3.3 Tamed non-renormalizability
For the reasons explained in the Appendices B.2.2 and B.3 we now study the quantum theory based on
the sigma-model Lagrangian (3.16) in the setting of the covariant background field expansion. Since this is
a well-tested formalism (see Section B.3) it has the additional advantage that any unexpected findings
cannot be blamed on the use of an untested formalism. Technically it is also convenient to use dimensional
regularization and minimal subtraction. The analysis can then be done to all orders of sigma-model
perturbation theory.
Our first goal thus is to construct the infinite cut-off limit of the background effective action
in the covariant background field formalism to all orders of the loop expansion. It
turns out that this can be done only if infinitely many essential couplings are allowed, so even the
trunctated functional integral based on Equation (3.16) is not renormalizable in the strict sense. However,
once one allows for infinitely couplings strict cutoff independence (
) can be achieved. Remarkably,
for Equation (3.1) the generalized beta function for a generating functional of these couplings can be found
in closed form (see Equation (3.88) below). This allows one to study their RG flow in detail and to prove
the existence of a non-Gaussian UV stable fixed point. One also finds a Gaussian fixed point which is not
UV stable.
The main principle guiding the renormalization are generalized Ward identities for the Noether
currents (3.26) and the conformal currents (3.27). The general solution of these generalized
Ward identities suggests a space of actions which is stable under the renormalization flow. One
finds that by suitable redefinitions the general solution can always be brought into the form
Here
is the generating functional for an infinite set of couplings; it defines a function of one real variable
whose argument in Equation (3.56) is the field
. The fact that the renormalization flow does not force
one to leave the (infinite-dimensional) space of actions, Equation (3.56) of course has to be justified by
explicit construction. Below we shall show this in renormalized perturbation theory to all loop orders. Since
the ansatz (3.56) is based on a symmetry characterization is seems plausible, however, that also a
nonperturbatively constructed flow would not force one to leave the space (3.56), though one
could certainly start off with a more general ansatz containing for example higher derivative
terms.
As described above in the sigma-model perturbation theory we use dimensional regularization and
minimal subtraction. Counter terms will then have poles in
rather than containing positive powers
of the cutoff
. The role of the scale
is played by the renormalization scale
, the fields and the
couplings at the cutoff scale are called the “bare” fields and the “bare” couplings, while the fields and
couplings at scale
are referred to as “renormalized”. The fact that the ansatz (3.56) ‘works’ is
expressed in the following result:
Result (Generalized renormalizability) [154
, 155
]:
To all orders in the sigma-model loop expansion there exist nonlinear field renormalizations
such that for any prescribed bare generating coupling functional
there
exists a renormalized
such that
Both coupling functionals are related by
The
are explicitly known functionals of the renormalized
(see Equation (3.70) below). The
nonlinear field renormalizations are likewise explicitly known and given in Equations (3.65,
3.70).
A subscript ‘
’ denotes the bare fields while the plain symbols refer to the renormalized ones and
similarly for
. Notably no higher order derivative terms are enforced by the renormalization process;
strict cutoff independence can be achieved without them. However the fact that
and
differ
marks the deviation from conventional renormalizability. The pre-factor
also has a physical
interpretation: To lowest order it is for pure gravity the conformal factor in a Weyl transformation
of a generic four-dimensional metric
with two Killing vectors in adapted
coordinates.
The derivation of this result is based on a reformulation of the class of QFTs based on Equation (3.56)
as a Riemannian sigma-model in the sense of Friedan [84
]. This is a class of two-dimensional QFTs
which is also (perturbatively) renormalizable only in a generalized sense, namely by allowing for
infinitely many relevant couplings. The generating functional for these couplings in this case is a
(pseudo-)Riemannian metric
on a “target manifold
” of arbitrary dimension
and field
coordinates
, where
is the two-dimensional “base manifold”. The renormalization theory
of these systems is well understood. A brief summary of the results relevant here is given in
Appendix B.3.
The systems (3.56) can be interpreted as Riemannian sigma models where the target manifold of a
special class of “warped products” (see Equation (3.59) below) and the fields are
. The
relation between the quantum theory of these Riemannian sigma-models and the QFT based on
Equation (3.56) will roughly be that one performs an infinite reduction of couplings in a sense similar
to [236
, 160
, 174
]. The generating functional
is parameterized by
functions of
variables, while the generating functional
in Equation (3.56) amounts to one function of one variable.
Thus “
” many couplings are reduced to “
” many couplings. As always
in a reduction of couplings the nontrivial point is that this reduction can be done in a way
compatible with the RG dynamics. The original construction in [236, 160] was in the context of
strictly renormalizable QFTs with a finite number of relevant couplings. In a QFT with infinitely
many relevant couplings (QCD in a lightfront formulation) the reduction principle was used by
Perry–Wilson [174]. A general study of an ‘infinite reduction’ of couplings has been performed
in [11].
