Given this value of $\tau$, the charged lepton masses can be reproduced by
adjusting $\alpha_{e}$, $\beta_{e}$ and $\gamma_{e}$, only three real
parameters $\Lambda_{2}/\Lambda_{1}$, $\Lambda_{3}/\Lambda_{1}$ and
$g^{2}v^{2}_{u}/\Lambda_{1}$ describe the entire neutrino sector including the
three neutrino masses and the lepton mixing matrix. The best fit values of the
free parameters are found to be

$\displaystyle\beta_{u}/\alpha_{u}=16.8133\,,~{}~{}\gamma_{u1}/\alpha_{u}=3993.01\,,~{}~{}\gamma_{u2}/\alpha_{u}=-1981.34\,,~{}~{}\alpha_{u}v_{u}=0.00019~{}\text{GeV}$
$\displaystyle\beta_{d}/\alpha_{d}=91.7322\,,~{}~{}\gamma_{d1}/\alpha_{d}=0.3093\,,~{}~{}\gamma_{d2}/\alpha_{d}=-2.0306\,,~{}~{}\alpha_{d}v_{d}=0.00034~{}\text{GeV}\,,$
$\displaystyle\beta_{e}/\alpha_{e}=12.0121\,,~{}~{}\gamma_{e}/\alpha_{e}=79.5546\,,~{}~{}\alpha_{e}v_{d}=0.2065\text{
MeV}\,,~{}~{}\Lambda_{2}/\Lambda_{1}=0.9517\,,$
$\displaystyle\Lambda_{3}/\Lambda_{1}=4.6129\,,~{}~{}g^{2}v^{2}_{u}/\Lambda_{1}=5.999~{}\text{eV}\,.$
(128)

The masses and mixing parameters of quarks and leptons are predicted to be

$\displaystyle\theta^{q}_{12}=0.23035\,,~{}~{}\theta^{q}_{13}=0.00327\,,~{}~{}\theta^{q}_{23}=0.03929\,,~{}~{}\delta^{q}_{CP}/^{\circ}=69.58\,,$
$\displaystyle
m_{u}/m_{c}=0.001919\,,~{}~{}m_{c}/m_{t}=0.00272\,,~{}~{}m_{d}/m_{s}=0.04950\,,~{}~{}m_{s}/m_{b}=0.01767\,,$
$\displaystyle\sin^{2}\theta^{l}_{12}=0.3347\,,~{}~{}\sin^{2}\theta^{l}_{13}=0.02253\,,~{}~{}\sin^{2}\theta^{l}_{23}=0.5810\,,$
$\displaystyle\delta^{l}_{CP}=1.118\pi\,,~{}~{}\alpha_{21}=1.067\pi\,,~{}~{}\alpha_{31}=1.163\pi\,,$
$\displaystyle
m_{1}=12.7458~{}\text{meV}\,,~{}~{}m_{2}=15.3738~{}\text{meV}\,,~{}~{}m_{3}=52.1731~{}\text{meV}\,,$
$\displaystyle\sum_{i}m_{i}=80.2927~{}\text{meV}\,,~{}~{}|m_{ee}|=2.3570~{}\text{meV}\,.$
(129)

We see that the solar mixing angle $\theta^{l}_{12}$ is within the $2\sigma$
experimental region, and all other observables fall in the $1\sigma$ ranges.
The sum of neutrino masses is determined to be 80.293 meV, this is compatible
with the latest bound $\sum_{i}m_{i}<120$ meV at $95\%$ confidence level from
Planck [72].

##  8 Conclusion

The homogeneous finite modular group $\Gamma^{\prime}_{N}$ provides new
opportunity for understanding the flavor structure of quarks and leptons based
on modular invariance. $\Gamma^{\prime}_{2}$ is identical to $\Gamma_{2}\cong
S_{3}$, $\Gamma^{\prime}_{N}$ is the double covering of the inhomogeneous
finite modular group $\Gamma_{N}$ for $N>2$, and $\Gamma_{N}$ is isomorphic to
the quotient of $\Gamma^{\prime}_{N}$ over its center $\\{I,-I\\}$, i.e.,
$\Gamma_{N}\cong\Gamma^{\prime}_{N}/\\{I,-I\\}$. It is notable that texture
zeros of fermion mass matrices can be naturally obtained from
$\Gamma^{\prime}_{N}$, and $\Gamma^{\prime}_{3}\cong T^{\prime}$ has been
studied in [23, 44]. In the present work, we have considered the modular group
$\Gamma^{\prime}_{4}\equiv S^{\prime}_{4}$ in the setup of modular invariance
approach.

The weight 1 modular forms of level 4 are constructed in terms of the Dedekind
eta function, and they can be arranged into a triplet
$Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}(\tau)=(Y_{1}(\tau),Y_{2}(\tau),Y_{3}(\tau))^{T}$
which transforms as $\mathbf{\hat{3}^{\prime}}$ of $S^{\prime}_{4}$. The
higher weight modular forms up to weight 6 are built from the tensor products
of $Y^{(1)}_{\mathbf{\hat{3}^{\prime}}}(\tau)$, and they are homogeneous
polynomials of $Y_{1,2,3}$. The odd weight modular forms can be decomposed
into the hatted representations $\mathbf{\hat{1}}$,
$\mathbf{\hat{1}}^{\prime}$, $\mathbf{\hat{2}}$, $\mathbf{\hat{3}}$ and
$\mathbf{\hat{3}}^{\prime}$ of $S^{\prime}_{4}$ while the even weight modular
forms can be organized into the other representations $\mathbf{1}$,
$\mathbf{1}^{\prime}$, $\mathbf{2}$, $\mathbf{3}$ and $\mathbf{3}^{\prime}$ in
common with $S_{4}$. The results are summarized in table 1. Solving the
consistency condition, we find the generalized CP transformation corresponding
to $\tau\rightarrow-\tau^{*}$ is $X_{\mathbf{r}}=\rho_{\mathbf{r}}(S)$ which
is a combination of the modular symmetry transformation $S$ and the canonical
CP transformation. All couplings in the Lagrangian would be real if the
generalized CP symmetry is imposed.

We perform a systematical analysis of $S^{\prime}_{4}$ modular models for
lepton masses and mixing with/without generalized CP. We assume that the left-
handed leptons transform as triplet of $S^{\prime}_{4}$, and the right-handed
charged leptons are assigned to singlets under $S^{\prime}_{4}$, and we
consider both the case where neutrino masses are described by the Weinberg
operator and the case where neutrino masses arise from the type I seesaw
mechanism. The charged lepton mass matrix can only take four possible forms in
table 2 and the forms of the neutrino mass matrices are summarized in table 3
if the weights of the involved modular forms are less than 4. The charged
lepton masses $m_{e}$, $m_{\mu}$ and $m_{\tau}$ are in a one-to-one
correspondence with the parameters $\alpha$, $\beta$ and $\gamma$ which can be
taken real without loosing generality. We look for phenomenologically viable
models with a minimal amount of free parameters. We find fifteen predictive
lepton models which can describe the neutrino masses, mixing angles and CP
violation phases in terms of five real parameters $|g_{2}/g_{1}|$,
$\arg{(g_{2}/g_{1})}$, $\texttt{Re}(\tau)$, $\texttt{Im}(\tau)$ and the
overall scale $g^{2}_{1}v_{u}^{2}/\Lambda$. If generalized CP symmetry is
imposed, the phase $\arg{(g_{2}/g_{1})}$ is restricted to be $0$ or $\pi$.
Thus only four real input parameters $g_{2}/g_{1}$, $\texttt{Re}(\tau)$,
$\texttt{Im}(\tau)$ and $g^{2}_{1}v_{u}^{2}/\Lambda$ are left, and we find six
out of the fifteen models can fit the charged lepton masses and neutrino
oscillation data very well, as shown in table 5. A remarkable feature of these
models is that the light neutrino masses can be very tiny, while the neutrino
masses are typically quasi-degenerate in previous models based on $\Gamma_{N}$
modular group.

We have extended the $S^{\prime}_{4}$ modular symmetry to quark sector,
different possible assignments (triplet, the direct sum of a doublet and a
singlet, or the direct sum of three singlets) of the quark fields under
$S^{\prime}_{4}$ are considered. Because of the rich structure of the
$S^{\prime}_{4}$ modular group, we find many models can accommodate the
observed patterns of quark masses and CKM mixing matrix. For illustration, we
select eight benchmark models in which all the best fit values of observables
fall in the $1\sigma$ experimental ranges. It is notable that the hierarchical
quark masses, quark mixing angles and CP violation phase can even be described
very well by models with only nine real parameters including real and
imaginary parts of the modulus $\tau$. Note that models at level 3 achieve
this with ten [33] or more free parameters [18, 21, 35].

Finally we present a quark-lepton unified model which can explain the masses
and mixing of quarks and leptons simultaneously for a common value of the
complex modulus $\tau$. The value of $\tau$ is mainly fixed by the precisely
measured quark masses and mixing, then the entire neutrino sector including
the three neutrino masses as well as the lepton mixing matrix only depends on
three real parameters $\Lambda_{2}/\Lambda_{1}$, $\Lambda_{3}/\Lambda_{1}$ and
$g^{2}v^{2}_{u}/\Lambda_{1}$. We summarize that $S^{\prime}_{4}$ modular
symmetry is a promising framework to understand the flavor structure of quarks
and leptons.

Note added: During the final preparations of this work, a paper [74] dealing
with the same topic appeared on the arXiv. We use different representation
basis of $S^{\prime}_{4}$, the Clebsch-Gordan coefficients in our basis is
simpler, and this basis is convenient to classify the $S^{\prime}_{4}$ modular
models. Modular forms of level $N=4$ up to weight 6 are constructed in this
work, and higher weight modular forms until weight 10 are given in [74]. The
authors of [74] present one Weinberg operator model and one type I seesaw
model, and the right-handed charged leptons $E^{c}$ are assigned to a triplet
of $S^{\prime}_{4}$ in [74]. We perform a systematical classification of
modular $S^{\prime}_{4}$ symmetry models for leptons with/without generalized
CP symmetry, $E^{c}$ are assumed to transform as singlets under
$S^{\prime}_{4}$ in this work. We also apply the $S^{\prime}_{4}$ modular
symmetry to the quark sector, and construct a quark-lepton unified model. Our
work significantly extends the model construction of [74].

## Acknowledgements

XGL and GJD are supported by the National Natural Science Foundation of China
under Grant Nos 11975224, 11835013 and 11947301. CYY acknowledges the China
Post-doctoral Science Foundation under Grant No. 2018M641621.

## Appendix

##  Appendix A Group theory of $S_{4}^{\prime}$

The double covering group of $S_{4}$ has 48 elements, and it can be generated
by three generators $S,T$ and $R$ satisfying the multiplication rules:

$S^{2}=R,\quad T^{4}=(ST)^{3}=1,\quad R^{2}=1,\quad RT=TR\,.$ (A.1)

After we input these multiplication rules in GAP [75], its group ID can be
determined as [48, 30]. Notice that $S_{4}$ is not a subgroup of
$S^{\prime}_{4}$, it is isomorphic to the quotient group of $S^{\prime}_{4}$
over $Z_{2}^{R}$, i.e. $S_{4}\cong S_{4}^{\prime}/Z_{2}^{R}$, where
$Z_{2}^{R}=\\{1,R\\}$ is a normal subgroup of $S_{4}^{\prime}$. All the
elements of $S_{4}^{\prime}$ group can be divided into 10 conjugacy classes:

$\displaystyle 1C_{1}$ $\displaystyle=$ $\displaystyle\\{1\\}\,,$
$\displaystyle 1C_{2}$ $\displaystyle=$ $\displaystyle\\{R\\}=(1C_{1})\cdot
R\,,$ $\displaystyle 3C_{2}$ $\displaystyle=$
$\displaystyle\left\\{T^{2},~{}ST^{2}S^{3},~{}(ST^{2})^{2}\right\\}\,,$
$\displaystyle 3C_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\left\\{T^{2}R,~{}ST^{2}S^{3}R,~{}(ST^{2})^{2}R\right\\}=(3C_{2})\cdot
R\,,$ $\displaystyle 8C_{3}$ $\displaystyle=$
$\displaystyle\left\\{ST,~{}TS,~{}(ST)^{2},~{}(TS)^{2},~{}TS^{3}T^{2},~{}T^{2}ST^{3},~{}T^{2}S^{3}T,~{}T^{3}ST^{2}\right\\}\,,$
$\displaystyle 6C_{4}$ $\displaystyle=$
$\displaystyle\left\\{S,~{}TST^{3},~{}T^{2}ST^{2},~{}T^{3}ST,~{}TST^{2}S^{3},~{}ST^{2}S^{3}T\right\\}\,,$
$\displaystyle 6C_{4}^{\prime}$ $\displaystyle=$
$\displaystyle\left\\{T,~{}ST^{2},~{}T^{2}S,~{}T^{3}S^{2},~{}TST,~{}STS^{3}\right\\}\,,$
$\displaystyle 6C_{4}^{\prime\prime}$ $\displaystyle=$
$\displaystyle\left\\{SR,~{}TST^{3}R,~{}T^{2}ST^{2}R,~{}T^{3}STR,~{}TST^{2}S^{3}R,~{}ST^{2}S^{3}TR\right\\}=(6C_{4})\cdot
R\,,$ $\displaystyle 6C_{4}^{\prime\prime\prime}$ $\displaystyle=$
$\displaystyle\left\\{TR,~{}ST^{2}R,~{}T^{2}SR,~{}T^{3}S^{2}R,~{}TSTR,~{}STS^{3}R\right\\}=(6C_{4}^{\prime})\cdot
R\,,$ $\displaystyle 8C_{6}$ $\displaystyle=$
$\displaystyle\left\\{STR,~{}TSR,~{}(ST)^{2}R,~{}(TS)^{2}R,~{}TS^{3}T^{2}R,~{}T^{2}ST^{3}R,~{}\right.$
(A.2) $\displaystyle\left.T^{2}S^{3}TR,~{}T^{3}ST^{2}R\right\\}=(8C_{3})\cdot
R\,,$

where $kC_{n}$ denotes a conjugacy class with $k$ elements of order $n$. Note
that one half of these conjugacy classes can be written as the product of the
other half with $R$. There are four one-dimensional irreducible
representations $\mathbf{1},\mathbf{1}^{\prime},\mathbf{\hat{1}}$ and
$\mathbf{\hat{1}}^{\prime}$, two two-dimensional irreducible representations
$\mathbf{2}$ and $\mathbf{\hat{2}}$, and four three-dimensional irreducible
representations $\mathbf{3},\mathbf{3}^{\prime},\mathbf{\hat{3}}$ and
$\mathbf{\hat{3}}^{\prime}$. We have summarized the explicit matrix
representations in table 7. In the representations $\mathbf{1}$,
$\mathbf{1}^{\prime}$, $\mathbf{2}$, $\mathbf{3}$ and $\mathbf{3}^{\prime}$,
the generator $R=1$ is identity matrix, the representation matrices of $S$ and
$T$ coincide with those of $S_{4}$ [29], consequently $S^{\prime}_{4}$ can not
be distinguished from $S_{4}$ in these representations since they are
represented by the same set of matrices. In the representations
$\mathbf{\hat{1}}$, $\mathbf{\hat{1}}^{\prime}$, $\mathbf{\hat{2}}$,
$\mathbf{\hat{3}}$ and $\mathbf{\hat{3}}^{\prime}$, the generator $R=-1$. The
character table of $S_{4}^{\prime}$ can be obtained directly as shown in table
8. Moreover, the Kronecker products between all irreducible representations
are given as follows:

