# Breakthrough On Dynamical Higgs Mechanism For dRGT Gravity: Example In Which Graviton Gains Mass Through Electroweak Phase Transition

Emmanuel Kanambaye

► To cite this version:

Emmanuel Kanambaye. Breakthrough On Dynamical Higgs Mechanism For dRGT Gravity: Example In Which Graviton Gains Mass Through Electroweak Phase Transition. 2025. hal-04663145v8

HAL Id: hal-04663145

https://hal.science/hal-04663145v8

Preprint submitted on 18 Sep 2025 (v8), last revised 19 Dec 2025 (v9)

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Breakthrough On Dynamical Higgs Mechanism For dRGT Gravity: Example In Which Graviton Gains Mass Through Electroweak Phase Transition

Emmanuel Kanambaye, Independent Researcher

Adress: Librairie BAH SARL, Hall du Grand Hôtel de Bamako-Mali;

Postal Code: BP 104 Bamako

E-mail: wadouba@gmail.com

**Abstract.** Massive gravity is an interesting theoretical framework capable of explaining among others things cosmological IR (infra-red) phenomena like late-time cosmic acceleration.

Unfortunately, it turned out to be very difficult of getting fundamental massive gravity theory due among other things to ghost and strong coupling break-down problems etc...
Of course, if since the work of de Rham, Gabadadze and Tolley; the ghost problem appears overcame; things remain unchanged for the strong coupling break-down problem insofar as it seems require dynamical Higgs mechanism for gravity in order to be overcame; a challenge that appeared difficult to be achieved.

It is this difficulty that I overcome in the present paper by showing that it is quite possible of getting a four-dimensional modified gravity which is massless before electroweak phase transition and massive after electroweak phase transition, more precisely a gravity theory for which the graviton gain mass dynamically through the dynamical standard model electroweak phase transition which endows with mass most of the fields of Standard Model of Particle Physics.

Said otherwise, I propose the first example of dynamical Higgs mechanism for dRGT gravity.

# 1. Introduction

Since Fierz and Pauli work [1], the question of whether gravity can be made dynamically massive in a consistent way has roughly always arisen.

Indeed contrary to interactions mediated by vector fields which could be made dynamically massive in a consistent way through a spontaneous symmetry breaking of the kind Higgs mechanism [2, 3, 4]; gravitation gives generically rise to difficulty.

In fact, even to have a viable (i.e. ghost free) Lorentz Invariant massive gravity has been a challenge until de Rham, Gabadadze and Tolley find acceptable trick [5].

Nevertheless, in spite of this advance [5]; neither dRGT Lorentz Invariant Massive Gravity [5] nor Lorentz Violating Massive Gravity [6, 7] are brought through dynamical phase transition endowing gravity with mass (below some characteristic energy-scale)
as do Higgs field [2, 3, 4] in Standard Model of Particle Physics [8]; i.e. both in dRGT Lorentz Invariant Massive Gravity [5] as well as in Lorentz Violating Massive Gravity [6, 7]; it is not a massless gravity theory which is made dynamically massive below some characteristic energy-scale through dynamical spontaneous symmetry breaking; what could be among the best way of handling the strong-coupling breakdown problem of massive gravity.

Unfortunately, as we know [9]; it turned out to be roughly difficult to find reasonable trick permitting such an advance.

In the present paper, it is this difficulty that I propose to overcome; more precisely by involving some auxiliary fields and Lagrange multipliers; I succeeded to achieve a consistent massless gravity theory which becomes (dynamically) massive through the standard model electroweak phase transition which takes place below some characteristic energy-scale [2, 3, 4].

Said otherwise, I achieve the first example of dynamical Higgs mechanism for dRGT gravity.

Before to start, let's emphasize that our aim is to show that it is in principle, quite possible to get/construct from variational principle a consistent gravitational theory which (irrespective of its justification from the point of view of some underlying more fundamental physics or principle) is massless before electroweak phase transition and massive after electroweak phase transition; as consequence we will in the present paper, not try to justify or motivate the action/lagrangian which yields our theory from the point of view of some underlying more fundamental physics or principle like quantum gravity or other which anyway is not known yet.

The paper is organised as follow, in the section 2, I make a brief recall of original dRGT massive gravity [5] and Higgs mechanism [2, 3, 4]; In the section 3, I derive the model that I propose from variational principle; In the section 4, I show that the model is a massive gravity after electroweak phase transition and massless before electroweak phase transition, while I devote Section 5 for conclusion.

The signature convention (–+++) will be assumed for the metric; likewise Einstein's summation convention will be assumed.

## 2. Brief Recall

### 2.1. Brief Recall of dRGT Massive Gravity

In [5], de Rham, Gabadadze and Tolley proposed among the simplest viable Lorentz Invariant massive gravity from the action:

$$S = \frac{M_P^2}{2} \int \left[ R + 2\kappa L_M + m^2 \mathcal{U}(g, \varphi^\rho) \right] \sqrt{-g} d^4 x \quad (1)$$

where $M_P$ denotes Planck mass; R the Ricci scalar; $\kappa$ the Einstein gravitational con-
stant; $L_M$ the lagrangian of standard matter [15]; $m$ a constant; $\varphi^\rho = (\varphi^0, \varphi^j)$ a four-components Stückelberg scalar field, while [5, 10]:

$$ \mathcal{U}(g, \varphi^\rho) = \mathcal{U}_2 + \alpha_3 \mathcal{U}_3 + \alpha_4 \mathcal{U}_4 \qquad (2) $$

$$ \mathcal{U}_2 = [K]^2 - [K^2] \qquad (3) $$

$$ \mathcal{U}_3 = [K]^3 - 3[K][K^2] + 2[K^3] \qquad (4) $$

$$ \mathcal{U}_4 = [K]^4 - 6[K^2][K^2] + 8[K][K^3] + 3[K^2]^2 - 6[K^4] \qquad (5) $$

$$ K_{\nu}^{\mu} = \delta_{\nu}^{\mu} - M_{\nu}^{\mu} \qquad (6) $$

$$ M_{\sigma}^{\mu} M_{\nu}^{\sigma} = g^{\mu\sigma} f_{\sigma\nu} = g^{\mu\sigma} \partial_{\sigma} \varphi^{\rho} \partial_{\nu} \varphi^{\beta} f_{\rho\beta} \qquad (7) $$

with $[K] = K_{\mu}^{\mu}$; $[K^2] = K_{\nu}^{\mu} K_{\mu}^{\nu}$; $[K^3] = K_{\nu}^{\mu} K_{\rho}^{\nu} K_{\mu}^{\rho}$; $[K^4] = K_{\nu}^{\mu} K_{\rho}^{\nu} K_{\sigma}^{\rho} K_{\mu}^{\sigma}$; $f_{\rho\beta}$ a fudicial metric that can be Minkowski metric, while $\alpha_3$ and $\alpha_4$ are free constants that can be zero.

