text:
\section{Introduction}

The search for a theoretically consistent Lorentz-invariant massive graviton has been a challenging issue since 1939, when Fierz and Pauli (FP) proposed a linear theory of massive spin-2 field \cite{Fierz:1939ix}.
Once the FP mass term are taken into account in general relativity,
one would naively expect that it recovers the results of general relativity as the mass of the graviton goes to zero. However, a non-vanishing degree of freedom in the massless limit would lead to
discontinuity found by van Dam and Veltman \cite{vanDam:1970vg} and
 Zakharov \cite{Zakharov:1970cc}. Although Vainshtein claimed that this problem can be cured by taking into account nonlinear interactions \cite{Vainshtein:1972sx}, Boulware and Daser pointed out the scalar degree of freedom responsible for Vainshtein's argument carries an extra ghost degree of freedom so-called BD ghost \cite{Boulware:1972aa}. Remarkably, this unwanted degree of freedom associated with higher derivatives can be eliminated by adding fully nonlinear graviton's mass terms such that those higher derivative terms vanish due to the total derivative terms, and this theoretically consistent massive
gravity theory is now known as de Rham-Gabadadze-Tolley (dRGT) theory \cite{deRham:2010kj,deRham:2010ik}.

In recent years, there have been a number of attempts for constructing a broad class of theories of massive gravity, e.g., mass-varying massive gravity \cite{Huang:2012pe}, quasi-dilation theory \cite{DAmico:2012hia},  and massive bi-gravity \cite{Hassan:2011zd}. The most interesting one would be new kinetic interactions for a massive graviton without introducing any extra degree of freedom.
Such a kinetic interaction embedded in dRGT theory has been first found in the context of a pseudo-linear theory \cite{Hinterbichler:2013eza}, however, its nonlinear completions can not be included in dRGT theory in a consistent way due to the reappearance of BD ghost at non-linear level \cite{Kimura:2013ika,deRham:2013tfa}. The crucial problem of these derivative interactions was the appearance of higher derivatives of the scalar mode in Euler-Lagrange equations, and this, in general, leads to the Ostrogradsky's ghost \cite{Ostrogradsky1850}.

Meanwhile it has been recently argued that higher derivatives in Euler-Lagrange equations are not essentially problematic as long as the appropriate number of constraints exists in a theory. Such concrete examples are found in the context of point particles~\cite{Langlois:2015cwa,Motohashi:2016ftl,Klein:2016aiq,Motohashi:2017eya,Kimura:2017gcy,Motohashi:2018pxg}, their field theoretical application~\cite{Crisostomi:2017aim,Kimura:2018sfs},  scalar-tensor theories~\cite{Gleyzes:2014dya,Langlois:2015cwa, 2015PhRvL.114u1101G,Crisostomi:2016czh, BenAchour:2016fzp}, vector-tensor theories~\cite{Kimura:2016rzw}.
The key point of these theories is the degeneracy of the kinetic matrix, which provides an associated primary constraint (and subsequent constraint depending on the spin of a field), and an unwanted degree of freedom can be then successfully eliminated. In fact, in such a theory, higher time derivatives appearing in Euler-Lagrange equations should be removed by combining Euler-Lagrange equations and their time derivatives,
therefore, initial conditions to solve the resultant differential equations are consistent with the number of degrees of freedom in ghost-free theories.

This fact opens up a new direction of study for searching new theories of a massive graviton, and it is therefore worth revisiting the pioneering attempt by Fierz and Pauli as a starting point of constructing a theoretically consistent massive gravity. To this end, in the present paper, we construct the most general quadratic theory of a massive spin-2 field and a massless spin-2 field with Lorentz invariance in flat spacetime, based on the Hamiltonian analysis.
This paper is organized as follows.
In Sec.~\ref{sec:theory}, we introduce the linear theory of a rank-2 symmetric tensor field
 and decompose it into tensors expressed by scalar quantities, those by vector quantities and those by tensor quantities, where each quantity is defined based on transformation properties with respect to a $3-$dimensional rotation in Minkowski spacetime.
The Hamiltonian formalism in Fourier space is also summarized. In Sec.~\ref{Sec:3}, we first derive the action for the transverse traceless tensor mode and a condition for avoiding instability. Then we find a condition to eliminate ghosty degrees of freedom for the scalar and vector modes and classify theories based on the Hamiltonian analysis. In Sec.~\ref{Sec:4}, we investigate the properties of obtained theories under the field redefinition to see relations with the known theories.
Sec.~\ref{sec:summary} is devoted to summary.
In Appendix~\ref{sec:other cases}, we perform the complete Hamiltonian analysis of the special cases. In Appendix~\ref{BDproof}, the explicit proof of the existence of ghost degree of freedom is given if a theory has six or more degrees of freedom. In Appendix~\ref{App:gauge}, we derive gauge transformation and construct gauge invariant variables for each case. We also derive conditions for avoiding ghost and gradient instabilities for the scalar mode from reduced Lagrangian.

\section{Set up}
\label{sec:theory}

In this section, we introduce the most general Lorentz-invariant action for a rank-2 symmetric tensor field, which contains up to the Lagrangian quadratic in the tensor field and two derivatives with respect to spacetime.
Since a theory for the rank-2 symmetric tensor field in general contains ten degrees of freedom, some of them might be ghost modes, which are unwanted degrees of freedom in a theory.
To this end, we then apply scalar, vector, and tensor decomposition to the rank-2 symmetric tensor field which is defined based on transformation properties with respect to a $3-$dimensional rotation in Minkowski spacetime. We will also provide an overview of the Hamiltonian formalism in Fourier space.

\subsection{action}
Let us consider a generic Lorentz-invariant action for a rank-2 symmetric tensor field
$h_{\mu\nu}$ up to the quadratic order in Minkowski spacetime,
\begin{align}
	S [h_{\mu \nu}] = \int {\rm d}^4 x \Bigl(-
	{\cal K}^{\alpha \beta | \mu \nu \rho \sigma} h_{\mu \nu, \alpha} h_{\rho \sigma, \beta}
	-{\cal M}^{\mu \nu \rho \sigma} h_{\mu \nu} h_{\rho \sigma} \Bigr) \,,
	\label{Lagrangian}
\end{align}
where ${\cal K}^{\alpha \beta | \mu \nu \rho \sigma}$ and ${\cal M}^{\mu \nu \rho \sigma}$ are  the most general combinations of the Minkowski metric
$\eta_{\mu\nu}$,
\begin{align}
	{\cal K}^{\alpha \beta | \mu \nu \rho \sigma}
	&= \kappa_1 \eta^{\alpha \beta} \eta^{\mu \rho} \eta^{\nu \sigma}
	+ \kappa_2 \eta^{\mu \alpha} \eta^{\rho \beta} \eta^{\nu \sigma}
	+ \kappa_3 \eta^{\alpha \mu} \eta^{\nu \beta} \eta^{\rho \sigma}
	+ \kappa_4 \eta^{\alpha \beta} \eta^{\mu \nu} \eta^{\rho \sigma} \,,\label{Ktensor}\\
	{\cal M}^{\mu \nu \rho \sigma}
	&= \mu_1 \eta^{\mu \rho} \eta^{\nu \sigma}
	+ \mu_2 \eta^{\mu \nu} \eta^{\rho \sigma} \label{Mtensor}\,,
\end{align}
and $\kappa_{1,2,3,4}$ and $\mu_{1,2}$ are constant parameters.
Contracting all the Minkowski metric, the action can be rewritten, after integration by parts, as
\begin{align}
	S [h_{\mu \nu}] = -\int {\rm d}^4 x \Bigl[ \kappa_1 h_{\mu \nu \,, \alpha} h^{\mu \nu \,, \alpha}
	+ \kappa_2 h^\alpha{}_{ \mu \,, \alpha} h^{\beta \mu}{}_{, \beta}
	+ \kappa_3 h^{\alpha \beta}{}_{, \alpha} h_{, \beta}
	+ \kappa_4 h_{, \alpha} h^{, \alpha}
	+ \mu_1 h_{\mu \nu} h^{\mu \nu} + \mu_2 h^2 \Bigr] \,,
\label{reLagrangian}
\end{align}
where the indices of $h_{\mu \nu}$ are raised by $\eta^{\mu \nu}$ and $h$ is the trace of $h_{\mu\nu}$ contracted with the Minkowski metric $\eta_{\mu\nu}$.
A comma denotes a partial derivative with respect to spatial coordinates.
The linearized Einstein-Hilbert action can be reproduced by setting
$\kappa_2 = -\kappa_3 = 2\kappa_4 = -2 \kappa_1$ up to overall factor,
and the kinetic term of \eqref{Lagrangian} is then invariant under the
gauge transformation $h_{\mu\nu} \to h_{\mu\nu} + \partial_{\mu}\xi_{\nu}+ \partial_{\nu}\xi_{\mu}$, where $\xi_\mu$ is a gauge parameter.
In addition to this choice of the parameters, when the mass parameters satisfy $ \mu_1 = -\mu_2 \neq 0$,
the Lagrangian \eqref{Lagrangian} reproduces the Fierz-Pauli theory \cite{Fierz:1939ix}. Although the Fierz-Pauli theory respects the gauge invariance in the kinetic term,
it is not necessary for a generic massive spin-2 field that we consider in the present paper.

\subsection{SVT decomposition}
In order to simplify analysis,
we decompose the rank-2 symmetric tensor field $h_{\mu\nu}$ into
a transverse-traceless tensor, tensors expressed by transverse vectors, and tensors expressed by scalars
where scalar, vector and tensors are defined based on transformation properties with respect to a $3-$dimensional rotation in Minkowski spacetime:
\begin{align}
	h_{0 0} &= h^{0 0} = - 2 \alpha \,, \qquad
	h_{0 i} = - h^{0 i} = {\widehat{\beta}_{, i}} + B_i \quad (B^i{}_{, i} = 0) \\
	h_{i j} &= h^{i j}
	= 2 {\cal R} \delta_{i j} + 2\, {\widehat{{\cal E}}_{, i j}} + F_{i \,, j} + F_{j \,, i}
	+ 2 H_{i j} \quad ( F^i{}_{, i} = 0 \,, \quad H^i{}_i = H^{i j}{}_{, j} = 0) \,.
\end{align}
Here the transverse-traceless tensor $H_{ij}$, two transverse vectors $B_i$ and $F_i$, and four scalars $\alpha, \widehat\beta, {\cal R}$, and $\widehat{\cal E}$ respectively have two, four, and four components in total. Therefore, to obtain a theory whose
number of degrees of freedom is up to five including the degree of freedom of transverse-traceless tensor, we need to eliminate two components in the vector sector
and three components in the scalar sector respectively. Otherwise, ghost degree(s) of freedom appears as shown in Appendix~\ref{BDproof}.
Below we split the action into three parts and each of them is solely composed by single type of perturbations, that is, scalar, vector and tensor perturbations, which is always possible at the level of linear perturbation,
\begin{align}
	S [h_{\mu \nu}]
	&= S^S [\alpha \,, \widehat\beta \,, {\cal R} \,, \widehat{\cal E}\,] + S^V[B_i \,, F_i] + S^T [H_{i j}] \,.
\end{align}
In the next section, based on this separated action, we will look for theories with at most $5$ degrees of freedom finding the appropriate degeneracy conditions for the scalar and vector sectors, respectively.

\subsection{Hamiltonian formalism in Fourier space}
In this subsection, we briefly summarize the Hamiltonian formalism in the Fourier space.
We, for convenience, work in the Fourier space, and the Fourier component of a field,  $A (t, \mathbf{k})$, is given by
\begin{align}
	A (t, \mathbf{k}) =
	\int {\rm d}^3 x A(t,\mathbf{x}) \mathrm{e}^{i \, \delta_{j k} k^j x^k}\,,
\end{align}
The Hamiltonian is defined by
\begin{eqnarray}
H (t) = \int {\rm d}^3 k \, {\cal H} (t, {\bf k}) \,,
\end{eqnarray}
where ${\cal H}$ is the Hamiltonian density in the Fourier space,
\begin{eqnarray}
{\cal H} (t, {\bf k})
= \sum_I\dot s^I (t, {\bf k})  \, \pi_{s^I} (t,{\bf k}) - {\cal L} [s^I(t, {\bf k}), \dot{s}^I(t, {\bf k})] \,,
\end{eqnarray}
where $s^I$ and $\pi_{s^I}$ are respectively sets of canonical fields and their conjugate momenta.
If the system has $n$ primary constraints ${\cal C}_i$, the total Hamiltonian and its density are given by
\begin{eqnarray}
 H_T (t) = \int {\rm d}^3 k \, {\cal H}_T (t, {\bf k}) \,, \qquad
{\cal H}_T (t, {\bf k})  = {\cal H} (t, {\bf k})  + \sum_{a=1}^n \lambda_{a} (t, {\bf k}) \, {\cal C}_a (t, {\bf k})\,,
\end{eqnarray}
where $\lambda_i$ are Lagrange multipliers associated with each primary constraint ${\cal C}_i$.
The Poisson bracket between ${\cal A}$ and ${\cal B}$ is defined by
\begin{eqnarray}
\{{\cal A} (t, {\bf k}), {\cal B} (t, {\bf k}')\} = \int d^3k''
\sum_{I}
\left[
{\delta {\cal A} (t, {\bf k}) \over \delta s^I(t, {\bf k}'')} {\delta {\cal B} (t, {\bf k}') \over \delta \pi_{s^I}(t, {\bf k}'')}
-{\delta {\cal A} (t, {\bf k}) \over \delta \pi_{s^I}(t, {\bf k}'')} {\delta {\cal B} (t, {\bf k}') \over \delta s^I(t, {\bf k}'')}
\right] \,.
\end{eqnarray}
The time-evolution of the function ${\cal A}(t, {\bf k})$ is given by
\begin{eqnarray}
\dot{{\cal A}} (t, {\bf k}) = \{{\cal A} (t, {\bf k}), H_T (t)\}
=  \int {\rm d}^3 k' \left[\{{\cal A} (t, {\bf k}), {\cal H} (t, {\bf k}')\}
+ \sum_{a=1}^n \lambda_{a} (t, {\bf k}') \{{\cal A} (t, {\bf k}),  {\cal C}_a (t, {\bf k}') \} \right] \,.
\end{eqnarray}

\section{Hamiltonian analysis and classification}
\label{Sec:3}

In this section, we perform the Hamiltonian analysis for the theory (\ref{Lagrangian}) and classify the theory based on the number of degrees of freedom and constraint structures.
Since the number of degrees of freedom in the theory is ten in general, one needs to adequately eliminate extra degrees of freedom in
each mode decomposed in the previous section. We first take a look at the tensor mode and derive the condition for avoiding a ghost mode.
Then we seek conditions to eliminate unwanted modes for vector and scalar modes and conditions to have subsequent constraints.
The existence of ghost degrees of freedom in the scalar sector is proved in the Appendix~\ref{BDproof} when the total number of degrees of freedom is more than five.

