### B.1 Standard effective action and its perturbative construction

We begin with a quick reminder on the standard effective action: After coupling to a source one has an (initially formal) functional integral representation for the Euclidean generating functional of the connected Schwinger functions: . Source dependent normalized expectation values of some (smooth) observable are defined by
and it is assumed that (at least for vanishing source) the map is a positive functional in the sense that positive functions have positive expectation values. In order to make the functional integral well-defined a UV cutoff is needed; for example one could replace by a -dimensional lattice with lattice spacing . The flat reference functional measure would then be proportional to . In the following we implicitly assume such a UV regularization but leave the details unspecified and use a continuum notation for the fields and their Fourier transforms. The dependence on the cutoff will be specified only when needed. We will also omit overall normalizations in the functional integrals. Whenever in addition to being positive is also a normalized measure (on a suitable space of functions ) it follows from Hölder’s inequality that is a convex functional of the source, i.e. , for , . Taking , and expanding in powers of gives
This means the second functional derivative is a kernel of positive type; under suitable falloff conditions it defines a positive bounded integral operator on the space of the functions . Kernels of positive type allow one to (re-)construct a Hilbert space such that schematically is recovered as an inner product (“a two-point function”). A fully fledged reconstruction of the operator picture requires knowledge of all multipoint functions and is roughly the content of the Osterwalder–Schrader reconstruction theorem.

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  [237]. 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

where Equation (B.6) is short for the fact that the -dependent integral operators with kernels and are inverse to each other. Since is a kernel of positive type, so is .

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

In a situation without spontaneous symmetry breaking the are independent of the choice of . The original connected Greens functions can be reconstructed from the and by purely algebraic means, as can be seen by repeated differentiation of Equation (B.4). Finally we remark that both the ‘connectedness property’ of the multipoint functions and the ‘irreducibility property’ of the vertex functions can be characterized intrinsically [55], 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:

(see e.g. [223]). 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

where we momentarily use DeWitt’s ‘condensed index notation’ [68], 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
Note that the term linear in drops out without assuming that is an extremizing configuration. Equation (B.9) is now re-inserted into Equation (B.8) and the exponentials involving positive powers of are expanded. This reduces the evaluation of the functional integral on the right-hand-side to the evaluation of correlators with respect to the Gaussian measure with covariance . By the source-free condition the ones with an odd number of ’s vanish, so that also the right-hand-side gives an expansion in integer powers of . Matching both sides of Equation (B.9) then gives a recursive algorithm for the computation of the , . The first two equations are
The expression for is the familiar regularized one-loop determinant. Inserting this into the second equation the reducible parts cancel and one can verify the equivalence to the two-loop result in [237] (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

where , determines the parameter functions at the next order. The “counter term” action is then added to the bare action to produce . Re-computing with this new bare action, the limit turns out to be finite, that is subdivergencies cancel as well. The traditional proof of this result (e.g. for scalar field theories) involves the analysis of Feynman diagrams and their (sub-)divergencies. More elegant is the use of flow equations (see [123] 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.