#  On Lorentz-invariant bi-spin-2 theories

Rampei Kimura  Waseda Institute for Advanced Study, Waseda University, 1-6-1
Nishi-Waseda, Shinjuku, Tokyo 169-8050, Japan  Atsushi Naruko  Center for
Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto
University, Kyoto 606-8502, Japan  Daisuke Yamauchi  Faculty of Engineering,
Kanagawa University, Kanagawa-ku, Yokohama-shi, Kanagawa, 221-8686, Japan

###### Abstract

We investigate a Lorentz invariant action which is quadratic in two rank-2
symmetric tensor fields in Minkowski spacetime. We apply a scalar-vector-
tensor decomposition to two tensor fields by virtue of 3-dimensional rotation-
invariance of Minkowski spacetime and classify theories with seven degrees of
freedom based on the Hamiltonian analysis. We find two new theories, which
cannot be mapped from the linearized Hassan-Rosen bigravity. In these
theories, the new mass interactions can be allowed thanks to the transverse
diffeomorphism invariance of action.

††preprint: YITP-20-144

##  I Introduction

The attempt to seek ghost-free massive gravity theories has again attracted
considerable attention by the discovery of de Rham-Gabadadze-Tolley (dRGT)
massive gravity de Rham and Gabadadze (2010). The first attempt of
constructing massive spin-2 theory has been carried out by Fierz and Pauli,
and it is the quadratic action for a massive spin-2 particle in a flat
spacetime Fierz and Pauli (1939). Once we embed this into a curved spacetime,
the behavior of the massive spin-2 field does not smoothly connect to the
well-known massless one, i.e., the linearized general relativity Zakharov
(1970); van Dam and Veltman (1970). The discontinuity found by van Dam,
Veltman, and Zhakalov turned out to be an artifact of the truncation at linear
order, and the massive spin-2 theory in fact has the continuous massless limit
when taking into account nonlinearities as pointed out by Vainshtein
Vainshtein (1972). Nonetheless, an unwanted degree of freedom (DOF), Boulware-
Deser ghost Boulware and Deser (1972), which is absent at linear order,
reappears at nonlinear level, and it unfortunately behaves as Ostrogradsky’s
ghost Ostrogradsky (1850). In dRGT massive gravity, such an unwanted degree of
freedom is successfully eliminated by the careful choice of nonlinear
potential terms de Rham and Gabadadze (2010); de Rham  _et al._ (2011).
Although the dRGT massive gravity possess the cosmological constant solution
in a cosmological background Gumrukcuoglu  _et al._ (2011), it is
perturbatively unstable De Felice  _et al._ (2012); Gumrukcuoglu  _et al._
(2012). For this reason, one needs to seek a ghost-free extension of massive
gravity which should be at least cosmologically viable and stable. Such an
attempt without introducing an extra DOF has been investigated, taking into
account derivative interactions Hinterbichler (2013); Kimura and Yamauchi
(2013); de Rham  _et al._ (2014) and metric transformation Gümrükçüoglu and
Koyama (2019), but most of them are not successful unfortunately. Recently, by
breaking the translation invariance of the Stückelberg field, new extended
theories of massive gravity have found, and their cosmological perturbations
are stable around cosmological backgrounds De Rham  _et al._ (2014); Kenna-
Allison  _et al._ (2019); Gumrukcuoglu  _et al._ (2020).

Another way to extend massive gravity is to introduce the second dynamical
symmetric tensor field. In massive gravity theories, to give mass to graviton,
in addition to the metric $g_{\mu\nu}$, one needs to introduce the so-called
reference metric $f_{\mu\nu}$, which is usually taken to be a Minkowski
metric. In massive bigravity theories, the reference metric can be promoted to
be a dynamical tensor field by introducing its kinetic terms. The simplest
extension of dRGT massive gravity is proposed by Hassan and Rosen by adding
the Einstein-Hilbert kinetic terms even for the second metric Hassan and Rosen
(2012a). In Hassan-Rosen bigravity, the total number of physical DOFs is
seven, which consists of two from a massless graviton and five from a massive
graviton. This fact can be easily seen by expanding both metric around
Minkowski spacetime, that is, $g_{\mu\nu}\to\eta_{\mu\nu}+h_{\mu\nu}/M_{g}$
and $f_{\mu\nu}\to\eta_{\mu\nu}+f_{\mu\nu}/M_{f}$, where $M_{g}$ and $M_{f}$
are respectively the Planck mass for the metric $g_{\mu\nu}$ and $f_{\mu\nu}$.
Then the quadratic Lagrangian is given by Hassan and Rosen (2012a)

$\displaystyle{\cal L}^{\rm(2)}_{\rm HR}$ $\displaystyle=$
$\displaystyle-\left(h_{\mu\nu}\widehat{\cal
E}^{\mu\nu\alpha\beta}h_{\alpha\beta}+f_{\mu\nu}\widehat{\cal
E}^{\mu\nu\alpha\beta}f_{\alpha\beta}\right)-\frac{m^{2}M_{\rm
eff}^{2}}{4}\Biggl{[}\left(\frac{h^{\mu}{}_{\nu}}{M_{g}}-\frac{f^{\mu}{}_{\nu}}{M_{f}}\right)^{2}-\left(\frac{h}{M_{g}}-\frac{f}{M_{f}}\right)^{2}\Biggr{]}\,.$
(1)

Here $\widehat{\cal E}^{\mu\nu\alpha\beta}$ is the linearized Einstein-Hilbert
kinetic operator defined as

$\displaystyle\widehat{\cal
E}^{\mu\nu}_{~{}~{}\alpha\beta}=\left[\eta^{(\mu}_{~{}\alpha}\eta^{\nu)}_{~{}\beta}-\eta^{\mu\nu}\eta_{\alpha\beta}\right]\square-2\partial^{(\mu}\partial_{(\alpha}\eta^{\nu)}_{~{}\beta)}+\partial^{\mu}\partial^{\nu}\eta_{\alpha\beta}+\partial_{\alpha}\partial_{\beta}\eta^{\mu\nu}\,,$
(2)

where the round brackets denote the symmetrization of indices, $m$ is the mass
of graviton, and the effective Planck mass is given by $M_{\rm
eff}^{2}=(1/M_{g}^{2}+1/M_{f}^{2})^{-1}$. The mixing terms between $h$ and $f$
in the mass terms can be removed by introducing the linear combination of two
metric,

$\displaystyle\frac{1}{M_{\rm
eff}}u_{\mu\nu}\equiv\frac{1}{M_{f}}h_{\mu\nu}+\frac{1}{M_{g}}f_{\mu\nu}\,,\qquad\frac{1}{M_{\rm
eff}}v_{\mu\nu}\equiv\frac{1}{M_{f}}h_{\mu\nu}-\frac{1}{M_{g}}f_{\mu\nu}\,.$
(3)

Then the Lagrangian becomes

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\left(u_{\mu\nu}\widehat{\cal
E}^{\mu\nu\alpha\beta}u_{\alpha\beta}+v_{\mu\nu}\widehat{\cal
E}^{\mu\nu\alpha\beta}v_{\alpha\beta}\right)-\frac{m^{2}}{4}\left(v^{\mu\nu}v_{\mu\nu}-v^{2}\right)\,.$
(4)

This clearly shows that Hassan-Rosen bigravity at linear order consists of the
lienarized general relativity for $u_{\mu\nu}$ and the Fierz-Pauli theory for
$v_{\mu\nu}$. The absence of the Boulware-Deser ghost has been proved in
Hassan and Rosen (2012b).

Since the construction of bigravity theory is inspired by the dRGT theory, it
is not trivial whether the mass interaction of Hassan-Rosen bigravity is
unique or not. As for the uniqueness of dRGT mass term in linear massive
gravity theories, see also Naruko  _et al._ (2019). For this reason, one might
be able to find a new type of mass interactions in bimetric gravity theories.
However, such a construction would be extremely difficult to start with a
curved spacetime. To this end, in this paper, we investigate a theory with bi-
spin-2 particle in a flat spacetime, which could represent the linear
expansion of a certain nonlinear massive bigravity.

This paper is organized as follows. In Sec. II, we give an action for two
rank-$2$ tensor fields in our setup and decompose them into scalar, vector,
and tensor sectors based on transformation properties of tensors with respect
to a $3$-dimensional spatial rotation. In Sec. III, we give ghost-free
conditions for the tensor mode. In Sec. IV, we perform Hamiltonian analysis
and derive the conditions to have $2$ physical DOFs in the vector sector. In
Sec. V, we investigate the scalar sector and classify theories with $1$ scalar
DOFs. Sec. VI is devoted to the summary. In Appendix A, we introduce a linear
field redefinition and investigate the reduction of the model parameter space.
In Appendix B, we provide an explicit expression of the Lagrangian in the
scalar sector. In Appendix C, we perform the Hamiltonian analysis of the
vector sector with $2$ primary constraints. In Appendix D, we investigate the
scalar sector with $2$ primary constraints.

##  II Setup

In the present paper, we consider a Lorentz invariant action for two rank-2
symmetric tensor fields, $h_{\mu\nu}$ and $f_{\mu\nu}$, and consider the most
general quadratic action which contains up to two derivatives with respect to
spacetime for each term111Strictly speaking, one can also include a Lorentz-
invariant scalar ${\eta_{\mu\nu}x^{\mu}x^{\nu}=-t^{2}+{\bf x}^{2}}$ for
theories invariant under a global Lorentz transformation. Once introducing
this scalar quantity, the analysis would be more complicated. For simplicity,
we here do not consider such a possibility.. In general, these symmetric
tensor fields possess 20 DOFs in total, and therefore we should impose some
conditions to eliminate unwanted DOFs, which could behave as ghost. Due to the
complexity of the analysis, we only focus on theories with 7 physical DOFs,
namely $2\times 2$ (tensor) + $2$ (vector) + $1$ (scalar) DOFs, as in the
Hassan-Rosen bigravity Hassan and Rosen (2012a) that consists of massless and
massive spin-2 fields in the linearized limit. As preparation for later
analysis, in this section, we introduce the generic action for bi-spin-$2$
tensor field and scalar-vector-tensor decomposition of it. For the Hamiltonian
analysis in Fourier space, we follow the procedure developed in Sugano and
Kamo (1982); Sugano  _et al._ (1986); Sugano and Kimura (1990) and adopt the
notation in Naruko  _et al._ (2019).

###  II.1 Double spin-2 theory

Let us consider a generic action for two rank-2 symmetric tensor fields up to
the quadratic order in fields around Minkowski spacetime,

$\displaystyle S$ $\displaystyle=\int{\rm d}^{4}x\Bigl{(}-{\cal
K}_{h}^{\alpha\beta|\mu\nu\rho\sigma}h_{\mu\nu,\alpha}h_{\rho\sigma,\beta}-{\cal
K}_{f}^{\alpha\beta|\mu\nu\rho\sigma}f_{\mu\nu,\alpha}f_{\rho\sigma,\beta}-{{\cal
G}^{\alpha\beta|\mu\nu\rho\sigma}}h_{\mu\nu,\alpha}f_{\rho\sigma,\beta}$
$\displaystyle\qquad\qquad\qquad-{\cal
M}_{h}^{\mu\nu\rho\sigma}h_{\mu\nu}h_{\rho\sigma}-{\cal
M}_{f}^{\mu\nu\rho\sigma}f_{\mu\nu}f_{\rho\sigma}-{\cal
N}^{\mu\nu\rho\sigma}h_{\mu\nu}f_{\rho\sigma}\Bigr{)}\,,$ (5)

where the coefficients ${\cal K}_{\Phi}$, ${\cal G}$, ${\cal N}$, and ${\cal
M}_{\Phi}$ consist of all the possible combinations with the Minkowski metric
$\eta_{\mu\nu}$,

$\displaystyle{\cal K}_{\Phi}^{\alpha\beta|\mu\nu\rho\sigma}$
$\displaystyle=\kappa_{\Phi
1}\eta^{\alpha\beta}\eta^{\mu\rho}\eta^{\nu\sigma}+\kappa_{\Phi
2}\eta^{\mu\alpha}\eta^{\rho\beta}\eta^{\nu\sigma}+\kappa_{\Phi
3}\eta^{\alpha\mu}\eta^{\nu\beta}\eta^{\rho\sigma}+\kappa_{\Phi
4}\eta^{\alpha\beta}\eta^{\mu\nu}\eta^{\rho\sigma}\,,$ (6) $\displaystyle{\cal
G}^{\alpha\beta|\mu\nu\rho\sigma}$
$\displaystyle=(l_{1}\eta^{\alpha\beta}\eta^{\mu\rho}+l_{2}\eta^{\mu\alpha}\eta^{\rho\beta})\eta^{\nu\sigma}+(l_{3}\eta^{\alpha\mu}\eta^{\nu\beta}+l_{4}\eta^{\alpha\beta}\eta^{\mu\nu})\eta^{\rho\sigma}+l_{5}\eta^{\mu\nu}\eta^{\beta\sigma}\eta^{\alpha\rho}\,,$
(7) $\displaystyle{\cal M}_{\Phi}^{\mu\nu\rho\sigma}$ $\displaystyle=\mu_{\Phi
1}\eta^{\mu\rho}\eta^{\nu\sigma}+\mu_{\Phi
2}\eta^{\mu\nu}\eta^{\rho\sigma}\,,$ (8) $\displaystyle{\cal
N}^{\mu\nu\rho\sigma}$
$\displaystyle=n_{1}\eta^{\mu\rho}\eta^{\nu\sigma}+n_{2}\eta^{\mu\nu}\eta^{\rho\sigma}\,,$
(9)

and we defined the label $\Phi=(h,f)$. A comma denotes a partial derivative
with respect to coordinates. Here, $\kappa_{\Phi 1,\Phi 2,\Phi 3,\Phi 4}$,
$l_{1,2,3,4,5}$, $\mu_{\Phi 1,\Phi 2}$, and $n_{1,2}$ are constant parameters.
The linearized Hassan-Rosen bigravity corresponds to

$\displaystyle\kappa_{h2}=-\kappa_{h3}=2\kappa_{h4}=-2\kappa_{h1}\,,\qquad\kappa_{f2}=-\kappa_{f3}=2\kappa_{f4}=-2\kappa_{f1}\,,\qquad
l_{1}=l_{2}=l_{3}=l_{4}=l_{5}=0\,,$
$\displaystyle\mu_{h2}=-\mu_{h1}\,,\qquad\mu_{f2}=-\mu_{f1}\,,\qquad
n_{2}=-n_{1}=2\sqrt{\mu_{h1}\mu_{f1}}\,,$ (10)

as shown in (2) and this theory is invariant under the gauge transformation,

$\displaystyle h_{\mu\nu}\to h_{\mu\nu}+{1\over
2\sqrt{\mu_{h1}}}(\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu})\,,\qquad
f_{\mu\nu}\to f_{\mu\nu}+{1\over
2\sqrt{\mu_{f1}}}(\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu})\,.$ (11)

Alternatively, one can diagonalize the mass terms to remove $n_{1}$ and
$n_{2}$ without changing the kinetic terms by taking linear combinations of
$h_{\mu\nu}$ and $f_{\mu\nu}$, and then the resultant theory satisfies

$\displaystyle\kappa_{h2}=-\kappa_{h3}=2\kappa_{h4}=-2\kappa_{h1}\,,\qquad\kappa_{f2}=-\kappa_{f3}=2\kappa_{f4}=-2\kappa_{f1}\,,\
$ $\displaystyle\mu_{h2}=-\mu_{h1}\,,\qquad\mu_{f1}=\mu_{f2}=0\,,\qquad
l_{1}=l_{2}=l_{3}=l_{4}=l_{5}=n_{1}=n_{2}=0\,,$ (12)

as found in (4). Then this theory with (12) is invariant under the gauge
transformation

$\displaystyle h_{\mu\nu}\to h_{\mu\nu}\,,\qquad f_{\mu\nu}\to
f_{\mu\nu}+\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}\,.$ (13)

Thus, this is nothing but the Fierz-Pauli massive spin-2 field for
$h_{\mu\nu}$ and the linearized general relativity for $f_{\mu\nu}$.

###  II.2 Scalar-vector-tensor decomposition

Following Naruko  _et al._ (2019), we decompose the rank-2 symmetric tensor
fields $h_{\mu\nu}$ and $f_{\mu\nu}$ into transverse-traceless tensors,
transverse vectors, and scalars as

$\displaystyle\Phi_{00}$
$\displaystyle=\Phi^{00}=-2\alpha_{\Phi}\,,\qquad\Phi_{0i}=-\Phi^{0i}={\beta^{\Phi}_{,i}}+B^{\Phi}_{i}\quad(B^{i}{}_{\Phi,i}=0)\,,$
(14) $\displaystyle\Phi_{ij}$ $\displaystyle=\Phi^{ij}=2{\cal
R}_{\Phi}\delta_{ij}+2\,{{{\cal
E}}^{\Phi}_{,ij}}+F^{\Phi}_{i\,,j}+F^{\Phi}_{j\,,i}+2H^{\Phi}_{ij}\quad(F^{i}{}_{\Phi,i}=0\,,\quad
H^{i}{}_{\Phi i}=H^{ij}{}_{\Phi,j}=0)\,.$ (15)

Here, scalar, vector and tensors are defined based on transformation
properties with respect to a $3$-dimensional rotation in Minkowski spacetime,
and the transverse-traceless tensors ${H^{\Phi}_{ij}}$, two transverse vectors
$B^{\Phi}_{i}$ and ${F^{\Phi}_{i}}$, and four scalars
$\alpha_{\Phi},{\beta_{\Phi}},{\cal R}_{\Phi}$, and ${{\cal E}_{\Phi}}$
respectively have two, four, and four components in each $\Phi$. Since we
focus on theories with 7 DOFs, to be more precise $2\times 2$ (tensor) + $2$
(vector) + $1$ (scalar) DOFs, we need to eliminate 6 components of the
transverse-vectors and 7 components of the scalars, and then the final DOFs
becomes $20-6-7=7$ DOFs. Under this decomposition, the quadratic action can be
always separated into three parts which solely consists of scalar, vector, and
tensor perturbations respectively :

$\displaystyle S[h_{\mu\nu},f_{\mu\nu}]$
$\displaystyle=S^{S}[\alpha_{\Phi}\,,{\beta_{\Phi}}\,,{\cal R}_{\Phi}\,,{{\cal
E}_{\Phi}}\,]+S^{V}[{B^{\Phi}_{i}}\,,{F^{\Phi}_{i}}]+S^{T}[{H^{\Phi}_{ij}}]\,.$
(16)

In the following section, we will examine each sector and derive conditions to
have theories with $7$ DOFs by the Hamiltonian analysis. Hereafter, we replace
all the spatial derivatives as $\partial^{2}\to-k^{2}$ after integrating by
parts, where $k$ is the wavenumber in the Fourier space.

