Abstract
We consider the vacuum partition function of a 4d scalar QFT in a curved background as function of bare marginal and relevant couplings. A local UV cutoff Lambda (x) transforming under Weyl rescalings allows to construct Weyl invariant kinetic terms including Wilsonian cutoff functions. The local cutoff can be absorbed completely by a rescaling of the metric and the bare couplings. The vacuum partition function satisfies consistency conditions which follow from the Abelian nature of local redefinitions of the cutoff, and which differ from Weyl rescalings. These imply a gradient flow for beta functions describing the cutoff dependence of rescaled bare couplings. The consistency conditions allow to satisfy all but one Hamiltonian constraints required for a holographic description of the flow of bare couplings with the cutoff.
Highlights
In quantum field theories (QFTs) coupling constants including masses can be promoted to local functions of space-time in which case they become sources for corresponding operators
The introduction of a local ultraviolet cutoff in the kinetic terms of a generic QFT together with local couplings in a curved background allows to gain new insight into properties of the bare vacuum partition function as function of bare couplings. It allows to realize local Weyl transformations under which the local cutoff transforms; here the local component of the cutoff plays the role of a compensator allowing for a local renormalization group equations (RGEs) satisfied by the bare vacuum partition function
This local RGE can be solved in terms of a rescaled spacetime metric, and rescaled bare couplings such that they are dimensionless. (For marginal bare couplings, this rescaling is trivial.) one can study the variation of the bare vacuum partition function as function of the rescaled space-time metric and rescaled bare couplings under local variations of the cutoff
Summary
In quantum field theories (QFTs) coupling constants including masses can be promoted to local functions of space-time in which case they become sources for corresponding operators. One finds that the local cutoff dependence of the bare vacuum partition function can be absorbed completely by rescalings of the space-time metric and bare relevant couplings, a particular feature of a Wilsonian cutoff. They lead to consistency conditions similar to (but different from) local Weyl transformations These consistency conditions imply a gradient flow for the beta functions describing the cutoff dependence of bare marginal and rescaled relevant couplings. The remaining Hamiltonian constraints allow to reconstruct a suitable 5d bulk Lagrangian for a generic (massive) 4d QFT provided one additional condition (within the considered order of an expansion in derivatives) is satisfied This is the main result of the second part of this paper.
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