Indeed the action (1) would yield the famous dRGT massive gravity field equations [10]:

$$ G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \kappa T_{\mu\nu} - m^2 \mathcal{U}_{\mu\nu} \qquad (8) $$

$$ \mathcal{E}_{\rho} = \partial_{\mu} \mathcal{U}_{\rho}^{\mu} = \nabla_{\mu} \left[ \frac{\partial \mathcal{U}(g, \varphi^{\rho})}{\partial (\partial_{\mu} \varphi^{\rho})} \right] = \nabla^{\mu} \mathcal{U}_{\mu\nu} = 0 \qquad (9) $$

where $T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta(\sqrt{-g}L_M)}{\delta g^{\mu\nu}}$ denotes the energy-momentum tensor of standard matter, and (9) the equation of motion for the Stückelberg fields $\varphi^\rho$, while [5, 9, 10]:

$$ \mathcal{U}_{\mu\nu} = \frac{\delta(\mathcal{U}(g, \varphi^\rho)\sqrt{-g})}{\sqrt{-g}\delta g^{\mu\nu}} = K_{\mu\nu} - [K]g_{\mu\nu} - [1 + 3\alpha_3]\left[K_{\mu\nu}^2 - [K]K_{\mu\nu} - \frac{1}{2}\mathcal{U}_2 g_{\mu\nu}\right] \\ + 3[4\alpha_4 + \alpha_3]\left[K_{\mu\nu}^3 - [K]K_{\mu\nu}^2 + \frac{1}{2}K_{\mu\nu}\mathcal{U}_2 - \frac{1}{6}\mathcal{U}_3 g_{\mu\nu}\right] \qquad (10) $$

Of course, we are not going to do a review of dRGT model in the present work (we recommend the papers [5, 9, 10] to reader interested by further details on the model); nevertheless it is worth to emphasize that dRGT gravity is among the first example of ghost free Lorentz-Invariant massive gravity; unfortunately, it fails above the strong-coupling scale $\Lambda_3 = [M_P m^2]^{1/3}$ well below Planck scale.

This comes amongst other things from the fact that the model is not achieved from a massless gravity theory which becomes dynamically massive through a dynamical phase
transition endowing the graviton with mass below some characteristic energy-scale as do Higgs field [2, 3, 4] in Standard Model of Particle Physics [8]; a goal proved to be difficult to achieve [9].

In the next sections, we are going to see how this difficulty can be overcame; more precisely we are going to see how for example a consistent massless gravity theory that becomes dynamically dRGT massive gravity through the dynamical "standard model" Higgs mechanism can be achieved.

But before, let's make a brief recall of the basic of standard model Higgs mechanism [2, 3, 4].

## 2.2. Brief Recall On Electroweak Phase Transition

As we well know; the Higgs field $\Phi$ of the standard model is a doublet of complex scalar fields $\phi^+ = \frac{\varphi_1+i\varphi_2}{\sqrt{2}}$ and $\phi^0 = \frac{\varphi_3+i\varphi_4}{\sqrt{2}}$, charged under the $SU(2)_L \times U(1)_Y$ group characterized by two coupling constants $g$ and $g'$ [11] i.e.:

$$ \Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} \varphi_1 + i\varphi_2 \\ \varphi_3 + i\varphi_4 \end{pmatrix} \qquad (11) $$

The lagrangian of this Higgs field $\Phi$ reads [2, 3, 4, 11]:

$$ L_H = -[D_\mu \Phi]^\dagger [D^\mu \Phi] - V(\Phi^\dagger, \Phi) \qquad (12) $$

where:

$$ D_{\mu} = \partial_{\mu} + ig\frac{Y_H}{2}B_{\mu} + ig'\tau_a W_{\mu}^a \qquad (13) $$

with $B_\mu$ and $W_\mu^a$ the electroweak gauge bosons; $Y_H$ the Higgs weak hypercharge and $\tau_a$ the generators of the $SU(2)$ group; also the index $\mu$ runs from 0 to 3 i.e. $\mu = 0, 1, 2, 3$, while the index $a$ runs from 1 to 3 i.e. $a = 1, 2, 3$.

Likewise usually, the potential $V(\Phi^\dagger, \Phi)$ is chosen to be:

$$ V(\Phi^\dagger, \Phi) = m_0^2 \Phi^\dagger \Phi + \lambda [\Phi^\dagger \Phi]^2 \qquad (14) $$

where $m_0$ denotes the effective mass parameter of $\Phi$ and $\lambda > 0$ a positive constant.

Indeed, the interest of this above potential (14) is that for $m_0^2 < 0$, there will be a spontaneous symmetry breaking endowing the field $\Phi$ with a non-zero vacuum expectation value [2, 3, 4, 11]:

$$ \langle \Phi \rangle = \langle \phi^0 \rangle = v = \sqrt{-\frac{m_0^2}{\lambda}} \qquad (15) $$

As consequence, the perturbation of $\Phi$ around its minimum could be written as [2, 3,
4, 11]:

$$ \Phi = \frac{1}{\sqrt{2}}(v+H) \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ v+H \end{pmatrix} \quad (16) $$

where $H$ is a real scalar field, the Higgs boson! what would give [2, 3, 4, 11]:

$$ \Phi^{\dagger}\Phi = \frac{1}{2}[v+H]^2 = \frac{1}{2}v^2 + \frac{1}{2}[2v+H]H \quad (17) $$

whence the lagrangian (12) would become [2, 3, 4, 11]:

$$ L_H = -\frac{1}{2}\partial_{\mu}H\partial^{\mu}H - V(H) - \frac{1}{4}[v^2 + 2vH + H^2][g'^2W_{\mu}^{+}W^{-\mu} + \frac{m_Z^2}{v^2}Z_{\mu}Z^{\mu}] \quad (18) $$

where [2, 3, 4, 11]:

$$ V(H) = \lambda v^2 H^2 + \lambda v H^3 + \frac{\lambda}{4} H^4 - \frac{\lambda v^4}{4} \quad (19) $$

while $Z_\mu$ and $W_\mu^\pm$ are respectively the electrically neutral and charged weak bosons of masses $m_Z = \frac{v}{\sqrt{2}}\sqrt{g^2 + g'^2}$ and $m_W = \frac{v}{\sqrt{2}}g'$; whence the assertion that Higgs field $\Phi$ yields, through dynamical spontaneous electroweak symmetry breaking, masses for the weak interaction bosons $Z_\mu$ and $W_\mu^\pm$.

