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Renormalization, Conformal Ward Identities and the Origin of a Conformal Anomaly Pole

Claudio Coriano, Matteo Maria Maglio

TL;DR

This work investigates the origin of conformal anomaly poles in the TJJ vertex by combining a d-dimensional F-basis renormalization analysis with nonperturbative conformal Ward identities expressed in an A-basis. It identifies that the anomaly pole arises from renormalizing a single divergent form factor, F13, and demonstrates this mechanism in both abelian (QED) and non-abelian (QCD) theories, including implications for off-shell structures. The results bridge perturbative insights with nonperturbative CWI solutions, clarifying the physical role of massless exchanges associated with the conformal anomaly and connecting these ideas to phenomena in topological materials. The findings underscore the universality of anomaly poles as intrinsic signatures of conformal symmetry breaking, with potential relevance to transport phenomena in condensed-matter systems such as topological insulators and Weyl semimetals.

Abstract

We investigate the emergence of a conformal anomaly pole in conformal field theories in the case of the $TJJ$ correlator. We show how it comes to be generated in dimensional renormalization, using a basis of 13 form factors (the $F$-basis), where only one of them requires renormalization $(F_{13})$, extending previous studies. We then combine recent results on the structure of the non-perturbative solutions of the conformal Ward identities (CWI's) for the $TJJ$ in momentum space, expressed in terms of a 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.

Renormalization, Conformal Ward Identities and the Origin of a Conformal Anomaly Pole

TL;DR

This work investigates the origin of conformal anomaly poles in the TJJ vertex by combining a d-dimensional F-basis renormalization analysis with nonperturbative conformal Ward identities expressed in an A-basis. It identifies that the anomaly pole arises from renormalizing a single divergent form factor, F13, and demonstrates this mechanism in both abelian (QED) and non-abelian (QCD) theories, including implications for off-shell structures. The results bridge perturbative insights with nonperturbative CWI solutions, clarifying the physical role of massless exchanges associated with the conformal anomaly and connecting these ideas to phenomena in topological materials. The findings underscore the universality of anomaly poles as intrinsic signatures of conformal symmetry breaking, with potential relevance to transport phenomena in condensed-matter systems such as topological insulators and Weyl semimetals.

Abstract

We investigate the emergence of a conformal anomaly pole in conformal field theories in the case of the correlator. We show how it comes to be generated in dimensional renormalization, using a basis of 13 form factors (the -basis), where only one of them requires renormalization , extending previous studies. We then combine recent results on the structure of the non-perturbative solutions of the conformal Ward identities (CWI's) for the in momentum space, expressed in terms of a minimal set of 4 form factors ( basis), with the properties of the -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.

Paper Structure

This paper contains 10 sections, 33 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Poles in special and general kinematics
  3. The perturbative analysis
  4. The perturbative $TJJ$ vertex
  5. The structure of the (partially transverse) $F$-basis
  6. The transverse traceless basis
  7. The renormalization of $F_{13}$: $d$-dimensional analysis
  8. The anomaly pole from renormalization
  9. Implications in the non-abelian case
  10. Conclusions

Figures (1)

  • Figure 1: The complete $TJJ$ one-loop vertex (a) given by the sum of the 1PI contributions with triangle (b) and pinched topologies (c).