\subsection{Tensor modes}

The action in the tensor sector is given by
\begin{align}
	S^T [H_{i j}]
	&= 4 \int {\rm d} t \,{\rm d}^3 k \Bigl[  \kappa_1 \dot{H}_{ij}^2 - (\kappa_1 k^2 + \mu_1) H_{i j}^2 \Bigr] \,,
	\label{ST}
\end{align}
where a dot represents derivative with respect to time $t$.
As one can see from \eqref{ST}, the tensor modes are controlled by only two parameters $\kappa_1$ and $\mu_1$, and the existence of tensor modes and the condition for avoiding the ghost instability demand
\begin{align}
	``\text{Condition 1}"\ :\ \kappa_1
	>0\,.
	\label{condition:1}
\end{align}
Throughout this paper, we always assume the "Condition 1" \eqref{condition:1}, and then the number of the degrees of freedom in the tensor sector is two. Furthermore, the parameter $\mu_1$ should not be negative
in order for avoiding the tachyonic instability in the tensor sector.

\subsection{Vector modes}
\label{sec:vector}
In this subsection, we focus on the vector modes and find conditions to avoid extra ghost degrees of freedom, based on the Hamiltonian analysis.
Before proceeding analysis, let us comment on counting the number of physical degrees of freedom and constraints in vector modes.
Since a vector in vector modes, $V_i$, satisfy the transverse condition, $k^i V_i = 0$, degrees of freedom in $V_i$ are two while there are three components.
 Hence when we have primary constraints of vector type like $\pi_{V_i} = 0$, this should be understood as
 two primary constraints and not three since the transverse conditions put a constraint for one component in $V_i$ and primary constraints should be obtained for the remaining two degrees of freedom.

The action for the vector modes
can be written with the replacement $kF_i\to F_i$,
\begin{align}
	S^V [B_i \,, F_i]
	&= \int {\rm d} t \,{\rm d}^3 k \Bigl[
	-(2  \kappa_1+\kappa_2) \dot{B}_i^2 + 2 \kappa_1 \dot{F}_i^2
      + 2 \kappa_2 k B_i \dot{F}_i
	+ 2 \left(\kappa_1 k^2+\mu_1\right) B_i^2 -\left(k^2 (2 \kappa_1+\kappa_2)
	+ 2 \mu_1\right) F_i^2
	\Bigr] \,.
	\label{SV}
\end{align}
One may immediately notice from (\ref{SV}) that there are the appearance of either ghost or gradient instabilities in $B_i$ or $F_i$ modes, depending on the sign of $2\kappa_1+\kappa_2$ as well as $\kappa_1$. This concludes that one needs to at least eliminate either $B_i$ or $F_i$ in order to have two degrees of freedom. The existence of the tensor modes \eqref{condition:1} leads to the unique option to have a primary constraint for $B_i$, that is,
\begin{align}
	``\text{Condition 2}"\ :\
	2 \kappa_1 + \kappa_2 = 0 \qquad
	\Longleftrightarrow  \qquad
	\kappa_2 = -2\kappa_1 \,.
	\label{condition:2}
\end{align}
With this condition \eqref{condition:2}, the kinetic term of $B_i$ vanishes, which implies $B_i$ are manifestly non-dynamical.
Then the action for the vector mode can be recast as
\begin{align}
	S^V [B_i \,, F_i]
	&= \int{\rm d} t\, {\rm d}^3 k\,{\cal L}^V
	= \int {\rm d} t\, {\rm d}^3 k \Bigl[
	 2 \kappa_1 \dot{F}_i^2 - 4 \kappa_1 k B_i \dot{F}_i
	+ 2 \left(\kappa_1 k^2
	+ \mu_1\right) B_i^2 - 2 \mu_1 F_i^2
	\Bigr] \,.
\end{align}
Apparently, the action for the vector modes depends on only two parameters,
$\kappa_1$ and $\mu_1$, as in the tensor modes.
The conjugate momenta for $B_i$ and $F_i$ are given by
\begin{align}
\pi_{B_i} &\equiv \frac{\delta {\cal L}^V}{\delta \dot{B}_i} = 0 \,,\\
\pi_{F_i} &\equiv \frac{\delta {\cal L}^V}{\delta \dot{F}_i} =  4 \kappa_1 (\dot{F}_i - k B_i) \,,
\end{align}
and we therefore have two primary constraints instead of three as mentioned at the beginning of this subsection, which are defined by
\begin{align}
	{\cal C}_1^{B_i} = \pi_{B_i} = 0 \,.
	\label{primaryBi}
\end{align}
Then Hamiltonian and the total Hamiltonian densities read
\begin{align}
	{\cal H}^V &= \dot{B}_i \pi_{B_i} + \dot{F}_i \pi_{F_i} - {\cal L}^V
	\approx \frac{\pi_{F_i}^2}{8 \kappa_1} + k \pi_{F_i} B_i - 2 \mu_1 B_i^2 + 2 \mu_1 F_i^2\,,\nonumber\\
	{\cal H}_T^V & = {\cal H}^V + \lambda_{B_i} \pi_{B_i} \,,
\end{align}
where $\lambda_{B_i}$ are Lagrange multipliers.
We have suppressed the terms $\dot{B}_i \pi_{B_i}$ in the final expression of ${\cal H}^V$ since they vanish once the primary constraints (\ref{primaryBi}) are imposed.
One can easily check that the evolution of the primary constraints automatically yields secondary constraints,
\begin{align}
	{\cal C}_2^{B_i} \equiv \dot{{\cal C}}_1^{B_i} = \{ {\cal C}_1^{B_i} \,, H_T^V \}
	 = \{ {\cal C}_1^{B_i} \,, H^V \}
	=  k \pi_{F_i} + 4 \mu_1 B_i \approx 0\,.
\end{align}
Then the time-evolution of the secondary constraints are given by\footnote{
To be precise, one needs the integral over the Fourier space
in front of the Lagrange multipliers $\lambda_{B_j}$, which can be always
integrable because of the appearance of the Dirac's delta function. For simplicity, we omit this integral and arguments of each variable since
results do not change.
}
\begin{align}
	\dot{{\cal C}}_2^{B_i} = \{ {\cal C}_2^{B_i} \,, H_T^V \} = \{ {\cal C}_2^{B_i} \,, H^V \}
	+ \{ {\cal C}_2^{B_i} \,, {\cal C}_1^{B_j} \} \lambda_{B_j} \approx 0\,,
	\label{dotC2B}
\end{align}
where the coefficients of $\lambda_{B_i}$ are given by
\begin{align}
	\{ {\cal C}_2^{B_i} \,, {\cal C}_1^{B_j} \} &= 4\mu_1 \,\delta_{ij} \,.
\label{C2BC1B}
\end{align}
Therefore, we have two cases :

\vspace{2mm}

{\bf Case V1 :} $\mu_1 = 0$ \\
In this case, in addition that the Poisson bracket \eqref{C2BC1B} vanishes, $\{ {\cal C}_2^{B_i}\,, H^V\}$ is trivially zero. Then, there is no more constraint.
Thus there are two primary constraints ${\cal C}_1^{B_i}$ and two secondary constraints ${\cal C}_2^{B_i}$, and
all of them are first-class since all the Poisson brackets between these constraints vanish. Therefore,
\begin{eqnarray}
\textrm{vector DOF} = \frac{4\times 2 - 4 \,(\textrm{2 primary \& 2 secondary}) \times 2 \,(\textrm{first-class})}{2}
=0 \,.
\end{eqnarray}
This case is exactly the same as the linearized Einstein's gravity, and thus the Lagrangian is invariant under the gauge transformation, $B_i \to B_i + {\dot \zeta}_i$ and $F_i \to F_i + \zeta_i$, where $\zeta_i$ is 3-vector satisfying the transverse condition $\partial^i \zeta_i =0$.
Note that a theory with only one degree of freedom in the vector sector is prohibited by spatial covariance of the theory.

\vspace{2mm}

{\bf Case V2 :} $\mu_1 \neq 0$ \\
When $\mu_1 \neq 0$, the last equation \eqref{dotC2B} can be used to determine the Lagrange multipliers $\lambda_{B_i}$,
\begin{align}
\lambda_{B_i} \approx
{1 \over 4\mu_1}\{ {\cal C}_2^{B_i} \,, H^V \} \,.
\end{align}
Thus, there are two primary constraints ${\cal C}_1^{B_i}$ and two secondary constraints ${\cal C}_2^{B_i}$, and
all of them are second-class since the Poisson brackets between these constraints are non-vanishing.
Therefore, the number of the degrees of freedom for the vector modes is given by
\begin{eqnarray}
\textrm{vector DOFs} = \frac{4\times 2 - 4 \,(\textrm{2 primary \& 2 secondary})}{2}
=2\,.
\end{eqnarray}
This case includes the FP theory.

\subsection{Scalar modes}
\label{sec:scalar}
In this subsection, we investigate the scalar modes and classify theories by finding condition to avoid appearing extra ghost degrees of freedom. In the Appendix~\ref{BDproof}, we see that there are dangerous degrees of freedom if the system has two or more degrees of freedom. We also derive gauge transformation and conditions for avoiding instabilities of obtained theories in the Appendix~\ref{App:gauge}.

As for scalar perturbations, by introducing dimensionless quantities $\beta$ and ${\cal E}$
 which are defined by $\beta \equiv k \widehat{\beta}$ and ${\cal E} \equiv k^2 \widehat{{\cal E}}$, respectively,
 the action reduces to
\begin{align}
	S^S [\alpha \,, \beta \,, {\cal R} \,, {\cal E}]
	&=\int{\rm d} t\, {\rm d}^3 k\,{\cal L}^S
	= \int {\rm d} t \, {\rm d}^3 k\, \Bigl( {\cal L}_\mathrm{kin}^S + {\cal L}_\mathrm{cross}^S + {\cal L}_\mathrm{mass}^S \Bigr) \,,
	\label{SS}
\end{align}
where
\begin{align}
	{\cal L}_\mathrm{kin}^S &=
	 4 (\kappa_1+\kappa_2+\kappa_3+\kappa_4) \dot{\alpha}^2
	- (2 \kappa_1+\kappa_2) \dot\beta^2
	+ 12 (\kappa_1+3 \kappa_4) \dot{{\cal R}}^2
	+ 4 (\kappa_1+\kappa_4) \dot{{\cal E}}^2
	\notag\\
	& \qquad
	- 4 (\kappa_3+2 \kappa_4)\left( -3\dot{{\cal R}} + \dot{{\cal E}}
	\right) \dot{\alpha}
	-8 (\kappa_1+3 \kappa_4) \dot{{\cal R}} \dot{{\cal E}}
	\,,
	\label{Skin}\\
	{\cal L}_\mathrm{cross}^S &=
	-4
	\left[  (\kappa_2+\kappa_3) \dot{\alpha}
	+  (\kappa_2+3 \kappa_3) \dot{{\cal R}}
	-  (\kappa_2+\kappa_3) \dot{\cal E}
	\right] k \beta \,,
	\\
	{\cal L}_\mathrm{mass}^S &=
	-4 \Bigl[ k^2 (\kappa_1+\kappa_4)+\mu_1+\mu_2\Bigr] \alpha^2
	+ \Bigl[  k^2 (2 \kappa_1+\kappa_2) +2 \mu_1 \Bigr] \beta^2
	\notag\\
	& \qquad
	- 4\Bigl[ k^2 (3 \kappa_1+\kappa_2+3 \kappa_3+9 \kappa_4)+3 (\mu_1+3 \mu_2) \Bigr] {\cal R}^2
	- 4 \Bigl[ k^2 (\kappa_1+\kappa_2+\kappa_3+\kappa_4)+\mu_1+\mu_2 \Bigr] {\cal E}^2
	\notag\\
	& \qquad
	- 4\Bigl[ \Bigl( k^2 (\kappa_3+6 \kappa_4)+6 \mu_2 \Bigr) {\cal R}
	- \Bigl( k^2 (\kappa_3+2 \kappa_4)+2 \mu_2 \Bigr) {\cal E}
	\Bigr] \alpha
	\notag\\
	& \qquad
	+8 \Bigl[ k^2 (\kappa_1+\kappa_2+2 \kappa_3+3 \kappa_4)+ (\mu_1+3 \mu_2) \Bigr] {\cal R} {\cal E}
	\,.\label{LS mass term}
\end{align}
Now under the ``Condition 2'' \eqref{condition:2}, $2 \kappa_1 + \kappa_2 = 0$, the time derivative of $\beta$ vanishes in the Lagrangian. Then $\beta$ becomes non-dynamical.