$\displaystyle\mathbf{1}\otimes\mathbf{1}=\mathbf{1}^{\prime}\otimes\mathbf{1}^{\prime}=\mathbf{\hat{1}}\otimes\mathbf{\hat{1}}^{\prime}=\mathbf{1},\quad\mathbf{1}\otimes\mathbf{\hat{1}}=\mathbf{1}^{\prime}\otimes\mathbf{\hat{1}}^{\prime}=\mathbf{\hat{1}},$
$\displaystyle\mathbf{1}\otimes\mathbf{1}^{\prime}=\mathbf{\hat{1}}\otimes\mathbf{\hat{1}}=\mathbf{\hat{1}}^{\prime}\otimes\mathbf{\hat{1}}^{\prime}=\mathbf{1}^{\prime},\quad\mathbf{1}\otimes\mathbf{\hat{1}}^{\prime}=\mathbf{1}^{\prime}\otimes\mathbf{\hat{1}}=\mathbf{\hat{1}}^{\prime},$
$\displaystyle\mathbf{1}\otimes\mathbf{2}=\mathbf{1}^{\prime}\otimes\mathbf{2}=\mathbf{\hat{1}}\otimes\mathbf{\hat{2}}=\mathbf{\hat{1}}^{\prime}\otimes\mathbf{\hat{2}}=\mathbf{2},\quad\mathbf{1}\otimes\mathbf{\hat{2}}=\mathbf{1}^{\prime}\otimes\mathbf{\hat{2}}=\mathbf{\hat{1}}\otimes\mathbf{2}=\mathbf{\hat{1}}^{\prime}\otimes\mathbf{2}=\mathbf{\hat{2}},$
$\displaystyle\mathbf{1}\otimes\mathbf{3}=\mathbf{1}^{\prime}\otimes\mathbf{3}^{\prime}=\mathbf{\hat{1}}\otimes\mathbf{\hat{3}}^{\prime}=\mathbf{\hat{1}}^{\prime}\otimes\mathbf{\hat{3}}=\mathbf{3},\quad\mathbf{1}\otimes\mathbf{\hat{3}}=\mathbf{1}^{\prime}\otimes\mathbf{\hat{3}}^{\prime}=\mathbf{\hat{1}}\otimes\mathbf{3}=\mathbf{\hat{1}}^{\prime}\otimes\mathbf{3}^{\prime}=\mathbf{\hat{3}},$
$\displaystyle\mathbf{1}\otimes\mathbf{3}^{\prime}=\mathbf{1}^{\prime}\otimes\mathbf{3}=\mathbf{\hat{1}}\otimes\mathbf{\hat{3}}=\mathbf{\hat{1}}^{\prime}\otimes\mathbf{\hat{3}}^{\prime}=\mathbf{3}^{\prime},\quad\mathbf{1}\otimes\mathbf{\hat{3}}^{\prime}=\mathbf{1}^{\prime}\otimes\mathbf{\hat{3}}=\mathbf{\hat{1}}\otimes\mathbf{3}^{\prime}=\mathbf{\hat{1}}^{\prime}\otimes\mathbf{3}=\mathbf{\hat{3}}^{\prime},$
$\displaystyle\mathbf{2}\otimes\mathbf{2}=\mathbf{\hat{2}}\otimes\mathbf{\hat{2}}=\mathbf{1}\oplus\mathbf{1}^{\prime}\oplus\mathbf{2},\quad\mathbf{2}\otimes\mathbf{\hat{2}}=\mathbf{\hat{1}}\oplus\mathbf{\hat{1}}^{\prime}\oplus\mathbf{\hat{2}},$
$\displaystyle\mathbf{2}\otimes\mathbf{3}=\mathbf{2}\otimes\mathbf{3}^{\prime}=\mathbf{\hat{2}}\otimes\mathbf{\hat{3}}=\mathbf{\hat{2}}\otimes\mathbf{\hat{3}}^{\prime}=\mathbf{3}\oplus\mathbf{3}^{\prime},\quad\mathbf{2}\otimes\mathbf{\hat{3}}=\mathbf{2}\otimes\mathbf{\hat{3}}^{\prime}=\mathbf{\hat{2}}\otimes\mathbf{3}=\mathbf{\hat{2}}\otimes\mathbf{3}^{\prime}=\mathbf{\hat{3}}\oplus\mathbf{\hat{3}}^{\prime},$
$\displaystyle\mathbf{3}\otimes\mathbf{3}=\mathbf{3}^{\prime}\otimes\mathbf{3}^{\prime}=\mathbf{\hat{3}}\otimes\mathbf{\hat{3}}^{\prime}=\mathbf{1}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3}^{\prime},\quad\mathbf{3}\otimes\mathbf{\hat{3}}=\mathbf{3}^{\prime}\otimes\mathbf{\hat{3}}^{\prime}=\mathbf{\hat{1}}\oplus\mathbf{\hat{2}}\oplus\mathbf{\hat{3}}\oplus\mathbf{\hat{3}}^{\prime},$
$\displaystyle\mathbf{3}\otimes\mathbf{3}^{\prime}=\mathbf{\hat{3}}\otimes\mathbf{\hat{3}}=\mathbf{\hat{3}}^{\prime}\otimes\mathbf{\hat{3}}^{\prime}=\mathbf{1}^{\prime}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3}^{\prime},\quad\mathbf{3}\otimes\mathbf{\hat{3}}^{\prime}=\mathbf{3}^{\prime}\otimes\mathbf{\hat{3}}=\mathbf{\hat{1}}^{\prime}\oplus\mathbf{\hat{2}}\oplus\mathbf{\hat{3}}\oplus\mathbf{\hat{3}}^{\prime}.$
(A.3)

| $S$ | $T$ | $R$
---|---|---|---
$\mathbf{1},\mathbf{1^{\prime}}$ | $\pm 1$ | $\pm 1$ | $1$
$\mathbf{\hat{1}},\mathbf{\hat{1}^{\prime}}$ | $\pm i$ | $\mp i$ | $-1$
$\mathbf{2}$ | $\begin{pmatrix}{0}&{1}\\\ {1}&{0}\end{pmatrix}$ | $\begin{pmatrix}{0}&{\omega^{2}}\\\ {\omega}&{0}\end{pmatrix}$ | $\begin{pmatrix}{1}&{0}\\\ {0}&{1}\end{pmatrix}$
$\mathbf{\hat{2}}$ | $i\begin{pmatrix}{0}&{1}\\\ {1}&{0}\end{pmatrix}$ | $-i\begin{pmatrix}{0}&{\omega^{2}}\\\ {\omega}&{0}\end{pmatrix}$ | $-\begin{pmatrix}{1}&{0}\\\ {0}&{1}\end{pmatrix}$
$\mathbf{3},\mathbf{3^{\prime}}$ | $\pm\dfrac{1}{3}\begin{pmatrix}{1}&{-2}&{-2}\\\ {-2}&{-2}&{1}\\\ {-2}&{1}&{-2}\end{pmatrix}$ | $\pm\dfrac{1}{3}\begin{pmatrix}{1}&{-2\omega^{2}}&{-2\omega}\\\ {-2}&{-2\omega^{2}}&{\omega}\\\ {-2}&{\omega^{2}}&{-2\omega}\end{pmatrix}$ | $\begin{pmatrix}{1}&{0}&{0}\\\ {0}&{1}&{0}\\\ {0}&{0}&{1}\end{pmatrix}$
$\mathbf{\hat{3}},\mathbf{\hat{3}^{\prime}}$ | $\pm\dfrac{i}{3}\begin{pmatrix}{1}&{-2}&{-2}\\\ {-2}&{-2}&{1}\\\ {-2}&{1}&{-2}\end{pmatrix}$ | $\mp\dfrac{i}{3}\begin{pmatrix}{1}&{-2\omega^{2}}&{-2\omega}\\\ {-2}&{-2\omega^{2}}&{\omega}\\\ {-2}&{\omega^{2}}&{-2\omega}\end{pmatrix}$ | $-\begin{pmatrix}{1}&{0}&{0}\\\ {0}&{1}&{0}\\\ {0}&{0}&{1}\end{pmatrix}$
Table 7: The representation matrices of the generators $S,T$ and $R$ in the irreducible representations of $S_{4}^{\prime}$ in our working basis, where $\omega=e^{2\pi i/3}$. Classes | $1C_{1}$ | $1C_{2}$ | $3C_{2}$ | $3C_{2}^{\prime}$ | $8C_{3}$ | $6C_{4}$ | $6C_{4}^{\prime}$ | $6C_{4}^{\prime\prime}$ | $6C_{4}^{\prime\prime\prime}$ | $8C_{6}$
---|---|---|---|---|---|---|---|---|---|---
$G$ | $1$ | $R$ | $T^{2}$ | $T^{2}R$ | $ST$ | $S$ | $T$ | $SR$ | $TR$ | $STR$
$\mathbf{1}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$\mathbf{1^{\prime}}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-1$ | $1$
$\mathbf{\hat{1}}$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $i$ | $-i$ | $-i$ | $i$ | $-1$
$\mathbf{\hat{1}^{\prime}}$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $-i$ | $i$ | $i$ | $-i$ | $-1$
$\mathbf{2}$ | $2$ | $2$ | $2$ | $2$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $-1$
$\mathbf{\hat{2}}$ | $2$ | $-2$ | $-2$ | $2$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $1$
$\mathbf{3}$ | $3$ | $3$ | $-1$ | $-1$ | $0$ | $-1$ | $1$ | $-1$ | $1$ | $0$
$\mathbf{3^{\prime}}$ | $3$ | $3$ | $-1$ | $-1$ | $0$ | $1$ | $-1$ | $1$ | $-1$ | $0$
$\mathbf{\hat{3}}$ | $3$ | $-3$ | $1$ | $-1$ | $0$ | $-i$ | $-i$ | $i$ | $i$ | $0$
$\mathbf{\hat{3}^{\prime}}$ | $3$ | $-3$ | $1$ | $-1$ | $0$ | $i$ | $i$ | $-i$ | $-i$ | $0$
Table 8: Character table of $S_{4}^{\prime}$, the representative element of
each conjugacy class is given in the second row.

Corresponding to the above direct product rule, we give the Clebsch-Gordan
(CG) coefficients of $S_{4}^{\prime}$ one by one in our working basis. All CG
coefficients can be expressed in the form of $\alpha\otimes\beta$, we use
$\alpha_{i}(\beta_{i})$ to denote the component of the left (right) basis
vector $\alpha(\beta)$. For direct products involving singlet
$\mathbf{\mathstrut\mkern 2.5mu\hat{1}\mkern-11.0mu\raise
7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}^{(\prime)}$, their CG coefficients are as
follows

$\begin{array}[]{llll}\mathbf{\mathstrut\mkern 2.5mu\hat{1}\mkern-11.0mu\raise
7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}^{(\prime)}\otimes\mathbf{\mathstrut\mkern
2.5mu\hat{2}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}\rightarrow\mathbf{\mathstrut\mkern
2.5mu\hat{2}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}&\left\\{\begin{array}[]{c}{}\hfil\\\\[8.61108pt]
p=\mathrm{even}\\\\[43.05542pt] p=\mathrm{odd}\\\\[8.61108pt]
{}\hfil\end{array}\right.&\left.\begin{array}[]{l}\mathbf{1}\otimes\mathbf{2}\rightarrow\mathbf{2}\\\
\mathbf{\hat{1}}^{\prime}\otimes\mathbf{\hat{2}}\rightarrow\mathbf{2}\\\
\mathbf{1}\otimes\mathbf{\hat{2}}\rightarrow\mathbf{\hat{2}}\\\
\mathbf{\hat{1}}\otimes\mathbf{2}\rightarrow\mathbf{\hat{2}}\\\\[8.61108pt]
\mathbf{1}^{\prime}\otimes\mathbf{2}\rightarrow\mathbf{2}\\\
\mathbf{\hat{1}}\otimes\mathbf{\hat{2}}\rightarrow\mathbf{2}\\\
\mathbf{1}^{\prime}\otimes\mathbf{\hat{2}}\rightarrow\mathbf{\hat{2}}\\\
\mathbf{\hat{1}}^{\prime}\otimes\mathbf{2}\rightarrow\mathbf{\hat{2}}\\\
\end{array}\right\\}&\mathbf{\mathstrut\mkern 2.5mu\hat{2}\mkern-11.0mu\raise
7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}\sim\alpha\begin{pmatrix}(-1)^{p}\beta_{1}\\\
\beta_{2}\\\ \end{pmatrix}\\\ &&&\\\ \mathbf{\mathstrut\mkern
2.5mu\hat{1}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}^{(\prime)}\otimes\mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}^{(\prime)}\rightarrow\mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}^{(\prime)}&\left\\{\begin{array}[]{c}{}\hfil\\\\[81.8053pt]
p=\mathrm{even}\\\\[81.8053pt]
{}\hfil\end{array}\right.&\left.\begin{array}[]{l}\mathbf{1}\otimes\mathbf{3}\rightarrow\mathbf{3}\\\
\mathbf{1}^{\prime}\otimes\mathbf{3}^{\prime}\rightarrow\mathbf{3}\\\
\mathbf{\hat{1}}\otimes\mathbf{\hat{3}}^{\prime}\rightarrow\mathbf{3}\\\
\mathbf{\hat{1}}^{\prime}\otimes\mathbf{\hat{3}}\rightarrow\mathbf{3}\\\
\mathbf{1}\otimes\mathbf{3}^{\prime}\rightarrow\mathbf{3}^{\prime}\\\
\mathbf{1}^{\prime}\otimes\mathbf{3}\rightarrow\mathbf{3}^{\prime}\\\
\mathbf{\hat{1}}\otimes\mathbf{\hat{3}}\rightarrow\mathbf{3}^{\prime}\\\
\mathbf{\hat{1}}^{\prime}\otimes\mathbf{\hat{3}}^{\prime}\rightarrow\mathbf{3}^{\prime}\\\
\mathbf{1}\otimes\mathbf{\hat{3}}\rightarrow\mathbf{\hat{3}}\\\
\mathbf{1}^{\prime}\otimes\mathbf{\hat{3}}^{\prime}\rightarrow\mathbf{\hat{3}}\\\
\mathbf{\hat{1}}\otimes\mathbf{3}\rightarrow\mathbf{\hat{3}}\\\
\mathbf{\hat{1}}^{\prime}\otimes\mathbf{3}^{\prime}\rightarrow\mathbf{\hat{3}}\\\
\mathbf{1}\otimes\mathbf{\hat{3}}^{\prime}\rightarrow\mathbf{\hat{3}}^{\prime}\\\
\mathbf{1}^{\prime}\otimes\mathbf{\hat{3}}\rightarrow\mathbf{\hat{3}}^{\prime}\\\
\mathbf{\hat{1}}\otimes\mathbf{3}^{\prime}\rightarrow\mathbf{\hat{3}}^{\prime}\\\
\mathbf{\hat{1}}^{\prime}\otimes\mathbf{3}\rightarrow\mathbf{\hat{3}}^{\prime}\\\
\end{array}\right\\}&\mathbf{\mathstrut\mkern 2.5mu\hat{3}\mkern-11.0mu\raise
7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}^{(\prime)}\sim\alpha\begin{pmatrix}\beta_{1}\\\
\beta_{2}\\\ \beta_{3}\\\ \end{pmatrix}\end{array}$

where we have introduced the notation $p$ to distinguish between different
products, it makes the results more compact. The CG coefficients for the
direct product involving doublet $\mathbf{\mathstrut\mkern
2.5mu\hat{2}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}$ are as follows