##  III tensor sector

The action in the tensor sector is found to be

$\displaystyle S^{T}[H_{ij}^{h},H_{ij}^{f}]$ $\displaystyle=$ $\displaystyle
4\int
dtd^{3}k\Big{[}\kappa_{h1}({\dot{H}}^{h}_{ij})^{2}-(\kappa_{h1}k^{2}+\mu_{h1})(H^{h}_{ij})^{2}+\kappa_{f1}({\dot{H}}^{f}_{ij})^{2}-(\kappa_{f1}k^{2}+\mu_{f1})(H^{f}_{ij})^{2}$
(17)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{-l_{1}{\dot{H}}_{ij}^{h}{\dot{H}}_{f}^{ij}-(k^{2}l_{1}+n_{1})H_{ij}^{h}H_{f}^{ij}}\Big{]}\,,~{}~{}$

where a dot denotes the time derivative. It is manifest that the action is
symmetric under the replacement $h$ and $f$, and hence the result will be
applied to both modes in parallel. Throughout this paper, assuming
$\kappa_{h1}\neq 0$, we set

$\displaystyle l_{1}=0\,,$ (18)

which can be achieved by a field redefinition of $f$ without loss of
generality (See Appendix A). Thanks to $l_{1}=0$ by the field redefinition,
the kinetic matrix composed of $h$ and $f$ is diagonal, and the existence and
ghost-free conditions of both the tensor modes requires

$\displaystyle\kappa_{h1}>0\,,\qquad\kappa_{f1}>0\,.$ (19)

Hereafter, we impose the condition (19), and it is manifest that the physical
degrees of freedom in the tensor sector is two for each field.

##  IV Vector sector

In this section, we perform the Hamiltonian analysis for the vector variables.
In order to have a theory with $7$ DOFs in total, the vector sector should
have $2$ physical DOFs, which means the reduction of the phase space is
necessary in the view point of the Hamiltonian analysis. We first rescale
$F^{\Phi}_{i}$ as $F^{\Phi}_{i}\to F^{\Phi}_{i}/k$ for convenience. Then, the
action in the vector sector is given by

$\displaystyle S^{V}[{B^{\Phi}_{i}},{F^{\Phi}_{i}}]=\int
dtd^{3}k\,\Big{(}{\cal L}^{V}_{\rm kin}+{\cal L}^{V}_{\rm cross}+{\cal
L}^{V}_{\rm mass}\Big{)}\,,$ (20)

where each Lagrangian is given by

$\displaystyle{\cal L}^{V}_{\rm kin}$
$\displaystyle=-(2\kappa_{h1}+\kappa_{h2})({\dot{B}}^{h}_{i})^{2}+2\kappa_{h1}({\dot{F}}^{h}_{i})^{2}-(2\kappa_{f1}+\kappa_{f2})({\dot{B}}^{f}_{i})^{2}+2\kappa_{f1}({\dot{F}}^{f}_{i})^{2}-l_{2}{\dot{B}}^{h}_{i}{\dot{B}}^{f}_{i}\,,$
(21) $\displaystyle{\cal L}^{V}_{\rm cross}$
$\displaystyle=2k\kappa_{h2}B^{h}_{i}{\dot{F}}^{h}_{i}+2k\kappa_{f2}B^{h}_{i}{\dot{F}}^{f}_{i}+kl_{2}(B^{h}_{i}{\dot{F}}^{f}_{i}+B^{f}_{i}{\dot{F}}^{h}_{i})\,,$
(22) $\displaystyle{\cal L}^{V}_{\rm mass}$
$\displaystyle=2(k^{2}\kappa_{h1}+\mu_{h1})(B^{h}_{i})^{2}-\left(k^{2}(2\kappa_{h1}+\kappa_{h2})+2\mu_{h1}\right)(F^{h}_{i})^{2}+2(k^{2}\kappa_{f1}+\mu_{f1})(B^{f}_{i})^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left(k^{2}(2\kappa_{f1}+\kappa_{f2})+2\mu_{f1}\right)(F^{f}_{i})^{2}+2n_{1}B^{h}_{i}B^{f}_{i}-\left(k^{2}l_{2}+2n_{1}\right)F^{h}_{i}F^{f}_{i}\,.$
(23)

The relation between conjugate momentum $\pi_{\Phi}\equiv\partial{\cal
L}/\partial{\dot{\Phi}}$ and the time derivatives of canonical variables of
$B^{\Phi}_{i}$ and $F^{\Phi}_{i}$ is found to be

$\displaystyle\begin{pmatrix}\pi_{B^{h}_{i}}\\\ \pi_{B^{f}_{i}}\\\
\pi_{F^{h}_{i}}\\\
\pi_{F^{f}_{i}}\end{pmatrix}=\begin{pmatrix}-2(2\kappa_{h1}+\kappa_{h2})&-l_{2}&0&0\\\
-l_{2}&-2(2\kappa_{f1}+\kappa_{f2})&0&0\\\ 0&0&4\kappa_{h1}&0\\\
0&0&0&4\kappa_{f1}\end{pmatrix}\begin{pmatrix}\dot{B}^{h}_{i}\\\
\dot{B}^{f}_{i}\\\ \dot{F}^{h}_{i}\\\
\dot{F}^{f}_{i}\end{pmatrix}+\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
2k\kappa_{h2}&kl_{2}&0&0\\\
kl_{2}&2k\kappa_{f2}&0&0\end{pmatrix}\begin{pmatrix}B^{h}_{i}\\\ B^{f}_{i}\\\
F^{h}_{i}\\\ F^{f}_{i}\end{pmatrix}\,.$ (24)

Then the Hamiltonian is defined by

$\displaystyle{\cal H}^{V}$
$\displaystyle={\dot{B}_{i}^{h}\pi_{B_{i}^{h}}+\dot{B}_{i}^{f}\pi_{B_{i}^{f}}+\dot{F}_{i}^{h}\pi_{F_{i}^{h}}+\dot{F}_{i}^{f}\pi_{F_{i}^{f}}}-{\cal
L}^{V}\,.$ (25)

As one can see from (21), the kinetic parts of $B^{\Phi}_{i}$ and
$F^{\Phi}_{i}$ are completely decoupled, i.e., the kinetic matrix is block
diagonalized and the kinetic terms for $F^{\Phi}_{i}$ indeed exist since we
have imposed (19). Therefore, in order to see the degeneracy of the vector
sector, it allows us to consider only the kinetic matrix of $B^{\Phi}_{i}$,
which is

$\displaystyle{\cal
K}_{V}=\begin{pmatrix}-2(2\kappa_{h1}+\kappa_{h2})&-l_{2}\\\
-l_{2}&-2(2\kappa_{f1}+\kappa_{f2})\end{pmatrix}\,.$ (26)

The eigenvalue equation ${\cal F}_{V}(\lambda)$ of the kinetic matrix ${\cal
K}_{V}$ is found to be

$\displaystyle{{\cal F}_{V}(\lambda)\equiv\det({\cal K}_{V}-\lambda
I)=(4\kappa_{f1}+2\kappa_{f2}+\lambda)(4\kappa_{h1}+2\kappa_{h2}+\lambda)-l_{2}^{2}}=0\,.$
(27)

The determinant of the kinetic matrix is simply given by $\det{\cal
K}_{V}={\cal F}_{V}(0)$.

Now we would like to classify the cases based on the number of primary
constraints as follows :

$\displaystyle\textrm{2 primary constraints : }\quad{\cal
F}_{V}(0)=0\quad\&\quad{\cal F}^{\prime}_{V}(0)\neq
0\quad\longleftrightarrow\quad\kappa_{f2}=-2\kappa_{f1}+\frac{l_{2}^{2}}{4(2\kappa_{h1}+\kappa_{h2})}\,,$
(28) $\displaystyle\textrm{4 primary constraints : }\quad{\cal
F}_{V}(0)=0\quad\&\quad{\cal
F}^{\prime}_{V}(0)=0\quad\longleftrightarrow\quad\kappa_{h2}=-2\kappa_{h1}\quad\&\quad\kappa_{f2}=-2\kappa_{f1}\quad\&\quad
l_{2}=0\,.$

Here, at this point, $h$ and $f$ are symmetric, therefore, the case with
$2\kappa_{h1}+\kappa_{h2}=0$ in the case of 2 primary constraints can be
obtained by simply replacing $h$ and $f$. When there are only two primary
constraints, the Hamiltonian analysis shows that the number of the final
physical DOFs can be at least 4, and it is the undesired number. The analysis
for two primary case is summarized in the Appendix C. For this reason,
hereafter we only consider 4 primary case, where both $B^{h}_{i}$ and
$B^{f}_{i}$ becomes non-dynamical.

In this case, we have four primary constraints, which are given by

$\displaystyle{\cal C}^{(1)}_{B^{h}_{i}}$ $\displaystyle\equiv$
$\displaystyle\pi_{B^{h}_{i}}\approx 0\,,$ (30) $\displaystyle{\cal
C}^{(1)}_{B^{f}_{i}}$ $\displaystyle\equiv$
$\displaystyle\pi_{B^{f}_{i}}\approx 0\,,$ (31)

and we define the total Hamiltonian by adding the Lagrange multipliers
$\lambda_{B^{h}_{i}}$ and $\lambda_{B^{f}_{i}}$,

$\displaystyle{\cal H}^{V}_{T}={\cal H}^{V}+\lambda_{B^{h}_{i}}{\cal
C}^{(1)}_{B^{h}_{i}}+\lambda_{B^{f}_{i}}{\cal C}^{(1)}_{B^{f}_{i}}\,.$ (32)

Then the time-evolution of the primary constraints generates the secondary
constraints

$\displaystyle{\cal C}^{(2)}_{B^{h}_{i}}$ $\displaystyle\equiv$
$\displaystyle\\{{\cal C}^{(2)}_{B^{h}_{i}}\,,{\cal
H}^{V}_{T}\\}=4\mu_{h1}B^{h}_{i}+2n_{1}B^{f}_{i}-k\pi_{F^{h}_{i}}\approx 0\,,$
(33) $\displaystyle{\cal C}^{(2)}_{B^{f}_{i}}$ $\displaystyle\equiv$
$\displaystyle\\{{\cal C}^{(2)}_{B^{f}_{i}}\,,{\cal
H}^{V}_{T}\\}=2n_{1}B^{h}_{i}+4\mu_{f1}B^{f}_{i}-k\pi_{F^{f}_{i}}\approx 0\,,$
(34)

and the time-evolution of the secondary constraints gives

$\displaystyle\begin{pmatrix}{\dot{\cal C}}^{(2)}_{B^{h}_{i}}\\\ {\dot{\cal
C}}^{(2)}_{B^{f}_{i}}\end{pmatrix}=\begin{pmatrix}\\{{\cal
C}^{(2)}_{B^{h}_{i}}\,,{\cal H}^{V}_{T}\\}\\\ \\{{\cal
C}^{(2)}_{B^{f}_{i}}\,,{\cal
H}^{V}_{T}\\}\end{pmatrix}=\begin{pmatrix}\\{{\cal
C}^{(2)}_{B^{h}_{i}}\,,{\cal H}^{V}\\}\\\ \\{{\cal
C}^{(2)}_{B^{f}_{i}}\,,{\cal
H}^{V}\\}\end{pmatrix}+\begin{pmatrix}4\mu_{h1}&2n_{1}\\\
2n_{1}&4\mu_{f1}\end{pmatrix}\begin{pmatrix}\lambda_{B^{h}_{i}}\\\
\lambda_{B^{f}_{i}}\end{pmatrix}\approx 0\,.$ (35)

Therefore, when $n_{1}^{2}-4\mu_{f1}\mu_{h1}\neq 0$, namely the coefficient
matrix in front of the Lagrange multipliers is not degenerate, all the
Lagrange multipliers $\lambda_{B^{h}_{i}}$ and $\lambda_{B^{f}_{i}}$ are
determined by the above equations, and all the primary and secondary
constraints are second class. In this case, the total number of physical DOFs
is $(8\times 2-8)/2=4$, and thus we disregard this option.

On the other hand, when $n_{1}^{2}-4\mu_{f1}\mu_{h1}=0$, the coefficient
matrix in front of the Lagrange multipliers is degenerate, and two out of four
Lagrange multipliers cannot be determined. Hereafter we assume $\mu_{h1}\neq
0$ and solve $n_{1}^{2}-4\mu_{f1}\mu_{h1}=0$ for $\mu_{f1}$222For $\mu_{h1}=0$
case, one can simply switch all the notation of $h$ and $f$. When
$\mu_{h1}=\mu_{f1}=n_{1}=0$, all the constraints becomes first-class, implying
that the physical degrees of freedom is zero in the vector sector.. It is
convenient to redefine the primary constraints associated with ${B^{f}_{i}}$
as a linear combination of the original primary constraints :

$\displaystyle\widetilde{\cal C}^{(1)}_{B^{h}_{i}}$ $\displaystyle\equiv$
$\displaystyle\pi_{B^{h}_{i}}\approx 0\,,$ (36) $\displaystyle\widetilde{\cal
C}^{(1)}_{B^{f}_{i}}$ $\displaystyle\equiv$
$\displaystyle\pi_{B^{f}_{i}}-\frac{n_{1}}{2\mu_{h1}}\pi_{B^{h}_{i}}\approx
0\,.$ (37)

We also redefine the total Hamiltonian

$\displaystyle{\cal H}^{V}_{T}={\cal
H}^{V}+\widetilde{\lambda}_{B^{h}_{i}}\widetilde{\cal
C}^{(1)}_{B^{h}_{i}}+\widetilde{\lambda}_{B^{f}_{i}}\widetilde{\cal
C}^{(1)}_{B^{f}_{i}}\,.$ (38)

Then the secondary constraints become

$\displaystyle\widetilde{\cal C}^{(2)}_{B^{h}_{i}}$ $\displaystyle\equiv$
$\displaystyle\\{\widetilde{\cal C}^{(1)}_{B^{h}_{i}}\,,{\cal
H}^{V}_{T}\\}=4\mu_{h1}B^{h}_{i}+2n_{1}B^{f}_{i}-k\pi_{F^{h}_{i}}\approx 0\,,$
(39) $\displaystyle\widetilde{\cal C}^{(2)}_{B^{f}_{i}}$ $\displaystyle\equiv$
$\displaystyle\\{\widetilde{\cal C}^{(1)}_{B^{f}_{i}}\,,{\cal
H}^{V}_{T}\\}=k\left(\frac{n_{1}}{2\mu_{h1}}{\pi_{F^{h}_{i}}}-\pi_{F^{f}_{i}}\right)\approx
0\,.$ (40)

The time-evolution of the secondary constraints yields

$\displaystyle{\dot{\widetilde{\cal
C}}}{}^{(2)}_{B^{h}_{i}}=\\{\widetilde{\cal C}^{(2)}_{B^{h}_{i}}\,,{\cal
H}^{V}_{T}\\}={2k\left(2\mu_{h1}F^{h}_{i}+n_{1}F^{f}_{i}\right)}+4\mu_{h1}\widetilde{\lambda}_{B^{h}_{i}}\approx
0\,,$ (41) $\displaystyle{\dot{\widetilde{\cal
C}}}{}^{(2)}_{B^{f}_{i}}=\\{\widetilde{\cal C}^{(2)}_{B^{f}_{i}}\,,{\cal
H}^{V}_{T}\\}\approx 0\,.$ (42)

Here, the time-evolution of the secondary constraints $\widetilde{\cal
C}^{(2)}_{B^{f}_{i}}$ is trivially zero. Therefore, two of the Lagrange
multipliers can be determined by the time evolution of the secondary
constraints, and the rest of them are undetermined. Since the constraints
$\widetilde{\cal C}^{(1)}_{B^{f}_{i}}$ and $\widetilde{\cal
C}^{(2)}_{B^{f}_{i}}$ commute with all the constraints including themselves,
they are first-class constraints. To summarize, we find

$\displaystyle\textrm{Vector DOF}=\frac{8\times 2-4\,(\textrm{2 primary \& 2
secondary )}-4\,(\textrm{2 primary \& 2 secondary})\times 2\,(\textrm{first-
class})}{2}=2\,.$ (43)

The choice of the coefficients are

$\displaystyle\kappa_{h2}=-2\kappa_{h1}\neq
0\,,\qquad\kappa_{f2}=-2\kappa_{f1}\neq 0\,,\qquad
l_{2}=0\,,\qquad\mu_{f1}=\frac{n_{1}^{2}}{4\mu_{h1}}\,,\qquad\mu_{h1}\neq
0\,.$ (44)

In the analysis for the scalar sector in the next section, the conditions (19)
and (44) are imposed.

##  V Scalar sector

In this section, we focus on the scalar sector. Here, we need to eliminate $7$
DOFs in the scalar sector in order to have $1$ physical DOF. Introducing
dimensionless variables, $\beta_{\Phi}\to\beta_{\Phi}/k$ and ${\cal
E}_{\Phi}\to{\cal E}_{\Phi}/k^{2}$, the Lagrangian reduces to

$\displaystyle{\cal L}^{S}={\cal L}^{S}_{\rm kin}+{\cal L}^{S}_{\rm
cross}+{\cal L}^{S}_{\rm mass}\,,$ (45)

where the explicit form of the first part reads

$\displaystyle{\cal L}_{\mathrm{kin}}^{S}$ $\displaystyle=$ $\displaystyle
4(\kappa_{h1}+\kappa_{h2}+\kappa_{h3}+\kappa_{h4})\dot{\alpha}_{h}^{2}-(2\kappa_{h1}+\kappa_{h2})\dot{\beta}_{h}^{2}+12(\kappa_{h1}+3\kappa_{h4})\dot{{\cal
R}}_{h}^{2}+4(\kappa_{h1}+\kappa_{h4})\dot{{\cal E}}_{h}^{2}$ (46)
$\displaystyle-4(\kappa_{h3}+2\kappa_{h4})\left(-3\dot{{\cal
R}}_{h}+\dot{{\cal
E}}_{h}\right)\dot{\alpha}_{h}-8(\kappa_{h1}+3\kappa_{h4})\dot{{\cal
R}}_{h}\dot{{\cal E}}_{h}$
$\displaystyle+4(\kappa_{f1}+\kappa_{f2}+\kappa_{f3}+\kappa_{f4})\dot{\alpha}_{f}^{2}-(2\kappa_{f1}+\kappa_{f2})\dot{\beta}_{f}^{2}+12(\kappa_{f1}+3\kappa_{f4})\dot{{\cal
R}}_{f}^{2}+4(\kappa_{f1}+\kappa_{f4})\dot{{\cal E}}_{f}^{2}$
$\displaystyle-4(\kappa_{f3}+2\kappa_{f4})\left(-3\dot{{\cal
R}}_{f}+\dot{{\cal
E}}_{f}\right)\dot{\alpha}_{f}-8(\kappa_{f1}+3\kappa_{f4})\dot{{\cal
R}}_{f}\dot{{\cal E}}_{f}$
$\displaystyle+4\left(l_{2}+l_{3}+l_{4}+l_{5}\right)\dot{\alpha}_{h}\dot{\alpha}_{f}-l_{2}\dot{\beta}_{h}\dot{\beta}_{f}+36l_{4}\dot{\cal
R}_{h}\dot{\cal R}_{f}+4l_{4}\dot{\cal E}_{h}\dot{\cal
E}_{f}+12\left(l_{3}+l_{4}\right)\dot{\alpha}_{h}\dot{\cal
R}_{f}+12\left(l_{4}+l_{5}\right)\dot{\alpha}_{f}\dot{\cal R}_{h}$
$\displaystyle-4\left(l_{3}+l_{4}\right)\dot{\alpha}_{h}\dot{\cal
E}_{f}-4\left(l_{4}+l_{5}\right)\dot{\alpha}_{f}\dot{\cal
E}_{h}-12l_{4}\left(\dot{\cal R}_{h}\dot{\cal E}_{f}+\dot{\cal R}_{f}\dot{\cal
E}_{h}\right)\,.$

and the explicit expression for the remaining parts can be found in the
Appendix B. Once we impose the condition (44), the time derivative of
$\beta_{h}$ and $\beta_{f}$ vanishes in the Lagrangian, hence
$\beta_{h}\,,\beta_{f}$ can be treated as the non-dynamical variables. By
utilizing the field redefinition summarized in Appendix A, we can further
impose without loss of generality

