text:
\section{Introduction}

The empirical success of Einstein's theory of gravity does not imply that
the gravitational action must contain only the Hilbert term,
nor that any other terms must be ``small''.
For example, consider terms quadratic in curvature.
Neglecting indices and setting the cosmological constant to zero,
the Lagrangian will be of the form
\bea
\label{Lsymb}
{\cal L}=m_P^2 R+c R^2\ ,
\ena
where $m_P$ is the Planck mass, ``$R^2$'' stands for generic quadratic combinations
of curvature tensors and $c$ stands for dimensionless couplings.
In the weak field limit we can expand around flat space and write in momentum space
\bea
\label{psymb}
{\cal L}=m_P^2 p^2+c p^4\ .
\ena
The agreement of Einstein's theory with observation puts bounds on the $c$'s: $c\ll (m_P/p)^2$.
The strongest bound comes the highest momenta.
{}From the validity of Newton's law in the millimeter range, one gets the bound $c\ll 10^{62}$.
This ridiculously weak bound is just a consequence of our inability
of testing gravity at high energy, and is another way of seeing how good
the effective field theory of gravity really is \cite{donoghue,burgess}.
If one believes that gravity must be described by some quantum theory at some level,
then the logic of effective field theories leads us to expect that ${\cal L}$
will contain terms quadratic in curvatures, with coefficients $c$ presumably of order unity.
In fact, one expects to find in the action all possible diffeomorphism--invariant terms
constructed with the metric and its derivatives.

In this non-renormalizable effective field theory of gravity, the expansion parameter
is $p/m_P=p\sqrt{G}=\tilde G$, the propagator is given
by the Einstein--Hilbert term and higher derivative terms can be treated as perturbations.
On the other hand, it had been shown long ago that if one uses as propagator
the inverse of (\ref{psymb}), then the theory is perturbatively renormalizable \cite{Stelle1}.
The price one pays is that perturbative unitarity is lost.
More precisely, if the coefficients $c$ are chosen so as to make the theory
renormalizable, there is an unphysical ghost state, and if the coefficients $c$
are such that there is no ghost, the theory is not renormalizable.
Several possible ways out of this dilemma have been proposed, and we will mention some later on.
Here we would like to point out that even when this theory is renormalizable,
it is not UV complete, because Newton's constant is an irrelevant coupling
and $\tilde G$ blows up.
The only way it could be UV complete is if $\tilde G$ tended to a finite limit,
in other words if there was a fixed point (FP).
This is called asymptotic safety~\cite{Weinberg}.

The beta functions for higher derivative gravity in four dimensions have been
calculated at one loop long ago. After some early attempts~\cite{julve,ft1},
the correct beta functions for the dimensionless couplings $c$ have been obtained
by Avramidi and Barvinsky~\cite{Av}.
The calculation has been reviewed and extended to $4-\epsilon$ dimensions
in \cite{BS1,BS2}.
The running of Newton's constant and of the cosmological constant were also derived,
but only taking into account the contribution from the universal
logarithmically divergent terms (or equivalently, in dimensional regularization,
from the simple pole at $D=4$).
The non-universal terms, corresponding to quadratic and quartically divergent terms
in the effective action, were calculated later in \cite{CP,CPR}
and they were shown to lead to a FP for Newton's constant
and for the cosmological constant.
The calculation was repeated using different techniques (and hence with slightly
different results for the non-universal part) in \cite{niedermaier,sgrz}.
Some calculations that go beyond the one-loop approximation have been performed
in \cite{lauscher,bms}, and find also for the $c$ coefficients a finite but nonzero
FP value. The significance of this result is not entirely clear at the moment.

Higher derivative gravity is an interesting subject also in three dimensions,
where Einstein theory is trivial (in the sense that it does not have any propagating
degrees of freedom).
If one adds to the Einstein--Hilbert action a Chern--Simons term,
which contains three derivatives, then it contains a propagating massive graviton
whose helicity depends on the sign of the ``topological mass'' \cite{Top}.
The renormalization group (RG) of this ``topologically massive gravity''
has been studied in \cite{ps}, and its supergravity extension in \cite{ppps}.

These theories, however, violate parity.
More recently it has been shown that the addition of specific four-derivative
terms to three-dimensional gravity (with wrong sign of Einstein--Hilbert term)
makes the theory unitary \cite{BHT1}.
The addition of these terms introduces propagating massive graviton
around flat Minkowski and curved maximally symmetric spacetimes (anti-de Sitter
(AdS) and de Sitter (dS) spacetimes).
In contrast to topologically massive gravity, this so--called ``new massive gravity''
preserves parity. This is very interesting in that we have really dynamical theory of
gravity that is unitary even though higher derivative terms are included.
Various aspects of this theory have been later investigated.
Linearized excitations in the field equations were studied in \cite{liu}.
Unitarity is proven for Minkowski spacetime in \cite{nakasone,deser,gullu},
whereas it is discussed in \cite{BHT2} for maximally symmetric spacetimes.
A complete classification of the unitary theories for the most general action
with arbitrary coefficients of all possible terms is given in \cite{Ohta1}.
The partial result of unitarity condition on the flat Minkowski spacetime
was known for the usual sign of the Einstein theory \cite{nishino}.
Unfortunately, the new massive gravity turned out to
be non-renormalizable though the general theory with arbitrary coefficients for the quadratic
curvature terms is renormalizable \cite{deser,BHT3,Ohta2}.
As in the four--dimensional case, perturbative renormalizability is not
sufficient for UV completeness. For that, a FP is needed.

To establish whether new massive gravity, or some other three--dimensional gravity theory,
has a nontrivial FP, it is necessary to study its beta functions.
It is well known that Einstein gravity, in spite of not having any propagating degrees
of freedom, has such a FP, and it was shown in \cite{ps}
that also topologically massive gravity does.
Much less is known in the case when four--derivative terms are present.
The running of Newton's constant and of the cosmological constant has been
studied in \cite{Ohta3} and it has been shown that it has a FP
with the desired properties, but the running of the four--derivative couplings
has not been studied so far.
One of the main motivations of this paper is to derive the RG flow of
generic four--derivative gravities in three dimensions.
The main questions are whether a FP exists (we will see that it does)
and whether the special ratio of couplings that defines new massive gravity
is stable under RG flow (it is not).

Since the derivation of the beta functions changes little in different dimensions,
we will perform most of the calculations in arbitrary dimension and discuss
in detail only the cases $D=4$ and $D=3$, with additional results for $D=5$ and 6.

This paper is organized as follows. In sect.~2, we summarize the gravity theory we consider
with the Einstein term, cosmological constant and quadratic curvatures,
and give definitions of coupling constants.
In sect.~3, we give quadratic expansion of the action necessary for our calculation
of beta functions.
In sect.~4, we give derivation of the beta functions from functional renormalization group
equation (FRGE) in arbitrary dimensions.
Though we have the results on beta functions for arbitrary dimensions, they are very
complicated and the explicit form is not so illuminating.
So in sects.~5 to 9, we discuss the beta functions and their fixed points explicitly
only for dimensions 4, near 4, 3, 5 and 6, respectively,
and show that the theories have nontrivial UV FPs and are asymptotically safe
in all these dimensions.
By analysing beta functions near four dimensions, we find an indication
that Weyl-invariant FP exists in four dimensions.
We also find that the new massive gravity does not correspond to any FP.
Sect.~10 is devoted to our conclusions.

\section{Four--derivative Gravity}

We will consider actions of the general form
\bea
S=\int d^D x \sqrt{- g} \Big[ \frac{1}{\kappa^2}(\s R - 2 \Lambda) + \a R^2
+\b R_{\mu\nu}^2 + \c R_{\mu\nu\rho\la}^2 \Big] ,
\label{action}
\ena
where $\kappa^2=16\pi G$ is the $D$-dimensional gravitational constant,
$\sigma=\pm1$ is the sign of the Hilbert action, $\Lambda$ is the cosmological constant,
$\alpha$, $\beta$, $\gamma$ are the higher derivative couplings.
It is sometimes more convenient to use a different basis for the higher derivative terms,
namely $R^2$, the square of the Weyl tensor
\bea
C^2 = R_{\mu\nu\a\b}^2-\frac{4}{D-2} R_{\mu\nu}^2+ \frac{2}{(D-1)(D-2)} R^2,
\ena
and the Gauss--Bonnet combination
\bea
E = R_{\mu\nu\a\b}^2- 4 R_{\mu\nu}^2+ R^2,
\ena
which is topological for $D=4$ and vanishes identically for $D=3$.
Then the action has the alternative form
\bea
S = \int d^D x \sqrt{- g} \Big[ \frac{1}{\kappa^2}(\s R - 2 \Lambda)
+ \frac{1}{2\la} C^2 - \frac{1}{\rho} E + \frac{1}{\xi} R^2 +\tau \Box R \Big] ,
\label{action1}
\ena
where
\bea
\lambda = \frac{2(D-3)}{(D-2)(\beta+4\gamma)}, ~~
\rho = \frac{4(D-3)}{(D-2)\beta+4\gamma}, ~~
\xi = -\frac{4(D-1)}{4(D-1)\alpha+D\beta+4\gamma}\ .
\ena
or conversely
\bea
\a = -\frac{1}{\rho}+\frac{1}{\xi} +\frac{1}{(D-1)(D-2)\la}, ~~
\b = \frac{4}{\rho} -\frac{2}{(D-2)\la}, ~~
\c = -\frac{1}{\rho} +\frac{1}{2\la}.
\ena
Note that in $D=3$, $C^2$ and $E$ both vanish identically and the form (\ref{action1})
is not appropriate.
The couplings $\lambda$, $\rho$ and $\xi$ have mass dimension $4-D$.
In dimensions higher than three, it is customary to define the dimensionless combinations
\bea
\omega \equiv -\frac{(D-1)\la}{\xi},~~~
\theta \equiv \frac{\la}{\rho}\ .
\ena
Our conventions are summarized in Appendix A.

The variation of each term gives the field equations:
\bea
\s G_{\mu\nu} +\Lambda g_{\mu\nu} + \kappa^2(\a E_{\mu\nu}^{(1)} + \b E_{\mu\nu}^{(2)}
 + \c E_{\mu\nu}^{(3)})
=0,
\ena
where $G_{\mu\nu}$ is the Einstein tensor and
\bea
E_{\mu\nu}^{(1)} &=& 2RR_{\mu\nu}-2\nabla_\mu \nabla_\nu R
 + g_{\mu\nu}\Big(2\Box R -\frac12 R^2\Big), \nn
E_{\mu\nu}^{(2)} &=& 2R_{\mu\la}R_\nu^\la - 2 \nabla^\la \nabla_{(\mu} R_{\nu)\la}
 + \Box R_{\mu\nu} + \frac12 (\Box R -R_{\la\rho}^2 )g_{\mu\nu}, \nn
E_{\mu\nu}^{(3)} &=& 2R_{\mu\rho\la\s}R_\nu{}^{\rho\la\s}+ 4\nabla_{(\rho}\nabla_{\la)}
R_\mu{}^\rho{}_\nu{}^\la -\frac12 g_{\mu\nu} R_{\rho\la\s\tau}^2.
\ena

\section{Quadratic expansion of the action}

We will apply the standard background field method, expanding the metric as
\bea
g_{\mu\nu}= \bg_{\mu\nu} + h_{\mu\nu}\ .
\label{fluc}
\ena
In order to derive the effective action at the one-loop level,
or to calculate the one-loop beta functions,
we need the expansion of the action to second order in $h_{\mu\nu}$.
This calculation is discussed in detail in Appendix B.
The most complicated part comes from the expansion of the terms quadratic in curvature.
We report here the final form, where we have dropped terms with linear derivatives acting
on the fluctuation and terms with two derivatives acting on a background curvature
(omitting indices, these are of the form $h(\nabla\br)\nabla h$ and $h(\nabla\nabla\br)h$;
such terms do not contribute to the final results, see for example \cite{Av,BS2}).
The terms proportional to $\alpha$ can be written in the form
\bea
&&
\a h^{\mu\nu} \Big[ \nabla_\mu \nabla_\nu \nabla_\a \nabla_\b
- \bg_{\mu\nu} \Box \nabla_\a \nabla_\b - \bg_{\a\b} \nabla_\mu \nabla_\nu \Box
+\bg_{\mu\nu} \bg_{\a\b} \Box^2 -\br \bg_{\nu\b} \nabla_\a \nabla_\mu \nn
&& - (2 \br_{\mu\nu} - \br \bg_{\mu\nu})\nabla_\a \nabla_\b
+ 2 \br_{\mu\nu} \bg_{\a\b}\Box +\frac12 \br (\bg_{\mu\a}\bg_{\nu\b}
- \bg_{\mu\nu}\bg_{\a\b})\Box -\br \br_{\a\b}\bg_{\mu\nu} \nn
&&
+ 2\br \br_{\mu\a} \bg_{\b\nu} + \br_{\mu\nu} \br_{\a\b}
-\frac{1}{4}J_{\mu\nu\alpha\beta}\br^2
\Big] h^{\a\b}\ .
\label{r1}
\ena
Here and in what follows, a bar indicates that the quantity is evaluated on the background;
the indices are raised, lowered and contracted by the background metric $\bg$,
the covariant derivative $\nabla$ is constructed with the background metric.
The tensor $J$ is defined by
\bea
J_{\mu\nu\a\b}=\d_{\mu\nu,\a\b}-\frac12 \bg_{\mu\nu} \bg_{\a\b}
\label{defJ}
\ena
where
\bea
\d_{\mu\nu,\a\b} = \frac12 (\bg_{\mu\a} \bg_{\nu\b} + \bg_{\mu\b} \bg_{\nu\a})
\equiv \hat 1,
\label{id}
\ena
is the identity in the space of symmetric tensors.
We should note that due to the presence of the external factors of $h$,
the expression in the square bracket is automatically symmetrized under the interchanges
$\mu\leftrightarrow\nu$, $\alpha\leftrightarrow\beta$ and
$(\mu,\nu)\leftrightarrow (\alpha,\beta)$.

