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The local Callan-Symanzik equation: structure and applications

Florent Baume, Boaz Keren-Zur, Riccardo Rattazzi, Lorenzo Vitale

TL;DR

This paper develops a comprehensive framework for the local Callan-Symanzik equation in four dimensions using the background-source method, revealing a manifestly consistent Weyl anomaly structure and a dilaton-based route to compute off-critical correlators of the trace of the energy-momentum tensor. It demonstrates that near a conformal fixed point, both UV and IR asymptotics are conformal, and derives a gradient-flow relation for the anomaly coefficient $\tilde{a}$ that yields monotonic RG-flow behavior with a positive metric in coupling space. The work elucidates the decomposition of the Weyl anomaly into a set of covariant objects, clarifies scheme-dependence, and connects the local anomaly analysis to non-local dilaton interactions that generate constrained dilaton scattering amplitudes. Altogether, the paper provides a systematic method to constrain RG flows, relate anomaly data to coupling-space geometry, and suggests avenues for extending the approach to parity-violating and supersymmetric theories.

Abstract

The local Callan-Symanzik equation describes the response of a quantum field theory to local scale transformations in the presence of background sources. The consistency conditions associated with this anomalous equation imply non-trivial relations among the $β$-function, the anomalous dimensions of composite operators and the short distance singularities of correlators. In this paper we discuss various aspects of the local Callan-Symanzik equation and present new results regarding the structure of its anomaly. We then use the equation to systematically write the n-point correlators involving the trace of the energy-momentum tensor. We use the latter result to give a fully detailed proof that the UV and IR asymptotics in a neighbourhood of a 4D CFT must also correspond to CFTs. We also clarify the relation between the matrix entering the gradient flow formula for the $β$-function and a manifestly positive metric in coupling space associated with matrix elements of the trace of the energy momentum tensor.

The local Callan-Symanzik equation: structure and applications

TL;DR

This paper develops a comprehensive framework for the local Callan-Symanzik equation in four dimensions using the background-source method, revealing a manifestly consistent Weyl anomaly structure and a dilaton-based route to compute off-critical correlators of the trace of the energy-momentum tensor. It demonstrates that near a conformal fixed point, both UV and IR asymptotics are conformal, and derives a gradient-flow relation for the anomaly coefficient that yields monotonic RG-flow behavior with a positive metric in coupling space. The work elucidates the decomposition of the Weyl anomaly into a set of covariant objects, clarifies scheme-dependence, and connects the local anomaly analysis to non-local dilaton interactions that generate constrained dilaton scattering amplitudes. Altogether, the paper provides a systematic method to constrain RG flows, relate anomaly data to coupling-space geometry, and suggests avenues for extending the approach to parity-violating and supersymmetric theories.

Abstract

The local Callan-Symanzik equation describes the response of a quantum field theory to local scale transformations in the presence of background sources. The consistency conditions associated with this anomalous equation imply non-trivial relations among the -function, the anomalous dimensions of composite operators and the short distance singularities of correlators. In this paper we discuss various aspects of the local Callan-Symanzik equation and present new results regarding the structure of its anomaly. We then use the equation to systematically write the n-point correlators involving the trace of the energy-momentum tensor. We use the latter result to give a fully detailed proof that the UV and IR asymptotics in a neighbourhood of a 4D CFT must also correspond to CFTs. We also clarify the relation between the matrix entering the gradient flow formula for the -function and a manifestly positive metric in coupling space associated with matrix elements of the trace of the energy momentum tensor.

Paper Structure

This paper contains 36 sections, 195 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The local Callan-Symanzik equation
  3. General set-up
  4. The structure of Weyl symmetry
  5. The global CS equation, dilations and conformal transformations
  6. The local CS equation and the operator algebra
  7. Ward identities and ambiguities
  8. Source reparametrization and the form of $\Delta_\sigma$
  9. Consistency conditions
  10. Dimension 2 covariant functions
  11. Limiting cases
  12. The structure of the Weyl anomaly
  13. Comments on the $R^2$ anomaly
  14. Weyl consistency conditions and gradient flows
  15. Gradient flow for the vector $\beta$-functions?
  16. ...and 21 more sections

Figures (4)

  • Figure 1: Our discussion concerns RG flows in the vicinity of a conformal fixed point, where the $\beta$-function and the anomalous dimensions can be treated as small perturbations.
  • Figure 2: The contour $C$ in the complex $s$ plane.
  • Figure 3: The 2-2 and 3-1 cuts of the on-shell dilaton scattering amplitude.
  • Figure 4: The different configurations for the diagrams with 2-2 cut.