Of course to capture the whole Higgs mechanism, we have to consider amongst other things, Yukawa coupling of the Higgs field with fermions; Kinetic term of the gauge bosons etc... [2, 3, 4, 11].

## 2.3. Useful Notes

Before starting the next section; let's introduce the function $k(\Phi)$ of the Higgs doublet $\Phi$, reading:

$$ k(\Phi) = [2B(e^{\Phi_1}, 1 + e^{\Phi_2}) - B(e^{\Phi_2}, e^{\Phi_1})]^2 \quad (20) $$

where $e$ is the Euler's number or Napier's constant; $B(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ the beta function; while $\Phi_1 = \frac{\sqrt{v^2 - [2v - \sqrt{2\Phi^\dagger\Phi}]\sqrt{2\Phi^\dagger\Phi}}}{4v}$ and $\Phi_2 = \frac{\sqrt{2\Phi^\dagger\Phi}}{4v} - \frac{1}{4}$, with $v$ the constant given by (15).

Of course, we can verify that by construction of $\Phi_1$ and $\Phi_2$; it is only in the case of (17) or more precisely, only when $\Phi$ develops the non-zero vacuum expectation value $v$, i.e. only after electroweak phase transition, that we could have:

$$ \Phi_1 = \Phi_2 = \frac{H}{4v} \quad (21) $$

Now since beta function satisfies the identities [12]:
$$B(x, y + 1) = \frac{y}{x+y} B(x, y) \qquad (22)$$

$$2B(y, y + 1) = B(y, y) \qquad (23)$$

then we will have $k(\Phi) = 0$ only in the case of (21) where $\Phi_1 = \Phi_2 = \frac{H}{4v}$.

Said otherwise, by construction, the function $k(\Phi)$ given by (20), becomes zero only when $\Phi$ develops the non-zero vacuum expectation value $v$ i.e. only after electroweak phase transition; what makes $k(\Phi)$ an interesting function as we will see later.

Now, by writing the fudicial metric as:

$$f_{\rho\beta} = \begin{pmatrix} f_{00} & f_{01} & f_{02} & f_{03} \\ f_{10} & f_{11} & f_{12} & f_{13} \\ f_{20} & f_{21} & f_{22} & f_{23} \\ f_{30} & f_{31} & f_{32} & f_{33} \end{pmatrix} \qquad (24)$$

we can, by using the Stückelberg scalar fields $\varphi_\rho$ and the function $k(\Phi)$, construct a new tensor $\Phi_{\rho\beta}$ defined as:

$$\Phi_{\rho\beta} = \begin{pmatrix} e^{-[\varphi_0 k(\Phi)\varphi_0]} f_{00} & e^{[\varphi_0 k(\Phi)\varphi_1]} f_{01} & e^{[\varphi_0 k(\Phi)\varphi_2]} f_{02} & e^{[\varphi_0 k(\Phi)\varphi_3]} f_{03} \\ e^{[\varphi_1 k(\Phi)\varphi_0]} f_{10} & e^{-[\varphi_1 k(\Phi)\varphi_1]} f_{11} & e^{[\varphi_1 k(\Phi)\varphi_2]} f_{12} & e^{[\varphi_1 k(\Phi)\varphi_3]} f_{13} \\ e^{[\varphi_2 k(\Phi)\varphi_0]} f_{20} & e^{[\varphi_2 k(\Phi)\varphi_1]} f_{21} & e^{-[\varphi_2 k(\Phi)\varphi_2]} f_{22} & e^{[\varphi_2 k(\Phi)\varphi_3]} f_{23} \\ e^{[\varphi_3 k(\Phi)\varphi_0]} f_{30} & e^{[\varphi_3 k(\Phi)\varphi_1]} f_{31} & e^{[\varphi_3 k(\Phi)\varphi_2]} f_{32} & e^{-[\varphi_3 k(\Phi)\varphi_3]} f_{33} \end{pmatrix} \qquad (25)$$

where e is the Euler's number, and $f_{00}$, $f_{01}$ etc... the components of the fudicial metric $f_{\rho\beta}$.

Now what is interesting with this tensor $\Phi_{\rho\beta}$, is that it becomes the fudicial metric tensor $f_{\rho\beta}$ when $k(\Phi) = 0$ since in the case of $k(\Phi) = 0$, we will have:

$$\Phi_{\rho\beta} = \begin{pmatrix} e^0 f_{00} & e^0 f_{01} & e^0 f_{02} & e^0 f_{03} \\ e^0 f_{10} & e^0 f_{11} & e^0 f_{12} & e^0 f_{13} \\ e^0 f_{20} & e^0 f_{21} & e^0 f_{22} & e^0 f_{23} \\ e^0 f_{30} & e^0 f_{31} & e^0 f_{32} & e^0 f_{33} \end{pmatrix} = \begin{pmatrix} f_{00} & f_{01} & f_{02} & f_{03} \\ f_{10} & f_{11} & f_{12} & f_{13} \\ f_{20} & f_{21} & f_{22} & f_{23} \\ f_{30} & f_{31} & f_{32} & f_{33} \end{pmatrix} \qquad (26)$$

while, in the case of $k(\Phi) \neq 0$; $\Phi_{\rho\beta}$ will not be a fudicial metric tensor, since incapable of providing a fixed reference background metric alongside the physical spacetime metric $g_{\mu\nu}$.

Said otherwise, if in the case of $k(\Phi) = 0$, the tensor $\Phi_{\rho\beta}$ becomes the fudicial metric tensor $f_{\rho\beta}$; in the case of $k(\Phi) \neq 0$, $\Phi_{\rho\beta}$ is not a fudicial metric tensor at all.