$\begin{array}[]{llll}\mathbf{\mathstrut\mkern 2.5mu\hat{2}\mkern-11.0mu\raise
7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}\otimes\mathbf{\mathstrut\mkern
2.5mu\hat{2}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}\rightarrow\mathbf{\mathstrut\mkern
2.5mu\hat{1}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}\oplus\mathbf{\mathstrut\mkern
2.5mu\hat{1}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}^{\prime}\oplus\mathbf{\mathstrut\mkern
2.5mu\hat{2}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}&\left\\{\begin{array}[]{c}{}\hfil\\\\[-4.30554pt]
p=\mathrm{even}\\\ \\\\[2.15277pt] p=\mathrm{odd}\\\
\end{array}\right.&\left.\begin{array}[]{l}\mathbf{2}\otimes\mathbf{2}\rightarrow\mathbf{1}\oplus\mathbf{1^{\prime}}\oplus\mathbf{2}\\\
\mathbf{2}\otimes\mathbf{\hat{2}}\rightarrow\mathbf{\hat{1}}\oplus\mathbf{\hat{1}}^{\prime}\oplus\mathbf{\hat{2}}\\\\[8.61108pt]
\mathbf{\hat{2}}\otimes\mathbf{\hat{2}}\rightarrow\mathbf{1}\oplus\mathbf{1^{\prime}}\oplus\mathbf{2}\end{array}\right\\}&\begin{array}[]{l}\mathbf{\mathstrut\mkern
2.5mu\hat{1}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}\sim\alpha_{1}\beta_{2}+(-1)^{p}\alpha_{2}\beta_{1}\\\
\mathbf{\mathstrut\mkern 2.5mu\hat{1}\mkern-11.0mu\raise
7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}^{\prime}\sim\alpha_{1}\beta_{2}-(-1)^{p}\alpha_{2}\beta_{1}\\\
\mathbf{\mathstrut\mkern 2.5mu\hat{2}\mkern-11.0mu\raise
7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}\sim\begin{pmatrix}\alpha_{2}\beta_{2}\\\
(-1)^{p}\alpha_{1}\beta_{1}\\\ \end{pmatrix}\\\ \end{array}\\\\[8.61108pt]
&&&\\\ \mathbf{\mathstrut\mkern 2.5mu\hat{2}\mkern-11.0mu\raise
7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}\otimes\mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}^{(\prime)}\rightarrow\mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}\oplus\mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}^{\prime}&\left\\{\begin{array}[]{c}{}\hfil\\\\[8.61108pt]
p=\mathrm{even}\\\\[47.36096pt] p=\mathrm{odd}\\\\[17.22217pt]
\end{array}\right.&\left.\begin{array}[]{l}\mathbf{2}\otimes\mathbf{3}\rightarrow\mathbf{3}\oplus\mathbf{3}^{\prime}\\\
\mathbf{\hat{2}}\otimes\mathbf{\hat{3}}^{\prime}\rightarrow\mathbf{3}\oplus\mathbf{3}^{\prime}\\\
\mathbf{2}\otimes\mathbf{\hat{3}}\rightarrow\mathbf{\hat{3}}\oplus\mathbf{\hat{3}}^{\prime}\\\
\mathbf{\hat{2}}\otimes\mathbf{3}\rightarrow\mathbf{\hat{3}}\oplus\mathbf{\hat{3}}^{\prime}\\\\[8.61108pt]
\mathbf{2}\otimes\mathbf{3}^{\prime}\rightarrow\mathbf{3}\oplus\mathbf{3}^{\prime}\\\
\mathbf{\hat{2}}\otimes\mathbf{\hat{3}}\rightarrow\mathbf{3}\oplus\mathbf{3}^{\prime}\\\
\mathbf{2}\otimes\mathbf{\hat{3}}^{\prime}\rightarrow\mathbf{\hat{3}}\oplus\mathbf{\hat{3}}^{\prime}\\\
\mathbf{\hat{2}}\otimes\mathbf{3}^{\prime}\rightarrow\mathbf{\hat{3}}\oplus\mathbf{\hat{3}}^{\prime}\\\
\end{array}\right\\}&\begin{array}[]{l}\mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}\sim\begin{pmatrix}\alpha_{2}\beta_{3}+(-1)^{p}\alpha_{1}\beta_{2}\\\
\alpha_{2}\beta_{1}+(-1)^{p}\alpha_{1}\beta_{3}\\\
\alpha_{2}\beta_{2}+(-1)^{p}\alpha_{1}\beta_{1}\\\
\end{pmatrix}\\\\[21.52771pt] \mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}^{\prime}\sim\begin{pmatrix}\alpha_{2}\beta_{3}-(-1)^{p}\alpha_{1}\beta_{2}\\\
\alpha_{2}\beta_{1}-(-1)^{p}\alpha_{1}\beta_{3}\\\
\alpha_{2}\beta_{2}-(-1)^{p}\alpha_{1}\beta_{1}\\\ \end{pmatrix}\\\
\end{array}\end{array}$

The last case involves the direct product of $\mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}^{(\prime)}\otimes\mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 7.3194pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.99994pt$\scalebox{0.3}{\bf)}}}^{(\prime)}$, the CG coefficients are given as
follows

$\begin{array}[]{llll}\begin{array}[]{l}\mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 5.85553pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.19995pt$\scalebox{0.3}{\bf)}}}^{(\prime)}\otimes\mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 5.85553pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.19995pt$\scalebox{0.3}{\bf)}}}^{(\prime)}\rightarrow\\\
~{}~{}~{}\mathbf{\mathstrut\mkern 2.5mu\hat{1}\mkern-11.0mu\raise
5.85553pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.19995pt$\scalebox{0.3}{\bf)}}}^{(\prime)}\oplus\mathbf{\mathstrut\mkern
2.5mu\hat{2}\mkern-11.0mu\raise 5.85553pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.19995pt$\scalebox{0.3}{\bf)}}}\oplus\mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 5.85553pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.19995pt$\scalebox{0.3}{\bf)}}}\oplus\mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 5.85553pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.19995pt$\scalebox{0.3}{\bf)}}}^{\prime}\end{array}&\left\\{\begin{array}[]{c}{}\hfil\\\\[6.88889pt]
p=\mathrm{even}\\\\[51.66664pt] p=\mathrm{odd}\\\\[20.66666pt]
\end{array}\right.&\left.\begin{array}[]{l}\mathbf{3}\otimes\mathbf{3}\rightarrow\mathbf{1}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3}^{\prime}\\\
\mathbf{3}^{\prime}\otimes\mathbf{3}^{\prime}\rightarrow\mathbf{1}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3}^{\prime}\\\
\mathbf{\hat{3}}\otimes\mathbf{\hat{3}}^{\prime}\rightarrow\mathbf{1}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3}^{\prime}\\\
\mathbf{3}\otimes\mathbf{\hat{3}}\rightarrow\mathbf{\hat{1}}\oplus\mathbf{\hat{2}}\oplus\mathbf{\hat{3}}\oplus\mathbf{\hat{3}}^{\prime}\\\
\mathbf{3}^{\prime}\otimes\mathbf{\hat{3}}^{\prime}\rightarrow\mathbf{\hat{1}}\oplus\mathbf{\hat{2}}\oplus\mathbf{\hat{3}}\oplus\mathbf{\hat{3}}^{\prime}\\\\[6.88889pt]
\mathbf{3}\otimes\mathbf{3}^{\prime}\rightarrow\mathbf{1}^{\prime}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3}^{\prime}\\\
\mathbf{\hat{3}}\otimes\mathbf{\hat{3}}\rightarrow\mathbf{1}^{\prime}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3}^{\prime}\\\
\mathbf{\hat{3}}^{\prime}\otimes\mathbf{\hat{3}}^{\prime}\rightarrow\mathbf{1}^{\prime}\oplus\mathbf{2}\oplus\mathbf{3}\oplus\mathbf{3}^{\prime}\\\
\mathbf{3}\otimes\mathbf{\hat{3}}^{\prime}\rightarrow\mathbf{\hat{1}}^{\prime}\oplus\mathbf{\hat{2}}\oplus\mathbf{\hat{3}}\oplus\mathbf{\hat{3}}^{\prime}\\\
\mathbf{3}^{\prime}\otimes\mathbf{\hat{3}}\rightarrow\mathbf{\hat{1}}^{\prime}\oplus\mathbf{\hat{2}}\oplus\mathbf{\hat{3}}\oplus\mathbf{\hat{3}}^{\prime}\\\
\end{array}\right\\}&\begin{array}[]{l}\mathbf{\mathstrut\mkern
2.5mu\hat{1}\mkern-11.0mu\raise 5.85553pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.19995pt$\scalebox{0.3}{\bf)}}}^{(\prime)}\sim\alpha_{1}\beta_{1}+\alpha_{2}\beta_{3}+\alpha_{3}\beta_{2}\\\\[3.44444pt]
\mathbf{\mathstrut\mkern 2.5mu\hat{2}\mkern-11.0mu\raise
5.85553pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.19995pt$\scalebox{0.3}{\bf)}}}\sim\begin{pmatrix}(-1)^{p}(\alpha_{1}\beta_{3}+\alpha_{2}\beta_{2}+\alpha_{3}\beta_{1})\\\
\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}+\alpha_{3}\beta_{3}\\\
\end{pmatrix}\\\\[8.6111pt] \mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 5.85553pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.19995pt$\scalebox{0.3}{\bf)}}}\sim\begin{pmatrix}\alpha_{1}\beta_{1}-\alpha_{2}\beta_{3}+(-1)^{p}(\alpha_{3}\beta_{2}-\alpha_{1}\beta_{1})\\\
\alpha_{3}\beta_{3}-\alpha_{1}\beta_{2}+(-1)^{p}(\alpha_{2}\beta_{1}-\alpha_{3}\beta_{3})\\\
\alpha_{2}\beta_{2}-\alpha_{3}\beta_{1}+(-1)^{p}(\alpha_{1}\beta_{3}-\alpha_{2}\beta_{2})\\\
\end{pmatrix}\\\\[15.49998pt] \mathbf{\mathstrut\mkern
2.5mu\hat{3}\mkern-11.0mu\raise 5.85553pt\hbox{\scalebox{0.3}{\bf(}$\kern
3.19995pt$\scalebox{0.3}{\bf)}}}^{\prime}\sim\begin{pmatrix}\alpha_{1}\beta_{1}-\alpha_{2}\beta_{3}-(-1)^{p}(\alpha_{3}\beta_{2}-\alpha_{1}\beta_{1})\\\
\alpha_{3}\beta_{3}-\alpha_{1}\beta_{2}-(-1)^{p}(\alpha_{2}\beta_{1}-\alpha_{3}\beta_{3})\\\
\alpha_{2}\beta_{2}-\alpha_{3}\beta_{1}-(-1)^{p}(\alpha_{1}\beta_{3}-\alpha_{2}\beta_{2})\\\
\end{pmatrix}\\\ \end{array}\end{array}$

