Conformal symmetry and differential regularization of the three-gluon vertex

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 δ-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 Г μvp(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 vet clear why the expected breaking due to gauge fixing is absent. The conformal property implies that the gluon, ghost, and quark loop contributions to Г μvp are each purely numerical combinations of two universal conformal tensors D μvp ( x, y, z) and C μvp ( x, y, z) whose explicit form is given in the text. Only D μvp has an ultraviolet divergence, although C μvp 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. Other results which are relevant to the conformal property include the following: 1. (1) An analytic calculation shows that the linear deviation from the Feynman gauge is not conformal invariant, and a separate computation using symbolic manipulation confirms that among D μ b μ background gauges, only the Feynman gauge is conformal invariant. 2. (2) The conventional (i.e., non-background) gluon vertex function is not conformal invariant because the Slavnov-Taylor identity it satisfies is more complicated than the simple Ward identity for the background vertex, and a superposition of D μvp and C μvp necessarily satisfies a simple Ward identity. However, the regulated conventional vertex can be expressed as a multiple of the tensor D μvp plus an ultraviolet finite non-conformal remainder. Mixed vertices with both external background and quantum gluons have similar properties.