The canonical momenta for $Q = \{ \alpha\,, \beta \,, {\cal R} \,, {\cal E} \}$ are defined by $\pi_Q \equiv \delta {\cal L}^S/\delta \dot{Q}$ and read
\begin{align}
\begin{pmatrix}
\pi_\alpha \\
\pi_\beta \\
\pi_{\cal R} \\
\pi_{\cal E} \\
\end{pmatrix}
 = 4 \, {\cal K}^S \,
\begin{pmatrix}
 \dot{\alpha} \\
 \dot{\beta} \\
 \dot{{\cal R}} \\
 \dot{{\cal E}} \\
\end{pmatrix}
 + 4
\begin{pmatrix}
2\kappa_1 - \kappa_3 \\
0 \\
2\kappa_1-3\kappa_3 \\
- 2\kappa_1+\kappa_3
\end{pmatrix}
k\beta \,,
\end{align}
 where
\begin{align}
 {\cal K}^S \equiv
\begin{pmatrix}
2(- \kappa_1 +\kappa_3 +\kappa_4) & 0 &
3(\kappa_3+2\kappa_4) &
- (\kappa_3+2\kappa_4) \\
0 & 0 & 0 & 0 \\
3 (\kappa_3+2\kappa_4) & 0 &
6 (\kappa_1+3\kappa_4) &
- 2 (\kappa_1+3\kappa_4) \\
- (\kappa_3+2\kappa_4) & 0 &
- 2 (\kappa_1+3\kappa_4) &
2 (\kappa_1+\kappa_4)
\end{pmatrix}
 \,.
\label{KineticMatrix33}
\end{align}
Let us calculate the determinant of the kinetic matrix, ${\cal K}^S$, which is given by
\begin{align}
	|{\cal K}^S| =
	- 4 \kappa_1 \left(4 \kappa_1^2-4 \kappa_1 \kappa_3+8 \kappa_1
	\kappa_4+3 \kappa_3^2\right) \,.
\end{align}
Taking into account the condition for having the tensor mode, $\kappa_1\neq0$,
the determinant vanishes only when
\begin{align}
	``\text{Condition 3}" \ :\
	4 \kappa_1^2-4 \kappa_1 \kappa_3+8 \kappa_1
	\kappa_4+3 \kappa_3^2 =0
	\qquad \Longleftrightarrow \qquad
	\kappa_4 = - \frac{4 \kappa_1^2 - 4 \kappa_1 \kappa_3 + 3 \kappa_3^2}{8 \kappa_1} \,.
	\label{condition:3}
\end{align}
Then, the degeneracy of the kinetic matrix \eqref{KineticMatrix33} leads to an additional primary constraint in addition to the one for $\beta$, $\pi_\beta = 0$.
Note that the linearized Einshtein-Hilbert kinetic term satisfies the ``Condition 3" \eqref{condition:3}.
When this degeneracy condition is satisfied, one of the eigenvalues of the kinetic matrix vanishes
 and the remaining eigenvalues, $\lambda$, will be a solution of the following eigen equation:
\begin{eqnarray}
 \lambda^2 + \left( 10 \kappa_1 - 26 \kappa_3 + \frac{33 \kappa_3^2}{2 \kappa_1} \right) \lambda
 = 8 \Bigl[ 4 (\kappa_1 - \kappa_3)^2 + \kappa_3^2 \Bigr] > 0\,.
\end{eqnarray}
As long as $\kappa_{1,3}$ are real, other eigenvalues will be non-vanishing.

\subsubsection{
one primary constraint :  $4 \kappa_1^2-4 \kappa_1 \kappa_3+8 \kappa_1\kappa_4+3 \kappa_3^2 \neq 0$}
\label{sec:VII}

Let us consider the case with only one primary constraint, i.e., the ``Condition 3'' (\ref{condition:3}) is not imposed.
Thus, we have a primary constraint :
\begin{align}
{\cal C}^\beta_1 = \pi_\beta = 0 \,.
\label{primaryconstraint1}
\end{align}
The Hamiltonian and the total Hamiltonian densities in the scalar sector reads
\begin{align}
{\cal H}^S &= \dot{\alpha} \pi_\alpha + \dot{\beta} \pi_\beta + \dot{{\cal R}} \pi_{\cal R} + \dot{{\cal E}} \pi_{\cal E} - {\cal L}^S  \nonumber \\
&\approx
4\left[\mu_1+\mu_2 + k^2(\kappa_1+ \kappa_4)\right]\alpha^2 -2\mu_1 \beta^2
+4\left[\mu_1 +\mu_2 -k^2(\kappa_1-\kappa_3-\kappa_4)\right] {\cal E}^2\nonumber \\
&~~~
-8\left[\mu_1+3\mu_2 -k^2(\kappa_1-2\kappa_3-3\kappa_4)\right]{\cal E}{\cal R}
+4\left[3\mu_1 +9\mu_2 +k^2(\kappa_1+3\kappa_3 +9\kappa_4)\right]{\cal R}^2\nonumber \\
&~~~
-\left[4\left(2\mu_2 + k^2 (\kappa_3+2\kappa_4)\right){\cal E} -4 \left(6\mu_2+k^2(\kappa_3+6\kappa_4)\right){\cal R}\right]\alpha\nonumber \\
&~~~
+{1 \over 32 \kappa_1} \left(2\pi_{\cal R} + 3\pi_{\cal E}  \right) \pi_{\cal E}+(\pi_\alpha + \pi_{\cal E})k\beta
-\frac{8{\kappa_1}(\kappa_1+3\kappa_4) \pi_\alpha^2 -8{\kappa_1}(\kappa_3+2\kappa_4)\pi_\alpha \pi_{\cal R}
	- (2\kappa_1 - \kappa_3)^2 \pi_{\cal R}{^2}
}{32{\kappa_1}(4\kappa_1^2-4\kappa_1\kappa_3+3\kappa_3^2+8\kappa_1\kappa_4)}
\,,  \\
{\cal H}_T^S&
= {\cal H}^S + \lambda_\beta \pi_\beta \,.
\end{align}
We have suppressed the term $\dot{\beta} \pi_\beta$ in the final expression of ${\cal H}^S$ since it vanishes once the primary constraint (\ref{primaryconstraint1}) is imposed.
The evolution of the primary constraint automatically yields a secondary constraint
\begin{align}
{\cal C}^\beta_2 \equiv \dot{{\cal C}}^\beta_1 = \{ {\cal C}^\beta_1 \,, H_T^S \} = \{ {\cal C}^\beta_1 \,, H^S \}
=  -k\pi_\alpha -  k\pi_{\cal E} +4 \mu_1 \beta \approx 0\,.
\label{secondaryconstraint}
\end{align}
Then, the Poisson bracket between the secondary and primary constraints of $\beta$ is given by
\begin{align}
\{ {\cal C}^\beta_2 \,, {\cal C}^\beta_1 \} &= 4\mu_1 \,.
\end{align}
If $\mu_1 \neq 0$, then no more constraint will be generated and the last equation can be used to determine the value of $\lambda_\beta$, that is
$\lambda_\beta \approx - \{ {\cal C}^\beta_2 \,, H^S \} / \{ {\cal C}^\beta_2 \,, {\cal C}^\beta_1 \} $.
In this case, we have one primary constraint ${\cal C}_1^\beta$, and one secondary constraint ${\cal C}_2^\beta$, all of which are
second-class. Therefore, the number of degrees of freedom in the scalar sector is $(8-2)/2=3$,
signalling the existence of an extra degree of freedom. The explicit proof of the existence of a ghost degree of freedom is
given in the Appendix \ref{BDproof}.
Therefore, one has to impose an extra condition $\mu_1 = 0$ in order to
eliminate this extra degree of freedom.

\vspace{2mm}
{\bf Case SI :} $\mu_1 = 0$, $4\kappa_1^2-4\kappa_1\kappa_3+3\kappa_3^2+8\kappa_1\kappa_4 \neq 0$ \\
If $\mu_1=0$, the consistency of the secondary constraint ${\cal C}_2^\beta$ generates a tertiary constraint,
\begin{eqnarray}
{\cal C}_3^\beta \equiv \dot{{\cal C}}_2^\beta
= \{ {\cal C}^\beta_2 \,, H_T^S \} = \{ {\cal C}^\beta_2 \,, H^S \}
= 4k^3 \bigl[
(2\kappa_1 - \kappa_3) \alpha +(2\kappa_1 -3 \kappa_3) {\cal R} -(2\kappa_1-\kappa_3){\cal E}
\bigr] \approx 0\,.
\end{eqnarray}
Now one can check that all the constraints commute each other, i.e., $\{ {\cal C}^\beta_i \,, {\cal C}^\beta_j \}=0$ where ($i,j = 1,2,3$). One can also check that $\dot{{\cal C}}_3^\beta=k^2 {\cal C}_2^\beta$, implying no more constraint.
There are one primary constraint ${\cal C}_1^\beta$, one secondary constraint ${\cal C}_2^\beta$, and one tertiary constraint ${\cal C}_3^\beta$, and all of them
are first-class. Therefore,
\begin{eqnarray}
\textrm{scalar DOF} = \frac{4\times 2 - 3\,(\textrm{1 primary \& 1 secondary \& 1 tertiary})\times 2 \,(\textrm{first class})}{2}
=1
\end{eqnarray}
Since $\mu_1=0$ corresponds to Case V1, the vector sector does not have any degree of freedom, and the total number of degrees of freedom in this case is three.

\subsubsection{two primary constraints : $4 \kappa_1^2-4 \kappa_1 \kappa_3+8 \kappa_1
	\kappa_4+3 \kappa_3^2=0$}
Hereafter we impose the ``Condition 3'' (\ref{condition:3}), implying the existence of two primary constraints.
As one can easily see, the canonical momenta $\pi_\alpha$ and $\pi_{{\cal R}}$ are
not independent because of the degeneracy condition \eqref{condition:3},
i.e.,
\begin{eqnarray}
 (2\kappa_1 - 3\kappa_3) \pi_\alpha - (2\kappa_1-\kappa_3) \pi_{{\cal R}} = 0 \,.
\label{primaryconstraint2}
\end{eqnarray}
This clearly shows that there are two primary constraints in this class of theories, namely \eqref{primaryconstraint1} and \eqref{primaryconstraint2}.
We classify theories depending on vanishing or non-vanishing of the coefficient of $\pi_\alpha$,
$2\kappa_1 - 3\kappa_3$.
Hereafter, we will focus only on the case of $2\kappa_1-3\kappa_3\neq 0$.
The Hamiltonian analysis in the class of models with $2\kappa_1 - 3\kappa_3=0$ is presented
 in Appendix \ref{sec:other cases}.

In this case, we have the following two primary constraints :
\begin{align}
	{\cal C}^\alpha_1 &= \pi_\alpha -{2\kappa_1-\kappa_3 \over 2\kappa_1-3\kappa_3} \pi_{\cal R} = 0 \,, \qquad
	{\cal C}^\beta_1 = \pi_\beta = 0 \,.
	\label{primaryalphabeta}
\end{align}
Here it is clear that ${\cal C}_1^\alpha$ and ${\cal C}_1^\beta$ commute each other.
Then the Hamiltonian and the total Hamiltonian densities read
\begin{align}
	{\cal H}^S &=\dot{\alpha} \pi_\alpha + \dot{\beta} \pi_\beta + \dot{{\cal R}} \pi_{\cal R} + \dot{{\cal E}} \pi_{\cal E} - {\cal L}^S \,,\nonumber \\
	&\approx \left[\frac{k^2 (2\kappa_1-\kappa_3)(2\kappa_1+3\kappa_3)}{2\kappa_1}+4(\mu_1+\mu_2) \right] \alpha^2
	- 2\mu_1 \beta^2
	+ \left[- \frac{3k^2(2\kappa_1-\kappa_3)^2}{2\kappa_1}+4(\mu_1+\mu_2)\right]{\cal E}^2\nonumber\\
	&~~~
	+ \left[\frac{k^2(2\kappa_1-\kappa_3)(10\kappa_1-9\kappa_3)}{\kappa_1}-8(\mu_1+3\mu_2)\right]{\cal E}{\cal R}
	+\left[- \frac{k^2(14\kappa_1-9\kappa_3)(2\kappa_1-3\kappa_3)}{2\kappa_1}+12(\mu_1+3\mu_2)\right]{\cal R}^2 \nonumber\\
	&~~~
	+ \left[\frac{k^2(2\kappa_1-\kappa_3)(2\kappa_1-3\kappa_3)}{\kappa_1}-8\mu_2\right]\alpha {\cal E}
	+ \left[ - \frac{k^2 (12\kappa_1^2 -16\kappa_1\kappa_3 + 9\kappa_3^2)}{\kappa_1} + 24\mu_2\right]\alpha{\cal R}  \nonumber\\
	&~~~
	+k \beta\pi_{\cal E}+\frac{3}{32 \kappa_1}\pi_{\cal E}^2
	+\frac{2\kappa_1-\kappa_3}{2\kappa_1-3\kappa_3}k\beta\pi_{\cal R}
	+{1 \over 16 \kappa_1}\pi_{\cal R} \pi_{\cal E}
	-\frac{(2\kappa_1-\kappa_3)(2\kappa_1+3\kappa_3)}{32 \kappa_1 (2\kappa_1 -3\kappa_3)^2}\pi_{\cal R}^2
	\,,
\end{align}
 and
\begin{align}
	{\cal H}_T^S&= {\cal H}^S + \lambda_\alpha {\cal C}_1^\alpha + \lambda_\beta {\cal C}_1^\beta \,.
\end{align}
where we have suppressed the terms proportional to the primary constraints (\ref{primaryalphabeta}) in the final expression of ${\cal H}^S$.

The evolution of the primary constraints requires
\begin{align}
	{\cal C}_2^\alpha &\equiv \dot{{\cal C}}_1^\alpha
	= \{ {\cal C}^\alpha_1 \,, H_T^S \} = \{ {\cal C}^\alpha_1 \,, H^S \} = c_{2}^\alpha\, \alpha + c_{2}^{\cal R}\, {\cal R} + c_{2}^{\cal E}\, {\cal E} \approx 0\,,
\label{const:c2alpha}
	\\
	{\cal C}_2^\beta &\equiv\dot{{\cal C}}_1^\beta
	= \{ {\cal C}^\beta_1 \,, H_T^S \} = \{ {\cal C}^\beta_1 \,, H^S \} = -\frac{2\kappa_1 - \kappa_3}{2\kappa_1-3\kappa_3}k \pi_{\cal R}
	- k\pi_{\cal E} +4\mu_1 \beta \approx 0\,,
\label{const:c2beta}
\end{align}
where we have defined the coefficients,
\begin{eqnarray}
c_2^\alpha&=&-\frac{8\left[\mu_1 (2\kappa_1 -3\kappa_3) -4\mu_2 \kappa_1 + 2k^2(2\kappa_1-\kappa_3)(\kappa_1-\kappa_3)\right]}{2\kappa_1 -3\kappa_3} \,,
\label{coe:c2a}
\\
c_2^{\cal R}&=&-\frac{8\left[-3\mu_1 (2\kappa_1 -\kappa_3) -12\mu_2 \kappa_1 + 2k^2(2\kappa_1-3\kappa_3)(\kappa_1-\kappa_3)\right]}{2\kappa_1 -3\kappa_3} \,,
\label{coe:c2r}
\\
c_2^{\cal E}&=&-\frac{8\left[\mu_1 (2\kappa_1 -\kappa_3) +4\mu_2 \kappa_1 - 2k^2(2\kappa_1-\kappa_3)(\kappa_1-\kappa_3)\right]}{2\kappa_1 -3\kappa_3} \,.
\label{coe:c2e}
\end{eqnarray}
In general we will have two secondary constraints, but there exists an exceptional case where only one secondary constraint exists if $\kappa_1 = \kappa_3$. With this condition satisfied, the terms proportional to $k^2$ in the numerator of
 Eqs.~(\ref{coe:c2a}, \ref{coe:c2r}, \ref{coe:c2e}) vanish and hence ${\cal C}_2^\alpha$ reduces to a trivial equation depending on the value of $\mu_1$ and $\mu_2$. This special case happens either $\mu_1 = \mu_2 =0$ or $\mu_1+4\mu_2 = 0$ is satisfied, which will be investigated in the end of this subsection.
 In the meantime, we consider the case with two non-trivial secondary constraints assuming $\kappa_1 \neq \kappa_3$.