## References

  * [1] Particle Data Group Collaboration, M. Tanabashi et al., “Review of Particle Physics,” Phys. Rev. D98 no. 3, (2018) 030001.
  * [2] T2K Collaboration, K. Abe et al., “Constraint on the matter–antimatter symmetry-violating phase in neutrino oscillations,” Nature 580 no. 7803, (2020) 339–344, arXiv:1910.03887 [hep-ex].
  * [3] F. Feruglio and A. Romanino, “Neutrino Flavour Symmetries,” arXiv:1912.06028 [hep-ph].
  * [4] F. Feruglio, C. Hagedorn, and R. Ziegler, “Lepton Mixing Parameters from Discrete and CP Symmetries,” JHEP 07 (2013) 027, arXiv:1211.5560 [hep-ph].
  * [5] M. Holthausen, M. Lindner, and M. A. Schmidt, “CP and Discrete Flavour Symmetries,” JHEP 04 (2013) 122, arXiv:1211.6953 [hep-ph].
  * [6] J.-N. Lu and G.-J. Ding, “Alternative Schemes of Predicting Lepton Mixing Parameters from Discrete Flavor and CP Symmetry,” Phys. Rev. D 95 no. 1, (2017) 015012, arXiv:1610.05682 [hep-ph].
  * [7] C.-C. Li, J.-N. Lu, and G.-J. Ding, “Toward a unified interpretation of quark and lepton mixing from flavor and CP symmetries,” JHEP 02 (2018) 038, arXiv:1706.04576 [hep-ph].
  * [8] J.-N. Lu and G.-J. Ding, “Quark and lepton mixing patterns from a common discrete flavor symmetry with a generalized CP symmetry,” Phys. Rev. D98 no. 5, (2018) 055011, arXiv:1806.02301 [hep-ph].
  * [9] J.-N. Lu and G.-J. Ding, “Dihedral flavor group as the key to understand quark and lepton flavor mixing,” JHEP 03 (2019) 056, arXiv:1901.07414 [hep-ph].
  * [10] F. Feruglio, “Are neutrino masses modular forms?,” in From My Vast Repertoire …: Guido Altarelli’s Legacy, A. Levy, S. Forte, and G. Ridolfi, eds., pp. 227–266.  2019\.  arXiv:1706.08749 [hep-ph].
  * [11] T. Kobayashi, K. Tanaka, and T. H. Tatsuishi, “Neutrino mixing from finite modular groups,” Phys. Rev. D98 no. 1, (2018) 016004, arXiv:1803.10391 [hep-ph].
  * [12] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T. H. Tatsuishi, and H. Uchida, “Finite modular subgroups for fermion mass matrices and baryon/lepton number violation,” arXiv:1812.11072 [hep-ph].
  * [13] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, “Modular $S_{3}$ invariant flavor model in SU(5) GUT,” arXiv:1906.10341 [hep-ph].
  * [14] H. Okada and Y. Orikasa, “Modular $S_{3}$ symmetric radiative seesaw model,” Phys. Rev. D 100 no. 11, (2019) 115037, arXiv:1907.04716 [hep-ph].
  * [15] J. C. Criado and F. Feruglio, “Modular Invariance Faces Precision Neutrino Data,” arXiv:1807.01125 [hep-ph].
  * [16] T. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, “Modular $A_{4}$ invariance and neutrino mixing,” arXiv:1808.03012 [hep-ph].
  * [17] F. J. de Anda, S. F. King, and E. Perdomo, “$SU(5)$ Grand Unified Theory with $A_{4}$ Modular Symmetry,” arXiv:1812.05620 [hep-ph].
  * [18] H. Okada and M. Tanimoto, “CP violation of quarks in $A_{4}$ modular invariance,” arXiv:1812.09677 [hep-ph].
  * [19] P. P. Novichkov, S. T. Petcov, and M. Tanimoto, “Trimaximal Neutrino Mixing from Modular A4 Invariance with Residual Symmetries,” arXiv:1812.11289 [hep-ph].
  * [20] T. Nomura and H. Okada, “A modular $A_{4}$ symmetric model of dark matter and neutrino,” arXiv:1904.03937 [hep-ph].
  * [21] H. Okada and M. Tanimoto, “Towards unification of quark and lepton flavors in $A_{4}$ modular invariance,” arXiv:1905.13421 [hep-ph].
  * [22] T. Nomura and H. Okada, “A two loop induced neutrino mass model with modular $A_{4}$ symmetry,” arXiv:1906.03927 [hep-ph].
  * [23] X.-G. Liu and G.-J. Ding, “Neutrino Masses and Mixing from Double Covering of Finite Modular Groups,” JHEP 08 (2019) 134, arXiv:1907.01488 [hep-ph].
  * [24] G.-J. Ding, S. F. King, and X.-G. Liu, “Modular A4 symmetry models of neutrinos and charged leptons,” JHEP 09 (2019) 074, arXiv:1907.11714 [hep-ph].
  * [25] H. Okada and Y. Orikasa, “A radiative seesaw model in modular $A_{4}$ symmetry,” arXiv:1907.13520 [hep-ph].
  * [26] T. Nomura, H. Okada, and O. Popov, “A modular $A_{4}$ symmetric scotogenic model,” Phys. Lett. B 803 (2020) 135294, arXiv:1908.07457 [hep-ph].
  * [27] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, “$A_{4}$ lepton flavor model and modulus stabilization from $S_{4}$ modular symmetry,” Phys. Rev. D 100 no. 11, (2019) 115045, arXiv:1909.05139 [hep-ph]. [Erratum: Phys.Rev.D 101, 039904 (2020)].
  * [28] T. Asaka, Y. Heo, T. H. Tatsuishi, and T. Yoshida, “Modular $A_{4}$ invariance and leptogenesis,” JHEP 01 (2020) 144, arXiv:1909.06520 [hep-ph].
  * [29] G.-J. Ding, S. F. King, X.-G. Liu, and J.-N. Lu, “Modular $S_{4}$ and $A_{4}$ Symmetries and Their Fixed Points: New Predictive Examples of Lepton Mixing,” arXiv:1910.03460 [hep-ph].
  * [30] D. Zhang, “A modular $A_{4}$ symmetry realization of two-zero textures of the Majorana neutrino mass matrix,” Nucl. Phys. B 952 (2020) 114935, arXiv:1910.07869 [hep-ph].
  * [31] T. Nomura, H. Okada, and S. Patra, “An Inverse Seesaw model with $A_{4}$-modular symmetry,” arXiv:1912.00379 [hep-ph].
  * [32] T. Kobayashi, T. Nomura, and T. Shimomura, “Type II seesaw models with modular $A_{4}$ symmetry,” arXiv:1912.00637 [hep-ph].
  * [33] J.-N. Lu, X.-G. Liu, and G.-J. Ding, “Modular symmetry origin of texture zeros and quark lepton unification,” arXiv:1912.07573 [hep-ph].
  * [34] S. J. King and S. F. King, “Fermion Mass Hierarchies from Modular Symmetry,” arXiv:2002.00969 [hep-ph].
  * [35] H. Okada and M. Tanimoto, “Quark and lepton flavors with common modulus $\tau$ in $A_{4}$ modular symmetry,” arXiv:2005.00775 [hep-ph].
  * [36] J. T. Penedo and S. T. Petcov, “Lepton Masses and Mixing from Modular $S_{4}$ Symmetry,” Nucl. Phys. B939 (2019) 292–307, arXiv:1806.11040 [hep-ph].
  * [37] P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, “Modular $S_{4}$ Models of Lepton Masses and Mixing,” arXiv:1811.04933 [hep-ph].
  * [38] I. de Medeiros Varzielas, S. F. King, and Y.-L. Zhou, “Multiple modular symmetries as the origin of flavour,” arXiv:1906.02208 [hep-ph].
  * [39] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, “New $A_{4}$ lepton flavor model from $S_{4}$ modular symmetry,” JHEP 02 (2020) 097, arXiv:1907.09141 [hep-ph].
  * [40] S. F. King and Y.-L. Zhou, “Trimaximal TM1 mixing with two modular $S_{4}$ groups,” Phys. Rev. D 101 no. 1, (2020) 015001, arXiv:1908.02770 [hep-ph].
  * [41] J. C. Criado, F. Feruglio, and S. J. King, “Modular Invariant Models of Lepton Masses at Levels 4 and 5,” JHEP 02 (2020) 001, arXiv:1908.11867 [hep-ph].
  * [42] X. Wang and S. Zhou, “The minimal seesaw model with a modular S4 symmetry,” JHEP 05 (2020) 017, arXiv:1910.09473 [hep-ph].
  * [43] P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, “Modular $A_{5}$ Symmetry for Flavour Model Building,” arXiv:1812.02158 [hep-ph].
  * [44] G.-J. Ding, S. F. King, and X.-G. Liu, “Neutrino Mass and Mixing with $A_{5}$ Modular Symmetry,” arXiv:1903.12588 [hep-ph].
  * [45] G.-J. Ding, S. F. King, C.-C. Li, and Y.-L. Zhou, “Modular Invariant Models of Leptons at Level 7,” arXiv:2004.12662 [hep-ph].
  * [46] G.-J. Ding and F. Feruglio, “Testing Moduli and Flavon Dynamics with Neutrino Oscillations,” arXiv:2003.13448 [hep-ph].
  * [47] P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, “Generalised CP Symmetry in Modular-Invariant Models of Flavour,” arXiv:1905.11970 [hep-ph].
  * [48] A. Baur, H. P. Nilles, A. Trautner, and P. K. S. Vaudrevange, “Unification of Flavor, CP, and Modular Symmetries,” arXiv:1901.03251 [hep-th].
  * [49] B. S. Acharya, D. Bailin, A. Love, W. A. Sabra, and S. Thomas, “Spontaneous breaking of CP symmetry by orbifold moduli,” Phys. Lett. B357 (1995) 387–396, arXiv:hep-th/9506143 [hep-th]. [Erratum: Phys. Lett.B407,451(1997)].
  * [50] T. Dent, “CP violation and modular symmetries,” Phys. Rev. D64 (2001) 056005, arXiv:hep-ph/0105285 [hep-ph].
  * [51] J. Giedt, “CP violation and moduli stabilization in heterotic models,” Mod. Phys. Lett. A17 (2002) 1465–1473, arXiv:hep-ph/0204017 [hep-ph].
  * [52] A. Baur, H. P. Nilles, A. Trautner, and P. K. Vaudrevange, “A String Theory of Flavor and $\mathscr{CP}$,” Nucl. Phys. B 947 (2019) 114737, arXiv:1908.00805 [hep-th].
  * [53] R. de Adelhart Toorop, F. Feruglio, and C. Hagedorn, “Finite Modular Groups and Lepton Mixing,” Nucl. Phys. B858 (2012) 437–467, arXiv:1112.1340 [hep-ph].
  * [54] J. H. Bruinier, G. V. D. Geer, G. Harder, and D. Zagier, The 1-2-3 of Modular Forms.  Universitext. Springer Berlin Heidelberg, 2008.
  * [55] D. Schultz, “Notes on Modular Forms.” https://faculty.math.illinois.edu/~schult25/ModFormNotes.pdf, 2015.
  * [56] F. Diamond and J. M. Shurman, A first course in modular forms, vol. 228 of Graduate Texts in Mathematics.  Springer, 2005.
  * [57] S. Lang, Introduction to modular forms, vol. 222 of Graduate Texts in Mathematics.  Springer, 2012.
  * [58] S. Ferrara, D. Lust, A. D. Shapere, and S. Theisen, “Modular Invariance in Supersymmetric Field Theories,” Phys. Lett. B225 (1989) 363.
  * [59] S. Ferrara, D. Lust, and S. Theisen, “Target Space Modular Invariance and Low-Energy Couplings in Orbifold Compactifications,” Phys. Lett. B 233 (1989) 147–152.
  * [60] H. P. Nilles, S. Ramos-Sanchez, and P. K. Vaudrevange, “Eclectic Flavor Groups,” JHEP 02 (2020) 045, arXiv:2001.01736 [hep-ph].
  * [61] H. P. Nilles, S. Ramos-Sanchez, and P. K. Vaudrevange, “Lessons from eclectic flavor symmetries,” arXiv:2004.05200 [hep-ph].
  * [62] H. Ohki, S. Uemura, and R. Watanabe, “Modular Flavor Symmetry on Magnetized Torus,” arXiv:2003.04174 [hep-th].
  * [63] I. Esteban, M. C. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni, and T. Schwetz, “Global analysis of three-flavour neutrino oscillations: synergies and tensions in the determination of $\theta_{23},\delta_{CP}$, and the mass ordering,” arXiv:1811.05487 [hep-ph].
  * [64] G. Ross and M. Serna, “Unification and fermion mass structure,” Phys. Lett. B664 (2008) 97–102, arXiv:0704.1248 [hep-ph].
  * [65] KamLAND-Zen Collaboration, A. Gando et al., “Search for Majorana Neutrinos near the Inverted Mass Hierarchy Region with KamLAND-Zen,” Phys. Rev. Lett. 117 no. 8, (2016) 082503, arXiv:1605.02889 [hep-ex]. [Addendum: Phys.Rev.Lett. 117, 109903 (2016)].
  * [66] CUORE Collaboration, C. Alduino et al., “First Results from CUORE: A Search for Lepton Number Violation via $0\nu\beta\beta$ Decay of 130Te,” Phys. Rev. Lett. 120 no. 13, (2018) 132501, arXiv:1710.07988 [nucl-ex].
  * [67] EXO Collaboration, J. Albert et al., “Search for Neutrinoless Double-Beta Decay with the Upgraded EXO-200 Detector,” Phys. Rev. Lett. 120 no. 7, (2018) 072701, arXiv:1707.08707 [hep-ex].
  * [68] GERDA Collaboration, M. Agostini et al., “Improved Limit on Neutrinoless Double-$\beta$ Decay of 76Ge from GERDA Phase II,” Phys. Rev. Lett. 120 no. 13, (2018) 132503, arXiv:1803.11100 [nucl-ex].
  * [69] SNO+ Collaboration, S. Andringa et al., “Current Status and Future Prospects of the SNO+ Experiment,” Adv. High Energy Phys. 2016 (2016) 6194250, arXiv:1508.05759 [physics.ins-det].
  * [70] LEGEND Collaboration, N. Abgrall et al., “The Large Enriched Germanium Experiment for Neutrinoless Double Beta Decay (LEGEND),” AIP Conf. Proc. 1894 no. 1, (2017) 020027, arXiv:1709.01980 [physics.ins-det].
  * [71] nEXO Collaboration, J. Albert et al., “Sensitivity and Discovery Potential of nEXO to Neutrinoless Double Beta Decay,” Phys. Rev. C 97 no. 6, (2018) 065503, arXiv:1710.05075 [nucl-ex].
  * [72] Planck Collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” arXiv:1807.06209 [astro-ph.CO].
  * [73] S. Antusch and V. Maurer, “Running quark and lepton parameters at various scales,” JHEP 11 (2013) 115, arXiv:1306.6879 [hep-ph].
  * [74] P. Novichkov, J. Penedo, and S. Petcov, “Double Cover of Modular $S_{4}$ for Flavour Model Building,” arXiv:2006.03058 [hep-ph].
  * [75] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.10.1 (2019),.  http://www.gap-system.org/.


# An effective field theory of holographic dark energy

Chunshan Lin

###### Abstract

A general covariant local field theory of the holographic dark energy model is
presented. It turns out the low energy effective theory of the holographic
dark energy is the massive gravity theory whose graviton has 3 polarisations,
including one scalar mode and two tensor modes. The Compton wavelength is the
size of the future event horizon of the universe. The UV-IR correspondence in
the holographic dark energy model stems from the scalar graviton’s strong
coupling at the energy scale that marks the breaking down of the effective
field theory.

##  1 Introduction

The holographic dark energy model [1][2] is based on the simple idea that the
vacuum energy density arising from the quantum fluctuation of the UV-cut-off
quantum field theory should relate to the boundary surface of a system in the
way

$\displaystyle\rho_{\Lambda}\sim M_{p}^{2}L^{-2},$ (1.1)

where $M_{p}^{2}=8\pi G$ is the reduced Planck mass, $L$ is the size of a
system (thus $L^{2}$ is essentially the area of the boundary surface). This
relation between the energy density in a bulk and the area of its space-time
boundary surface is rooted in the holographic principle [6]. The argument that
leads to the above relation is the following. The zero-point energy diverges
quartically and thus the energy density scales as
$\rho_{\Lambda}\sim\Lambda^{4}$ if we simply cut-off the divergence at the UV
scale $\Lambda$ 111Noted that there are some works [3][4][5] that challenge
this perspective.. However, this simple scaling is violated when the total
energy in the system with size $L$, i.e. $L^{3}\Lambda^{4}$, approaches to the
mass of the black hole in the same size $LM_{p}^{2}$. In fact, the total
energy must be bounded by the mass of the black hole from above, as the
effective quantum field theory breaks down at the Schwarzschild radius scale.
This energy bound is stronger than the Bekenstein entropy bound [7] which
inspired the proposal of the holographic principle [6]. Nowadays it is widely
believed that the holographic principle is one of the most important
cornerstones of quantum gravity.

Assuming that the energy bound is saturated on the cosmological background,
one obtains the important UV-IR correspondence

$\displaystyle\Lambda\sim\sqrt{M_{p}/L}$ (1.2)

and the vacuum energy $\rho_{\Lambda}$ comparable to our current critical
energy density. The related idea was discussed in Ref. [8][9] with the Hubble
radius as the IR cut-off. However, as pointed out in Ref. [10], the resultant
equation of state is greater than $-1/3$ and thus the vacuum energy fails to
accelerate the cosmic expansion, if we simply take the Hubble radius $L=1/H$
as the IR cut-off. Later in the same year, by adopting the future event
horizon as the IR cut-off, an accelerated expanding solution in the Friedman
equation was eventually obtained in Ref.[2]. Interestingly, the cosmic
coincidence problem can also be resolved by inflation in this scenario,
provided the minimal number of inflationary e-foldings. Since then, the
holographic dark energy has drawn a lot of attention and has been widely
studied, see Ref. [11] for a comprehensive review on the topic. In passing, I
shall mention that it was pointed out very recently that the holographic dark
energy model may alleviate the Hubble tension problem [12].

Given the phenomenological success, however, one of key pieces for our
holographic jigsaw puzzle is still missing. Namely we do not know how to write
down the general covariant action for the holographic dark energy model. An
attempt was made in the unpublished work [13] in which a mini superspace
action was given. Nevertheless, due to the absence of the general covariant
action, one may easily spot several conceptual problems. For instance, the
evolution of our universe in the past is dependent on the one in the future.
This apparent causality violation may not imply the pathology of the model,
but rather that the low energy effective field theory is missing. This
causality violation may be partially addressed at the cosmological background
evolution level, given the mini superspace action [13]. However, it is still
not quite clear whether the local physics violates the causality, as the
perturbation theory is also missing222A preliminary analysis on the
perturbative stability was conducted in Ref. [14].

