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Exact correlators from conformal Ward identities in momentum space and the perturbative TJJ vertex. (English) Zbl 1407.81125

In conformal field theory, an \(n\)-point function is a function that we assign to an \(n\)-tuple of tensor field operators of the conformal symmetry group. These functions satisfy certain “Ward identities” induced by the conformal symmetry. A momentum space framework to study three-point functions of conformal field theory has been introduced in [A. Bzowski et al., J. High Energy Phys. 2014, No. 3, Paper No. 111, 105 p. (2014; Zbl 1406.81082)]. This essentially amounts to restating the \(n\)-point functions and the Ward identities in term of their Fourier transforms.
In the paper under review the authors study three-point functions in momentum space. In the first part of the paper, the authors elaborate on the consequences of translational invariance in coordinate space which results in momentum conservation implying that an \(n\)-point function is a function of \(n-1\) independent momenta. The authors study contributions which appear after Fourier transforming the conformal transformations and show that such contributions do not cancel out, but lead to specific forms of the conformal generators in momentum space which are in agreement with those presented in [loc. cit.].
In the second part, the authors study scalar and tensor correlators and the solutions of the Ward identities. They elaborate on the apparent violation of the Leibnitz rule for the special conformal generator which emerges whenever the momentum conservation is imposed. The authors proceed with an analysis of the \(TJJ\) correlator, presenting a detailed re-derivation of the conformal equations, study the resulting constraints on the form factors \(A_i\) introduced in [loc. cit.] by using Ward identities correponding to Lorentz transformations, and confirm the results of [loc. cit.].
In the position space translational invariance can be used to put each of the operators included in the three-point function on the origin and treat it as a singlet under Lorentz transformations. The authors show how different choices for the singlet operator leads to equivalent outcomes. They also compute the form factors by investigating the Fuchsian structure of the equations, providing a new method of solution which differs from the one based on triple-\(K\) integrals presented in [loc. cit.]. The authors show that the number of integration constants obtained in this method does not necessarily coincide with those presented in [loc. cit.]; more important, they find evidence that the Fuchsian exponents are universal and characterize the entire system of equations.
The authors compute \(TJJ\) correlator in the transverse traceless basis both in QED and in scalar QED by perturbation and by comparing the \(A_i\) basis with the \(F_i\)-basis introduced in [M. Giannotti and E. Mottola, “The trace anomaly and massless scalar degrees of freedom in gravity”, Phys. Rev. D 79, No. 4, Article ID 045014, 33 p. (2009; doi:10.1103/PhysRevD.79.045014)] show that the \(TJJ\) correlator is affected by one anomaly pole in the graviton line, induced by renormalization. This result implies that the origin of the anomaly, in this correlator can be attributed to the exchange of a massless effective degree of freedom.
Finally the authors show how the perturbative solutions for the form factor \(A_i\), which are given in the appendix, reproduce the exact results of [Bzowski et al., loc. cit.] in a simplified way. This correspondence is studied by fixing an appropriate normalization of the photon two-point functions and shows that the choice of different perturbative sectors (scalar, fermion) in both cases are sufficient to reproduce the entire nonperturbative result. This implies that there should be significant cancellations among the contributions of the triple-\(K\) integrals or those given by the authors in such a way that they can be expressed in terms of two elementary master integrals.

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T18 Feynman diagrams
81V17 Gravitational interaction in quantum theory
83E15 Kaluza-Klein and other higher-dimensional theories

Citations:

Zbl 1406.81082
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References:

