Conformal Anomaly Actions and Dilaton Interactions

TL;DR

This work develops a comprehensive framework tying conformal and Weyl symmetries to quantum anomalies in curved spacetime, with a focus on the trace (conformal) anomaly and its nonlocal and local dilaton realizations. It performs a thorough one-loop analysis of the three-graviton vertex for free conformal field theories, derives Ward identities, and demonstrates how renormalization via $F$ (Weyl-squared) and $G$ (Euler) counterterms reproduces the trace anomaly, including the emergence of anomaly poles suggestive of a dilaton degree of freedom. It then establishes two general methods to map correlators between position and momentum space, clarifying when a Lagrangian description exists and how logarithmic structures signal nonlocality or the need for auxiliary fields, with applications to $ ext{TOO}$, $ ext{TVV}$, $ ext{VVV}$, and $ ext{TTT}$ vertices. Building on this, the thesis analyzes dilaton interactions in the Standard Model: radiative electroweak corrections to dilaton–gauge–gauge vertices, off-shell dilaton–gluon–gluon couplings in QCD, and the role of scale-invariant extensions, where anomaly poles can be interpreted as a (composite) dilaton that couples to the trace anomaly and can yield enhanced decays into photons and gluons. The results provide a coherent picture linking anomaly-induced effective actions (Riegert/Wess-Zumino) with perturbative correlators, offering a framework for exploring dilaton phenomenology in high-energy experiments and potential connections to infrared gravity corrections.

Abstract

A number of computational results concerning quantum conformal symmetry is presented. After a review of the connection between conformal symmetry for a Lagrangian field theory in flat space and Weyl symmetry for the same system embedded in a gravitational background, which is discussed in chapter 1, in chapter 2 the 3 energy momentum tensors correlation function is explicitly computed in three free field theories in 4 dimensions; the result is given for two of the three operators on the mass-shell. In chapter 3 a general method to map Green functions built in position space on the ground of symmetry requirements to momentum space, where they can be computed in terms of Feynman diagrams, is developed and discussed: an "integrability" condition, allowing to decide whether a certain correlator can exist within a Lagrangian theory, is derived. Chapter 4 discusses the possible phenomenological implications of the conformal anomaly pole which shows up in the 3 point Green function of one energy momentum tensor with two gauge currents and is interpreted as the perturbative signature of the pseudo-Goldstone boson of conformal symmetry, the dilaton. In chapter 5 we present the computation of the completely traced 4 point function of the energy momentum tensor with a method that exploits the relation between 1-loop counterterms and conformal anomalies, completely bypassing perturbative computations with Feynman diagrams. Later in chapter 6, an algorithm is developed which allows to compute recursively the completely traced Green functions of any number of energy momentum tensors in any renormalization scheme, starting from the dilaton Wess-Zumino action for conformal anomalies. This is derived by applying the Weyl-gauging procedure to the 1-loop counterterms in dimensional regularization. The result is explicitly derived and tested in 2, 4 and 6 dimensions.

Figures (13)

• Figure 1: One loop expansion of the 3-graviton vertex in terms of the triangle and the self-energy-type contributions. Shown here are the diagrams for the scalar particle. The expansion for the other two CFT's investigated can be obtained by replacing the scalar by a fermion or a photon in the loops. In the former case one has to consider, for the triangle case, two inequivalent contributions, distinguished by the direction of flow of the momentum flow of the fermion; for the latter, ghost corrections follow the same topologies.
• Figure 2: $TTT$ and its counterterms generated with the choice of the square of the Weyl ($F$) tensor in 4 dimensions and the Euler density ($G$).
• Figure 3: The contributions to the renormalized $TTT$ vertex from the square of the Weyl tensor in $d$-dimensions ($F_dd$) and the Euler density ($G$).
• Figure 4: The relation between the counterterm generated by $F_d$ and the same obtained from $F$. The difference is a finite renormalization ($F_{fin}$) term in the counterterm Lagrangian, which generates the local contribution to the trace anomaly.
• Figure 5: The fermion and the scalar sectors contributing to the conformal VVV vertex in any dimension.