Thus for example, by replacing into the tensor $U_{\mu\nu}$ given by (10), the fudicial metric $f_{\rho\beta}$ by the tensor $\Phi_{\rho\beta}$, we will get the tensor $U_{\mu\nu}(\Phi)$ reading:
$$ \mathcal{U}_{\mu\nu}(\Phi) = \mathcal{K}_{\mu\nu} - [\mathcal{K}]g_{\mu\nu} - [1 + 3\alpha_3] \left[ \mathcal{K}_{\mu\nu}^2 - [\mathcal{K}] \mathcal{K}_{\mu\nu} - \frac{1}{2} \mathcal{U}_2 g_{\mu\nu} \right] \\ + 3[4\alpha_4 + \alpha_3] \left[ \mathcal{K}_{\mu\nu}^3 - [\mathcal{K}] \mathcal{K}_{\mu\nu}^2 + \frac{1}{2} \mathcal{K}_{\mu\nu} \mathcal{U}_2 - \frac{1}{6} \mathcal{U}_3 g_{\mu\nu} \right] \quad (27) $$

where $\mathcal{K}_{\nu}^{\mu} = \delta_{\nu}^{\mu} - \tilde{M}_{\nu}^{\mu}$, and $\tilde{M}_{\sigma}^{\mu} \tilde{M}_{\nu}^{\sigma} = g^{\mu\sigma} \Phi_{\sigma\nu} = g^{\mu\sigma} \partial_{\sigma} \varphi^{\rho} \partial_{\nu} \varphi^{\beta} \Phi_{\rho\beta}$, while $[\mathcal{K}] = \mathcal{K}_{\mu}^{\mu}$; $\mathcal{K}_{\mu\nu}^2 = \mathcal{K}_{\mu\sigma} \mathcal{K}_{\nu}^{\sigma}$; $\mathcal{U}_2 = [\mathcal{K}]^2 - [\mathcal{K}^2]$; $\mathcal{U}_3 = [\mathcal{K}]^3 - 3[\mathcal{K}][\mathcal{K}^2] + 2[\mathcal{K}^3]$; $[\mathcal{K}^2] = \mathcal{K}_{\nu}^{\mu} \mathcal{K}_{\mu}^{\nu}$ etc...

This means that the tensor $\mathcal{U}_{\mu\nu}(\Phi)$ will become the tensor $\mathcal{U}_{\mu\nu}$ when $k(\Phi) = 0$, while when $k(\Phi) \neq 0$, $\mathcal{U}_{\mu\nu}(\Phi)$ will be a tensor not involving a fiducial metric; more precisely a tensor incapable of providing a fixed reference background metric alongside the physical spacetime metric $g_{\mu\nu}$.

As far as we are concerned, $k(\Phi)$ and $\mathcal{U}_{\mu\nu}(\Phi)$ are enough to understand the rest of the paper.

### 3. A New Gravity Theory: Derivation From Variational Principle

To get the paradigm allowing gravity to become dynamically massive through electroweak phase transition; let's begin by modifying the action (1) as follows:

$$ S = \frac{M_P^2}{2} \int \left[ R + 2\kappa L_M + m^2 \mathcal{U}(g, \varphi^\rho) - L_{\chi_0} - L_{\chi_1} - L_{\chi_2} - \frac{\chi_2}{m} \left[ \mathcal{E}_{\mu\nu} \mathcal{U}^{\mu\nu} \right]^2 \right] \sqrt{-g} d^4 x \quad (28) $$

1) where $L_{\chi_0}$, $L_{\chi_1}$ and $L_{\chi_2}$ are additional lagrangians respectively reading:

$$ L_{\chi_0} = \omega_0 \left[ \partial_{\mu} \chi_0 \partial^{\mu} \chi_0 + 1 \right] \quad (29) $$

$$ L_{\chi_1} = \omega_1 \left[ \partial_{\mu} \chi_1 \partial^{\mu} \chi_1 + 1 \right] + \varepsilon_1 \left[ \omega_1 - m^2 \right] - m^2 \partial_{\mu} \chi_1 \partial^{\mu} \chi_1 \quad (30) $$

$$ L_{\chi_2} = \omega_2 \left[ \partial_{\mu} \chi_2 \partial^{\mu} \chi_2 + 1 \right] + \varepsilon_2 \left[ \omega_2 + m^2 \right] + m^2 \partial_{\mu} \chi_2 \partial^{\mu} \chi_2 \quad (31) $$

in which $\omega_0$, $\chi_0$, $\omega_1$, $\chi_1$, $\varepsilon_1$, $\omega_2$, $\chi_2$ and $\varepsilon_2$ are auxiliary variable, with m a dimensioned constant.

2) while $\mathcal{U}_{\mu\nu}$ is the tensor given by (10) and $\mathcal{E}_{\mu\nu}$ a tensor reading:

$$ \mathcal{E}_{\mu\nu} = \omega_0 \partial_{\mu} \chi_0 \partial_{\nu} \chi_0 - m^2 \mathcal{U}_{\mu\nu} - k(\Phi) A_{\mu} A_{\nu} + m^2 \mathcal{U}_{\mu\nu}(\Phi) \quad (32) $$

where $A_{\mu}$ is an auxiliary vector field; $k(\Phi)$ the function given by (20) and $\mathcal{U}_{\mu\nu}(\Phi)$ the tensor given by (27).
It is worth mentioning that we consider the terms $k(\Phi)$ and $\mathcal{U}_{\mu\nu}(\Phi)$ only because of their ability to yield the tensor $\mathcal{U}_{\mu\nu}$ after electroweak phase transition; we will give no other justification of their use insofar as our aim is not to justify our action (28) from the point of view of some underlying more fundamental physics or principle; we just require that it be a consistent and stable action; which is the case.