The $\b$ terms can be written in the form
\bea
\b h^{\mu\nu}\hs{-7}&& \Big[ \frac12 \nabla_\mu \nabla_\nu \nabla_\a \nabla_\b
- \frac12 \bg_{\mu\nu} \nabla_\a \Box \nabla_\b
-\frac12 \bg_{\nu\b} \nabla_\mu \Box \nabla_\a +\frac14( \bg_{\mu\a} \bg_{\nu\b}
+\bg_{\mu\nu} \bg_{\a\b})\Box^2 +\frac12 \br_{\nu\b} \nabla_\a \nabla_\mu \nn
&& -2 \br_\mu^\rho \bg_{\nu\b} \nabla_\rho\nabla_\a
+ \frac32 \bg_{\a\b}\br_{\rho\mu} \nabla^\rho \nabla_\nu
+ \br_{\mu\a\nu\b} \Box +\frac14 (2 \bg_{\mu\a} \bg_{\nu\b}
- \bg_{\mu\nu} \bg_{\a\b}) \br^{\rho\la}\nabla_\rho \nabla_\la
-\frac32 \br^\rho_\b \br_{\rho\mu\nu\a} \nn
&& + \br_{\mu\rho\la\nu} \br_\a{}^{\rho\la}{}_\b
 +\frac12 \bg_{\nu\b} \br_{\mu\rho}\br^{\rho}_\a
- \bg_{\a\b} \br_{\mu\rho} \br^\rho_\nu
-\frac{1}{4}J_{\mu\nu\alpha\beta}\br_{\rho\sigma}^2
\Big] h^{\a\b}\ ,
\label{r2}
\ena
and the terms proportional to $\c$ are
\bea
\c h^{\mu\nu} \hs{-5}&& \hs{-2} \Big[ \nabla_\mu \nabla_\nu \nabla_\a \nabla_\b
+ \bg_{\mu\a} \bg_{\nu\b} \Box^2
-2 \bg_{\nu\b}\nabla_\mu \Box \nabla_\a
- 2 \bg_{\nu\b} \br_{\mu\rho\la\a} \nabla^\rho \nabla^\la
+ 3 \br_{\mu\a\nu\b}\Box \nn
&&
- 4 \br_{\mu \a\rho\nu} \nabla^\rho \nabla_\b
- 4 \bg_{\nu\b} \br_{\rho\mu} \nabla^\rho \nabla_\a
-2\bg_{\nu\b}\br_{\mu\a}\Box
+ \bg_{\mu\a} \bg_{\nu\b} \br_{\rho\la} \nabla^\rho \nabla^\la \nn
&& -2\bg_{\mu\nu} \br_{\rho\a\la\b}\nabla^\la\nabla^\rho
+ 4 \br_{\mu\a} \nabla_\nu \nabla_\b
+ 2 \bg_{\nu\b} \br_{\mu\la\rho\s} \br_\a{}^{\la\rho\s}
-2 \bg_{\nu\b} \br^{\rho\la} \br_{\mu\rho\a\la}
\nn
&&
- \br_{\mu\la\rho\b} \br_\nu{}^{\rho\la}{}_\a
+ 3 \br_{\mu\a}{}^{\rho\la} \br_{\nu\rho\b\la}
-3 \br_{\mu\la\nu\rho} \br_\a{}^{\rho\la}{}_\b
-3 \br^\rho_\a \br_{\mu\b\nu\rho}
\nn
&&
+2 \br_{\mu\a} \br_{\nu\b}
- \bg_{\mu\nu} \br_\a{}^{\rho\la\s} \br_{\b\rho\la\s}
-\frac{1}{4}J_{\mu\nu\alpha\beta}\br_{\rho\sigma\lambda\tau}^2
\Big] h^{\a\b}.
\label{r3}
\ena
It can be checked that when arranged in the Gauss-Bonnet combination
($\gamma=\alpha$, $\beta=-4\alpha$)
and the background metric is maximally symmetric, one obtains a total derivative.
This gives a nontrivial check of the results.

The BRST transformation for the fields is found to be
\bea
\d_B g_{\mu\nu} \hs{-2}&=&\hs{-2} -\d \la [ g_{\rho\nu}\nabla_\mu c^\rho
 + g_{\rho\mu}\nabla_\nu c^\rho
] \equiv -\d\la{\cal D}_{\mu\nu,\rho}c^\rho, \nn
\d_B c^\mu \hs{-2}&=&\hs{-2} -\d\la c^\rho \nabla_\rho c^\mu,~~~
\d_B \bar c_\mu = i \d\la\, B_\mu, ~~~
\d_B B_\mu = 0,
\label{brst}
\ena
which is nilpotent. Here $c^\mu, \bar c_\mu$ and $B_\mu$ are the Faddeev-Popov ghost,
anti-ghost and an auxiliary field, respectively, and $\d\la$ is an anticommuting parameter.
The gauge fixing term and the Faddeev-Popov ghost terms are concisely written as
\bea
{\cal L}_{GF+FP}/\sqrt{-\bg} \hs{-2}&=&\hs{-2} i \d_B [\bar c_\mu Y^{\mu\nu}
(\chi_\nu-\frac{a}{2} B_\nu)]/\d\la \nn
\hs{-2} &=&\hs{-2} - B_\mu Y^{\mu\nu} \chi_\nu
+ i \bar c_\mu Y^{\mu\nu} ( \nabla^\la{\cal D}_{\la\nu,\rho}
+ b \nabla_\nu{\cal D}_{\la,\rho}^\la) c^\rho +\frac{a}{2} B_\mu Y^{\mu\nu} B_\nu \nn
&\simeq& -\frac{1}{2a} \chi_\mu Y^{\mu\nu} \chi_\nu
+ i \bar c_\mu Y^{\mu\nu} [ g_{\nu\rho} \Box +(2b+1)\nabla_\nu \nabla_\rho +\br_{\nu\rho}
] c^\rho,
\label{gfgh}
\ena
where the auxiliary field $B_{\mu}$ is integrated out in the last line.
Here
\bea
\chi_\mu &\equiv& \nabla^\la h_{\la\mu} + b \nabla_\mu h, \nn
Y_{\mu\nu} &\equiv& \bg_{\mu\nu} \Box+ c \nabla_\mu \nabla_\nu - d \nabla_\nu \nabla_\mu.
\ena
where $a$, $b$, $c$ and $d$ are gauge parameters.
We choose them such that the non-minimal four derivative terms
$\nabla_\mu \nabla_\nu \nabla_\a \nabla_\b$, $\bg_{\mu\nu} \Box\nabla_\a \nabla_\b$
and $\bg_{\nu\b} \nabla_\mu \Box \nabla_\a$ cancel. This leads to the choice~\cite{BS2}
\bea
a= \frac{1}{\b+4\c},~~~
b= \frac{4\a+\b}{4(\c-\a)}, ~~~
c-d = \frac{2(\c-\a)}{\b+4\c}-1.
\ena
In order to simplify the gauge-fixing term, we will further choose $d=1$.
With these choices, the ghost operator is
\bea
\Delta_{gh}=\delta^\mu{}_\nu\Box+(1+2b)\nabla_\mu\nabla^\nu+\br^\mu{}_\nu\ .
\ena
Then, the quadratic terms in the action can be written in the form
$h^{\mu\nu} {\cal K}_{\mu\nu,\a\b} h^{\a\b}$,
where
\bea
{\cal K}=K \Box^2 + D_{\rho\la} \nabla^\rho \nabla^\la + W.
\ena
The explicit forms of the coefficients are
\bea
(K)_{\mu\nu,\a\b} = \frac{\b+4\c}{4} \Big(\bg_{\mu\a}\bg_{\nu\b}
+\frac{4\a+\b}{4(\c-\a)}\bg_{\mu\nu}\bg_{\a\b} \Big),
\ena
\bea
(D_{\rho\la})_{\mu\nu,\a\b} \hs{-2}&=& \hs{-2}
-2\c \bg_{\nu\b}\br_{\a\rho\la\mu}
+4\c \bg_{\rho\nu} \br_{\la\a\mu\b}
+(\b + 3\c) \bg_{\rho\la} \br_{\mu\a\nu\b}
-(2\b+4\c) \bg_{\a\rho}\bg_{\nu\b} \br_{\mu\la} \nn
&& \hs{-2} -2\c \bg_{\nu\b} \br_{\mu\a} \bg_{\rho\la}
+ \b \bg_{\mu\nu}\bg_{\b\rho} \br_{\a\la}
-2\a \bg_{\a\rho} \bg_{\b\la} \br_{\mu\nu}
+2\a \bg_{\mu\nu} \bg_{\rho\la} \br_{\a\b}
+2 \c \bg_{\mu\nu} \br_{\a\rho\la\b} \nn
&& \hs{-2}
+ \Big(\frac{\a}{2}\br +\frac{\s}{4\kappa^2} \Big) (\bg_{\mu\a} \bg_{\nu\b} \bg_{\rho\la}
- \bg_{\mu\nu} \bg_{\a\b} \bg_{\rho\la}
- 2 \bg_{\nu\b} \bg_{\mu\rho} \bg_{\a\la}
+ 2 \bg_{\mu\nu} \bg_{\a\rho} \bg_{\b\la}) \nn
&& \hs{-2}  +2\c \bg_{\nu\rho} \bg_{\b\la} \br_{\mu\a}
+\Big(\frac{\b}{2}+\c \Big) \bg_{\mu\a} \bg_{\nu\b} \br_{\rho\la}
-\frac{\b}{4} \bg_{\mu\nu} \bg_{\a\b} \br_{\rho\la},
\label{der}
\ena
\bea
(W)_{\mu\nu,\a\b} \hs{-2}&=& \hs{-2}
\c \bg_{\nu\b} \br_\mu{}^{\rho\la\s} \br_{\a\rho\la\s}
+ 5\c \br_{\rho\a\mu\la} \br_{\nu\b}{}^{\rho\la}
- \c \br_{\rho\a\mu\la} \br^\rho{}_{\nu\b}{}^\la
+(\b+3\c) \br_{\rho\mu\la\nu} \br^\rho{}_\a{}^\la{}_\b \nn
&& \hs{-2} +(\b+7\c) \br^\rho_\mu \br_{\rho\a\b\nu}
+ \Big(\frac{\b}{2}+2\c \Big) \br_{\mu\a} \br_{\nu\b}
+\Big(\a\br + \frac{\s}{2\kappa^2}\Big)(\br_{\mu\a\nu\b}+
\bg_{\b\nu}\br_{\mu\a} -  \bg_{\mu\nu}\br_{\a\b}) \nn
&& \hs{-2} + \a \br_{\mu\nu} \br_{\a\b}
+ \frac{1}{8} \Big(\a \br^2 +\b \br_{\rho\la}^2 +\c \br_{\rho\la\s\tau}^2
+\frac{1}{\kappa^2}(\s\br-2\Lambda) \Big)
(\bg_{\mu\nu} \bg_{\a\b}-2\bg_{\mu\a} \bg_{\nu\b} ) \nn
&& \hs{-2} +\Big(\frac{5}{2}\b+4\c \Big) \bg_{\nu\b} \br_{\mu\rho} \br^\rho_\a
- \c \bg_{\mu\nu} \br_\a{}^{\rho\la\s} \br_{\b\rho\la\s}
- \b \bg_{\a\b} \br_{\mu\rho} \br^\rho_\nu
- 2\c \bg_{\nu\b} \br^{\rho\la} \br_{\mu\rho\a\la},
\label{cur}
\ena
where we have used the identity~\p{combo1} and dropped terms with two derivatives acting
on a background curvature.

Since the explicit factors of $h$ have been removed, one should now explicitly
perform the symmetrizations $\mu\leftrightarrow\nu$, $\alpha\leftrightarrow\beta$ and
$(\mu,\nu)\leftrightarrow (\alpha,\beta)$.
The latter involves partial integrations,
where we also drop irrelevant terms linear in $\nabla h^{\a\b}$.
After this symmetrization, the first, second and fourth terms as well as the third term
in the third line, and the first in the fourth line in ~\p{der} produce additional terms,
modifying \p{cur} to
\bea
(W)_{\mu\nu,\a\b} \hs{-2}&=& \hs{-2}
\frac32 \c \bg_{\nu\b} \br_\mu{}^{\rho\la\s} \br_{\a\rho\la\s}
+ 4\c \br_{\rho\a\mu\la} \br_{\nu\b}{}^{\rho\la}
- \c \br_{\rho\a\mu\la} \br^\rho{}_{\nu\b}{}^\la
+(\b+5\c) \br_{\rho\mu\la\nu} \br^\rho{}_\a{}^\la{}_\b \nn
&& \hs{-2} + 6\c \br^\rho_\mu \br_{\rho\a\b\nu}
+ \Big(\frac{\b}{2}+\c \Big) \br_{\mu\a} \br_{\nu\b}
+\Big(\a\br + \frac{\s}{2\kappa^2}\Big)\Big(\frac12\br_{\mu\a\nu\b}+
\frac32\bg_{\b\nu}\br_{\mu\a} -  \bg_{\mu\nu}\br_{\a\b}\Big) \nn
&& \hs{-2} + \a \br_{\mu\nu} \br_{\a\b}
+ \frac{1}{8} \Big(\a \br^2 +\b \br_{\rho\la}^2 +\c \br_{\rho\la\s\tau}^2
+\frac{1}{\kappa^2}(\s\br-2\Lambda) \Big)
(\bg_{\mu\nu} \bg_{\a\b}-2\bg_{\mu\a} \bg_{\nu\b} ) \nn
&& \hs{-2} +\Big(\frac{5}{2}\b+4\c \Big) \bg_{\nu\b} \br_{\mu\rho} \br^\rho_\a
- \c \bg_{\mu\nu} \br_\a{}^{\rho\la\s} \br_{\b\rho\la\s}
- \b \bg_{\a\b} \br_{\mu\rho} \br^\rho_\nu
- (\b+4\c) \bg_{\nu\b} \br^{\rho\la} \br_{\mu\rho\a\la},\nn
\label{curm}
\ena
where the symmetrization $\a\lra\b$ and $\mu\lra\nu$ should still be understood
and we have used the identity~\p{cid}.
As a final step, we factorize the tensor $K$ in the operator ${\cal K}$:
\bea
{\cal K}=K{\cal H}\ ;\qquad
{\cal H}=\Box^2 + V_{\rho\la} \nabla^\rho \nabla^\la +U\ .
\label{hami}
\ena
The form of the coefficients $V_{\rho\lambda}$ and $U$
is complicated and is reported in Appendix~C.
We are now ready to discuss the beta functions in the RG equation.

\section{Derivation of beta functions from functional renormalization group equation}

In the Wilsonian RG, we consider the effective action $\G_k$
describing physical phenomena at momentum scale $k$, which can be regarded as
the lower limit of the functional integration and the infrared cutoff.
The dependence of the effective action on $k$ gives the RG flow,
which can be written as a FRGE~\cite{wetterich} having on the r.h.s.
a trace of functions of the kinetic operators.

