As mentioned, in perturbation theory one initially only aims at defining the expectations (A.1) as a
formal power series in the loop counting parameter
, where the sum of all
-loop contributions to a
quantity is assigned a factor
. For the reasons explained in [106
] the loop expansion does not
necessarily coincide with an expansion in powers of Planck’s constant
. For example when massless fields
are involved, 1-loop diagrams can contribute to the classical limit
. The loop counting parameter
refers to a set of free fields of mass
such that the formal expansion of the exponential in
Equation (A.1
) gives expectations whose computation can be reduced to the evaluation of Gaussians. We
denote the (quadratic) action of this set of free fields by
. The interaction is described by a set of
monomials
,
, which are “power counting renormalizable”. The latter means that their
mass dimension
is such that
. It is also assumed that the
are functionally
independent, so that the corresponding couplings are essential. The so-called “bare” action
functional then is
, where
are the essential “bare” couplings
(including masses) corresponding to the interaction monomials
. Inessential parameters are
generated by subjecting
to a suitable class of field redefinitions. In more detail one writes
The normalizations in Equation (2.1) can be chosen such that
and
,
but one is really interested in the regime where
. Inserting these parameterizations into
gives an expression of the form
So far the counterterms in Equation (2.1) have been left unspecified. The point of introducing them is of
course as a means to absorb the cut-off dependence generated by the regularized functional
integral in Equation (A.1
). Specifically, one replaces the Boltzmann factor by its power series
expansion in
, i.e.
, and aims at an
evaluation of multipoint functions
as formal power series in
. After
inserting Equation (2.1
) and the expansion of
this reduces the problem to an evaluation
of the free multipoint functions
computed with the quadratic
action
on the field space with cutoff
. The free multipoint functions will contain
contributions which diverge in the limit
. On the other hand via the parameterization (2.1
,
2.2
) the coefficients carry an adjustable
dependence. In a renormalizable QFT the
dependence in the coefficients can be chosen such as to cancel (for
) that generated
by the multipoint functions
. With this adjustment the limits
Since the renormalization scale is arbitrary, changing its value must not affect the values of
observables. The impact of a change in
can most readily be determined from Equation (2.1
). The
left-hand-sides are
independent, so by differentiating these relations with respect to
and extracting
the coefficients in a power series in (say)
and/or
consistency conditions arise for the derivatives
and
. The ones obtained from the leading order are the most interesting relations. For the
couplings one obtains a system of ordinary differential equations which define their renormalization
flow under a change of
. As usual it is convenient to work with dimensionless couplings
, where
is the mass dimension of
. The flow equations then take the form
By construction the perturbative beta functions have a fixed point at , which is called the
perturbative Gaussian fixed point. Nothing prevents them from having other fixed points, but the Gaussian
one is built into the construction. This is because a free theory has vanishing beta functions and the
couplings
have been introduced to parameterize the deviations from the free
theory with action
. Not surprisingly the stability matrix
of the
perturbative Gaussian fixed point just reproduces the information which has been put in. The
eigenvalues come out to be
modulo corrections in the loop coupling parameter, where
are the mass dimensions of the corresponding interaction monomials. For the eigenvectors one
finds a one-to-one correspondence to the unit vectors in the ‘coupling direction’
, again
with power corrections in the loop counting parameter. One sees that the couplings
not
irrelevant with respect to the stability matrix
computed at the perturbative Gaussian fixed
point are the ones with mass dimensions
, i.e. just the power counting renormalizable
ones.
