 fourth rank tensors satisfying the vector current conservation on the symmetric current. Also in this case it is suggestive to interpret external lines with momenta p and q. such exchanges as due to interpolating effective axion/dilaton/ax- μναβ iti (p, q) ino interactions. Obviously, it remains an open issue whether such   1 k2 gμν − kμkν uαβ (p.q) behaviour is an indication of the existence of a phase of the the-   ory in which supersymmetry is nonlinearly realized. In such a case 2 k2 gμν − kμkν wαβ (p.q)   such composite intermediate states could become asymptotic, be- 3 p2 gμν − 4pμ pν uαβ (p.q) ing the Goldstone modes of a broken superconformal symmetry.   4 p2 gμν − 4pμ pν wαβ (p.q) It is quite interesting that recent analysis in solid state theory   5 q2 gμν − 4qμqν uαβ (p.q) have suggested that such massless exchanges in the chiral and con-   formal anomaly actions play an important role in the theory of 6 q2 gμν − 4qμqν wαβ (p.q) topological materials [18,19], confirming previous analysis in high 7 [p · qgμν − 2(qμ pν + pμqν )] uαβ (p.q) energy [1–3,20]. Such universal behaviour is related to the funda- 8 [p · qgμν − 2(qμ pν + pμqν )] wαβ (p.q) mental role played by the anomalies in quantum field theory.     9 p · qpα − p2qα pβ qμ pν + pμqν − p · q (gβν pμ + gβμ pν )     10 p · qqβ − q2 pβ qα qμ pν + pμqν − p · q (gανqμ + gαμqν ) 1.3. The perturbative TJJvertex   11 p · qpα − p2qα 2 qβ qμqν − q2(gβνqμ + gβμqν )   The perturbative cases discussed in the past, concerning this 12 p · qqβ − q2 pβ 2 pα pμ pν − p2(gαν pμ + gαμpν )     vertex, cover QED, QCD and the neutral currents sector of the Stan- pμqν + pνqμ gαβ + p · q gαν gβμ + gαμgβν − gμν uαβ dard Model [8], where the features described above are evident 13     − gβν pμ + gβμ pν qα − gανqμ + gαμqν pβ at one-loop. Even in the presence of a broken (massive) phase, in a mass-independent scheme such as dimensional regularization, it is still possible to identify anomaly poles in this where the invariant amplitudes Fi are functions of the kinematic correlator, which are present in each gauge-invariant sector. = 2 = + 2 = 2 = 2 μ1ν1μ2μ3 invariants s p1 (p2 p3) , s1 p2, s2 p3, and the ti In QCD, for instance, the two gauge invariant sectors involve define the basis of the independent tensor structures. at one-loop either quarks or gluons, and the pattern that we are On this basis, which is built by imposing on the TJJvertex all going to describe is separately present in each of these two sec- the Ward identities derived from diffeomorphism invariance and tors. We refer to [5]for a general and combined analysis of such gauge invariance, it is possible to use Bose symmetry and conser- features and to [7]for a complete analysis of the neutral currents vation WI’s to reduce the number of form factors. sector of the Standard Model. We briefly summarize the status of this analysis in the case of QED. 1.4. The structure of the (partially transverse) F -basis The TJJ vertex, in QED, describes the coupling of a graviton to two and is a source of the conformal anomaly (Fig. 1). The set of the 13 tensors ti is linearly independent for generic Perturbative investigations of this correlator have shown that the k2, p2, q2 different from zero. Five of the 13 are Bose symmetric, pole contribution is described, in the 1-particle irreducible effec- tive action, by the term μναβ = μνβα = ti (p, q) ti (q, p), i 1, 2, 7, 8, 13 , (1.5) e2 S =− while the remaining eight tensors are Bose symmetric pairwise pole 2 36π μναβ μνβα   t (p, q) = t (q, p), (1.6) − 3 5 × d4xd4 y h(x) − ∂ ∂ hμν(x)  1 F (x)F αβ (y) μ ν xy αβ μναβ = μνβα t4 (p, q) t6 (q, p), (1.7) (1.3) μναβ = μνβα t9 (p, q) t10 (q, p), (1.8) which can be extracted from the 1-loop expression of the vertex, μναβ μνβα t (p, q) = t (q, p). (1.9) using a suitable decomposition. In fact, the amplitude for the TJJ 11 12 can be expanded in the basis proposed by [1], in terms of 13 in- In the set are present two tensor structures dependent tensors structures given in Table 1. It can be written αβ αβ α β as u (p, q) ≡ (p · q)g − q p , (1.10a) αβ 2 2 αβ α β 2 α β 13 w (p, q) ≡ p q g + (p · q)p q − q p p μ ν μ μ μ1ν1μ2μ3  1 1 2 3 (p , p ) = F (s; s , s , 0)t (p , p ), (1.4) 2 α β 2 3 i 1 2 i 2 3 − p q q , (1.10b) i=1 286 C. Corianò, M.M. Maglio / Physics Letters B 781 (2018) 283–289