Up to this point, we have considered the action in Minkowski space.
In the following derivation of beta functions, we make Euclideanization.
That changes the signs for the Einstein and cosmological terms,
so we should change $\s\to -\s$
and $\Lambda\to -\Lambda$ in our results for the Minkowskian case.
Also in the discussion of fixed points one has to make the couplings
dimensionless by the definitions
\bea
\tilde\alpha=\alpha k^{4-D}\ ;\quad
\tilde\beta=\beta k^{4-D}\ ;\quad
\tilde\gamma=\gamma k^{4-D}\ ;\quad
\tilde\Lambda=\Lambda/k^2\ ;\quad
\tilde G\equiv \kappa^2 k^{D-2}/16\pi,\ .
\ena

In our quadratic action, we have the three operators: ${\cal H}$
acting on the graviton $h_{\mu\nu}$, the ghost operator $\Delta_{gh}$
and the third ghost operator $Y^{\mu\nu}$.
Let us choose cutoffs for the graviton, ghost and third ghost to be functions
of these operators, respectively:
$K R_k({\cal H})$ for the graviton,
$R_k(\Delta_{gh})$ for the ghosts and
$R_k(Y)$ for the third ghost.
The FRGE says that
\bea
\dot\Gamma_k=\frac{1}{2}\mbox{Tr}\frac{\dot R_k({\cal H})}{P_k({\cal H})}
-\mbox{Tr}\frac{\dot R_k(\Delta_{gh})}{P_k(\Delta_{gh})}
-\frac{1}{2}\mbox{Tr}\frac{\dot R_k(Y)}{P_k(Y)},
\ena
where we define $P_k(z)=z+R_k(z)$, and the dot represents the derivative with
respect to $\ln k$.
One can obtain the beta functions of $\Lambda$, $G$, $\alpha$, $\beta$, $\gamma$
by calculating the terms in the r.h.s. proportional to $\int dx\sqrt{g}$,
$\int dx\sqrt{g}\br$, $\int dx\sqrt{g}\br^2$,
$\int dx\sqrt{g}\br_{\mu\nu}\br^{\mu\nu}$.
$\int dx\sqrt{g}\br_{\mu\nu\rho\sigma}\br^{\mu\nu\rho\sigma}$.

In \cite{Ohta3}, the first two terms have been computed in $D=3$
using spectral sums on the sphere.
In order to separate the beta functions of $\alpha$ and $\beta$,
it is not enough to compute the traces on the sphere,
and in more general spaces we do not know the spectrum.
However, we can compute the r.h.s. using the following
general formulas for the trace of a function of an operator.
Calling $\tilde W$ the Laplace transform of $W$, we have for
a differential operator of order $p$ in $D$ dimensions:
\bea
\mathrm{Tr}[W(\Delta)] &=& \sum_n W(\lambda_n)=
\sum_n \int_{0}^{\infty}\!\! ds \, e^{-\lambda_n s} \tilde{W}(s)
=\int_{0}^{\infty}\!\! ds\,\tilde{W}(s)\,\mathrm{Tr}\, e^{-s\Delta}
\nonumber
\\
&=&
\sum_{n=0}^{\infty}B_{2n}(\Delta)
\int_{0}^{\infty}\!\! ds \, \tilde{W}(s) s^{-\frac{D}{p}+\frac{2n}{p}} =
\sum_{n=0}^{\infty} B_{2n}(\Delta)\, Q_{\frac{D-2n}{p}}(W),
\ena
where $B_{2n}$ are the coefficients appearing in the expansion of the heat kernel of the operator
\bea
\mathrm{Tr}\, e^{-s\Delta} =
\sum_{n=0}^{\infty}B_{2n}(\Delta)s^{-\frac{D}{p}+\frac{2n}{p}},
\ena
and the $Q$-functionals are given (for $m>0$) by
\bea
Q_m(W)=\int_{0}^{\infty}\!\! ds \, \tilde{W}(s) s^{-m}
=\frac{1}{\Gamma(m)}\int_{0}^{\infty}\!\! dz z^{m-1} W(z).
\ena
The last form is the more useful one.
(It is obtained more easily going from right to left.
Insert the Laplace expansion of $W$ in the r.h.s.,
exchange the order of the integrations over $s$ and $z$, and then use the integral
representation of the Gamma function.)
For $m=0$, one has $Q_0(W)=W(0)$.

With this formula the FRGE, expanded up to terms quadratic in curvature, is
\bea
\label{master}
\dot\Gamma_k
&=&
\frac{1}{2}B_0({\cal H})Q\left(4,D/4\right)
+\frac{1}{2}B_2({\cal H})Q\left(4,(D-2)/4\right)
+\frac{1}{2}B_4({\cal H})Q\left(4,(D-4)/4\right)
\nonumber
\\
&&-B_0(\Delta_{gh})Q\left(2,D/2\right)
-B_2(\Delta_{gh})Q\left(2,(D-2)/2\right)
-B_4(\Delta_{gh})Q\left(2,(D-4)/2\right)
\nonumber
\\
&&-\frac{1}{2}B_0(Y)Q\left(2,D/2\right)
-\frac{1}{2}B_2(Y)Q\left(2,(D-2)/2\right)
-\frac{1}{2}B_4(Y)Q\left(2,(D-4)/2\right).
\ena
We have to calculate the $Q$-functionals
$Q\left(p,m\right)=Q_m\left(\frac{\dot R_k}{P_k}\right)$, for an operator of order $p$.
For convenience, we choose the cutoff profile \cite{Litim}
$R_k(z)=(k^p-z)\theta(k^p-z)$, where $z$ is a differential operator of order $p$.
Define $z=yk^p$ and then we have
\bea
R_{k}(z)&=&(k^p-z)\theta (k^p-z) = k^p (1-y)\theta(1-y), \\
\dot{R}_{k}(z)&=&pk^p\theta (k^p-z) = pk^p\theta(1-y), \\
P_k (z)&=& z+R_{k}(z) = k^p\quad \mbox{for }\ z<k^p, \\
\frac{\dot R_k}{P_k}&=&p\,\theta(1-y).
\ena
For $m>0$, we find
\bea
Q(p,m) &=&
\frac{1}{\Gamma(m)}\int_{0}^{\infty}\!\! dz z^{m-1} \frac{\dot{R}_{k}(z)}{P_k (z)}
= \frac{k^{mp}}{\Gamma(m)}\int_{0}^{\infty}\!\! dy y^{m-1} p\theta(1-y)
\nonumber \\
&=&  \frac{pk^{mp}}{\Gamma(m)}\int_0^1\! dy y^{m-1}
=  \frac{pk^{mp}}{m\Gamma(m)}.
\label{charlie}
\ena
Furthermore $Q(p,0)=p$.
In $D=3$, we will need also
\bea
Q(p,-1/2)=\frac{p}{\sqrt{\pi}k^{p/2}}\ ;\qquad
Q(p,-1/4)=\frac{p}{\Gamma(3/4)k^{p/4}}\ .
\ena
which derive from \p{charlie} by analytic continuation.

Next we list the necessary heat kernel coefficients.
{}From \cite{Gusynin1990}, we have
\bea
B_4({\cal H})
\hs{-2}&=& \hs{-2}
\frac{1}{(4\pi)^{D/2}}\frac{\Gamma(D/4)}{2\Gamma((D-2)/2)}
\int d^Dx\,\mathrm{tr}\Bigl[
\frac{\hat 1}{90} \br_{\rho\la\s\tau}^2 -\frac{\hat 1}{90} \br_{\rho\la}^2
+\frac{\hat 1}{36}\br^2 +\frac{1}{6} {\cal R}_{\rho\la}{\cal R}^{\rho\la} \nn
&& \hs{-7} - \frac{2}{D-2} U - \frac{1}{6(D-2)} (2 \br_{\rho\la}V^{\rho\la}
- \br V^\rho{}_\rho) + \frac{1}{4(D^2-4)} (V^\rho{}_\rho V^\la{}_\la
+ 2 V_{\rho\la} V^{\rho\la}) \Bigr],~~
\label{grav}
\ena
where $\hat 1$ is the identity defined in \p{id} and
${\cal R}_{\rho\la}$ is the commutator of the covariant derivatives acting
on the tensor $h^{\a\b}$: ${\cal R}_{\rho\la} = [\nabla_\rho, \nabla_\la ]$.
The partial results for the traces in this formula are collected in Appendix~\ref{traces}.
Collecting, we find
\bea
B_4({\cal H})
\hs{-2}&=&\hs{-2}
\frac{1}{(4\pi)^{D/2}}\frac{\Gamma(D/4)}{2\Gamma((D-2)/2)}
\int d^Dx\, \Biggl[
\br_{\mu\nu\rho\la}^2\left(\frac{D(D+1)}{180}-\frac{D+2}{6}
-\frac{2}{D-2}A_1+\frac{12}{D^2-4}D_1\right)
\nn
&&
-\br_{\mu\nu}^2\left(\frac{D(D+1)}{180}+\frac{2}{D-2}A_2
+\frac{1}{3(D-2)}C_1-\frac{12}{D^2-4}D_2\right)
\nn
&&
+\br^2 \left(\frac{D(D+1)}{72}-\frac{2}{D-2}A_3+\frac{1}{6(D-2)}B_1
-\frac{1}{3(D-2)}C_2+\frac{12}{D^2-4}D_3\right)
\nn
&&
+\frac{\s}{\kappa^2}\br\left(-\frac{2}{D-2}A_4+\frac{1}{6(D-2)}B_2
-\frac{1}{3(D-2)}C_3+\frac{12}{D^2-4}D_4\right)
\nn
&&
+\frac{1}{\kappa^2}\left(-\frac{2\Lambda}{D-2}A_5+\frac{1}{\kappa^2}\frac{12}{D^2-4}D_5\s^2\right)
\Biggr],
\label{fre}
\ena
where the constants $A_i, B_i,C_i$ and $D_i$ are defined in Appendix~\ref{traces}.

{}From \cite{GK1999}, we have for the Euclidean operator
\bea
Y_{\mu\nu}=-\bg_{\mu\nu}\Box+\sigma_Y\nabla_\mu\nabla_\nu+\br_{\mu\nu},
\ena
with $\sigma_Y=1-2\frac{\gamma-\alpha}{\beta+4\gamma}$,
\bea
B_4(Y)
\hs{-2}&=&\hs{-2}
\frac{1}{(4\pi)^{D/2}}\int d^D x \sqrt{-\bg}
\Big[
\frac{D-16+(1-\sigma_Y)^{\frac{4-D}{2}}}{180} \br_{\mu\nu\rho\la}^2
\nn
&& \qquad\qquad
-\frac{D-91+(1-\sigma_Y)^\frac{4-D}{2}}{180} \br_{\mu\nu}^2
+\frac{D-13+(1-\sigma_Y)^\frac{4-D}{2}}{72} \br^2
\Big],
\ena
whereas for the Euclidean ghost operator
\bea
\Delta_{gh\mu\nu}=-\bg_{\mu\nu}\Box+\sigma_g\nabla_\mu\nabla_\nu-\br_{\mu\nu},
\ena
with $\sigma_g=-(1+2b)=-\left(1+2\frac{\beta+4\alpha}{4(\gamma-\alpha)}\right)$,
\bea
B_4(\Delta_{gh})
\hs{-2}&=&\hs{-2}
\frac{1}{(4\pi)^{D/2}}\int d^D x \sqrt{-\bg} \Big[
\frac{D-16+(1-\sigma_g)^{\frac{4-D}{2}}}{180} \br_{\mu\nu\rho\la}^2
\nn
&&
-\frac{(1-\sigma_g )^{-D/2}}{180D(D^2-4)\sigma_g}\Big(
D(D^2-4)\sigma_g^3
-2D(D+2)(D+58)\sigma_g^2
\nn
&&
+(D+2)(D^2+118D+720)\s_g
-1440D
\nn
&&
+(1-\sigma_g )^{D/2}
\{ (D^4-91D^3+596D^2-596D-1440 )\s_g+1440D \}
\Big)\br_{\mu\nu}^2
\nn
&&
+\frac{(1-\sigma_g )^{-D/2}}{72D(D^2-4)\sigma_g}\Big(
D(D^2-4)\sigma_g^3
-2D(D+2)(D+10)\sigma_g^2
\nn
&&
+(D+2)(D^2+22D+144) \s_g
-576
\nn
&&
+ (1-\sigma_g )^{D/2} \{(D^4+11 D^3-28D^2+52D-288)\s_g+576 \}
\Big)
\br^2
\Big],
\ena
The $B_4$ agree with the formulas in \cite{BS2} for $D=4$, but the dependence on $D$ is more
complicated than appears there.

We also need the lower heat kernel coefficients:
\bea
B_2({\cal H})
\hs{-2}&=&\hs{-2}
\frac{1}{(4\pi)^{D/2}}\frac{\Gamma((D-2)/4)}{2\Gamma((D-2)/2)}
\int d^Dx\,\sqrt{-\bg}\,\mathrm{tr}\left[\frac{\br}{6}+\frac{1}{2D}V^\mu_\mu\right],
\\
B_2(\Delta_{gh})
\hs{-2}&=&\hs{-2}
\frac{1}{(4\pi)^{D/2}}\frac{D+5+(1-\sigma_g)^{1-D/2}+\frac{12}{D}((1-\sigma_g)^{-D/2}-1)}{6}
\int d^Dx\,\sqrt{-\bg}\,\br ,
\\
B_2(Y)
\hs{-2}&=&\hs{-2}
\frac{1}{(4\pi)^{D/2}}\frac{D-7+(1-\sigma_Y)^{1-D/2}}{6}
\int d^Dx\,\sqrt{-\bg}\,\br ,
\ena

\bea
B_0({\cal H})
\hs{-2}&=&\hs{-2}
\frac{1}{(4\pi)^{D/2}}\frac{\Gamma(D/4)}{2\Gamma(D/2)}\frac{D(D+1)}{2}
\int d^Dx\,\sqrt{-\bg}\;,
\\
B_0(\Delta_{gh})
\hs{-2}&=&\hs{-2}
\frac{1}{(4\pi)^{D/2}}
(1-\sigma_g)^{-D/2}\left(1+(D-1)(1-\sigma_g)^{D/2}\right)\int d^Dx\,\sqrt{-\bg}\;,
\\
B_0(Y)
\hs{-2}&=&\hs{-2}
\frac{1}{(4\pi)^{D/2}}
(1-\sigma_Y)^{-D/2}\left(1+(D-1)(1-\sigma_Y)^{D/2}\right)\int d^Dx\,\sqrt{-\bg}\;.
\ena

Substituting the heat kernel coefficients in equation (\ref{master}),
and extracting the coefficients of $1$, $\br$, $\br^2$, $\br_{\mu\nu}\br^{\mu\nu}$,
and $\br_{\mu\nu\rho\sigma}\br^{\mu\nu\rho\sigma}$,
we obtain the beta functions of $\Lambda$, $G$, $\alpha$, $\beta$, $\gamma$.
It is possible to write the formulas for general $D$ but they are very complicated
and not especially illuminating.
In the following we shall consider the cases $D=3$, $4$, $5$, $6$ as well as
continuous dimensions close to $4$.