[1] Osborn, H.; Petkou, A. C., Implications of conformal invariance in field theories for general dimensions, Ann. Phys., 231, 311-362 (1994) · Zbl 0795.53073
[2] Erdmenger, J.; Osborn, H., Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions, Nucl. Phys. B, 483, 431-474 (1997) · Zbl 0925.81340
[3] Capper, D. M.; Duff, M. J., Conformal anomalies and the renormalizability problem in quantum gravity, Phys. Lett. A, 53, 361 (1975)
[4] Giannotti, M.; Mottola, E., The trace anomaly and massless scalar degrees of freedom in gravity, Phys. Rev. D, 79, Article 045014 pp. (2009)
[5] Armillis, R.; Corianò, C.; Delle Rose, L., Anomaly poles as common signatures of chiral and conformal anomalies, Phys. Lett. B, 682, 322 (December 2009)
[6] Armillis, R.; Corianò, C.; Delle Rose, L.; Guzzi, M., Anomalous U(1) models in four and five dimensions and their anomaly poles, J. High Energy Phys., 12, Article 029 pp. (2009)
[7] Armillis, R.; Corianò, C.; Delle Rose, L., Conformal anomalies and the gravitational effective action: the TJJ correlator for a Dirac fermion, Phys. Rev. D, 81, Article 085001 pp. (2010)
[8] Chernodub, M. N.; Cortijo, A.; Vozmediano, M. A.H., A Nernst current from the conformal anomaly in Dirac and Weyl semimetals, Phys. Rev. Lett., 120, Article 206601 pp. (2018)
[9] Rinkel, P.; Lopes, P. L.S.; Garate, I., Signatures of the chiral anomaly in phonon dynamics, Phys. Rev. Lett., 119, 10, Article 107401 pp. (2017)
[10] Bzowski, A.; McFadden, P.; Skenderis, K., Holographic predictions for cosmological 3-point functions, J. High Energy Phys., 03, Article 091 pp. (2012) · Zbl 1309.81146
[11] Bzowski, A.; McFadden, P.; Skenderis, K., Implications of conformal invariance in momentum space, J. High Energy Phys., 03, Article 111 pp. (2014) · Zbl 1406.81082
[12] Bzowski, A.; McFadden, P.; Skenderis, K., Evaluation of conformal integrals, J. High Energy Phys., 02, Article 068 pp. (2016) · Zbl 1388.81643
[13] Bzowski, A.; McFadden, P.; Skenderis, K., Renormalised 3-point functions of stress tensors and conserved currents in CFT · Zbl 1404.81219
[14] Bzowski, A.; McFadden, P.; Skenderis, K., Implications of conformal invariance in momentum space, J. High Energy Phys., 3, Article 111 pp. (2014) · Zbl 1406.81082
[15] Bastianelli, F.; Corradini, O.; Davila, J. M.; Schubert, C., Photon-graviton amplitudes from the effective action, Phys. Part. Nucl., 43, 630-634 (2012)
[16] Bastianelli, F.; Corradini, O.; Dávila, J. M.; Schubert, C., On the low-energy limit of one-loop photon-graviton amplitudes, Phys. Lett. B, 716, 345-349 (2012)
[17] Bonora, L.; Cvitan, M.; Dominis Prester, P.; Duarte Pereira, A.; Giaccari, S.; Stemberga, T., Axial gravity, massless fermions and trace anomalies, Eur. Phys. J. C, 77, 8, 511 (2017)
[18] Bastianelli, F.; Martelli, R., On the trace anomaly of a Weyl fermion, J. High Energy Phys., 11, Article 178 pp. (2016) · Zbl 1390.83323
[19] Bonora, L.; Giaccari, S.; Lima de Souza, B., Trace anomalies in chiral theories revisited, J. High Energy Phys., 07, Article 117 pp. (2014) · Zbl 1317.81237
[20] Corianò, C.; Delle Rose, L.; Mottola, E.; Serino, M., Graviton vertices and the mapping of anomalous correlators to momentum space for a general conformal field theory, J. High Energy Phys., 08, Article 147 pp. (2012) · Zbl 1397.81296
[21] Maldacena, J. M.; Pimentel, G. L., On graviton non-Gaussianities during inflation, J. High Energy Phys., 09, Article 045 pp. (2011) · Zbl 1301.81147
[22] Corianò, C.; Maglio, M. M., Renormalization, conformal Ward identities and the origin of a conformal anomaly pole, Phys. Lett. B, 781, 283 (2018) · Zbl 1398.81205
[23] Corianò, C.; Delle Rose, L.; Serino, M., Three and four point functions of stress energy tensors in \(\text{D} = 3\) for the analysis of cosmological non-gaussianities, J. High Energy Phys., 12, Article 090 pp. (2012) · Zbl 1397.83085
[24] Corianò, C.; Delle Rose, L.; Mottola, E.; Serino, M., Solving the conformal constraints for scalar operators in momentum space and the evaluation of Feynman’s master integrals, J. High Energy Phys., 1307, Article 011 pp. (2013) · Zbl 1342.81138
[25] Corianò, C.; Costantini, A.; Delle Rose, L.; Serino, M., Superconformal sum rules and the spectral density flow of the composite dilaton (ADD) multiplet in \(N = 1\) theories, J. High Energy Phys., 06, Article 136 pp. (2014)
[26] Broadhurst, David J.; Kataev, A. L., Connections between deep inelastic and annihilation processes at next to next-to-leading order and beyond, Phys. Lett. B, 315, 179-187 (1993)
[27] Kataev, A. L., The generalized Crewther relation: the peculiar aspects of the analytical perturbative QCD calculations, (Proceedings of the Conference on Continuous Advances in QCD 1996. Proceedings of the Conference on Continuous Advances in QCD 1996, Minneapolis, USA, March 28-31 (1996)), 107-132
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