B.3 Renormalization of Riemannian sigma-models
Here we summarize the results on the UV renormalization of Riemannian sigma-models needed
for Section 3. We largely follow the thorough treatment of Osborn [162
] and use the results
of [84
, 201
, 57
, 110
, 220
, 80
].
The goal will be to construct the renormalized effective action based on a geodesic
background-fluctuation split and as discussed in Section 3.2.2. The variant where the normal
coordinate field is the dynamical variable is used as well as dimensional regularization. The
latter has the advantage that the additional curvature dependent terms in the
measure in
Equation (B.28) do not contribute. The explicit counterterms have been computed in minimal
subtraction and all scheme dependent quantities will refer to this scheme. The dynamical scalar fields
will be denoted by
, the background fields by
and the normal coordinate fields by
.
For the purposes of renormalization it is useful to consider an extended Lagrangian of the form
Here
, is a collection of generalized couplings/sources (of the tensor type
indicated by the index structure) that depend both on the fields
and explicitly on the point
in the “base space”. The latter is a two-dimensional Riemannian space with a classical
background metric
, extended to
dimensions in the sense of dimensional regularization,
and
. The action functional is
. The
explicit
-dependence of the sources
allows one to define local composite operators via
functional differentiation after renormalization. In addition the scalar source
provides an
elegant way to compute the nonlinear renormalizations of the quantum fields in the background
expansion [110
].
Recall that the geodesic background-fluctuation split involves decomposing the fields
into a
background field configuration
and a power series in the fluctuation fields
whose coefficients are
functions of
. The series is defined in terms of the unique geodesic
from the point
(with
, say) in the target manifold to the (nearby) point
(with
), where
is the
tangent to
vector at
. We shall write
for this series, and refer to
,
, and
as the full field, the background field, and the quantum field, respectively.
On the bare level one starts with fields
which upon renormalization are
replaced by
. The expansion of the Lagrangian (B.57) can be computed from
The
derivatives can be reduced to background covariant derivatives
,
where
are the Christoffel symbols of
evaluated at
. One easily verifies that along with
and
also
transform as vectors under background field
transformations. Hence all monomials built from a covariant tensor
in
,
by contracting it with combinations of
,
,
, will be invariant under background field
transformations. These are precisely the monomials entering the expansion of a reparameterization invariant
Lagrangian like
. For example for the metric part
one finds from
Equation (B.58)
etc., with
of order
in
. Here
is the Riemann tensor of
evaluated at
.
Both
and
transform as vectors under reparameterizations of the background fields
. The
-th order term
in Equation (B.58) is of order
in
and at most quadratic
in
. The generalized couplings/sources
are evaluated at
and transform according to their
respective tensor type under
. In total this renders each term in Equation (B.58) individually
invariant under these background field transformations. The term
quadratic in
is used
to define the propagators, all other terms are treated as interactions. Due to the expansion
around a nontrivial background field configuration
does not have a standard kinetic term,
e.g. the first term in Equation (B.57) gises rise to
To get standard
kinetic terms one introduces the components
, with respect to a frame field
satisfying
. Rewriting
in terms of the frame fields
a potential is
generated which combines with the other terms in
, but the
have a standard kinetic term
and are formally massless. These fields are given a mass
which sets the renormalization
scale. The development of perturbation theory then by-and-large follows the familar lines, the
main complication comes from the complexity of the interaction Lagrangians
. In
addition nonlinear field renormalizations are required; the transition function
can be
computed from the differential operator
in Equation (B.57) below. For our purposes we
in addition have to allow for a renormalization
of the background fields. As usual
we adopt the convention that the fields
remain dimensionless for base space dimension
.
With this setting the bare couplings/sources
have dimension
and are expressed as a
dimensionless sum of the renormalized
and covariant counter tensors built from
. A
suitable parameterization is
Here
denotes differentiation with respect to
at fixed
. The quantities
and
contain poles in
(but no other type of singularities) whose coefficients are defined
by minimal subtraction. Except for
they depend on
only;
in addition depends quadratically
on
and
, but the quadratic forms with which they are contracted again only depend on
.