The attribute “perturbative Gaussian” indicates that whenever in a nonperturbative construction of the
renormalization flow in the same ‘basis’ of interaction monomials is also a fixed point (called the
Gaussian fixed point), the perturbatively defined expectations are believed to provide an asymptotic
(nonconvergent) expansion to the expectations defined nonperturbatively based on the Gaussian fixed point,
schematically
Nevertheless, except for some special cases, it is difficult to give a mathematically precise meaning to the
‘’ in Equation (2.5
). Ideally one would be able to prove that perturbation theory is asymptotic to the
(usually unknown) exact answer for the same quantity. For lattice theories on a finite lattice this
is often possible; the problems start when taking the limit of infinite lattice size (see [159]
for a discussion). In the continuum limit a proof that perturbation theory is asymptotic has
been achieved in a number of low-dimensional quantum field theories: the superrenormalizable
and
theories [69, 43] and the two-dimensional Gross–Neveu model, where the
correlation functions are the Borel sum of their renormalized perturbation expansion [87, 89]. Strong
evidence for the asymptotic correctness of perturbation theory has also been obtained in the O(3)
nonlinear sigma-model via the form factor bootstrap [22]. In four or higher-dimensional theories
unfortunately no such results are available. It is still believed that whenever the above
is
asymptotically free in perturbation theory, that the corresponding series is asymptotic to the
unknown exact answer. On the other hand, to the best of our knowledge, a serious attempt to
establish the asymptotic nature of the expansion has never been made, nor are plausible strategies
available. The pragmatic attitude usually adopted is to refrain from the attempt to theoretically
understand the domain of applicability of perturbation theory. Instead one interprets the ‘
’ in
Equation (2.5
) as an approximate numerical equality, to a suitable loop order
and in a benign
scheme, as long as it works, and attributes larger discrepancies to the ‘onset of nonperturbative
physics’. This is clearly unsatisfactory, but often the best one can do. Note also that some of
the predictive power of the QFT considered is wasted by this procedure and that it amounts
to a partial immunization of perturbative predictions against (experimental or theoretical)
refutation.
So far the discussion was independent of the nature of the running of (which was traded for
). The chances that the vague approximate relation ‘
’ in Equation (2.5
) can be promoted to the
status of an asymptotic expansion are of course way better if
is driven towards
by the
perturbative flow. Only then is it reasonable to expect that an asymptotic relation of the form (2.5
)
holds, linking the perturbative Gaussian fixed point to a genuine Gaussian fixed point defined
by nonperturbative means. The perturbatively and the nonperturbatively defined coupling
can then be identified asymptotically and lie in the unstable manifold of the fixed point
. On the other hand the existence of a Gaussian fixed point with a nontrivial unstable
manifold is thought to entail the existence of a genuine continuum limit in the sense discussed
before. In summary, if
is traded for a running
, a perturbative criterion for the
existence of a genuine continuum limit is that the perturbative flow of
is regular with
. Since the beta functions of the other couplings are formal power series in
without constant coefficients, the other couplings will vanish likewise as
, and one
recovers the local quadratic action
at the fixed point. The upshot is that the coupling
with respect to which the perturbative expansion is performed should be asymptotically free
in perturbation theory in order to render the existence of a nonperturbative continuum limit
plausible.
The reason for going through this discussion is to highlight that is applies just as well to a
perturbative non-Gaussian fixed point. This sounds like a contradiction in terms, but it is not.
Suppose that in a situation with several couplings the perturbative beta functions
(which are formal power series in
without constant coefficients) admit a nontrivial zero,
. Suppose in addition that all the couplings lie in the unstable manifold of that
zero, i.e. the flows
are regular and
. We shall call a coupling with
this property asymptotically safe, so that the additional assumption is that all couplings are
asymptotically safe. As before one must assign
a numerical value in order to define the flow. Since
the series in
anyhow has zero radius of convergence, the ‘smallness’ of
is not off-hand
a measure for the reliability of the perturbative result (the latter intuition in fact precisely
presupposes Equation (2.5
)). Any one of the deviations
, which is of order
at
some
can be used as well to parameterize the original loop expansion. By a relabeling or
reparameterization of the couplings we may assume that this is the case for
. The original loop
expansion can then be rearranged to read
. However, if there is an underlying
nonperturbative structure at all, it is reasonable to assume that it refers to a non-Gaussian fixed point,
Summarizing: In perturbation theory the removal of the cutoff can be done independently of the properties of the coupling flow, while in a non-perturbative setting both aspects are linked. Only if the coupling flow computed from the perturbative beta functions meets certain conditions is it reasonable to assume that there exists an underlying non-perturbative framework to whose results the perturbative series is asymptotic. Specifically we formulate the following criterion:
Criterion (Continuum limit via perturbation theory):
(PTC1) Existence of a formal continuum limit, i.e. removal of the UV cutoff is possible and the
renormalized physical quantities are independent of the scheme and of the choice of interpolating fields, all
in the sense of formal power series in the loop counting parameter.
(PTC2) The perturbative beta functions have a Gaussian or a non-Gaussian fixed point and the dimension of its unstable manifold (as computed from the perturbative beta functions) equals the number of independent essential couplings. Equivalently, all essential couplings are asymptotically safe in perturbation theory.
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