$\displaystyle\kappa_{h3}=2\kappa_{h1},\qquad\kappa_{f3}=2\kappa_{f1},\qquad
l_{3}+l_{4}+l_{5}=0\,,$ (47)

in addition to $l_{1}=0$. Hereafter, we assume these conditions to simplify
the discussion. The conjugate momenta can be written as

$\displaystyle\begin{pmatrix}\pi_{\alpha_{h}}\\\ \pi_{{\cal R}_{h}}\\\
\pi_{{\cal E}_{h}}\\\ \pi_{\alpha_{f}}\\\ \pi_{{\cal R}_{f}}\\\ \pi_{{\cal
E}_{f}}\\\ \end{pmatrix}=4{\cal K}_{S}\begin{pmatrix}\dot{\alpha}_{h}\\\
\dot{\cal R}_{h}\\\ \dot{\cal E}_{h}\\\ \dot{\alpha}_{f}\\\ \dot{\cal
R}_{f}\\\ \dot{\cal E}_{f}\\\ \end{pmatrix}+4\begin{pmatrix}0\\\
-4\kappa_{h1}\\\ 0\\\ -l_{3}\\\ -3l_{3}\\\ l_{3}\\\
\end{pmatrix}k\beta_{h}+4\begin{pmatrix}-l_{5}\\\ -3l_{5}\\\ l_{5}\\\ 0\\\
-4\kappa_{f1}\\\ 0\\\ \end{pmatrix}k\beta_{f}\,,$ (48)

where the kinetic matrix for the scalar variables $\\{\alpha_{h},{\cal
R}_{h},{\cal E}_{h}\,,\alpha_{f},{\cal R}_{f},{\cal E}_{f}\\}$ is given by

$\displaystyle{\cal
K}_{S}=\begin{pmatrix}2(\kappa_{h1}+\kappa_{h4})&6(\kappa_{h1}+\kappa_{h4})&-2(\kappa_{h1}+\kappa_{h4})&0&-3l_{5}&l_{5}\\\
*&6(\kappa_{h1}+3\kappa_{h4})&-2(\kappa_{h1}+3\kappa_{h4})&-3l_{3}&-9(l_{3}+l_{5})&3(l_{3}+l_{5})\\\
*&*&2(\kappa_{h1}+\kappa_{h4})&l_{3}&3(l_{3}+l_{5})&-(l_{3}+l_{5})\\\
*&*&*&2(\kappa_{f1}+\kappa_{f4})&6(\kappa_{f1}+\kappa_{f4})&-2(\kappa_{f1}+\kappa_{f4})\\\
*&*&*&*&6(\kappa_{f1}+3\kappa_{f4})&-2(\kappa_{f1}+3\kappa_{f4})\\\
*&*&*&*&*&2(\kappa_{f1}+\kappa_{f4})\\\ \end{pmatrix}\,,$ (49)

and

$\displaystyle\pi_{\beta_{h}}=\pi_{\beta_{f}}=0\,.$ (50)

Note that there are at least two primary constraints from
$\pi_{\beta_{h}}{=0}$ and $\pi_{\beta_{f}}{=0}$. The Hamiltonian is given by

$\displaystyle{\cal H}^{S}$ $\displaystyle=$
$\displaystyle\dot{\alpha}_{h}\pi_{\alpha_{h}}+\dot{\alpha}_{f}\pi_{\alpha_{f}}+\dot{\cal
R}_{h}\pi_{{\cal R}_{h}}+\dot{\cal R}_{f}\pi_{{\cal R}_{f}}+\dot{\cal
E}_{h}\pi_{{\cal E}_{h}}+\dot{\cal E}_{f}\pi_{{\cal
E}_{f}}+\dot{\beta}_{h}\pi_{\beta_{h}}+\dot{\beta}_{f}\pi_{\beta_{f}}-{\cal
L}^{S}\,.$ (51)

###  V.1 Classification of primary constraints

Now, we would like to classify the cases based on the number of primary
constraints. As performed in the analysis of the vector sector, we consider
the eigenvalue equation,

$\displaystyle{\cal F}_{S}(\lambda)\equiv{\rm det}({\cal K}_{S}-\lambda I)\,,$
(52)

The eigenvalue equation with $\lambda=0$, namely the determinant of the
kinetic matrix, reads

$\displaystyle{\rm det}\,{\cal K}_{S}={\cal
F}_{S}(0)=16\kappa_{h1}\kappa_{f1}\Bigl{[}8(\kappa_{f1}+\kappa_{f4})\kappa_{h1}+3l_{3}^{2}\Bigr{]}\Bigl{[}8(\kappa_{h1}+\kappa_{h4})\kappa_{f1}+3l_{5}^{2}\Bigr{]}\,.$
(53)

When the above determinant is non-zero, that is, $\det{\cal K}_{S}\neq 0$,
there are only two primary constraints, which can be defined by (50). In this
case, the number of the physical DOFs is four as proved in Appendix D.
Therefore, we disregard this option. The case of $3$ primary constraint can be
obtained by demanding ${\cal F}_{S}(0)=0$,

$\displaystyle\textrm{3 primary constraints : }\quad{\cal
F}_{S}(0)=0\quad\longleftrightarrow\quad\kappa_{f4}=-\kappa_{f1}-\frac{3l_{3}^{2}}{8\kappa_{h1}}\qquad{\rm
or}\qquad\kappa_{h4}=-\kappa_{h1}-\frac{3l_{5}^{2}}{8\kappa_{f1}}\,.$ (54)

Using the conditions above, we have

$\displaystyle{\cal
F}_{S}^{\prime}(0)=\begin{cases}\,-8\kappa_{f1}(5l_{3}^{2}+8\kappa_{h1}^{2})(3l_{5}^{2}+8\kappa_{f1}(\kappa_{h1}+\kappa_{h4}))\quad{\rm
for}\quad\displaystyle{\kappa_{f4}=-\kappa_{f1}-\frac{3l_{3}^{2}}{8\kappa_{h1}}}\,,\\\
\\\
\,-8\kappa_{h1}(5l_{5}^{2}+8\kappa_{f1}^{2})(3l_{3}^{2}+8\kappa_{h1}(\kappa_{f1}+\kappa_{f4}))\quad{\rm
for}\quad\displaystyle{\kappa_{h4}=-\kappa_{h1}-\frac{3l_{5}^{2}}{8\kappa_{f1}}}\,.\end{cases}$
(55)

Now, ${\cal F}_{S}^{\prime}(0)=0$ gives only one solution of $4$ primary case
due to the symmetric property under $h$ and $f$ in (54),

$\displaystyle\textrm{4 primary constraints : }\quad{\cal
F}_{S}(0)=0\quad\&\quad{\cal
F}_{S}^{\prime}(0)=0\quad\longleftrightarrow\quad\kappa_{h4}=-\kappa_{h1}-\frac{3l_{5}^{2}}{8\kappa_{f1}}\quad\&\quad\kappa_{f4}=-\kappa_{f1}-\frac{3l_{3}^{2}}{8\kappa_{h1}}\,.$
(56)

The absence of the case with five primary constraints can be proved as
follows. In addition to ${\cal F}_{S}(0)=0$ and ${\cal F}_{S}^{\prime}(0)=0$,
we further need to impose ${\cal F}_{S}^{\prime\prime}(0)=0$, which is given
by

$\displaystyle{\cal
F}_{S}^{\prime\prime}(0)=32(l_{5}^{2}+16\kappa_{f1}^{2})\left(\kappa_{h1}-\frac{9l_{3}l_{5}\kappa_{f1}}{l_{5}^{2}+16\kappa_{f1}^{2}}\right)^{2}+\frac{8l_{3}^{2}(5l_{5}^{2}+8\kappa_{f1}^{2})^{2}}{l_{5}^{2}+16\kappa_{f1}^{2}}=0\,.$
(57)

It is manifest that there is no real solution for this equation under the
assumption (19), and therefore, the scalar sector cannot have 5 or more
primary constraints.

###  V.2 3 primary constraints

In this subsection, we consider the case with $3$ primary constraints.
Although there are two options as in (54), they are essentially equivalent
since they are transformed each other as shown in Appendix A.2, which
satisfies :

$\displaystyle\kappa_{h2}=-\kappa_{h3}=-2\kappa_{h1}\neq
0\,,\qquad\kappa_{f2}=-\kappa_{f3}=-2\kappa_{f1}\neq
0\,,\qquad\kappa_{f4}=-\kappa_{f1}-\frac{3l_{3}^{2}}{8\kappa_{h1}}\,,$
$\displaystyle\ l_{1}=l_{2}=l_{4}=\mu_{f1}=n_{1}=0\,,\qquad l_{5}=-l_{3}\,.$
(58)

In this case, we have three primary constraints, which are defined by :

$\displaystyle{\cal C}^{(1)}_{\alpha_{f}}\equiv\pi_{\alpha_{f}}-{l_{3}\over
4\kappa_{h1}}(\pi_{{\cal R}_{h}}-3\pi_{\alpha_{h}})\approx 0\,,\qquad{\cal
C}^{(1)}_{\beta_{h}}\equiv\pi_{\beta_{h}}\approx 0\,,\qquad{\cal
C}^{(1)}_{\beta_{f}}\equiv\pi_{\beta_{f}}\approx 0\,.$ (59)

The total Hamiltonian is given by

$\displaystyle{\cal H}_{T}^{S}={\cal H}^{S}+\lambda_{\alpha_{f}}{\cal
C}^{(1)}_{\alpha_{f}}+\lambda_{\beta_{h}}{\cal
C}^{(1)}_{\beta_{h}}+\lambda_{\beta_{f}}{\cal C}^{(1)}_{\beta_{f}}\,.$ (60)

The evolution of the primary constraints yields the secondary constraints

$\displaystyle{\cal C}^{(2)}_{\alpha_{f}}\equiv\\{{\cal
C}^{(1)}_{\alpha_{f}},{\cal
H}_{T}^{S}\\}{=-2\left(2k^{2}l_{3}+2n_{2}+{3l_{3}\mu_{h1}\over\kappa_{h1}}\right)\alpha_{h}+2\left(2k^{2}l_{3}-6n_{2}+{3l_{3}\mu_{h1}\over\kappa_{h1}}\right){\cal
R}_{h}+2\left(2k^{2}l_{3}+2n_{2}-{l_{3}\mu_{h1}\over\kappa_{h1}}\right){\cal
E}_{h}}$
$\displaystyle\qquad\qquad\qquad\qquad~{}~{}~{}+4\left({k^{2}l_{3}^{2}\over\kappa_{h1}}-2\mu_{f2}\right)(\alpha_{f}-{\cal
E}_{f})+4\left({k^{2}(3l_{3}^{2}+4\kappa_{f1}\kappa_{h1})\over\kappa_{h1}}-6\mu_{f2}\right){\cal
R}_{f}\approx 0\,,$ (61) $\displaystyle{\cal
C}^{(2)}_{\beta_{h}}\equiv\\{{\cal C}^{(1)}_{\beta_{h}},{\cal
H}_{T}^{S}\\}=-k\left(\pi_{\alpha_{h}}+\pi_{{\cal
E}_{h}}\right)+4\mu_{h1}\beta_{h}\approx 0\,,$ (62) $\displaystyle{\cal
C}^{(2)}_{\beta_{f}}\equiv\\{{\cal C}^{(1)}_{\beta_{f}},{\cal
H}_{T}^{S}\\}=-k\pi_{{\cal
E}_{f}}+\frac{kl_{3}}{4\kappa_{h1}}(3\pi_{\alpha_{h}}-\pi_{{\cal
R}_{h}})\approx 0\,.$ (63)

Here, none of the above constraints can be trivially zero with any choice of
the parameters under the assumption (19). Then, the time evolution of the
secondary constraint ${\cal C}^{(2)}_{\beta_{f}}$ gives the tertiary
constraint

$\displaystyle{\cal C}^{(3)}_{\beta_{f}}\equiv\\{{\cal
C}^{(2)}_{\beta_{f}},{\cal H}_{T}^{S}\\}=k\,{\cal
C}^{(2)}_{\alpha_{f}}+4k^{3}\Big{[}l_{3}(\alpha_{h}-{\cal E}_{h}+3{\cal
R}_{h})-4\kappa_{f1}{\cal R}_{f}\Big{]}\approx 0\,,$ (64)

and its time evolution demands

$\displaystyle{\dot{\cal C}}^{(3)}_{\beta_{f}}=k({\dot{\cal
C}}^{(2)}_{\alpha_{f}}+k{\cal C}^{(2)}_{\beta_{f}})\approx 0\,.$ (65)

Since ${\dot{\cal C}}^{(3)}_{\beta_{f}}$ does not generate an independent
equation, there is no more constraint from $\beta_{f}$. The evolution of the
rest of the secondary constraints are given by

$\displaystyle\begin{pmatrix}{\dot{\cal C}}^{(2)}_{\alpha_{f}}\\\ {\dot{\cal
C}}^{(2)}_{\beta_{h}}\end{pmatrix}=\begin{pmatrix}\\{{\cal
C}^{(2)}_{\alpha_{f}}\,,{{\cal H}^{S}_{T}}\\}\\\ \\{{\cal
C}^{(2)}_{\beta_{h}}\,,{{\cal
H}^{S}_{T}}\\}\end{pmatrix}=\begin{pmatrix}\\{{\cal
C}^{(2)}_{\alpha_{f}}\,,{{\cal H}^{S}}\\}\\\ \\{{\cal
C}^{(2)}_{\beta_{h}}\,,{{\cal
H}^{S}}\\}\end{pmatrix}+2\begin{pmatrix}\displaystyle{-4\mu_{f2}-{3l_{3}^{2}\mu_{h1}\over\kappa_{h1}^{2}}}&0\\\
0&2\mu_{h1}\end{pmatrix}\begin{pmatrix}\lambda_{\alpha_{f}}\\\
\lambda_{\beta_{h}}\end{pmatrix}\approx 0\,.$ (66)

Since $\mu_{h1}\neq 0$, the Lagrange multiplier $\lambda_{\beta_{h}}$ is
determined by ${\dot{\cal C}}^{(2)}_{\beta_{h}}=0$. When the Poisson bracket
$\\{{\cal C}^{(2)}_{\alpha_{f}},{\cal C}^{(1)}_{\alpha_{f}}\\}$, i.e., the
coefficient of $\lambda_{\alpha_{f}}$ in ${\dot{\cal C}}^{(2)}_{\alpha_{f}}$,
is non-vanishing, the evolution of ${\cal C}^{(2)}_{\alpha_{f}}$ determines
the Lagrange multiplier $\lambda_{\alpha_{f}}$ and no more constraint from
${\cal C}^{(2)}_{\alpha_{f}}$ will be generated. Now we redefine the following
constraints :

$\displaystyle\widetilde{{\cal C}}^{(2)}_{\beta_{f}}\equiv{\cal
C}^{(2)}_{\beta_{f}}-k{\cal C}^{(1)}_{\alpha_{f}}\approx
0\,,\qquad\widetilde{{\cal C}}^{(3)}_{\beta_{f}}\equiv{\cal
C}^{(3)}_{\beta_{f}}-k{\cal C}^{(2)}_{\alpha_{f}}\approx 0\,.$ (67)

The constraints, ${\cal C}^{(1)}_{\beta_{f}}$ and $\widetilde{{\cal
C}}^{(2,3)}_{\beta_{f}}$ commute with all constraints, therefore, these are
first class. The rest of constraints are second class. In summary, the number
of the physical DOFs is $(8\times 2-4-3\times 2)/2=3$.

In order to eliminate extra DOFs, one has to impose an extra condition
$\\{{\cal C}^{(2)}_{\alpha_{f}}\,,{\cal C}^{(1)}_{\alpha_{f}}\\}=0$, namely

$\displaystyle\mu_{f2}=-{3l_{3}^{2}\mu_{h1}\over 4\kappa_{h1}^{2}}\,.$ (68)

In this case, the evolution of ${\cal C}^{(2)}_{\alpha_{f}}$ yields the
tertiary constraint when $l_{3}\neq 0$ or $n_{2}\neq 0$,

$\displaystyle{\cal C}^{(3)}_{\alpha_{f}}$ $\displaystyle\equiv$
$\displaystyle\\{{\cal C}^{(2)}_{\alpha_{f}},{\cal H}_{T}^{S}\\}\approx 0\,,$
(69)

where the explicit expression of ${\cal C}^{(3)}_{\alpha_{f}}$ is given in
Appendix E. When $l_{3}=n_{2}=0$, ${\cal C}^{(3)}_{\alpha_{f}}=-k{\cal
C}^{(2)}_{\beta_{f}}$, implying that there is no more constraint. Therefore,
we, hereafter, consider the case with $l_{3}\neq 0$ or $n_{2}\neq 0$. Now, the
evolution of ${\cal C}^{(3)}_{\alpha_{f}}$ yields the quaternary constraint
${\cal C}^{(4)}_{\alpha_{f}}=\\{{\cal C}^{(3)}_{\alpha_{f}},{\cal
H}^{S}_{T}\\}\approx 0$. Since ${\cal C}^{(4)}_{\alpha_{f}}$ contains
$\lambda_{\beta_{h}}$, it is useful to define the following linear combination
of constraints :

$\displaystyle\widetilde{{\cal C}}^{(3)}_{\alpha_{f}}$ $\displaystyle\equiv$
$\displaystyle{{\cal C}}^{(3)}_{\alpha_{f}}+{2kl_{3}\over\kappa_{h1}}{{\cal
C}}^{(2)}_{\beta_{h}}{+k{\cal C}^{(2)}_{\beta_{f}}}\approx 0\,,$ (70)
$\displaystyle\widetilde{\cal C}^{(4)}_{\alpha_{f}}$ $\displaystyle\equiv$
$\displaystyle\\{\widetilde{\cal C}^{(3)}_{\alpha_{f}},{\cal
H}^{S}_{T}\\}\approx 0\,.$ (71)

When $\\{{\widetilde{\cal C}^{(4)}_{\alpha_{f}}},{\cal
C}^{(1)}_{\alpha_{f}}\\}\neq 0$, the Lagrange multiplier
$\lambda_{\alpha_{f}}$ is determined by the time evolution of
${\widetilde{\cal C}^{(4)}_{\alpha_{f}}}$. In this case, the constraints,
${\cal C}^{(1)}_{\beta_{f}}$ and $\widetilde{{\cal C}}^{(2,3)}_{\beta_{f}}$
still commute with all constraints, and hence, these are first class. The rest
of constraints are second class, therefore, the number of the physical DOFs is
given by $(8\times 2-6-3\times 2)/2=2$.