Now as all the elements of the additional lagrangians (29), (30), (31) and tensor $\mathcal{E}_{\mu\nu}$ are specified; then we can apply variational principle; indeed:

a) varying the action (28) with respect to $\delta g^{\mu\nu}$ would yield:

$$G_{\mu\nu} = \kappa T_{\mu\nu} - m^2 \mathcal{U}_{\mu\nu} + \frac{1}{\sqrt{-g}} \left[ \frac{\delta(\sqrt{-g}L_{\chi_0})}{\delta g^{\mu\nu}} + \frac{\delta(\sqrt{-g}L_{\chi_1})}{\delta g^{\mu\nu}} + \frac{\delta(\sqrt{-g}L_{\chi_2})}{\delta g^{\mu\nu}} + \frac{\delta(\sqrt{-g}\frac{\chi_2}{m}[\mathcal{E}_{\mu\nu}\mathcal{U}^{\mu\nu}]^2)}{\delta g^{\mu\nu}} \right] \quad (33)$$

where of course $T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta(\sqrt{-g}L_M)}{\delta g^{\mu\nu}}$ denotes the energy-momentum tensor of standard matter and $\mathcal{U}_{\mu\nu}$ the tensor given by (10), while.

$$\frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g}L_{\chi_0})}{\delta g^{\mu\nu}} = \omega_0 \partial_{\mu} \chi_0 \partial_{\nu} \chi_0 - \frac{\omega_0}{2} g_{\mu\nu} \left[ \partial_{\sigma} \chi_0 \partial^{\sigma} \chi_0 + 1 \right] \quad (34)$$

$$\frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g}L_{\chi_1})}{\delta g^{\mu\nu}} = [\omega_1 - m^2] \left[ \partial_{\mu} \chi_1 \partial_{\nu} \chi_1 - \frac{1}{2} g_{\mu\nu} \partial_{\sigma} \chi_1 \partial^{\sigma} \chi_1 \right] - \frac{\omega_1}{2} g_{\mu\nu} \quad (35)$$

$$\frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g}L_{\chi_2})}{\delta g^{\mu\nu}} = [\omega_2 + m^2] \left[ \partial_{\mu} \chi_2 \partial_{\nu} \chi_2 - \frac{1}{2} g_{\mu\nu} \partial_{\sigma} \chi_2 \partial^{\sigma} \chi_2 \right] - \frac{\omega_2}{2} g_{\mu\nu} \quad (36)$$

$$\frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g}\frac{\chi_2}{m}[\mathcal{E}_{\mu\nu}\mathcal{U}^{\mu\nu}]^2)}{\delta g^{\mu\nu}} = \frac{\chi_2}{m} \left[ 2[\mathcal{U}^{\mu\nu} \frac{\delta \mathcal{E}_{\mu\nu}}{\delta g^{\mu\nu}} + \mathcal{E}_{\mu\nu} \frac{\delta \mathcal{U}^{\mu\nu}}{\delta g^{\mu\nu}}] - \frac{1}{2} g_{\mu\nu} \mathcal{E}_{\rho\sigma} \mathcal{U}^{\rho\sigma} \right] \mathcal{E}_{\rho\sigma} \mathcal{U}^{\rho\sigma} \quad (37)$$

b) varying (28) with respect to $\delta\varphi^\rho$; $\delta\omega_0$ and $\delta\chi_0$ would respectively yield:

$$m^2 \nabla_{\mu} \left[ \frac{\partial \mathcal{U}(g, \varphi^{\rho})}{\partial (\partial_{\mu} \varphi^{\rho})} \right] + 2 \nabla^{\mu} \left[ \frac{\chi_2}{m} \mathcal{E}_{\mu\nu} \mathcal{U}^{\mu\nu} \left[ \mathcal{U}^{\mu\nu} \frac{\partial \mathcal{E}_{\mu\nu}}{\partial (\partial^{\mu} \varphi^{\rho})} + \mathcal{E}_{\mu\nu} \frac{\partial \mathcal{U}^{\mu\nu}}{\partial (\partial^{\mu} \varphi^{\rho})} \right] \right] = 0 \quad (38)$$

$$\partial_{\mu} \chi_{0} \partial^{\mu} \chi_{0} + 1 + 2 \frac{\chi_{2}}{m} \mathcal{E}_{\mu\nu} \mathcal{U}^{\mu\nu} \mathcal{U}^{\nu\mu} \frac{\partial \mathcal{E}_{\mu\nu}}{\partial \omega_{0}} = 0 \quad (39)$$

$$\nabla^{\mu} \left[ 2\omega_0 \partial_{\mu} \chi_0 + 2 \frac{\chi_2}{m} \mathcal{E}_{\mu\nu} \mathcal{U}^{\mu\nu} \mathcal{U}^{\nu\mu} \frac{\partial \mathcal{E}_{\mu\nu}}{\partial (\partial^{\mu} \chi_0)} \right] = 0 \quad (40)$$

c) varying (28) with respect to $\delta\omega_1$, $\delta\varepsilon_1$ and $\delta\chi_1$ would respectively yield:
$$ \partial_{\mu} \chi_{1} \partial^{\mu} \chi_{1} + 1 + \varepsilon_{1} = 0 \qquad (41) $$

$$ \omega_{1} - m^{2} = 0 \qquad (42) $$

$$ \nabla^{\mu} [2(\omega_{1} - m^{2}) \partial_{\mu} \chi_{1}] = 0 \qquad (43) $$

d) varying (28) with respect to $\delta\omega_2$, $\delta\varepsilon_2$, $\delta\chi_2$ and $\delta A_\mu$ would respectively yield:

$$ \partial_{\mu} \chi_{2} \partial^{\mu} \chi_{2} - 1 + \varepsilon_{2} = 0 \qquad (44) $$

$$ \omega_{2} + m^{2} = 0 \qquad (45) $$

$$ \nabla^{\mu} [2(\omega_{2} + m^{2}) \partial_{\mu} \chi_{2}] - \frac{1}{m} [\varepsilon_{\mu\nu} \mathcal{U}^{\mu\nu}]^{2} = 0 \qquad (46) $$

$$ 2 \frac{\chi_2}{m} \varepsilon_{\mu\nu} \mathcal{U}^{\mu\nu} \mathcal{U}^{\nu\mu} \frac{\partial \varepsilon_{\mu\nu}}{\partial A_{\mu}} = 0 \qquad (47) $$

From there, it is clear that because of (45); the above equation (46) would enforce:

$$ \frac{1}{m} [\varepsilon_{\mu\nu} \mathcal{U}^{\mu\nu}]^2 = 0 \qquad (48) $$

Now since $m$ is a constant and that $\mathcal{U}^{\mu\nu}$ is not always zero; then one infers that the above (48) enforces in its turn:

$$ \varepsilon_{\mu\nu} = \omega_0 \partial_{\mu} \chi_0 \partial_{\nu} \chi_0 - m^2 \mathcal{U}_{\mu\nu} - k(\Phi) A_{\mu} A_{\nu} + m^2 \mathcal{U}_{\mu\nu}(\Phi) = 0 \qquad (49) $$

