Since is convex, the effective action can be introduced as the Legendre transform , which is a convex functional of . Although is always convex it may not be differentiable everywhere. In fact, has ‘cusps’ in the case of spontaneous symmetry breaking. Even on the subspace of homogeneous solutions the supremum in the definition of may then be reached for several configurations and is ‘flat’ in these directions. If admits a series expansion in powers of , a formal inversion of the series defines a unique with the property . Often this extra assumption isn’t needed and we shall write for any configuration on which the supremum in the Legendre transform is reached; functional derivatives with respect to evaluated at will be denoted by . The defining properties of the Legendre transform then read
To switch off the source one selects configurations such that . As defined by series inversion one can directly take as the source-free condition. This is typical in the absence of spontaneous symmetry breaking, otherwise one should use the defining relation for to switch off the source. The vertex functions are defined for by, i.e. without going through the above construction.
Inserting Equation (B.3) into the definition of one sees that the effective action is characterized by the following functional integro-differential equation:). In itself of course Equation (B.8) is useless because one still has to perform a functional integral. It can be made computationally useful, however, in two ways. First as a tool to generate a recursive algorithm to compute perturbatively, and second as a starting point to derive a functional differential equation for . For the latter we refer to Appendix C.2, here we briefly recap the perturbative construction.
It is helpful to restore the implicit dependence on the UV cutoff and we write from now on. In outline, the perturbative algorithm based on Equation (B.8) involves the following steps. One introduces the loop counting parameter as follows: , , , where is the bare action depending on . After the rescaling one expands the exponent on the right-hand-side in powers of . Schematically the result is, that is, functional differentiation is denoted by and the index contraction is short for a -integration. Further we used an ansatz for of the form  (Equation (A6.12)). The presence of the UV cutoff in renders the expressions for , , well-defined.
The removal of the cutoff is done by a recursive procedure which is based on the following crucial fact: In a perturbatively (quasi) renormalizable QFT the divergent (as ) part of the loop contribution to the effective action is local, i.e. it equals a single integral over a local function in the fields and their derivatives. Moreover this divergent part has the same structure as the bare action (2.2) with specific parameter functions . This can be used to compute the parameter functions in the bare action (2.2) recursively in the number of loops.
Schematically one proceeds as follows. Let denote the bare action (2.2) at loop order, with the parameter functions , , known. In particular the limits is finite and defines the -loop renormalized action. Further the parameter functions are such that has a finite limit as . Using this bare action one computes the effective action at the next order. According to the above result it can be decomposed as for a recent review). Once the renormalized is known (to exist), the vertex functions (B.7) and the S-matrix elements are in principle likewise known to all loop orders. The latter can then be shown to be invariant under local redefinitions of the fields.
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