To obtain a theory with $1$ DOF in the scalar sector, one more DOF has to be
eliminated. Then, we would like to consider the following case

$\displaystyle\\{{\widetilde{\cal C}^{(4)}_{\alpha_{f}}},{\cal
C}^{(1)}_{\alpha_{f}}\\}={\frac{4}{\kappa_{h1}^{2}}\left(-\frac{3l_{3}^{2}\mu_{h1}^{2}}{\kappa_{h1}}+\frac{\kappa_{f1}(2\kappa_{h1}n_{2}+3l_{3}\mu_{h1})^{2}}{8\kappa_{f1}(\kappa_{h1}+\kappa_{h4})+3l_{3}^{2}}\right)}=0\,.$
(72)

Solving the above equation, we obtain

$\displaystyle\kappa_{h4}={-\kappa_{h1}-\frac{3l_{3}^{2}}{8\kappa_{f1}}+\frac{\kappa_{h1}(2\kappa_{h1}n_{2}+3l_{3}\mu_{h1})^{2}}{24l_{3}^{2}\mu_{h1}^{2}}}\,.$
(73)

In this case, we have two additional constraints :

$\displaystyle\widetilde{\cal C}^{(5)}_{\alpha_{f}}\equiv\\{\widetilde{\cal
C}^{(4)}_{\alpha_{f}},{\cal H}^{S}_{T}\\}\approx 0\,,\qquad\widetilde{\cal
C}^{(6)}_{\alpha_{f}}\equiv\\{\widetilde{\cal C}^{(5)}_{\alpha_{f}},{\cal
H}^{S}_{T}\\}\approx 0\,.$ (74)

Again, $\widetilde{\cal C}^{(6)}_{\alpha_{f}}$ contains the Lagrange
multiplier $\lambda_{\beta_{h}}$, we redefine the constraint as

$\displaystyle\overline{{\cal C}}{}^{(5)}_{\alpha_{f}}\equiv{\widetilde{\cal
C}}^{(5)}_{\alpha_{f}}+x\,{{\cal C}}^{(2)}_{\beta_{h}}\approx
0\,,\qquad\overline{\cal C}{}^{(6)}_{\alpha_{f}}\equiv\\{\widetilde{\cal
C}^{(5)}_{\alpha_{f}},{\cal H}^{S}_{T}\\}\approx 0\,,$ (75)

where

$\displaystyle
x=\frac{kl_{3}}{\kappa_{h1}^{3}}\biggl{[}{-2}k^{2}\kappa_{h1}^{2}+\frac{\mu_{h1}(8n_{2}\kappa_{f1}\kappa_{h1}^{2}-9l_{3}^{3}\mu_{h1})}{\kappa_{f1}(2\kappa_{h1}n_{2}+3l_{3}\mu_{h1})}\biggr{]}\,.$
(76)

If $\\{\overline{\cal C}{}^{(6)}_{\alpha_{f}},{\cal
C}^{(1)}_{\alpha_{f}}\\}\neq 0$, the Lagrange multiplier
$\lambda_{\alpha_{f}}$ can be determined by $\dot{\overline{\cal
C}}{}^{(6)}_{\alpha_{f}}=\\{\overline{\cal C}^{(6)}_{\alpha_{f}},{\cal
H}^{S}_{T}\\}\approx 0$ and no further constraint is generated.

$\displaystyle\rm{first\,class}\quad$ $\displaystyle:$
$\displaystyle\quad{\cal C}^{(1)}_{\beta_{f}}\,,\qquad\widetilde{\cal
C}^{(2)}_{\beta_{f}}\,,\qquad\widetilde{\cal C}^{(3)}_{\beta_{f}}\,,$ (77)
$\displaystyle\rm{second\,class}\quad$ $\displaystyle:$
$\displaystyle\quad{\cal C}^{(1)}_{\alpha_{f}}\,,\qquad{\cal
C}^{(2)}_{\alpha_{f}}\,,\qquad\widetilde{\cal
C}^{(3)}_{\alpha_{f}}\,,\qquad\widetilde{\cal
C}^{(4)}_{\alpha_{f}}\,,\qquad\overline{\cal
C}^{(5)}_{\alpha_{f}}\,,\qquad\overline{\cal
C}^{(6)}_{\alpha_{f}}\,,\qquad{\cal C}^{(1)}_{\beta_{h}}\,,\qquad{\cal
C}^{(2)}_{\beta_{h}}\,,$ (78)

we finally have

Scalar DOF $\displaystyle=$ $\displaystyle\frac{1}{2}\times\Bigl{[}8\times
2-8\,(\textrm{2 primary \& 2 secondary \& 1 tertiary \& 1 quaternary + 2
more)}$ (79) $\displaystyle~{}~{}~{}~{}~{}~{}~{}-3\,(\textrm{1 primary \& 1
secondary \& 1 tertiary)}\times 2\,(\textrm{first-class})\Bigr{]}=1\,.$

To summarize, we find a novel class of theory,
(Class Ia) :

$\displaystyle\kappa_{h2}=-\kappa_{h3}=-2\kappa_{h1}\neq
0\,,\qquad\kappa_{f2}=-\kappa_{f3}=-2\kappa_{f1}\neq
0\,,\qquad\kappa_{f4}=-\kappa_{f1}-{3l_{3}^{2}\over
8\kappa_{h1}}\,,\qquad{\mu_{f2}=-\frac{3l_{3}^{2}\mu_{h1}}{4\kappa_{h1}^{2}}}\,,$
$\displaystyle\ l_{1}=l_{2}=l_{4}=\mu_{f1}=n_{1}=0\,,\qquad
l_{5}=-l_{3}\,,\qquad\kappa_{h4}={-\kappa_{h1}-\frac{3l_{3}^{2}}{8\kappa_{f1}}+\frac{\kappa_{h1}(2\kappa_{h1}n_{2}+3l_{3}\mu_{h1})^{2}}{24l_{3}^{2}\mu_{h1}^{2}}\,.}$
(80)

The Lagrangian for class with 1 DOF in the scalar sector Ia is given by

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\left(\kappa_{h1}h_{\mu\nu}\widehat{\cal
E}^{\mu\nu\alpha\beta}h_{\alpha\beta}+\kappa_{f1}f_{\mu\nu}\widehat{\cal
E}^{\mu\nu\alpha\beta}f_{\alpha\beta}\right)+\delta\kappa_{h4}h_{,\mu}f^{,\mu}+{3l_{3}^{2}\over
8\kappa_{h1}}f_{,\mu}f^{,\mu}+l_{3}\left(h_{,\nu}f^{\mu\nu}_{~{}~{},\mu}-h^{\mu\nu}_{~{}~{},\mu}f_{,\nu}\right)$
(81)
$\displaystyle-\mu_{h1}h_{\mu\nu}h^{\mu\nu}-\mu_{h2}h^{2}+\left({{3l_{3}^{2}\mu_{h1}\over
4\kappa_{h1}^{2}}}f-n_{2}h\right)f\,,$

where

$\displaystyle\delta\kappa_{h4}=\frac{3l_{3}^{2}}{8\kappa_{f1}}-\frac{\kappa_{h1}(2\kappa_{h1}n_{2}+3l_{3}\mu_{h1})^{2}}{24l_{3}^{2}\mu_{h1}^{2}}\,.$
(82)

One can check that this theory is invariant under the gauge transformation

$\displaystyle h_{\mu\nu}$ $\displaystyle\to$
$\displaystyle{\widetilde{h}}_{\mu\nu}=h_{\mu\nu}\,,$ (83) $\displaystyle
f_{\mu\nu}$ $\displaystyle\to$
$\displaystyle{\widetilde{f}}_{\mu\nu}=f_{\mu\nu}+\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}\qquad{\rm
with}\qquad\partial^{\mu}\xi_{\mu}=0\,.$ (84)

As one can see from the transverse condition in the gauge transformation, this
class is totally distinct from the linearized Hassan-Rosen bigravity, and
there are non-trivial kinetic terms for $h$, derivative and mass interactions.

Let us finally discuss the final option where the time-evolution of the
tertiary constraint $\dot{\widetilde{\cal C}}{}^{(3)}_{\alpha_{f}}$ does not
yield new constraint. Such a case can be found by rewriting
$\dot{\widetilde{\cal C}}{}^{(3)}_{\alpha_{f}}$ in terms of other constraints,
${\cal C}^{(1)}_{\beta_{f}}$, $\widetilde{\cal C}^{(2)}_{\beta_{f}}$,
$\widetilde{\cal C}^{(3)}_{\beta_{f}}$, ${\cal C}^{(1)}_{\alpha_{f}}$, ${\cal
C}^{(2)}_{\alpha_{f}}$, ${\cal C}^{(1)}_{\beta_{h}}$, and ${\cal
C}^{(2)}_{\beta_{h}}$, and setting it to be zero. Then, we obtain two
conditions : Eq. (73) and

$\displaystyle\frac{24l_{3}^{2}\mu_{h1}^{3}(4n_{2}^{2}\kappa_{h1}^{2}+3l_{3}^{2}\mu_{h1}(\mu_{h1}+4\mu_{h2}))}{\kappa_{h1}^{4}(2n_{2}\kappa_{h1}+3l_{3}\mu_{h1})^{2}}=0\,,$
(85)

which can be solved for $\mu_{h2}$,

$\displaystyle\mu_{h2}=-\frac{n_{2}^{2}\kappa_{h1}^{2}}{3l_{3}^{2}\mu_{h1}}-{\mu_{h1}\over
4}\,,$ (86)

since we assumed $l_{3}\neq 0$ and $\mu_{h1}\neq 0$. Note that the case where
$\\{\overline{\cal C}{}^{(6)}_{\alpha_{f}},{\cal C}^{(1)}_{\alpha_{f}}\\}$
vanishes in the Class Ia reduces to this option. In this case, as shown in
(207), the time-evolution of both the tertiary constraints $\widetilde{\cal
C}^{(3)}_{\alpha_{f}}$ and $\widetilde{\cal C}^{(3)}_{\beta_{f}}$ can be
written in terms of the linear combination of the primary and second class
constraints, implying no further constraints. Redefining ${\cal
C}^{(2)}_{\alpha_{f}}$ as

$\displaystyle\widetilde{\cal C}^{(2)}_{\alpha_{f}}\equiv{\cal
C}^{(2)}_{\alpha_{f}}+{2kl_{3}\over\kappa_{h1}}{\cal C}^{(1)}_{\beta_{h}}\,,$
(87)

we find

$\displaystyle\rm{first\,class}\quad$ $\displaystyle:$
$\displaystyle\quad{\cal C}^{(1)}_{\alpha_{f}}\,,\qquad\widetilde{\cal
C}^{(2)}_{\alpha_{f}}\,,\qquad\widetilde{\cal
C}^{(3)}_{\alpha_{f}}\,,\qquad{\cal
C}^{(1)}_{\beta_{f}}\,,\qquad\widetilde{\cal
C}^{(2)}_{\beta_{f}}\,,\qquad\widetilde{\cal C}^{(3)}_{\beta_{f}}\,,$ (88)
$\displaystyle\rm{second\,class}\quad$ $\displaystyle:$
$\displaystyle\quad{\cal C}^{(1)}_{\beta_{h}}\,,\qquad{\cal
C}^{(2)}_{\beta_{h}}\,,$ (89)

and

Scalar DOF $\displaystyle=$ $\displaystyle\frac{1}{2}\times\Bigl{[}8\times
2-2\,(\textrm{1 primary \& 1 secondary)}$ (90)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-6\,(\textrm{2 primary \& 2 secondary \& 2
tertiary)}\times 2\,(\textrm{first-class})\Bigr{]}=1\,.$

In this case, the number of the physical DOF is the same as the Class Ia, and
the resultant theory is invariant under the gauge transformation (84). For
this reason, this case can be considered as the special case of the Class I by
choosing (86) although an additional gauge symmetry is present. To summarize,
(Class Ib) :

$\displaystyle\kappa_{h2}=-\kappa_{h3}=-2\kappa_{h1}\neq
0\,,\qquad\kappa_{f2}=-\kappa_{f3}=-2\kappa_{f1}\neq
0\,,\qquad\kappa_{f4}=-\kappa_{f1}-{3l_{3}^{2}\over
8\kappa_{h1}}\,,\qquad{\mu_{f2}=-\frac{3l_{3}^{2}\mu_{h1}}{4\kappa_{h1}^{2}}}\,,$
$\displaystyle\ l_{1}=l_{2}=l_{4}=\mu_{f1}=n_{1}=0\,,\qquad
l_{5}=-l_{3}\,,\qquad\kappa_{h4}=-\kappa_{h1}-\frac{3l_{3}^{2}}{8\kappa_{f1}}+\frac{\kappa_{h1}(2\kappa_{h1}n_{2}+3l_{3}\mu_{h1})^{2}}{24l_{3}^{2}\mu_{h1}^{2}}\,,$
$\displaystyle\mu_{2}=-\frac{n_{2}^{2}\kappa_{h1}^{2}}{3l_{3}^{2}\mu_{h1}}-{\mu_{h1}\over
4}\,.$ (91)

The Lagrangian for the Class Ib is given by

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\left(\kappa_{h1}h_{\mu\nu}\widehat{\cal
E}^{\mu\nu\alpha\beta}h_{\alpha\beta}+\kappa_{f1}f_{\mu\nu}\widehat{\cal
E}^{\mu\nu\alpha\beta}f_{\alpha\beta}\right)+\delta\kappa_{h4}h_{,\mu}f^{,\mu}+{3l_{3}^{2}\over
8\kappa_{h1}}f_{,\mu}f^{,\mu}+l_{3}\left(h_{,\nu}f^{\mu\nu}_{~{}~{},\mu}-h^{\mu\nu}_{~{}~{},\mu}f_{,\nu}\right)$
(92)
$\displaystyle{-\mu_{h1}\left(h_{\mu\nu}h^{\mu\nu}+\frac{1}{4}h^{2}\right)+\frac{1}{3\mu_{h1}}\left(\frac{\kappa_{h1}n_{2}}{l_{3}}h-\frac{3l_{3}\mu_{h1}}{2\kappa_{h1}}f\right)^{2}}\,.$

###  V.3 $4$ primary constraints

Next, let us consider the case with four primary constraints. As you can see
from (56), there are kinetic interactions between $h$ and $f$ fields, which
will make the Hamiltonian analysis involved in general. It is interesting to
note that we can always map this theory into a simpler theory with two
Einstein-Hilbert terms without kinetic interactions between them as explicitly
shown in A.3. Hereinafter we will perform the Hamiltonian analysis in this
simple model :

$\displaystyle\kappa_{h2}=-\kappa_{h3}=2\kappa_{h4}=-2\kappa_{h1}\neq
0\,,\qquad\kappa_{f2}=-\kappa_{f3}=2\kappa_{f4}=-2\kappa_{f1}\neq 0\,,$
$\displaystyle\ l_{1}=l_{2}=l_{3}=l_{4}=l_{5}=0\,,\qquad\mu_{f1}=n_{1}=0\,.$
(93)

Now we have the following four primary constraints:

$\displaystyle{\cal C}^{(1)}_{\alpha_{h}}\equiv\pi_{\alpha_{h}}{\approx
0}\,,\qquad{\cal C}^{(1)}_{\alpha_{f}}\equiv\pi_{\alpha_{f}}{\approx
0}\,,\qquad{\cal C}^{(1)}_{\beta_{h}}\equiv\pi_{\beta_{h}}{\approx
0}\,,\qquad{\cal C}^{(1)}_{\beta_{f}}\equiv\pi_{\beta_{f}}{\approx 0}\,.$ (94)

The total Hamiltonian can be expressed as

$\displaystyle{\cal H}_{T}^{S}={\cal H}^{S}+\lambda_{\alpha_{h}}{\cal
C}^{(1)}_{\alpha_{h}}+\lambda_{\alpha_{f}}{\cal
C}^{(1)}_{\alpha_{f}}+\lambda_{\beta_{h}}{\cal
C}^{(1)}_{\beta_{h}}+\lambda_{\beta_{f}}{\cal C}^{(1)}_{\beta_{f}}\,.$ (95)

The evolution of the primary constraints is given by

$\displaystyle{\cal C}^{(2)}_{\alpha_{h}}\equiv\\{{\cal
C}^{(1)}_{\alpha_{h}},{\cal
H}_{T}^{S}\\}=-8(\mu_{h1}+\mu_{h2})\alpha_{h}+8(2k^{2}\kappa_{h1}-3\mu_{h2}){\cal
R}_{h}+8\mu_{h2}{\cal E}_{h}{-4n_{2}(\alpha_{f}+3{\cal R}_{f}-{\cal
E}_{f})}\approx 0\,,$ (96) $\displaystyle{\cal
C}^{(2)}_{\alpha_{f}}\equiv\\{{\cal C}^{(1)}_{\alpha_{f}},{\cal
H}_{T}^{S}\\}=-8\mu_{f2}\alpha_{f}+8(2k^{2}\kappa_{f1}-3\mu_{f2}){\cal
R}_{f}+8\mu_{f2}{\cal E}_{f}{-4n_{2}(\alpha_{h}+3{\cal R}_{h}-{\cal
E}_{h})}\approx 0\,,$ (97) $\displaystyle{\cal
C}^{(2)}_{\beta_{h}}\equiv\\{{\cal C}^{(1)}_{\beta},{\cal
H}^{S}_{T}\\}=-k\pi_{{\cal E}_{h}}+4\mu_{h1}\beta_{h}\approx 0\,,$ (98)
$\displaystyle{0\approx}{\cal C}^{(2)}_{\beta_{f}}\equiv\\{{\cal
C}^{(1)}_{\beta_{f}},{\cal H}^{S}_{T}\\}=-k\pi_{{\cal E}_{f}}\approx 0\,.$
(99)

Here, all the secondary constraints cannot be trivially zero with any choice
of the coefficients since $\kappa_{h1}\neq 0$ and $\kappa_{f1}\neq 0$. First,
let us take a look at the time evolution of the other primary constraints,
that is,

$\displaystyle\begin{pmatrix}{\dot{\cal C}}^{(2)}_{\alpha_{h}}\\\ {\dot{\cal
C}}^{(2)}_{\alpha_{f}}\\\ {\dot{\cal
C}}^{(2)}_{\beta_{h}}\end{pmatrix}=\begin{pmatrix}\\{{\cal
C}^{(2)}_{\alpha_{h}}\,,{{\cal H}^{S}_{T}}\\}\\\ \\{{\cal
C}^{(2)}_{\alpha_{f}}\,,{{\cal H}^{S}_{T}}\\}\\\ \\{{\cal
C}^{(2)}_{\beta_{h}}\,,{{\cal
H}^{S}_{T}}\\}\end{pmatrix}=\begin{pmatrix}\\{{\cal
C}^{(2)}_{\alpha_{h}}\,,{{\cal H}^{S}}\\}\\\ \\{{\cal
C}^{(2)}_{\alpha_{f}}\,,{{\cal H}^{S}}\\}\\\ \\{{\cal
C}^{(2)}_{\beta_{h}}\,,{{\cal
H}^{S}}\\}\end{pmatrix}+\begin{pmatrix}-8(\mu_{h1}+\mu_{h2})&-4n_{2}&0\\\
-4n_{2}&-8\mu_{f2}&0\\\
0&0&4\mu_{h1}\end{pmatrix}\begin{pmatrix}\lambda_{\alpha_{h}}\\\
\lambda_{\alpha_{f}}\\\ \lambda_{\beta_{h}}\end{pmatrix}\approx 0\,.$ (100)

When $n_{2}^{2}-4\mu_{f2}(\mu_{h1}+\mu_{h2})\neq 0$, all the Lagrange
multipliers, $\lambda_{\alpha_{h}}$, $\lambda_{\alpha_{f}}$, and
$\lambda_{\beta_{h}}$, are determined by the above equations. As for ${\cal
C}^{(2)}_{\beta_{f}}$, it commutes with all the primary constraints and the
consistency of ${\cal C}^{(2)}_{\beta_{f}}$ gives the tertiary constraint,

$\displaystyle{\cal C}^{(3)}_{\beta_{f}}=\\{{{\cal C}^{(2)}_{\beta_{f}}},{\cal
H}_{T}^{S}\\}={-8k\mu_{f2}(\alpha_{f}+3{\cal R}_{f}-{\cal
E}_{f})-4kn_{2}(\alpha_{h}+3{\cal R}_{h}-{\cal E}_{h})}\approx 0\,.$ (101)

Now we redefine the secondary and tertiary constraints for $\beta_{f}$ as

$\displaystyle\widetilde{\cal C}^{(2)}_{\beta_{f}}$ $\displaystyle=$
$\displaystyle{\cal C}^{(2)}_{\beta_{f}}-k{\cal
C}^{(1)}_{\alpha_{f}}=-k(\pi_{\alpha_{f}}+\pi_{{\cal E}_{f}})\approx 0\,.$
(102) $\displaystyle\widetilde{\cal C}^{(3)}_{\beta_{f}}$ $\displaystyle=$
$\displaystyle{\cal C}^{(3)}_{\beta_{f}}-k{\cal
C}^{(2)}_{\alpha_{f}}=-16k^{3}\kappa_{f1}{\cal R}_{f}\approx 0\,.$ (103)

Then, $\widetilde{{\cal C}}^{(3)}_{\beta_{f}}$ does commute with ${\cal
C}^{(1)}_{\alpha_{h}}$ and ${\cal C}^{(1)}_{\alpha_{f}}$, and one can see
$\dot{\widetilde{\cal C}}{}^{(3)}_{\beta_{f}}=k^{2}{\cal
C}^{(2)}_{\beta_{f}}\approx 0$, implying no more constraint is generated. In
addition, one can also check that the constraints, ${\cal
C}^{(1)}_{\beta_{f}}$ and $\widetilde{\cal C}^{(2,3)}_{\beta_{f}}$, commute
with all the constraints and hence these are first class while the rest of
constraints are second class. Therefore, we conclude the number of the
physical DOFs is $(8\times 2-6-3\times 2)/2=2$ when
$n_{2}^{2}-4\mu_{f2}(\mu_{h1}+\mu_{h2})\neq 0$.