\section{Four dimensions}

We begin with the case $D=4$, which is the most studied in the
literature and provides a test of our techniques.
In four dimensions, the couplings $\alpha$, $\beta$ and $\gamma$
(or equivalently $\lambda$, $\xi$ and $\rho$) are all dimensionless.
We use the couplings $\tilde\Lambda=\Lambda/k^2$, $\tilde G=G\,k^2$,
$\lambda$, $\omega$ and $\theta$, as defined in sections 2 and 4.
The beta functions of the three latter couplings form a closed subsystem:
\bea
\beta_{\lambda} & =&  -\frac{1}{(4\pi)^{2}}\frac{133}{10}\lambda^{2},
\\
\beta_{\omega} & =&  -\frac{1}{(4\pi)^{2}}\frac{25 + 1098\,\omega+ 200\,\omega^2}{60}\lambda,
\\
\beta_{\theta} & = & \frac{1}{(4\pi)^{2}}\frac{7(56-171\,\theta)}{90}\lambda\ .
\ena
They agree with the results of \cite{BS2}.
The coupling $\lambda$ has the usual logarithmic approach to asymptotic freedom.
The condition $\lambda=0$ also guarantees the vanishing of the other two beta functions.
In order to find preferred values for $\omega$ and $\theta$, it is customary to define
a rescaled renormalization group time $\bt$ such that $d\bt=\lambda\, dk/k$.
With this variable, one finds
\bea
\frac{d\omega}{d\bt} & =&  -\frac{1}{(4\pi)^{2}}\frac{25 + 1098\,\omega+ 200\,\omega^2}{60}\ ,
\\
\frac{d\theta}{d\bt} & = & \frac{1}{(4\pi)^{2}}\frac{7(56-171\,\theta)}{90}\ .
\ena
which have real FPs
$$
\mbox{FP}_1:(\omega_*,\theta_*)\approx(-5.46714,0.327485)\ ;\qquad
\mbox{FP}_2:(\omega_*,\theta_*)\approx(-0.0228639,0.327485)\ .
$$
This flow is shown in Fig.~\ref{f1}.
Note that $\frac{d\lambda}{d\bt}=-\frac{1}{(4\pi)^{2}}\frac{133}{10}\lambda$,
so also with this variable one sees the FP at $\lambda=0$.

\begin{figure}[htp]
\centering
\resizebox{0.4\textwidth}{!}{
\includegraphics{omegatheta4.eps}
}
\caption{The flow in the $\omega$-$\theta$ plane in $D=4$
with the fixed points FP$_1$ and FP$_2$.}
\label{f1}
\end{figure}

The beta functions for $\tilde G$ and $\tilde\Lambda$ can be written as
\bea
\label{fleq1}
\beta_{\tilde \Lambda} & = &
-2\tilde\Lambda
+p(\lambda,\omega)\tilde\Lambda
-q(\omega)\tilde G \tilde\Lambda
+r(\omega)\tilde G
+s(\lambda,\omega)
+\frac{t(\lambda,\omega)}{\tilde G}\ ,
\\
\beta_{\tilde G} & = & 2 \tilde G-u(\lambda,\omega)\tilde G
-q(\omega)\tilde G^2
\ ,
\label{fleq2}
\ena
where
\bea
p(\lambda,\omega)&=&\frac{1}{(4\pi)^{2}}\frac{1+86\omega+40\omega^2}{12\omega}\lambda,\\
q(\omega)&=&\frac{171+298\omega+152\omega^2+16\omega^3}{36\pi\s (1+\omega)},\\
r(\omega)&=&\frac{283+664\omega+204\omega^2-128\omega^3-32\omega^4}{144\pi(1+\omega)^2},\\
s(\lambda,\omega)&=&-\frac{\s}{(4\pi)^{2}}\frac{1+10\omega}{4\omega}\lambda,\\
t(\lambda,\omega)&=&\frac{\s^2}{(4\pi)^{2}}\frac{1+20\omega^2}{256\pi\omega^2}\lambda^2,\\
u(\lambda,\omega)&=&\frac{1}{(4\pi)^{2}}\frac{3+26\omega-40\omega^2}{12\omega}\lambda.
\ena

To picture the flow of $\tilde\Lambda$ and $\tilde G$,
we set the remaining variables to their FP values $\omega=\omega_*$, $\theta=\theta_*$,
and $\lambda=\lambda_*=0$. Then, defining
$r_*=r(\omega_*)$,
$q_*=q(\omega_*)$,
the flow equations~\p{fleq1} and \p{fleq2} become very simple:
\bea
\beta_{\tilde \Lambda} & = &
-2\tilde\Lambda
+r_*\tilde G
-q_*\tilde G \tilde\Lambda,
\\
\beta_{\tilde G} & = & 2\tilde G-q_*\tilde G^2.
\ena

\begin{figure}[t]
\begin{center}
\begin{minipage}{60mm}
\includegraphics[width=6cm]{d4flow1.eps}
\put(-90,-15){(a)}
\end{minipage}
\hs{20}
\begin{minipage}{60mm}
\includegraphics[width=6cm]{d4flow2.eps}
\put(-90,-15){(b)}
\end{minipage}
\caption{The flow in the $(\tilde\Lambda,\tilde G)$--plane
(a) for the FP $(\tilde{\Lambda}_{*},\tilde{G}_{*})
\approx (0.293, -2.148)$ and
(b) for $(\tilde{\Lambda}_{*},\tilde{G}_{*})
\approx (0.209, 1.346)$.}
\label{fig1}
\end{center}
\end{figure}

The resulting flow in the $(\tilde\Lambda,\tilde G)$--plane is shown in Fig.~\ref{fig1}
for $\s=1$.
It has two FPs for each FP of $\omega$ and $\theta$.
At FP$_1$, we have $r_*\approx-0.545$, $q_*\approx -0.931/\s$.
Then there is the Gaussian FP at $\tilde{\Lambda}=\tilde{G}=0$
and another nontrivial FP at
\bea
\tilde{\Lambda}_{*}=\frac{r_*}{2 q_*}\approx 0.293 \s \ ,\ \ \ \ \
\tilde{G}_{*}=\frac{2}{q_*}\approx -2.148 \s\ .
\ena
At FP$_2$, one has
$r_*\approx 0.620$,
$q_*\approx 1.486/\s$ and again there is a Gaussian FP
and a non-Gaussian one at
\bea
\tilde{\Lambda}_{*}=\frac{r_*}{2 q_*}\approx 0.209 \s \ ,\ \ \ \ \
\tilde{G}_{*}=\frac{2}{q_*}\approx 1.346 \s\ .
\ena
We note that with the normal choice $\sigma=1$, the FP occurs
for positive $G$ in the case of FP$_2$ and for negative $G$ in the case of FP$_1$.

The attractivity properties of these FPs are determined by the stability matrix
$$
M_{ij}=\frac{\partial\tilde{\beta}_{i}}{\partial\tilde{g}_{j}}=
\left( \begin{array}{cc}
-2-q_* \tilde G_* & r_*-q_*\tilde\Lambda \\
0 & 2-2q_* \tilde G_*  \end{array} \right).
$$
At the Gaussian FP the eigenvalues of $M$ are $(-2,2)$;
the attractive eigenvector points along the $\tilde\Lambda$ axis and
the repulsive eigenvector has components $(r_*/4,1)$.
At the non--Gaussian FP the eigenvalues of $M$ are $(-4,-2)$
with the same eigenvectors as before.

Finally we mention that the beta functions of $\lambda$, $\omega$ and $\theta$,
are universal, in the sense that they do not depend on the choice of the cutoff function $R_k$.
This follows from the fact that they are proportional to the universal coefficients $Q(p,0)$.
The beta functions of $\tilde\Lambda$ and $\tilde G$ are not universal.
There are however some contributions that are proportional to $Q(p,0)$
times $B_4$ coefficients, which are universal.
These are the functions $p$, $s$, $t$, $u$, which can be written in terms
of the coefficients given in Appendix~D:
\bea
p&=&\frac{1}{192\pi^2}\left(-12A_4-6A_5+B_2-2C_3+12D_4\right)\ ;
\nonumber\\
s&=&\frac{1}{128\pi^2} B_2\ ;\qquad
t=\frac{1}{512\pi^2}D_5\ ;\nonumber\\
u&=&\frac{1}{192\pi^2}\left(12A_4-B_2+2C_3-12D_4\right)\ .
\ena
On the other hand, the functions $q$ and $r$
are proportional to the scheme--dependent coefficients,
$Q(p,1)$ and $Q(p,2)$, multiplying heat kernel coefficients $B_2$ and $B_0$.
This can be confirmed also by looking at equations (3.2) in \cite{sgrz}.\footnote{
A factor $1+10\omega^2$ in equation (8a) in \cite{CP}
contains a misprint and should read $1+10\omega$.}
It is important to stress that although the value of $Q(p,1)$ and $Q(p,2)$ can be
changed by changing the function $R_k$, they always remain positive,
so that the existence of a nontrivial FP for $\tilde G$ is universal.

\section{Near four dimensions}

Before considering the more interesting cases of three, five and six dimensions,
let us discuss some properties of the theory in $D=4+\epsilon$ dimensions.
This case had been considered previously in \cite{BS2},
but the $D$--dependence of the heat kernel coefficients which we took from
\cite{Gusynin1990,GK1999}
is more complicated than theirs, and so are the beta functions.
Also note that we do not treat the dimension as a device to regulate
divergences: the beta functions obtained by our procedure are automatically finite.
Therefore, we can treat the dimension as an external continuous parameter
and ask questions about the structure of the FPs as this parameter
changes continuously.
Since we have the complete beta functions for any $D$, we could in principle
study the $D$-dependence without any approximation.
However, the beta functions for general $D$ are quite unwieldy and therefore we limit ourselves
to the first order of a Taylor expansion around four dimensions.
The main question we are interested in here is:
what becomes of the fixed points FP$_1$ and FP$_2$ when one moves away from
four dimensions?

To first order in $\epsilon$, the beta functions are
\bea
\beta_{\lambda} & =&  -\frac{133}{160\pi^2}\lambda^{2}
+\epsilon\,\Biggl[\lambda
+\frac{\lambda^2}{17280\pi^2\omega(1-2\omega)}
\Biggl(90
\nonumber\\
&&
+\omega\left(9094-7182\gamma+15475\log2+7182\log\pi-1111\log\frac{4(1+\omega)}{3}\right)
\nonumber\\
&&
-6\omega^2\left(3058-2394\gamma+4505\log2+2394\log\pi+283\log\frac{4(1+\omega)}{3}\right)
\nonumber\\
&&
-360\omega^3\left(1+2\log\frac{2(1+\omega)}{3}\right)
-80\omega^4\left(1+2\log\frac{2(1+\omega)}{3}\right)
\nonumber\\
&&
+60(1-2\omega)(1+70\omega)\theta
-6480\omega(1-2\omega)\theta^2
\Biggr)\Biggr]\ ,
\\
\beta_{\omega} & =&  -\frac{25 + 1098\,\omega+ 200\,\omega^2}{960\pi^2}\lambda
\nonumber\\
&&
+\epsilon\,
\frac{\lambda}{34560\pi^2\omega(1-2\omega)}
\Biggl[
\omega\left(1484-450\gamma+695\log2+450\log\pi+205\log\frac{4(1+\omega)}{3}\right)
\nonumber\\
&&
+4\omega^2\left(7055-4716\gamma+10310\log2+4716\log\pi-878\log\frac{4(1+\omega)}{3}\right)
\nonumber\\
&&
-12\omega^3\left(4618-2994\gamma+5525\log2+2994\log\pi+463\log\frac{4(1+\omega)}{3}\right)
\nonumber\\
&&
-80\omega^4\left(173-90\gamma+152\log2+90\log\pi+28\log\frac{4(1+\omega)}{3}\right)
\nonumber\\
&&
-160\omega^5\left(1+2\log\frac{2(1+\omega)}{3}\right)
\nonumber\\
&&
+60(1-2\omega)(5+22\omega+80\omega^2)\theta
-12960\omega^2(1-2\omega)\theta^2
\Biggr]\ ,
\\
\beta_{\theta} & = & \frac{7(56-171\,\theta)}{1440\pi^2}\lambda
+\epsilon\,
\frac{\lambda}{34560\pi^2\omega(1-2\omega)}
\Biggl[
\nonumber\\
&&
-\omega\left(8086-4704\gamma+10525\log2+4704\log\pi-1117\log\frac{4(1+\omega)}{3}\right)
\nonumber\\
&&
+6\omega^2\left(2662-1568\gamma+2855\log2+1568\log\pi+281\log\frac{4(1+\omega)}{3}\right)
\nonumber\\
&&
+360\omega^3\left(1+2\log\frac{2(1+\omega)}{3}\right)
+80\omega^4\left(1+2\log\frac{2(1+\omega)}{3}\right)
\nonumber\\
&&
+\Biggl(-60+2\omega\left(7954-7182\gamma+15475\log2+7182\log\pi-1111\log\frac{4(1+\omega)}{3}\right)
\nonumber\\
&&
-12\omega^2\left(2598-2394\gamma+4505\log2+2394\log\pi+283\log\frac{4(1+\omega)}{3}\right)
\nonumber\\
&&
-720\omega^3\left(1+2\log\frac{2(1+\omega)}{3}\right)
-160\omega^4\left(1+2\log\frac{2(1+\omega)}{3}\right)\Biggr)\theta
\nonumber\\
&&
+120(1-2\omega)(1+70\omega)\theta^2
-12960\omega(1-2\omega)\theta^3
\Biggr]\ .
\ena

All the terms in these beta functions contain at least one power of $\lambda$,
so $\lambda=0$ is a FP.
However, in this way the values of $\omega$ and $\theta$ remain undetermined.
In order to find FPs for the latter, we proceed as in four dimensions
and we use the rescaled parameter $\bt$.
Since $\beta_\omega$ and $\beta_\theta$ are linear in $\lambda$,
the equations $\frac{d\omega}{d\bt}=0$ and $\frac{d\theta}{d\bt}=0$
do not contain any power of $\lambda$ and can be solved numerically.
Since the terms of order $\epsilon$ contain $\log(1+\omega)$,
the equations are real only for $\omega>-1$.
The fixed point FP$_1$ has $\omega<-1$ and therefore can only
be real for $\epsilon$ strictly equal to zero.
We will not discuss this FP further.