μ1ν1 μ2 μ3 μ1ν1 μ2 μ3 which appear in t1 and t2 respectively. Each of them satisfies the T J J  = t j j  + local terms. (1.17) Bose symmetry requirement, Here we have switched to a symmetric notation for the exter- αβ βα ≡ u (p, q) = u (q, p), (1.11a) nal momenta, with (p1, p2, p3) (k, p, q), and with the transverse traceless parts expanded in terms of a set of the form factors A j wαβ (p, q) = wβα(q, p), (1.11b) mentioned above and vector current conservation, μ1ν1 μ2 μ3 t (p1) j (p2) j (p3) αβ αβ p u (p, q) = 0 = qβ u (p, q), (1.12a) α μ1ν1 μ2 μ3 α1 β1 α2 α3 α2α3 α1 β1 = 1 π2 π3 A1 p p p p + A2 δ p p αβ αβ α1β1 α2 α3 2 2 3 1 2 2 pα w (p, q) = 0 = qβ w (p, q). (1.12b) β α β α + A δα1α2 p 1 p 3 + A (p ↔ p )δα1α3 p 1 p 2 They are obtained from the variation of gauge invariant quantities 3 2 1 3 2 3 2 3 μν μ νλ + α1α3 α2β1 Fμν F and (∂μ F λ)(∂ν F ) A4 δ δ . (1.18)   2 μν μ1ν1 1 · + · δ {Fμν F (0)} In this expression 1 is a transverse and traceless projector αβ =− 4 4 ip x iq y α1β1 u (p, q) d x d ye , (1.13) μ2 μ3 4 δ Aα(x)Aβ (y) built out of momentum p1, while π2α and π3α denote trans-   2 3 2{ μ νλ } verse projectors respect to the momenta p2 and p3. αβ 1 4 4 ip·x+iq·y δ ∂μ F λ∂ν F (0) w (p, q) = d x d ye . Coming to the explicit form of the A j , these can be determined, 2 δ Aα(x)Aβ (y) modulo some constants, by the solution of the primary Ward iden- (1.14) tities. Primary Ward identities are second order (vector) differential constraints on a tensor correlator which are reformulated as a set All the ti ’s are transverse in their indices of scalar equations [16]. They are obtained by the action of the α μναβ = β μναβ = generators of the special conformal transformations (K κ ) on the q ti 0 p ti 0. (1.15) TJJamplitude. = t2 ...t13 are traceless, t1 and t2 have trace parts in d 4. With this In momentum space, after an involved analysis, one obtains a decomposition, the two vector Ward identities are automatically set of scalar equations for the A j , whose primary Ward identities satisfied by all the amplitudes, as well as the Bose symmetry. are formulated in terms of a set of second order scalar operators Diffeomorphism invariance, instead, is automatically satisfied 2 + − (separately) by the two tensor structures t1 and t2, which are com- ∂ d 1 2 i ∂ Ki = + , i = 1, 2, 3 (1.19) 2 p ∂ p pletely transverse, while it has to be imposed on the second set ∂ pi i i (t ...t ). Such identities are 3 13 Kij = Ki − K j, (1.20) − 2 + 2 + · + 2 + · − 2 2 p F3 (3q 4p q)F5 (2p p q)F7 p q F10 where i is the conformal dimension of the i-th operator in the 2 2 2 2 3-point function under consideration. In our case, for the TJJ, − p (p + p · q)F9 + p q F11 = 0 , 1 = d, 2 = 3 = d − 1, and the primary CWI’s take the form 2 2 2 p F4 − (3q + 4p · q)F6 − (2p + p · q)F8 − p · qF10 2 0 = K13 A1 0 = K23 A1 + (q + 2p · q)F11 = 0 , 2 2 2 2 0 = K13 A2 + 2A1 0 = K23 A2 − p · q (p + p · q)F9 − q (q + p · q)F11 + F13 + (p ) = 0 , (1.16) 0 = K13 A3 − 4A1 0 = K23 A3 − 4A1

2 = ↔ = ↔ + with (p ) being the scalar 2-point function of momentum p, and 0 K13 A3(p2 p3) 0 K23 A3(p2 p3) 4A1 a symmetric set of 3 equations obtained from (1.16)by exchanging 0 = K13 A4 − 2A3(p2 ↔ p3) 0 = K23 A4 + 2A3 − 2A3(p2 ↔ p3). p with q and using the pair relations (1.6). In this way it is possi- ble to extract from the 9 traceless tensor structures a (completely) (1.21) transverse and traceless set of 5 amplitudes, which will be given The solutions of such equations are expressed in terms of linear below, two of them related by the bosonic symmetry. combinations of generalized hypergeometric functions of two vari- To summarize, from the original 13 tensor structures ti , split ables (Appel’s functions F4), recently solved in terms of parametric into a set of two transverse and trace components and a remaining integrals of three Bessel functions (3-K integrals), as discussed in = set of 11 partially transverse but traceless ones (in d 4), one is [16,21]. A direct analysis of the solutions using the Fuchsian prop- left with 7 form factors after imposing the pairing conditions (1.6). erties of these equations will be presented elsewhere [22]. Finally, imposing the conservations WI’s (1.16)these are reduced The tensor nature of the correlator