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.