All purely
-dependent counter tensors are algebraic functions of
, its covariant derivatives, and its
curvature tensors.
and
specifically are linear differential operators acting on scalars and
vectors on the target manifold, respectively. The combined pole and loop expansion takes the form
for any of the quantities
. The residue of the simple pole is denoted by
. We do not include explicitly powers of the loop counting parameter
in Equation (B.62). For
the purely
-dependent counter terms of interest here they are easily restored by inserting
and utilizing the scaling properties listed below. However once
is ‘deformed’ into a
nontrivial function of
the ‘scaling decomposition’ (B.62) no longer coincides with the expansion
in powers of
and the former is the fundamental one. Under a constant rescaling of the
metric the purely
-dependent counter term coefficients transform homogeneously as follows
In principle the higher order pole terms
,
, are determined recursively by the residues
of the first order poles via “generalized pole equations”. The latter can be worked out in
analogy to the quantum field theoretical case (see [57
, 162
]). Taking the consistency of the
cancellations for granted one can focus on the residues of the first order poles, which we shall do
throughout.
Explicit results for them are typically available up to and including two loops [84, 220
, 57
, 110
, 162
].
For the metric
and the dilaton
beta functions also the three-loop results are
known:
where the three-loop term has been computed independently in [80] and [96]. For
the results
are [162
, 57
, 220
]
where
is the dimension of the target manifold. For the other quantities one has [162
, 110
, 220
]
The expressions for
and
are likewise known [162
] but are not needed here.
Some explanatory comments should be added. First, in addition to the minimal subtraction scheme the
above form of the counter tensors refers to the background field expansion in terms of Riemannian normal
coordinates. If a different covariant expansion is used the counter tensors change. Likewise the
standard form of the higher pole equations [57
, 162
] is only valid in a preferred scheme. For
instance for the metric counter terms in this scheme additive contributions to
of the form
are absent [110
]. Note that adding such a term for
leaves the metric beta
function in Equation (B.76) below unaffected, provided
is functionally independent of
.
So far only the full fields entered,
on the bare and
on the renormalized level. Their split into
background and quantum contributions is however likewise subject to renormalization. A convenient way to
determine the transition function
from the bare to the renormalized quantum fields was found by
Howe, Papadopolous, and Stelle [110
]. In effect one considers the inversion
of the normal
coordinate expansion
of the renormalized fields. If
in Equation (B.61, B.66) is
regarded as a differential operator acting on the second argument of this function, i.e. on
,
one obtains the desired
relation by inversion. To lowest order
yields
At each loop order the coefficient is a power series in
whose coefficients are covariant expressions built
from the metric
at the background point.
With all these renormalizations performed the result can be summarized in the proposition [110, 162
]
that the source-extended background functional
defines a finite perturbative measure to all orders of the loop expansion. The additional source
here is constrained by the requirement that
. The key properties of
are:
- It is invariant under reparameterizations of the background fields
.
- It obeys a simple renormalization group equation (which would not be true without the
F-source).
- A generalized action principle holds that allows one to construct local composite operators of
dimension
, by variation with respect to the renormalized sources.
Let
,
,
be a scalar, a vector, and a symmetric tensor on the target manifold,
respectively. ‘Pull-back’ composite operators of dimension 0,1,2 are defined by [162
]
The functional derivatives here act on functionals on the target manifold at fixed
,
e.g.
. For
in addition the dependence of the counter terms on
has to be taken into account, so that
. Further
is the bare Lagrangian regarded as a function of the renormalized quantities. The
contractions on the base space are with respect to the background metric
. The additional total
divergence in the last relation of Equation (B.70) reflects the effect of operator mixing. The normal
products as given in Equation (B.70) still refer to the functional measure as defined by the source-extended
Lagrangian. After all differentiations have been performed the sources should be set to zero
or rendered
-independent again to get the composite operators e.g. for the purely metric
sigma-model.
