Functional Callan-symanzik Equation for Qed

An exact evolution equation, the functional generalization of the CallanSymanzik method, is given for the effective action of QED where the electron mass is used to turn the quantum fluctuations on gradually. The usual renormalization group equations are recovered in the leading order but no Landau pole appears. Introduction: The idea of dimensional transmutation [1] is that a dimensionless parameter is traded for a dimensionful one. This replacement can be used to generate a renormalization flow where the field amplitude which is a quantity related to the size of the quantum fluctuations is evolved. These flows (”fluctuation flows”) prove to be equivalent with the usual momentum flows at one-loop order. This scheme is developed in this paper for QED by constructing the evolution of the one-particle irreducible (1PI) generator functional where a mass parameter controls the quantum fluctuations. We derive an exact equation describing this evolution and will recover the usual one-loop momentum flows. We conclude with some observations on the connection between the two renormalization schemes. 1 The functional approach to renormalization group started with the use of a ’sharp cut-off’ k, i.e. the elimination of the Fourier modes with |p| > k was performed [2]. An infinitesimal change k → k + ∆k produces the renormalization group (RG) equation for the Wilsonian, blocked action. This procedure has been developed later [3], as well as other schemes involving a ’smooth cut-off’, i.e. where the Fourier modes are suppressed by means of a smooth, regulated version of the step-function and the evolution of the effective action is sought [4, 5]. The work presented here, although being quite different from a formal point of view, is actually motivated by the smooth cut-off scheme for the evolution of the effective action. The difference compared to that method is that the suppression depends on the amplitude rather than the characteristic wave-length of the modes. This method has been developed for a scalar model [6] where the usual RG equations were recovered in the U.V. regime, i.e. where the mass is close to the cut-off. Finally, the same functional method was also used in QED with an external field [7] where the evolution of the generator functional of the one-particle irreducible (1PI) graphs with the amplitude of the external field led to the dependence of the 1PI graphs on the external gauge. Evolution equation: We start with the following bare Lagrangian in dimension d = 4− ε L = 1 2e2με Aμ2 (T μν + αL)Aν +Ψ (i / ∂ − / A− zm0)Ψ, (1) where T μν and L are respectively the transverse and longitudinal projectors in the inverse photon propagator, and the gauge parameter α characterizes the gauge fixing. The parameter z is introduced to control the amplitude of the fluctuations. For z >> 1 the theory is dominated by a free mass term contribution and is perturbative. As z decreases the interaction with the gauge field becomes more important and quantum corrections increase in amplitude. Our aim is to study the evolution in z of the generator functional Γz of the 1PI diagrams, the effective action. The functional Wz of the connected diagrams is given by expWz[η, η, j ] (2) = ∫