In order to remove an extra DOF, we need to impose an additional constraint
for the parameter :

$\displaystyle n_{2}^{2}-4\mu_{f2}(\mu_{h1}+\mu_{h2})$ $\displaystyle=$
$\displaystyle 0\,,$ (104)

which yields two branches :

$\displaystyle\mu_{h2}$ $\displaystyle=$
$\displaystyle-\mu_{h1}+\displaystyle{\frac{n_{2}^{2}}{4\mu_{f2}}}\qquad{\rm(Class~{}II)}\,,$
(105) $\displaystyle\mu_{f2}$ $\displaystyle=$ $\displaystyle
n_{2}=0\qquad\qquad\quad{\rm(Class~{}III)}\,.$ (106)

Note that ${\cal C}^{(3)}_{\beta_{f}}$ trivially vanishes in the second case
(Class III).

####  V.3.1 Class II

Let us consider Class II first. For convenience, we redefine the primary
constraint for $\alpha_{h}$ with a linear combination of those for
$\alpha_{h}$ and $\alpha_{f}$. Then the four primary constrains reads :

$\displaystyle{\cal C}^{(1)}_{\alpha_{h}}\equiv\pi_{\alpha_{h}}-{n_{2}\over
2\mu_{f2}}\pi_{\alpha_{f}}\approx 0\,,\qquad{\cal
C}^{(1)}_{\alpha_{f}}\equiv\pi_{\alpha_{f}}\approx 0\,,\qquad{\cal
C}^{(1)}_{\beta_{h}}\equiv\pi_{\beta_{h}}\approx 0\,,\qquad{\cal
C}^{(1)}_{\beta_{f}}\equiv\pi_{\beta_{f}}\approx 0\,.$ (107)

We have the same constrains from the evolution of the primary constraints for
$\alpha_{f}$, $\beta_{h}$ and $\beta_{f}$ as in (97), (98) and (99)
respectively. Due to the condition (104), only one of the Lagrange
multipliers, $\lambda_{\alpha_{h}}$ or $\lambda_{\alpha_{f}}$, is determined
by the evolution of ${\cal C}^{(2)}_{\alpha_{h}}$ or ${\cal
C}^{(2)}_{\alpha_{f}}$. Suppose that $\lambda_{\alpha_{f}}$ has been
determined by the evolution of ${\cal C}^{(2)}_{\alpha_{f}}$ though
$\lambda_{\alpha_{h}}$ has not. The evolution of the primary constraint for
$\alpha_{h}$ demands

$\displaystyle{\cal C}^{(2)}_{\alpha_{h}}\equiv\\{{\cal
C}^{(1)}_{\alpha_{h}},{\cal H}_{T}^{S}\\}=-8\mu_{h1}{\cal
E}_{h}-{8k^{2}n_{2}\kappa_{f1}\over\mu_{f2}}{\cal
R}_{f}+8(2k^{2}\kappa_{h1}+3\mu_{h1}){\cal R}_{h}\approx 0\,,$ (108)
$\displaystyle{\cal C}^{(3)}_{\alpha_{h}}\equiv\\{{\cal
C}^{(2)}_{\alpha_{h}},{\cal
H}^{S}_{T}\\}=-8k\mu_{h1}\beta_{h}-{k^{2}n_{2}\over 2\mu_{f2}}\pi_{{\cal
E}_{f}}+k^{2}\pi_{{\cal E}_{h}}-{\mu_{h1}\over 2\kappa_{h1}}\pi_{{\cal
R}_{h}}\approx 0\,,$ (109)

Since ${\cal C}^{(3)}_{\alpha_{h}}$ does not commute with ${\cal
C}^{(1)}_{\beta_{h}}$, it is coveninent to introduce a linear combination of
${\cal C}^{(3)}_{\alpha_{h}}$ and ${\cal C}^{(2)}_{\beta_{h}}$ as

$\displaystyle\widetilde{\cal C}^{(3)}_{\alpha_{h}}={\cal
C}^{(3)}_{\alpha_{h}}+2k{\cal C}^{(2)}_{\beta_{h}}\,.$ (110)

The evolution of $\widetilde{\cal C}^{(3)}_{\alpha_{h}}$ yields the constraint
$\widetilde{\cal C}^{(4)}_{\alpha_{h}}=\\{\widetilde{\cal
C}^{(3)}_{\alpha_{h}},{\cal H}^{S}_{T}\\}\approx 0$. Since $\\{\widetilde{\cal
C}^{(4)}_{\alpha_{h}},{\cal
C}^{(1)}_{\alpha_{h}}\\}=-12\mu_{h1}^{2}/\kappa_{h1}\neq 0$, the evolution of
$\widetilde{\cal C}^{(4)}_{\alpha_{f}}$ determines the Lagrange multiplier
$\lambda_{\alpha_{h}}$ and no more constraint is generated. It can be easily
checked that $\widetilde{\cal C}^{(3)}_{\alpha_{h}}$ and $\widetilde{\cal
C}^{(4)}_{\alpha_{h}}$ cannot be trivially zero. Since in this case

$\displaystyle\rm{first\,class}\quad$ $\displaystyle:$
$\displaystyle\quad{\cal C}^{(1)}_{\beta_{f}}\,,\quad\widetilde{\cal
C}^{(2)}_{\beta_{f}}\,,\qquad\widetilde{\cal C}^{(3)}_{\beta_{f}}\,,$ (111)
$\displaystyle\rm{second\,class}\quad$ $\displaystyle:$
$\displaystyle\quad{\cal C}^{(1)}_{\alpha_{h}}\,,\qquad{\cal
C}^{(2)}_{\alpha_{h}}\,,\qquad\widetilde{\cal
C}^{(3)}_{\alpha_{h}}\,,\qquad\widetilde{\cal
C}^{(4)}_{\alpha_{f}}\,,\qquad{\cal C}^{(1)}_{\alpha_{f}}\,,\qquad{\cal
C}^{(2)}_{\alpha_{f}}\,,\qquad{\cal C}^{(1)}_{\beta_{h}}\,,\qquad{\cal
C}^{(2)}_{\beta_{h}}\,,$ (112)

therefore we have :

Scalar DOF $\displaystyle=$ $\displaystyle\frac{1}{2}\biggl{[}8\times
2-8\,(\textrm{3 primary \& 3 secondary \& 1 tertiary \& 1 quaternary\,)}$
(113) $\displaystyle\quad-3\,(\textrm{1 primary \& 1 secondary \& 1
tertiary)}\times 2\,(\textrm{first-class})\biggr{]}=1\,.$

To summarize, we find another novel class of theory with a single DOF in the
scalar sector :
Class II :

$\displaystyle\kappa_{h2}=-\kappa_{h3}=2\kappa_{h4}=-2\kappa_{h1}\neq
0\,,\qquad\kappa_{f2}=-\kappa_{f3}=2\kappa_{f4}=-2\kappa_{f1}\neq 0\,,$
$\displaystyle\ l_{1}=l_{2}=l_{3}=l_{4}=l_{5}=0\,,\qquad\mu_{f1}=0\,,\qquad
n_{1}=0\,,\qquad\mu_{h1}+\mu_{h2}-{n_{2}^{2}\over 4\mu_{f2}}=0\,.$ (114)

The Lagrangian for class II is given by

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\left(\kappa_{h1}h_{\mu\nu}\widehat{\cal
E}^{\mu\nu\alpha\beta}h_{\alpha\beta}+\kappa_{f1}f_{\mu\nu}\widehat{\cal
E}^{\mu\nu\alpha\beta}f_{\alpha\beta}\right)-\mu_{h1}(h_{\mu\nu}h^{\mu\nu}-h^{2})-\frac{1}{4\mu_{f2}}\left(n_{2}h+2\mu_{f2}f\right)^{2}\,.$
(115)

One can check that this theory is invariant under the gauge transformation

$\displaystyle h_{\mu\nu}$ $\displaystyle\to$
$\displaystyle{\widetilde{h}}_{\mu\nu}=h_{\mu\nu}\,,$ (116) $\displaystyle
f_{\mu\nu}$ $\displaystyle\to$
$\displaystyle{\widetilde{f}}_{\mu\nu}=f_{\mu\nu}+\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}\qquad{\rm
with}\qquad\partial^{\mu}\xi_{\mu}=0\,.$ (117)

Again, due to the transverse condition in the gauge transformation, this
theory is different from the linearized Hassan-Rosen bigravity.

####  V.3.2 Class III

In this case, we have the same primary constraints as well as the same
Hamiltonian as before with the only exception that $\mu_{f2}=n_{2}=0$ and
hence the subsequent constraints are the same. To summarize, we have

$\displaystyle\eqref{HT4p}\quad:\quad$ $\displaystyle{\cal H}_{T}^{S}={\cal
H}^{S}+\lambda_{\alpha_{h}}{\cal
C}^{(1)}_{\alpha_{h}}+\lambda_{\alpha_{f}}{\cal
C}^{(1)}_{\alpha_{f}}+\lambda_{\beta_{h}}{\cal
C}^{(1)}_{\beta_{h}}+\lambda_{\beta_{f}}{\cal C}^{(1)}_{\beta_{f}}\,,$ (118)

and

$\displaystyle\eqref{primary4p}\quad:\quad$ $\displaystyle{\cal
C}^{(1)}_{\alpha_{h}}\equiv\pi_{\alpha_{h}}\approx 0\,,\quad{\cal
C}^{(1)}_{\alpha_{f}}\equiv\pi_{\alpha_{f}}\approx 0\,,\quad{\cal
C}^{(1)}_{\beta_{h}}\equiv\pi_{\beta_{h}}\approx 0\,,\quad{\cal
C}^{(1)}_{\beta_{f}}\equiv\pi_{\beta_{f}}\approx 0\,.$ (119)
$\displaystyle(\ref{4pC2ah}-\ref{4pC2bf})\quad:\quad$ $\displaystyle{\cal
C}^{(2)}_{\alpha_{h}}\approx 0\,,\qquad{\cal C}^{(2)}_{\alpha_{f}}\approx
0\,,\qquad{\cal C}^{(2)}_{\beta_{h}}\approx 0\,,\qquad{\cal
C}^{(2)}_{\beta_{f}}\approx 0\,.$ (120)

The Lagrange multipliers $\lambda_{\alpha_{h}}$ and $\lambda_{\beta_{h}}$ are
determined from the time-evolution of ${\cal C}^{(2)}_{\alpha_{h}}$ and ${\cal
C}^{(2)}_{\beta_{h}}$. This is because $\\{{\cal C}^{(2)}_{\alpha_{h}},{\cal
C}^{(1)}_{\alpha_{h}}\\}=-8(\mu_{h1}+\mu_{h2})$ and $\\{{\cal
C}^{(2)}_{\beta_{h}},{\cal C}^{(1)}_{\beta_{h}}\\}=4\mu_{h1}\neq 0$ with the
fact that other Poisson brackets with the primary constraints vanish. In
addition, the evolution of ${\cal C}^{(2)}_{\alpha_{f}}$ and ${\cal
C}^{(2)}_{\beta_{f}}$ does not yield a new constraint since

$\displaystyle\dot{{\cal C}}^{(2)}_{\alpha_{f}}=\\{{\cal
C}^{(2)}_{\alpha_{f}},{\cal H}^{S}_{T}\\}=-k{\cal C}^{(2)}_{\beta_{f}}\approx
0\,,$ (121)

and $\dot{{\cal C}}^{(2)}_{\beta_{f}}=\\{{\cal C}^{(2)}_{\beta_{f}},{\cal
H}^{S}_{T}\\}\approx 0$. Therefore, we find two DOFs in the scalar sector:

Scalar DOF $\displaystyle=$ $\displaystyle\frac{1}{2}\biggl{[}8\times
2-4\,(\textrm{2 primary \& 2 secondary)}-4\,(\textrm{2 primary \& 2
secondary)}\times 2\,(\textrm{first-class})\biggr{]}=2\,,$

since

$\displaystyle\rm{first\,class}\quad$ $\displaystyle:$
$\displaystyle\quad{\cal C}^{(1)}_{\alpha_{f}}\,,\qquad{\cal
C}^{(2)}_{\alpha_{f}}\,,\qquad{\cal C}^{(1)}_{\beta_{f}}\,,\qquad{\cal
C}^{(2)}_{\beta_{f}}\,,$ (123) $\displaystyle\rm{second\,class}\quad$
$\displaystyle:$ $\displaystyle\quad{\cal C}^{(1)}_{\alpha_{h}}\,,\qquad{\cal
C}^{(2)}_{\alpha_{h}}\,,\qquad{\cal C}^{(1)}_{\beta_{h}}\,,\qquad{\cal
C}^{(2)}_{\beta_{h}}\,.$ (124)

Now the only possible option to have a single DOF is to impose

$\displaystyle\mu_{h1}+\mu_{h2}=0\,,$ (125)

so that we obtain the tertiary constraint from $\alpha_{h}$. In this case the
tertiary constraint reads :

$\displaystyle{\cal C}^{(3)}_{\alpha_{h}}=\\{{\cal C}^{(2)}_{\alpha_{h}},{\cal
H}^{S}_{T}\\}=-8k\mu_{h1}\beta_{h}+k^{2}\pi_{{\cal E}_{h}}-{\mu_{h1}\pi_{{\cal
R}_{h}}\over 2\kappa_{h1}}\approx 0\,.$ (126)

Since ${\cal C}^{(3)}_{\alpha_{h}}$ does not commute with ${\cal
C}^{(1)}_{\beta_{h}}$, let us define

$\displaystyle\widetilde{\cal C}^{(3)}_{\alpha_{h}}={\cal
C}^{(3)}_{\alpha_{h}}+2k{\cal C}^{(2)}_{\beta_{h}}\,.$ (127)

The evolution of this constraint gives the quaternary constraint:

$\displaystyle\widetilde{\cal C}^{(4)}_{\alpha_{h}}$ $\displaystyle=$
$\displaystyle\\{\widetilde{\cal C}^{(3)}_{\alpha_{h}},{\cal
H}^{S}_{T}\\}={4\mu_{h1}\over\kappa_{h1}}\left[-3\mu_{h1}\alpha_{h}+2\mu_{h1}{\cal
E}_{h}+{2(k^{2}\kappa_{h1}-3\mu_{h1})}{\cal R}_{h}\right]\approx 0\,.$ (128)

The time-evolution of $\widetilde{\cal C}^{(4)}_{\alpha_{h}}$ determines the
Lagrange multiplier $\lambda_{\alpha_{h}}$. On the other hand the evolution of
the secondary constraints for $\alpha_{f}$, $\beta_{h}$ and $\beta_{f}$ do not
yield a new constraint. The evolution of ${\cal C}^{(2)}_{\alpha_{f}}$ and
${\cal C}^{(2)}_{\beta_{f}}$ are trivial since $\dot{{\cal
C}}^{(2)}_{\alpha_{f}}=\\{{\cal C}^{(2)}_{\alpha_{f}},{\cal
H}^{S}_{T}\\}=-k{\cal C}^{(2)}_{\beta_{f}}$ and $\dot{{\cal
C}}^{(2)}_{\beta_{f}}=\\{{\cal C}^{(2)}_{\beta_{f}},{\cal H}^{S}_{T}\\}=0$.
The time-evolution of ${\cal C}^{(2)}_{\beta_{h}}$ can be used to determine
the Lagrange multiplier, $\lambda_{\beta_{h}}$. Since

$\displaystyle\rm{first\,class}\quad$ $\displaystyle:$
$\displaystyle\quad{\cal C}^{(1)}_{\alpha_{f}}\,,\qquad{\cal
C}^{(2)}_{\alpha_{f}}\,,\qquad{\cal C}^{(1)}_{\beta_{f}}\,,\qquad{\cal
C}^{(2)}_{\beta_{f}}\,,$ (129) $\displaystyle\rm{second\,class}\quad$
$\displaystyle:$ $\displaystyle\quad{\cal C}^{(1)}_{\alpha_{h}}\,,\qquad{\cal
C}^{(2)}_{\alpha_{h}}\,,\qquad\widetilde{\cal
C}^{(3)}_{\alpha_{h}}\,,\qquad\widetilde{\cal
C}^{(4)}_{\alpha_{h}}\,,\qquad{\cal C}^{(1)}_{\beta_{h}}\,,\qquad{\cal
C}^{(2)}_{\beta_{h}}\,,$ (130)

we find

Scalar DOF $\displaystyle=$ $\displaystyle\frac{1}{2}\biggl{[}8\times
2-6\,(\textrm{2 primary \& 2 secondary \& 1 tertiary \& 1 quaternary\,)}$
(131) $\displaystyle\quad-4\,(\textrm{2 primary \& 2 secondary)}\times
2\,(\textrm{first-class})\biggr{]}=1\,.$

In this case
Class III :

$\displaystyle\kappa_{h2}=-\kappa_{h3}=2\kappa_{h4}=-2\kappa_{h1}\neq
0\,,\qquad\kappa_{f2}=-\kappa_{f3}=2\kappa_{f4}=-2\kappa_{f1}\neq 0\,,$
$\displaystyle
l_{1}=l_{2}=l_{3}=l_{4}=l_{5}=n_{1}=n_{2}=\mu_{f1}=\mu_{f2}=0\,,\qquad\mu_{h2}=-\mu_{h1}\,.$
(132)

The Lagrangian for class III is given by

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\left(\kappa_{h1}h_{\mu\nu}\widehat{\cal
E}^{\mu\nu\alpha\beta}h_{\alpha\beta}+\kappa_{f1}f_{\mu\nu}\widehat{\cal
E}^{\mu\nu\alpha\beta}f_{\alpha\beta}\right)-\mu_{h1}(h_{\mu\nu}h^{\mu\nu}-h^{2})\,.$
(133)

It is clear that this case corresponds to the linearized Hassan-Rosen
bigravity, (10).