In the current work, I aim at finding this important missing piece for our
holographic jigsaw puzzle. As we will see in the remain of this paper, the low
energy effective field theory (EFT) for the holographic dark energy model is
actually a massive gravity theory. A graviton has three polarisations, namely
one scalar mode, and two tensor modes. The UV-IR correspondence eq. (1.2)
stems from the strong coupling of the scalar mode above the energy scale where
our effective field theory breaks down. One may get puzzled at this point as
the Poincare symmetry in 4 dimensional space-time implies a massive spin-2
particle has 5 polarisations (including helicity 0, $\pm 1$, $\pm 2$).
However, the massive gravity theory that we are going to engage is different
from the conventional one [15][16], as the global Lorentz invariance is broken
on the cosmological background and thus a graviton does not necessary to be
well equipped with 5 polarisations [17][18]. See Ref. [19][20] for two
comprehensive reviews on the topic of massive gravity.

This paper is organised in the following. I will start from the mini
superspace action in the section 2, and then write down the covariant action
by adopting the Stueckelberg trick. Upon writing down the general covariant
action, an Ostrogradsky ghost is spotted in the decoupling limit. The ghost is
eliminated in the section 3, in which the linear perturbation and Hamiltonian
structure are analyzed in detail. The conclusive remarks and outlooks are
given in the section 4.

##  2 From mini superspace to general covariance

We take the flat FLRW ansatz $ds^{2}=-N^{2}dt^{2}+a^{2}d\textbf{x}^{2}$ and
the following mini superspace action [13] as our starting point,

$\displaystyle S=\frac{M_{p}^{2}}{2}\int
dt\left[\sqrt{-g}\left(\mathcal{R}-\frac{2c}{a^{2}L^{2}}\right)-\lambda\left(\dot{L}+\frac{N}{a}\right)\right]+S_{m},$
(2.1)

where $M_{p}^{2}\equiv 8\pi G$ is the reduced Planck mass, $\mathcal{R}$ is
the 4 dimensional Ricci scalar, and $L$ is the variable with length dimension
and subject to the constraint equation enforced by the Lagrangian multiplier
$\lambda$, i.e. $\dot{L}=-N/a$. We may integrate this constraint equation from
the infinite past $-\infty$ to nowadays, namely

$\displaystyle
L=\int_{-\infty}^{t}\frac{-Ndt^{\prime}}{a(t^{\prime})}+L(-\infty),$ (2.2)

where $L(-\infty)$ is the initial condition in the infinite past, which
requires the input from new physics (such as quantum gravity) to fully
determine its value, in light of the Hawking-Penrose’s singularity theorem
[21]. Therefore $L(-\infty)$ remains unknown to me due to my ignorance and
obtuseness. On the other hand, one may try to take the integration from the
other side, namely

$\displaystyle
L=\int^{\infty}_{t}\frac{Ndt^{\prime}}{a(t^{\prime})}+L(+\infty),$ (2.3)

given the asymptotic solution in the infinite future $t=+\infty$. It turns out
$L(+\infty)=0$ for the asymptotic solution derived in Ref. [13] and later is
confirmed by the numerical computation in Ref. [22]. Therefore, $R_{h}\equiv
aL$ is exactly the size of the future event horizon333Mathematically a more
general solution to $\dot{L}=-N/a$ is
$L=\int_{t_{0}}^{t}\frac{-Ndt^{\prime}}{a(t^{\prime})}+L(t_{0}).$ We may input
the value of $L$ at the moment $t_{0}$ in the past by hand, and then treat it
as the initial condition of the equations of motion, which is equivalent to
introducing an assumption about the UV completion of the theory.. It follows
the energy density of the dark sector

$\displaystyle\rho_{\text{dark}}=M_{p}^{2}\left(\frac{c}{a^{2}L^{2}}+\frac{\lambda}{2a^{4}}\right),$
(2.4)

where the first term is the holographic term, and the second term is the dark
radiation as it scales as $a^{-4}$. In passing, I shall mention that in some
cases, for instance, a cyclic universe realised by means of some exotic matter
content, the future event horizon may not exist as the integral in the eq.
(2.3) may not converge to a finite and definite value. The effective field
theory of the holographic dark energy, which will be developed in the remains
of the this paper, does not apply to these cases. Therefore, we will only
focus on the universe models whose future event horizon has finite, definite
and non-vanishing size.

The general covariance can be recovered by Stueckelberging the mini superspace
action [23]. We may follow the following dictionary,

$\displaystyle\frac{1}{a^{2}}\to\frac{1}{3}g^{ij}\delta_{ij}\to
g^{\mu\nu}\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{b}\delta_{ab},$
$\displaystyle\qquad\text{where}\qquad\langle\phi^{a}\rangle=\frac{x^{i}\delta_{ia}}{\sqrt{3}},$
$\displaystyle L\to\varphi(t,\textbf{x}),$
$\displaystyle-\frac{\dot{L}^{2}}{N^{2}}\to
g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi,$
$\displaystyle\qquad\text{where}\qquad\langle\varphi\rangle=\varphi(t),$ (2.5)

where $\delta_{ab}$ is the metric of the Stueckelberg scalar field space,
$\delta_{ia}$ is the pullback mapping between the physical space-time and the
scalar field space, and $i,j,k...$ are adopted as 3 dimensional spatial
coordinate indices, while $a,b,c...$ are adopted as the field space indices.
These 4 Stueckelberg fields are not yet canonically normalized and thus they
are of the length dimension. The field space is flat, and respects the $SO(3)$
rotational invariance. In passing, I shall mention that the global Lorentz
invariance is broken in this scalar configuration. Nevertheless, one should
not be worried about this Lorentz-violating vacuum expectation values (VEVs)
of scalars as the Lorentz invariance is broken anyway on the cosmological
background. For instance, CMB is not invariant under a Lorentz boost.

Before writing down the general covariant action, which is pretty easy and
straightforward at this point, I shall remind readers the hierarchical
structure of the theory. We have the Planck scale as the genetic fundamental
scale in the first place, and a secondary fundamental scale for our EFT of the
dark sector, which is expected to be generated via the non-trivial VEV of the
time-like Stueckelberg field $\varphi$, namely
$\Lambda\sim\sqrt{M_{p}/\varphi}$. The low energy effective field theory that
I begin with is the following,

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left\\{\frac{1}{2}M_{p}^{2}\mathcal{R}-M_{p}^{2}\varphi^{-2}\left[\left(c+\lambda\right)\cdot\partial^{\mu}\phi^{a}\partial_{\mu}\phi^{b}\delta_{ab}+\lambda\partial^{\mu}\varphi\partial_{\mu}\varphi\right]\right\\}+S_{m},$
(2.6)

where $c$ is a constant, $\lambda$ is a Lagrangian multiplier, and $S_{m}$ is
the action of the matter sector which minimally couples to gravity. The theory
respects the $SO(3)$ rotational invariance, $Z_{2}$ symmetry, and the
rescaling invariance

$\displaystyle\phi^{a}\to\ell\cdot\phi^{a},\qquad\varphi\to\ell\cdot\varphi,$
(2.7)

where $\ell$ is a constant. The Lagrangian multiplier $\lambda$ generates a
constraint between the temporal Stueckelberg field and the spatial
Stueckelberg fields. A similar example of this sort can be found in the
generalized unimodular gravity [24]. The hierarchical structure of the theory
is manifest at the action level. One would expect that the hierarchy
disappears if we trace the cosmic evolution all the way backward in time to
the infinite past, and it was generated later as the universe expands. The
physical significance of the energy scale $\Lambda\sim\sqrt{M_{p}/\varphi}$
will be shown in the next section.

I adopt the FLRW ansatz, and the four Stueckelberg scalar fields take the
space-time VEVs in the eq. (2). The background equations of motion read,

$\displaystyle 3M_{p}^{2}H^{2}$ $\displaystyle=$
$\displaystyle\frac{c}{a^{2}\varphi^{2}}+\frac{\lambda}{2a^{4}}+\rho_{m},$
$\displaystyle-M_{p}^{2}\dot{H}$ $\displaystyle=$
$\displaystyle\frac{c}{3a^{2}\varphi^{2}}+\frac{\lambda}{3a^{4}}+\frac{1}{2}\left(\rho_{m}+p_{m}\right),$
(2.8)

and

$\displaystyle\dot{\varphi}=-\frac{1}{a},\qquad\qquad\dot{\lambda}=\frac{-4ca}{\varphi^{3}},$
(2.9)

where $\rho_{m}$ and $p_{m}$ are the energy density and pressure of the matter
sector, the lapse has been absorbed into the redefinition of time, namely
$Ndt\to dt$, and I have rescaled the Lagrangian multiplier
$\lambda\to\lambda\cdot\frac{\varphi^{2}}{4a^{2}M_{p}^{2}}$ for convenience.
All equations of motion presented in Ref. [13] have been reproduced444 The
constraint equation associated with the Lagrangian multiplier $\lambda$ has
two solutions $\dot{\varphi}=\pm 1/a$, which are equivalent due to the $Z_{2}$
symmetry of the theory. Without losing generality, we adopt the solution
$\dot{\varphi}=-1/a$.. Some detailed analyses about this set of equations,
including the analytical one and the numerical one, have already been
conducted in Ref. [13] and Ref. [22], and I shall not repeat it in my current
work.

Before moving on to study the local dynamics, I shall briefly comment on the
causality. The $apparent$ causality violation stems from our ignorance of the
UV completion of the theory, which compels us to integrate the differential
equation (2.9) backward from $t=\infty$ to the moment of interest, as the
asymptotic value of $\varphi$ at the infinite future is known to be
$\varphi(\infty)=0$ [13][22]. Assuming we had enough knowledge about the
quantum gravity and we were able to specify the initial condition at
$t=-\infty$, we might just integrate the differential equation from
$t=-\infty$ to the moment of interest, as we normally do, and there is no
apparent causality violation in this setup. The result obtained from this
setup should be equivalent to the one integrated backward from $t=\infty$. In
other words, the $apparent$ causality violation appears as an artifactual and
mathematical treatment for our equations of motion, rather than a real
phenomenon.

It is quite remarkable to see that the holographic dark energy, which is
$seemingly$ non local and causality-violating, actually originates from a
simple and well defined local field theory. The trick of the game is the
spontaneous symmetry breaking, namely that I start from the general covariant
theory, then 4 scalar fields take the non-trivial space-time VEVs and break
the spatial and temporal diffeomorphism invariance around this vacuum
solution. According the Goldstone theorem, there is a massless boson for each
generator of the symmetry that is broken. I shall start to introduce the
Goldstone excitations around this asymmetric state,

$\displaystyle\phi^{a}=\frac{1}{\sqrt{3}}\left(x^{i}\delta_{i}^{a}+\pi^{a}\right),\qquad\qquad\varphi=\varphi(t)+\pi^{0},$
(2.10)

where $\pi^{a}$ is the space-like Nambu-Goldstone boson, and $\pi^{0}$ is the
time-like one. Due to the $SO(3)$ rotational symmetry, we can decompose the
helicity $\pi^{a}=\delta^{i}_{a}\partial_{i}\pi+\hat{\pi}^{a}$, where $\pi$ is
the longitudinal mode, and $\hat{\pi}^{a}$ are two transverse modes satisfying
the transverse condition $\partial_{i}\hat{\pi}^{i}=0$.

In a gauge field theory, the Goldstone pions decouple from the gauge bosons at
the scale much shorter than the Compton wavelength. The physics becomes very
transparent in the decoupling limit. A similar decoupling limit can also be
adopted, where the Goldstone bosons decouple from the graviton, which is
regarded as a gauge boson in this case. The limit is defined by setting
$M_{p}^{2}\to\infty$ while keeping $\Lambda=\sqrt{M_{p}/\varphi}$ fixed (I
neglect the time dependence of $\varphi$ for the time being for the schematic
analysis). It turns out this decoupling limit is a very illuminating
perspective allowing us to take a quick peek at the microscopic dynamics of
the dark sector. Perturbatively expanding the action eq. (2.6) up to quadratic
order in the Goldstone excitations, I get the following Goldstone action in
the decoupling limit,

$\displaystyle
S_{\pi}\simeq\Lambda^{4}\int\frac{\lambda}{4}\left(\dot{\pi}^{0}\dot{\pi}^{0}-\partial_{i}\pi^{0}\partial_{i}\pi^{0}\right)+\frac{4c+\lambda}{12}\left(\partial_{i}\dot{\pi}\partial_{i}\dot{\pi}-\partial^{2}\pi\partial^{2}\pi\right)+\frac{1}{6}\delta\lambda\left(3\dot{\pi}^{0}-\partial^{2}\pi\right),$
(2.11)

where $\partial^{2}\equiv\partial_{i}\partial_{j}\delta^{ij}$. Seemingly so
far so good, but unfortunately there is a pitfall lies in the Goldstone
action. Taking variation w.r.t $\delta\lambda$ we get the constraint
$\partial^{2}\pi=3\dot{\pi}^{0}$. Inserting it back into the eq. (2.11), we
get a higher order temporal derivative term 555The operator $1/\partial^{2}$
should be understood in the momentum space.

$\displaystyle
S_{\pi}\supset\frac{12c+3\lambda}{4}\int\ddot{\pi}^{0}\frac{1}{\partial^{2}}\ddot{\pi}^{0}+...$
(2.12)

This term yields to a fourth order equation of motion, which requires 4
initial conditions to fully determine its evolution. It implies there are 2
scalar degrees of freedom in the system, instead of one. One of them is
actually the infamous Ostrogradsky ghost [25][26]. The ghost instability
spoils the validity of the low energy effective field theory, and renders the
theory inconsistent. I will devote the next section to kill the ghost.

##  3 ghost elimination and perturbation analysis

###  3.1 Stueckelberg fields and gauge transformation

I define the linear perturbation of our FLRW metric as follows,

$\displaystyle g_{00}$ $\displaystyle=$
$\displaystyle-N(t)^{2}\left(1+2\alpha\right),$ $\displaystyle g_{0i}$
$\displaystyle=$ $\displaystyle N(t)a(t)\left(\partial_{i}\beta+S_{i}\right),$
$\displaystyle g_{ij}$ $\displaystyle=$ $\displaystyle
a(t)^{2}\left[\delta_{ij}+2\psi\delta_{ij}+\partial_{i}\partial_{j}E+\frac{1}{2}\left(\partial_{i}F_{j}+\partial_{j}F_{i}\right)+\gamma_{ij}\right].$
(3.1)

where $\alpha,\beta,\psi$ and $E$ are the scalar perturbations, $S_{i}$ and
$F_{i}$ are the vector perturbations satisfying the transverse condition
$\partial_{i}S_{i}=\partial_{i}F_{i}=0$, and $\gamma_{ij}$ is the tensor
perturbation satisfying the transverse and traceless condition
$\gamma_{ii}=\partial_{i}\gamma_{ij}=0$.

Under the linear gauge transformation

$\displaystyle x^{\mu}\to x^{\mu}+\xi^{\mu}\left(t,\textbf{x}\right),$ (3.2)

these four Stueckelberg fields transform accordingly,

$\displaystyle\pi^{0}\to\pi^{0}+\dot{\varphi}\xi^{0},\qquad\pi^{a}\to\pi^{a}+\xi^{i}\delta_{i}^{a}.$
(3.3)

On the other hand, the vector $Z^{\mu}$ defined by [27]

$\displaystyle Z^{0}\equiv\frac{-a}{N}\beta+\frac{a^{2}}{2N^{2}}\dot{E},\qquad
Z^{i}\equiv\frac{1}{2}\delta^{ij}\left(\partial_{j}E+F_{j}\right),$ (3.4)

transforms in the same manner

$\displaystyle Z^{\mu}\to Z^{\mu}+\xi^{\mu}.$ (3.5)

Therefore, the combination $Z^{i}-\pi^{i}$ and $\dot{\varphi}Z^{0}-\pi^{0}$
are gauge invariant. It is very convenient to transform to the unitary gauge
by simply muting all Goldstone bosons, while keeping all perturbation
variables in the eq. (3.1). It can be achieved by adopting a proper
diffeomorphism $\xi^{\mu}(t,\textbf{x})$. In this gauge, the Goldstone bosons
are eaten by the graviton and the graviton develops a mass gap at the low
energy spectrum. The graviton turns into a massive spin-2 particle with at
most 5 polarizations (it can be less), if the Ostrogradsky ghost had been
eliminated (which will be done soon!).

