| Summary: | We construct a modified version of the renormalized tree expansion developed by Gallavotti
and Nicolo, the loop regularized running covariance (LRC) tree expansion for expressing
the connected Green's functions of the imaginary charge QED model in perturbation
theory. By integrating out a sequence of slice fields, the LRC scheme generates a flow of
effective potentials V⁸. Here we do not demand that the flow of V⁸ be gauge invariant
but only that the Ward Identities hold at the end of the flow.
From the flow of V⁸, we obtain a flow of the couplings ʎ⁸ of the local parts of Vs. Using
a fixed point analysis in a suitable Banach space whose norm captures the asymptotic
form of ʎ⁸, we determine the asymptotic behavior of ʎ⁸ satisfying boundary conditions
partially fixed by the Ward Identities. At each step of the flow, the slice covariance is
transformed by shifting the local quadratic terms of V⁸ to the Gaussian measure. In
this way, the corresponding ʎ⁸ is governed by a flow of an effective coupling ζ⁸which
is ultra-violet asymptotically free around the origin. The UV asymptotic freedom of ζ⁸
provides the stability of ʎ⁸ so that the asymptotic form of ʎ⁸ can be obtained from a
primitive flow corresponding to only a few low order diagrams of the LRC expansion.
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