implies that also some first to 4, which are related to the 4 form factors Ai ’s introduced by order differential constraints need to be imposed (called secondary BMS in their reconstruction method [16]. CWI’s in [16]). The solution of such constraints, however, can be performed at special kinematic points (for instance at equal in- 1.5. The transverse traceless basis 2 = 2 variant mass of the two photons, p2 p3, or, alternatively, in the massless limit of the graviton line), which constrain the undeter- It is possible to show that these amplitudes are in a one-to-one mined constants of the general solutions of the primary CWI’s = correspondence with the form factors A j ( j 1, ...4) introduced (1.21). Secondary CWI’s are related to longitudinal/trace terms of in the parameterization of the TJJ correlator presented in [16]. the correlators, and henceforth to contact terms. In that work the full 3-point function is parameterized in terms of transverse (with respect to all the external momenta) trace- 2. The renormalization of F13: d-dimensional analysis less components plus extra terms identified via longitudinal Ward identities of the TJJ (the so-called local or contact terms) charac- In order to clarify the connection between the appearance terised by pinched topologies of a pole and the process of renormalization, we consider the C. Corianò, M.M. Maglio / Physics Letters B 781 (2018) 283–289 287

d-dimensional structure of the CWI’s of the TJJ in the F -basis. Notice that the result for the Ai ’s presented in [17]shows, by QED provides a realization of this behaviour at one-loop and we an analysis of the 3-K integrals, that the singularities of the Ai ’s will stick to this example for definiteness, in order to clarify our are those of A2, A3 and A4. This is consistent with the mapping discussion. (2.2)since those are exactly the combinations in which the diver- The TJJ correlator in QED is conformal in d dimension, with gent form factor F13 appears. finite form factors which are dimensionally regulated and there- This specific origin of the singularity, which can be directly fore it does not develop any conformal anomaly. We can use the identified in the F-basis, is not directly manifest in the A-basis. F -basis to parameterize the correlator, now in d dimensions, in The Ai ’ s, on the other hand, describe a minimal set of form fac- terms of the same 13 form factors Fi introduced before and of the tors which are suitable for resolving the CWI’s of the correlators, corresponding tensor structures ti . but shadow the origin of the singular behaviour, since 3 out of 4 Notice that the separation of these 13 structures into trace- of them manifest UV singularities and need to be renormalized. By free and trace parts is valid only in d = 4for most of the struc- using the F -basis, instead we know where to look for singularities tures, except for t9, t10, t11 and t12, which remain traceless in d in a rather simple way, this is F13. dimensions. We are assuming that the contractions with the met- ric tensor is performed in d dimensions with a metric gμν (d). The 2.1. The anomaly pole from renormalization 4-dimensional metric, instead, will be denoted as gμν (4). For instance, a contraction of t1 and t2 in d-dimensions will In order to trace back the origin of the anomaly pole in TJJ, give starting from d-dimension and using the F -basis, we request that this correlator has no trace (i.e. be anomaly free). The anomaly will μν μναβ = − 2 αβ → g (d)t1 (d 4)k u (p, q) emerge in dimensional regularization as we take the d 4 limit. μναβ The trace WI’s provide the two key conditions that we need. In gμν(d)t = (d − 4)k2 wαβ (p, q), (2.1) 2 fact we obtain and similarly for all the other structures, except for those men- −  (d 4) 2 2 tioned above, which are trace-free in any dimensions. F1 = F13 − p F3 − p F5 − p2 · p3 F7 (2.4) p2(d − 1) 2 3 Using the completeness of the F -basis and by a direct analysis 1 of the CWI’s which will be detailed elsewhere, we can identify the and mapping between the form factors of such basis and those of the (d − 4)  F = p2 F + p2 F + p · p F . (2.5) A-basis. They are conveniently expressed in terms of the momenta 2 2 − 2 4 3 6 2 3 8 p1(d 1) (p1, p2, p3) in the form Both equations are crucial in order to understand