##  VI summary

In this paper, we investigated a Lorentz invariant action for two rank-2
symmetric tensor fields $h_{\mu\nu}$ and $f_{\mu\nu}$. Based on the
Hamiltonian analysis, we classified theories with seven physical degrees of
freedom whose action consists of the most generic quadratic terms containing
up to two derivatives with respect to spacetime for each term. To simplify the
problem, we have utilized a field redefinition to reduce the model parameter
space. We then found three distinct classes of theories, which are not
connected by a linear field redefinition. In any cases, the Hamiltonian
structure in the tensor and vector sectors are the same, that is, one of the
fields behaves as massless, and the other has non-vanishing mass in dispersion
relations. The first theory, the Class I, contains three primary constraints
in the scalar sector and is invariant under the transverse diffeomorphism.
Furthermore, the kinetic terms for both fields do not take the form of
Einstein-Hilbert term even by the field redefinition, and the mass term no
longer has the Fierz-Pauli tuning. The Class II is also invariant under the
transverse diffeomorphism but contains four primary constraints differently
from the Class I. The kinetic terms for both fields are described by the
Einstein-Hilbert terms, and a new tuning parameter enters in the mass matrix
thanks to the transverse condition in the gauge transformation, which was
absent in the linearized Hassan-Rosen bigravity. The Class III is nothing but
the linearized Hassan-Rosen bigravity, which is invariant under the standard
diffeomorphism. Since we have reduced the model parameter space by the linear
field redefinition before the Hamiltonian analysis, a broader class of
theories can be obtained by the field redefinition, which could be different
theories depending on the matter coupling, although their Hamiltonian
properties and physical degrees of freedom does not change.

The transverse diffeomorphism appeared in the Class I and II can be
nonlinearized by introducing the unimodular condition $\det g=1$, where $g$ is
one of the metrics in bimetric gravity. Therefore, the first two class of
theories, Class I and Class II, might open a new window of finding extended
theories of massive bimetric gravity. In fact, if we linearize the Hassan-
Rosen bigravity with the unimodular condition, one is able to obtain a part of
the Class II, where all the mixing terms are switched off. Although such a
case is trivial because the unimodular condition brings just a cosmological
constant in the Einstein equation as the (massless) unimodular gravity, it
would be interesting to investigate weather nonlinear completions of the Class
II itself can be possible or not. Moreover, the nonlinearization of the Class
I would be also interesting.

###### Acknowledgements.

We would be grateful to Norihiro Tanahashi for the initial collaboration in
the early stage of this work. A.N. would also like to thank Takahiro Tanaka
for fruitful discussion and useful suggestion. This work was supported in part
by JSPS Grant-in-Aid for Scientific Research Nos. JP17K14304 (D.Y.),
JP19H01891 (A.N. and D.Y.) and 20H05852 (A.N.).

##  Appendix A Linear field redefinition

In this appendix, we consider the transformation of the action for the fields
$h_{\mu\nu}$ and $f_{\mu\nu}$ under a redefinition of them. The most generic
transformation linear in the fields is :

$\displaystyle
h_{\mu\nu}=\Omega_{h}\overline{h}_{\mu\nu}+\omega_{h}\overline{f}_{\mu\nu}+\left(\Gamma_{h}\overline{h}+\gamma_{h}\overline{f}\right)\eta_{\mu\nu}\,,$
(134) $\displaystyle
f_{\mu\nu}=\Omega_{f}\overline{f}_{\mu\nu}+\omega_{f}\overline{h}_{\mu\nu}+\left(\Gamma_{f}\overline{f}+\gamma_{f}\overline{h}\right)\eta_{\mu\nu}\,,$
(135)

where $\Omega_{h,f}$ and $\Gamma_{h,f}$ are constants,
$\overline{h},\overline{f}$ are the trace of
$\overline{h}_{\mu\nu}\,,\overline{f}_{\mu\nu}$ contracted by $\eta_{\mu\nu}$.
Since $\Omega_{h}$ and $\Omega_{f}$ only change the normalization for each
Lagrangian, we hereafter set $\Omega_{h}=\Omega_{f}=1$. Applying the
transformation to the generic action, one obtains :

$\displaystyle S$ $\displaystyle=\int{\rm d}^{4}x\Bigl{(}-{\overline{\cal
K}}_{h}^{\alpha\beta|\mu\nu\rho\sigma}h_{\mu\nu,\alpha}h_{\rho\sigma,\beta}-{\overline{\cal
K}}_{f}^{\alpha\beta|\mu\nu\rho\sigma}f_{\mu\nu,\alpha}f_{\rho\sigma,\beta}-{\overline{\cal
G}}^{\alpha\beta\mu\nu\rho\sigma}h_{\mu\nu,\alpha}f_{\rho\sigma,\beta}$
$\displaystyle\qquad\qquad\qquad-{\overline{\cal
M}}_{h}^{\mu\nu\rho\sigma}h_{\mu\nu}h_{\rho\sigma}-{\overline{\cal
M}}_{f}^{\mu\nu\rho\sigma}f_{\mu\nu}f_{\rho\sigma}-{\overline{\cal
N}}^{\mu\nu\rho\sigma}h_{\mu\nu}f_{\rho\sigma}\Bigr{)}\,,$ (136)

where the coefficients of the transformed Lagrangian read

$\displaystyle\overline{\kappa}_{h1}=\kappa_{h1}+\omega_{f}\left(l_{1}+\omega_{f}\kappa_{f1}\right)\,,$
(137)
$\displaystyle\overline{\kappa}_{h2}=\kappa_{h2}+\omega_{f}\left(l_{2}+\omega_{f}\kappa_{f2}\right)\,,$
(138)
$\displaystyle\overline{\kappa}_{h3}=2\Gamma_{h}\kappa_{h2}+\left(1+4\Gamma_{h}\right)\kappa_{h3}+\omega_{f}\Bigl{[}2\gamma_{f}\kappa_{f2}+\left(\omega_{f}+4\gamma_{f}\right)\kappa_{f3}\Bigr{]}$
$\displaystyle\quad\quad\quad+\left(\gamma_{f}+\Gamma_{h}\omega_{f}\right)l_{2}+\left(\omega_{f}+4\gamma_{f}\right)l_{3}+\omega_{f}\left(1+4\Gamma_{h}\right)l_{5}\,,$
(139)
$\displaystyle\overline{\kappa}_{h4}=2\Gamma_{h}\left(1+2\Gamma_{h}\right)\kappa_{h1}+\Gamma_{h}^{2}\kappa_{h2}+\Gamma_{h}\left(1+4\Gamma_{h}\right)\kappa_{h3}+\left(1+4\Gamma_{h}\right)^{2}\kappa_{h4}$
$\displaystyle\quad\quad\quad+2\gamma_{f}\left(\omega_{f}+2\gamma_{f}\right)\kappa_{f1}+\gamma_{f}^{2}\kappa_{f2}+\gamma_{f}\left(\omega_{f}+4\Gamma_{f}\right)\kappa_{f3}+\left(\omega_{f}+4\gamma_{f}\right)^{2}\kappa_{f4}$
$\displaystyle\quad\quad\quad+\Bigl{[}\gamma_{f}\left(1+4\Gamma_{h}\right)+\omega_{f}\Gamma_{h}\Bigr{]}l_{1}+\gamma_{f}\Gamma_{h}l_{2}+\Gamma_{h}\left(\omega_{f}+4\gamma_{f}\right)l_{3}+\left(1+4\Gamma_{h}\right)\left(\omega_{f}+4\gamma_{f}\right)l_{4}+\gamma_{f}\left(1+4\Gamma_{h}\right)l_{5}\,,$
(140)
$\displaystyle{\bar{\mu}}_{h1}=\mu_{h1}+\omega_{f}^{2}\mu_{f1}+\omega_{f}n_{1}\,,$
(141)
$\displaystyle{\bar{\mu}}_{h2}=2\Gamma_{h}\left(1+2\Gamma_{h}\right)\mu_{h1}+\left(1+4\Gamma_{h}\right)^{2}\mu_{h2}+\Bigl{[}2\gamma_{f}\left(\omega_{f}+2\gamma_{f}\right)\mu_{f1}+\left(\omega_{f}+4\gamma_{f}\right)^{2}\mu_{f2}\Bigr{]}$
$\displaystyle\quad\quad\quad+\Bigl{[}\Gamma_{h}\omega_{f}+(1+4\Gamma_{f})\gamma_{f}\Bigr{]}n_{1}+\left(1+4\Gamma_{h}\right)\left(\omega_{f}+4\gamma_{f}\right)n_{2}\,,$
(142)

and ${\bar{\kappa}}_{f1,f2,f3,f4}$ and ${\bar{\mu}}_{f1,f2}$ can be obtained
by replacing the labels $h$ and $f$. And also we find

$\displaystyle\overline{l}_{1}=\left(1+\omega_{h}\omega_{f}\right)l_{1}+2\omega_{h}\kappa_{h1}+2\omega_{f}\kappa_{f1}\,,$
(143)
$\displaystyle\overline{l}_{2}=\left(1+\omega_{h}\omega_{f}\right)l_{2}+2\omega_{h}\kappa_{h2}+2\omega_{f}\kappa_{f2}\,,$
(144)
$\displaystyle\overline{l}_{3}=\left(\Gamma_{f}+\omega_{f}\gamma_{h}\right)l_{2}+\left(1+4\Gamma_{f}\right)l_{3}+\omega_{f}\left(\omega_{h}+4\gamma_{h}\right)l_{5}$
$\displaystyle\quad\quad\quad+2\gamma_{h}\kappa_{h2}+\left(\omega_{h}+4\gamma_{h}\right)\kappa_{h3}+2\omega_{f}\Gamma_{f}\kappa_{f2}+\omega_{f}\left(1+4\Gamma_{f}\right)\kappa_{f3}\,,$
(145)
$\displaystyle\overline{l}_{4}=\left(\Gamma_{h}+\Gamma_{f}+4\Gamma_{h}\Gamma_{f}+\omega_{h}\gamma_{f}+\omega_{f}\gamma_{h}+4\gamma_{h}\gamma_{f}\right)l_{1}+\left(\Gamma_{h}\Gamma_{f}+\gamma_{h}\gamma_{f}\right)l_{2}+\Bigl{[}\Gamma_{h}\left(1+4\Gamma_{f}\right)+\gamma_{h}\left(\omega_{f}+4\gamma_{f}\right)\Bigr{]}l_{3}$
$\displaystyle\quad\quad\quad+\Bigl{[}\left(1+4\Gamma_{h}\right)\left(1+4\Gamma_{f}\right)+\left(\omega_{h}+4\gamma_{h}\right)\left(\omega_{f}+4\gamma_{f}\right)\Bigr{]}l_{4}+\Bigl{[}\Gamma_{f}\left(1+4\Gamma_{h}\right)+\gamma_{f}\left(\omega_{h}+4\gamma_{h}\right)\Bigr{]}l_{5}$
$\displaystyle\quad\quad\quad+2\Bigl{[}\gamma_{h}\left(1+4\Gamma_{h}\right)+\omega_{h}\Gamma_{h}\Bigr{]}\kappa_{h1}+2\gamma_{h}\Gamma_{h}\kappa_{h2}+\Bigl{[}\gamma_{h}\left(1+8\Gamma_{h}\right)+\omega_{h}\Gamma_{h}\Bigr{]}\kappa_{h3}+2\left(1+4\Gamma_{h}\right)\left(\omega_{h}+4\gamma_{h}\right)\kappa_{h4}$
$\displaystyle\quad\quad\quad+2\Bigl{[}\gamma_{f}\left(1+4\Gamma_{f}\right)+\omega_{f}\Gamma_{f}\Bigr{]}\kappa_{f1}+2\gamma_{f}\Gamma_{f}\kappa_{f2}+\Bigl{[}\gamma_{f}\left(1+8\Gamma_{f}\right)+\omega_{f}\Gamma_{f}\Bigr{]}\kappa_{f3}+2\left(1+4\Gamma_{f}\right)\left(\omega_{f}+4\gamma_{f}\right)\kappa_{f4}\,,$
(146)
$\displaystyle\overline{l}_{5}=\left(\Gamma_{h}+\omega_{h}\gamma_{f}\right)l_{2}+\omega_{h}\left(\omega_{f}+4\gamma_{f}\right)l_{3}+\left(1+4\Gamma_{h}\right)l_{5}$
$\displaystyle\quad\quad\quad+2\gamma_{f}\kappa_{f2}+\left(\omega_{f}+4\gamma_{f}\right)\kappa_{f3}+2\omega_{h}\Gamma_{h}\kappa_{h2}+\omega_{h}\left(1+4\Gamma_{h}\right)\kappa_{h3}\,,$
(147)
$\displaystyle\overline{n}_{1}=\left(1+\omega_{h}\omega_{f}\right)n_{1}+2\omega_{h}\mu_{h1}+2\omega_{f}\mu_{f1}\,,$
(148)
$\displaystyle\overline{n}_{2}=\left(\Gamma_{h}+\Gamma_{f}+4\Gamma_{h}\Gamma_{f}+\omega_{h}\gamma_{f}+\omega_{f}\gamma_{h}+4\gamma_{h}\gamma_{f}\right)n_{1}+\Bigl{[}\left(1+4\Gamma_{h}\right)\left(1+4\Gamma_{f}\right)+[\left(\omega_{h}+4\gamma_{h}\right)\left(\omega_{f}+4\gamma_{f}\right)\Bigr{]}n_{2}$
$\displaystyle\quad\quad\quad+2\Bigl{[}\gamma_{h}\left(1+4\Gamma_{h}\right)+\omega_{h}\Gamma_{h}\Bigr{]}\mu_{h1}+2\left(1+4\Gamma_{h}\right)\left(\omega_{h}+4\gamma_{h}\right)\mu_{h2}$
$\displaystyle\quad\quad\quad+2\Bigl{[}\gamma_{f}\left(1+4\Gamma_{f}\right)+\omega_{f}\Gamma_{f}\Bigr{]}\mu_{f1}+2\left(1+4\Gamma_{f}\right)\left(\omega_{f}+4\gamma_{f}\right)\mu_{f2}\,.$
(149)

The inverse transformation of the fields is given by

$\displaystyle{\overline{f}}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{\frac{4h_{\mu\nu}-4\omega_{h}f_{\mu\nu}-(h-\omega_{h}f)\eta_{\mu\nu}}{1-\omega_{h}\omega_{f}}+\frac{(1+4\Gamma_{f})h-(\omega_{h}+4\gamma_{h})f}{(1+4\Gamma_{h})(1+4\Gamma_{f})-(\omega_{f}+4\gamma_{f})(\omega_{h}+4\gamma_{h})}\eta_{\mu\nu}\right\\}\,$
(150) $\displaystyle{\overline{f}}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{\frac{4f_{\mu\nu}-4\omega_{f}h_{\mu\nu}-(f-\omega_{f}h)\eta_{\mu\nu}}{1-\omega_{h}\omega_{f}}+\frac{(1+4\Gamma_{h})f-(\omega_{f}+4\gamma_{f})h}{(1+4\Gamma_{h})(1+4\Gamma_{f})-(\omega_{f}+4\gamma_{f})(\omega_{h}+4\gamma_{h})}\eta_{\mu\nu}\right\\}\,,$
(151)

and that for the trace of the fields

$\displaystyle{\bar{h}}$ $\displaystyle=$
$\displaystyle\frac{(1+4\Gamma_{f})h-(\omega_{h}+4\gamma_{h})f}{1+4\Gamma_{h}+4\Gamma_{f}(1+4\Gamma_{h})-(\omega_{f}+4\gamma_{f})(\omega_{h}+4\gamma_{h})}\,,$
(152) $\displaystyle{\bar{f}}$ $\displaystyle=$
$\displaystyle\frac{(1+4\Gamma_{h})f-(\omega_{f}+4\gamma_{f})h}{1+4\Gamma_{h}+4\Gamma_{f}(1+4\Gamma_{h})-(\omega_{f}+4\gamma_{f})(\omega_{h}+4\gamma_{h})}\,.$
(153)

where the inverse transformation exists only when

$\displaystyle 1-\omega_{h}\omega_{f}$ $\displaystyle\neq$ $\displaystyle
0\,,$ (154) $\displaystyle
1+4\Gamma_{h}+4\Gamma_{f}(1+4\Gamma_{h})-(\omega_{f}+4\gamma_{f})(\omega_{h}+4\gamma_{h})$
$\displaystyle\neq$ $\displaystyle 0\,.$ (155)

###  A.1 Transformation under vector conditions

In this appendix, we show that one can impose (47) by using the field
redefinition, without loss of generality. Here, we consider a specific field
redefinition under the vector condition (44) to simplify the analysis. Let us
first consider the following transformation,

$\displaystyle
h_{\mu\nu}=\overline{h}_{\mu\nu}-\frac{l_{1}}{2\kappa_{h1}}\overline{f}_{\mu\nu}\,,\qquad
f_{\mu\nu}=\overline{f}_{\mu\nu}\,,$ (156)

and then one can find $\overline{l}_{1}=0$ in the transformed theories.
Moreover, in the case of $\kappa_{h3}\neq\kappa_{h1}$ and
$\kappa_{f3}\neq\kappa_{f1}$, if one considers the field transformation
defined as

$\displaystyle
h_{\mu\nu}=\overline{h}_{\mu\nu}-\frac{2\kappa_{h1}-\kappa_{h3}}{2(\kappa_{h1}-\kappa_{h3})}\overline{h}\,\eta_{\mu\nu}\,,\qquad
f_{\mu\nu}=\overline{f}_{\mu\nu}-\frac{2\kappa_{f1}-\kappa_{f3}}{2(\kappa_{f1}-\kappa_{f3})}\overline{f}\,\eta_{\mu\nu}\,,$
(157)

one can transform to the theories with
$\overline{\kappa}_{h3}=2\overline{\kappa}_{h1}$ and
$\overline{\kappa}_{f3}=2\overline{\kappa}_{f1}$ with the use of the first two
conditions of (44). Next, when one considers the following transformation:

$\displaystyle
h_{\mu\nu}=\overline{h}_{\mu\nu}+\left(-\frac{l_{3}}{\kappa_{h1}}\gamma_{f}\overline{h}+\gamma_{h}\overline{f}\right)\eta_{\mu\nu}\,,$
(158) $\displaystyle
f_{\mu\nu}=\overline{f}_{\mu\nu}+\left(-\frac{l_{5}}{\kappa_{f1}}\gamma_{h}\overline{f}+\gamma_{f}\overline{h}\right)\eta_{\mu\nu}\,,$
(159)

one can check that the transformed parameters still satisfy the conditions:
$\overline{\kappa}_{h3}=2\overline{\kappa}_{h1}$ and
$\overline{\kappa}_{f3}=2\overline{\kappa}_{f1}$. Under this conditions, using
the transformaltion Eqs. (158) and (159) in which only $\gamma_{f}$ is
considered, one finds

$\displaystyle\overline{l}_{3}+\overline{l}_{4}+\overline{l}_{5}=l_{3}+l_{4}+l_{5}+\biggl{[}8\left(\kappa_{f1}+\kappa_{f4}\right)-\frac{l_{3}(l_{3}+4l_{4}+4l_{5})}{\kappa_{h1}}\biggr{]}\gamma_{f}\,.$
(160)

Hence, performing the transformation Eqs. (158) and (159) with

$\displaystyle\gamma_{f}=-\frac{\kappa_{h1}(l_{3}+l_{4}+l_{5})}{l_{3}(l_{3}+4l_{4}+4l_{5})-8\kappa_{h1}(\kappa_{f1}+\kappa_{f4})}\,,\qquad\gamma_{h}=0\,,$
(161)

one can transform to the theories with
$\overline{\kappa}_{h3}=2\overline{\kappa}_{h1},\overline{\kappa}_{f3}=2\overline{\kappa}_{f1}$
and $\overline{l}_{3}+\overline{l}_{4}+\overline{l}_{5}=0$.