The fixed point FP$_2$ is real in a neighborhood of $D=4$,
however, for $\epsilon<-0.0663$ it becomes complex.
When one follows the evolution of this FP for increasing $\epsilon$,
one finds that the complex solution branches into two real solutions.
The more negative one is FP$_2$, the more positive one is a new FP
that we shall call FP$_3$.
When we consider the equation $\frac{d\lambda}{d\bt}=0$,
we find that all terms are linear in $\lambda$ except for the
first term in the square bracket, which is independent of $\lambda$.
Therefore, $\lambda=0$ is not a solution of $\frac{d\lambda}{d\bt}=0$.
There is however a nontrivial solution which can be found numerically
by using the FP values of $\omega$ and $\theta$.
The values of these FPs as functions of $\epsilon$ are shown in Fig.~\ref{f:ripart}.

\begin{figure}[htp]
\centering
\resizebox{0.9\textwidth}{!}{
\includegraphics{reomega.eps}
\qquad\qquad
\includegraphics{imomega.eps}
}
\vskip1cm
\resizebox{0.9\textwidth}{!}{
\includegraphics{retheta.eps}
\qquad\qquad
\includegraphics{imtheta.eps}
}
\vskip1cm
\resizebox{0.9\textwidth}{!}{
\includegraphics{relambda.eps}
\qquad\qquad
\includegraphics{imlambda.eps}
}
\caption{The real and imaginary parts of $\omega_*$ (top),
$\theta_*$ (middle) and $\lambda_*$ (bottom) as functions of $\epsilon$.
Gray curves: FP$_2$, black curves: FP$_3$.}
\label{f:ripart}
\end{figure}

The analysis of the linearized flow near the FPs shows that
$\lambda$ is always an UV--attractive eigendirection.
For $\epsilon>0$, both FP$_2$ and FP$_3$ are UV--attractive.
(This can happen because they are separated by a repulsive singular line.)
For $\epsilon<0$, FP$_2$ is attractive and FP$_3$ is mixed,
with a relevant direction lying mostly in the $\lambda$-$\theta$ plane
and an irrelevant direction lying mostly in the $\lambda$-$\omega$ plane.
A picture of the flow in the $\omega$-$\theta$ plane is shown in Fig.~\ref{f:epsilon}.

\begin{figure}[htp]
\centering
\resizebox{0.9\textwidth}{!}{
\includegraphics{omegathetanegepsilon.eps}
\qquad\qquad
\includegraphics{omegathetaposepsilon.eps}
}
\caption{The flow in the $\omega$-$\theta$ plane for $\epsilon=-0.05$ (left)
and $\epsilon=0.05$ (right). In both cases the line $\omega=0$ is a singularity
of the beta functions. Both FPs are UV-attractive for $\epsilon>0$;
FP$_2$ is UV attractive and FP$_3$ is mixed for $\epsilon<0$.
}
\label{f:epsilon}
\end{figure}

The most interesting result of this analysis is that FP$_3$
has a natural continuation at $D=4$ where
$$
\mbox{FP}_3:(\lambda_*,\omega_*,\theta_*)=(0,0,0.325296)\ .
$$
This FP cannot be seen when one directly solves the equations in $D=4$.
The reason is that (in the absence of lower derivative terms, that do not affect the
evolution of the four--derivative terms anyway) $\omega=0$ corresponds to a Weyl--invariant
four--derivative sector.
Quantization of this theory requires that also Weyl--invariance be gauge fixed and gives rise
to different beta functions from the ones discussed in the preceding section.
When one continues to $D\not=4$, the theory is no longer Weyl--invariant and the general
quantization scheme discussed in this paper is valid.
This is a strong indication for the existence of a Weyl--invariant FP
in four dimensions. We plan to study this case separately in the future.

\section{Three dimensions}

Using that in three dimensions $\br_{\mu\nu\rho\sigma}^2=4\br_{\mu\nu}^2-\br^2$,
$\c$ automatically disappears from the expressions
of the beta functions when one makes the replacement $\a\to \a-\c, \b\to \b-4\c$.
We then obtain for $\tilde\alpha=k\alpha$ and $\tilde\beta=k\beta$:
\bea
\beta_{\tilde\alpha}
\hs{-2}&=&\hs{-2}
\tilde\alpha-
\frac{1}{960 \pi ^2}
\Bigg[
\left(1+\frac{47\tilde\alpha}{\tilde\beta}
+\frac{64\tilde\alpha^2}{\tilde\beta^2}
-\frac{128\tilde\alpha}{2 \tilde\a+\tilde\beta}
\right)\sqrt{-\frac{2\tilde\beta}{\tilde\alpha}}
\nn
&&
+8\Bigg( 129+\frac{46\tilde\alpha}{\tilde\beta}
-\frac{114\tilde\alpha^2}{\tilde\beta^2}
+\frac{32\tilde\a}{2\tilde\a+\tilde\b}
+\tilde\alpha\frac{131\tilde\alpha+49\tilde\beta}{(8\tilde\alpha+3\tilde\beta)^2}
\Bigg)\Bigg],
\ena
\bea
\beta_{\tilde\beta}
\hs{-2}&=&\hs{-2}
\tilde\beta+
\frac{1}{1440\pi^2}
\Bigg[
-3\left(1
-\frac{113\tilde\a}{\tilde\b}
+\frac{64\tilde\a^2}{\tilde\b^2}
+\frac{192\tilde\a}{2\tilde\a+\tilde\b}
\right) \sqrt{-\frac{2\tilde\beta}{\tilde\alpha}}
\nn
&&
+8\Bigg(
413 - \frac{93\tilde\a}{\tilde\b}
+\frac{72\tilde\a^2}{\tilde\b^2}
+\frac{144\tilde\a}{2\tilde\alpha+\tilde\beta}
+\tilde\a\frac{652\tilde\alpha+243\tilde\beta}{(8\tilde\alpha+3\tilde\beta)^2}\Bigg)
\Bigg].
\ena
These form a closed system of equations with a FP at
$\tilde\alpha=0.0496$, $\tilde\beta=-0.1381$.

The beta functions of $\tilde\Lambda\equiv\Lambda/k^2$ and
$\tilde G\equiv G\,k$ are
\bea
\beta_{\tilde \Lambda} & = &
-2\tilde\Lambda
+p(\tilde\alpha,\tilde\beta)\tilde\Lambda
-q(\tilde\alpha,\tilde\beta)\tilde G \tilde\Lambda
+r(\tilde\alpha,\tilde\beta)\tilde G
+s(\tilde\alpha,\tilde\beta)
+\frac{t(\tilde\alpha,\tilde\beta)}{\tilde G}\ ,
\\
\beta_{\tilde G} & = & \tilde G-u(\tilde\alpha,\tilde\beta)\tilde G
-q(\tilde\alpha,\tilde\beta)\tilde G^2
\ ,
\ena
where
\bea
p(\tilde\alpha,\tilde\beta)
\hs{-2}&=&\hs{-2}
\frac{-2304\tilde\alpha^3+1120\tilde\alpha^2 \tilde\beta
+1778\tilde\alpha\tilde\beta^2 +387\tilde\beta^3
}{48\pi^2(8\tilde\a+3\tilde\b)^2 \tilde\beta^2},
\\
q(\tilde\alpha,\tilde\beta)
\hs{-2}&=&\hs{-2}
\frac{1}{3\pi\s}
\left[37 + \frac{\tilde\b}{8\tilde\a+3\tilde\b}
-\frac{16\tilde\a}{\tilde\b}
+\sqrt{-\frac{2\tilde\b}{\tilde\a}}
+4 \left(1-\frac{8\tilde\a}{\tilde\b} \right)\sqrt{\frac{-2\tilde\a}{\tilde\b}}
\right],~~~~
\\
r(\tilde\alpha,\tilde\beta)
\hs{-2}&=&\hs{-2}
\frac{1}{3\pi^2}\left[24-\frac12 \left(\frac{-2\tilde\b}{\tilde\a}\right)^{3/2}
-8 \left(\frac{-2\tilde\a}{\tilde\b}\right)^{3/2} \right],
\\
s(\tilde\alpha,\tilde\beta)
\hs{-2}&=&\hs{-2}
-\s\frac{16\tilde\alpha+5\tilde\beta}{8\pi^2(8\tilde\alpha+3\tilde\beta)\tilde\beta},
\\
t(\tilde\alpha,\tilde\beta)
\hs{-2}&=&\hs{-2}
3\s^2\frac{128\tilde\alpha^2+96\tilde\alpha\tilde\beta+19\tilde\beta^2}
{1024\pi^3(8\tilde\alpha+3\tilde\beta)^2 \tilde\beta^2},\\
u(\tilde\alpha,\tilde\beta)
\hs{-2}&=&\hs{-2}
\frac{2304\tilde\alpha^3 +2912\tilde\alpha^2\tilde\beta
+1174\tilde\alpha\tilde\beta^2 +153\tilde\beta^3}
{48\pi^2(8\tilde\alpha+3\tilde\beta)^2 \tilde\beta^2}.
\ena

If we choose $\tilde\alpha$ and $\tilde\beta$ at their FP,
one finds a FP at $\tilde\Lambda=0.5355\s$ and $\tilde G=0.1758\s$,
with eigenvalues $-2.866$ and $-1.235$. Thus this is a UV--stable FP.
This flow is depicted in Fig.~\ref{fig2}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=6cm]{d3flow.eps}
\caption{The flow in the $(\tilde\Lambda,\tilde G)$--plane
when $\tilde\alpha$ and $\tilde\beta$ are at FP$_1$}
\label{fig2}
\end{center}
\end{figure}
Note that there is no Gaussian FP in this plane because $\tilde\alpha$ and $\tilde\beta$
are already at a nontrivial FP.

Let us change coordinates and consider the beta functions of the couplings
$\lambda_3=1/\b$ and $\omega_3=-2\a/\b$.
These coupling constants are similar but different from those defined in the action~\p{action1}.
We find that they are given by
\bea
\b_{\la_3}
\hs{-2}&=&\hs{-2}
-\la_3 - \frac{\la_3^2}{480\pi^2 \sqrt{\omega_3}(\omega_3-1)}
\Big[ 2-81\omega_3-81\omega_3^2-32\omega_3^3 \nn
&&\hs{-2} +\frac{8\sqrt{\omega_3}(-1239+4660\omega_3-5720\omega_3^2+2219\omega_3^3
+8\omega_3^4+96\omega_3^5)}{(4\omega_3-3)^2} \Big], \\
\b_{\omega_3}
\hs{-2}&=&\hs{-2}
-\frac{\la_3}{480\pi^2 \sqrt{\omega_3}(\omega_3-1)}
\Big[ (1+\omega_3)(2+79\omega_3-81\omega_3^2-32\omega_3^3) \nn
&&\hs{-2} +\frac{2\sqrt{\omega_3}(4644-23486\omega_3+43055\omega_3^2-32209\omega_3^3+5788\omega_3^4
+1856\omega_3^5+384\omega_3^6)}{(4\omega_3-3)^2} \Big].~~
\ena
We see that the beta functions vanish for $\la_3=0$, leaving $\omega_3$ undetermined.
In this variable, we see that we can have the Gaussian FP for $\la_3=0$ with fixed $\omega_3$,
corresponding to $\tilde\b\to \infty$ with $\tilde\a/\tilde\b$ = fixed and $s=t=0$.
As in four dimensions, using the variable $\bt$ defined by $d\bt=\lambda_3 dk/k$
removes one power of $\lambda_3$ from the beta functions and
one finds a FP at $(\la_3,\omega_3)=(-7.240, 0.7187)$.
Another important result is that $8\tilde\a+3\tilde\b$ does not vanish at the FP.
This implies that the new massive gravity does not correspond to the FP.
This was suggested in \cite{Ohta3}, and we confirm it.

