Conformal Symmetry and Differential Regularization of the Three-Gluon Vertex

TL;DR

This paper demonstrates that the one-loop three-gluon vertex in massless QCD exhibits conformal invariance in a specific gauge and formalism, revealing new insights into the structure and regularization of gauge theory amplitudes.

Contribution

It shows that the one-loop three-gluon vertex function is conformally invariant in the background field formalism with Feynman gauge, and provides explicit conformal tensor structures for the amplitude.

Findings

The three-gluon vertex is conformally invariant at one-loop in this formalism.

The vertex decomposes into universal conformal tensors D and C.

Regularization and renormalization preserve Ward identities and allow beta-function extraction.

Abstract

The conformal symmetry of the QCD Lagrangian for massless quarks is broken both by renormalization effects and the gauge fixing procedure. Renormalized primitive divergent amplitudes have the property that their form away from the overall coincident point singularity is fully determined by the bare Lagrangian, and scale dependence is restricted to $\delta$-functions at the singularity. If gauge fixing could be ignored, one would expect these amplitudes to be conformal invariant for non-coincident points. We find that the one-loop three-gluon vertex function $\Gamma_{\mu\nu\rho}(x,y,z)$ is conformal invariant in this sense, if calculated in the background field formalism using the Feynman gauge for internal gluons. It is not yet clear why the expected breaking due to gauge fixing is absent. The conformal property implies that the gluon, ghost and quark loop contributions to $\Gamma_{\mu\nu\rho}$ are each purely numerical combinations of two universal conformal tensors $D_{\mu\nu\rho}(x,y,z)$ and $C_{\mu\nu\rho}(x,y,z)$ whose explicit form is given in the text. Only $D_{\mu\nu\rho}$ has an ultraviolet divergence, although $C_{\mu\nu\rho}$ requires a careful definition to resolve the expected ambiguity of a formally linearly divergent quantity. Regularization is straightforward and leads to a renormalized vertex function which satisfies the required Ward identity, and from which the beta-function is easily obtained. Exact conformal invariance is broken in higher-loop orders, but we outline a speculative scenario in which the perturbative structure of the vertex function is determined from a conformal invariant primitive core by interplay of the renormalization group equation and Ward identities.