###  A.2 $3$ primary case in the scalar sector

In this appedix, we show that the conditions (58) can be imposed by the field
redefinition, without loss of generality. Let us first consider the first case
of (54) for the original theory described by $h_{\mu\nu}$. In order to simply
the Lagrangian, we impose

$\displaystyle\overline{l}_{1}=\overline{l}_{4}=0,\qquad\overline{l}_{3}+\overline{l}_{5}=0,\qquad\overline{\kappa}_{h3}=2\overline{\kappa}_{h1},\qquad\overline{\kappa}_{f3}=2\overline{\kappa}_{f1},\qquad\overline{n}_{1}=0\,.$
(162)

These conditions determine the coefficients of the field redefinition as
follows :

$\displaystyle\omega_{h}=-{n_{1}\over
2\mu_{h1}}\,,\qquad\omega_{f}={n_{1}\kappa_{h1}\over 2\mu_{h1}\kappa_{f1}}\,,$
$\displaystyle\Gamma_{h}=-\frac{-2l_{3}\mu_{h1}(l_{3}l_{5}n_{1}-2(l_{3}+l_{5})\mu_{h1}\kappa_{f1})+l_{3}n_{1}(l_{3}n_{1}+2\mu_{h1}\kappa_{f1})\kappa_{h1}+n_{1}^{2}\kappa_{f1}\kappa_{h1}^{2}}{8\mu_{h1}(-l_{3}n_{1}+2\mu_{h1}\kappa_{f1})(l_{3}l_{5}-\kappa_{f1}\kappa_{h1})}\,,\quad\Gamma_{f}=\frac{n_{1}(2l_{3}\mu_{h1}+n_{1}\kappa_{h1})}{8\mu_{h1}(2\mu_{h1}\kappa_{f1}-l_{3}n_{1})}\,,$
$\displaystyle\gamma_{h}={n_{1}\over
8\mu_{h1}}\,,\qquad\gamma_{f}=-\frac{\kappa_{h1}}{\kappa_{f1}}\frac{l_{3}^{2}l_{5}n_{1}^{2}+2l_{3}\mu_{h1}\kappa_{f1}(2\mu_{h1}\kappa_{f1}-l_{5}n_{1})+\kappa_{f1}(4l_{5}\mu_{h1}^{2}\kappa_{f1}+n_{1}(l_{5}n_{1}+2\mu_{h1}{\kappa_{f1}})\kappa_{h1})}{8\mu_{h1}(l_{3}n_{1}{-}2\mu_{h1}\kappa_{f1})(l_{3}l_{5}-\kappa_{f1}\kappa_{h1})}\,.$
(163)

Then the transformed Lagrangian satisfies

$\displaystyle\kappa_{h2}=-\kappa_{h3}=-2\kappa_{h1}\neq
0\,,\qquad\kappa_{f2}=-\kappa_{f3}=-2\kappa_{f1}\neq
0\,,\qquad\kappa_{f4}=-\kappa_{f1}-\frac{3l_{3}^{2}}{8\kappa_{h1}}$
$\displaystyle\ l_{1}=l_{2}=l_{4}=\mu_{f1}=n_{1}=0\,,\qquad l_{5}=-l_{3}\,.$
(164)

Here the bars are omitted. Thus the kinetic term for $f_{\mu\nu}$ is Einstein-
Hilbert term, and all kinetic interactions between $h$ and $f$ are absent in
this frame. Since these conditions are same as (58), we conclude that the
theories having the the first option of (54) are transformed to the theories
with (58).

Next let us consider the second case of (54). Imposing the same conditions
(162), we find the transformation,

$\displaystyle\omega_{h}=-{n_{1}\over
2\mu_{h1}}\,,\qquad\omega_{f}={n_{1}\kappa_{h1}\over 2\mu_{h1}\kappa_{f1}}\,,$
$\displaystyle\Gamma_{h}=-\frac{4l_{3}l_{5}^{2}\mu_{h1}^{2}+2l_{3}\mu_{h1}(-l_{5}n_{1}+2\mu_{h1}\kappa_{f1})\kappa_{h1}+n_{1}((l_{3}+l_{5})n_{1}+2\mu_{h1}\kappa_{f1})\kappa_{h1}^{2}}{8\mu_{h1}(2l_{5}\mu_{h1}-n_{1}\kappa_{h1})(l_{3}l_{5}-\kappa_{f1}\kappa_{h1})}\,,\qquad\Gamma_{f}=-{1\over
4}\,,$
$\displaystyle\gamma_{h}=\frac{l_{5}n_{1}+2\mu_{h1}\kappa_{f1}}{8l_{5}\mu_{h1}-4n_{1}\kappa_{h1}}\,,\quad\gamma_{f}=-\frac{\kappa_{h1}}{\kappa_{f1}}\frac{2l_{5}^{2}\mu_{h1}(l_{3}n_{1}-2\mu_{h1}\kappa_{f1})-(l_{5}(l_{3}+l_{5})n_{1}^{2}+2l_{5}\mu_{h1}n_{1}\kappa_{f1}+4\mu_{h1}^{2}\kappa_{f1}^{2})\kappa_{h1}}{8\mu_{h1}(2l_{5}\mu_{h1}-n_{1}\kappa_{h1})(l_{3}l_{5}-\kappa_{f1}\kappa_{h1})}\,,$
(165)

where the transformed Lagrangian satisfies (164) equivalent to (58).
Therefore, both the first and second option of the $3$ primary case (54) can
be mapped into (58).

###  A.3 $4$ primary case in the scalar sector

In this appendix, we show that the conditions (93) can be imposed using the
field redefinition without loss of generality. Let us now consider the $4$
primary case (56) for the original theory described by $h_{\mu\nu}$. In order
to simplify the Lagrangian, we here impose

$\displaystyle\overline{l}_{1}=0,\qquad\overline{\kappa}_{h3}=2\overline{\kappa}_{h1},\qquad\overline{\kappa}_{f3}=2\overline{\kappa}_{f1},\qquad\overline{\kappa}_{h4}=-\overline{\kappa}_{h1},\qquad\overline{\kappa}_{f4}=-\overline{\kappa}_{f1},\qquad\overline{\mu}_{f1}=0\,.$
(166)

These conditions provides the field redefinition with the following
coefficients :

$\displaystyle\omega_{h}=-{n_{1}\over
2\mu_{h1}},\qquad\omega_{f}={n_{1}\kappa_{h1}\over
2\mu_{h1}\kappa_{f1}},\qquad\Gamma_{h}=-\frac{2l_{3}l_{5}\mu_{h1}-l_{3}n_{1}\kappa_{h1}}{8\mu_{h1}(l_{3}l_{5}-\kappa_{f1}\kappa_{h1})},\qquad\Gamma_{f}=-\frac{2l_{3}l_{5}\mu_{h1}+l_{5}n_{1}\kappa_{h1}}{8\mu_{h1}(l_{3}l_{5}-\kappa_{f1}\kappa_{h1})},$
(167)
$\displaystyle\gamma_{h}=\frac{l_{3}l_{5}n_{1}+2l_{3}\mu_{h1}\kappa_{f1}}{8\mu_{h1}(l_{3}l_{5}-\kappa_{f1}\kappa_{h1})},\qquad\gamma_{f}=-\frac{\kappa_{h1}(l_{3}l_{5}n_{1}-2l_{5}\kappa_{f1}\mu_{h1})}{8\mu_{h1}\kappa_{f1}(l_{3}l_{5}-\kappa_{f1}\kappa_{h1})}\,.$
(168)

Then the transformed Lagrangian satisfies

$\displaystyle\kappa_{h2}=-\kappa_{h3}=2\kappa_{h4}=-2\kappa_{h1}\neq
0\,,\qquad\kappa_{f2}=-\kappa_{f3}=2\kappa_{f4}=-2\kappa_{f1}\neq 0\,,$
$\displaystyle\ l_{1}=l_{2}=l_{3}=l_{4}=l_{5}=0\,,\qquad\mu_{f1}=n_{1}=0\,.$
(169)

which is obviously equivalent to (93). Here we omit the bar of the
coefficients. Thus in the $4$ primary case, we can always map them into two
Einstein-Hilbert term with no kinetic interactions between $h$ and $f$.

##  Appendix B Lagrangian in the scalar sector

The Lagrangian for $h$ scalar perturbations is given by

$\displaystyle{\cal L}_{hh,\mathrm{kin}}^{S}$ $\displaystyle=$ $\displaystyle
4(\kappa_{h1}+\kappa_{h2}+\kappa_{h3}+\kappa_{h4})\dot{\alpha}_{h}^{2}-(2\kappa_{h1}+\kappa_{h2})\dot{\beta}_{h}^{2}+12(\kappa_{h1}+3\kappa_{h4})\dot{{\cal
R}}_{h}^{2}+4(\kappa_{h1}+\kappa_{h4})\dot{{\cal E}}_{h}^{2}$ (170)
$\displaystyle-4(\kappa_{h3}+2\kappa_{h4})\left(-3\dot{{\cal
R}}_{h}+\dot{{\cal
E}}_{h}\right)\dot{\alpha}_{h}-8(\kappa_{h1}+3\kappa_{h4})\dot{{\cal
R}}_{h}\dot{{\cal E}}_{h}\,,$ $\displaystyle{\cal L}_{hh,\mathrm{cross}}^{S}$
$\displaystyle=$
$\displaystyle-4\left[(\kappa_{h2}+\kappa_{h3})\dot{\alpha}_{h}+(\kappa_{h2}+3\kappa_{h3})\dot{{\cal
R}}_{h}-(\kappa_{h2}+\kappa_{h3})\dot{\cal E}_{h}\right]k\beta_{h}\,,$ (171)
$\displaystyle{\cal L}_{hh,\mathrm{mass}}^{S}$ $\displaystyle=$
$\displaystyle-4\Bigl{[}k^{2}(\kappa_{h1}+\kappa_{h4})+\mu_{h1}+\mu_{h2}\Bigr{]}\alpha_{h}^{2}+\Bigl{[}k^{2}(2\kappa_{h1}+\kappa_{h2})+2\mu_{h1}\Bigr{]}\beta_{h}^{2}$
(172)
$\displaystyle-4\Bigl{[}k^{2}(3\kappa_{h1}+\kappa_{h2}+3\kappa_{h3}+9\kappa_{h4})+3(\mu_{h1}+3\mu_{h2})\Bigr{]}{\cal
R}_{h}^{2}-4\Bigl{[}k^{2}(\kappa_{h1}+\kappa_{h2}+\kappa_{h3}+\kappa_{h4})+\mu_{h1}+\mu_{h2}\Bigr{]}{\cal
E}_{h}^{2}$
$\displaystyle-4\Bigl{[}\Bigl{(}k^{2}(\kappa_{h3}+6\kappa_{h4})+6\mu_{h2}\Bigr{)}{\cal
R}_{h}-\Bigl{(}k^{2}(\kappa_{h3}+2\kappa_{h4})+2\mu_{h2}\Bigr{)}{\cal
E}_{h}\Bigr{]}\alpha_{h}$
$\displaystyle+8\Bigl{[}k^{2}(\kappa_{h1}+\kappa_{h2}+2\kappa_{h3}+3\kappa_{h4})+(\mu_{h1}+3\mu_{h2})\Bigr{]}{\cal
R}_{h}{\cal E}_{h}\,.$

The Lagrangian for $f$ perturbations can be obtained by replacing the above
Lagrangian for $h$ with $f$.

$\displaystyle{\cal L}^{S}_{hf,{\rm kin}}$ $\displaystyle=$ $\displaystyle
4\left(l_{2}+l_{3}+l_{4}+l_{5}\right)\dot{\alpha}_{h}\dot{\alpha}_{f}-l_{2}\dot{\beta}_{h}\dot{\beta}_{f}+36l_{4}\dot{\cal
R}_{h}\dot{\cal R}_{f}+4l_{4}\dot{\cal E}_{h}\dot{\cal
E}_{f}+12\left(l_{3}+l_{4}\right)\dot{\alpha}_{h}\dot{\cal
R}_{f}+12\left(l_{4}+l_{5}\right)\dot{\alpha}_{f}\dot{\cal R}_{h}$ (173)
$\displaystyle-4\left(l_{3}+l_{4}\right)\dot{\alpha}_{h}\dot{\cal
E}_{f}-4\left(l_{4}+l_{5}\right)\dot{\alpha}_{f}\dot{\cal
E}_{h}-12l_{4}\left(\dot{\cal R}_{h}\dot{\cal E}_{f}+\dot{\cal R}_{f}\dot{\cal
E}_{h}\right)\,,$ $\displaystyle{\cal L}^{S}_{hf,{\rm cross}}$
$\displaystyle=$
$\displaystyle-2\left[\left(l_{2}+2l_{5}\right)\dot{\alpha}_{h}+\left(l_{2}+6l_{5}\right)\dot{\cal
R}_{h}-\left(l_{2}+2l_{5}\right)\dot{\cal E}_{h}\right]k\beta_{f}$ (174)
$\displaystyle-2\left[\left(l_{2}+2l_{3}\right)\dot{\alpha}_{f}+\left(l_{2}+6l_{3}\right)\dot{\cal
R}_{f}-\left(l_{2}+2l_{3}\right)\dot{\cal E}_{f}\right]k\beta_{h}\,,$
$\displaystyle{\cal L}^{S}_{hf,{\rm mass}}$ $\displaystyle=$
$\displaystyle-4\left(k^{2}l_{4}+n_{1}+n_{2}\right)\alpha_{h}\alpha_{f}+\left(k^{2}l_{2}+2n_{1}\right)\beta_{h}\beta_{f}-4\left[k^{2}(l_{2}+3l_{3}+9l_{4}+3l_{5})+3n_{1}+9n_{2}\right]{\cal
R}_{h}{\cal R}_{f}$ (175)
$\displaystyle-4\left[k^{2}(l_{2}+l_{3}+l_{4}+l_{5})+n_{1}+n_{2}\right]{\cal
E}_{h}{\cal E}_{f}-4\left[k^{2}(3l_{4}+l_{5})+3n_{2}\right)\alpha_{h}{\cal
R}_{f}-4\left(k^{2}(l_{3}+3l_{4})+3n_{2}\right]\alpha_{f}{\cal R}_{h}$
$\displaystyle+4\left[k^{2}(l_{4}+l_{5})+n_{2}\right]\alpha_{h}{\cal
E}_{f}+4\left[k^{2}(l_{3}+l_{4})+n_{2}\right]\alpha_{f}{\cal
E}_{h}+4\left[k^{2}(l_{2}+l_{3}+3l_{4}+3l_{5})+n_{1}+3n_{2}\right]{\cal
R}_{h}{\cal E}_{f}$
$\displaystyle+4\left[k^{2}(l_{2}+3l_{3}+3l_{4}+l_{5})+n_{1}+3n_{2}\right]{\cal
R}_{f}{\cal E}_{h}\,.$

##  Appendix C 2 primary case in vector sector

In this Appendix, we investigate the Hamiltonian analysis in the case of $2$
primary constraints in the vector sector, where

$\displaystyle\kappa_{f2}=-2\kappa_{f1}+\frac{l_{2}^{2}}{4(2\kappa_{h1}+\kappa_{h2})}\,,$
(176)

is satisfied. In this case, we have the following two primary constraints,
which is defined by

$\displaystyle{\cal
C}^{(1)}_{B^{f}_{i}}\equiv\pi_{B^{f}_{i}}-\frac{l_{2}}{2(2\kappa_{h1}+\kappa_{h2})}\pi_{B^{h}_{i}}\approx
0\,.$ (177)

Then we define the total Hamiltonian

$\displaystyle{\cal H}^{V}_{T}={\cal H}^{V}+\lambda_{B^{f}_{i}}{\cal
C}^{(1)}_{B^{f}_{i}}\,.$ (178)

The consistency of the primary constraints gives the secondary constraints

$\displaystyle{\cal C}^{(2)}_{B^{f}_{i}}$ $\displaystyle\equiv$
$\displaystyle\\{{\cal C}^{(2)}_{B^{f}_{i}}\,,{\cal
H}^{V}_{T}\\}={\left(4\mu_{f1}-\frac{{l_{2}}n_{1}}{2\kappa_{h1}+\kappa_{h2}}\right)B^{f}_{i}+2\left(n_{1}-\frac{{l_{2}}\mu_{h1}}{2\kappa_{h1}+\kappa_{h2}}\right)B^{h}_{i}-k\pi_{F^{f}_{i}}+\frac{kl_{2}}{2(2\kappa_{h1}+\kappa_{h2})}\pi_{F^{h}_{i}}}\,,$
(179)

and the time-evolution of the secondary constraints gives

$\displaystyle{\cal C}^{(3)}_{B^{f}_{i}}$ $\displaystyle\equiv$
$\displaystyle\\{{\cal C}^{(2)}_{B^{f}_{i}}\,,{\cal H}^{V}_{T}\\}=\\{{\cal
C}^{(2)}_{B^{f}_{i}}\,,{\cal H}^{V}\\}+\lambda_{B^{f}_{i}}\\{{\cal
C}^{(2)}_{B^{f}_{i}}\,,{\cal C}^{(1)}_{B^{f}_{i}}\\}\approx 0\,,$ (180)

where

$\displaystyle\\{{\cal C}^{(2)}_{B^{f}_{i}}\,,{\cal
C}^{(1)}_{B^{f}_{i}}\\}=4\mu_{f1}+\frac{l_{2}(l_{2}\mu_{h1}-2n_{1}(2\kappa_{h1}+\kappa_{h2}))}{(2\kappa_{h1}+\kappa_{h2})^{2}}\,.$
(181)

Therefore, when $\\{{\cal C}^{(2)}_{B^{f}_{i}}\,,{\cal
C}^{(1)}_{B^{f}_{i}}\\}\neq 0$, the Lagrange multipliers $\lambda_{B^{f}_{i}}$
are determined by the above equation, and the primary and secondary
constraints are second class. Therefore, the number of the physical DOFs in
the vector sector is $(8\times 2-4)/2=6$.