\section{Five dimensions}

In dimensions $D\not= 4$, $\omega$ and $\theta$ remain dimensionless but
$\lambda$ has dimension $4-D$, so we define $\tilde\lambda=\lambda\,k^{D-4}$.
We find the following beta functions:
\bea
\beta_{\tilde\lambda} &=& \tilde\lambda+\frac{\tilde\lambda ^2}{3360(4 \pi)^3 \omega ^2(\omega -1)}
\Big[  -60 (\omega-1)(621 \omega^2 - 42 \omega +1) \theta^2 \nn
&& + 80 ( 279 \omega-2)(\omega -1)  \omega\theta
-(270 \omega^4 + 1290 \omega^3 +131258 \omega^2 -112798 \omega +460)\omega
\nn
&&
+\sqrt{2} \omega ^2 \frac{243 \omega^4
+1872 \omega^3
+8922 \omega^2
+18304 \omega
+11619 }{\sqrt{3\omega+5}}
\Big], \\
\b_\omega &=&
\frac{\tilde\lambda}{3360(4 \pi)^3(\omega -1) \omega ^2}
\Big[ 60(1-\omega)(621 \omega^3 +147 \omega^2-17 \omega +9)\t^2 \nn
&& +40(1-\omega)( 9\omega^2 -446 \omega-63) \omega\t \nn
&&
-(270 \omega^4 +30810 \omega^3+130058 \omega^2-124370 \omega-4000)\omega^2
 \nn
&&
+\sqrt{2} \omega^2 \frac{243 \omega^5 +2925 \omega^4+15690 \omega^3+32326 \omega^2+20003 \omega
-5651}{\sqrt{3 \omega +5}} \Big], \nn
\b_\t
&=&
\frac{\tilde\lambda}{20160 (4\pi)^3 (\omega -1) \omega ^2}
\Big[ -360 (\omega-1) (621 \omega^2-42 \omega +1)\theta ^3 \nn
&&
-12 (135 \omega^4 +645 \omega^3 +70189 \omega^2 -60679 \omega-50) \t  \omega \nn
&&
+ 60 (\omega-1)(2799 \omega^2-106 \omega+3) \t^2
+ 10 (81 \omega^3 +387 \omega^2 +23103 \omega -17427) \omega^2 \nn
&&
+6 \sqrt{\frac{2}{3\omega+5}} \Big( 243 \omega^4 +1872 \omega^3
+8922 \omega^2 +18304 \omega +11619 \Big) \t \omega^2 \nn
&&
-3 \sqrt{\frac{2}{3\omega+5}} ( 243 \omega^4 +1872 \omega^3 +8698 \omega^2
+17856 \omega +12291) \omega ^2 \Big].
\ena
In the standard RG time $t=\log k$, these beta functions
have a FP for $\tilde\lambda=0$ and undetermined values of $\omega$ and $\theta$.
The flow in this sector is therefore studied more conveniently
using the variable $\bt$, which has the effect of removing one power of $\tilde\lambda$
from the beta functions.
There is then no FP with $\tilde\lambda=0$ but there are four FPs
with coordinates given in the following table:
\begin{center}
    \begin{tabular}{ | l | r | r | r |}
    \hline
     & $\tilde\lambda_*$\hs{5} & $\omega_*$\hs{3} & $\theta_*$\hs{3} \\
      \hline
    FP$_1$ & $1.2654$   & $1007.9$ & $-406.9$ \\ \hline
    FP$_2$ & $-0.61121$ & $1796.0$ & $841.4$ \\ \hline
    FP$_3$ & $-9.4483$  & $27.9$ & $0.54$ \\ \hline
    FP$_4$ & $53.53$    & $-0.082$ & $0.27$ \\ \hline
    \end{tabular}
\end{center}

For $\tilde\Lambda$ and $\tilde G$ we have the beta functions:
\bea
\b_{\tilde\Lambda}
&=&
-2 \tilde\Lambda
-\frac{\tilde\lambda  \sigma  (9 \omega +1)}{64 \pi^3 \omega}
+\frac{15 \tilde\lambda^2 \sigma^2 (9 \omega^2+1)}{32768 \pi^4 \tilde G \omega^2}
+\frac{\tilde\Lambda  \Big[ 6(9 \omega^2+1) \t + \omega( 135\omega^2+280 \omega+3) \Big]}
{1024 \pi^3 \omega^2} \tilde\lambda
\nn &&
-\tilde G \frac{ \sqrt{2} (3\omega +5)^5 -2304(3 \omega +5)^{5/2} +16384\sqrt{2}}
{960 \pi^2(3 \omega +5)^{5/2}}
-\frac{\tilde G \tilde\Lambda}{1440 \pi^2 \sigma  \omega}
\Big[144 \theta (9 \omega +1) \nn
&&
+ (2952 \omega+3240)\omega
+\sqrt{\frac{2}{3\omega+5}}(81 \omega^3+ 495 \omega^2 +975 \omega+625)\omega
+ \frac{2560\sqrt{2}\omega}{(3\omega+5)^{3/2}}
\Big] ,~~~~ \\
\b_{\tilde G}
&=&
3\tilde G + \frac{\tilde G}{1024 \pi^3 \omega^2}
\Big(6 \theta (9 \omega^2+1)
+ \omega (135 \omega^2-180 \omega -17) \Big)\tilde\lambda
-\frac{\tilde G^2}{1440 \pi^2 \sigma \omega} \Big[144 \theta (9 \omega +1) \nn
&&
+\omega (2952 \omega+3240) +\sqrt{\frac{2}{3\omega+5}}(81 \omega^3
+495 \omega^2+975\omega+625)\omega + \frac{2560\sqrt{2}\omega}{(3\omega+5)^{3/2}}
 \Big].
\ena
The Gaussian FP corresponds to $\tilde\lambda=\tilde G=\tilde\Lambda=0$,
with arbitrary values of $\omega$ and $\theta$.
One finds that the eigenvalues of the linearized flow equation at this point
are equal to $3$, $-2$, $1$, $0$, $0$, which are just the opposites of the
canonical dimensions of $G$, $\Lambda$, $\lambda$, $\omega$ and $\theta$,
as expected.

Next we consider FPs with $\tilde\lambda_*\not=0$.
We take the FP values of $\tilde\lambda$, $\omega$ and $\theta$
given in the table above and use them in the beta functions of $\tilde G$, $\tilde\Lambda$.
One finds various FPs with $\tilde G$ and $\tilde\Lambda$
either zero or nonzero.
If one chooses $\tilde G$ and $\tilde\Lambda$ equal to zero,
the matrix of the derivatives of the beta functions has
singularities in some off-diagonal elements
(more precisely the derivatives of $\tilde\Lambda$ with respect to $\tilde\lambda$,
$\omega$ and $\tilde G$).
We will not consider this case further.
The FPs with non-zero $\tilde\Lambda$ and $\tilde G$ are
\begin{center}
    \begin{tabular}{ | l | r | r | r |}
    \hline
      & $\tilde G_*$~~~~~~~ & $\tilde\Lambda_*$~~~ \\
      \hline
    FP$_1$ & $4.992\times 10^{-5}$ & $-0.4839$  \\ \hline
    FP$_2$ & $-3.976\times10^{-6}$ & $-0.5397$  \\ \hline
    FP$_3$ & $6.619\times10^{-2}$ & $-0.1201$ \\ \hline
    FP$_4$ & $13.74$ & $0.6038$ \\ \hline
    \end{tabular}
\end{center}
(These are the values for $\sigma=1$; for $\sigma=-1$ all the
elements of the table have opposite sign.)

Of these FPs, the second and the third are physically uninteresting
because of the sign of $\tilde\lambda_*$. The first three also have suspiciously
small values for $\tilde G$ and the first two also have one very large critical
exponent (of order $10^3$). It is possible that these are spurious FPs of the
type that often appear in polynomial truncations.
The fourth is more promising: the values of $\tilde G$ and $\tilde\Lambda$
are closer to those found in the Einstein-Hilbert truncation.
The critical exponents at this FP are as follows:
the cosmological constant is an eigendirection with eigenvalue $-4.635$;
the remaining eigenvalues are $-3.460$ in the direction of $\tilde G$, with a small
mixture of $\tilde\Lambda$; $-3.150$ in the direction of $\tilde G$, with a small
mixture of all the other operators;
$-1$ in the direction of $\tilde\lambda$ with a small mixture of $\tilde\Lambda$ and $\tilde G$;
$-0.9870$ in the direction of $\tilde\lambda$ with small mixtures of all the other operators.
This FP is therefore UV attractive in all five directions considered.

\section{Six Dimensions}

In dimension $D=6$, we have
\bea
\b_{\tilde\lambda}
&=&
2\tilde\lambda+
\frac{\tilde\lambda ^2}{243\times 10^3(4 \pi)^3 \omega ^2}
\Big[-1215 \theta ^2(24 \omega -1) (32 \omega -3)+2700 \theta  \omega (212 \omega -3)
\nn &&
+\frac{\omega}{4\omega+9} (2048 \omega^5 +20352 \omega^4 +181728 \omega^3 -7130808 \omega^2
-16720182 \omega +54675) \Big], \\
\beta_\omega
&=&
-\frac{\tilde\lambda}{243\times 10^3(4 \pi)^3 \omega ^2}
\Big[3645 \theta ^2 (\omega +1)\{16 \omega  (16 \omega -3)+21\}
+2700 \theta  \omega \{\omega  (124 \omega -387)-36\} \nn
&&
-\frac{\omega^2}{4\omega+9} (2048 \omega^5 +32640 \omega^4 -1510560 \omega^3
-10360440 \omega^2-15408630 \omega +7533) \Big], \\
\b_\t
&=&
\frac{\tilde\lambda}{486\times 10^3(4 \pi)^3 \omega ^2}
 \Big[-2430 \theta ^3 (24 \omega -1) (32 \omega -3)
+135 \theta ^2 \{4 \omega (2918 \omega -219)+27 \} \nn
&&
-32 \omega ^2 (16 \omega^3 +123\omega^2+1098 \omega -28134) \nn
&&
+\frac{2 \theta  \omega}{4\omega+9}
(2048 \omega^5 +20352 \omega^4 +181728 \omega^3 -7919208 \omega^2
-18526482 \omega -18225)
\Big].
\ena
Proceeding as in the five--dimensional cases one finds seven FPs:
\begin{center}
    \begin{tabular}{ | l | r | r | r |}
    \hline
     & $\tilde\lambda_*$\hs{5} & $\omega_*$\hs{3} & $\theta_*$\hs{3} \\
      \hline
    FP$_1$ & $148.358$ & $105.0$ & $-25.98$ \\ \hline
    FP$_2$ & $-37.24$ & $265.4$ & $103.0$ \\ \hline
    FP$_3$ & $55.54$ & $-33.17$ & $0.4701$\\ \hline
    FP$_4$ & $-106.647$ & $24.58$ & $0.5544$\\ \hline
    FP$_5$ & $502.2$ & $-4.545$ & $0.2493$\\ \hline
    FP$_6$ & $396.5$ & $-2.268$ & $0.1871$\\ \hline
    FP$_7$ & $513.725$ & $-0.1376$ & $0.2448$\\ \hline
    \end{tabular}
\end{center}

Upon substituting these values in the beta functions for $\tilde\Lambda$ and $\tilde G$.
\bea
\b_{\tilde\Lambda}
\hs{-2}&=&\hs{-2}
-2 \tilde\Lambda
-\frac{\tilde\lambda  \sigma  (28 \omega +3)}{768 \pi^3 \omega}
+\frac{\tilde\lambda ^2 \sigma ^2 (56 \omega^2+9)}{49152 \pi^4 \tilde G \omega^2}
+\frac{\tilde\Lambda  \Big[3(56 \omega^2+9)\t +(168 \omega^2+286 \omega+3) \omega \Big]}
{5760 \pi^3 \omega^2} \tilde\lambda \nn
&&
+\frac{\tilde G}{\pi^2}\Big[ \frac{9}{16}-\frac{1125}{16(4\omega +9)^3}
-\frac{(4\omega+9)^3}{81000}\Big]
-\frac{\tilde G\tilde\Lambda}{81000\pi^2\s\omega}\Big[
1350 \theta (28 \omega +3)
\nn &&
+\frac{\omega}{ (4 \omega +9)^2} (2048 \omega^5+26880 \omega^4 + 714240 \omega^3+3346920 \omega^2
+5175900 \omega + 2646513)\Big] ,~~~~~~ \\
\b_{\tilde G}
\hs{-2}&=&\hs{-2}
4\tilde G+ \frac{\tilde G}{5760 \pi^3 \omega^2}
\Big[3 \theta (56 \omega^2+9) +2 \omega (84 \omega^2-163 \omega -12) \Big]\tilde\lambda
-\frac{\tilde G^2}{81000 \pi^2 \sigma  \omega}
\Big[1350 \theta  (28 \omega +3) \nn
&&
+\frac{\omega}{(4\omega+9)^2} (2048 \omega^5+26880 \omega^4 + 714240 \omega^3+3346920 \omega^2
+5175900 \omega + 2646513)\Big].
\ena
one finds the following values of $\tilde G_*$ and $\tilde\Lambda_*$:
\begin{center}
    \begin{tabular}{ | l | r | r | r
     |}
    \hline
      & $\tilde G_*$\hs{7} & $\tilde\Lambda_*$\hs{6} \\
      \hline
    FP$_1$ & $0.0722003$ & $-0.679014$  \\ \hline
    FP$_2$ & $-0.00284264$ & $-0.211878$  \\ \hline
    FP$_3$ & $-0.372332$ & $-0.183136$ \\ \hline
    FP$_4$ & $0.374408$ & $-0.0119273$ \\ \hline
    FP$_5$ & $-6.84831$ & $-0.342368$ \\ \hline
    FP$_6$ & $0.0269567$ & $129.099$ \\ \hline
    FP$_7$ & $123.127$ & $1.16044$ \\ \hline
    \end{tabular}
\end{center}
(These are the values for $\sigma=1$; for $\sigma=-1$ all the
elements of the table have opposite sign.)

Of these fixed points, FP$_3$, FP$_4$, FP$_6$ and FP$_7$ are UV attractive in all directions;
the others have one or two irrelevant directions.

\section{Conclusions}

We have studied the one-loop beta functions in higher derivative gravity for general
background in arbitrary dimensions.
The results for $D=4$ agree with the literature, up to minor differences due to the
scheme--dependence of certain coefficients.
In the four--derivative sector, there are at one loop two well--known
fixed points FP$_1$ and FP$_2$.
To each of these there correspond two FPs in the $\tilde\Lambda$--$\tilde G$ planes.
There is a Gaussian FP, which is UV--attractive in the $\tilde\Lambda$ direction
and repulsive along a complementary direction, making $\tilde G$
(with a small mixture of $\tilde\Lambda$) an irrelevant coupling.
There is also a non-Gaussian FP that is UV--attractive in both directions.
Altogether the qualitative structure of the flow in the $\tilde\Lambda$--$\tilde G$ planes
is surprisingly similar to that of the theory in the Einstein--Hilbert truncation.
The main difference is the reality of the critical exponents,
to be contrasted with the complex conjugate critical exponents seen in the
Einstein--Hilbert truncation of the flow,
which lead to spiralling trajectories near the non-Gaussian FP.
It may be that the approximation used here is too crude.
On the other hand, recent studies of the Einstein--Hilbert truncation
taking into account the anomalous dimension of the fluctuation field $h_{\mu\nu}$
find real critical exponents in that case too \cite{clpr,cdp}.
(See also the general argument in \cite{bene}.)
To settle the question of the correct critical exponents, one may have to
perform more complicated calculations, for example
the full truncated RG of higher derivative gravity taking
into account the anomalous dimension of the fluctuation field.
Another issue that could be clarified by such a calculation is the position of the
FP of the dimensionless coupling $\tilde\lambda$, which was found to occur
at some non-zero value in \cite{bms}.

In this paper we have extended the calculation of one-loop beta functions of
higher derivative gravity to arbitrary dimension.
Insofar as the beta functions can be obtained directly as finite quantities,
without having to refer to divergences and regulators,
the dimension can be treated formally as a continuous parameter of the theory.
In this way the structure of the FPs can be reliably studied as a function of dimension.
There has been one previous study along these lines \cite{BS2}, but our results
differ significantly.
This can be traced to the structure of the heat kernel coefficients
of fourth--order and of non--minimal second order operators, that we have
taken from \cite{Gusynin1990,GK1999}.
We find that FP$_1$ becomes complex as soon as one departs from $D=4$,
but FP$_2$ remains real in a neighborhood of $D=4$.
Furthermore, for arbitrarily small $\epsilon$, there is another fixed point FP$_3$
which does not solve our beta functions in $D=4$.
We believe that this FP must correspond to a Weyl--invariant theory,
which requires a different quantization procedure.
We hope to return on this point in the future.

