The kernel defining is conveniently chosen such that both and define a trace-class operator [157] (see Equation (2.10)). Once the trace-class condition is satisfied one can adjust the other features of the kernel to account for the mode suppression. These features are arbitrary to some extent; what matters is the limiting behavior for and (with foresight) . In the simplest case we require that is smooth in all variables and of the factorized form

In the first condition is a smooth approximation to the delta distribution, normalized such that . In Fourier space the finiteness of the trace then amounts to . The condition (C.4) guarantees that the large momentum modes are integrated out in the usual way, while the behavior for small leads to a suppression of the small momentum modes by a soft mass-like IR cut-off. Indeed, if the bare action has the structure , the addition of a term as in Equation (2.10) leads to where is the renormalized mass and the dots indicate the interaction terms and terms which vanish for . Obviously the cutoff function has the interpretation of a momentum dependent mass square which vanishes for and assumes the constant value for . How is assumed to interpolate between these two regimes is a matter of calculational convenience. In practical calculations one often uses the exponential cutoff , but many other choices are possible [29, 146].Next one introduces the Legendre transform of ,

which is a convex functional of . Making the usual simplifying assumption that admits a series expansion in powers of , a formal inversion of the series defines a unique configuration with the property and . The actual effective average action is defined by The subtraction of the mode suppression term is essential for the properties listed below. The main properies of the effective average action are:- If the bare action is quadratic (free field theory) the action (but not ) is independent of and equals the bare one: , for .
- It satisfies the functional integro-differential equation for the standard effective action with playing the role of the bare action, i.e. Equation (C.8) readily follows by converting Equation (C.1) via the definitions using the relation , which is ‘dual’ to .
- It interpolates between the and the UV regularized standard effective action , according to The first relation, , follows trivially from Equation (B.8) and the fact that vanishes for all when . The limit of Equation (2.8) is more subtle. A formal argument for is as follows. Since approaches for , and large, the second exponential on the right-hand-side of Equation (2.8) becomes For this approaches a delta-functional , up to an irrelevant normalization. The integration can be performed trivially then and one ends up with , for large. In a more careful treatment [29] one shows that the saddle point approximation of the functional integral in Equation (2.8) about the point becomes exact in the limit . As a result and differ at most by the infinite mass limit of a one-loop determinant, which is ignored in Equation (2.7) since it plays no role in typical applications (see [188] for a more careful discussion).

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