The Poisson brackets between the primary and secondary constraints are given by
\begin{align}
	\{ {\cal C}_2^\alpha \,, {\cal C}_1^\alpha \} &=- \frac{32}{(2\kappa_1-3\kappa_3)^2}\left[\mu_1(4\kappa_1^2 -6\kappa_1 \kappa_3 + 3\kappa_3^2)+4\mu_2\kappa_1^2\right]\,,\label{C2aC1a} \\
	\{ {\cal C}_2^\beta \,, {\cal C}_1^\beta \} &= 4 \mu_1 \,. \label{C2bC1b}
\end{align}
Thus, as long as these Poisson brackets are non-vanishing,
no more constraints will be generated.
In this case, we have two primary constraints ${\cal C}_1^\alpha, {\cal C}_1^\beta$ and two secondary constraints ${\cal C}_2^\alpha, {\cal C}_2^\beta$ , and
all the constraints are second-class, which implies the number of degrees of freedom
is $(8-4)/2=2$. The explicit proof of the existence of a ghost degree of freedom is
given in the Appendix \ref{BDproof}.
Therefore, one needs to eliminate an extra degree of freedom.
In order to have an extra constraint, there are three options :
both $\mu_1$ and $\mu_2$ vanish ({\bf Case SIIa}), either \eqref{C2aC1a} ({\bf Case SIIc})
or \eqref{C2bC1b} ({\bf Case SIIb}) vanishes.

\vspace{2mm}

{\bf Case SIIa :} $\mu_1=\mu_2=0$,
\\
If $\mu_1=\mu_2=0$, both \eqref{C2aC1a} and \eqref{C2bC1b} vanish.
In addition,one can see
$\dot{\cal C}_2^\alpha \propto {\cal C}_2^\beta \approx 0$ and
$\dot{\cal C}_2^\beta \propto {\cal C}_2^\alpha \approx 0$\,,
which implies that no more constraint is therefore generated.
Since all the constraints commute, all the primary and secondary constraints are first-class.
Therefore,
\begin{eqnarray}
\textrm{scalar DOF} = \frac{4\times 2 - 4 \,(\textrm{2 primary \& 2 secondary})\times 2 \,(\textrm{first-class})}{2}
=0 \,.
\end{eqnarray}
In this case,
the scalar mode as well as the vector mode do not have any DOF (Case V1), and only the tensor mode
can propagate.
Since the linearized Einstein-Hilbert term satisfy the ``Condition 3'' \eqref{condition:3}, and the mass terms are absent, this class reduces to linearized general relativity when $2\kappa_1-\kappa_3=0$.
As we will see in the next section, the whole parameter family of this case SIIa can be mapped from linearized general relativity by a field redefinition.

\vspace{2mm}

{\bf Case SIIb :} $\mu_1 = 0$, $\mu_2 \neq 0$, \\
In this case, since \eqref{C2aC1a} is non-vanishing while \eqref{C2bC1b} vanishes,
$\lambda_\alpha$ is determined by $\dot{{\cal C}}^\alpha_2 \approx 0$,
that is  $\lambda_\alpha = -\{{\cal C}_2^\alpha, H^S\}/\{{\cal C}_2^\alpha, {\cal C}_1^\alpha\}$.
On the other hand as for ${\cal C}_2^\beta$,
since ${\cal C}_2^\beta$ does not commute with ${\cal C}_2^\alpha$,
we shall consider a linear combination of ${\cal C}_2^\beta$ and ${\cal C}_1^\alpha$
which commute with ${\cal C}_2^\alpha$
instead of the original ${\cal C}_2^\beta$:
\begin{eqnarray}
 \widetilde{{\cal C}}_2^\beta &\equiv {\cal C}_2^\beta - k \, {\cal C}_1^\alpha = - k (\pi_\alpha + \pi_{\cal E})\,.
\end{eqnarray}
The consistency of $\widetilde{{\cal C}}^\beta_2$ yields a tertiary constraint,
\begin{eqnarray}
{\cal C}_3^\beta &\equiv & \dot{\widetilde{{\cal C}}}{}_2^\beta
= \{ \widetilde{{\cal C}}^\beta_2 \,, H_T^S \} = \{ \widetilde{{\cal C}}^\beta_2 \,, H^S \}
=
4 k^3 \Bigl[ (2 \kappa_1 - \kappa_3) \alpha + (2 \kappa_1 - 3 \kappa_3) {\cal R}
 - (2 \kappa_1 - \kappa_3) {\cal E} \Bigr]
\approx 0 \,.
\label{eq:C3b}
\end{eqnarray}
The Poisson brackets between this constraint and primary constraints vanish
and one also find $\dot{\cal C}_3^\beta = k^2{\cal C}_2^\beta = k^2 (\widetilde{{\cal C}}_2^\beta + k {\cal C}_1^\alpha) \approx 0$
implying that no more constraint is generated.
One can straightforwardly show that the constraints ${{\cal C}}^\beta_1$, ${\widetilde {\cal C}}_2^\beta$, and
${{\cal C}}_3^\beta$
commute with all constraints including themselves.
Therefore,  there are two primary constraints ${\cal C}_1^\alpha, {\cal C}_1^\beta$, two secondary constraints ${\cal C}_2^\alpha, {\widetilde {\cal C}}_2^\beta$, and one tertiary constraint ${\cal C}_3^\beta$. The constraints ${{\cal C}}^\beta_1$, ${\widetilde {\cal C}}_2^\beta$, and ${\cal C}_3^\beta$ are first-class, and rest of them are second-class. Therefore,
\begin{eqnarray}
\textrm{scalar DOF} = \frac{4\times 2 - 3 \, (\textrm{1 primary \& 1 secondary \& 1 tertiary}) \times 2 \, (\textrm{first-class})- 2 \, (\textrm{1 primary \& 1 secondary})}{2}
=0 \,. \nonumber\\
\end{eqnarray}
In this case, the vector mode does not propagate (Case V1) and the total number of degrees of freedom is two.

\vspace{2mm}

{\bf Case SIIc :} $\mu_1(4\kappa_1^2 -6\kappa_1 \kappa_3 + 3\kappa_3^2)+4\mu_2\kappa_1^2 =0$, $\mu_1 \neq 0$, \\
In this case, since \eqref{C2bC1b} does not vanish,
$\lambda_\beta$ can be determined by imposing $\dot{{\cal C}}^\beta_2\approx 0$,
that is  $\lambda_\beta = -\{{\cal C}_2^\beta, H^S\}/\{{\cal C}_2^\beta, {\cal C}_1^\beta\}$,
and hence no more constraint will be generated as for ${\cal C}_1^\beta$.
On the other hand, we solve the condition for $\mu_2$ from $\{{\cal C}_2^\alpha, {\cal C}_1^\alpha\} = 0$, which is given by
\begin{eqnarray}
	``\text{Condition 4}" \ :\ \mu_2
	= - \frac{4 \kappa_1^2-6 \kappa_1 \kappa_3+3 \kappa_3^2}{4 \kappa_1^2} \mu_1 \,.
\label{condition:4}
\end{eqnarray}
Since ${\cal C}_2^\alpha$ does not commute with ${\cal C}_2^\beta$,
 we shall consider a linear combination of ${\cal C}_2^\alpha$ and ${\cal C}_1^\beta$
which commutes with ${\cal C}_2^\beta$
instead of the original ${\cal C}_2^\alpha$:
\begin{align}
 \widetilde{{\cal C}}_2^\alpha
 &\equiv {\cal C}_2^\alpha + k \frac{8 (\kappa_1 - \kappa_3)}{2 \kappa_1 - 3 \kappa_3} {\cal C}_1^\beta \,.
\end{align}
Then, the time consistency of $\widetilde{{\cal C}}^\alpha_2$ yields a tertiary constraint
\begin{align}
 {\cal C}_3^\alpha \equiv \dot{\widetilde{{\cal C}}}{}_2^\alpha
 = \{ \widetilde{{\cal C}}^\alpha_2 \,, H_T^S \} = \{ \widetilde{{\cal C}}^\alpha_2 \,, H^S \}
 =
 - \frac{4 (\kappa_1 - \kappa_3)}{2 \kappa_1 - 3 \kappa_3}
 \left( \frac{k^2 (2 \kappa_1 - \kappa_3) - 2 \mu_1}{2 \kappa_1 - 3 \kappa_3} \pi_{\cal R}
 + k^2 \pi_{\cal E} \right)
  \approx 0 \,.
\label{eq:C3a}
\end{align}
The Poisson bracket between this constraint and primary constraints vanishes and the
 time-consistency of ${\cal C}_3^\alpha$ yields a quaternary constraint
\begin{eqnarray}
{\cal C}_4^\alpha \equiv \dot{{\cal C}}_3^\alpha
= \{ {\cal C}^\alpha_3 \,, H_T^S \} =
\{ {\cal C}^\alpha_3 \,, H^S \}
=
c_4^\alpha\, \alpha + c_4^{\cal R}\, {\cal R} + c_4^{\cal E}\, {\cal E}
 \approx 0
\,,
\label{C4a}
\end{eqnarray}
where
\begin{eqnarray}
c_4^\alpha &=&
-\frac{8(\kappa_1-\kappa_3)}{\kappa_1^2 (2\kappa_1-3\kappa_3)^2} \left[
2k^4 \kappa_1^2 (2\kappa_1-\kappa_3)^2
+ k^2 \kappa_1 \mu_1
{(2\kappa_1-3\kappa_3)(2\kappa_1-\kappa_3)}
-6\mu_1^2(4\kappa_1^2-6\kappa_1\kappa_3 + 3\kappa_3^2)
\right]
\,,\\
c_4^{\cal R} &=& -
\frac{8(\kappa_1-\kappa_3)}{\kappa_1^2(2\kappa_1-3\kappa_3)}
\left[2k^4\kappa_1^2(2\kappa_1-\kappa_3)
+
k^2\kappa_1\mu_1
{(2\kappa_1-3\kappa_3)}
{-}6\mu_1^2(4\kappa_1-3\kappa_3)
\right]
\,, \\
c_4^{\cal E} &=&
\frac{8(\kappa_1-\kappa_3)}{\kappa_1^2 (2\kappa_1-3\kappa_3)^2} \left[
2k^4 \kappa_1^2 (2\kappa_1-\kappa_3)^2
{- k^2\kappa_1 \mu_1}
{(2\kappa_1+3\kappa_3)(2\kappa_1-\kappa_3)}
{-}2 \mu_1^2(2\kappa_1-3\kappa_3) (4\kappa_1-3\kappa_3)
\right]
\,.
\end{eqnarray}
The Poisson brackets between this constraint and primary constraints are
	\begin{eqnarray}
	\{{\cal C}_4^\alpha, {\cal C}_1^\alpha\}
	= - \frac{192 \mu_1^2 (\kappa_1 - \kappa_3)^2}{\kappa_1 (2 \kappa_1 - 3 \kappa_3)^2}
	\,, \qquad
	\{{\cal C}_4^\alpha, {\cal C}_1^\beta\} = 0\,.
	\end{eqnarray}
Then, the consistency of this constraint $\dot{{\cal C}}^\alpha_4\approx 0$ fixes the Lagrange multiplier $\lambda_\alpha = -\{{\cal C}_4^\alpha, H^S\}/\{{\cal C}_4^\alpha, {\cal C}_1^\alpha\}$.
There are six second class constraints ${\cal C}_1^\alpha, {\cal C}_1^\beta, \widetilde{{\cal C}}_2^\alpha, {\cal C}_2^\beta, {\cal C}_3^\alpha,$ and ${\cal C}_4^\alpha$. Therefore,
\begin{eqnarray}
\textrm{scalar DOF} = \frac{4\times 2 - 6 \, (\textrm{2 primary \& 2 secondary \& 1 tertiary \& 1 quaternary})}{2}
=1 \,.
\end{eqnarray}
In this case, the total number of degrees of freedom is five (Case V2).
It should be noted that the Fierz-Pauli theory is included in this class
since the linearized Einstein-Hilbert term satisfy the ``Condition 3'' \eqref{condition:3}, and the condition $\mu_1 = -\mu_2$ is included in the ``Condition 4'' \eqref{condition:4}.
Therefore, this is a wider class of Fierz-Pauli theory with five degrees of freedom for a massive spin-2 field.
The whole parameter family of this case can be mapped from Fierz-Pauli theory by a field redefinition as we will see in the next section.

\vspace{2mm}

{\bf Other case :} $\kappa_1 = \kappa_3$, $\kappa_4 = - (4 \kappa_1^2 - 4 \kappa_1 \kappa_3 + 3 \kappa_3^2)/(8 \kappa_1) = - (3/8) \kappa_1$ \\
Before the end of this section, we shall consider the case with $\kappa_1 = \kappa_3$
 and $\mu_1 = \mu_2 = 0$ or $\mu_1 = - 4 \mu_2 \neq 0$
 in which only one secondary constraint, ${\cal C}_2^\beta$ \eqref{const:c2beta}, exists
  while ${\cal C}_2^\alpha$ \eqref{const:c2alpha} vanishes
  and hence no more constraint will be generated as for ${\cal C}_2^\alpha$.