the way the = − − − 2 − 2 renormalization procedure works for such correlator. From Eq. (2.5) A1 4(F7 F3 F5) 2p2 F9 2p3 F10 it is clear that by sending d → 4, F vanishes, = 2 − 2 − 2 − − 2 A2 2(p1 p2 p3)(F7 F5 F3)  2 2 2 2 F = p F + p F + p · p F → 0, (2.6) − − + − 2 2 2 4 3 6 2 3 8 4p2 p3(F6 F8 F4) 2F13 − (d 1)p1 A = p2 p2 − p2 − p2 F − 2p2 p2 F − 2F 3 3( 1 2 3) 10 2 3 12 13 for all the form factors F4, F6 and F8 are finite for dimensional ↔ = 2 2 − 2 − 2 − 2 2 − reasons. In fact, from the scaling dimensions of the corresponding A3(p2 p3) p2(p1 p2 p3)F9 2p2 p3 F11 2F13 tensor structures t4, t6 and t8 one concludes that they are finite, = 2 − 2 − 2 A4 (p1 p2 p3)F13, (2.2) and therefore F2 is indeed zero in this limit, since the right hand side of (2.5) has no poles in ≡ d − 4. which are transverse and traceless, with A , A and A symmetric. 1 2 4 At this stage, after the limiting procedure, at d = 4we are left Given the correspondence (2.2), it is worth noticing that the in the F -basis with 4 independent combinations of form factors form factor A and its corresponding symmetric A (p ↔ p ) are 3 3 2 3 from the original 7 (those given in (2.2)), which are sufficient to consistently defined in terms of the Fi , since in the F-set describe the (complete) transverse traceless sector of the theory, plus an additional form factor F1. Therefore, by taking the d → 4 F (s; s , s , 0) = F (s; s , s , 0), 10 1 2 9 2 1 limit, the F -set contains only one single tensor structure (and an F12(s; s1, s2, 0) = F11(s; s2, s1, 0), (2.3) associated form factor) with a nonzero trace, which should account for the anomaly in d = 4. This result is obviously confirmed in per- which shows the consistency of the mapping between the F and turbation theory in QED [3]. A basis. As already mentioned, F is the only form factor that needs = 13 The presence of two tensor structures of nonzero trace in d 4 to be renormalized in the F -set and it is characterized by the ap- in the F -basis, however, is at first sight slightly puzzling, since pearance of a single pole in 1/ in dimensional regularization. The the correspondence between the appearance of an anomaly pole fact that such singularity will be at all orders of the form 1/ and (and henceforth of a trace) and the process of renormalization not higher is a crucial ingredient in the entire construction. Such does not seem to be unique. We are looking for a single (anomaly) assumption is expected to be consistent with the analysis in con- pole whose origin should be traced back to renormalization. The expansion may allow extra poles, but they will be unrelated to formal field theory since the only available counterterm to regulate the theory is given by renormalization. We are going to show that indeed there are no  extra poles sharing such a feature. 1 √ d4x gF F μν (2.7) The two sets A j and Fi differ in several ways, and emphasize μν different properties of the same TJJ correlator. On the one hand,   the F -basis sheds light on the origin of the anomaly pole, as we which renormalizes the 2-point function JJ and henceforth F13. are going to show below, by linking its origin to the single form Explicit computations in QED show that factor F13 which exhibits a divergence and requires renormaliza- 2 2 3 1 2 2 tion. F13 = G0(p , p , p ) − [ (p ) + (p )] (2.8) 1 2 3 2 2 3 288 C. Corianò, M.M. Maglio / Physics Letters B 781 (2018) 283–289

with G0 a lengthy expression which remains finite as d → 4, with the case of QED, the relation between the prefactor in front of the the origin of the singularity traced back to the scalar form factor 1/s pole and its relation to the QED β-function has been exten- (p2) of the photon 2-point function. For this purpose, we just re- sively discussed in [1,3], to which we refer for further details. In call that the structure of the two-point function of two conserved performing the limit we have used the finiteness of the remaining form factors. vector currents of scaling dimensions η1 and η2 is given by [23]   d/2 α β αβ π (d/2 − η1) p p 3. Implications in the non-abelian case G (p) = δ c ηαβ − V η1η2 V 12 η −d/2 2 4 1 (η1) p − Further comparisons with the non-perturbative solutions of the × (p2)η1 d/2 , (2.9) CWI’s, in the approach presented in [17]can be made using the