The definition (B.70) of the normal products is consistent with redefinitions of the couplings/sources
that change the Lagrangian only by a total divergence. The operative identities are
for a scalar
. They entail
Moreover the invariance of the regularization under reparameterizations of the target manifold allows one to
convert the reparameterization invariance of the basic Lagrangian (B.57) into a “diffeomorphism Ward
identity” [201
, 162
]:
with
. The Lie derivative terms on the right-hand-side are the response of the
couplings/sources under an infinitesimal diffeomorphism
. Thus
may be viewed as
a “diffeomorphism current”. The last term on the right-hand-side is the (by itself finite) “equations of
motion operator”. In deriving Equation (B.73) the following useful consistency conditions arise
So far the renormalization was done at a fixed normalization scale
. The scale dependence of the
renormalized couplings/sources
is governed by a set of renormalization functions
which follow from Equation (B.61). For a counter tensor of the form (B.62) it is convenient to introduce
which in view of Equation (B.62) can be regarded as a parametric derivative of
. Then
The associated renormalization group operator is
For example the dimension 0 composite operators in Equation (B.70) obey
and similar equations hold for the dimension 1, 2 composite operators.
An important application of this framework is the determination of the Weyl anomaly as an ultraviolet
finite composite operator. We shall only need the version without vector and scalar functionals. The result
then reads [201
, 220
, 162
]
Here the so-called Weyl anomaly coefficients enter:
where
and
are the renormalization group functions of Equation (B.76) and
These expressions hold in dimensional regularization, minimal subtraction, and the backgound field
expansion in terms of normal coordinates. Terms proportional to the equations of motion operator
have been omitted. The normal-products (B.70) are normalized such that the expectation value of an
operator contains as its leading term the value of the corresponding functional on the background,
, where the subleading terms are in general nonlocal and depend on the
scale
. For the expectation value of the trace anomaly this produces a rather cumbersome
expression (see e.g. [220]). As stressed in [201
] the result (B.70), in contrast, allows one to
use
as a simple criterion to select functionals which ‘minimize’ the conformal
anomaly.
The Weyl anomaly coefficients (and the anomaly itself) can be shown to be invariant under field
redefinitions of the form
with
functionally independent of the metric. Roughly speaking
Equation (B.82) changes the beta function by a Lie derivative term that is compensated by a contribution
of the diffeomorphism current to the anomaly which amounts to
[201
]. It is important to
distinguish these diffeomorphisms from field renormalizations like Equation (3.65, 3.70) that depend on the
metric. Although formally Equation (B.82) amounts to
in
Equation (3.65); clearly one cannot cancel one against the other. The distinction is also highlighted by
considering the change in the metric counter terms
under Equation (B.82). Without further specifications this would not be legitimate for a
-dependent
vector. Although the Lie derivative term in Equation (B.83) drops out when recomputing
directly as
a parametric derivative, in combinations like
the term
in the metric beta function of Equation (B.76) induces an effective shift
Similarly
is shifted to
and the Weyl anomaly coefficients are invariant.
In the context of Riemannian sigma-models
is usually interpreted as a “string dilaton” for the
systems (B.57) defined on a curved base space. If one is interested in the renormalization of
Equation (B.57) on a flat base space,
on the other hand plays the role of a potential for the
improvement term of the energy momentum tensor. This role of
can be made manifest by rewriting
Equation (B.79) by means of the diffeomorphism Ward identity. Returning to a flat base space one
finds [201, 162]
where again terms proportional to the equations of motion operator have been omitted. Here
is the ‘naive’ improvement term while the additional total divergence is induced by operator
mixing.
The functions
and
are linked by an important consistency condition, the Curci–Paffuti
relation [57]. We present it in two alternative versions,
The first version displays the fact that the identity relates various
-dependent counter terms without
entering. In the second version
is introduced in a way that yields an identity among the Weyl
anomaly coefficients. It has the well-known consequence that
is constant when
vanishes:
where
is the central charge of energy momentum tensor derived from Equation (B.57).