One of the main motivations of this work was the structure of higher derivative
gravity in $D=3$. Due to the fact that in $D=3$, $E=0$ identically, there are
only two independent beta functions in the four--derivative sector,
and the FP structure is different from $D=4$.
In fact, when $D\lesssim 3.93$ the fixed points FP$_2$ and FP$_3$
merge and become complex.
Nevertheless, there is also in $D=3$ a non-Gaussian FP with
non--vanishing values of $\tilde\alpha$ and $\tilde\beta$ (recall that the
four--derivative couplings are dimensionful in $D=3$).
The corresponding flow in the $\tilde\Lambda$--$\tilde G$ planes
has a non--Gaussian FP that is somewhat similar to the
one that is found in the Einstein--Hilbert truncation or
in the Chern--Simons theory, except that it has a relatively large
value of $\tilde\Lambda$.
An important point is that new massive gravity does not
correspond to a FP, confirming the result in \cite{Ohta3}.

On the other hand, when one goes to dimensions greater than four
the fixed points FP$_2$ and FP$_3$ remain real but there appear
further FPs.
There is always a Gaussian FP with $\tilde\lambda=\tilde\Lambda=\tilde G=0$
and unspecified $\omega$ and $\theta$, but there are also FPs
with $\tilde\lambda\not=0$.
It is difficult, within the present analysis, to establish which ones of these are physical
and which ones are mere truncation artifacts.

\section*{Acknowledgment}

We would like to thank Ilya Shapiro for valuable discussions.
Part of this work was carried out while the authors were attending
the workshops ``Topics in Holography, Supersymmetry and Higher Derivatives''
at Texas A\&M University, and ``2nd Mediterranean Conference on Classical and Quantum Gravity''
at Veli Lo\v{s}inj, Croatia.
We thank the organizers for invitation and providing stimulating atmosphere.
This work was supported in part by the Grant-in-Aid for
Scientific Research Fund of the JSPS (C) No. 24540290, and (A) No. 22244030.

redpajama_set_name: RedPajamaArXiv


text:
<?php

namespace Peast\Syntax;

/**
 * Encoding exception class. Encoding errors while scanning the source are
 * thrown using this using this exception class.
 *
 * @author Marco Marchiò <marco.mm89@gmail.com>
 *
 * @codeCoverageIgnore
 */
class EncodingException extends \Exception
{
}

redpajama_set_name: RedPajamaGithub


text:
A SimpleDB (via AWS::Record::Model) adapter for the orm_adapter gem.

[![Gem Version](https://badge.fury.io/rb/orm_adapter_aws.png)](http://badge.fury.io/rb/orm_adapter_aws)
[![Code Climate](https://codeclimate.com/repos/52f5dd11e30ba05c5200820a/badges/01e7c4925a3dcf8c01cd/gpa.png)](https://codeclimate.com/repos/52f5dd11e30ba05c5200820a/feed)

## Installation

Add this line to your application's Gemfile:

    gem 'orm_adapter_aws'

And then execute:

    $ bundle

Or install it yourself as:

    $ gem install orm_adapter_aws

## Usage

This gem was created as a means of using Devise with SimpleDB.  Please see _article coming soon about how to use this gem with devise_.

## Thanks

Development of this gem was generously sponsored by _company name
withheld pending approval from the legal department_.

## Testing

This gem relies on the `example_app_shared` set of examples from
`orm_adapter`.  Those examples test some functionality that is not
supplied by SimpleDB, so those tests are failing.  Be aware of this
when running the specs.  Again, *some test failures are expected*. See
`spec/spec_helper.rb` for a list of specs that are expedted to fail.

Run the tests with :

```bash
$ rspec spec
```

## Contributing

1. Fork it
2. Create your feature branch (`git checkout -b my-new-feature`)
3. Commit your changes (`git commit -am 'Add some feature'`)
4. Push to the branch (`git push origin my-new-feature`)
5. Create new Pull Request


We Deeply LOVE our Country and Proudly Support our USA Troops. Our Soldiers are protecting our Freedom by giving up so much of their own. Each and Every Soldier is out there to give everything they have including their life, so, our support to them is just a way of Appreciation, Respect, Courtesy and a way to show our Love and Care, as they really need it.
ATTENTION our Dear Soldiers: If you are at Any Given APO Address, we will Ship your Order Free of Charge. Please email us to get your Unique Coupon Code, which you will be able to apply at checkout to get Free Shipping and Handling for your order. At the same time, if you will not catch this notice before placing your order, then please be rest assured that we will issue Refund for the Shipping and Handling Charge on your Order. Stay Safe and Protected.


Cooperstown, New York is a small village that has a population of approximately 2000 people in an area that is less than two square miles. Reason why most people know of Cooperstown is because it is the location of the Baseball Hall of Fame where the greats of the game of baseball are immortalized. There are four major categories in the Baseball Hall of Fame player, manager, umpire, and executive. There are currently 34 inductees that are classified as executives which makes up 10.9% of the population of the hall of famers. But there is one executive who would need to be in consideration for this honor and his name is George M. Steinbrenner III.
Save on new Slabbed Trading Cards at SteinerSports.com!
Given the moniker "The Boss" during his time as owner of the Yankees, Steinbrenner had periods of being maligned and also periods of adoration from his own fan base. To understand why an owner would be detested just examine Steinbrenner's relationship with Yankee managers. Billy Martin had five different stints as manager of the Yankees under boss Steinbrenner and each time Martin was relieved of his duties by the orders of Steinbrenner. Yogi Berra who is a hall of fame player and helped the Yankees win 10 World Series was removed as Yankee manager in such a distasteful fashion that Berra refused to enter Yankee Stadium for over two decades.
Steinbrenner also had the dubious distinction of being suspended for baseball twice during his reign as Yankee principal owner. The first suspension was for the duration of two years with the causation being Steinbrenner provided illegal funds to Richard Nixon's presidential campaign. The second suspension was initially a lifetime ban because a man named Howie Spira received a payment of 40,000 from Steinbrenner to find information to extort/blackmail Dave Winfield.
While a bleak portrait of Steinbrenner has been painted with these aforementioned negative and sometimes illegal moments the New York Yankees were extremely successful on the field during the Boss's tenure. From 1973 to his passing in 2010 the Bronx Bombers won 7 World Series Championships, 10 pennants, and the American League East crown 16 times. The criticism of the dysfunctional Yankees in the 1970s turned to venom directed towards Steinbrenner in the 1980s but transformed to adoration during the latest edition of the Bronx bombers dynasty. This is demonstrated by the outpouring of emotion the Boss received when he delivered the balls to home plate during the 2008 All-Star Game. The once fierce lion appeared to be much less boisterous and calm as his former players greeted him near the mound and the baseball fans provided an ovation for George M. Steinbrenner III.
Steinbrenner also altered the game. First, his unquenchable thirst to win allowed the Yankee organization to spend millions of dollars to acquire free agents to assist the yearly goal under Steinbrenner: win the World Series. A side effect of this action was an increase in players salaries from the time Steinbrenner purchased the Yankees in 1973 till now with the average baseball players' salary being roughly 36,000 dollars in 1973 to an average of 4 million dollars in 2015.
Steinbrenner was also an innovator by creating his own sports network that was primarily dedicated towards his own franchise. Many teams have followed suit and mimicked this idea. Making similar shrewd business moves like this predicated the initial 10 million dollar investment in the Yankees in 1973 to now having the pinstripes worth an estimated 2.5 billion dollars.
While there are arguments for both sides a decision still needs to be rendered and the decision should be Steinbrenner deserves to be in the Baseball Hall of Fame. While Steinbrenner has made his errors other inductees in Cooperstown have character flaws. Ty Cobb poor behavior off the field has been well documented and Kenesaw Mountain Landis has been speculated to have prevented desegregation of baseball. Second, Steinbrenner awoke the dormant franchise and turned it into the evil empire that fans love to hate by outspending and defeating their opponents. As time goes on the tarnish of George Steinbrenner's infractions will slowly deteriorate with only the glowing success radiating off his resume. Because of this Steinbrenner will eventually, and rightfully so, be enshrined into the Baseball Hall of Fame.


The sheik: We are lost! He is sending for the world's greatest detective - Sherlock Jr.!
Not only is this Buster Keaton's best film, but it is among the greatest achievements in the history of cinema, period. While it is not a feature-length film--and thus barred from most critics' lists of great films--it invented just about every single basic special effect known to movies (except for morphing). The story itself, about a film projectionist who desires to become part of the movies, and then does, by walking right onto the screen, made palpable the desire that we all have to be in the movies: To get the girl, to be an action hero, to outsmart the bad guys. Keaton invented meta-cinema before anyone even had a phrase for it.
This movie has entered our dreams.


text:
Court of Arbitration
The Court of Arbitration of the Vienna Commodity Exchange
Among the services offered, the Vienna Commodity Exchange has a Court of Arbitration, which was established by law and resolves disputes swiftly, competently and inexpensively and may also be called on to secure evidence in the form of expert opinions.
The Decree issued by the Federal Minister of Finance and of the Federal Minister for the Economy and Labor in agreement with the Federal Minister of Justice on the implementation of Art. XIII ILCCP - Introductory Law to the Code of Civil Procedure Rules of the Court of Arbitration of the Vienna Stock Exchange Federal Law Gazette II No. 230 of 25 July 2000 (pdf-file 15 KB).
Fourth Decree issued by the Federal Minister of Finance and the Federal Minister of Digital and Economic Affairs in agreement with the Federal Minister of Constitutional Affairs, Reforms, Deregulation and Justice on the implementation of Article XIII of the Introductory Law to the Code of Civil Procedure Schedule of Fees of the Court of Arbitration (pdf-file 13 KB).
Conditions for calling on the court of arbitration
Anyone who carries on a business regardless of whether such person is an individual, an Austrian or non-Austrian shall have the right to call the Court of Arbitration. It is not required that a business be registered in the Companies Register. However, one of the parties to the dispute (usually, the party filing the complaint) must be a member of the Vienna Commodity Exchange. To become a member, one has to apply and pay a one-time registration fee of EUR 200. The annual fees are EUR 100. One may resign from membership at any time.
The Vienna Commodity Exchange may be called for claims arising out of contracts for the purchase of goods as well as for claims for payment (These cases are processed fast, because defendants do not attempt to construe, for example unfounded quality complaints due to the expertise of the members on the arbitration panel.)
In which cases?
The competence of the Court of Arbitration must be agreed-on in writing by the disputing parties. It is recommended that the following clause be added to sales contracts (contract notes): "Arbitration clause: Both parties hereby agree that all disputes arising out of this contract or any other transactions concluded between the two parties to this contract shall be subject to the Rules of Arbitration and the Court of Arbitration of the Vienna Commodity Exchange, admitting no appeal, said court shall apply Austrian law."
In which cases should an expert opinion be obtained (taking samples)?
If the state of the goods will not improve by further storage.
If after a few weeks it will no longer be possible to clearly determine the state of the goods at the time of delivery.
If it cannot be examined whether or not a foreign expert who might be commissioned has the expert knowledge required and, above all, a knowledge of Austrian rules and regulations.
In cases in which without an expert opinion it would not be possible to furnish evidence because the goods have meanwhile been sold or processed.
In cases of unfounded claims of deficient quality, these are usually withdrawn after a date has been set for taking a sample.
Ursula Schönthaler
ursula.schoenthaler(a)wienerboerse.at
T +43 1 531 65-255
Timber Price List
Rules of the Court of Arbitration of the Vienna Stock Exchange (pdf-file 170 KB)
Schedule of Fees of the Court of Arbitration (pdf-file 15 KB)

redpajama_set_name: RedPajamaCommonCrawl


text:
Albrecht von Heyden-Linden (* 25. Oktober 1872 in Stettin; † 24. Mai 1946 in Tützpatz) war Rittergutsbesitzer und Mitglied des Provinziallandtags der Provinz Pommern.

Leben
Albrecht von Heyden-Linden war der älteste Sohn von Wilhelm von Heyden-Linden (1842–1877) und dessen Frau Agathe von Trotha (1848–1917). Nach dem frühen Tod des Vaters erbte er das Fideikommiss Tützpatz mit den Vorwerken Idashof und Philippshof, das sein Onkel Friedrich von Heyden-Linden bis zu seiner Volljährigkeit verwaltete. Er besuchte seit 1888 mit seinem jüngeren Bruder Otto Thomson Wilhelm (1874–1900) die Ritterakademie in Dom Brandenburg und nahm ein Studium an der Universität Bonn auf, das er 1895 in Greifswald als Referendar beendete. Während seines Studiums wurde er 1891 Mitglied des Corps Borussia Bonn. Seinen Militärdienst leistete er beim Garde-Kürassier-Regiment, sein letzter Dienstgrad war Rittmeister.

Danach widmete er sich der Bewirtschaftung seiner Güter. 1908 ließ er das abgebrannte Gutshaus Tützpatz neu aufbauen. Er war bis 1945 Vorsitzender des Landwirtschaftlichen Ein- und Verkaufsvereins in Altentreptow. Nachdem die Fideikommisse 1919 per Gesetz aufgehoben worden waren, übereignete er Ende der 1920er Jahre das Gut Philippshof seinem Cousin Rüdiger von Heyden-Linden.

Albrecht von Heyden-Linden war Mitglied der Konservativen Partei und nach 1918 der Deutschnationalen Volkspartei. Von 1921 bis 1933 gehörte er dem Provinziallandtag der Provinz Pommern an.

Während des Ersten Weltkriegs war er Ordonnanzoffizier beim Oberkommando der 4. Armee. Er wurde mit dem Eisernen Kreuz 2. Klasse ausgezeichnet.

Albrecht von Heyden-Linden war Erblandmundschenk von Altvorpommern und seit 1914 Rechtsritter des Johanniterordens. 1939 umfasste sein Gut Tützpatz 1067 ha, davon 178 ha Wald.

Albrecht von Heyden-Linden war seit 1898 mit Hertha von Ploetz (1876–1949) verheiratet. Die beiden hatten zwei Töchter. Nach dem Ende des Zweiten Weltkriegs blieb er mit seiner Frau in Tützpatz. Bei der Bodenreform wurde er enteignet, musste aber das Gut wegen der Fürsprache der Landarbeiter nicht verlassen. Er starb 1946 und wurde in Tützpatz begraben.