Clearly, because of this above constraint (49); the equations (38), (39), (40) and (47) would respectively simplify as:

$$ m^2 \nabla_{\mu} \left[ \frac{\partial \mathcal{U}(g, \varphi^{\rho})}{\partial (\partial_{\mu} \varphi^{\rho})} \right] = 0 \qquad (50) $$

$$ \partial_{\mu} \chi_{0} \partial^{\mu} \chi_{0} + 1 = 0 \qquad (51) $$

$$ \nabla^{\mu} [2 \omega_{0} \partial_{\mu} \chi_{0}] = 0 \qquad (52) $$

$$ 0 = 0 \qquad (53) $$

In the same way, because of (42), (45), (49) and (51); the equations (34), (35), (36) and
(37) would simplify as:

$$ \frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g} L_{\chi_0})}{\delta g^{\mu\nu}} = \omega_0 \partial_{\mu} \chi_0 \partial_{\nu} \chi_0 \qquad (54) $$

$$ \frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g} L_{\chi_1})}{\delta g^{\mu\nu}} = -\frac{1}{2} m^2 g_{\mu\nu} \qquad (55) $$

$$ \frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g} L_{\chi_2})}{\delta g^{\mu\nu}} = +\frac{1}{2} m^2 g_{\mu\nu} \qquad (56) $$

$$ \frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g} \frac{\chi_2}{m} [\mathcal{E}_{\mu\nu} \mathcal{U}^{\mu\nu}]^2)}{\delta g^{\mu\nu}} = 0 \qquad (57) $$

whence the gravitational equation (33) would simplify as:

$$ G_{\mu\nu} = \kappa T_{\mu\nu} - m^2 \mathcal{U}_{\mu\nu} + \omega_0 \partial_{\mu} \chi_0 \partial_{\nu} \chi_0 \qquad (58) $$

From there, since the constraint (49) implies also:

$$ \omega_0 \partial_{\mu} \chi_0 \partial_{\nu} \chi_0 = m^2 \mathcal{U}_{\mu\nu} + k(\Phi) A_{\mu} A_{\nu} - m^2 \mathcal{U}_{\mu\nu}(\Phi) \qquad (59) $$

then we will finally get the gravitational equation reading:

$$ G_{\mu\nu} = \kappa T_{\mu\nu} - m^2 \mathcal{U}_{\mu\nu}(\Phi) + k(\Phi) A_{\mu} A_{\nu} \qquad (60) $$

while the equation of motion of the Stückelberg scalar fields $\phi^\rho$ still remains (50).

Also, it is worth mentioning that because of the enforced constraint (49); we can verify that the Euler-Lagrange equation of motion of any given field $\Psi$ of the standard model of particle physics (including the Higgs field $\Phi$) would remain the usual one:

$$ \nabla^{\mu} \left[ 2\kappa \frac{\partial L_M}{\partial(\partial^{\mu} \Psi)} - \frac{\chi_2}{m} \frac{\partial [\mathcal{E}_{\mu\nu} \mathcal{U}^{\mu\nu}]^2}{\partial(\partial^{\mu} \Psi)} \right] - 2\kappa \frac{\partial L_M}{\partial \Psi} + \frac{\chi_2}{m} \frac{\partial [\mathcal{E}_{\mu\nu} \mathcal{U}^{\mu\nu}]^2}{\partial \Psi} = \nabla^{\mu} \left[ 2\kappa \frac{\partial L_M}{\partial(\partial^{\mu} \Psi)} \right] - 2\kappa \frac{\partial L_M}{\partial \Psi} = 0 \qquad (61) $$

i.e. standard model of particle physics remains unchanged in our paradigm (28).

Now, since as we clearly see, none of the auxiliary variable $\omega_0$, $\chi_0$, $\omega_1$, $\chi_1$, $\epsilon_1$, $\omega_2$, $\chi_2$ and $\epsilon_2$, contribute neither in the equation of motion (61) of standard matter, nor in the equation of motion (50) for the Stückelberg fields $\phi^\rho$, while only $A_\mu$ contribute explicitly in the gravitation equation (60); then excepted $A_\mu$, all the other auxiliary variables $\omega_0$, $\chi_0$, $\omega_1$, $\chi_1$, $\epsilon_1$, $\omega_2$, $\chi_2$ and $\epsilon_2$ could be seen as just pure mathematical fields (i.e. non-physical fields) helping to achieve the Euler-Lagrange equations of motion (50), (60) and (61).

It is also worth to mention that because of (61); applying the Bianchi identity $\nabla^\mu G_{\mu\nu} = 0$ to (60) would impose:
$$ \nabla^{\mu} \left[ k(\Phi) A_{\mu} A_{\nu} - m^2 \mathcal{U}_{\mu\nu}(\Phi) \right] = 0 \qquad (62) $$

which is the equation that enforces the auxiliary variable $A_μ$; an equation which of course could also be obtained from (52) by considering (50), (51) and (59); whence the gravitational theory yielded by our action (28), would in summary be:

$$ G_{\mu\nu} = \kappa T_{\mu\nu} - m^2 \mathcal{U}_{\mu\nu}(\Phi) + k(\Phi) A_{\mu} A_{\nu} \qquad (63) $$

$$ \nabla_{\mu} \left[ \frac{\partial \mathcal{U}(g, \varphi^{\rho})}{\partial (\partial_{\mu} \varphi^{\rho})} \right] = \nabla^{\mu} \mathcal{U}_{\mu\nu} = 0 \qquad (64) $$

$$ \nabla^{\mu} \left[ k(\Phi) A_{\mu} A_{\nu} - m^2 \mathcal{U}_{\mu\nu}(\Phi) \right] = 0 \qquad (65) $$

where of course $G_{\mu\nu}$ denotes the Einstein tensor; $\kappa$ the Einstein gravitational constant; $T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta(\sqrt{-g}L_M)}{\delta g^{\mu\nu}}$ the energy-momentum tensor of standard matter; $m$ a constant; $k(\Phi)$ the function given by (20); $A_μ$ an auxiliary variable; $\mathcal{U}_{\mu\nu}$ the tensor given by (10) and $\mathcal{U}_{\mu\nu}(\Phi)$ the tensor given by (27).