On the other hand, we can also transform back to the Goldstone bosons’ gauge
by muting the vector $Z^{\mu}$, which can also be done by adopting a proper
diffeomorphism $\xi^{\mu}$. Noted that the way back to the Goldstone bosons’
gauge is not unique, we may choose to mute some other perturbation variables
in the eq. (3.1), based on their transformations under the diffeomprhism.

###  3.2 Hamiltonian analysis in the unitary gauge

The Hamiltonian analysis was introduced by P. Dirac in the 50s and 60s [28],
as a way of counting dynamical degrees of freedom, and quantizing mechanical
systems such as gauge theories (for busy/lazy readers see the appendix A in
the Ref. [29] for a digested version of the method).

It is more convenient to perform the Hamiltonian analysis in the unitary gauge
where $\pi^{0}=\pi^{i}=0$, as all terms introduced by Stueckelberg fields
appear only in the potential sector in the action and thus the Legendre
transformation can be easily done. In the unitary gauge, the action eq. (2.6)
reduces to (I set $M_{p}^{2}=1$ in this subsection)

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}\mathcal{R}-\frac{c}{3}\Lambda_{1}(t)g^{ij}\delta_{ij}+\frac{\lambda}{4}\left(\frac{\Lambda_{2}(t)}{N^{2}}-\frac{1}{3}g^{ij}\delta_{ij}\right)\right].$
(3.6)

where $\Lambda_{1}=\varphi(t)^{-2}$ and $\Lambda_{2}(t)=\dot{\varphi}(t)^{2}$,
namely $\varphi$ and $\dot{\varphi}$ should not be treated as a canonical
variable and its velocity, and instead, they should be treated as two time-
dependent functions subject to some constraint equations which will be derived
later. The reason is that In the unitary gauge, all 4 Stueckelberg fields are
eaten by the graviton, and turn themselves into ADM variables, or some space-
time functions, and cease to be the independent dynamical degrees of freedom.

The Hamiltonian can be obtained by performing the Legendre transformation. To
this end, I need to adopt the ADM decomposition,

$\displaystyle
ds^{2}=-N^{2}dt^{2}+h_{ij}\left(dx^{i}+N^{i}dt\right)\left(dx^{j}+N^{j}dt\right),$
(3.7)

the inverse of the metric reads,

$\displaystyle g^{00}=-\frac{1}{N^{2}},\qquad
g^{0i}=\frac{N^{i}}{N^{2}},\qquad g^{ij}=h^{ij}-\frac{N^{i}N^{j}}{N^{2}}.$
(3.8)

The conjugate momenta are defined in the following,

$\displaystyle\Pi^{ij}=\frac{\partial\mathcal{L}}{\partial\dot{h}_{ij}}=\frac{1}{2}\sqrt{h}\left(K_{ij}-Kh_{ij}\right),\qquad\Pi_{N}=\frac{\partial\mathcal{L}}{\partial\dot{N}}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\Pi_{i}=\frac{\partial\mathcal{L}}{\partial\dot{N}^{i}}=0,\qquad\Pi_{\lambda}=\frac{\partial\mathcal{L}}{\partial\dot{\lambda}}$
$\displaystyle=$ $\displaystyle 0.$ (3.9)

The Hamiltonian is obtained by performing the Legendre transformation,

$\displaystyle H$ $\displaystyle=$ $\displaystyle\int
d^{3}x\left(\Pi^{ij}\dot{h}_{ij}-\mathcal{L}+\varrho_{N}\Pi_{N}+\varrho^{i}\Pi_{i}+\varrho_{\lambda}\Pi_{\lambda}\right)$
(3.10) $\displaystyle=$ $\displaystyle\int
d^{3}x\sqrt{h}\left(\mathcal{H}+\varrho_{N}\Pi_{N}+\varrho^{i}\Pi_{i}+\varrho_{\lambda}\Pi_{\lambda}\right)$

where $\varrho_{N},\varrho^{i},\varrho_{\lambda}$ are Lagrangian multipliers,
and

$\displaystyle\Pi_{N}\approx 0,\qquad\Pi_{i}\approx
0,\qquad\Pi_{\lambda}\approx 0$ (3.11)

are 5 primary constraints, and

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{2N}{h}\left(\Pi^{ij}\Pi_{ij}-\frac{1}{2}\Pi^{2}\right)-\frac{NR}{2}+N\left(\frac{c}{3}\Lambda_{1}(t)+\frac{\lambda}{4}\right)\left(h^{ab}-\frac{N^{a}N^{b}}{N^{2}}\right)\delta_{ab}-\frac{\lambda}{4}\frac{\Lambda_{2}(t)}{N}$
(3.12)
$\displaystyle-2\nabla_{j}\left(\frac{\Pi^{j}_{~{}i}}{\sqrt{h}}\right)N^{i},$

where $R$ is the 3-dimensional Ricci scalar, $\nabla_{i}$ is the covariant
derivative compatible with 3-dimensional induced metric $h_{ij}$, and
$\approx$ denotes the “weak equivalence”, namely equalities hold on the
constraint surface. Caution should be paid to the terms with different types
of indices. For instance, in the 3 dimensional hyperspace $h^{ab}\equiv
h^{ij}\delta_{i}^{a}\delta_{j}^{b}$ and $N^{a}\equiv N^{i}\delta_{i}^{a}$ are
scalars (as their space-time indices are dummy indices), $\delta^{a}_{i}$ is a
vector, while $h^{ij}$ and $N^{i}$ are tensor and vector respectively. The
consistency conditions of these 5 primary constraints give rise to the
following 5 secondary constraints,

$\displaystyle\frac{d\Pi_{N}}{dt}$ $\displaystyle=$
$\displaystyle\\{\Pi_{N},H\\}=-\mathcal{H}_{0}\approx 0,$
$\displaystyle\frac{d\Pi_{i}}{dt}$ $\displaystyle=$
$\displaystyle\\{\Pi_{i},H\\}=-\mathcal{H}_{i}\approx 0,$
$\displaystyle\frac{d\Pi_{\lambda}}{dt}$ $\displaystyle=$
$\displaystyle\\{\Pi_{\lambda},H\\}=-\mathcal{C}_{\lambda}\approx 0,$ (3.13)

where $\mathcal{H}_{0}$ is the Hamiltonian constraint, $\mathcal{H}_{i}$ is
the momentum constraint, and $\mathcal{C}_{\lambda}$ is the constraint
introduced by hand at the action level,

$\displaystyle\mathcal{H}_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{2}{h}\left(\Pi^{ij}\Pi_{ij}-\frac{1}{2}\Pi^{2}\right)-\frac{R}{2}+\left(\frac{c}{3}\Lambda_{1}(t)+\frac{\lambda}{4}\right)\left(h^{ab}+\frac{N^{a}N^{b}}{N^{2}}\right)\delta_{ab}+\frac{\lambda}{4}\frac{\Lambda_{2}(t)}{N^{2}},$
$\displaystyle\mathcal{H}_{i}$ $\displaystyle\equiv$
$\displaystyle-2\nabla_{j}\left(\frac{\Pi^{j}_{~{}i}}{\sqrt{h}}\right)-\frac{2N^{b}\delta_{ab}\delta_{i}^{a}}{N}\left(\frac{c}{3}\Lambda_{1}(t)+\frac{\lambda}{4}\right),$
$\displaystyle\mathcal{C}_{\lambda}$ $\displaystyle\equiv$
$\displaystyle\frac{N}{4}\left(h^{ab}-\frac{N^{a}N^{b}}{N^{2}}\right)\delta_{ab}-\frac{\Lambda_{2}(t)}{4N}.$
(3.14)

These 5 secondary constraints must be conserved in time, which yields to the
following 5 consistency conditions,

$\displaystyle\frac{d\mathcal{H}_{0}}{dt}=\frac{\partial\mathcal{H}_{0}}{\partial
t}+\\{\mathcal{H}_{0},H\\}$ $\displaystyle\approx$ $\displaystyle 0,$
$\displaystyle\frac{d\mathcal{H}_{i}}{dt}=\partial_{t}\mathcal{H}_{i}+\\{\mathcal{H}_{i},H\\}$
$\displaystyle\approx$ $\displaystyle 0,$
$\displaystyle\frac{d\mathcal{C}_{\lambda}}{dt}=\partial_{t}\mathcal{C}_{\lambda}+\\{\mathcal{C}_{\lambda},H\\}$
$\displaystyle\approx$ $\displaystyle 0.$ (3.15)

Whether these 5 consistency conditions generate some tertiary constraints, or
they only fix the Lagrangian multipliers in the Hamiltonian, crucially depends
on the rank of the matrix

$\displaystyle\mathcal{M}_{AB}\equiv\\{\phi_{A},\phi_{B}\\},$ (3.16)

where $\phi_{A}$ is the whole set of constraints that we have so far, i.e.
$\phi_{A}=(\Pi_{N},\Pi_{i},\Pi_{\lambda},\mathcal{H}_{0},\mathcal{H}_{i},\mathcal{C}_{\lambda})$.
The commutation relations among these constraints are showed in the following,

$\displaystyle\\{\Pi_{N},\Pi_{N}\\}$ $\displaystyle\approx$ $\displaystyle
0,\qquad\\{\Pi_{N},\Pi_{i}\\}\approx
0,\qquad\\{\Pi_{N},\Pi_{\lambda}\\}\approx 0,$
$\displaystyle\\{\Pi_{N},\mathcal{H}_{0}\\}$ $\displaystyle\neq$
$\displaystyle 0,\qquad\\{\Pi_{N},\mathcal{H}_{i}\\}\neq
0,\qquad\\{\Pi_{N},\mathcal{C}\\}\neq 0,$ $\displaystyle\\{\Pi_{i},\Pi_{j}\\}$
$\displaystyle\approx$ $\displaystyle
0,\qquad\\{\Pi_{i},\Pi_{\lambda}\\}\approx
0,\qquad\\{\Pi_{i},\mathcal{H}_{0}\\}\neq 0,$
$\displaystyle\\{\Pi_{i},\mathcal{H}_{j}\\}$ $\displaystyle\neq$
$\displaystyle 0,\qquad\\{\Pi_{i},\mathcal{C}_{\lambda}\\}\neq 0,$
$\displaystyle\\{\Pi_{\lambda},\Pi_{\lambda}\\}$ $\displaystyle\approx$
$\displaystyle 0,\qquad\\{\Pi_{\lambda},\mathcal{H}_{0}\\}\neq
0,\qquad\\{\Pi_{\lambda},\mathcal{H}_{i}\\}\approx 0,$
$\displaystyle\\{\Pi_{\lambda},\mathcal{C}_{\lambda}\\}$
$\displaystyle\approx$ $\displaystyle 0,$
$\displaystyle\\{\mathcal{H}_{0},\mathcal{H}_{0}\\}$ $\displaystyle\neq$
$\displaystyle 0,\qquad\\{\mathcal{H}_{0},\mathcal{H}_{i}\\}\neq
0,\qquad\\{\mathcal{H}_{0},\mathcal{C}_{\lambda}\\}\neq 0,$
$\displaystyle\\{\mathcal{H}_{i},\mathcal{H}_{j}\\}$ $\displaystyle\neq$
$\displaystyle 0,\qquad\\{\mathcal{H}_{i},\mathcal{C}_{\lambda}\\}\neq 0,$
$\displaystyle\\{\mathcal{C}_{\lambda},\mathcal{C}_{\lambda}\\}$
$\displaystyle\approx$ $\displaystyle 0.$ (3.17)

The rank of the matrix $\mathcal{M}_{AB}$ is 10. Therefore the consistency
conditions eq. (3.2) only fix the Lagrangian multipliers $\varrho^{\prime}$s,
instead of generating new tertiary constraints. Let’s collect all of primary
and secondary constraints in the total Hamiltonian and treat them on the same
footing,

$\displaystyle H_{tot}=\int
d^{3}x\sqrt{h}\left(\mathcal{H}+\varrho_{N}\pi_{N}+\varrho^{i}\Pi_{i}+\varrho_{\lambda}\Pi_{\lambda}+\varrho^{0}\mathcal{H}_{0}+\tilde{\varrho}^{i}\mathcal{H}_{i}+\varrho_{\lambda}\mathcal{C}_{\lambda}\right)$
(3.18)

where
$\left(\varrho_{N},\varrho^{i},\varrho_{\lambda},\varrho^{0},\tilde{\varrho}^{i},\varrho_{\lambda}\right)$
are Lagrangian multipliers, and we have absorbed the shift vector into the
$\tilde{\varrho}^{i}$. The algebra closes here.

The rank of the matrix also implies all of these 10 constraints are second
class. This is the consequence of that the gauge symmetries in GR, namely the
space-time diffeomorphisms are all broken in the unitary gauge. Now let’s
count the number of degrees. In the phase space we have got 22 degrees in the
first place, where 20 degrees are from the 10 independent components of
$g^{\mu\nu}$ and their conjugate momenta, and 2 degrees are from Lagrangian
multiplier $\lambda$ and its conjugate momentum. On the other hand, those 10
second class constraints $\phi_{A}$ remove 10 degrees in the phase space, and
the leftover is

$\displaystyle 22-10=12~{}\text{ degrees in the phase space},$ (3.19)

which corresponds to 6 degrees of freedom in the physical space-time. The
sixth mode is the Ostrogradsky ghost.