First the condition for the mass parameter in Case SIIc, ``Condition 4", is equivalent to
\begin{eqnarray}
 \mu_2 = - \frac{4 \kappa_1^2-6 \kappa_1 \kappa_3+3 \kappa_3^2}{4 \kappa_1^2} \mu_1
 = - \frac{1}{4}{\mu_1} \,.
\end{eqnarray}
Since $\{{\cal C}_2^\beta\,, {\cal C}_1^\beta\}\propto\mu_1\neq 0$, $\lambda_\beta$ can be determined by imposing $\dot{{\cal C}}^\beta_2\approx 0$,
that is  $\lambda_\beta = -\{{\cal C}_2^\beta, H^S\}/\{{\cal C}_2^\beta, {\cal C}_1^\beta\}$.
Since ${\cal C}_1^\alpha$ is first-class while ${\cal C}_{1,2}^\beta$ are second-class, we have
\begin{eqnarray}
\textrm{scalar DOFs} = \frac{4\times 2 - 1 \,(\textrm{1 primary})\times 2 \,(\textrm{first-class})
 - 2 \,(\textrm{1 primary \& 1 secondary})}{2}
=2 \,.
\end{eqnarray}
As we see in Appendix \ref{BDproof}, one of the modes is a ghost and hence we will no longer consider this case.

\vspace{2mm}

{\bf Case SW :} $\kappa_1 = \kappa_3$, $\kappa_4 = - (4 \kappa_1^2 - 4 \kappa_1 \kappa_3 + 3 \kappa_3^2)/(8 \kappa_1) = - (3/8) \kappa_1$, $\mu_1 = \mu_2 = 0$, \\
In this case, which is similar to Case SIIa except for the additional condition, $\kappa_1 = \kappa_3$,
 the time consistency of ${{\cal C}}^\beta_2$ yields a tertiary constraint
 since $\{ {\cal C}_2^\beta \,, {\cal C}_1^\beta \}$ vanishes,
\begin{eqnarray}
 {\cal C}_3^\beta &\equiv & \dot{{\cal C}}_2^\beta
 = \{ {\cal C}^\beta_2 \,, H_T^S \} = \{ {\cal C}^\beta_2 \,, H^S \}
 = 4 k^3 \kappa_1 (\alpha - {\cal R} - {\cal E}) \approx 0 \,.
\end{eqnarray}
The Poisson brackets with primary constraints are trivially satisfied and also the time-evolution of ${\cal C}_3^\beta$ turns out to be $\dot{{\cal C}}_3^\beta \propto {\cal C}_2^\beta$.
Since all the constraints commute each other, two primary, secondary, and tertiary constraints are first-class. Therefore
\begin{eqnarray}
\textrm{scalar DOF} = \frac{4\times 2 - 4 \,(\textrm{2 primary \& 1 secondary \& 1 tertiary})\times 2 \,(\textrm{first-class})}{2}
=0 \,.
\end{eqnarray}
The total number of degrees of freedom is two with 2 tensor modes and without vector mode corresponding to Case V1.

\section{Theoretical Properties}
\label{Sec:4}
In this section, we investigate theoretical properties of the obtained theories in the previous section in detail. We first consider a field redefinition linear in $h_{\mu \nu}$, which helps us to understand structures and classification of the theories. Furthermore, we clarify the crucial differences between the case SI and SIIb, which cannot be obtained from the known theories (linearized general relativity and Fierz-Pauli theory)
by any invertible field redefinition.

\subsection{Linear field redefinition}
\label{subsec:LFD}
In this subsection, we consider a linear field redefinition of the rank-2 tensor $h_{\mu\nu}$, which respects Lorentz invariance.
A possible field redefinition as studied in \cite{Alvarez:2006uu,Bonifacio:2015rea} is
\begin{eqnarray}
h_{\mu\nu} = \Omega^2 \, {\overline h_{\mu\nu}} + \Gamma\, {\overline h} \, \eta_{\mu\nu} \,,
\label{eq:field redef}
\end{eqnarray}
where $\Omega$ and $\Gamma$ are constants, ${\overline h}$ is the trace of $\overline h_{\mu\nu}$ contracted by $\eta_{\mu\nu}$. Hereafter we set $\Omega=1$ without loss of generality since
it only changes the normalization of the Lagrangian.
The inverse transformation is given by
\begin{eqnarray}
{\overline h}_{\mu\nu} = h_{\mu\nu} - \frac{\Gamma}{1+ 4\Gamma}\, h\, \eta_{\mu\nu} \,.
\end{eqnarray}
When $\Gamma = - 1/4$, this transformation is not invertible since its determinant vanishes.

Now we apply this transformation to our theories.
After the field redefinition, the Lagrangian \eqref{Lagrangian} reads
\begin{align}
	S \Bigl[ h_{\mu \nu} = {\overline h_{\mu\nu}} + \Gamma\, {\overline h} \, \eta_{\mu\nu} \Bigr]
	= \int {\rm d}^4 x \Bigl(-
	{\overline {\cal K}}^{\alpha \beta | \mu \nu \rho \sigma} {\bar h}_{\mu \nu, \alpha} {\bar h}_{\rho \sigma, \beta}
	-{\overline {\cal M}}^{\mu \nu \rho \sigma} {\bar h}_{\mu \nu} {\bar h}_{\rho \sigma} \Bigr) \,,
	\label{Lagrangian2}
\end{align}
where
\begin{align}
	{\overline {\cal K}}^{\alpha \beta | \mu \nu \rho \sigma}
	&= {\overline \kappa}_1 \eta^{\alpha \beta} \eta^{\mu \rho} \eta^{\nu \sigma}
	+ {\overline \kappa}_2 \eta^{\mu \alpha} \eta^{\rho \beta} \eta^{\nu \sigma}
	+ {\overline \kappa}_3 \eta^{\alpha \mu} \eta^{\nu \beta} \eta^{\rho \sigma}
	+ {\overline \kappa}_4 \eta^{\alpha \beta} \eta^{\mu \nu} \eta^{\rho \sigma} \,,\\
	{\overline {\cal M}}^{\mu \nu \rho \sigma}
	&= \overline\mu_1 \eta^{\mu \rho} \eta^{\nu \sigma}
	+ \overline\mu_2 \eta^{\mu \nu} \eta^{\rho \sigma} \,.
\end{align}
The straightforward calculation shows the relation between $\kappa_i, {\mu_i}$ and ${\overline \kappa}_i, {{\overline\mu}_i}$ :
\begin{eqnarray}
&&{\overline \kappa}_1 = \kappa_1\,, \qquad
{\overline \kappa}_2 = \kappa_2\,, \qquad
{\overline \kappa}_3 = {2\Gamma\kappa_2+(1+4\Gamma)\kappa_3},\\ &&{\overline \kappa}_4 = 2\Gamma(1+2\Gamma)\kappa_1+\Gamma^2\kappa_2+\Gamma(1+4\Gamma)\kappa_3+(1+4\Gamma)^2\kappa_4\,,
\\
&& \overline\mu_1 = \mu_1\,, \qquad
\overline\mu_2 = {2\mu_1\Gamma(1+2\Gamma)+\mu_2(1+4\Gamma)^2}
\,.
\end{eqnarray}
It should be noted that coefficients $\kappa_{1,2}$ and $\mu_1$ are invariant under this transformation as long as $\Omega = 1$.
We also find that the degeneracy conditions for the scalar and vector modes, Eqs.~\eqref{condition:2} and \eqref{condition:3},
are invariant under the field redefinition Eq.~\eqref{eq:field redef}.
Namely when $\kappa_i$ and $\mu_i$ satisfy the ``Condition 2'' and ``Condition 3'',
the transformed parameters still satisfy the same relations, that is
\begin{eqnarray}
	``\text{Condition 2}" : 2\overline\kappa_1+\overline\kappa_2=0 \,, \qquad
	``\text{Condition 3}" : 4\overline\kappa_1^2-4\overline\kappa_1\overline\kappa_3
	+3\overline\kappa_3^2
	+8\overline\kappa_1\overline\kappa_4 =0
	\,.
\end{eqnarray}
In addition to these, the additional condition to remove the extra degree of freedom for the scalar modes, $\mu_1=0$ for Case SI/SIIa\,(S${\overline{\rm IIa}}$)/SIIb\,(S${\overline{\rm IIb}}$) or
the ``Condition 4'' \eqref{condition:4} for Case SIIc, is also an invariant quantity.
This immediately leads that linear degenerate theories can be transformed to different linear degenerate theories
with a different parameter set through the field redefinition Eq.~\eqref{eq:field redef}.
In particular, when one chooses
\begin{eqnarray}
	\Gamma ={-\frac{2\kappa_1-3\kappa_3}{12(\kappa_1-\kappa_3)}}
	\,,\label{TransG}
\end{eqnarray}
one can obtain the degenerate theories with $\overline\kappa_3=2\overline\kappa_1/3$ like in Case S${\overline{\rm IIa}}$ from the degenerate theories with $\kappa_3\neq 2\kappa_1/3$ like in Case SIIa.
We also note that the same kinetic structure as linearized
Einstein-Hilbert term can be obtained for all the theories in Case SII by setting $\overline\kappa_3=2\overline\kappa_1$ via the field redefinition with
\begin{eqnarray}
	\Gamma =-\frac{2\kappa_1-\kappa_3}{4(\kappa_1-\kappa_3)}
	\,.\label{GammaToGR}
\end{eqnarray}
Hence, upon the field redefinition \eqref{eq:field redef}, one can show that
\begin{eqnarray}
\textrm{Case SIIa and S${\overline{\rm IIa}}$} ~&\longleftrightarrow&~ \textrm{Linearlized general relativity} \,, \\
\textrm{Case SIIb} ~ &\longleftrightarrow& ~ \textrm{Case S$\overline{{\rm IIb}}$} \,, \\
\textrm{Case SIIc and S${\overline{\rm IIc}}$} ~ &\longleftrightarrow & ~ \textrm{Fierz-Pauli theory} \,.
\end{eqnarray}

\subsection{Symmetry of Lagrangian}
It is interesting to see the symmetry of Lagrangian
 as also studied in \cite{Alvarez:2006uu,Bonifacio:2015rea}.
When $2 \kappa_1 + \kappa_2 = 0$ is satisfied,
Lagrangian \eqref{reLagrangian} can be rewritten as assuming $\kappa_1 \neq \kappa_3$,
\begin{align}
S [h_{\mu \nu}]
= \kappa_1 S^\mathrm{LGR} [\widetilde{h}_{\mu \nu} ]
 &- \int {\rm d}^4 x \, \kappa_1
 \frac{4 \kappa_1^2 - 4 \kappa_1 \kappa_3 + 3 \kappa_3^2 + 8 \kappa_1 \kappa_4}{8
 (\kappa_1 - \kappa_3)^2} \, \widetilde{h}_{, \alpha} \widetilde{h}^{, \alpha} \notag\\
 & - \int {\rm d}^4 x \left[
 \mu_1 \, \widetilde{h}_{\mu \nu} \widetilde{h}^{\mu \nu}
+\frac{(2\kappa_1-\kappa_3)\kappa_3\mu_1+4\kappa_1^2\mu_2}{4(\kappa_1-\kappa_3)^2}\,
\widetilde{h}^2 \right] \,,
\label{lagwithC1}
\end{align}
 where
\begin{align}
 \widetilde h_{\mu \nu} \equiv h_{\mu \nu} + \frac{\kappa_3- 2\kappa_1}{4\kappa_1} \, h \, \eta_{\mu \nu} \,,
\end{align}
and $S^\mathrm{LGR}$ stands for action of linearlized general relativity:
\begin{align}
 S^\mathrm{LGR} [h_{\mu \nu} ]
  \equiv - \int {\rm d}^4 x \Bigl( h_{\mu \nu \,, \alpha} h^{\mu \nu \,, \alpha}
 - 2 h^\alpha{}_{ \mu \,, \alpha} h^{\beta \mu}{}_{, \beta}
+ 2 h^{\alpha \beta}{}_{, \alpha} h_{, \beta}
- h_{, \alpha} h^{, \alpha} \Bigr) \,.
 \end{align}
First we should note that the kinetic term of tensorial parts in $\widetilde{h}_{\mu \nu}$
 is nicely summarized in a single term, that is $S^\mathrm{LGR}$.
 This is not possible with a generic $\kappa_2$.
The difference between $\widetilde{h}_{\mu \nu}$ and $h_{\mu \nu}$ is subtle
 since $\widetilde{h}_{\mu \nu}$ reduces to $h_{\mu \nu}$ once $\kappa_3 = 2 \kappa_1$ holds,
 which can be realized by a suitable field redefinition as studied in the last subsection.
 Hence its difference is not essential in discussion below at least in the absence of matter.

Now let us study the symmetry of Lagrangian. The first term $S^\mathrm{LGR}$ is invariant under
a transformation with diffeomorphisms for $\widetilde{h}_{\mu \nu}$:
\begin{align}
 \widetilde{h}_{\mu \nu} \to \widetilde{h}_{\mu \nu} + \partial_\mu \xi_\nu + \partial_\nu \xi_\mu\,,
\end{align}
while other terms do not in general.
However, once the gauge parameters satisfy a condition such that $\partial_\mu \xi^\mu = 0$,
the kinetic term enjoys this restricted
gauge symmetry since $\widetilde{h}$ is invariant under this gauge transformation.
Moreover, if $\mu_1$ also disappears, this symmetry becomes the symmetry of the whole Lagrangian.
This subgroup of the diffeomorphisms is known as transverse diffeomorphisms~\cite{Alvarez:2006uu}.

It is now clear that Case SI and SIIb (S$\overline{\rm IIb}$) possess such the transverse gauge symmetry
with $3$ gauge parameters, $1$ for scalar sector and $2$ for vector sector,
since $\mu_1$ is vanishing as studied in detail in Appendix \ref{App:gauge}.
On the other hand, Case SIIa (S$\overline{\rm IIa}$) respects diffeomorphisms
with $4$ gauge parameters since $\mu_1 = \mu_2 =0$ as in linearlized general relativity.
There is no gauge symmetry in Case SIIc (S$\overline{\rm IIc}$) since both $\mu_1$ and $\mu_2$ are present.
As for the remaining case SW, the Lagrangian can be written as
\begin{align}
	S^\mathrm{SW} [h_{\mu \nu}]
	&= - \kappa_1 \int {\rm d}^4 x \left[ h_{\mu \nu \,, \alpha} h^{\mu \nu \,, \alpha}
	- 2 h^\alpha{}_{ \mu \,, \alpha} h^{\beta \mu}{}_{, \beta}
	+ h^{\alpha \beta}{}_{, \alpha} h_{, \beta}
	- \frac{3}{8} h_{, \alpha} h^{, \alpha} \right] \notag\\
	&= - \kappa_1 \int {\rm d}^4 x \Bigl[ \widehat{h}_{\mu \nu \,, \alpha} \widehat{h}^{\mu \nu \,, \alpha}
	- 2 \widehat{h}^\alpha{}_{ \mu \,, \alpha} \widehat{h}^{\beta \mu}{}_{, \beta} \Bigr]
	= \kappa_1 S^\mathrm{LGR}
	\Bigl[ \, \widehat{h}_{\mu \nu}
	\equiv h_{\mu \nu} - \frac{1}{4} h \, \eta_{\mu \nu} \Bigr] \,.
\end{align}
Since the Lagrangian can be described in the form of lineralized general relativity
 with traceless tensor $\widehat{h}_{\mu \nu} \equiv h_{\mu \nu} - (1/4) h \, \eta_{\mu \nu}$,
  this theory enjoys diffeomorphism invariance as well as the invariance under a field redefinition \eqref{eq:field redef}.
 This enhanced symmetry is called as Weyl-invariant transverse diffeomorphism invariance \cite{Alvarez:2006uu}.