Now, what is interesting with this gravitational theory (63-65), is that, it is not only a "gravitational theory" derivable from variational principle but also, it is (as we are going to see in the next section), the first example of four-dimensional modified gravity which is massless before electroweak phase transition and massive after electroweak phase transition i.e. a gravity theory for which the graviton gain mass dynamically through electroweak spontaneous symmetry breaking.

Let's see how.

# 4. Massive After ElectroWeak Phase Transition, Massless Before: Proof

## 4.1. After Electroweak Phase Transition

To verify that our gravitational theory (63-65) is a massive gravity theory after the electroweak phase transition; it suffices to notice that the equation (20) guarantees that after the electroweak phase transition we have:

$$ k(\Phi) = 0 \qquad (66) $$

As consequence, after the electroweak phase transition we will have (see subsection 2.3.):

$$ \mathcal{U}_{\mu\nu}(\Phi) = \mathcal{U}_{\mu\nu} \qquad (67) $$
whence after the electroweak phase transition where $k(\Phi) = 0$, the equations (63-65) will transform/simplify into:

$$G_{\mu\nu} = \kappa T_{\mu\nu} - m^2 U_{\mu\nu} \qquad (68)$$

$$\nabla_{\mu} \left[ \frac{\partial U(g, \varphi^{\rho})}{\partial(\partial_{\mu} \varphi^{\rho})} \right] = \nabla^{\mu} U_{\mu\nu} = 0 \qquad (69)$$

This latter (68-69) being nothing than exactly the dRGT massive gravity (8-9); then we can quite affirm that after the electroweak phase transition, our gravitational theory (63-65) yields/becomes explicitly a massive gravity, more precisely the famous dRGT massive gravity [5, 10].

## 4.2. Before Electroweak Phase Transition

If as we have clearly seen, our model (63-65) yields/becomes after the electroweak phase transition, a massive gravity, more precisely the famous dRGT massive gravity (8-9); we are going to see in this subsection that before the electroweak phase transition, (63-65) is "well and truly" a massless gravity theory.

Indeed since before the electroweak phase transition; the field $\Phi$ not have the non-zero vacuum expectation value $v$; then it results that before electroweak phase transition, we have:

$$k(\Phi) \neq 0 \qquad (70)$$

and thus:

$$U_{\mu\nu}(\Phi) \neq U_{\mu\nu} \qquad (71)$$

Better, $U_{\mu\nu}(\Phi)$ will in this case $k(\Phi) \neq 0$, be (see subsection 2.3.) a tensor incapable of providing a fixed reference background metric alongside the physical spacetime metric $g_{\mu\nu}$, and thus a tensor incapable of yielding mass-term for the graviton.

This can be verified more easily by linearizing (63-65).

Indeed as well known [13]; in the linearized regime characterized by $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, with $h_{\mu\nu} \ll 1$ a perturbation around Minkowski metric $\eta_{\mu\nu}$; the Einstein tensor $G_{\mu\nu}$ could be simplified as [13]:

$$G_{\mu\nu} = -\frac{1}{2} \Box \tilde{h}_{\mu\nu} \qquad (72)$$

where $\Box = \nabla_{\mu} \nabla^{\mu}$ denotes the d'Alembert operator and $\tilde{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}h\eta_{\mu\nu}$ the trace-reversed perturbation variable (graviton).

whence in the linearized regime $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ characterized by (72); our gravity equa-
tion (63) would simplify as:

$$ \Box \tilde{h}_{\mu\nu} = -2\kappa T_{\mu\nu} + 2m^2 \mathcal{U}_{\mu\nu}(\Phi) - 2k(\Phi)A_{\mu}A_{\nu} \quad (73) $$

From there let's for simplicity, consider case $\alpha_3 = \alpha_4 = 0$ and $f_{\rho\beta} = \eta_{\rho\beta}$ i.e. the fudicial metric is Minkowski metric.

In this case, it is well known that the tensor $\mathcal{U}_{\mu\nu}$ would in the linearized regime $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, simplify as [5, 9, 10]:

$$ \mathcal{U}_{\mu\nu} = \frac{1}{2} \tilde{h}_{\mu\nu} \quad (74) $$

Now since we have $\mathcal{U}_{\mu\nu}(\Phi) = \mathcal{U}_{\mu\nu}$ when $k(\Phi) = 0$; then it results that in regime $k(\Phi) = 0$ i.e. after the electroweak phase transition; the equation (73) for the graviton would become:

$$ [\Box - m^2] \tilde{h}_{\mu\nu} = -2\kappa T_{\mu\nu} \quad (75) $$

which is nothing than the equation of motion of a massive graviton; more precisely the linearized regime of the dRGT massive gravity (68-69); whence after the electroweak phase transition where $k(\Phi) = 0$; our model (63-65) is as already discussed, a massive gravity theory; more precisely the famous dRGT massive gravity [5, 9, 10].

However since in the regime $k(\Phi) \neq 0$ i.e. before the electroweak phase transition, the tensor $\mathcal{U}_{\mu\nu}(\Phi)$ could for $\alpha_3 = \alpha_4 = 0$, write:

$$ \mathcal{U}_{\mu\nu}(\Phi) = g_{\mu\nu} [6 + 2f_{\varphi} \Phi - f_{\varphi} \sqrt{\Phi_{\rho}^{\sigma} \Phi_{\sigma}^{\rho}}] - f_{\varphi} \Phi_{\mu\nu} - [3 - 2\sqrt{f_{\varphi} \Phi}] \sqrt{f_{\varphi} g_{\mu\nu} \Phi_{\mu\nu}} \quad (76) $$

where $f_{\varphi} = \partial_{\rho}\varphi^{\rho}\partial_{\rho}\varphi^{\rho}$, with $\varphi^{\rho}$ the four Stückelberg scalars fields; $\Phi = g^{\mu\nu} \Phi_{\mu\nu}$ and $\Phi_{\mu\nu}$ the tensor given by (25).

then it results that before the electroweak phase transition where $k(\Phi) \neq 0$; the equation (73) for the graviton would read:

$$ \Box \tilde{h}_{\mu\nu} = -2\kappa T_{\mu\nu} - 2m^2 \mathcal{T}_{\mu\nu} \quad (77) $$

where:

$$ \mathcal{T}_{\mu\nu} = \frac{k(\Phi)}{m^2} A_{\mu} A_{\nu} + f_{\varphi} \Phi_{\mu\nu} - [6 + 2f_{\varphi} \Phi - f_{\varphi} \sqrt{\Phi_{\rho}^{\sigma} \Phi_{\sigma}^{\rho}}] g_{\mu\nu} + [3 - 2\sqrt{f_{\varphi} \Phi}] \sqrt{f_{\varphi} g_{\mu\nu} \Phi_{\mu\nu}} \quad (78) $$