The reduction technique used here is different, but essentially Equation (3.68) below plays the role of
the reduction equation. Apart from the different derivation and the fact that the reduction is performed on
the level of generating functionals, the main difference to a usual reduction is that Equation (3.68) also
involves nonlinear field redefinitions without which the reduction could not be achieved here. The
reduction equation (3.68) thus mixes field redefinitions and couplings. From the viewpoint of
Riemannian sigma-models this amounts to the use of metric dependent diffeomorphisms on the
target manifold, a concept neither needed nor used in the context of Riemannian sigma-models
otherwise.
Since Riemannian sigma-models have been widely used in the context of “strings in curved spacetimes”
it may be worthwhile to point out the differences to their use here:
- First, the scalar fields
in Equation (3.56) parameterize a 4D spacetime metric
with 2 Killing vectors (not the position of a string in target space) while the target space
metric
here (see Equation (3.60) below) has 4 Killing vectors. It is auxiliary and not
interpreted as a physical spacetime metric. From the viewpoint of “strings in curved spacetime”
the system (3.56) (without matter), on the other hand, describes strings moving on a spacetime
with 4 Killing vectors and signature
.
- The aim in the renormalization process here is to preserve the conformal geometry in target
space, not conformal invariance on the worldsheet (base space)
. To achieve this one needs
metric dependent diffeomorphisms in target space which, as mentioned before, neither need to
be nor have been considered before in the context of Riemannian sigma-models.
- As a consequence of Difference 2 the renormalized fields
and
become scale dependent
and their renormalization flow backreacts on the coupling flow (see Equations (3.75, 3.84)
below). This aspect is absent if one naively specializes the renormalization theory of a generic
Riemannian sigma-model to a target space geometry which is a warped product (see [221
]).
- As will become clear later in the class of warped product sigma-models considered here the
Weyl anomaly is overdetermined at the fixed point of the coupling flow. In contrast to a generic
Riemannian sigma-model one is therefore not free to adjust the renormalized target space
metric
such that the Weyl anomaly vanishes and the system is a conformally invariant
2D field theory.
- The renormalization flow in Riemannian sigma-models is of the form
, where
is the renormalized generating coupling functional (“target space metric”) with the
renormalized quantum fields
inserted. Conceptually the highly nonlinear but local
on the right-hand-side thus is a (very special) composite operator, whose finiteness is guaranteed
by the construction (see Section B.3). The fact that this very special composite operator is
finite does of course not entail that any other nonlinear composite operator built from
or
is finite (without introducing additional counterterms). For example
or a
curvature combination of
not occuring in
is simply not defined off-hand. This
is true no matter how
is chosen, so the folklore that one can restrict attention
to functionals
for which the trace or Weyl anomaly vanishes and get a “finite” QFT is
incorrect (see [201
] for a discussion). Moreover the Weyl anomaly is itself a (very special)
composite operator and the condition for its vanishing is not equivalent to a partial differential
equation of the same form for any classical metric. By expanding the quantum fields
around a classical background configuration one can convert the condition for a vanishing
Weyl anomaly into a condition formulated in terms of a classical metric [220
]. However beyond
lowest order (that is, beyond the Ricci term) nonlocal terms are generated, and the resulting
cumbersome equations are rarely used. As a consequence beyond leading order (beyond Ricci
flatness modulo an improvement term) most of the “consistent string backgrounds” (defined
by ad-hoc replacing the composite operator
by a classical metric in the formula for
the Weyl anomaly as a composite operator) are actually not consistent, in the sense that the
corresponding metric re-interpreted (ad-hoc) as one with the quantum fields re-inserted does
not guarantee the vanishing of the Weyl anomaly in its operator form.
- Even the Ricci flow equations arising at lowest order have the property that for a generic
smooth target space metric the flow is often singular towards the ultraviolet [52]. For generic
target spaces the Riemannian sigma-models are therefore unlikely to give rise to genuine (not
merely effective) quantum field theories.
The situation changes drastically if one considers Riemannian sigma-models where the target manifold is
one the warped products (3.59) below. The Problem 5 is absent on the basis of the following
Non-renormalization Lemma, the Problem 6 is evaded because the Ricci-type flow arising at first
order is constant [61] while to higher orders the asymptotic safety property to be described
strikes:
Non-renormalization Lemma [154
]:
The field
is nonlinearly renormalized but once it is renormalized arbitrary powers thereof (defined by
multiplication pointwise on the base manifold) are automatically finite, without the need of additional
counterterms. In terms of the normal product defined in Appendix B.3.
for an
arbitary (analytic) function
.