\subsection{New theories}
We have seen that the case SI, SIIb and S$\overline{\rm{IIb}}$ are the new theories of a spin-2 field, which cannot be mapped into linearized general relativity and Fierz-Pauli theory by the field redefinition as studied in the previous subsection \ref{subsec:LFD}. In this subsection, we take a closer look at the Lagrangian of these theories particularly focusing on its scalar sectors.

Now let us first consider linearized general relativity to understand the structure of theories, which can be simply obtained by setting $2\kappa_1 - \kappa_3 =0$ in the case SIIa or equivalently performing a field redefinition with a special $\Gamma$ \eqref{GammaToGR}. Rewriting the Lagrangian in the scalar sector in terms of the gauge invariant variables \eqref{GIVIIb}, we obtain up to total derivative terms
\begin{eqnarray}
{\cal L}^S_{\rm LGR} &=& -3 \dot{{\cal R}}^2 + k^2 {\cal R}^2 + 2k^2 \widetilde{\alpha} {\cal R}\,, \qquad
 \widetilde{\alpha} = \alpha + \frac{\dot{\beta}}{k} - \frac{\ddot{\cal E}}{k^2} \,,
 \label{lag:LGR}
\end{eqnarray}
where we have set the overall factor $\kappa_1$ to be $1/8$ for simplicity. Then the variation with respect to $\widetilde{\alpha}$ gives the constraint ${\cal R}=0$, and it is manifest that the Lagrangian becomes zero after substituting the constraint. Thus we have confirmed that the number of degree of freedom in the scalar sector is zero in the Lagrangian formalism, that is consistent with the Hamiltonian analysis in the previous section.

\vspace{2mm}
	{\bf Case SIIb \& S$\overline{\rm \bf IIb}$}\ \
Now we would like to perform the same analysis for the case IIa. Again, we can set  $2\kappa_1 - \kappa_3 =0$ and  $\kappa_1 = 1/8$ without loss of generality. Then the Lagrangian in the scalar sector in terms of the gauge invariant variables \eqref{GIVI} is given by
\begin{eqnarray}
{\cal L}^S_{\rm IIb} &=& -3 \dot{{\cal R}}^2 + k^2 {\cal R}^2 + 2k^2 {\widetilde \alpha} {\cal R}
 -4\mu_2 \, {\widetilde {\cal E}}^2 \,, \qquad
 \widetilde{{\cal E}} = {\cal E} - \alpha - 3 {\cal R} \,.
\end{eqnarray}
One can clearly see that the first three terms are the exactly same as in the case of SIIa (S$\overline{\rm IIa}$).
However, in this case, there is an extra term, $\mu_2\,{\widetilde {\cal E}}^2 =\mu_2({\rm Tr}\, h_{\mu\nu})^2$
 where $- \widetilde{{\cal E}}$ is the trace of a spin-2 field.
Since it is completely decoupled from ${\cal R}$ and $\widetilde{\alpha}$ and has no kinetic term,  it is actually a non-dynamical degree of freedom.
 The case SIIb (S$\overline{\rm IIb}$) cannot be obtained from any gauge fixing of the case SIIa (S$\overline{\rm IIa}$), and these theories are therefore independent each other.  Since $\widetilde{\alpha}$ and $\widetilde{{\cal E}}$ are non-dynamical, the Lagrangian becomes zero after integrating out these variables just as in the case SIIa (S$\overline{\rm IIa}$).

\vspace{2mm}
	{\bf Case SI}\ \
Finally, we consider the case SI, whose number of scalar degree of freedom is one.
In this case, we can also choose $2\kappa_1 - \kappa_3 =0$ and $\kappa_1 = 1/8$ without loss of generality.
Furthermore, we set $\kappa_4=1/8$ to simplify the coefficient of the kinetic term for $\widetilde{{\cal E}}$.
Then the Lagrangian in terms of the gauge invariant variables \eqref{GIVI} is given by
\begin{eqnarray}
{\cal L}^S_{\rm I} &=& -3 \dot{{\cal R}}^2 + k^2 {\cal R}^2 + 2k^2 {\widetilde \alpha} {\cal R}
+\dot{\widetilde {\cal E}}{}^2-(k^2+4\mu_2) {\widetilde {\cal E}}^2 \,.
\end{eqnarray}
In this case, the trace of $h_{\mu\nu}$, namely $- \widetilde{\cal E}$, which again decouples from other variables ${\cal R}$ and $\widetilde \alpha$, becomes dynamical since it has now the kinetic term, differently from the case SIIb. Thus after plugging the constraint of $\widetilde \alpha$, one scalar degree of freedom remains in the case SI, which is also consistent with the Hamiltonian analysis.

\section{Summary \& Discussion}
\label{sec:summary}

\begin{table}[!t]
	\begin{center}
		\begin{tabular}{|c||c|c|c|c|} \hline
			Case & DOF  & Conditions & Free parameters & Comments \\ \hline \hline
			SI \& V1 & ~$3=2+0+1$~ & $\mu_1 = 0$  & $\kappa_3, \kappa_4, \mu_2$ & New theories \\
			\hline
			SIIa \& V1 & $2=2+0+0$ &  ``Condition 3'' \& $\mu_1=\mu_2=0$   & $\kappa_3$ & General relativity is included \\
			\hline
			SIIb \& V1 & $2=2+0+0$ &   ``Condition 3'' \& $\mu_1 = 0$ \& $\mu_2\neq 0$ & $\kappa_3, \mu_2$ & New theories  \\
			\hline
			SIIc \& V2 & $5=2+2+1$ &  ``Condition 3 \& 4" \& $\mu_1 \neq 0$  & $\kappa_3, \mu_1$ & Fierz-Pauli is included\\
			\hline\hline
			S$\overline{\rm{IIa}}$ \& V1 & $2=2+0+0$ & ``Condition 5'' \& $\mu_1=\mu_2=0$   & {\rm None}&  ~~$-2\kappa_1 + 3\kappa_3 = 0$ limit of  SIIa~~\\
			\hline
			S$\overline{\rm{IIb}}$ \& V1 & $2=2+0+0$ &  ``Condition 5'' \& $\mu_1 = 0$ \& $\mu_2\neq 0$  & $\mu_2$ & ~~$-2\kappa_1 + 3\kappa_3 = 0$ limit of SIIb~~  \\
			\hline
			S$\overline{\rm{IIc}}$ \& V2 & $5=2+2+1$ & ``Condition 5'' \& $\mu_1 + 3\mu_2= 0$ \& $\mu_1 \neq 0$ & $\mu_1$ & $-2\kappa_1 + 3\kappa_3 = 0$ limit of SIIc \\
			\hline
		\end{tabular}
		\caption{The number of the degrees of freedom, the conditions, free parameters and comments for each case is shown. For any case, the ``Condition 1'' (\ref{condition:1}) and  the ``Condition 2'' (\ref{condition:2})
			are always imposed. ``Condition 3,4,5'' are shown in \eqref{condition:3}, \eqref{condition:4}, and \eqref{condition:5} respectively. Among free parameters, $\kappa_1$ is not included since it only changes the normalization of the Lagrangian if its sign is appropriately chosen.}
		\label{table}
	\end{center}
\end{table}

{\bf Summary }
In the present paper, we constructed the most generic spin-2 field theories in a flat spacetime
with at most five degrees of freedom without ghost mode, whose Lagrangian consists of the quadratic terms of the field and its first derivatives.
By decomposing the spin-2 field $h_{\mu\nu}$ into the transverse-traceless tensor, tensor composed by transverse vectors, and tensor composed by scalar components, we classified theoretically consistent theories based on the Hamiltonian analysis in a systematic manner.

We found that the existence of the tensor degrees of freedom is always controlled by one parameter $\kappa_1$, which is assumed to be non-zero while we imposed the degeneracy conditions in order to eliminate extra problematic degrees of freedom for the vector and scalar modes.
Under the degeneracy conditions,
we found two classes in the vector sector : 2 propagating vector degrees of freedom and no degrees of freedom. As in the vector sector, we have also classified theories in the scalar sector based on the Hamiltonian analysis.
The classification of the obtained theories is summarized in Table.~\ref{table}. The case SIIa and SIIc are a wider class of the known theories : linearized general relativity and Fierz-Pauli theory, and we have shown that the case SIIa and SIIc can be mapped from these known theories by field redefinition. On the other hand, the case SI and SIIb are new theories, which cannot be mapped from the known theories. The case SIIb has the same number of degrees of freedom in linearized general relativity, however, it has less gauge degrees of freedom, and it contains the trace of the spin-2 field, which is non-dynamical. On the other hand, the case I has the dynamical degree of freedom coming from the trace of the spin-2 field while the vector degrees of freedom is absent. The other remaining three cases: S$\overline{\rm IIa}$, S$\overline{\rm IIb}$, and S$\overline{\rm IIc}$, are the subset of the cases SIIa, SIIb, and SIIc and hence the case S$\overline{\rm IIb}$ is also a new theory. We provided the conditions for avoiding the ghost, gradient, and tachyonic instabilities by calculating reduced Lagrangian in Appendix \ref{App:gauge}.

\vspace{2mm}
{\bf Non-linear extension to gravity theory}
It is interesting to consider a non-linear extension to (massive) gravity theory of the obtained theories for a spin-2 field. Let us consider the following
Lagrangian\footnote{
The effect of the background curvature in the Lagrangian has been discussed in the literature, e.g. \cite{Akagi:2018ufw}, which is beyond the scope of the present paper.},
\begin{eqnarray}
S = \int {\rm d}^4x \sqrt{-g}
\left[
a_ 1 R + a_2 hR + a_3 h_{\mu\nu} R^{\mu\nu}  +
{\mathfrak K}^{\alpha \beta | \mu \nu \rho \sigma} \nabla_\alpha h_{\mu \nu} \nabla_\beta h_{\rho \sigma}
+
{\mathfrak M}^{\mu \nu \rho \sigma} h_{\mu \nu} h_{\rho \sigma}  + {\cal O} (h^2R, h^3)
\right] \,,
\label{MGaction}
\end{eqnarray}
where the fluctuation tensor is defined by $h_{\mu\nu} = g_{\mu\nu} -\eta_{\mu\nu}$. The coefficients of the kinetic and mass terms, ${\mathfrak K}^{\alpha \beta | \mu \nu \rho \sigma}$ and $
{\mathfrak M}^{\mu \nu \rho \sigma}$, are functions of the metric
$g_{\mu\nu}$,
\begin{align}
{\mathfrak K}^{\alpha \beta | \mu \nu \rho \sigma}
&= b_1 g^{\alpha \beta} g^{\mu \rho} g^{\nu \sigma}
+ b_2 g^{\mu \alpha} g^{\rho \beta} g^{\nu \sigma}
+ b_3 g^{\alpha \mu} g^{\nu \beta} g^{\rho \sigma}
+ b_4 g^{\alpha \beta} g^{\mu \nu} g^{\rho \sigma} \,,\\
{\mathfrak M}^{\mu \nu \rho \sigma}
&= \mu_1 g^{\mu \rho} g^{\nu \sigma}
+ \mu_2 g^{\mu \nu} g^{\rho \sigma} \,,
\end{align}
and $a_{1,2,3}$ and $b_{1,2,3,4}$ are constant parameters.
By expanding the action up to the quadratic terms in $h_{\mu\nu}$ and identifying this field as the (massive) spin-2 field introduced in this paper,
one can find that the relations between these constant parameters and the $\kappa$ parameters introduced in the Lagrangian \eqref{Lagrangian} are given by
\begin{eqnarray}
\kappa_1 = b_1 -{1 \over 2} a_1 + {1 \over 2}a_3 \,, \qquad
\kappa_2 = b_2 + a_1 -a_3 \,, \qquad
\kappa_3 = b_3 -a_1 -a_2 -{1 \over 2}a_3 \,, \qquad
\kappa_4 = b_4 + {1 \over 2}a_1 + a_2 \,.
\end{eqnarray}
Then the gravitational action \eqref{MGaction} perturbed around Minkowski space-time reproduces the Lagrangian \eqref{Lagrangian} at the linear level. Therefore it is for sure that this Lagrangian \eqref{MGaction} with the conditions that we have obtained does not have an extra degrees of freedom at linear order.
However, one might need some extra tuning of the parameters $a_i$ and $b_i$ for avoiding the appearance of Boulware-Daser ghost at nonlinear level, and it has to be carefully examined just as in the construction of the dRGT theory.