In order to further reduce the variable in the phase space, we need to impose
an extra condition. The only option here is $\\{{\cal
C}^{(2)}_{B^{f}_{i}}\,,{\cal C}^{(1)}_{B^{f}_{i}}\\}=0$, i.e.,

$\displaystyle\mu_{f1}=-\frac{l_{2}(l_{2}\mu_{h1}-2n_{1}(2\kappa_{h1}+\kappa_{h2}))}{4(2\kappa_{h1}+\kappa_{h2})^{2}}\,.$
(182)

Then, ${\cal C}^{(3)}_{B^{f}_{i}}$ serves as the tertiary constraints,

$\displaystyle{\cal C}^{(3)}_{B^{f}_{i}}$ $\displaystyle=$
$\displaystyle\frac{\left[n_{1}(2\kappa_{h1}+\kappa_{h2})-l_{2}\mu_{h1}\right]\left[{k}l_{2}F^{f}_{i}+2k(2\kappa_{h1}+\kappa_{h2})F^{h}_{i}-\pi_{B^{h}_{i}}\right]}{(2\kappa_{h1}+\kappa_{h2})^{2}}\approx
0\,,$ (183)

and, since $\\{{\cal C}^{(3)}_{B^{f}_{i}},{\cal C}^{(1)}_{B^{f}_{i}}\\}=0$,
subsequently we have quaternary constraints,

$\displaystyle{\cal C}^{(4)}_{B^{f}_{i}}$ $\displaystyle\equiv$
$\displaystyle\\{{\cal C}^{(3)}_{B^{f}_{i}}\,,{\cal
H}^{V}_{T}\\}=-\frac{\left[n_{1}(2\kappa_{h1}+\kappa_{h2})-l_{2}\mu_{h1}\right]\left[\left({k^{2}}l_{2}+2n_{1}\right)B^{f}_{i}+2\left(k^{2}(2\kappa_{h1}+\kappa_{h2})+2\mu_{h1}\right)B^{h}_{i}-k\pi_{F^{h}_{i}}\right]}{(2\kappa_{h1}+\kappa_{h2})^{2}}\approx
0\,.$

Now the time-evolution of the quaternary constraints gives

$\displaystyle{\dot{\cal C}^{(4)}_{B^{f}_{i}}}=\\{{\cal
C}^{(4)}_{B^{f}_{i}}\,,{\cal H}^{V}_{T}\\}=\\{{\cal
C}^{(4)}_{B^{f}_{i}}\,,{\cal H}^{V}\\}+\lambda_{B_{f,i}}\\{{\cal
C}^{(4)}_{B_{f,i}}\,,{\cal C}^{(1)}_{B^{f}_{i}}\\}\approx 0\,,$ (185)

where

$\displaystyle\\{{\cal C}^{(4)}_{B^{f}_{i}}\,,{\cal
C}^{(1)}_{B^{f}_{i}}\\}=-\frac{2\left(n_{1}(2\kappa_{h1}+\kappa_{h2})-l_{2}\mu_{h1}\right)^{2}}{(2\kappa_{h1}+\kappa_{h2})^{3}}\,.$
(186)

Therefore, as long as $\\{{\cal C}^{(4)}_{B^{f}_{i}}\,,{\cal
C}^{(1)}_{B^{f}_{i}}\\}$ is non-vanishing, the Lagrange multiplier are
determined by the above equations, and the number of the physical DOFs is
$(8\times 2-8)/2=4$.

When $n_{1}(2\kappa_{h1}+\kappa_{h2})-l_{2}\mu_{h1}=0$, the above tertiary
constraint trivially vanishes and its time-evolution does not generate the
independent constraint. In this case, we only have the primary and secondary
constraints, but now they are first class since all the primary and secondary
commute each other. Thus, the number of the physical DOFs is $(8\times
2-4\times 2)/2=4$. Therefore, the case with $2$ primary constraints in the
vector sector cannot have $2$ physical DOFs.

##  Appendix D 2 primary case : ${\rm det}\,{\cal K}^{S}\neq 0$ in scalar
sector

Let us consider the case with two primary constraints, namely the degenerate
condition for the scalar components is not imposed, and the parameters only
satisfies the vector conditions (44). We define the following two primary
constraints for convenience,

$\displaystyle{\cal C}^{(1)}_{\beta_{h}}\equiv\pi_{\beta_{h}}\approx
0\,,\qquad{\cal
C}^{(2)}_{\beta_{f}}\equiv\pi_{\beta_{f}}-\frac{n_{1}}{2\mu_{h1}}\pi_{\beta_{h}}\approx
0\,.$ (187)

The total Hamiltonian is defined as

$\displaystyle{\cal H}_{T}^{S}={\cal H}+\lambda_{\beta_{h}}{\cal
C}^{(1)}_{\beta_{h}}+\lambda_{\beta_{f}}{\cal C}^{(1)}_{\beta_{f}}\,.$ (188)

The evolution of the two primary constraints yields two secondary constraints:

$\displaystyle{\cal C}^{(2)}_{\beta_{h}}=\\{{\cal C}^{(1)}_{\beta_{h}},{\cal
H}_{T}^{S}\\}=-k\left(\pi_{\alpha_{h}}+\pi_{{\cal
E}_{h}}\right)+4\mu_{h1}\beta_{h}+2n_{1}\beta_{f}\approx 0\,,$ (189)
$\displaystyle{\cal C}^{(2)}_{\beta_{f}}=\\{\widetilde{\cal
C}^{(1)}_{\beta_{f}},{\cal H}_{T}^{S}\\}=-k(\pi_{\alpha_{f}}+\pi_{{\cal
E}_{f}})+\frac{kn_{1}}{2\mu_{h1}}(\pi_{\alpha_{h}}+\pi_{{\cal E}_{h}})\approx
0\,.$ (190)

Since $\\{{\cal C}^{(2)}_{\beta_{h}},{\cal
C}^{(1)}_{\beta_{h}}\\}=4\mu_{h1}\neq 0$, the Lagrange multiplier
$\lambda_{\beta_{h}}$ is determined by imposing $\dot{\cal
C}^{(2)}_{\beta_{h}}\approx 0$, namely $\lambda_{\beta_{h}}\approx-\\{{\cal
C}^{(2)}_{\beta_{h}},{\cal H}\\}/\\{{\cal C}^{(2)}_{\beta_{h}},{\cal
C}^{(1)}_{\beta_{h}}\\}$. The evolution of the remaining secondary constraint
yields the tertiary constraint:

$\displaystyle{\cal C}^{(3)}_{\beta_{f}}\equiv\\{{\cal
C}^{(2)}_{\beta_{f}},{\cal
H}_{T}^{S}\\}={2k^{3}\Biggl{[}-2l_{5}\alpha_{h}+\frac{l_{3}n_{1}}{\mu_{h1}}\alpha_{f}-2\left(3l_{5}-\frac{2n_{1}\kappa_{h1}}{\mu_{h1}}\right){\cal
R}_{h}-\left(8\kappa_{f1}-\frac{3l_{3}n_{1}}{\mu_{h1}}\right){\cal
R}_{f}+2l_{5}{\cal E}_{h}-\frac{l_{3}n_{1}}{\mu_{h1}}{\cal
E}_{f}\Biggr{]}}\approx 0\,.$

Here, the tertiary constraint cannot be trivially zero since $\kappa_{f1}\neq
0$. One can also check that $\dot{\cal C}^{(3)}_{\beta_{f}}=k^{2}{\cal
C}^{(2)}_{\beta_{f}}\approx 0$, implying no more constraint is generated. The
constraints ${\cal C}^{(1,2,3)}_{\beta_{f}}$ commute with all other
constraints, and therefore, we have three first-class constraints ${\cal
C}^{(1,2,3)}_{\beta_{f}}$ and two second-class constraints ${\cal
C}^{(1,2)}_{\beta_{h}}$. Hence, the number of the physical DOFs is $(8\times
2-2-3\times 2)/2=4$. Since there is no further option to eliminate DOFs, one
cannot obtain $1$ DOF theory in this case.

##  Appendix E Explicit expression of constraints

In this appendix, we give explicit expression of the constraints in the case
of the Class I. Eq. (69) is given by

$\displaystyle{\cal C}^{(3)}_{\alpha_{f}}$ $\displaystyle\equiv$
$\displaystyle\\{{\cal C}^{(2)}_{\alpha_{f}},{\cal
H}_{T}^{S}\\}=-\frac{8kl_{3}\mu_{h1}}{\kappa_{h1}}\beta_{h}+c_{3}^{\alpha_{h}}\pi_{\alpha_{h}}+c_{3}^{{\cal
R}_{h}}\pi_{{\cal R}_{h}}+c_{3}^{{\cal R}_{f}}\pi_{{\cal
R}_{f}}+{k^{2}l_{3}\over\kappa_{h1}}\pi_{{\cal E}_{h}}+k^{2}\pi_{{\cal
E}_{f}}\approx 0\,,$ (192)

where

$\displaystyle c_{3}^{\alpha_{h}}$ $\displaystyle=$
$\displaystyle\frac{1}{4\kappa_{h1}}\biggl{[}l_{3}\left(k^{2}+\frac{6\mu_{h1}}{\kappa_{h1}}\right){-}(16\kappa_{h1}\kappa_{f1}+9l_{3}^{2})y\biggr{]}\,,$
(193) $\displaystyle c_{3}^{{\cal R}_{h}}$ $\displaystyle=$
$\displaystyle\frac{l_{3}}{4\kappa_{h1}}\left(k^{2}-\frac{2\mu_{h1}}{\kappa_{h1}}+3l_{3}y\right)\,,$
(194) $\displaystyle c_{3}^{{\cal R}_{f}}$ $\displaystyle=$ $\displaystyle-
l_{3}y\,,$ (195)

and

$\displaystyle
y=\frac{2n_{2}\kappa_{h1}+3l_{3}\mu_{h1}}{{2}\kappa_{h1}(8\kappa_{f1}(\kappa_{h1}+\kappa_{h4})+3l_{3}^{2})}\,.$
(196)

After rescaling, Eq. (70) is given by

$\displaystyle\widetilde{\cal C}^{(3)}_{\alpha_{f}}$ $\displaystyle\equiv$
$\displaystyle\\{{\cal C}^{(2)}_{\alpha_{f}},{\cal
H}_{T}^{S}\\}=\widetilde{c}_{3}^{\alpha_{h}}\pi_{\alpha_{h}}+\widetilde{c}_{3}^{{\cal
R}_{h}}\pi_{{\cal R}_{h}}+c_{3}^{{\cal R}_{f}}\pi_{{\cal
R}_{f}}-{k^{2}l_{3}\over\kappa_{h1}}\pi_{{\cal E}_{h}}\approx 0\,,$ (197)

where

$\displaystyle\widetilde{c}_{3}^{\alpha_{h}}=c_{3}^{\alpha_{h}}-\frac{5k^{2}l_{3}}{4\kappa_{h1}}=\frac{1}{4\kappa_{h1}}\biggl{[}l_{3}\left(-{4}k^{2}+\frac{6\mu_{h1}}{\kappa_{h1}}\right){-}(16\kappa_{h1}\kappa_{f1}+9l_{3}^{2})y\biggr{]}\,,$
(198) $\displaystyle\widetilde{c}_{3}^{{\cal R}_{h}}=c_{3}^{{\cal
R}_{h}}-\frac{k^{2}l_{3}}{4\kappa_{h1}}=\frac{l_{3}}{4\kappa_{h1}}\left(-\frac{2\mu_{h1}}{\kappa_{h1}}+3l_{3}y\right)\,.$
(199)

The time evolution of $\widetilde{\cal C}^{(3)}_{\alpha_{f}}$ yields the
constraint

$\displaystyle\dot{\widetilde{\cal
C}}{}^{(3)}_{\alpha_{f}}\equiv\widetilde{\cal
C}^{(4)}_{\alpha_{f}}=\widetilde{c}_{4}^{\alpha_{h}}\alpha_{h}+\widetilde{c}_{4}^{\alpha_{f}}\alpha_{f}+\widetilde{c}_{4}^{{\cal
E}_{h}}{\cal E}_{h}+\widetilde{c}_{4}^{{\cal E}_{f}}{\cal
E}_{f}+\widetilde{c}_{4}^{{\cal R}_{h}}{\cal R}_{h}+\widetilde{c}_{4}^{{\cal
R}_{f}}{\cal R}_{f}\approx 0\,,$ (200)

where

$\displaystyle\widetilde{c}_{4}^{\alpha_{h}}=8\left(4\kappa_{f1}(\kappa_{h1}+\kappa_{h4})+l_{3}^{2}\right)k^{2}y+2\left(\frac{3l_{3}}{\kappa_{h1}}(2\kappa_{h1}n_{2}+3l_{3}\mu_{h1})+16\kappa_{f1}(\mu_{h1}+\mu_{h2})\right)y-\frac{12l_{3}\mu_{h1}^{2}}{\kappa_{h1}^{2}}\,,$
(201)
$\displaystyle\widetilde{c}_{4}^{\alpha_{f}}=\frac{1}{\kappa_{h1}^{2}}\biggl{[}-2l_{3}(4\kappa_{h1}\kappa_{f1}+3l_{3}^{2})(2\kappa_{h1}k^{2}+3\mu_{h1})y+8\kappa_{f1}(2\kappa_{h1}n_{2}+3l_{3}\mu_{h1})y-2k^{2}l_{3}^{2}(2k^{2}\kappa_{h1}-\mu_{h1})\biggr{]}\,,$
(202) $\displaystyle\widetilde{c}_{4}^{{\cal
E}_{h}}=-8\left(4\kappa_{f1}(\kappa_{h1}+\kappa_{h4})+l_{3}^{2}\right)k^{2}y+2\left(\frac{3l_{3}}{\kappa_{h1}}(-2\kappa_{n1}n_{2}+l_{3}\mu_{h1})-16\kappa_{f1}\mu_{h2}\right)y+\frac{4l_{3}(2k^{2}\kappa_{h1}-\mu_{h1})\mu_{h1}}{\kappa_{h1}^{2}}\,,$
(203) $\displaystyle\widetilde{c}_{4}^{{\cal
E}_{f}}=\frac{4l_{3}(4\kappa_{h1}\kappa_{f1}+3l_{3}^{2})}{\kappa_{h1}}k^{2}y+2\left(-8\kappa_{f1}n_{2}+\frac{9l_{3}^{3}\mu_{h1}}{\kappa_{h1}^{2}}\right)y+\frac{2k^{2}l_{3}^{2}(2k^{2}\kappa_{h1}-\mu_{h1})}{\kappa_{h1}^{2}}\,,$
(204) $\displaystyle\widetilde{c}_{4}^{{\cal
R}_{h}}=8\left(4\kappa_{f1}(\kappa_{h1}+3\kappa
h4)-4l_{3}^{2}\right)k^{2}y+2\left(\frac{9l_{3}}{\kappa_{h1}}(2\kappa_{h1}n_{2}-l_{3}\mu_{h1})+48\kappa_{f1}\mu_{h2}\right)y$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-\frac{4l_{3}(4k^{4}\kappa_{h1}^{2}-2k^{2}\kappa_{h1}\mu_{h1}-3\mu_{h1}^{2})}{\kappa_{h1}^{2}}\,,$
(205) $\displaystyle\widetilde{c}_{4}^{{\cal
R}_{f}}=-\frac{4l_{3}(8\kappa_{h1}\kappa_{f1}+9l_{3}^{2})}{\kappa_{h1}}k^{2}y+6\left(8\kappa_{f2}n_{2}-\frac{9l_{3}^{2}\mu_{h1}}{\kappa_{h1}^{2}}\right)y-\frac{6k^{2}l_{3}^{2}(2k^{2}\kappa_{h1}-\mu_{h1})}{\kappa_{h1}^{2}}\,.$
(206)

Using other constraints, we then rewrite $\widetilde{\cal
C}^{(4)}_{\alpha_{f}}$ as

$\displaystyle\widetilde{\cal C}^{(4)}_{\alpha_{f}}=$
$\displaystyle\frac{2n_{2}\kappa_{h1}+3l_{3}\mu_{h1}}{\kappa_{h1}(8\kappa_{f1}(\kappa_{h1}+\kappa_{h4})+3l_{3}^{2})}\Biggl{[}\frac{l_{3}}{2}{\cal
C}^{(2)}_{\alpha_{f}}-\frac{6l_{3}^{3}\mu_{h1}^{2}+\kappa_{f1}^{2}\kappa_{f1}(k^{2}(2\kappa_{h1}n_{2}+3l_{3}\mu_{h1})-4n_{2}\mu_{h1})}{3kl_{3}^{2}\mu_{h1}^{2}}{\cal
C}^{(3)}_{\beta_{f}}$ (207)
$\displaystyle\quad-16\kappa_{f1}\left(\mu_{h2}+\frac{\kappa_{h1}^{2}n_{2}^{2}}{3l_{3}^{2}\mu_{h1}}+\frac{1}{4}\mu_{h1}\right)\left({\cal
E}_{h}-\alpha_{h}-3{\cal R}_{h}\right)\Biggr{]}$
$\displaystyle\quad+\frac{1}{\kappa_{h1}^{2}(2k^{2}\kappa_{h1}+3\mu_{h1})}\left(1-\frac{\kappa_{h1}\kappa_{f1}(2n_{2}\kappa_{h1}+3l_{3}\mu_{h1})^{2}}{3l_{3}^{2}\mu_{h1}^{2}(8\kappa_{f1}(\kappa_{h1}+\kappa_{h4})+3l_{3}^{2})}\right)\biggl{[}k\kappa_{h1}^{2}\left(2k^{2}\kappa_{h1}-\mu_{h1}\right){\cal
C}^{(3)}_{\beta_{f}}$
$\displaystyle\quad\quad+4\mu_{h1}\left(4k^{2}\kappa_{h1}^{2}n_{2}+3l_{3}\mu_{h1}^{2}\right)\left({\cal
E}_{h}-\alpha_{h}-3{\cal
R}_{h}\right)-48l_{3}\mu_{h1}^{2}\left(2k^{2}\kappa_{h1}{\cal
R}_{h}-\mu_{h1}\alpha_{h}\right)\biggr{]}\,.$

When we further impose the two additional conditions (73) and (86), the right-
hand-side of the above equation reduces to the linear combination of the
constraints, namely the time evolution of $\widetilde{\cal
C}^{(3)}_{\alpha_{f}}$ becomes trivial.

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