perturbative results of [4]forQCD, the pattern described above be- with cV 12 being an arbitrary constant. It requires the two cur- rents to share the same dimensions and manifests only a sin- ing still valid also in this case, although only the structure of the gle pole in 1/ . In dimensional regularization, in fact, the diver- on-shell vertex, with s arbitrary, but with the two gluons on shell (s = s = 0) is available for a direct comparison. In the QCD case, gence can be regulated with d → d − 2 . Expanding the product 1 2 − as already mentioned, there are two anomaly poles, one for each (d/2 − η) (p2)η d/2, which appears in the two-point function, in gauge invariant sector. While for the fermion loop the anomaly a Laurent series around d/2 − η =−n (integer) gives the single pole generated follows exactly the same pattern discussed above pole in 1/ behaviour [23]   with minimal changes (modulo extra colour factors), for the gluon − n loops another pole is present and shares quite similar features. In 2 η−d/2 ( 1) 1  (d/2 − η) (p ) = − + ψ(n + 1) + O ( ) fact, in the gluon sector only one form factor gets renormalized, by n! choosing an appropriate basis, [4] and the pattern that emerges in 2 n+ × (p ) , (2.14)is similar, with the obvious changes. Notice that in the non- (2.10) abelian case the number of form factors in the transverse traceless sector of the correlator is still 4, and their expressions gets mod- where ψ(z) is the logarithmic derivative of the Gamma function, ified just by simple colour factors, with 4 of them affected by a and takes into account the divergence of the two-point correlator single polar divergence in 1/(d −4), as pointed out in [17]. One can for particular values of the scale dimension and of the space- η show that such divergences are again associated to the renormal- time dimension d. Therefore, the divergence in F13 is then given ization of the gluon 2-point function, appearing in a single form by a single pole in is of the form factor (φ3, in the notations of [4]). = 1 ¯ + F13 F13 F13 f . (2.11) 4. Conclusions d − 4

In QED, for instance, one finds by an explicit computation that We have proven that conformal anomaly poles are not the ¯ =− 2 2 ¯ F13 e /(6π ) at one-loop and F13 f is finite [3] and gets renor- result of specific parameterizations of anomaly vertices, but are malized into F13R only in its photon self-energy contributions [3] the natural signature of the anomaly, being related to renormal- = 2 = 2 = 2 (s p1, s1 p2, s2 p3) ization. The solution of CWI’s in momentum space, presented in recent analysis, are completely consistent with previous studies 1 in abelian and non-abelian theories [1,3,4]. The phenomenon is F13,R (s; s1, s2, 0) =− [ R (s1, 0) + R (s2, 0)] + G0(s, s1, s2) 2 therefore generic, and it is surely not limited to the high energy (2.12) domain, but wherever anomaly actions are at work. Recent stud- with ies have underlined the important role played by such massless   exchanges [18,19] and of the anomaly in general [24,25], in the 2 e 5 s context of the transport properties of topological insulators and of R (s, 0) =− − log − , (2.13) 12 π 2 3 μ2 Weyl semimetals. In particular [18,19]suggest physical realizations of the observations of [1,3,26] concerning the structure of the chi- denoting the renormalized scalar form factor of the JJ correlator ral and conformal anomaly actions. at one-loop and with G defined in (2.8). 0 A more detailed technical discussion of the results of this work Inserting (2.11)into (2.4)we obtain  will be presented by us elsewhere. (d − 4) 1 = ¯ + − 2 − 2 F1 F13 F13 f p F3 p F5 Acknowledgements p2(d − 1) d − 4 2 3 1  We thank Emil Mottola, Paul McFadden, Luigi Delle Rose and −p · p F , (2.14) 2 3 7 Kostas Skenderis for discussions. C.C. thanks Fiorenzo Bastianelli and Olindo Corradini for discussions and hospitality at the Uni- which in the d → 4 limit gives, in general versities of Bologna and Modena and Maxim Chernodub at the ¯ University of Tours (LMTP) for hospitality and discussions. Finally, F13 F1 = (2.15) he thanks the High Energy Theory group at ETH-Zurich for hos- 3p2 1 pitality. This work is performed as part of the HEP-QFT research and specifically, in QED activity of INFN.

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