###  3.3 Ghost elimination in the Goldstone action in the decoupling limit

The Goldstone action in the decoupling limit is a very convenient perspective
for us to hunt and eventually execute the ghost. Taking a closer look at the
Goldstone action eq. (2.11), I notice that the ghost arises from the kinetic
term of the Goldsteon boson $\pi$, and the constraint equating the temporal
derivative of $\pi^{0}$ and the gradient of the $\pi$, namely

$\displaystyle\int\partial_{i}\dot{\pi}\partial_{i}\dot{\pi}~{}\to~{}\int\ddot{\pi}^{0}\frac{1}{\partial^{2}}\ddot{\pi}^{0}\qquad\text{as}\qquad\partial^{2}\pi=3\dot{\pi}^{0}.$
(3.20)

Therefore, a direct way to eliminate the higher order temporal derivative term
is to eliminate the kinetic term $\partial_{i}\dot{\pi}\partial_{i}\dot{\pi}$,
which can be achieved by introducing the symmetry [17]

$\displaystyle\pi^{i}(t,\textbf{x})\to\pi^{i}(t,\textbf{x})+\xi^{i}\left(t\right),$
(3.21)

where $\xi^{i}$ is an arbitrary function of time. This symmetry prohibits all
temporal derivative terms of the Goldstone pion $\pi^{i}$, including the ones
of the longitudinal mode $\pi$ and the ones of the transverse mode
$\hat{\pi}^{i}$. At the lowest dimensional operator level, the building block
which respects this symmetry is the following,

$\displaystyle
Z^{ab}\equiv\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{b}-\frac{\left(\partial_{\mu}\varphi\partial^{\mu}\phi^{a}\right)\left(\partial_{\nu}\varphi\partial^{\nu}\phi^{b}\right)}{\partial_{\mu}\varphi\partial^{\mu}\varphi}.$
(3.22)

In the unitary gauge where all Goldstone bosons are muted, we have
$Z^{ab}=\frac{1}{3}h^{ij}\delta_{i}^{a}\delta_{j}^{b}$, namely the residual
symmetry eq. (3.21) strips the shift off the $g^{ij}$, and leaves us with only
3 dimensional induced metric $h^{ij}$. Now let’s look at the leftover action,
which reads

$\displaystyle
S\simeq\Lambda^{4}\int-c\dot{\pi}^{0}\dot{\pi}^{0}-\frac{1}{3}\left(\lambda+c\right)\partial_{i}\pi^{0}\partial_{i}\pi^{0},$
(3.23)

where $c$ and $\lambda$ are assumed to be positive, to ensure that the energy
density of dark energy and dark radiation are positive. Therefore, the kinetic
term of the leftover scalar mode $\pi^{0}$ has a wrong sign. Unfortunately
after removing the problematic higher order temporal derivative term, the
theory still contains a ghost. This ghost arises from the gradient term of the
Goldstone boson $\pi$, namely

$\displaystyle\int-\frac{4c+\lambda}{12}\partial^{2}\pi\partial^{2}\pi~{}\to~{}\int-\left(3c+\frac{3\lambda}{4}\right)\dot{\pi}^{0}\dot{\pi}^{0}\qquad\text{as}\qquad\partial^{2}\pi=3\dot{\pi}^{0}.$
(3.24)

I have to flip the sign of the term $\partial^{2}\pi\partial^{2}\pi$ to cure
this ghost pathology. The remedy is the operator introduced in Ref. [30],

$\displaystyle\bar{\delta}Z^{ab}\equiv
Z^{ab}-3\frac{Z^{ac}Z^{db}\delta_{cd}}{Z^{cd}\delta_{cd}},$ (3.25)

which is traceless up to the linear perturbation level, and thus it does not
contribute to the background evolution. Additionally I shall stick to the
$Z_{2}$ symmetry and the global scaling invariance eq. (2.7) to tighten up the
structure and reduce the arbitrariness of the theory. At the lowest
dimensional operator level, I write down the general covariant and ghost free
action which respects the residual symmetry eq. (3.21), the global symmetry
eq. (2.7) and the $SO(3)$ rotational symmetry in the field space as follows,

$\displaystyle S=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left\\{\frac{M_{p}^{2}}{2}\mathcal{R}-M_{p}^{2}\varphi^{-2}\left[\left(c+\lambda\right)\cdot
Z+\lambda\partial^{\mu}\varphi\partial_{\mu}\varphi+\frac{3d}{8Z}\cdot\bar{\delta}Z^{ab}\bar{\delta}Z^{cd}\delta_{ac}\delta_{bd}\right]\right\\}$
(3.26) $\displaystyle\qquad\qquad+S_{m}\,,$

where $Z\equiv Z^{ab}\delta_{ab}$, the coefficient $3/8$ is chosen for the
later convenience and $S_{m}$ is the action of the matter sector which
minimally couples to gravity. This is the main result of my current work.
Compared with the original action eq. (2.6), in the unitary gauge the
$g^{ab}$, which is produced by the spatial Stueckelberg fields
$\partial^{\mu}\phi^{a}\partial_{\mu}\phi^{b}\delta_{ab}$, is replaced by one
of our building blocks $h^{ab}$ defined in the eq. (3.22). Take the note of
that $h^{ab}$ differs from $g^{ab}$ only at the perturbation level. On the
other hand, the operator $\bar{\delta}Z^{ab}$ only appears at the perturbation
level too as it is traceless. Therefore, all modifications are operated at the
perturbation level, and do not alter the background evolution which is subject
to the set of equations in the eq. (2) (2.9). The Goldstone action for the
scalar graviton reads

$\displaystyle
S\supset\int-\left(c+d\right)\dot{\pi}^{0}\dot{\pi}^{0}-\frac{1}{3}\left(\lambda+c\right)\partial_{i}\pi^{0}\partial_{i}\pi^{0},$
(3.27)

with the ghost free condition,

$\displaystyle c+d<0.$ (3.28)

This ghost free condition will be reproduced later in the full perturbation
analysis in the unitary gauge, where the Goldstone bosons are muted.

###  3.4 Full perturbation analysis in the unitary gauge

We adopt the metric perturbation decomposition in the eq. (3.1). Due to the
background $SO(3)$ rotational invariance, the scalar perturbations, the vector
perturbations and the tensor perturbations completely decouple from each other
at the linear perturbation level. We also adopt the unitary gauge and mute all
Goldstone bosons.

The massive gravity is an analog of the Higgs mechanism in the particle
physics, where the Higgs field resides in an asymmetric state. In the unitary
gauge, the Goldstone pions in this symmetry broken phase are eaten by the
gauge bosons and consequently the gauge bosons become massive. We expect the
same phenomena should also occur in the massive gravity, namely once we mute
the Goldstone bosons, the scalar graviton and gravitational waves are massive
in the unitary gauge.

I will only focus on the perturbation analysis of the pure gravity, while the
cases with matter included will be discussed in the future work.

####  3.4.1 Scalar perturbation

The quadratic action of the scalar perturbation is obtained after a
straightforward computation,

$\displaystyle S_{\text{scalar}}^{(2)}=\int
d^{4}x\left(\mathcal{L}_{\text{EH}}+\mathcal{L}_{\text{mass}}\right),$ (3.29)

where $\mathcal{L}_{\text{EH}}$ is the contribution from the Einstein-Hilbert
action,

$\displaystyle\frac{\mathcal{L}_{EH}}{M_{p}^{2}a^{3}}$ $\displaystyle=$
$\displaystyle-3\dot{\psi}^{2}+\frac{k^{2}}{a^{2}}\left[\psi^{2}-\frac{1}{2}a^{2}\dot{E}\left(H\psi-2\dot{\psi}\right)-\frac{1}{2}a^{2}HE\left(\dot{\psi}+3H\psi\right)\right]$
(3.30) $\displaystyle+\alpha\left[\frac{2k^{2}\beta
H}{a}+\frac{k^{2}}{a^{2}}\left(2\psi-a^{2}H\dot{E}\right)+6H\dot{\psi}-3H^{2}\alpha\right]-\frac{2k^{2}\beta\dot{\psi}}{a},$

and the $\mathcal{L}_{\text{mass}}$ is the contribution from the dark sector,
or in other word the graviton mass term,

$\displaystyle\mathcal{L}_{\text{mass}}$ $\displaystyle=$ $\displaystyle
M_{p}^{2}\varphi^{-2}\left[-\frac{3c+d}{36}k^{4}aE^{2}-\frac{1}{6}ck^{2}aE\left(2\alpha+\psi\right)+ca\psi\left(2\alpha+\psi\right)\right]$
(3.31)
$\displaystyle+\lambda\left[-\frac{k^{4}E^{2}}{48a}+\frac{k^{2}E\left(\psi-\alpha\right)}{12a}+\frac{\left(\alpha+\psi\right)^{2}}{4a}\right]+\delta\lambda\left(\frac{\psi-\alpha}{2a}-\frac{k^{2}E}{12a}\right).$

I have absorbed the lapse into the redefinition of time, and rescaled the
Lagrangian multiplier
$\lambda\to\lambda\cdot\frac{\varphi^{2}}{4a^{2}M_{p}^{2}}$ for convenience
once again. Taking the variation of the action with respect to the non-
dynamical variables $\alpha$, $\delta\lambda,$ and $~{}\beta$, I get the
following three constraint equations,

$\displaystyle
H\left(6\dot{\psi}-k^{2}\dot{E}\right)+\frac{2c\psi}{a^{2}\varphi^{2}}+\frac{k^{2}}{a^{2}}\left(2\psi-\frac{cE}{3a^{2}\varphi^{2}}\right)+\frac{\lambda}{12a^{4}M_{p}^{2}}\left(6\psi-k^{2}E\right)$
$\displaystyle-\frac{\delta\lambda}{2a^{4}M_{p}^{2}}+\alpha\left(\frac{\lambda}{2a^{4}M_{p}^{2}}-6H^{2}\right)+\frac{2k^{2}H\beta}{a}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
k^{2}E+6{\color[rgb]{1,0,0}\alpha}-6\psi$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle H\alpha-\dot{\psi}$ $\displaystyle=$ $\displaystyle 0.$ (3.32)

Substituting the solution of the above equations into the action, the
quadratic action of the scalar perturbation takes the following form,

$\displaystyle
S_{\text{scalar}}^{(2)}=\int-\frac{\left(c+d\right)aM_{p}^{2}}{H^{2}\varphi^{2}}\dot{\psi}^{2}+...$
(3.33)

where the eclipse denotes the potential term and gradient term. The ghost free
condition for the scalar perturbation reads

$\displaystyle c+d<0.$ (3.34)

We have reproduced the ghost free condition eq. (3.28), in the different
gauge. Let me redefine a new constant $b$ via

$\displaystyle b\equiv-2\left(c+d\right),\qquad\text{and we demand}\qquad
b>0.$ (3.35)

The scalar graviton is canonically normalised as

$\displaystyle\psi^{c}\equiv\frac{\sqrt{b}M_{p}}{aH\varphi}\psi.$ (3.36)

The canonically normalised scalar action reads

$\displaystyle S_{\text{scalar}}^{(2)}=\frac{1}{2}\int
dtd^{3}ka^{3}\left(\dot{\psi}^{c}\dot{\psi}^{c}-\frac{c_{s}^{2}k^{2}}{a^{2}}\psi^{c}\psi^{c}-M^{2}_{s}\psi^{c}\psi^{c}\right),$
(3.37)

where the sound speed reads

$\displaystyle
c_{s}^{2}=\frac{c}{b}\cdot\frac{\rho_{\text{dark}}}{\rho_{\text{hde}}}\cdot\left(1+w_{\text{dark}}\right),$
(3.38)

where $\rho_{\text{dark}}$ is energy density of the dark sector defined in the
eq. (2.4), $w_{\text{dark}}$ is its equation of state, and
$\rho_{hde}=cM_{p}^{2}/a^{2}\varphi^{2}$ is the energy density of the
holographic dark energy. It is clear that the $c_{s}^{2}$ becomes negative
when the equation of state of the dark sector drops below $-1$, this is
precisely the case for $c<6$ in the asymptotic future [13][22]. Therefore, if
we demand the freeness of the gradient instability throughout the whole cosmic
expansion history, the parameter region with $c<6$ needs to be excluded.
However, $c<6$ is still allowed, if we only demand the freeness of the
gradient instability at present and in the past, and allow the instability to
occur in the future. The mass of the scalar mode reads

$\displaystyle M_{s}^{2}$ $\displaystyle=$ $\displaystyle
2H^{2}\left[\frac{6c}{b}-\Omega_{\text{hde}}^{2}+\Omega_{\text{hde}}\left(-\frac{1}{2}-\frac{3}{c}+\frac{3c}{b}-4\Omega_{\text{rad}}\right)+\sqrt{\frac{3\Omega_{\text{hde}}}{c}}\left(1-\frac{2c}{b}+2\Omega_{\text{rad}}\right)\right.$
(3.39)
$\displaystyle\qquad\left.+\left(1+\frac{8c}{b}\right)\Omega_{\text{rad}}+\Omega_{\text{rad}}^{2}\left(\frac{4c}{b\Omega_{\text{hde}}}-4\right)\right],$

where $\Omega_{rad}$ is the fraction of the dark radiation defined by
$\Omega_{rad}\equiv\rho_{rad}/3M_{p}^{2}H^{2}$, and $\Omega_{\text{hde}}$ is
the fraction of the holographic dark energy defined by
$\Omega_{\text{hde}}\equiv\rho_{\text{hde}}/3M_{p}^{2}H^{2}$. The Compton
wavelength of the scalar mode is about Hubble radius size, up to a coefficient
whose value is dependent on the parameters $b$ and $c$.

One may wonder whether the mass of scalar graviton is always positive along
the cosmic evolution to ensure the stability in the scalar sector. This
problem may require a thorough numerical analysis, and thus it will not be
carried out in the current work. Instead, I would like to show the tachyon
freeness condition in the asymptotic de-Sitter phase where $c=6$ [13] and
$w_{\text{dark}}\to-1$. The scalar mass in the asymptotic phase reads

$\displaystyle M_{s}^{2}\to 8H^{2}\left[\frac{6}{b}-1\right].$ (3.40)

The tachyon freeness condition translates to

$\displaystyle 0<b<6.$ (3.41)

The stability analysis for general space-time solution will be performed in
the future work.

####  3.4.2 Tensor perturbation

The quadratic action for the tensor perturbation reads

$\displaystyle S_{\text{tensor}}^{(2)}=\frac{M_{p}^{2}}{8}\int
dtd^{3}ka^{3}\left[\dot{\gamma}^{ij}\dot{\gamma}_{ij}-\left(\frac{k^{2}}{a^{2}}+M^{2}_{GW}\right)\gamma_{ij}\gamma^{ij}\right],$
(3.42)

where

$\displaystyle
M_{GW}^{2}=\frac{1}{6R_{h}^{2}}\left[4c-b+3c\frac{\rho_{\text{dark}}}{\rho_{\text{hde}}}\left(1+w_{dark}\right)\right]$
(3.43)

The Compton wavelength of the tensor graviton is about the size of the future
event horizon. The following tachyon freeness condition is required

$\displaystyle
4c-b+3c\frac{\rho_{\text{dark}}}{\rho_{\text{hde}}}\left(1+w_{dark}\right)>0$
(3.44)

to ensure the stability of the tensor sector. In the asymptotic de-Sitter
phase where $c=6,~{}w_{\text{dark}}=-1$, the stability condition translates to
$0<b<24$, which is weaker than the stability condition eq. (3.41) derived in
the preceding subsection of the scalar mode analysis.

The non-vanishing mass modifies the dispersion relation of the gravitational
waves, and the group velocity is constrained with high precision $-3\cdot
10^{-15}<c_{\text{gw}}-1<7\cdot 10^{-16}$ at low redshift regime by the multi-
messenger observation GW170817 [31]. It gives us an upper bound on the
graviton mass around $m_{g}<10^{-23}\text{eV}$. Our graviton mass is much
lower than this upper bound by around 10 orders of magnitude. It remains
challenging to directly probe the non-vanishing graviton mass effects at the
late time epoch. However, the size of the future event horizon is much shorter
during the early universe, and thus it leads to a much larger graviton mass,
which may give rise to some interesting observational effects on the
stochastic gravitational waves background.

