C.1 Definition and basic properties
The construction of
starts out from a modified form,
, of the standard generating
functional
:
The extra factor
suppresses the “IR modes” with
. The modified
is easily seen to be still a convex functional of the source. The corresponding
,
, and
-dependent expectations of some (smooth) observable
are defined as in Equation (B.1)
The cutoff functional
is a quadratic form in the fields and has already been displayed in
Equation (2.10). In the literature it is often denoted by
to indicate that it should be thought of as
modifying the bare action.
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).