Now since the tensor $\Phi_{\mu\nu}$ is not a fudicial metric when $k(\Phi) \neq 0$; then it results that the equation (77) describes "well and trully" a massless graviton $\tilde{h}_{\mu\nu}$, since neither the first energy-momentum tensor $T_{\mu\nu}$ of standard matter, nor the second energy-momentum tensor $\mathcal{T}_{\mu\nu}$ carry/yield mass-term for the graviton $\tilde{h}_{\mu\nu}$; whence our model (63-65) is in
regime $k(\Phi) \neq 0$ i.e. before the electroweak phase transition, a massless gravity theory, more precisely, a modified massless gravity theory departing from General Relativity [13] by the presence of the tensor $J_{\mu\nu}$ given by (78).

It is interesting to note that $J_{\mu\nu}$ is a tensor which carries only the three propagating physical degrees of freedom yielded by the equation of motion (64) of the four Stückelberg scalar fields $\phi^\rho$ [10, 9, 5].

This comes from the fact that the auxiliary variable $A_\mu$ enters non-dynamically i.e. without kinetic term in the action (28); whence it not gives rise to additional propagating physical degree of freedom.

This mean that, both before the electroweak phase transition where it is a massless gravity, as well as after the electroweak phase transition where it is a massive gravity, our model (63-65) has five propagating physical degrees of freedom if we not count the degrees of freedom carried by $T_{\mu\nu}$; the two yielded by the metric tensor $g_{\mu\nu}$ plus the three yielded by the four Stückelberg scalar fields $\phi^\rho$ [10, 9, 5]; whence we can say that the graviton of our gravitational theory (63-65) gains mass dynamically through the dynamical standard model spontaneous electroweak symmetry breaking, and thus that we achieved consistent dynamical Higgs mechanism for dRGT gravity.

Another exciting feature of our theory (63-65) is that, although its graviton becomes massive through electroweak phase transition; the model leaves standard model of particle physics roughly unchanged i.e. our model (63-65) is a modified gravity theory with auxiliary variable which leaves standard model of particle physics roughly unchanged; particularly, the Higgs sector of standard model of particle physics undergoes no special change and remains the known one [2, 3, 4, 11]; whence at this stage, no new discussion on Higgs sector is necessary in our paradigm (63-65).

Before concluding, it is worth mentioning that the trick applied in the present work to "dRGT massive gravity" to get dynamical Higgs mechanism for dRGT gravity, could be applied to other massive gravities like Mass-varying massive gravity [14]; Quasi-dilaton massive gravity [15] and Mimetic massive gravity [6] etc...

In other words, we found a consistent "trick/method" permitting to make graviton consistently and dynamically massive through spontaneous symmetry breaking.

## 5. Conclusion

In this paper, we succeeded in showing that, it is clearly possible to achieve from variational principle a four-dimensional gravity theory (63-65) which is massless before electroweak phase transition and massive after electroweak phase transition; more precisely, we succeeded in getting the first example of four-dimensional gravity theory for which the graviton gains mass dynamically and consistently through spontaneous symmetry breaking!

In short, we achieved the first example of dynamical and consistent Higgs mechanism for dRGT gravity!
## 6. References

[1] Fierz, Markus; Pauli Wolfgang (1939). "On relativistic wave equations for particles of arbitrary spin in an electromagnetic field". Proc. R. Soc. Lond. 173 (953): 211-232.

[2] James D. Wells (2009). "Lectures on Higgs Boson Physics in the Standard Model and Beyond". arXiv:hep-ph/0909.4541.

[3] Higgs, P.W. (1964). "Broken symmetries and the masses of gauge bosons". Physical Review Letters. 13 (16): 508-509.

[4] Serguei Chatrchyan et al. (CMS collaboration) (2012). "Observation of a new boson at a mass of 125 GeV with CMS experiment at the LHC". Physics Letters B 716: 30-61. arXiv:hep-exp/1207.7235.

[5] de Rham, Claudia; Gabadadze, Gregory; Tolley, Andrew J. (2011). "Resummation of massive gravity". Physical Review Letters. 106 (23): 231101. arXiv:hep-th/1011.1232.

[6] A.H. Chamseddine; V. Mukhanov (2018). "Ghost Free Mimetic Massive Gravity". J. High Energy Phys. 06 (060). arXiv:hep-th/1805.06283.

[7] A.R. Solomon; V. Vardanyan; Y. Akrami (2019). "Massive mimetic cosmology". Phys. Lett. B 794: 135-142. arXiv:astro-ph.CO/1902.08533.

[8] R. Oerter (2006), "The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics". (Kindle ed.) Penguin Group. ISBN 978-0-13-236678-6.

[9] Claudia de Rham (2014). "Massive Gravity". Living Rev. Relativity 17, 7. arXiv:hep-ph/1401.4173.

[10] de Rham, Claudia; Gabadadze, Gregory; Tolley, Andrew J. (2012). "Ghost Free Massive Gravity in the Stückelberg language". Physics Letters B. 711 (2): 190-195. arXiv:hep-th/1107.3820.

[11] V. A. Bednyakov; N. D. Giokaris; A. V. Bednyakov (2008). "On Higgs mass generation mechanism in the Standard Model". Phys.Part.Nucl. 39: 13-36. arXiv:0703280 [hep-ph].

[12] Arfken, G. (1985). "The Beta Function". in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp.560-565.

[13] Sean M. Carroll (1997). "Lecture Notes on General Relativity". arXiv:9712019 [gr-qc].

[14] Qing-Guo Huang, Yun-Song Piao, Shuang-Yong Zhou (2012). "Mass-Varying Massive Gravity". Phys.Rev. D86, 124014. arXiv:1206.5678 [hep-th].

[15] D'Amico, Guido; Gabadadze, Gregory; Hui, Lam; Pirtskhalava, David (2013). "Quasidilaton: Theory and cosmology". Phys.Rev.; D87, 064037. arXiv:1206.4253 [hep-th].