Needless to say that the same is not true for
or any other of the quantum fields
. As a
consequence of this Non-renormalization Lemma the renormalization flow equations for the generating
functional
(the counterpart of
) can be consistently interpreted as an equation for a classical field,
which we also denote by
since the quantum field can be manipulated as if it was a classical field. The
resulting flow equations then take the form of a recursive system of nonlinear partial integro-differential
equations, which are studied in Section 3.4.
We now describe the derivation of these results in outline; the full details can be found
in [154
, 155
]. The class of warped product target manifolds relevant for Equation (3.56) is of the form
where
is the metric (3.11) on the symmetric space
,
is the ‘warp factor’, and
is
a flat two-dimensional space with Lorentzian metric given by the lower
block in the metric
If
is the scalar curvature of the
normalized as in Equation (3.12) the metric (3.60) has scalar
curvature
so that the warp function parameterizes the inverse curvature radius of the target space.
Here we combined the field vector (3.9) with
to a
-dimensional
vector
, and the metric (3.60) refer to this coordinate system. Further
are real
parameters kept mainly to illustrate that they drop out in the quantities of interest. The metric is chosen
such that for the parameter values
,
the Lagrangian (3.56) can be written in the form
In addition to the Killing vectors associated with
the metric (3.60) possesses two conformal
Killing vectors
and
, which together with
generate the isometries of
, i.e.
,
. Conversely any
metric with these conformal isometries can be brought into the above form. Each Killing vector
of
of course gives rise to a Noether current; the conformal Killing vectors
and
give rise to currents analogous to those in Equation (3.27). The counterpart of on-shell the
relations (3.27) is
,
. Upon quantization
in
Equation (3.62) plays the role of the loop counting parameter. In dimensional regularization
the
-loop counter terms contain poles of order
in
. We
denote the coefficient of the
-th order pole by
. In principle the higher order pole
terms are determined recursively by the residues
of the first order poles. Taking
the consistency of the cancellations for granted one can focus on the residues of the first oder
poles, which we shall do throughout. One can show [154
] that they have the following structure:
It should be stressed that this is not trivially a consequence of the block-diagonal form of Equation (3.60),
rather the properties (3.12) enter in an essential way.
The
are constants defined through the curvature scalars of
. The
are differential
polynomials in
invariant under constant rescalings of
and normalized to vanish for constant
.
The first three are
The counter terms (3.63) ought to be absorbed by nonlinear field renormalizations,
and a renormalization of the function
,
where
is the renormalization scale. Note that on both sides of Equation (3.66) the argument is the
renormalized field. The renormalized
function is allowed to depend on
; specifically we assume it to
have the form
where the first term ensures standard renormalizability at the 1-loop level – and is determined by this
requirement except for the power
. The power has no intrinsic significance; one could have chosen a
parameterization of the 4D spacetime metric
such that the action (3.10) with
replaced by
was the outcome of the classical reduction procedure. In particular the sectors
and
are
equivalent and we assume
throughout.
Combining Equations (3.60, 3.65, 3.66) and (3.63) one finds that the first order poles cancel in the
renormalized Lagrangian iff the following “reduction condition” holds:
where
and
is the Lie derivative of
,
. The
-dependence of
marks the deviation from
conventional renormalizability. Guided by the structure of Equations (3.60) and (3.63) we
search for a solution with
, where here and later on we also
use
,
for the index labeling. The Lie derivative term with this
is
The reduction condition (3.68) then is equivalent to a simple system of differential equations whose solution
is
Here we set
and slightly adjusted the notation to stress the functional dependence on
. Possibly
-dependent
integration constants have been absorbed into the lower integration boundaries of the integrals. Throughout
these solutions should be read as shorthands for their series expansions in
with
of the form (3.67).
For example
For the derivation of Equations (3.68, 3.70) we fixed a coordinate system in which the target space
metric takes the form (3.60). Under a change of parameterization
the reduction
condition (3.68) should transform covariantly, and indeed it does. The constituents transform as
The covariance of the counter terms as a function of the full field is nontrivial [110
, 41] and is one of the
main advantages of the covariant background field expansion. The relations (3.73) can be used to convert
the solutions (3.70) of the finiteness condition into any desired coordinate system on the target space. The
coordinates
and
used in Equation (3.60) are adapted to the Killing vector
and the conformal
Killing vectors
,
.