Literatur
Harald von Heyden: Beständig im Wandel. Berichte aus sechs Generationen der Familie von Heyden/von Heyden-Linden von 1800–1989. Heyden'sche Familienstiftung (Hrsg.), Borgwedel, S. 124–126.

Einzelnachweise

Mitglied des Provinziallandtages von Pommern
Kavallerist (Preußen)
Rechtsritter (Johanniterorden)
Mundschenk
Corpsstudent (19. Jahrhundert)
Mitglied der Konservativen Partei
DNVP-Mitglied
Person im Ersten Weltkrieg (Deutsches Reich)
Albrecht
Deutscher
Geboren 1872
Gestorben 1946
Mann

redpajama_set_name: RedPajamaWikipedia


text:
Płaczków (od 31 XII 1961 Mroczków) – dawna gromada, czyli najmniejsza jednostka podziału terytorialnego Polskiej Rzeczypospolitej Ludowej w latach 1954–1972.

Gromady, z gromadzkimi radami narodowymi (GRN) jako organami władzy najniższego stopnia na wsi, funkcjonowały od reformy reorganizującej administrację wiejską przeprowadzonej jesienią 1954 do momentu ich zniesienia z dniem 1 stycznia 1973, tym samym wypierając organizację gminną w latach 1954–1972.

Gromadę Płaczków z siedzibą GRN w Płaczkowie utworzono – jako jedną z 8759 gromad na obszarze Polski – w powiecie kieleckim w woj. kieleckim, na mocy uchwały nr 13c/54 WRN w Kielcach z dnia 29 września 1954. W skład jednostki weszły obszary dotychczasowych gromad Górki, Mroczków i Sobótka ze zniesionej gminy Bliżyn w powiecie kieleckim oraz obszar dotychczasowej gromady Płaczków ze zniesionej gminy Odrowąż w powiecie koneckim. Dla gromady ustalono 20 członków gromadzkiej rady narodowej.

31 grudnia 1959 gromadę włączono do powiatu koneckiego w tymże województwie.

Gromadę zniesiono 31 grudnia 1961 przez przeniesienie siedziby GRN z Płaczkowa do Mroczkowa i zmianę nazwy jednostki na gromada Mroczków.

Przypisy

Plxaczkozzxw


text:
Politics out of joint.
I wrote an article some time back entitled political parties can die. It was an opinion piece highlighting what was a trend I perceived about the electorate's rejection almost worldwide of the post-war condition reflex of voters' allegiance to one of the two traditional vanilla-flavoured parties irrespective of whatever democracy you lived in.
In a nutshell, it was about the electorate dumping that tired old Marxist analysis of politics as a class war between the exploiters and exploited, because for them the establishment of neither of those two sides of polity were actually representing the interests of anyone but themselves.
I'd like to explore that trend further by examining in detail its causes in the two mainstream parties in the upcoming presidential election in America.
We have the Democratic party which is nominally left-wing and on the side of the exploited. Their nominee for president is widely seen as tainted by too many scandals and whose only self-declared qualification for the job appears to be she's a woman, albeit one of that multitude her on/off husband fucked over the course of their marriage. However, she's totally Washington establishment, which is to say she's making deals in all directions – anything for power.
Both she and the Clinton Foundation reek of corruption and they're both sly recipients of large wads of money from Wall Street and some of the most distasteful regimes in the world. We live in an information rich age; perhaps to the point of information overload, but leaked emails reveal the Democratic establishment pushing Clinton to the fore while at the same time plotting to sabotage the Sander's campaign. They even kicked around ideas about how to use his faith against him. He's Jewish, if you didn't already know that, like most of the founders of left-wing political theory.
It's a worrying aspect of modern left-wing politics that there is a pervasive anti-Semitic element to it that is by now almost systemic. If you've any doubts about that, consider the sight of people inside the convention waving a Palestinian flag while outside they were burning the Stars and Bars, and topping off that little ritual by burning an Israeli flag. One of the fools in that little ceremony managed to set himself alight while dancing atop the burning flags.
In a sense, Sanders was a Trump who didn't have the nerve to ignite. When you go up against the king, you have to be audacious rather than noble. L'audace, l'audace, et toujours de l'audace otherwise the field is lost. The democratic establishment took him out and re-established control over the machine but they'd be wise to think over how far he got and what he might represent.
Bernie Sanders, if anything, was trying to force it back to doing more than nominally addressing its original blue-collar support base. The fact that anyone openly proclaiming themselves to be a socialist could do so well in the primaries is a sign of how disaffected that party's traditional support base is. In America, declaring yourself to be a socialist used to be equivalent to saying you're a rabid Marxist which was right down there in the pits with child molesting.
He spoke for the next generation of their support base, and that dissatisfaction with the party as currently configured is not going to go away.
To a certain extent, the popular support for Sanders was also a rejection of the race politics of the left-wing, who're determined to secure the non-white vote by stoking various grievances, real or otherwise, for the next generation or two. This is exemplified by the utterances, or more significantly the tepid silences of President Obama on things like murdered white policemen, and is seen by insiders as regaining Democratic Party control of the South in the aftermath of the civil rights legislation in the sixties.
Contrary to LBJ's prediction that they'd lose democratic support for a generation, they actually lost it for two, but Bill recouped a lot of it, with Obama, the most racially divisive president ever, completing the job. In essence, it was simply converting what was a poor white cracker support base to a poor black one. The problem with that simplistic analysis is that while I'm sure there are still a number of crackers and shanty blacks knocking around, that electoral demographic has largely disappeared to be replaced by mildly prosperous lower/middle classers who just want some stability and prosperity to raise their families.
They go to work together every day without much in the way of a race war breaking out, and anyway everyone knows that'd be the quick way to get themselves fired.
The anger displayed at the Democratic convention was really a reaction to the political establishment bending or breaking every rule possible to get their candidate nominated, which they succeeded in doing.
What happened at the Republican one was the reverse because the political establishment were outmanoeuvred, but the rage and anger, while subsumed beneath a grudging acceptance of the overwhelming will of the foot soldiers' popular choice, was palpable.
We are now presented with the hitherto unprecedented spectacle of a political party actively trying to sabotage the presidential run of their own nominee. This is being spun as a matter of principle, or some sort of concern for that higher abstraction of country, but what it's actually about is nothing more than petulance, and a real fear that the days of the Republican dinosaurs of the Jurassic period are over, and they are.
A telling sign, because it's about money, of how much they've lost not only the control but respect of their traditional supporters is the funding of the Trump campaign. The usual bulge-bracket party donors were discouraged from contributing and it was widely thought that Clinton would simply outspend him in the run off.
In response, the Trump campaign asked for donations online; ten, twenty or thirty dollars, whatever you could afford. Within a month, they've caught up with Clinton's estimated ninety million dollars a month in donations. A lot of ordinary people kicking in a few bucks and none if any Saudi-sized donors.
A right-wing party being funded by the little people and a left-wing one funded by big business interests and several filthy regimes – you couldn't make this stuff up. Politics out of joint.
When politicians and commentators complain of electorates being irrational, you know they're the ones being grossly out of touch with the common sentiment of the people. Believe it or not, they're there to represent the will of the people, not to impose their will on voters. The longer they ignore that simple reality, the harsher the electorate's reaction will be.
These stresses in mainstream parties are not going to go away. It's not some momentary fad. They signal a tectonic shift in both the style of how politics will be conducted in the future and what will be acceptable to the public in terms of mainstream policy.
The Jack is well and truly out of the box and in no mood to get back in.
Related articles by Pointman:
Political fracture points and power vacuums.
People are pissed off.
The loss of faith in the political class.
UK Election 2015 – The car crash of the chatterati against reality.
Click for a list of other articles.
Filed under Article · Tagged with Clinton, Politics, Trump
← Your brothers will come after them and make it right.
The importance of playing. →
15 Responses to "Politics out of joint."
Timbotoo says:
Quite agree that the system is groaning under its own weight. Neither party is in touch with the electorate, however a portion of Saunder's supporters also seem to be members of the Free Sh1t Army. It will be very interesting to see what happens if Trump the iconoclast is elected.
harrydhuffman (@harrydhuffman) says:
You're missing the point: Your narrative above fits the Right but not the Left. The entire Left is insane (Sanders merely more so; Clinton is more criminal), both the leaders and the electorate. The system is not groaning, it is thoroughly broken, with half the electorate refusing to confront the blatant lies being told them, because they can only see that they are being given attention (translation: being pandered to falsely). The re-election of Obama in 2012 proved this..
Martin A says:
Brexit is another example of the rejection of the dominance of the political class.
Can anybody make a contribution to Donald Trump's campaign? If so where do I find its tip box?
catweazle666 says:
Try this.
http://wc4.net/t?r=4248&c=1234&l=31&ctl=448D:28D07A4C2A28BC1BA4A3DC4DCCFF30F049CA475AD5540ABF&
gary turner says:
There's a radically new service called Google. Have you tried it? https://www.google.com/search?q=donate+to+trump+campaign&ie=utf-8&oe=utf-8
Graeme No.3 says:
In Australia we changed Prime Minister a year ago. We, the public had no say, it was a party room coup because the politicians thought the new leader would appeal to those more likely to vote with the opposition, and gain more favourable comments from the media. It turned out to be true, the vote among the inner city left leaning went up 1.2%, but it didn't capture any seats. Meanwhile the party's vote among its usual supporters plunged and they barely scrambled back into power, thanks partly to an opponent showing himself as quite unsuited for the job.
So now they think they have a 'mandate' to push those policies that nearly lost them the election, and ignore the problems that worry their former supporters.
Apparently after a few months in parliament politicians lose touch with reality, and start thinking that they know everything and the public are completely out of touch.
Spot on G3. Britain and the vote for Brexit by people who were ignored, the support for Senator Cory Bernardi's conservative movement here in Oz and the support for "The Donald" in the US are good examples of Pointies argument. "The great unwashed public" the professional politicians and "self appointed elites" so despise have had enough and have lost the complacency which has dogged the political scene in the west and allowed the rise of people like Clinton.
erl happ says:
Some, perhaps a great deal of the present economic malaise can be put down to distortions in the marketplace introduced at the behest of the environmental lobby. Many of these distortions are indirect acting to increase the price of housing to make it more 'energy efficient'. Many operate via changes to land use planning.
There is a great deal of stupidity required in order to meet the dictates of those who cry 'sustainability'. Can these eco-environmental Nazis not see that the planet is greening? Doesn't that greening represent increased economic capacity, enhanced sustainability?
Why has the oil price collapsed? Haven't the proponents of the 'peak oil' argument noticed? Don't they look? Do they live in a cocoon of self indulgent wistfulness?
The ordinary man has a modicum of common sense. White is not black. Black is not white. The wheel turns.
NZPete says:
Fascinating times we live in. Pointy's analysis IMHO is a good one.
Check out the article linked to in this text I sent out to my personal distribution list yesterday:
"Be wary of the MSM" is the view of this writer…
"All of the media believes the American people are yokels, and that they are fools, and they're there to be manipulated, so that they – the press, the better people – can set the tone."
http://www.breitbart.com/2016-presidential-race/2016/08/01/pat-caddell-pro-clinton-media-bias-terrifying-crisis-democracy/
http://tinyurl.com/hym33x5
We see this in the NZ media too, sadly.
Pete – The Clintons (all 3 of them), have taken criminal racketeering out of the primitive realms of Sicilian peasant mafiosi and into every facet of government around the world.
Australian journalist and blogger Michael Smith has spent months trying to track down the tens of millions of our taxpayer dollars that have been handed over to the Clintons. What he has found is extraordinary, and in this rather long and detailed blog post he lays out chapter and verse. The Australian Govt, at best, refuses to answer his enquiries under Freedom of Information laws and, at worst, simply have zero idea of what happened to over $80 million.
If the purpose of our foreign aid program is to show that Australia is a decent global benefactor, how is it that we permit the Clinton Foundation to take our money, to use a program we developed and paid for in 2008 – and then to add insult injury watch the Clinton Foundation take the entire credit for the venture as if we had nothing to do with it?
http://www.michaelsmithnews.com/2016/08/significant-progress-on-australias-taxpayer-donations-to-the-clinton-foundation-thank-you-for-bearin.html
The convoluted twists and turns of the Clintons' financial machinations are nothing more than criminal money-laundering … mega millions fed into the machine one end, 90% of it evaporates without trace, and 10% might (on the books) be accounted for, all dressed up as something it isn't.
Their political adversaries won't call them out on it because it might expose their own shady slush funds and dodgy deals.
And sorry Donald, but calling her "crooked" and "a liar" isn't going to do it for you. Get specific and quickly … cite a few of the many examples of the Clintons' criminal racketeering, get the bastards indicted and in jail, where they belong.
Years ago a disaffected Liberal Party politician by the name of Don Chipp (now deceased) began a breakaway political party, the Australian Democrats.
Don's very successful catchphrase was that he would "Keep the bastards honest."
Pauline Hanson, with Don in mind, once said "I want to get the bastards out of here."
Electorates are simply fed up with being lied to, their hard-earned tax dollars being squandered, and Party factions deciding who stands for election and who will be Leader.
Brexit and our election of a range of Senators to keep our faux Prime Minister (Goldman Sachs) Turnbull from exercising absolute power in the Lower House, is a good indicator that we are finally wising up.
We can only hope that irrational cult-like support for Clinton will be swamped by the Right and those who want to regain control over what their nation has become. Sadly, even comic book heroes can no longer fight for "Truth, Justice and the American way" because America has lost its way.
Perhaps a new Trump Administration that hasn't sold out to subversive corruption can turn their Ship of State around. We certainly hope so, because the outcome will reverberate in every Democracy around the world.
The bottom dwellers that comprise the mainstream media in the US are as maggot infested as the slimy rotten scum bags that make up the government. I for one will be cheering when they are all circling the toilet bowl on their way down to the sewer they most richly deserve.
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These samples have a clean and punchy sound with very low noise. The live sounds have been recorded with a close mic position. Each sound has been meticulously edited and neatly organized in to separate folders. As with all New Loops sample packs, once purchased 'Percussion' is 100% royalty free to use in your songs and audio productions. We are sure you will find this percussion library a great resource for years to come.
Shaker and Tambourine Loops are in various tempos ranging from 100 to 170 bpm. Also included are some hand drum fills and fx.
The sampled instruments are arranged into 4 folders. The total number of each sound are show in brackets.