\vspace{2mm}
{\bf Matter coupling }
Although we have obtained the interesting theories of the spin-2 field,
one carefully needs to introduce a coupling to matter fields.
To clarify this, let us consider the case SIIc with the matter coupling $h_{\mu\nu} T^{\mu\nu}/M$, where $T^{\mu\nu}$ is the energy-momentum tensor of an external source, and $M$ is a mass parameter, which corresponds to the Planck's mass in general relativity.
Ignoring the vector modes\footnote{A careful analysis shows that the decoupling limit Lagrangian of the vector field becomes massless $U(1)$ field, which completely decouples from other modes. },
the $\Lambda_3$ decoupling limit with the Stuckelberg decomposition, $h_{\mu\nu}  = {\widehat h}_{\mu\nu} +2m^2 \partial_\mu \partial_\nu \pi +b \,m^2\,  \eta_{\mu\nu}\square \pi$, and the conformal transformation,
${\widehat h}_{\mu\nu} = {\frak h}_{\mu\nu} - ({c_1}/{\kappa_1-\kappa_3}) \pi \eta_{\mu\nu}$
yields the following Lagrangian,
\begin{eqnarray}
{\cal L}^{\rm (DL)}&=&
{\cal L}^{\rm (DL)}_{\rm tensor}[{\frak h}]
-  {6c_1 \over \kappa_1} (\partial_\mu \pi)^2
+ {1 \over M} \left( {\frak h}_{\mu\nu} T^{\mu\nu}
- \frac{c_1}{\kappa_1-\kappa_3}\pi T\right)
+ {1 \over \Lambda_3^3} \Bigl[ 2 (\partial_\mu \partial_\nu \pi) T^{\mu\nu} + b (\square \pi) T \Bigr]\,,
\end{eqnarray}
where ${\cal L}^{\rm (DL)}_{\rm tensor}[{\widetilde h}]$ is the kinetic Lagrangian of
the case IIc, $\mu_1 = c_1 m^2$, $m$ is a mass parameter, $T=\eta_{\mu\nu} T^{\mu\nu}$
and $\Lambda_3 = (M m^2)^{1/3}$.
Here, the new interaction appears if $b\neq 0$ and $T \neq 0$
which cannot be appeared in the Fierz-Pauli theory with minimally coupled matter.
The equation of motion for the scalar mode $\pi$ naively contains
the second derivatives of the energy-momentum tensor, which might lead
to the higher derivatives, depending on matter fields.
Whether this matter coupling introduce an extra degree of freedom associated with higher derivatives or not will be reported in future work (see also \cite{Arnowitt:1962hi}).

\acknowledgments
We would like to thank Claudia de Rham, Kazuya Koyama and Nobuyoshi Ohta for fruitful discussions and useful suggestions.
This work was supported in part by JSPS Grant-in-Aid for Scientific Research
Nos.~JP17K14276 (RK), JP25287054 (RK), 17K14304 (DY).
A.N. is supported in part by a JST grant ``Establishing a Consortium for the Development of Human Resources in Science and Technology''. The work of A.N. is also partly supported by the JSPS Grant-in-Aid for Scientific Research No.16H01092.

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HE Dr. Omar Bin Sulaiman,…
HE Dr. Omar Bin Sulaiman, Governor of DIFC to speak tomorrow at FT/DIFC World Financial Centres Summit in London
Press Release 19 Oct 2008 06:07 pm
Changing global economic dynamic to be focus of His Excellency's Address
HE Dr. Omar Bin Sulaiman, Governor, Dubai International Financial Centre (DIFC) and Vice Chairman of the Central Bank of the UAE, will speak at the second FT/DIFC World Financial Centres Summit taking place in London on October 20 and 21, 2008. The focus of His Excellency's address to the Summit will be the factors driving change in the economy.
The Summit, organised by the Financial Times in partnership with the Dubai International Financial Centre (DIFC), brings together senior financial executives, corporate decision-makers and regulatory and government officials to discuss the current re-appraisal of risk, opportunity and strategic interest, and the impact on financial centres around the world.
Building on the success of the inaugural event in Dubai in November 2007, the conference - to be held at Gibson Hall near the Bank of England -- will be one of the year's most important high-level gatherings of policy-makers and strategists, at a time when the world financial system and its key intermediaries are undergoing significant change.
HE Dr. Omar Bin Sulaiman will be joined at the Summit by global financial leaders such as Stephen Green, Group Chairman, HSBC Holdings; Terrence Checki, Executive Vice President, Federal Reserve Bank of New York; Zhu Min, Group Executive Vice President, Bank of China; Stephen Pagliuca, Managing Director, Bain Capital; Norbert Walter, Chief Economist, Deutsche Bank Group; and Gerard Lyons, Chief Economist and Group Head of Global Research, Standard Chartered.
The Summit comes at a time when the sub-prime crisis and its aftermath are sending shockwaves through financial markets transforming the landscape for financial institutions around the world.
His Excellency Dr Omar Bin Sulaiman said: "The Summit is an opportunity to discuss the various forces driving the creation of a new global economic structure. The last few years have seen remarkable changes in global capital flows. China, other leading Asian economies and Gulf countries have become net exporters of capital. The effects of the financial crisis in the West are opening up opportunities for the GCC region that weren't available earlier. The Summit will generate several insights into the opportunities and challenges these economic changes are bringing."
Themes for discussion at the Summit include a new balance for world capital flows; the future of global financial centres; the challenges to the world financial system; the race for talent; the regulatory system; and new strategies for investment banks and private equity.
The annual FT/DIFC World Financial Centre Summits bring together some of the biggest names in international finance to discuss the challenges facing the world's developed and emerging financial hubs.

redpajama_set_name: RedPajamaCommonCrawl


for You to feel my glee?
for You to sense My depth immensely?
in order to be compared?
to prove I have a real dimple?
to take Dr. Seuss out of rhyme?
my actions to be of derilect?
just because I've never had a facelift?
if I want to make the good list?
if at first You can't believe?
when You had nine months to be mothered?


text:
Q: Cannot get Basic HTTP Authentication to work using Restlet 2.1 I'm trying to implement basic HTTP authentication using Restlet 2.1, and I just can't get it to work .. ! I'm using a ChallengeAuthenticator to setup Basic HTTP authentication. I have just one URI \test, for which I'm trying to get authentication working.
I build my code, then run it as a Web Application and then browse to http://localhost:8888/test to see if I get a prompt for a username/password, but I don't get any. I just get a blank screen.
Also when I browse to http://localhost:8888/test, I get the following written in Eclipse's Console:
WARNING: A response with a 200 (Ok) status should have an entity. Make sure that resource "http://localhost:8888/test" returns one or sets the status to 204 (No content).

When when I browse to http://user:password@localhost:8888/test, the result is exactly the same.
The HTTP headers (from Chrome) are as follows:
Request:
GET /test HTTP/1.1
Host: localhost:8888
Connection: keep-alive
Cache-Control: max-age=0
Accept: text/html,application/xhtml+xml,application/xml;q=0.9,image/webp,*/*;q=0.8
User-Agent: Mozilla/5.0 (Windows NT 6.3; WOW64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/35.0.1916.153 Safari/537.36
Accept-Encoding: gzip,deflate,sdch
Accept-Language: en-US,en;q=0.8,ms;q=0.6

Response (no mention of Basic authentication):
HTTP/1.1 200 OK
Date: Fri, 13 Jun 2014 11:21:05 GMT
Accept-Ranges: bytes
Server: Development/1.0
Content-Length: 0
Cache-Control: no-cache
Expires: Fri, 01 Jan 1990 00:00:00 GMT

Here is the Java code:
package com.poc.hw7;

import org.restlet.*;
import org.restlet.data.ChallengeScheme;
import org.restlet.data.Cookie;
import org.restlet.data.MediaType;
import org.restlet.routing.Router;
import org.restlet.security.*;
import org.restlet.util.Series;

public class AuthTestApp extends Application {

    private ChallengeAuthenticator authenticator;

    private ChallengeAuthenticator createAuthenticator() {
        Context context = getContext();
        boolean optional = false;
        ChallengeScheme challengeScheme = ChallengeScheme.HTTP_BASIC;
        String realm = "Example site";

        MapVerifier verifier = new MapVerifier();
        verifier.getLocalSecrets().put("user", "password".toCharArray());

        ChallengeAuthenticator auth = new ChallengeAuthenticator(context, optional, challengeScheme, realm, verifier) {
            @Override
            protected boolean authenticate(Request request, Response response) {
                if (request.getChallengeResponse() == null) {
                    return false;
                } else {
                    return super.authenticate(request, response);
                }
            }
        };
        return auth;
    }

    @Override
    public Restlet createInboundRoot() {
        this.authenticator = createAuthenticator();

        Restlet hw_restlet = new Restlet(getContext())
        {
        public void handle(Request request, Response response)
            {
            String message = "Hello World!";
            response.setEntity(message,MediaType.TEXT_PLAIN);
            }
        };

        Router router = new Router();
        router.attach("/test", hw_restlet);
        authenticator.setNext(router);
        return authenticator;
    }

    public boolean authenticate(Request request, Response response) {
        if (!request.getClientInfo().isAuthenticated()) {
            authenticator.challenge(response, false);
            return false;
        }
        return true;
    }
}

Here is the web.xml file:
<?xml version="1.0" encoding="utf-8"?>
<web-app xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://java.sun.com/xml/ns/javaee" xmlns:web="http://java.sun.com/xml/ns/javaee/web-app_2_5.xsd" xsi:schemaLocation="http://java.sun.com/xml/ns/javaee http://java.sun.com/xml/ns/javaee/web-app_2_5.xsd" version="2.5">
    <display-name>Restlet URI Rewrite</display-name>
    <servlet>
        <servlet-name>RestletServlet</servlet-name>
        <servlet-class>org.restlet.ext.servlet.ServerServlet</servlet-class>
        <init-param>
            <param-name>org.restlet.application</param-name>
            <param-value>com.poc.hw7.AuthTestApp</param-value>
        </init-param>
    </servlet>
    <servlet-mapping>
        <servlet-name>RestletServlet</servlet-name>
        <url-pattern>/*</url-pattern>
    </servlet-mapping>
</web-app>

Can someone please tell me how to get basic HTTP authentication working ?

A: Your response was a 200 code, which means your server never asked for one. You would need to send a 401 response. For most web browsers, this will automatically pop up the basic-auth box for a username and password.
As for why this is happening... it's hard to say. restlet looks very... bad. There are easier frameworks to work with, where you can just annotate a function to listen to a certain URI and/or request method.

redpajama_set_name: RedPajamaStackExchange


Copyright © 2007 rosiepink. We have put a lot of time and effort into making the items you see pictured on this blog so please respect that images must not be copied and used without our express permission. Thank you.
... that Annie and I decided would look good in felt, so I had a think about how I could best make it. Paint and wool are very different mediums and each has its merits and its difficulties.
I wanted the fish to be very clear so I used the same technique as when I made the Sea Dragon.
To make the mackerel picture, I wet felted the background using merino wool. In the middle of the top layer of fibres I formed a bright yellow oval plate shape then surrounded it with sea colours placed down in a swirly fashion. The plate edges were defined with a boucle-type yarn that felts in easily.
The fish shapes were cut from specially made pieces of white, aqua and turquoise felt, then needle felted onto the background. I needled some blue and pistachio mohair yarn between the joins.
The dark blue markings, eyes, fine lines and colourings were added with needle felting.
The acrylic painting was very quick to do - only a couple of hours - whereas I estimate that the felted version took ten working hours to complete. But it wasn't a slog as I really loved the felting processes and seeing it all come together during the needle felting was so satisfying.
The photo at the top of the post shows how the mackerel would look in a frame, like its painted cousin, and here's the uncropped picture with its organic edge. It measures 50 x 33cm.
Painting is fun, but I'm no Leonardo da Vinci. It's felted art for me!
All of the ebooks above are PDF format (readable on any computer or device) and are available for instant purchase and download - please click on the images for more details.
'rosiepink' is me, Annie, and my mum, Lyn and we both live on the South Coast of England. We love textiles and fibres and have a passion for felt artwork. Stitching on felt gives such beautiful textures so we combine hand and free machine embroidery with our feltmaking to create wall art. We write this occasional blog to share some of the things we create and some of the lessons we learn along the way. Please feel free to comment on any post - we'd love to hear from you whatever your interest in our blog may be.