This completes the renormalization of the Lagrangian
. The nonlinear field redefinitions alluded to
in Equation (3.57) are explicitly given by Equation (3.70). The function
plays the role of a
generating function of an infinite set of essential couplings. In principle it could be expanded
with respect to a basis of
-independent functions of
with
-dependent coefficients,
the couplings. Technically the fact that these couplings are essential (in sense defined in the
introduction) follows from Equations (3.27). Since
is a nontrivial function on the base manifold,
the Lagrangian is a total divergence on shell if and only if
, or
. The
first case corresponds to the classical Lagrangian (3.16), the second case was studied (in a
different context) by Tseytlin [221]. In the case
the identity
reflects the
fact that the overall scale of the metric is an inessential parameter (see Appendix A). The
renormalization flow associated with the coupling functional
will be studied in the next
Section 3.4.
The fields themselves, here to be viewed as a collection of inessential parameters, are likewise subject to
flow equations. Recall from Equation (3.65) the relation between the bare and the renormalized fields,
where
, while
,
have been computed in Equation (3.70) and depend
on
. Since the bare fields are
-independent, the renormalized fields
have to carry
an implicit
-dependence through
. (This is analogous to the situation in an ordinary
multiplicatively renormalizable quantum field theory, where the coupling dependence of the
wave function renormalization induces a compensating
-dependence of the renormalized
fields governed by the anomalous dimension function.) The flow equations involve functional
derivatives with respect to the
field. For any functional
of
we set
Observe that for any differential or integral polynomial
in
which is homogeneous of degree
, the
functional derivative (3.74) just measures the degree,
. From Equations (3.65, 3.70) and the
-flow (3.83) below one derives
where
,
refer to Equation (3.76) with the solution of Equation (3.83) inserted for
. Note
that, conceptually, the problems decouple: One first solves the autonomous equation (3.83) to obtain the
coupling flow
which is then used to specify the right-hand-side of the
-flow equation whose
solution in turn determines the
-flow. The ‘
’ derivatives of the solution (3.70) of the reduction
condition come out as
In
we set
and anticipated in the notation that this is the
conventional beta functions of a
coset sigma-model without coupling to gravity. In
we
absorbed a
-dependent additive constant into the lower integration boundary and used
, as
.
We proceed with the renormalization of composite operators. Again we borrow techniques from
Riemannian sigma models (see Appendix B.3). The normal product of scalar, vector, and tensor
operators on the target manifold is defined in Equation (B.70). For generic composite operators
of course the bare operator viewed as a function of the bare couplings and of the bare fields
will have a different functional form from the renormalized one viewed as a function of the
renormalized couplings and fields. An important exception was already described in the above
‘Non-renormalization Lemma’ for functions of
only. This is specific to the system here. Another
class of operators for which similar non-renormalization results hold are conserved Noether
currents; this is a feature true in general. In the case at hand the relevant Noether currents are
Equation (3.26), the current
, and the “would-be” energy momentum tensor
. In
terms of the normal product (B.70) the corresponding non-renormalization results read [154
]
For the current
the identity
follows similarly on general grounds, while
the stronger identity
is a consequence of the non-renormalization Lemma. The result (3.78) will later turn out to
reflect a property of the generalized beta function. For the renormalization of constraints in
Equation (3.15) improvement terms are crucial. As in Equation (3.15) we wish to identify the
constraints
and
with the components of the “would be” energy momentum tensor
associated with the ‘deformed’ Lagrangian
. To this end we decompose the energy momentum
tensor for the Lagrangian
into a symmetric tracefree part
and an improvement term
with
. The improvement term
is trivially conserved but its trace
vanishes only on-shell. In contrast to
its functional form is not protected by the
conservation equation, and for the finiteness of the composite operator
the improvement
potential
has to be renormalized in a way that changes its functional form. This is to say, there is
no function
such that the bare and the renormalized improvement potential would
merely be related by substituting the bare field
and the renormalized one
, respectively,
into
. Rather we set
and
, where
is a potentially
-dependent constant,
is a function of the bare field
and
,
and
is a function of the renormalized field
and
. The finiteness of
can then be achieved by relating
and
(the functions, not their values) according to
Here
is a differential polynomial in
that can be computed from the counterterms in
Equation (B.61) of Appendix B.3. Note that as in Equation (3.66) the argument on both sides is the
renormalized
field. Starting from the fact that the right-hand-side of Equation (3.80) is
-independent one can derive an non-autonomous flow equation [155
]
where
is determined by
. For a given solution
the flow equation (3.81) in principle
determines
. Finally
can be shown to be a finite composite operator. Upon specification of initial data for
and
satisfying the proper boundary or fall-off conditions in
, the composite operator
is completely determined. This holds for an arbitrary coupling function
. In general the
operator
will not be trace free. The interpretation of the components of
as
quantum constraints, on the other hand, requires that the trace vanishes as
should be equal
to both
and
. We shall return to this condition below (see Equations (3.100),
etc.).