####  3.4.3 Vector perturbation

I do not expect to find the dynamical degrees of freedom in the vector sector
at the linear perturbation level, as the residual symmetry eq. (3.21) has
projected out all temporal derivative of the Goldstone vector bosons. I will
try to confirm it in this subsection. The quadratic action of the vector
perturbation reads

$\displaystyle S_{\text{V}}^{(2)}=\frac{M_{p}^{2}}{16}\int
dtd^{3}kk^{2}a^{3}\left[\dot{F}_{i}\dot{F}^{i}-\frac{4S_{i}\dot{F}^{i}}{a}+\frac{4S_{i}S^{i}}{a^{2}}+\left(\frac{b-6c}{6a^{2}\varphi^{2}}-\frac{\lambda}{3M_{p}^{2}a^{4}}\right)F_{i}F^{i}\right].$
(3.45)

Taking the variation with respect to the Lagrangian multiplier $S_{i}$, I get

$\displaystyle a\dot{F}_{i}-2S_{i}=0.$ (3.46)

Substituting the solution of this constraint equation back to the action, I
get

$\displaystyle S_{\text{V}}^{(2)}=\frac{1}{16}M_{p}^{2}\int
dtd^{3}kk^{2}a^{3}\left(\frac{b-6c}{6a^{2}\varphi}-\frac{\lambda}{3M_{p}^{2}a^{4}}\right)F_{i}F^{i}.$
(3.47)

The kinetic term of the vector mode has vanished, and the action is subject to
a new constraint $F_{i}=0$. After substituting this solution back to the
action, the whole action vanishes and I conclude that no dynamical degree of
freedom is found in the vector sector at the linear perturbation level.
Whether there exists dynamical degrees of freedom at the higher order level
requires a fully non-linear analysis, which will be provided in the next
subsection.

###  3.5 The Hamiltonian analysis confirms the total number of degrees

I have performed the Hamiltonian analysis in the subsection (3.2), and I found
that the original theory contains 6 modes. Compared to the Hamiltonian of the
original theory (2.6), the modified one (3.26) strips off the term that is
non-linear in the shift vector. Consequently the primary constraint
$\Pi_{i}\approx 0$ commutes with all primary and secondary constraints. The
consistency conditions for the momentum constraints conserve in time yield to
another 3 tertiary constraints,

$\displaystyle\mathcal{T}_{i}\equiv\\{\mathcal{H}_{i},H\\}\approx 0.$ (3.48)

Now let’s add these 3 new tertiary constraints into the constraint set
$\phi^{A}$, and remove the $\Pi_{i}\approx 0$ as it is first class, we have
the new set

$\displaystyle\phi_{A}=(\Pi_{N},\Pi_{\lambda},\mathcal{H}_{0},\mathcal{H}_{i},\mathcal{C}_{\lambda},\mathcal{T}_{i}).$
(3.49)

We can show that the rank of the following $10\times 10$ matrix

$\displaystyle\mathcal{M}_{AB}\equiv\\{\phi_{A},\phi_{B}\\}$ (3.50)

is 10, no new constraints are generated and the algebra closes here.
Therefore, all constraints in the eq. (3.49) are second class, and they remove
10 degrees in the phase space. On the other hand, the constraints
$\Pi_{i}\approx 0$ remove 6 degrees in the phase space as they are first
class. The total number of the leftover dynamical degrees of freedom in the
phase space is

$\displaystyle 22-3\times 2-10=6,$ (3.51)

which corresponds to 3 dynamical degrees of freedom in the physical space-
time. As we have shown in our perturbation analysis, 2 of them are tensor
modes, and the rest one is a scalar graviton. The additional degrees of
freedom are fewer than the broken symmetry generators, because the Lorentz-
invariance of the background space-time is broken and therefore the Goldstone
theorem needs to be understood in the extended sense.

###  3.6 Physical interpretation of the UV-IR correspondence

The UV-IR correspondence is based on the assumption that the vacuum energy
arising from the quartic divergence should not exceed the energy density of a
black hole in the same size, and the effective quantum field theory should
break down at the Schwarzschild radius scale. Take the note of that the Hubble
radius of our universe coincides with its Schwarzschild radius, and thus the
bound should be saturated on the cosmological background and we have the UV-IR
relation $\Lambda\sim\sqrt{M_{p}/L}$.

One may ask how this UV-IR correspondence appears naturally in our effective
field theory framework? The answer is that it should be interpreted as the
scale where the scalar graviton strongly couples to itself. Let’s look at the
4-votex interaction in the decoupling limit,

$\displaystyle\int
M_{p}^{2}\varphi^{-2}\partial^{2}\pi\partial^{2}\pi\partial^{2}\pi\partial^{2}\pi\sim\int
M_{p}^{2}\varphi^{-2}\dot{\pi}^{0}\dot{\pi}^{0}\dot{\pi}^{0}\dot{\pi}^{0}\sim\int\frac{\omega^{4}}{M_{p}^{2}\varphi^{-2}}\pi^{0}_{c}\pi^{0}_{c}\pi^{0}_{c}\pi^{0}_{c},$
(3.52)

where $\pi^{0}_{c}$ is the canonical normalised Goldstone boson, and $\omega$
is the frequency of $\pi^{0}_{c}$. The strength of the coupling exceeds unity
if $\omega>\sqrt{M_{p}/\varphi}$, and our effective field theory breaks down.
As shown in our perturbation analysis, we have learnt that $\varphi$ is the
graviton’s Compton’s wavelength (remember $\varphi$ has the length dimension,
and we set the scale factor $a=1$ in the decoupling limit). Therefore, a
sensible and physical interpretation of this UV-IR correspondence is that the
UV cut-off scale of our EFT is inversely proportional to the Compton
wavelength of our graviton.

##  4 Conclusion and Discussion

In the current work, a general covariant local field theory is proposed for
the holographic dark energy model. I started from the mini superspace action
proposed a few years ago [13], showed that the direct, and perhaps the
simplest way of covariantizing the action gives rise to additional 4 degrees
of freedom (6 in total including 2 degrees in the gravitational wave sector),
which leads to the Ostrogradsky ghost instability. To remedy the ghost
pathology, I introduced a new symmetry to prohibit the problematic terms in
the theory. It has turned out that the low energy effective field theory of
the holographic dark energy model is actually the Lorentz-violating massive
gravity theory, whose graviton has 3 polarisations including 1 helicity 0 mode
and 2 helicity 2 modes. The Compton wavelength of the graviton is about the
size of the future event horizon of our universe, as shown in the linear
perturbation analysis in the unitary gauge. To confirm the total number of
dynamical degrees of freedom of our theory at the fully non-linear level, I
have performed the Hamiltonian analysis and found that the momentum
constraints yield to another 3 tertiary constraints. These 6 constraints,
which are all second class, eliminate 3 out of 4 additional degrees and
therefore there are only 3 dynamical degrees of freedom in total in the
gravity sector, including a scalar mode and two tensor modes. Our effective
field theory breaks down at the scale $\Lambda\sim\sqrt{M_{p}/L}$, where $L$
is the graviton’s Compton wavelength, due to the strong coupling of the scalar
graviton above this scale, which offers a natural and physical interpretation
for the UV-IR correspondence in the holographic dark energy model.

Our effective field theory provides a general framework in which the
holographic dark energy model can be tested. For instance, the non-vanishing
mass of the gravitational waves, which is small at late time epoch but
sizeable during the early universe, leads to a modified stochastic
gravitational waves background. On the other hand, the existence of the scalar
graviton whose Compton wavelength is about the size of the Hubble radius (up
to an order $\mathcal{O}(1)\sim\mathcal{O}(10)$ coefficient) during inflation
may be tested by the cosmological collider physics [32]. It is also very
intriguing to ask what would happen if the Stueckelberg field, say the time
like one $\varphi$, couples to the standard model fields. Our EFT framework
also provides a general setup where the perturbation theory of the holographic
dark energy can be developed, which allows us to investigate its impacts on
the cosmological structure formation in detail. All these possibilities
warrant further scrutiny.

I shall end by commenting that the global Lorentz invariance is broken in our
scalar field configuration. As an analog to the Higgs mechanism in the
particle physics, one may expect a similar symmetry restoration to occur at
high energy scale, where the Lorentz invariance is recovered and the graviton
becomes massless again. However, this Higgs-like mechanism is till missing. In
fact, to my best knowledge it is still an open question for all massive
gravity theories.

## Acknowledgement

This work is supported by the grant No. UMO-2018/30/Q/ST9/00795 from the
National Science Centre, Poland. The author would like to thank Yi-fu Cai,
Alexander Ganz, A. Emir Gümrükçüoğlu, Qing-Guo Huang, Miao Li, Yin-zhe Ma, and
Yi Wang for the useful discussions and suggestions.

## References

  * [1] A. G. Cohen, D. B. Kaplan and A. E. Nelson, “Effective field theory, black holes, and the cosmological constant,” Phys. Rev. Lett. 82, 4971-4974 (1999) doi:10.1103/PhysRevLett.82.4971 [arXiv:hep-th/9803132 [hep-th]].
  * [2] M. Li, “A Model of holographic dark energy,” Phys. Lett. B 603, 1 (2004) doi:10.1016/j.physletb.2004.10.014 [arXiv:hep-th/0403127 [hep-th]].
  * [3] E. K. Akhmedov, [arXiv:hep-th/0204048 [hep-th]].
  * [4] G. Ossola and A. Sirlin, Eur. Phys. J. C 31, 165-175 (2003) doi:10.1140/epjc/s2003-01337-7 [arXiv:hep-ph/0305050 [hep-ph]].
  * [5] J. F. Donoghue, [arXiv:2009.00728 [hep-th]].
  * [6] G. ’t Hooft, “Dimensional reduction in quantum gravity,” Conf. Proc. C 930308, 284-296 (1993) [arXiv:gr-qc/9310026 [gr-qc]].
  * [7] J. D. Bekenstein, Black Holes and Entropy, Phys. Rev. D7, 2333 (1973); Generalized Second Law of Thermodynamics in Black Hole Physics, Phys. Rev. D9, 3292 (1974); A Universal Upper Bound On The Entropy to Energy Ratio for Bounded Systems, Phys. Rev. D23, 287 (1981); Entropy Bounds and Black Hole Remnants, gr-qc/9307035, Phys. Rev. D49, 1912 (1994).
  * [8] P. Horava and D. Minic, “Probable values of the cosmological constant in a holographic theory,” Phys. Rev. Lett. 85, 1610-1613 (2000) doi:10.1103/PhysRevLett.85.1610 [arXiv:hep-th/0001145 [hep-th]].
  * [9] S. Thomas, Phys. Rev. Lett. 89 (2002) 081301.
  * [10] S. D. H. Hsu, “Entropy bounds and dark energy,” Phys. Lett. B 594, 13-16 (2004) doi:10.1016/j.physletb.2004.05.020 [arXiv:hep-th/0403052 [hep-th]].
  * [11] S. Wang, Y. Wang and M. Li, “Holographic Dark Energy,” Phys. Rept. 696, 1-57 (2017) doi:10.1016/j.physrep.2017.06.003 [arXiv:1612.00345 [astro-ph.CO]].
  * [12] W. M. Dai, Y. Z. Ma and H. J. He, “Reconciling Hubble Constant Discrepancy from Holographic Dark Energy,” Phys. Rev. D 102, 121302 (2020) doi:10.1103/PhysRevD.102.121302 [arXiv:2003.03602 [astro-ph.CO]].
  * [13] M. Li and R. X. Miao, “A New Model of Holographic Dark Energy with Action Principle,” [arXiv:1210.0966 [hep-th]].
  * [14] M. Li, C. Lin and Y. Wang, JCAP 05, 023 (2008) doi:10.1088/1475-7516/2008/05/023 [arXiv:0801.1407 [astro-ph]].
  * [15] M. Fierz, W. Pauli, Proc. Roy. Soc. Lond. A173, 211-232 (1939).
  * [16] C. de Rham, G. Gabadadze and A. J. Tolley, “Resummation of Massive Gravity,” Phys. Rev. Lett. 106, 231101 (2011) doi:10.1103/PhysRevLett.106.231101 [arXiv:1011.1232 [hep-th]].
  * [17] S. L. Dubovsky, “Phases of massive gravity,” JHEP 10, 076 (2004) doi:10.1088/1126-6708/2004/10/076 [arXiv:hep-th/0409124 [hep-th]].
  * [18] D. Comelli, F. Nesti and L. Pilo, “Nonderivative Modified Gravity: a Classification,” JCAP 11, 018 (2014) doi:10.1088/1475-7516/2014/11/018 [arXiv:1407.4991 [hep-th]].
  * [19] K. Hinterbichler, “Theoretical Aspects of Massive Gravity,” Rev. Mod. Phys. 84, 671-710 (2012) doi:10.1103/RevModPhys.84.671 [arXiv:1105.3735 [hep-th]].
  * [20] C. de Rham, “Massive Gravity,” Living Rev. Rel. 17, 7 (2014) doi:10.12942/lrr-2014-7 [arXiv:1401.4173 [hep-th]].
  * [21] S. W. Hawking and R. Penrose, “The Singularities of gravitational collapse and cosmology,” Proc. Roy. Soc. Lond. A 314, 529 (1970).
  * [22] M. Li, X. D. Li, J. Meng and Z. Zhang, “Cosmological constraints on the new holographic dark energy model with action principle,” Phys. Rev. D 88, no.2, 023503 (2013) doi:10.1103/PhysRevD.88.023503 [arXiv:1211.0756 [astro-ph.CO]].
  * [23] N. Arkani-Hamed, H. Georgi and M. D. Schwartz, “Effective field theory for massive gravitons and gravity in theory space,” Annals Phys. 305, 96-118 (2003) doi:10.1016/S0003-4916(03)00068-X [arXiv:hep-th/0210184 [hep-th]].
  * [24] A. O. Barvinsky, N. Kolganov and A. Vikman, Phys. Rev. D 103, no.6, 064035 (2021) doi:10.1103/PhysRevD.103.064035 [arXiv:2011.06521 [gr-qc]].
  * [25] M. Ostrogradsky: Mem. Ac. St. Petersbourg VI 4, 385 (1850).
  * [26] R. P. Woodard, Lect. Notes Phys. 720, 403 (2007) [astro-ph/0601672].
  * [27] A. E. Gumrukcuoglu, C. Lin and S. Mukohyama, “Cosmological perturbations of self-accelerating universe in nonlinear massive gravity,” JCAP 03 (2012), 006 doi:10.1088/1475-7516/2012/03/006 [arXiv:1111.4107 [hep-th]].
  * [28] P. A. Dirac, Can. J. Math. 2 129 (1950); Proc. Roy. Soc. London, ser. A, 246, 326 (1958); P. A. M. Dirac, Lectures on Quantum Mechanics (Yeshiva University, New York 1964).
  * [29] C. Lin, S. Mukohyama, R. Namba and R. Saitou, “Hamiltonian structure of scalar-tensor theories beyond Horndeski,” JCAP 10, 071 (2014) doi:10.1088/1475-7516/2014/10/071 [arXiv:1408.0670 [hep-th]].
  * [30] C. Lin and L. Z. Labun, “Effective Field Theory of Broken Spatial Diffeomorphisms,” JHEP 03, 128 (2016) doi:10.1007/JHEP03(2016)128 [arXiv:1501.07160 [hep-th]].
  * [31] B. P. Abbott et al. [LIGO Scientific, Virgo, Fermi-GBM and INTEGRAL], “Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A,” Astrophys. J. Lett. 848, no.2, L13 (2017) doi:10.3847/2041-8213/aa920c [arXiv:1710.05834 [astro-ph.HE]].
  * [32] “Cosmological Collider Physics”, Nima Arkani-Hamed, Juan Maldacena [arXiv:1503.08043 [hep-th]].