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 $\beta$-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 $\beta$-function and a manifestly positive metric in coupling space associated with matrix elements of the trace of the energy momentum tensor.
Highlights
The source method is a well established tool for probing the structure of Quantum Field Theory (QFT)
The basic idea, as sketched in figure 1, is to turn on all the possible marginal deformations of the conformal field theory (CFT), which we describe by a set of independent couplings λI, I = 1, . . . , N, such that λI = 0 corresponds to the unperturbed CFT
Like for gauge symmetry, unambiguous physical information is carried by the invariants, which in our case these are given by begin only at order (BI) = βI − SATAλ I
Summary
The source method is a well established tool for probing the structure of Quantum Field Theory (QFT). The original goal of this paper was to illustrate all these details and to present a systematic method for computing correlation functions of T off-criticality We achieved this goal by studying and applying the local Callan-Symanzik equation. We show that most of these conditions can be explicitly solved and that the anomaly can be reformulated in a manifestly consistent form, with only 3 non-trivial consistency conditions remaining One combination of these is the famous equation (1.2), while two others, involve anomalies related to external gauge fields.
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