text:
Library of Articles
Library: Articles
A New Wave of Rock Fever
by Michael Snyder
Berkeley Barb
Welcome to a fresh start. A new wave, as it were. Non-sequiturs aside, 1978's product has yet to hit the racks, so we're indulging in a quasi-retrospective of a few recorded goods that went unheralded in these pages during '77. No sense in another Tops-Of-The-Year list anyway. You can see Blair Jackson's poll of local scribes in a recent Bay Guardian, if interested.
Backtracking briefly, 1977 was -- pardon the alliteration -- a rock 'n' roll revelation. The narcotic ecstacy, epileptic rhythm and emotional range of classic rock music has been rediscovered by the young, loud and snotty. It comes in all varieties. You can dance to most of it, but unlike disco-drek, it's got news for you. Depth. Humor. Pith. Less is more! No 10-minute solos! No symphonic la-de-dah! They make their point and get out quick.
The Clash, Talking Heads, DEVO, Elvis Costello and Tom Petty promised us anything and gave us the fever again. The most telling symptom of the divine disease was the return of those ravers, The Flamin' Groovies, who blew San Francisco last month to record in Wales. Be on the lookout for their new Sire LP, produced by a Stiff's best friend, Dave Edmunds. It'll be preceded by a super-snazzy, limited edition, 12-inch, four-song EP, supposedly featuring a Groovie-ized "Blue Turns to Grey." Strictly "Teenage Confidential."
Let's raise a can of beer on high to calculated primitivism! It's alive! Dr. Pretorius! Alive! Cross Lou Reed's rock 'n' roll heart. We saw the corpse kicking and thrashing at Winterland throughout the Pistols' show last weekend. We heard and felt the beat. Born again, brothers and sisters...
AEROSMITH -- Draw The Line (Columbia): They drew it, but they didn't snort it. The title cut is as awesome as the Stones' "Rocks Off," its genealogical ancestor. There's even a dreamy middle 8, sandwiched between two slabs of pure furor. Alas, the remainder of the LP is simply Aero-hash. Savage production by Jack Douglas, as on the preceding and more potent Aerosmith album, Rocks. All for naught, unless you pick up the 45 of "Draw The Line."
JACKSON BROWN -- Running On Empty (Asylum): And sputtering to a halt.
CERRONE -- Supernature (Cotillion): The cover shows our Mediterranean disco-hero, Cerrone, turning musicians into swine, and other beasties. The record inside is your standard electro-drone. Draw your own conclusions.
ERIC CLAPTON -- Slowhand (RSO): Ten years ago, the nimblefingered guitar god was nicknamed "Slowhand" in supreme irony. Nowadays, the irony is that the name fits. Listen to him nod out during J.J. Cale's "Cocaine." Good night, sweet prince.
Close Encounters Of The Third Kind (Arista): If a lump clumped your throat at the sight of Carlo Rambaldi's rubberoid alien robot with the watery, Walter Keane eyes, then, by all means, hop on this John Williams soundtrack. If you wonder why Richard Dreyfuss couldn't construct a model of Devil's Tower in his backyard instead of his living room, then you probably don't watch television. Before long, they'll be syndicating episodes of "Duddy Kravitz, First Sweathog In Outer Space."
LOL CREME/KEVIN GODLEY -- Consequences (Mercury): Mammoth, three-record boxed set, directed by two former members of 10 CC. Weird synthetic noises, two or three fair-to-middlin' songs (one with the silky voice of Sarah Vaughn) and tons of garbled rapping by English satirist Peter Cook. It's a fascinating demo for Creme and Godley's instrumental invention, The Gizmo, in the same way that the Bell System's presentation of the Picture-Phone was fascinating at the New York World's Fair in 1965. Without anybody worthwhile to talk to at the other end of the line, the novelty wears thin and you might as well hang up.
DAVE EDMUNDS -- Get It (Swan Song): Tall Paul at Rather Ripped Records thinks this LP could pass as a K-Tel Decade Of Hits mail-order special. I think it's among the finest releases of the past year. Edmunds is a superior producer and guitarist, who has a definite affection for the rock 'n' roll of the past. Initially the leader of the English progressive blues band, Love Sculpture, his first solo hit was a remake of "I Hear You Knockin'." On Get It, he's equally at home with "Back To Schooldays" by Graham Parker and "Where Or When" by Rodgers and Hart. His own compositions could have been written by anyone from Brian Wilson to Hank Williams. A delightfully eclectic showcase by and for the rock 'n' roller.
INTERGALACTIC TOURING BAND (Passport): Recorded over a year's time from summer of 1976 to summer of 1977, This New York/London supersession is science fiction-flatulence with a couple of stirring moments in execution, but virtually none in libretto and concept. Rod Argent, Marge Raymond from Flame, Dave Cousins of the Strawbs, Annie Halsam from Renaissance and Meat Loaf get off some fine vocals. Nonetheless, an implied musical continuity is non-existent.
MEAT LOAF -- Bat Out Of Hell (Epic/-Cleveland International): A better name for this concoction could not be found. Todd Rundgren's kamikaze guitar and Wagnerian wall-of-sound. Composer Jim Steinman's Springsteen-like teenage mythos. Chunky ol' Meat Loaf himself, a leatherlunged neo-Clayton Thomas, who once sang for Ted Nugent and also played a jovial moron in The Rocky Horror Picture Show. The ketchup on top is the voice of the Yankees, Phil Rizzuto, doing play-by-play during "Paradise By The Dashboard Light." Personally, I prefer London Broil. Chacun a son gout.
BETTE MIDLER -- Broken Blossom (Atlantic): Wherein Bette the brassy babe becomes a fragile flower. Bessie Smith's "Empty Bed Blues," the Ronette's' "Paradise," a duet with Tom Waits on "I Never Talk To Strangers" (the same take that's on Waits' latest LP, Foreign Affairs) and "La Vie En Rose." Midler's strongest record since her first, which was arranged by Super-Wimp Barry Manilow. His absence makes Broken Blossom twice as good by default.
JONI MITCHELL -- Don Juan's Reckless Daughter (Asylum): Boney Joni solidifies her claim to the title of America's Premier Twisted Sister. She's certainly the most articulate and beloved jazz-folk singer-songwriter-neurotic in memory. Back when she was a lady of the canyon, Joni sang, "They paved paradise and put up a parking lot." This time around, she takes a leak in that very lot.
It's all chronicled in "Talk To Me," an excellent song about inebriated interpersonal communication. Not only does this number boast Weather Report's Jaco Pistorius rolling out a deliciously sensual bass-line, it also offers a chance to hear Joni's impression of a chicken. Too little material for two discs, but some insightful moments, particularly for her fans.
Oh yeah. The superfly dude next to Ms. Mitchell in the cover photograph is Joni the Drag King wearing blackface. I'd suggest she stay out of Watts.
NAZARETH -- Expect No Mercy (A&M): Expect no quality. People enamored of Dan McCafferty's rich, growly vocals are advised to pick up Hot Tracks, a well-chosen assortment of white-hot Nazareth tracks on A&M. Without that solid repertoire, they're a bunch of loud, kilt-less Scotsmen. Ach!
THE RAMONES -- Rocket To Russia (Sire): There's no stoppin' the cretins from hoppin'. Especially while they feed on high-speed syncopation, hilariously-understated lyrics, and the stopped-up nasal cavities of Joey Ramone. Poor Jimi "You'll never hear surf music again" Hendrix. With covers of "Surfin' Bird" and "Do You Wanna Dance?," plus newies like "Rockaway Beach" and "Sheena Is A Punk Rocker," the Ramones are blithely riding the pipeline. Not a wipe-out in sight.
10 CC -- Live And Let Live (Mercury): Half of the original band (5 CC?) continues to churn it out the easy way: double-record live album of tried and true favorites. Bolstered by a crew of faceless sidemen, Eric Stewart and Graham Gouldman flawlessly recreate the clean studio sound they cherish. "Ships Don't Just Disappear In The Night" sports a funkier arrangement and "Art For Art's Sake" gives Stewart room to stretch out on lead guitar.
The rest of the material is identical to the earlier studio performances, except... (1) The humorous songs aren't as funny. (2) The love songs aren't as lush. (3) The vocals, without the departed Creme or Godley or overdubs, are merely adequate.
GEORGE THOROGOOD AND THE DELAWARE DESTROYERS (Rounder): Look! Zooming out of the wilds of Wilmington, Delaware! It's Bo Diddley! It's Slim Harpo! It's Dr. Feelgood! No! It's actually George Thorogood! Possessing R&B and rockabilly chops far beyond those of normal groups, George and his trusty rhythm section fight a never-ending battle of the bands for sweat, slippin' and slidin'. One of the rare aggregates that will comply when the audience screams "Play all night!"
No silly clothes or posturing. Macho flagrant jamming. The record is a true representation of their virtues, but they must be seen live to be fully appreciated. Hitline: "One Bourbon, One Scotch, One Beer" and Bo Diddley's "Ride On Josephine."
GARY WRIGHT -- Touch And Gone (Warner Brothers): Gary Wrong? Seals & Crofts swallowed by a ring modulator? Don't touch. Just go.
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text:
2021 Preview & Prediction: Rams vs. Bears
Doug Nelson
Overview: You never really know what to expect in season openers. These teams were playoff teams last year, albeit Chicago with a mediocre 8-8 mark. Chicago Bears head coach Matt Nagy's job depends on how his team performs this season.
Both teams feature new quarterbacks in this game. The Rams gave up a fortune in talent and draft choices to get former Lion's signal-caller Matt Stafford. No one doubts Stafford's arm talent, but he's never won anything worth mentioning in his NFL career. Was that because stuck in Detroit, the Siberia of the NFL? Or is he not a winner? We may find out in 2021.
The Bears, meanwhile, turn to NFL journeyman Andy Dalton. Dalton is "just a guy, let us be straight" He won't kill you with rookie mistakes, but he won't carry a team on his back either. He's keeping the seat warm for Bear's first-round pick Justin Fields, from Ohio State University. We may or may not see Fields take some snaps in this game, But Nagy insists that Dalton is his starter. The Rams have reportedly been preparing for both players in their defensive scheming for this game.
Rams Defense vs. Bears Offense: One thing many folks will be watching for in this game is a potential marquee matchup between Rams corner Jalen Ramsey and Bears' wideout Allen Robinson, who is easily Chicago's biggest offensive threat. Ramsey regularly battled DK Metcalf, DeAndre Hopkins, Davante Adams, and Terry McLaurin last season. While he may not have been perfect in those matchups, he won more often than he lost. Robinson is an excellent receiver, but Ramsey puts the "shut down" in the 'shut down corner." We like his chances of at least containing Robinson.
The other big matchup, of course, features Rams All-World defensive lineman Aaron Donald against Bears interior lineman Cody Whitehair and James Daniels. The Bears O-line widely regarded as the team's weakest link on offense. They will likely do what everybody does, go all out to contain Donald. Whether they are successful will have a lot to do with the outcome of this game.
The Rams got some good news on the injury front that defensive lineman A'Shawn Robinson will likely play. Robinson, a mountain of a man, will be partially counted on to replace the departed Micheal Brockers in the Rams defensive front. He's reportedly shed a few pounds to improve his pass-rush skills.
This game will also be a reunion, as Rams linebacker Leonard Floyd faces his old team. Floyd considered a bust when he left Chicago, a former first-round pick from Georgia who underachieved in his stint with the Bears. But he revived his career last year with the Rams, notching 10.5 sacks in route to a fat new contract in the offseason. We would think he might have a little extra motivation to play well in this game.
The Bears rushing attack runs through David Montgomery, a well-above-average NFL back with some game-breaking potential. The Rams linebackers, especially the inside backers, have to do their job on Sunday night. If there is a weakness in this defense, they may be it.
The Rams defense is lead by defensive coordinator Raheem Morris, Morris is a former head coach who is very well thought of around the league, but he has big shoes to fill in replacing Brandon Staley. The latter coached the league's top-rated defense in L.A. last season. The Rams likewise will have to replace Safety John Johnson, a long-time stalwart in the secondary who played at a Pro-Bowl level. It won't be easy, but the Rams still have many talents on defense.
Edge: Rams
Rams Offense vs. Bears Defense: Even though Nagy brought a reputation as an offensive genius to the Bears when he came to Chicago from Kansas City a few years back, this team's strength has been chiefly on this side of the ball. With a front seven that includes Khalil Mack, Akiem Hicks, Roquan Smith, and Bilal Nichols, these guys can wreck a game plan..and an offense.
We are a little concerned about the Rams going with Brian Allen at the center. He wasn't terrific in 2019 when he last started. Hopefully, he's stepped up his game since then. Bears tackle Akiem Hicks may be looking forward to seeing Allen across the line. Let's hope Hicks is in for a disappointing night. The ears' defensive tackle battle is battling a knee injury, and they may try to compensate by having Hicks try to take advantage of Allen in this game.
Khalil Mack is another game-wrecker, especially as a pass-rusher. He might not entirely be Aaron Donald, but he's not far behind A.D., and that is saying something. Rams offensive tackle Rob Havenstein needs to hold his own against Mack at least. The Rams think they can take advantage of a questionable Bears secondary, but if Matthew Stafford is flat on his back all night, that won't happen. The Rams held Mack to a sack and a forced fumble last year in a Los Angeles win. Those may not be superior numbers from Ram's point of view, but Mack can do much more damage if Havenstein doesn't bring his A-game.
The Bear's secondary is the relative weak link here. One reason the Rams added Stafford and DeSean Jackson and rookie Tutu Atwell, the speedster from Louisville, was to try to open up a deep passing game that was lackluster last season. Expect them to take their shots down the field. Of course, veterans Robert Woods and Cooper Kupp are already excellent receivers, so if the Rams can keep Stafford upright, they could score some points.
The Rams got a tough break when they lost running back Cam Akers in the preseason. Darrel Henderson and Sony Michel expected to carry the bulk of the load. Bears linebacker Roquon Smith is a significant force in stopping the run and covering receivers out of the backfield. Smith vs. Henderson/Michel is another crucial matchup.
It's hard to say precisely how the new pieces on this Rams offense will come together in week 1, and the Bears have some serious defensive talent. We're going to check the box for Los Angeles, but this should be an exciting matchup. If the Bears can pressure Stafford and contain the running game, they absolutely can win this game.
Edge: Rams (not by much)
Special Teams: The honest answer in week one of 2021 is who knows? The Rams seemed to find their groove last season in the kicking game when they signed Matt Gay, and punter Johnny Hekker returns despite being out-performed in the preseason by Cory Bojorquez, subsequently traded to the Green Bay Packers. But Hekker is still an excellent punter, and as a former high-school quarterback, he adds the dimension of a fake punt to the arsenal for the Rams.
Los Angeles just wasn't excellent in kickoff and punt coverage or return game last season. They brought in 33-year veteran Joe DeCamillis to run the special teams this year. Let's hope he can step up the play from last season when special team's units were a problem after mostly being a team strength in previous years.
The Bear's special teams aren't necessarily a team strength either. They have a history of missing big kicks in crucial games, going back to a missed field goal that cost them a playoff game against the Eagles a few years ago. Cairo Santos is the current placekicker and considered solid, if not extraordinary. Pat O'Donnell handles the punting choirs, and the Rams will see a familiar face returning punts in former Ram Nsimba Webster. The Bears are missing Tarik Cohen on the return game, who is injured. Cohen is an excellent player, so his loss hurts.
There are too many variables here to pick a clear favorite until we see some regular-season action from both squads, so we'll rate this as a toss-up for now.
Edge: EVEN
Coaching: Matt Nagy had a reputation as an offensive genius when he was the Chiefs, offensive coordinator. We haven't seen that in Chicago. Then again, going from Patrick Mahomes to (former Bears quarterback), Mitch Trubisky might have something to do with that.
Sean McVay's reputation as an offensive genius likewise may have taken something of a hit last season when the Rams offense was not very good. Will the change at quarterback and the influx of speed at wide-receiver change things? We don't know. But we think the Rams have the overall advantage here.
Final Prediction: The Rams are considered a Super Bowl contender, while the Bears generally are not. But this is not an easy matchup by any means. The Bears upset Los Angeles's last Super Bowl team in 2018, and they are capable of doing so again, especially if they can generate some takeaways on defense. Season openers are always a little dicey. You don't know.
But we'll take the Rams
Rams 27, Bears 20


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Children's & YA
After his plane crashes in the desert a pilot has an unexpected and unbelievable encounter. A small boy from a small faraway planet appears at the break of dawn. His dreamy stories about his adventures during his space travel, as well as his questions, lead the pilot and the reader, to philosophical quests on the deeper meaning of human existence.
Antoine de Saint Exupéry's classic masterpiece for young and old readers in an original and inspired graphic novel.
Turkey, Sırtlan Kitap (forthcoming), Beijing Lingdu Culture & Media Co., LTD (forthcoming)
Author: Antoine de Saint-Exupéry
Illustrator: Stella Stergiou
Publisher: Polaris Publications
Genre: Graphic novel
Format: Soft cover, 20,5 x 28,5 cm
Stella Stergiou was born in 1992 in Athens, where she works as an illustrator and comics artist, while completing her studies in the School of Architecture of the National Technical University of Athens. She travels with her light blue bicycle from planet to planet and starts drawing wherever she lands. She is inspired from all terrestrial and extraterrestrial forms of life. She believes that there is a star and a rose for everyone. Perhaps a fox as well.
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